The distribution of dark matter in galaxies
星系中暗物质的分布

Abstract

The distribution of the non-luminous matter in galaxies of different luminosity and Hubble type is much more than a proof of the existence of dark particles governing the structures of the Universe. Here, we will review the complex but well-ordered scenario of the properties of the dark halos also in relation with those of the baryonic components they host. Moreover, we will present a number of tight and unexpected correlations between selected properties of the dark and the luminous matter. Such entanglement evolves across the varying properties of the luminous component and it seems to unequivocally lead to a dark particle able to interact with the Standard Model particles over cosmological times. This review will also focus on whether we need a paradigm shift, from pure collisionless dark particles emerging from “first principles”, to particles that we can discover only by looking to how they have designed the structure of the galaxies.
不发光物质在不同亮度和哈勃类型星系中的分布远不止是存在控制宇宙结构的暗粒子的证明。在这里，我们将回顾暗晕的性质的复杂但有序的情景，以及它们与所承载的重子组分的关系。此外，我们将介绍一些暗物质和发光物质的选定性质之间的紧密且意想不到的相关性。这种纠缠随着发光组分的不同性质而演变，并且似乎毫无疑问地导致了一个能够与标准模型粒子在宇宙时间内相互作用的暗粒子。本评论还将重点讨论我们是否需要一个范式转变，从纯粹从“第一原理”中出现的无碰撞暗粒子，到我们只能通过观察它们如何设计星系结构来发现的粒子。

1 Introduction

The idea of the presence of large amounts of invisible matter in and around spirals, distributed differently from the stellar and gaseous disks, turned up in the 1970s (Roberts 1978; Faber and Gallagher 1979; Rubin et al. 1980; Bosma 1981a, see also Bertone and Hooper 2018). There were, in fact, published optical and 21-cm rotation curves (RCs) behaving in a strongly anomalous way. These curves were incompatible with the Keplerian fall-off we would predict from their outer distribution of luminous matter (see Fig. 1).
20 世纪 70 年代出现了大量不可见物质存在于螺旋星系内外的想法（Roberts 1978; Faber and Gallagher 1979; Rubin et al. 1980; Bosma 1981a，另请参阅 Bertone 和 Hooper 2018）。事实上，已经发表了光学和 21 厘米旋转曲线（RCs），其行为方式非常异常。这些曲线与我们从它们的外部发光物质分布预测的开普勒式衰减不兼容（见图 1）。

From there, this dark component has started to take a role always more important in cosmology, astrophysics and elementary particles physics. On the other hand, the nature and the cosmological history of such dark component has always become more mysterious and difficult to be derived from paradigms and first principles. We must remark that a dark massive component in the mass budget of the Universe is necessary to explain: the redshift dependence of the expansion of its scale factor, the relative heights of the peaks in the CMB cosmic fluctuations, the bottom-up growth of the cosmological structures to their nonlinear phases, the large-scale distribution of galaxies and the internal mass distribution of the biggest structures of the Universe. These theoretical issues and observational evidences (that will not be treated in this review) add phenomenal support to the paradigm of a massive dark particle, which, a fortiori, must lay beyond the zoo of the Standard Model of the elementary particles. This support is not able, however, to determine the kind, the nature and the mass of such a particle.
从那时起，这种黑暗成分在宇宙学、天体物理学和基本粒子物理学中开始扮演越来越重要的角色。另一方面，这种黑暗成分的性质和宇宙学历史变得越来越神秘和难以从范式和第一原则中推导出来。我们必须指出，宇宙的质量预算中存在一个黑暗的大质量成分是必要的，以解释：其尺度因子扩展的红移依赖性，CMB 宇宙波动中峰值的相对高度，宇宙结构的自底向上增长到非线性阶段，星系的大尺度分布以及宇宙最大结构的内部质量分布。这些理论问题和观测证据（本文不会讨论）为大质量暗粒子的范式提供了巨大支持，这种粒子必然超越基本粒子标准模型的动物园。然而，这种支持无法确定这种粒子的种类、性质和质量。

There is no doubt that dark matter connects, as no other issue, the different fields of study of cosmology, particle physics and astrophysics. In the current $\text{Λ}$ cold dark matter ($\text{Λ}$CDM) paradigm, the DM is non-relativistic since its decoupling time and can be described by a collisionless fluid, whose particles interact only gravitationally and very weakly with the Standard Model particles (Jungman et al. 1996; Bertone 2010).
毫无疑问，暗物质连接了宇宙学、粒子物理和天体物理等不同领域的研究，暗物质在当前的冷暗物质（CDM）范式中是非相对论的，因为它的脱耦时间，可以被描述为一种无碰撞流体，其粒子仅通过引力与标准模型粒子发生非常微弱的相互作用（Jungman 等，1996 年；Bertone，2010 年）。

Fig. 1

The image of M33 and the corresponding rotation curve (Corbelli and Salucci 2000). What exactly does this large anomaly of the gravitational field indicate? The presence of (i) a (new) non-luminous massive component around the stellar disk or (ii) new physics of a (new) dark constituent?
M33 的图像和相应的旋转曲线（Corbelli 和 Salucci 2000）。引起引力场大异常的究竟是什么？（i）恒星盘周围存在一个（新的）非发光的大质量组分，还是（ii）一个（新的）暗物质的新物理？

In the past 30 years, in the preferred $\text{Λ}$CDM scenario, the complementary approach of detecting messengers of the dark particle and creating it at colliders has brought over an extraordinary theoretical and experimental effort that, however, has not reached a positive result. Moreover, on the scales $<50\text{kpc}$, where great part of the DM resides, there is a growing evidence of increasingly quizzical properties of the latter are, so that, a complex and surprising scenario, of very difficult understanding, is emerging.
在过去的 30 年中，在首选的 $\text{Λ}$ CDM 方案中，探测暗粒子信使并在对撞机中创造暗粒子的互补方法带来了非凡的理论和实验努力，然而，尚未取得积极的结果。此外，在 $<50\text{kpc}$ 的尺度上，大部分暗物质存在的地方，越来越多的证据显示出后者具有越来越令人费解的特性，因此，一个复杂且令人惊讶的场景正在出现，非常难以理解。

1.1 Scope of the review

The distribution of matter in galaxies does not seem to be the final act of a simple and well-understood history which has developed itself over the whole age of the Universe. It seems, instead, to lead to one of the two following possibilities: (1) the dark particle is a WIMP; however, baryons enter, heavily and in a very tuned way, into the process of galaxy formation, modifying, rather than following, the original DM distribution (2) the dark particle is something else, likely interacting with SM particle(s) and very likely lying beyond our current ideas of physics.
星系中物质的分布似乎不是一个简单且被充分理解的历史的最终结果，这个历史在整个宇宙的年龄中发展。相反，它似乎导致以下两种可能性之一：（1）暗物质粒子是 WIMP；然而，重子以一种非常调谐的方式进入星系形成过程，修改而不是遵循原始的暗物质分布（2）暗物质粒子是其他东西，可能与标准模型粒子相互作用，并且很可能超出我们当前的物理学理念。

In both cases, investigating deeply the distribution of dark matter in galaxies is necessary and worthwhile. In the first case, the peculiar imprint that baryons leave on the original distribution of the dark particles can serve us as an indirect, but telling, investigation of the latter. In the second case, with no guidance from first principles, a most complete investigation of the dark matter distribution in galaxies is essential to grasp its nature.
在这两种情况下，深入研究星系中暗物质的分布是必要且值得的。在第一种情况下，重子物质在暗粒子原始分布上留下的特殊印记可以作为对后者间接但有力的调查。在第二种情况下，在没有第一原则的指导下，对星系中暗物质分布进行最完整的调查对于理解其本质是至关重要的。

In any case, it is now possible to investigate such issue in galaxies of various morphological types and luminosities. We are sure that this will help us to shed light on the unknown physics underlying the dark matter mystery.
无论如何，现在可以研究各种形态和亮度的星系中的这类问题。我们相信这将帮助我们揭示黑暗物质之谜背后的未知物理学。

There are no doubts that the topic of this review is related and, in some case, even entangled with other main topics of cosmology and astroparticle physics. However, this work will be kept focused on the properties of dark matter where it mostly resides. Then, a number of issues, yet linked to the dark matter in galaxies, will not be dealt here or will be dealt in a very schematic way. This, both because we sense that looking for the “naked truth” of the galactic dark matter is the best way to approach the related mystery and because there are recent excellent reviews, suitable to complete the whole picture of dark matter in galaxies. These include: “The Standard Cosmological Model: Achievements and Issues” (Ellis 2018), standard and exotic dark-matter candidate particles and their related searches and productions (Roszkowski et al. 2017; Lisanti 2017), the $\text{Λ}$CDM scenario and its observational challenges (Naab and Ostriker 2017; Somerville and Dave 2015; Bullock and Boylan-Kolchin 2017; Turner 2018), “The Connection Between Galaxies and Their Dark Matter Halos” (Hudson et al. 2015; Wechsler and Tinker 2018), “Status of dark matter in the universe” (Freese 2017), “Galaxy Disks” (van der Kruit and Freeman 2011) and “Chemical Evolution of Galaxies” (Matteucci 2012). In addition, in the next sections, when needed, I will indicate the readers the papers that extend and deepen the content here presented.
毫无疑问，本审查的主题与宇宙学和宇宙粒子物理学的其他主题相关，并且在某些情况下甚至纠缠在一起。然而，本文将专注于暗物质的性质，因为它主要存在于其中。因此，一些与星系中的暗物质有关的问题将不在此处讨论，或者将以非常概略的方式处理。这是因为我们感觉到寻找“银河暗物质的真相”是接近相关神秘的最佳方式，而且有最近的优秀审查文章，适合完整呈现星系中的暗物质整体图景。这些包括：“标准宇宙学模型：成就与问题”（Ellis 2018），标准和奇异暗物质候选粒子及其相关的搜索和产生（Roszkowski 等人 2017；Lisanti 2017）， $\text{Λ}$ CDM 情景及其观测挑战（Naab 和 Ostriker 2017；Somerville 和 Dave 2015；Bullock 和 Boylan-Kolchin 2017；Turner 2018），“星系与其暗物质晕之间的联系”（Hudson 等人 2015；Wechsler 和 Tinker 2018），“宇宙中暗物质的现状”（Freese 2017），“星系盘”（van der Kruit 和 Freeman 2011）和“星系的化学演化”（Matteucci 2012）。此外，在接下来的章节中，如有需要，我将指示读者查阅扩展和深化本文内容的论文。

Let us stress that, although in this review one can find several observational evidences that can be played in disfavour of the $\text{Λ}$CDM scenario, this review is not meant to be a collection of observational challenges to such scenario and several issues at such regard, e.g., Müller et al. (2018), will not be considered here.
让我们强调，尽管在这篇评论中可以找到几个观察证据，可以对 $\text{Λ}$ CDM 情景不利，但这篇评论并不意味着是对这种情景的观察挑战的集合，也不会考虑在这方面的几个问题，例如，Müller 等人（2018 年）的研究。

It is worth pointing out that here we do not consider the theories alternative to the DM, that is, theories that dispose of the dark particle. The main reasons are (1) space: an honest account of them will require to add many more pages to this longish review and (2) my personal bias: no success in explaining the observations at galactic scale can compensate the intrinsic inability that these theories have in conceiving the galaxy formation process and interpreting the corpus of the cosmological observations.
值得指出的是，我们在这里不考虑与 DM 相反的理论，即那些不包含暗物质粒子的理论。主要原因是：（1）空间：对它们的诚实描述将需要在这篇冗长的评论中添加更多页面；（2）我的个人偏见：在解释星系尺度观测方面没有成功，无法弥补这些理论在构想星系形成过程和解释宇宙观测数据方面的固有无能。

1.2 The presence of dark matter in galaxies
1.2 星系中暗物质的存在

Let us introduce the “phenomenon” of dark matter in galaxies as it follows:let M(r) be the mass distribution of the gravitating matter and $M_L(r)$ that of the sum of all the luminous components. Let us notice that the radial logarithmic derivative of both mass profiles can be obtained from observations. Then, we realize that in spirals, for $r>r_T$, they do not match, in detail: $\mathrm{d}\log M/\mathrm{d}\log r>\mathrm{d}\log M_L/\mathrm{d}\log r$ (see Fig. 1, where the transition radius $r_T\simeq 4\text{kpc}$). Then, we introduce a non luminous component whose mass profile $M_H(r)$ accounts for the disagreement:
让我们介绍星系中暗物质的“现象”如下：让 M(r)表示引力物质的质量分布， $M_L(r)$ 表示所有发光组分的总和。让我们注意到，两个质量分布的径向对数导数可以从观测中获得。然后，我们意识到在螺旋星系中，对于 $r>r_T$ ，它们不匹配，具体来说： $\mathrm{d}\log M/\mathrm{d}\log r>\mathrm{d}\log M_L/\mathrm{d}\log r$ （见图 1，其中过渡半径 $r_T\simeq 4\text{kpc}$ ）。然后，我们引入一个非发光组分，其质量分布 $M_H(r)$ 解释了这种不一致：

$$\begin{array}{r}\frac{\mathrm{d}\log M(r)}{\mathrm{d}\log r}=\frac{M_L(r)}{M(r)}\frac{\mathrm{d}\log M_L}{\mathrm{d}\log r}+\frac{M_H(r)}{M(r)}\frac{\mathrm{d}\log M_H}{\mathrm{d}\log r}.\end{array}$$

(1)

The above immediately shows that the phenomenon of the mass discrepancy in galaxies emerges from the discordance between the value of the radial logarithmic derivative of the total mass profile and that of the luminous mass profile. We need to insert in the r.h.s. of Eq. (1) an additional (dark) term. This also implies that the DM phenomenon emerges observationally and can be investigated only if we are able to accurately measure the distribution of luminous and gravitating matter. In fact, the rotation curves $V(r)\propto (M(r)/r{)}^{1/2}$ have a property which is rarely found in astrophysics. We start with the fact that a good determination of the logarithmic derivative $\mathrm{\nabla }\equiv \mathrm{d}\log V/\mathrm{d}\log r$ is essential to successfully mass model a galaxy. Now, the analysis of N individual RCs with the same value of $\mathrm{\nabla }={\mathrm{\nabla }}_0$ and with a large uncertainty, e.g., $\delta {\mathrm{\nabla }}_0=0.2$ gives much less information on the mass distribution than one single RC with $\delta {\mathrm{\nabla }}_0=\pm 0.2/\sqrt{N}$. In short, a RC with large uncertainties gives no information on the underlying galaxy mass distribution.
以上立即显示出星系中的质量差异现象是由总质量剖面的径向对数导数值与发光质量剖面之间的不一致引起的。我们需要在方程（1）的右侧插入一个额外的（暗）项。这也意味着 DM 现象是从观测中出现的，只有当我们能够准确测量发光和引力物质的分布时才能进行研究。事实上，旋转曲线具有一种在天体物理学中很少见的特性。我们从一个事实开始，即良好地确定对数导数是成功地对星系进行质量建模所必需的。现在，对具有相同 $\mathrm{\nabla }={\mathrm{\nabla }}_0$ 值且具有较大不确定性的 N 个单独 RC 的分析，例如 $\delta {\mathrm{\nabla }}_0=0.2$ ，提供的关于质量分布的信息要比具有 $\delta {\mathrm{\nabla }}_0=\pm 0.2/\sqrt{N}$ 的单个 RC 少得多。简而言之，具有较大不确定性的 RC 对底层星系质量分布没有提供任何信息。

There is, however, a way to exploit the information carried by the low-quality RCs, namely, to properly stack them in coadded curves, killing so large part of their random uncertainties.
然而，有一种方法可以利用低质量的 RCs 携带的信息，即适当地将它们堆叠在合并曲线中，从而消除它们的随机不确定性的很大部分。

The luminous components of galaxies show a striking variety in morphology and in the values of their structural quantities. The range in magnitudes and central surface brightness are 15 mag and 16 mag/${\text{arcsec}}^2$. The distribution of the luminous matter in spirals is given by a stellar disk + a stellar central bulge and an extended HI disk and in ellipticals and dSphs by a stellar spheroid.
星系的发光组件在形态和结构数量的值上显示出明显的多样性。星系的亮度范围和中心表面亮度为 15 等和 16 等/ ${\text{arcsec}}^2$ 。螺旋星系中发光物质的分布由恒星盘+恒星中心球状体和扩展的 HI 盘组成，而椭圆星系和 dSphs 由恒星球状体组成。

How will the variety of the properties of the luminous matter contrast with the organized uniformity of the dark matter? The phenomenological scenario of dark matter in galaxies that we discuss in this review has to be considered as a privileged way to understand what dark matter halos are made of and to approach the involved (new) laws or processes of Nature.
发光物质的性质多样性将如何与暗物质的有序统一形成对比？我们在这篇评论中讨论的星系暗物质的现象学场景必须被视为理解暗物质晕所由之物质以及接近所涉及的（新的）自然规律或过程的特权方式。

Freeman (1970), in its Appendix A, first drew the attention of the astrophysical community to a discrepancy between the kinematics and the photometry of the spiral galaxy NGC 300 that implied the presence of large amounts of non-luminous matter. Then, during the 1970s the contribution of Morton Roberts to the cause of DM in galaxies has been crucial (Bullock and Boylan-Kolchin 2017). A next topical moment was when Vera Rubin published 20 optical RCs extended out to 0.8 the optical radii $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ that were still rising or flattish at the last measured point (Rubin et al. 1980) and a decisive kinematics was obtained by means of several 21-cm rotation curves extended out to the optical radii (Bosma 1981a, b). Moreover, we have to mention the Faber and Gallagher (1979) review that played a very important role to spread the idea of a dark halo component in galaxies.¹
Freeman（1970 年）在其附录 A 中首次引起了天体物理学界对螺旋星系 NGC 300 的运动学和光度之间存在的不一致性的关注，这暗示存在大量非发光物质。然后，在 1970 年代，Morton Roberts 对星系中暗物质的贡献至关重要（Bullock 和 Boylan-Kolchin 2017）。下一个重要时刻是 Vera Rubin 发表了 20 个光学 RCs，延伸至 0.8 倍光学半径，最后一个测量点仍在上升或趋于平缓（Rubin 等人，1980 年），并通过几个 21 厘米旋转曲线延伸至光学半径获得了决定性的运动学（Bosma 1981a，b）。此外，我们还要提到 Faber 和 Gallagher（1979 年）的综述对传播星系中暗晕成分的概念起到了非常重要的作用。

In this brief historical account of the discovery of dark matter in galaxies, one point should still be made. Until to few years ago, the nature of dark matter was not meant to be determined by the properties of the galaxy gravitational field, but to come from first principles verified by large-scale observations. In this review, instead, we will follow also a reverse-engineering approach: the unknown nature of the DM is searched within the (complex) observational properties of the dark halos in galaxies.
在这个关于在星系中发现暗物质的简要历史记载中，仍然应该强调一点。直到几年前，暗物质的性质并不是通过星系引力场的特性来确定的，而是来自大尺度观测验证的第一原则。在这篇评论中，相反，我们也将采用一种逆向工程的方法：在星系暗晕的（复杂）观测特性中寻找暗物质的未知性质。

2 The invisible character, dark particles and co.
2 不可见的字符，暗粒子等。

It is worth starting this review with a brief account of the dark matter candidate particles presently in the ballpark; one has to keep on mind, however, that there are likely to risk not to be “the” DM particle.
值得开始这篇评论的是对目前在讨论范围内的暗物质候选粒子的简要介绍；然而，必须记住，很可能存在风险，这些并非“暗物质”粒子。

2.1 Collisionless and cold dark particles
无碰撞和冷暗物质

Let us start by recalling the motivations that have led to about 30 years of fascination with the Weakly-Interacting Massive Particles (WIMPs) and especially with the lightest supersymmetric particle (Steigman and Turner 1985; see also Kolb and Turner 1990). At high temperatures, ($T\gg m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}$), WIMPs are thermally created and destroyed. As the temperature of the Universe decreases due to its expansion, the density is exponentially suppressed ($\propto \exp [-m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}/T]$) and becomes no longer high enough to pair-create them. When the WIMP mean free path is comparable to the Hubble distance, the particles also cease to annihilate, leave the thermal equilibrium state and “freeze-out”. At this point, the co-moving density remains constant. The temperature for which the freeze-out occurs is about 5% of the WIMP mass. Therefore, the (relic) density becomes constant when the particles are non-relativistic. The value of the relic density ${\text{Ω}}_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}$ depends only on the total annihilation cross-section ${\text{σ}}_A$ and the particles’ relative velocity $|\mathbf{v}|$:
让我们从回顾导致大约 30 年对弱相互作用重粒子（WIMPs）以及特别是最轻的超对称粒子（Steigman 和 Turner 1985;另见 Kolb 和 Turner 1990）产生兴趣的动机开始。在高温下（ $T\gg m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}$ ），WIMPs 被热产生和破坏。随着宇宙由于膨胀而降温，密度呈指数抑制（ $\propto \exp [-m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}/T]$ ），不再足够高以进行成对产生。当 WIMP 的平均自由程与哈勃距离相当时，粒子也停止湮灭，离开热平衡状态并“冻结”。在这一点上，共动密度保持恒定。冻结发生的温度约为 WIMP 质量的 5%。因此，（残余）密度在粒子非相对论时变得恒定。残余密度的值 ${\text{Ω}}_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}$ 仅取决于总湮灭截面 ${\text{σ}}_A$ 和粒子的相对速度 $|\mathbf{v}|$ ：

$$\begin{array}{r}{\text{Ω}}_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}\simeq \frac{6\times {10}^{-27}{\mathrm{c}\mathrm{m}}^3/\mathrm{s}}{⟨{\text{σ}}_A|\mathbf{v}|⟩},\end{array}$$

(2)

The scale of weak interaction strength ($\sim {\alpha }^2/m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}^2$) implies that $<{\text{σ}}_A|\mathbf{v}|>{10}^{-25}{\mathrm{c}\mathrm{m}}^3/\mathrm{s}$, where ${\text{σ}}_A$ is the cross section and the WIMP mass is taken to be 100 GeV. The resulting relic density for such a particle would be within a factor 3 of the measured value of the dark matter density ${\text{Ω}}_m$ (e.g., Planck Collaboration et al. 2016). This remarkable coincidence is referred to as the “WIMP miracle.” This particle, today, should interact with ordinary matter only through weak interaction, in addition to the gravitational one. The former should occur via the exchange of a scalar particle, or a vector boson interaction. These interactions together with the particle–particle annihilations ongoing in the densest region of the Universe, would make the particle detectable.
弱相互作用强度的尺度（ $\sim {\alpha }^2/m_{\mathrm{W}\mathrm{I}\mathrm{M}\mathrm{P}}^2$ ）意味着 $<{\text{σ}}_A|\mathbf{v}|>{10}^{-25}{\mathrm{c}\mathrm{m}}^3/\mathrm{s}$ ，其中 ${\text{σ}}_A$ 是截面，WIMP 质量取为 100 GeV。这种粒子的结果遗留密度将在暗物质密度 ${\text{Ω}}_m$ 的测量值的 3 倍范围内。这种显著的巧合被称为“WIMP 奇迹”。这种粒子今天应该只通过弱相互作用与普通物质相互作用，除了引力作用。前者应该通过交换标量粒子或矢量玻色子相互作用发生。这些相互作用以及在宇宙最密集区域进行的粒子-粒子湮灭将使粒子可检测到。

It is known that this scenario reproduces a wealth of cosmological observations, particularly on scales $>10$ Mpc. On the other hand, WIMPs have so far escaped detection (see Figs. 2, 3) and, furthermore, there is a number of small-scale issues that put in question their being the dark particle in galaxies.
众所周知，这种情景再现了丰富的宇宙观测数据，特别是在 $>10$ Mpc 的尺度上。另一方面，迄今为止暗物质粒子 WIMPs 尚未被探测到（见图 2、3），而且还存在一些小尺度问题，质疑它们是否是星系中的暗物质粒子。

Fig. 2

Top: Current 90% CL exclusion plots to the effective WIMP–proton cross section, see Kang et al. (2018)
顶部：当前 90% CL 排除图显示有效 WIMP-质子截面，详见 Kang 等人（2018 年）。

Fig. 3

Current 90% CL exclusion plots to the effective WIMP–nucleon cross section. Image reproduced with permission from Aprile et al. (2018), copyright by APS
当前 90%置信水平排除图显示了有效 WIMP-核子截面。图像经授权从 Aprile 等人（2018）复制，版权由 APS 所有。

2.2 An unexpected new candidate for cold dark particles
2.2 冷暗物质的意外新候选者

There might be a connection between the dark matter in galaxies, in particular the cold DM and the gravitational waves produced by the merging of stellar-mass black holes and possibly detectable by LIGO-Virgo experiments. This is due to the intriguing possibility that DM consists of black holes created in the very early Universe. In this case, the detection of primordial black hole binaries could provide an unambiguous observational window to pin down the nature of dark matter (Green 2016). These objects are also detectable as effect of their continuous merging since recombination. This violent process can have generated a stochastic background of gravitational waves that could be detected by LISA and PTA (see also García-Bellido 2017).
宇宙中的暗物质，尤其是冷暗物质与由恒星质量黑洞合并产生的引力波之间可能存在联系，这些引力波可能可以被 LIGO-Virgo 实验所探测到。这是因为有趣的可能性是，暗物质可能由早期宇宙中产生的黑洞组成。在这种情况下，原始黑洞双星的探测可以为确定暗物质的本质提供一个明确的观测窗口（Green 2016）。这些物体也可以通过它们自复合以来持续合并的效果来被探测到。这种暴力过程可能产生了引力波的随机背景，可以被 LISA 和 PTA 所探测到（另见 García-Bellido 2017）。

It is known that massive primordial black holes form at rest with respect to the flow of the expanding Universe and then with zero spin. Moreover, they have negligible cross-section with the ordinary matter and constitute a right candidate for the $\text{Λ}$CDM scenario (see, however, Koushiappas and Loeb 2017). Of course, just substituting WIMPs with primordial BHs does not immediately relieve the severe tension with the observations at galactic scales that these particles have. However, the question whether these primordial BHs could have some sort of interaction with baryons, which is instead forbidden to WIMPs, is open .
众所周知，大量原始黑洞在相对于膨胀宇宙的流动静止时形成，然后具有零自旋。此外，它们与普通物质的交叉面积微乎其微，并且构成了 $\text{Λ}$ CDM 方案的合适候选者（但请参见 Koushiappas 和 Loeb 2017）。当然，仅仅用原始黑洞替代 WIMPs 并不能立即缓解这些粒子在星系尺度上与观测结果之间的严重矛盾。然而，关于这些原始黑洞是否可能与重子有某种交互作用的问题，而这对于 WIMPs 是被禁止的，目前尚未解答。

2.3 Self-interacting DM particles
2.3 自相互作用的暗物质粒子

Self-interacting dark matter (SIDM) particles were proposed by Spergel and Steinhardt (2000) (see also Boddy et al. 2014; Bode et al. 2001) to solve the core–cusp and missing satellites problems (see also Tulin et al. 2013; Bellazzini et al. 2013). DM particles scatter elastically with each other through 2–2 interactions and, as low-entropy particles, are heated by elastic collisions within the dense inner halo and leave the region: the central and nearby densities are then reduced, turning an original cusp into a core. The collision rate is:
自相互作用暗物质（SIDM）粒子是由 Spergel 和 Steinhardt（2000 年）提出的（另见 Boddy 等人 2014 年；Bode 等人 2001 年），旨在解决核心-尖峰和缺失卫星问题（另见 Tulin 等人 2013 年；Bellazzini 等人 2013 年）。暗物质粒子通过 2-2 相互作用彼此弹性散射，并作为低熵粒子，在密集的内部暗物质晕中通过弹性碰撞受热并离开该区域：然后中心和附近的密度减少，将原始尖峰转变为核心。碰撞速率为：

$$\begin{array}{r}{R}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}=\text{σ}{v}_{\mathrm{r}\mathrm{e}\mathrm{l}}{\rho }_{\mathrm{D}\mathrm{M}}/m\approx 0.1{\mathrm{G}\mathrm{y}\mathrm{r}}^{-1}\times \left(\frac{{\rho }_{\mathrm{D}\mathrm{M}}}{0.1M_{\odot }/{\mathrm{p}\mathrm{c}}^3}\right)\left(\frac{v_{\mathrm{r}\mathrm{e}\mathrm{l}}}{50\mathrm{k}\mathrm{m}/\mathrm{s}}\right)\left(\frac{\text{σ}/m}{1\mathrm{c}{\mathrm{m}}^2/\mathrm{g}}\right),\end{array}$$

(3)

where m is the DM particle mass, $\text{σ},v_{\mathrm{r}\mathrm{e}\mathrm{l}}$ are the cross section and relative velocity for scattering. Within the central region of a typical dwarf galaxy we have: ${\rho }_{\mathrm{D}\mathrm{M}}\sim 0.1M_{\odot }/{pc}^3$ and $v_{\mathrm{r}\mathrm{e}\mathrm{l}}\sim 50\mathrm{k}\mathrm{m}/\mathrm{s}$. Therefore, the cross section per unit mass ($\text{σ}/m$) must be at least:
其中 m 是 DM 粒子的质量， $\text{σ},v_{\mathrm{r}\mathrm{e}\mathrm{l}}$ 是散射的截面和相对速度。在典型矮星系的中心区域，我们有： ${\rho }_{\mathrm{D}\mathrm{M}}\sim 0.1M_{\odot }/{pc}^3$ 和 $v_{\mathrm{r}\mathrm{e}\mathrm{l}}\sim 50\mathrm{k}\mathrm{m}/\mathrm{s}$ 。因此，每单位质量的截面积（ $\text{σ}/m$ ）至少必须是：

$$\begin{array}{r}\text{σ}/m\sim 1\mathrm{c}{\mathrm{m}}^2/\mathrm{g}\approx 2\times {10}^{-24}\mathrm{c}{\mathrm{m}}^2/\mathrm{G}\mathrm{e}\mathrm{V}\end{array}$$

(4)

to have an effect; this corresponds to about one scattering per particle over 10 Gyr galactic timescales. With the above value of $\text{σ}/m$, $R_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$ is negligible during the early Universe when structures form. SIDM, therefore, retains the success of large-scale structure formation of the $\text{Λ}$CDM scenario and affects the dark structures on small scales only once they are already virialized.
对于产生影响；这相当于在 10 亿年的星系时间尺度上每个粒子约有一次散射。在早期宇宙形成结构时， $\text{σ}/m$ 的上述值是可以忽略的。因此，SIDM 保留了 $\text{Λ}$ CDM 方案大尺度结构形成的成功，并且只在小尺度上影响暗物质结构，一旦它们已经处于维里化状态。

The self-interacting dark matter is then a cusp–core density profile transformer (e.g., Vogelsberger et al. 2014; Zavala et al. 2013; Kaplinghat et al. 2015). As result of the annihilation among these particles in the denser inner regions of the galactic halos, the originally cuspy DM density becomes constant with radius. Outside the core region, the number of annihilations rapidly falls as ${\rho }_{\mathrm{D}\mathrm{M}}^2(r)$ and the halo profile remains identical to the original one.
自相互作用的暗物质是一个尖峰-核心密度剖面转换器（例如，Vogelsberger 等人 2014 年；Zavala 等人 2013 年；Kaplinghat 等人 2015 年）。由于这些粒子在星系暗物质晕内更密集的区域之间的湮灭，最初尖峰状的暗物质密度随半径变化而变得恒定。在核心区域之外，湮灭的数量迅速下降，如 ${\rho }_{\mathrm{D}\mathrm{M}}^2(r)$ ，而暗物质晕剖面保持与原始剖面相同。

2.4 FUZZY dark particles

The idea is that the dark matter is a scalar dark particle of mass $m_a\sim {10}^{-22}\mathrm{e}\mathrm{V}$. At large scales its coherent macroscopic excitations can mimic the behaviour of the cold dark matter (CDM). At the scale of galaxies, however, this particle has macroscopic wave-like properties that may explain the classic “discrepancies” of the standard DM scenario (Weinberg 1978; Hui et al. 2017; Bernal et al. 2017; Ringwald 2012).
这个想法是暗物质是一种质量为 $m_a\sim {10}^{-22}\mathrm{e}\mathrm{V}$ 的标量暗粒子。在大尺度上，它的相干宏观激发可以模拟冷暗物质（CDM）的行为。然而，在星系尺度上，这种粒子具有宏观波动特性，可以解释标准暗物质方案的经典“差异”（Weinberg 1978 年；Hui 等人 2017 年；Bernal 等人 2017 年；Ringwald 2012 年）。

Once in galaxies, these particles behave as Bose–Einstein condensate (BEC); in this model, the inter-particle distance is much smaller than their de Broglie wave length. The particles move collectively as a wave: their equation of state can lead to cored configuration like those observed. The capability to detect such Bose–Einstein-condensed scalar field dark matter with the LIGO experiment is under analysis (Li et al. 2017).
在星系中，这些粒子表现为玻色-爱因斯坦凝聚态（BEC）；在这个模型中，粒子之间的距离远小于它们的德布罗意波长。这些粒子像波一样集体移动：它们的状态方程可以导致类似于观察到的核心配置。利用 LIGO 实验来检测这种玻色-爱因斯坦凝聚的标量场暗物质的能力正在进行分析（Li 等，2017）。

2.5 Warm dark matter particles
2.5 暖暗物质粒子

Warm dark matter (WDM) particle decouples from the cosmological plasma when it is still mildly relativistic. These particles can be created in the early Universe in a variety of ways (Dodelson and Widrow 1994; Shi and Fuller 1999; Kusenko 2009). In the case where the WDM consists of thermal relics, the suppression of small-scale power in the linear power spectrum (e.g., Bringmann 2016) $P_{\mathrm{W}\mathrm{D}\mathrm{M}}$, can be conveniently parametrized by reference to the CDM power spectrum $P_{\mathrm{C}\mathrm{D}\mathrm{M}}$, see Fig. 4. In the more likely cases in which the WDM particle is a non-resonantly produced sterile neutrino, its mass $m_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{e}}$, can be related to the mass of the equivalent thermal relic (Viel et al. 2005).This conversion depends on the specific particle production mechanism.
暖暗物质（WDM）粒子在宇宙等离子体仍处于轻度相对论状态时脱离。这些粒子可以通过多种方式在早期宇宙中产生（Dodelson 和 Widrow 1994；Shi 和 Fuller 1999；Kusenko 2009）。在 WDM 由热力学遗物组成的情况下，线性功率谱中小尺度功率的抑制（例如，Bringmann 2016）可以方便地通过参考 CDM 功率谱来参数化，参见图 4。在更有可能的情况下，WDM 粒子是非共振产生的无共振中微子，其质量可以与等效热力学遗物的质量相关联（Viel 等人 2005）。这种转换取决于特定的粒子产生机制。

Fig. 4

Linear power spectra in $\text{Λ}$CDM (black line) and $\text{Λ}$WDM (coloured lines) scenarios. $\text{Λ}$WDM models are labelled by their thermal relic mass and value of the damping scale $\alpha$. We have $(\frac{P_{\mathrm{W}\mathrm{D}\mathrm{M}}}{P_{\mathrm{C}\mathrm{D}\mathrm{M}}}{)}^{1/2}=[1+(\alpha k{)}^{2/1.1}{]}^{-5/1.1}$, k is the wave-number. Image reproduced with permission from Kennedy et al. (2014), copyright by the authors
在 $\text{Λ}$ CDM（黑线）和 $\text{Λ}$ WDM（彩色线）情景中的线性功率谱。 $\text{Λ}$ WDM 模型通过它们的热弃子质量和阻尼尺度的数值进行标记 $\alpha$ 。我们有 $(\frac{P_{\mathrm{W}\mathrm{D}\mathrm{M}}}{P_{\mathrm{C}\mathrm{D}\mathrm{M}}}{)}^{1/2}=[1+(\alpha k{)}^{2/1.1}{]}^{-5/1.1}$ ，其中 k 是波数。图像经 Kennedy 等人（2014）许可复制，作者版权所有。

Given the mass of this particle being about 2 keV, its de Broglie length-scale is of the order of 30 kpc, so that, inside the optical region of galaxies a quantum pressure emerges (Destri et al. 2013; de Vega and Sanchez 2017) and plays a role in the equilibrium of the structures. The DM particles follow, then, a Fermi–Dirac distribution:
鉴于该粒子的质量约为 2 千电子伏特，其德布罗意长度尺度约为 30 千秒差距，因此，在星系的光学区域内会出现量子压力（Destri 等，2013 年；de Vega 和 Sanchez，2017 年），并在结构的平衡中发挥作用。然后，暗物质粒子遵循费米-狄拉克分布：

$$\begin{array}{r}{f}_{\mathrm{F}\mathrm{D}}(p;T,\mu )=\frac{g}{(2\pi \hslash {)}^3}\frac{1}{\exp [(E-\mu )/T]+1},\end{array}$$

(5)

where p and $E=p^2/(2m)$ are the momentum and the single-particle kinetic energy; T(r), expressed in terms of energy, is the average temperature of DM particles at a radius r: $T(r)\propto V^2(r)$ in spirals and $T(r)\propto {\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}^2(r)$ in pressure dominated systems. Noticeably, f(p) has an upper limit: $f(p)\le \frac{g}{(2\pi \hslash {)}^3}$, where g is the number of internal degrees of freedom. We have, in this case, that the quantum pressure and not the Gravity Force shapes the inner DM density profile. WDM particles can be detected: they can produce a monochromatic gamma ray line at $2m_{\mathrm{W}\mathrm{D}\mathrm{M}}\mathrm{k}\mathrm{e}\mathrm{V}$, which is constrained by X-ray measurements, e.g., Boyarsky et al. (2007).
其中 p 和 $E=p^2/(2m)$ 分别是动量和单粒子动能；T(r)用能量表示，是半径 r 处 DM 粒子的平均温度： $T(r)\propto V^2(r)$ 在螺旋中， $T(r)\propto {\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}^2(r)$ 在压力主导系统中。值得注意的是，f(p)有一个上限： $f(p)\le \frac{g}{(2\pi \hslash {)}^3}$ ，其中 g 是内部自由度的数量。在这种情况下，量子压力而不是引力力塑造了内部 DM 密度分布。WDM 粒子可以被探测到：它们可以在 $2m_{\mathrm{W}\mathrm{D}\mathrm{M}}\mathrm{k}\mathrm{e}\mathrm{V}$ 产生单色伽玛射线，这受到 X 射线测量的限制，例如 Boyarsky 等人（2007 年）。

The properties of WDM particles, their scientific case and cosmological role and the various strategies to detect them, have recently been presented in a White Paper (Adhikari et al. 2017).
WDM 粒子的性质、它们的科学案例和宇宙学角色以及检测它们的各种策略，最近在一份白皮书中被提出（Adhikari 等，2017 年）。

2.6 In search for dark matter

For 30 years, WIMPs have been the first target in our attempt to detect and identify the dark particle. During the past decades, the sensitivity of the experiments involved has improved by three to four orders of magnitude, but an evidence for their existence is yet to come. On the other hand, searches at hadron colliders (which attempts to produce WIMPs through the collision of high-energy protons and the subsequent formation of stable dark matter particles that can be identified through the production of quarks and gluons), have given no result (see Butler 2018).
在过去的 30 年里，WIMPs 一直是我们尝试探测和识别暗物质粒子的首要目标。在过去几十年里，涉及的实验灵敏度提高了三到四个数量级，但对它们存在的证据尚未出现。另一方面，在强子对撞机的搜索中（试图通过高能质子的碰撞产生 WIMPs，随后形成稳定的暗物质粒子，可以通过夸克和胶子的产生来识别），没有得到任何结果（参见 Butler 2018）。

It is agreed that no conclusive detection signal of the particle has yet arrived as result of a many year-long extensive search program that combined, in a complementary way, direct, indirect, and collider probes (see Arcadi et al. 2018 for a detailed review).
人们一致认为，作为一个长达多年的广泛搜索计划的结果，尚未出现粒子的确切探测信号，该计划以直接、间接和对撞机探测相结合的方式进行（详见 Arcadi 等人 2018 年的详细评论）。

However, it is worth discussing astrophysical aspects, related to the above searches that have an intrinsic importance and that are valid also for any particle investigation. In direct searches, the differential event rate $R_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$
然而，值得讨论与上述搜索相关的天体物理学方面，这些方面具有固有重要性，对于任何粒子研究也是有效的。在直接搜索中，差分事件率 $R_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$

$$\begin{array}{r}\frac{\mathrm{d}{R}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}}{\mathrm{d}E}\propto g(v_{min})\rho (R_{\odot })\end{array}$$

(6)

is proportional to $\rho (R_{\odot })$ the local (i.e., at the solar radius) dark matter density and to the function $g(v_{min})={\int }_{v>v_{min}}^{v_{\mathrm{e}\mathrm{s}\mathrm{c}}}\frac{f(\mathbf{v})}{v}{\mathrm{d}}^3\mathbf{v}$. $v_{min}$ is the minimum particle speed that can cause in the detector a recoil of energy E (Gondolo 2002).
与 $\rho (R_{\odot })$ 本地（即在太阳半径处）暗物质密度成正比，且与函数 $g(v_{min})={\int }_{v>v_{min}}^{v_{\mathrm{e}\mathrm{s}\mathrm{c}}}\frac{f(\mathbf{v})}{v}{\mathrm{d}}^3\mathbf{v}$ 成正比。 $v_{min}$ 是能够在探测器中引起能量 E 的最小粒子速度（Gondolo 2002）。

$v_{\mathrm{e}\mathrm{s}\mathrm{c}}$ is the escape velocity from the Milky Way: $v_{\mathrm{e}\mathrm{s}\mathrm{c}}=(570\pm 120)\mathrm{k}\mathrm{m}/\mathrm{s}$ (Nesti and Salucci 2013). A reference value of $\rho (R_{\odot })=0.3\mathrm{G}\mathrm{e}\mathrm{V}/{\mathrm{c}\mathrm{m}}^3$ is often adopted; however, recent accurate determinations indicate a rather higher value: $\rho (R_{\odot })=(0.43\pm 0.06)\mathrm{G}\mathrm{e}\mathrm{V}/{\mathrm{c}\mathrm{m}}^3$ (Salucci et al. 2010; Catena and Ullio 2010).
$v_{\mathrm{e}\mathrm{s}\mathrm{c}}$ 是从银河系逃逸速度： $v_{\mathrm{e}\mathrm{s}\mathrm{c}}=(570\pm 120)\mathrm{k}\mathrm{m}/\mathrm{s}$ （Nesti 和 Salucci 2013）。通常采用的参考值为 $\rho (R_{\odot })=0.3\mathrm{G}\mathrm{e}\mathrm{V}/{\mathrm{c}\mathrm{m}}^3$ ；然而，最近准确的测定表明一个相当更高的数值： $\rho (R_{\odot })=(0.43\pm 0.06)\mathrm{G}\mathrm{e}\mathrm{V}/{\mathrm{c}\mathrm{m}}^3$ （Salucci 等人 2010；Catena 和 Ullio 2010）。

To obtain $g(v_{min})$, one needs the whole DM density distribution; however, for the Milky Way, we can consider the galaxy halo as an isotropic isothermal sphere with density profile $\rho (r)\propto r^{-2}$. Then $f(\mathbf{v})=\frac{N}{2\pi {\text{σ}}_v^2}\exp (-\frac{{\mathbf{v}}^2}{2{\text{σ}}_v^2})$, where N is a normalization constant and ${\text{σ}}_v$ is the DM particles’ one-dimensional velocity dispersion, which in the present model is related to the circular velocity V(r) by: ${\text{σ}}_v=V(r)/\sqrt{2}$.
要获得 $g(v_{min})$ ，需要整个 DM 密度分布；然而，对于银河系，我们可以将星系暗物质晕视为具有密度分布 $\rho (r)\propto r^{-2}$ 的各向同性等温球体。然后 $f(\mathbf{v})=\frac{N}{2\pi {\text{σ}}_v^2}\exp (-\frac{{\mathbf{v}}^2}{2{\text{σ}}_v^2})$ ，其中 N 是一个归一化常数， ${\text{σ}}_v$ 是 DM 粒子的一维速度离散度，在当前模型中与圆速度 V(r)相关： ${\text{σ}}_v=V(r)/\sqrt{2}$ 。

The indirect searches of DM are based on astrophysical observations of the products of the DM particles’ self-annihilation (or decay) able to climb up the emissions coming from the likely astrophysical mechanisms also producing antiprotons and positrons. The photon spectrum $\frac{\mathrm{d}{N}_{\gamma }^f}{\mathrm{d}{E}_{\gamma }}$, with $E_{\gamma }$ the photon energy, is expected to be proportional to ${\int }_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}\mathrm{d}l{\rho }^2(r)$ for annihilations and ${\int }_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}\mathrm{d}l\rho (r)$ for decays; as usual, $\rho (r)$ is the DM density within the galaxy and the integrals are performed over the line of sight l. The dependence of $\rho (r)$ on the above fluxes leads to a dependence of the signal on the inner distribution of DM in galaxies, modulo the fraction between the size of the dark halo and that of the telescope beam both projected on the plane of the sky (for details including the application to the Galactic Center, see Gammaldi 2016). As consequence of that, indirect searches require an accurate knowledge of the halo density profiles and, in this perspective, one should also consider cored dark matter halo distributions, in performing the analysis on the $\gamma$ flux. Here, we do not further enter into this (important) issue (see, e.g., Gammaldi 2015).
暗物质的间接搜索基于对暗物质粒子自相互湮灭（或衰变）产物的天体物理观测，这些产物能够攀升出来自可能的天体物理机制产生的反质子和正电子的辐射。光子谱 $\frac{\mathrm{d}{N}_{\gamma }^f}{\mathrm{d}{E}_{\gamma }}$ ，其中 $E_{\gamma }$ 为光子能量，预计与湮灭有关的比例为 ${\int }_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}\mathrm{d}l{\rho }^2(r)$ ，与衰变有关的比例为 ${\int }_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}\mathrm{d}l\rho (r)$ ；通常情况下， $\rho (r)$ 是星系内的暗物质密度，积分是沿视线 l 进行的。 $\rho (r)$ 对上述通量的依赖性导致了信号对星系内暗物质的内部分布的依赖性，取决于暗暗物质晕和望远镜光束在天空平面上的投影尺寸之间的比例（有关详细信息，包括应用于银河中心的内容，请参见 Gammaldi 2016）。因此，间接搜索需要对晕密度分布有准确的了解，在这个视角下，应该考虑到核心暗物质晕分布，在对 $\gamma$ 通量进行分析时。在这里，我们不再深入讨论这个（重要）问题（参见，例如，Gammaldi 2015）。

3 Baryons in galaxies

The luminous components in galaxies show a striking variety in morphology and in dimensions. Noticeably, the total luminosity and the radius $R_{1/2}$ enclosing half of the latter are good tags of the objects.
星系中的发光组件在形态和尺寸上展现出令人惊人的多样性。值得注意的是，总发光度和包围其中一半的 $R_{1/2}$ 的半径是这些物体的良好标志。

3.1 Spirals, LSB and UDG

Caveat some occasional cases not relevant for the present topic, the stars are distributed in a thin disk with surface luminosity (Freeman 1970, for a study on 967 late type spirals, see Persic et al. 1996)
除了一些偶发情况与当前主题无关外，星星分布在一个薄盘中，具有表面亮度（Freeman 1970，对 967 个晚型螺旋星系的研究，请参见 Persic 等人 1996 年）。

$$\begin{array}{r}I(R)=I_0{\mathrm{e}}^{-R/R_D}=\frac{M_D}{2\pi R_D^2}{\mathrm{e}}^{-R/R_D}{\left(\frac{M_D}{L}\right)}^{-1},\end{array}$$

(7)

where $R_D=1/1.67R_{1/2}$ is the disk length scale, $I_0$ is the central value of the surface luminosity and $M_D$ is the disk mass. The light profile of late spirals does not depend on galaxy luminosity and the length scale $R_D$ sets a consistent reference spatial scale.²
其中 $R_D=1/1.67R_{1/2}$ 是盘长度尺度， $I_0$ 是表面亮度的中心值， $M_D$ 是盘质量。晚期螺旋星系的光线分布不依赖于星系亮度，长度尺度 $R_D$ 设定了一个一致的参考空间尺度。 ²

The contribution to the circular velocity from this stellar component is:
这颗恒星组件对圆周速度的贡献是：

$$\begin{array}{r}{V}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}^2(r)=\frac{G{M}_D}{2R_D}{x}^{2}B\left(\frac{x}{2}\right),\end{array}$$

(8)

where $x\equiv R/R_D$ and $B=I_0{K}_0-I_1{K}_1$, a combination of known Bessel functions.
在 $x\equiv R/R_D$ 和 $B=I_0{K}_0-I_1{K}_1$ 的情况下，是已知贝塞尔函数的组合。

Classical LSB galaxies usually have central surface brightness down to ${\mu }_B(0)\sim 22$–23 mag ${\text{arcsec}}^{-2}$ (Impey et al. 1988). Extremely low surface brightness (LSB) galaxies with unexpectedly large sizes, namely ultra-diffuse galaxies (UDGs), are found in nearby galaxy clusters (Bothun et al. 1991; Toloba et al. 2018). UDGs have much lower central surface brightness ($\mu (0)=24$–26 mag ${\text{arcsec}}^{-2}$ in g band and half-light radii $R_{1/2}>1.5\mathrm{k}\mathrm{p}\mathrm{c}$ that, in spirals, are found in objects with stellar masses more than 10 times higher (van Dokkum et al. 2015; Shi 2017). In LSBs/UDGs the stellar disks follow the Freeman exponential profile as in normal spirals, but their two structural parameters ($I_0$ and $R_D$) do not correlate as in the latter, where, approximately: $L_I\propto R_D^2$.
经典的低表面亮度（LSB）星系通常具有中心表面亮度下降到 ${\mu }_B(0)\sim 22$ -23 mag ${\text{arcsec}}^{-2}$ （Impey 等人 1988）。在附近的星系团中发现了具有意外大尺寸的极低表面亮度（LSB）星系，即超扩散星系（UDGs）（Bothun 等人 1991; Toloba 等人 2018）。 UDG 具有更低的中心表面亮度（ $\mu (0)=24$ -26 mag ${\text{arcsec}}^{-2}$ 在 g 波段和半光半径 $R_{1/2}>1.5\mathrm{k}\mathrm{p}\mathrm{c}$ 中，在螺旋星系中，这些星系的恒星质量超过 10 倍（van Dokkum 等人 2015; Shi 2017）。在 LSB / UDG 中，恒星盘遵循 Freeman 指数剖面，如同正常螺旋星系，但它们的两个结构参数（ $I_0$ 和 $R_D$ ）不像后者那样相关，大约为： $L_I\propto R_D^2$ 。

3.1.1 HI distribution in disk systems
3.1.1 盘系统中的 HI 分布

Spirals have a gaseous HI disk which usually is important only as tracer of the galaxy gravitational field. Only at the outer radii ($R>R_{\mathrm{o}\mathrm{p}\mathrm{t}}$) of low luminosity objects, such disk becomes the major baryonic component of the circular velocity and must be included in the galaxy velocity model.
螺旋星系有一个气态的 HI 盘，通常只作为星系引力场的追踪器。只有在低亮度物体的外半径（ $R>R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ）处，这种盘才成为圆速度的主要重子成分，并必须包括在星系速度模型中。

The HI disks show, very approximately, a Freeman distribution with a scale length about three times larger than that of the stellar disc (Evoli et al. 2011; Wang et al. 2014).
HI 盘显示，非常粗略地，一个 Freeman 分布，其尺度长度大约是恒星盘的三倍（Evoli 等人 2011; Wang 等人 2014）。

$$\begin{array}{r}{\mu }_{\mathrm{H}\mathrm{I}}(R)={\mu }_{\mathrm{H}\mathrm{I},0}{\mathrm{e}}^{-\frac{R}{3R_D}}\end{array}$$

(9)

A rough estimate of the contribution of the gaseous disc to the circular velocity is
气态盘对圆速度的贡献的粗略估计是

$$\begin{array}{r}{V}_{\mathrm{H}\mathrm{I}}(R{)}^2=1.3\left(\frac{M_{\mathrm{H}\mathrm{I}}}{9M_D}\right)V_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}^2\left(\frac{R}{3R_D}\right),\end{array}$$

(10)

where the coefficient 1.3 is due to the He contribution. Of course when the resolved HI surface density is available, one derives $V_{\mathrm{H}\mathrm{I}}(R{)}^2$ directly from the latter. Inner ${\text{H}}_2$ and CO disks are also present, but they are negligible with respect to the stellar and HI ones (Gratier et al. 2010; Corbelli and Salucci 2000).
其中系数 1.3 是由于氦的贡献。当可解析的 HI 表面密度可用时，可以直接从后者导出 $V_{\mathrm{H}\mathrm{I}}(R{)}^2$ 。内部 ${\text{H}}_2$ 和 CO 盘也存在，但它们相对于恒星和 HI 盘来说是可以忽略的（Gratier 等人 2010; Corbelli 和 Salucci 2000）。

3.2 Ellipticals

Ellipticals are more compact objects than spirals so that, in objects with same stellar mass $M_{\star }$, they probe inner regions of the DM halo than spirals. Their profiles are well represented by the Sersic Law:
椭圆体比螺旋体更紧凑，因此，在具有相同恒星质量 $M_{\star }$ 的物体中，它们探测比螺旋体更内部的 DM 暗物质晕。它们的轮廓很好地由 Sersic 定律表示：

$$\begin{array}{r}\ln \left[\frac{{\text{Σ}}_{\mathrm{S}}(R)}{{\text{Σ}}_{R_{\mathrm{e}}}}\right]=-q[{\left(\frac{R}{R_{\mathrm{e}}}\right)}^{\frac{1}{m}}-1],\end{array}$$

(11)

${\text{Σ}}_{\mathrm{S}}(0)={\text{Σ}}_{R_{\mathrm{e}}}{\mathrm{e}}^q$, where R is the projected radial coordinate in the plane of the sky, ${\text{Σ}}_{R_{\mathrm{e}}}$ is the line of sight (l.o.s.) projected surface brightness at a projected scale radius $R_{\mathrm{e}}\simeq R_{1/2}$ and $q=2m-1/3$ with m a free parameter. By deprojecting the surface density ${\text{Σ}}_{\mathrm{S}}(R/R_{\mathrm{e}},m)$, we obtain the luminosity density j(r) and by assuming a radially constant stellar mass-to-light ratio $(M/L{)}_{\star }$ we obtain the spheroid stellar density ${\rho }_{\star }(r)$.
${\text{Σ}}_{\mathrm{S}}(0)={\text{Σ}}_{R_{\mathrm{e}}}{\mathrm{e}}^q$ ，其中 R 是天空平面上的投影径向坐标， ${\text{Σ}}_{R_{\mathrm{e}}}$ 是投影尺度半径为 $R_{\mathrm{e}}\simeq R_{1/2}$ 和 $q=2m-1/3$ 的视线（l.o.s.）投影面亮度，并且 m 是一个自由参数。通过反投影表面密度 ${\text{Σ}}_{\mathrm{S}}(R/R_{\mathrm{e}},m)$ ，我们得到光密度 j(r)，并且通过假设径向恒定的恒星质量与光比 $(M/L{)}_{\star }$ ，我们得到球状体恒星密度 ${\rho }_{\star }(r)$ 。

3.3 Dwarf spheroids

The distribution of stars in dSph plays a major role in the analysis of their internal kinematics. The information we have comes from the bright stars detected by dedicated imaging or spectroscopy and, more recently, by surveys like the Sloan Sky Digital Survey and Gaia. The 3D stellar density is obtained from the deprojection of the 2D luminosity profile and an assumed mass-to-light ratio. The former is well reproduced by the Plummer density profile (Plummer 1915), characterized by a length scale $R_{\mathrm{e}}$ and a central density ${\nu }_0=3M_{\mathrm{s}\mathrm{p}\mathrm{h}}/(4\pi R_{\mathrm{e}}^3)$ with $M_{\mathrm{s}\mathrm{p}\mathrm{h}}$ the total stellar mass. The projected mass (luminosity) distribution is given by: $\text{Σ}(R)=\frac{M_{\mathrm{s}\mathrm{p}\mathrm{h}}}{\pi R_{\mathrm{e}}^2}{(1+x^2)}^{-2},x=R/R_{\mathrm{e}}$. Then, the 3D stellar density is given by
dSph 中恒星的分布在分析其内部运动学中起着重要作用。我们获得的信息来自专门成像或光谱检测到的明亮恒星，以及最近的调查，如斯隆天空数字调查和盖亚。 3D 恒星密度是通过 2D 亮度剖面的反投影和假定的质量-光比获得的。前者由普卢默密度剖面（Plummer 1915）很好地再现，其特征是长度尺度 $R_{\mathrm{e}}$ 和中心密度 ${\nu }_0=3M_{\mathrm{s}\mathrm{p}\mathrm{h}}/(4\pi R_{\mathrm{e}}^3)$ 其中 $M_{\mathrm{s}\mathrm{p}\mathrm{h}}$ 是总恒星质量。投影质量（亮度）分布为： $\text{Σ}(R)=\frac{M_{\mathrm{s}\mathrm{p}\mathrm{h}}}{\pi R_{\mathrm{e}}^2}{(1+x^2)}^{-2},x=R/R_{\mathrm{e}}$ 。然后，3D 恒星密度由

$$\begin{array}{r}\nu (x)={\nu }_0(1+x^2{)}^{-5/2}.\end{array}$$

(12)

4 Probing the gravitational potential in galaxies
探测星系中的引力势

4.1 Rotation curves

The rotation curves (RCs) of spirals are an accurate proxy of their gravitational potential. We measure recessional velocities by Doppler shifts, and from these (often 2D) data, we construct the RC V(R). This process estimates also the sky coordinates of the galaxy kinematical center, its systemic velocity, the degree of symmetry and, often, the inclination angle.
螺旋星系的旋转曲线（RCs）是它们引力势的准确代理。我们通过多普勒频移测量退行速度，并从这些（通常是 2D）数据中构建 RC V(R)。这个过程还估计了星系运动中心的天空坐标、系统速度、对称程度，通常还有倾斜角。

Notice that the effectiveness of the RC is proved in many ways: e.g., in systems with $M_I<-18$ in the innermost luminous matter dominated regions the gravitating mass (measured by V(R)) agrees with the predictions from the light distribution (Ratnam and Salucci 2000).
请注意，旋转曲线的有效性在许多方面得到证明：例如，在内部最亮物质主导区域具有 $M_I<-18$ 的系统中，引力质量（通过 V(R)测量）与光分布（Ratnam 和 Salucci 2000 的预测一致）。

The rotation curves in disk systems have three different components: the relationship with the total gravitational potentials ${\varphi }_{\mathrm{t}\mathrm{o}\mathrm{t}}={\varphi }_b+{\varphi }_{\mathrm{H}}+{\varphi }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}+{\varphi }_{\mathrm{H}\mathrm{I}}$ is
盘系统中的旋转曲线有三个不同的组成部分：与总引力势 ${\varphi }_{\mathrm{t}\mathrm{o}\mathrm{t}}={\varphi }_b+{\varphi }_{\mathrm{H}}+{\varphi }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}+{\varphi }_{\mathrm{H}\mathrm{I}}$ 的关系是

$$\begin{array}{r}{V}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2(r)=r\frac{\mathrm{d}}{\mathrm{d}r}{\varphi }_{\mathrm{t}\mathrm{o}\mathrm{t}}=V_b^2+V_{\mathrm{H}}^2+V_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}^2+V_{\mathrm{H}\mathrm{I}}^2.\end{array}$$

(13)

Then, the velocity fields $V_i$ are the solutions of the four separated equations: ${\mathrm{\nabla }}^2{\text{Φ}}_i=4\pi G{\rho }_i$ where ${\rho }_i$ are dark matter, stellar disk, stellar bulge, HI disk surface/volume densities (${\rho }_h(r),{\rho }_{bu}(r),{\mu }_d(r)\delta (z),{\mu }_{\mathrm{H}\mathrm{I}}(r)\delta (z)$ with $\delta (z)$ the Kronecker function, z the cylindrical coordinate) and ${\varphi }_i$ the gravitational potential.
然后，速度场 $V_i$ 是四个分离方程的解： ${\mathrm{\nabla }}^2{\text{Φ}}_i=4\pi G{\rho }_i$ 其中 ${\rho }_i$ 是暗物质、恒星盘、恒星球状、HI 盘面/体密度（ ${\rho }_h(r),{\rho }_{bu}(r),{\mu }_d(r)\delta (z),{\mu }_{\mathrm{H}\mathrm{I}}(r)\delta (z)$ 与 $\delta (z)$ 是 Kronecker 函数，z 是柱坐标）和 ${\varphi }_i$ 是引力势。

Recently, a new way to exploit the RC to obtain the DM halo density distribution has been devised (Salucci et al. 2010). We assume that spirals are composed by a stellar disk (Freeman 1970), a HI disk and an unspecified spherical DM halo with density profile ${\rho }_H(r)$. Other baryonic components can be added, if needed.³ From the radial derivative of the equation of centrifugal equilibrium we obtain
最近，已经设计出一种利用 RC 来获得 DM 暗物质晕密度分布的新方法（Salucci 等人，2010 年）。我们假设螺旋星系由恒星盘（Freeman，1970 年）、HI 盘和具有密度分布 ${\rho }_H(r)$ 的未指定的球形 DM 暗物质晕组成。如果需要，可以添加其他重子组分。从离心平衡方程的径向导数中我们获得

$$\begin{array}{r}{\rho }_H(r)=\frac{1}{4\pi G{r}^2}\frac{\mathrm{d}}{\mathrm{d}r}\left[r^2(\frac{V^2(r)}{r}-a_D(r))\right].\end{array}$$

(14)

We have $a_D(r)=\frac{G{M}_{D}r}{R_D^3}(I_0{K}_0-I_1{K}_1)$, where $I_n$ and $K_n$ are the modified Bessel functions computed at $\frac{r}{2R_D}$. Noticeably, the second term of the r.h.s. of Eq. (14) goes exponentially to zero for $r/R_D>2$ (see Fig. 5). Then, for $R>2R_D$, we can determine the DM density profile (see Fig. 5). On the other hand, for $R<R_D$, the DM distribution is negligible, so that, if we have a good spatial coverage of the inner RC, we can use Eq. (14) also to obtain the disk mass with good precision.
我们有 $a_D(r)=\frac{G{M}_{D}r}{R_D^3}(I_0{K}_0-I_1{K}_1)$ ，其中 $I_n$ 和 $K_n$ 是在 $\frac{r}{2R_D}$ 计算的修正贝塞尔函数。值得注意的是，方程（14）右侧的第二项在 $r/R_D>2$ 时指数级地趋近于零（见图 5）。然后，对于 $R>2R_D$ ，我们可以确定 DM 密度分布（见图 5）。另一方面，对于 $R<R_D$ ，DM 分布是可以忽略的，因此，如果我们对内部 RC 有很好的空间覆盖，我们也可以使用方程（14）来获得盘质量的良好精度。

Fig. 5

A test case: NGC 3198. Effective total density (points with errorbars). Contributions: stellar disk (blue), HI disk (magenta), dark matter (green line), all components (green). Regions in which the method: (1) is not applicable (pink), (2) provides us with a the value of disk mass (green), b the halo density profile (white). Image reproduced with permission from Karukes et al. (2015), copyright by ESO
一个测试案例：NGC 3198。有效总密度（带误差条的点）。贡献：恒星盘（蓝色），HI 盘（品红色），暗物质（绿线），所有组分（绿色）。方法不适用的区域（粉色），提供盘质量值的区域（绿色），提供暗物质密度分布的区域（白色）。图像经 ESO 许可复制自 Karukes 等人（2015 年）版权所有

4.2 A reference velocity for disk systems
4.2 盘系统的参考速度

In spite of the fact that V(R), the circular velocity, is a function of radius we often require a meaningful reference velocity to tag each disk system. In the literature there is no shortage of proposed reference velocities, among those: $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$, $V_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}$, the linewidths $W_{20}$, $W_{50}$ and the maximum velocity $V_{max}$. Obviously, if the RC of an object is not available, we are forced to choose one of these kinematical measurements as a reference velocity; however, we must stress that they are very biased: (a) a flat part of the RC occurs only a limited number of objects and only over a limited radial region (Persic et al. 1996; b) $V_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}$ depends on the distribution of HI in the galaxies and on the sensitivity of radio telescope used; (c) the linewidths are similar to the case (b) and furthermore they depend on the full RC profiles; (d) the significance of $V_{max}$ changes as galaxy luminosity changes, sometimes coinciding with the outermost available velocity, in other cases, with the innermost one. The best unbiased reference velocity for spirals is the quantity: $V(k{R}_D)$ that also involves the stellar disks length scale. We have $k=2.2$ or 3.2, according whether we are investigating the properties of the luminous or of the dark matter.
尽管 V(R)，即圆速度，是半径的函数，我们经常需要一个有意义的参考速度来标记每个盘系统。在文献中，提出了许多参考速度，其中包括： $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ ， $V_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}$ ，线宽 $W_{20}$ ， $W_{50}$ 和最大速度 $V_{max}$ 。显然，如果一个物体的 RC 不可用，我们就被迫选择这些动力学测量中的一个作为参考速度；然而，我们必须强调它们是非常有偏见的：(a) RC 的平坦部分只出现在有限数量的物体上，而且只在有限的径向区域上（Persic 等人，1996 年；b） $V_{\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}$ 取决于星系中 HI 的分布和所使用的射电望远镜的灵敏度；(c) 线宽类似于情况（b），而且还取决于完整的 RC 轮廓；(d) $V_{max}$ 的重要性随着星系的亮度变化而变化，有时与最外层可用速度重合，在其他情况下，与最内层速度重合。螺旋星系最好的无偏参考速度是数量： $V(k{R}_D)$ ，它还涉及到恒星盘的长度尺度。我们有 $k=2.2$ 或 3.2，具体取决于我们是研究发光物质的性质还是暗物质的性质。

4.3 Vertical motions

The main goal of the DiskMass Survey (Bershady et al. 2010a, b) was to determine the dynamical mass-to-light ratio of the galaxy disks $(M/L{)}_{\mathrm{d}\mathrm{y}\mathrm{n}}$ by a suitable use of the stellar and gas kinematics. At a radius R, for a locally isothermal disk, we have
DiskMass Survey (Bershady 等人 2010a, b) 的主要目标是通过合适地利用恒星和气体动力学来确定星系盘的动力质量与光比 $(M/L{)}_{\mathrm{d}\mathrm{y}\mathrm{n}}$ 。在半径 R 处，对于局部等温盘，我们有

$$\begin{array}{r}(M/L{)}_{\mathrm{d}\mathrm{y}\mathrm{n}}=\frac{{\text{σ}}_z^2}{\pi Gb{h}_{z}I(R)},\end{array}$$

(15)

where the value $b=1.5$ is a reasonable approximation for the composite (gas + stars) density distribution (van der Kruit 1988), I the surface luminosity obtained from the photometry, ${\text{σ}}_z$ the vertical component of the stellar velocity dispersion. Noticeably, with the advent of 2-dimensional spectroscopy using integral field units (IFU), the accuracy and the z-extension of the measurements of ${\text{σ}}_z$ has been dramatically increased; $h_z$ is the disk scale height (van der Kruit and Searle 1981; Bahcall 1984) that can be directly measured, and that well correlates with the disk scale length $R_D$ (Kregel et al. 2002; Bershady et al. 2010a).
其中值 $b=1.5$ 是复合 (气体 + 恒星) 密度分布的合理近似值 (van der Kruit 1988)，I 是从光度学获得的表面亮度， ${\text{σ}}_z$ 是恒星速度离散度的垂直分量。值得注意的是，随着使用积分场单元 (IFU) 进行二维光谱学的出现， ${\text{σ}}_z$ 的测量精度和 z-延伸性得到了显著提高； $h_z$ 是盘的尺度高度 (van der Kruit 和 Searle 1981; Bahcall 1984)，可以直接测量，并且与盘的尺度长度 $R_D$ 相关 (Kregel 等人 2002; Bershady 等人 2010a)。

Let us stress that this approach leading to Eq. (15) is certainly a new avenue for investigating dark matter in galaxies, but some warning must be raised in that it can be subject to relevant biases (Hessman 2017).
让我们强调，导致方程 (15) 的这种方法无疑是研究星系中暗物质的新途径，但必须提出一些警告，因为它可能受到相关偏见的影响 (Hessman 2017)。

4.4 Dispersion velocities

It is well known that in spheroids the kinematics is complex, the stars are in gravitational equilibrium by balancing the gravitational potential, they are subject to, with the pressure arisen from the r.m.s. of their 3D motions. Moreover, we cannot directly measure the radial/tangential velocity dispersions linked to the mass profile, but only their projected values (e.g., Coccato et al. 2009).
众所周知，在球状体中，动力学是复杂的，恒星通过平衡引力势与由其 3D 运动的 r.m.s. 产生的压力来保持引力平衡。此外，我们无法直接测量与质量分布相关的径向/切向速度离散度，而只能测量它们的投影值 (例如，Coccato 等人 2009)。

The SAURON (Bacon et al. 2001) Integral Field Spectroscopy survey (de Zeeuw et al. 2002) was the first project to map the two-dimensional stellar kinematics of a sample of 48 nearby ellipticals with $M_B<18$. This survey was followed by the ${\text{ATLAS}}^{3\mathrm{D}}$ project (Cappellari et al. 2011), a multiwavelength survey of 260 ETGs galaxies. In Cappellari (2016) one finds the details of these observations.
SAURON (Bacon 等人 2001) 积分场光谱调查 (de Zeeuw 等人 2002) 是第一个对 48 个附近椭圆星系进行二维恒星动力学映射的项目，采用 $M_B<18$ 。该调查后来被 ${\text{ATLAS}}^{3\mathrm{D}}$ 项目 (Cappellari 等人 2011) 所继续，这是对 260 个椭圆星系进行的多波段调查。在 Cappellari (2016) 中可以找到这些观测的详细信息。

The dispersion velocity is related to the gravitational potential of a galaxy by the Jeans equation that we express as (Mamon and Lokas 2005; Binney and Tremaine 2008)
离散速度与星系的引力势之间有关，我们将其表达为 Jeans 方程 (Mamon 和 Lokas 2005; Binney 和 Tremaine 2008)

$$\begin{array}{r}\frac{\mathrm{\partial }\ln {\text{σ}}_r^2}{\mathrm{\partial }\ln r}=-\frac{1}{{\text{σ}}_r^2}\frac{GM}{r}-{\gamma }_{\star }-2\beta .\end{array}$$

(16)

Here, G is the gravitational constant and M(r) is the enclosed mass. The velocity anisotropy $\beta =1-\frac{{\text{σ}}_{\theta }^2+{\text{σ}}_{\varphi }^2}{2{\text{σ}}_r^2}$, where ${\text{σ}}_{\theta ,\varphi ,r}$ are the velocity dispersions in the $r,\theta$ and $\varphi$ directions, can be a function of radius r (Binney and Tremaine 2008). It is useful to define: $\alpha =\mathrm{d}\log {\text{σ}}_r/\mathrm{d}\log r$. Almost always the motions in the $\theta$ and $\varphi$ directions are assumed to coincide.
这里，G 是引力常数，M(r) 是封闭质量。速度各向异性 $\beta =1-\frac{{\text{σ}}_{\theta }^2+{\text{σ}}_{\varphi }^2}{2{\text{σ}}_r^2}$ ，其中 ${\text{σ}}_{\theta ,\varphi ,r}$ 是 $r,\theta$ 和 $\varphi$ 方向的速度离散度，可以是半径 r 的函数（Binney 和 Tremaine 2008）。定义如下是有用的： $\alpha =\mathrm{d}\log {\text{σ}}_r/\mathrm{d}\log r$ 。几乎总是假定在 $\theta$ 和 $\varphi$ 方向上的运动是一致的。

${\nu }_{\star }$ is the 3D stellar density distribution, ${\gamma }_{\star }=\mathrm{d}\log {\nu }_{\star }/\mathrm{d}\log r$. Under the assumption of constant $\beta$, the radial velocity dispersion ${\text{σ}}_r(r)$ can be expressed as follows:
${\nu }_{\star }$ 是 3D 恒星密度分布， ${\gamma }_{\star }=\mathrm{d}\log {\nu }_{\star }/\mathrm{d}\log r$ 。在假设恒定 $\beta$ 的情况下，径向速度离散度 ${\text{σ}}_r(r)$ 可以表示如下：

$$\begin{array}{r}{\text{σ}}_r^2(r)=\frac{1}{{\nu }_{\star }(r)}{\int }_r^{\mathrm{\infty }}{\nu }_{\star }(r^{\prime }){\left(\frac{r^{\prime }}{r}\right)}^{2\beta }\frac{GM(r^{\prime })}{r^{\prime 2}}\mathrm{d}{r}^{\prime }.\end{array}$$

(17)

We can then determine the galaxy mass profile by means of Eq. (17) and the line-of-sight velocity dispersion ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}$ when the anisotropy factor $\beta (r)$ is known or assumed:
当各向异性因子 $\beta (r)$ 已知或假定时，我们可以通过方程（17）和视线速度离散度 ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}$ 来确定星系质量分布：

$$\begin{array}{r}{\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}^2(R)=\frac{1}{I(R)}{\int }_{R^2}^{\mathrm{\infty }}\mathrm{d}{r}^2\frac{{\nu }_{\star }}{\sqrt{r^2-R^2}}{\text{σ}}_r^2[1-\beta \frac{R^2}{r^2}],\end{array}$$

(18)

where I(R) and ${\nu }_{\star }(r)$ are related by $I(R)=2{\int }_R^{+\mathrm{\infty }}\frac{{\nu }_{\star }(r)r\mathrm{d}r}{\sqrt{r^2-R^2}}$. I(R) and ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(R)$ are directly measured.
其中 I(R) 和 ${\nu }_{\star }(r)$ 由 $I(R)=2{\int }_R^{+\mathrm{\infty }}\frac{{\nu }_{\star }(r)r\mathrm{d}r}{\sqrt{r^2-R^2}}$ 关联。I(R) 和 ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(R)$ 是直接测量的。

The Schwarzschild method can be seen as a (complex) extension of the Jeans method and it is especially applied to dSph galaxies where the stellar component is totally negligible (Cretton et al. 1999; Breddels et al. 2013). It is based, fixed a specific gravitational potential, on the integration of test particle orbits drawn from a grid of integrals of motions, i.e., the energy and the angular momentum. The main feature of this method is that, differently from the Jeans method, it can successfully use the observed second and fourth velocity moment profiles to break the mass-anisotropy degeneracy (Breddels et al. 2013).
史瓦西德方法可以看作是 Jeans 方法的（复杂）延伸，特别适用于 dSph 星系，其中恒星成分完全可以忽略不计（Cretton 等人 1999 年；Breddels 等人 2013 年）。它基于固定特定的引力势能，通过积分来自一组运动积分的测试粒子轨道，即能量和角动量。这种方法的主要特点是，与 Jeans 方法不同，它可以成功地利用观测到的第二和第四速度矩分布来打破质量-各向异性退化（Breddels 等人 2013 年）。

4.5 Fast spheroidal rotators

In the case of objects (e.g., S0 galaxies) in which the dispersion velocity combines with the rotation motions to balance the galaxy self-gravity, there is a simple and efficient anisotropic generalization of the axisymmetric Jeans formalism which is used to model the stellar kinematics of galaxies (see Cappellari 2016 for details). The following is assumed: (i) a constant mass-to-light ratio M / L and (ii) a velocity ellipsoid that is aligned with cylindrical coordinates (R, z) and characterized by the classic anisotropy parameter ${\beta }_z=1-{\text{σ}}_z^2/{\text{σ}}_R^2$. These simple models are fit to integral-field observations of the stellar kinematics of fast-rotator early-type galaxies. With only two free parameters (${\beta }_z$ and the inclination) the models generally provide remarkably good descriptions of the shape of the first (V) and second ($V_{\mathrm{r}\mathrm{m}\mathrm{s}}\equiv \sqrt{V^2+{\text{σ}}^2}$) velocity moments. The technique can be used to determine the dynamical mass-to-light ratios and angular momenta of early-type fast-rotators and it allows for the inclusion of dark matter, supermassive central black holes, spatially varying anisotropy and multiple kinematic components.
对于物体（例如 S0 星系）中，其中离散速度与旋转运动相结合以平衡星系自重力的情况，有一个简单而高效的轴对称 Jeans 形式的各向异性推广，用于建模星系的恒星动力学（详见 Cappellari 2016）。假设如下：（i）恒定的质量-光比 M / L 和（ii）速度椭球体与柱坐标（R，z）对齐，并由经典各向异性参数 ${\beta }_z=1-{\text{σ}}_z^2/{\text{σ}}_R^2$ 表征。这些简单模型适用于快速旋转早型星系的星体动力学的积分场观测。仅有两个自由参数（ ${\beta }_z$ 和倾斜角）的模型通常能够非常好地描述第一（V）和第二（ $V_{\mathrm{r}\mathrm{m}\mathrm{s}}\equiv \sqrt{V^2+{\text{σ}}^2}$ ）速度矩的形状。该技术可用于确定早型快速旋转体的动力学质量-光比和角动量，并允许包括暗物质、超大质量中心黑洞、空间变化的各向异性和多个动力学成分。

4.6 Dispersion velocities versus rotation curves
4.6 离散速度与旋转曲线

Here, it is worth making a comparison between the circular velocity V(r) and the radial (or line-of-sight) velocity dispersion of an irrotational gravitational tracer with distribution ${\nu }_{\star }(r)$ and with anisotropy $\beta (r)$. From Eq. (16) we get the following:
在这里，值得比较圆速度 V(r) 与无旋引力示踪者的径向（或视线）速度离散度，其分布为 ${\nu }_{\star }(r)$ ，且各向异性为 $\beta (r)$ 。从方程（16）中我们得到以下结果：

$$\begin{array}{r}(-{\gamma }_{\star }(r)+2(\beta (r)+\alpha (r))){\text{σ}}_r^2(r)=V^2(r)\end{array}$$

(19)

$\alpha (r)$ and ${\gamma }_{\star }(r)$ are the logarithmic derivatives of ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}$ and ${\nu }_{\star }$. Let us notice that, in dispersion velocity supported systems, even in the case of isotropic orbits: $\beta (r)=0$, it is necessary to know the spatial distribution of the tracers in order to make any inference on the DM distribution. Flat RC and flat dispersion velocity profiles do not necessarily indicate the same gravitational field.
$\alpha (r)$ 和 ${\gamma }_{\star }(r)$ 是 ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}$ 和 ${\nu }_{\star }$ 的对数导数。让我们注意，在色散速度支持系统中，即使在各向同性轨道的情况下： $\beta (r)=0$ ，也需要了解示踪剂的空间分布才能对 DM 分布进行任何推断。平坦的 RC 和平坦的色散速度轮廓不一定指示相同的引力场。

4.7 Masses in spheroids within half-light radii
4.7 半光半径内球状体的质量

We can measure the total mass enclosed within the half-light radius $R_{1/2}$ by measuring ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(R_{1/2})$ the line of sight velocity dispersion at this radius (Wolf et al. 2010). Since ${\text{σ}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2={\text{σ}}_r^2+{\text{σ}}_{\theta }^2+{\text{σ}}_{\varphi }^2=(3-2\beta ){\text{σ}}_r^2$ we can write the Jeans equation as $GM(r)r^{-1}={\text{σ}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2(r)+{\text{σ}}_r^2(r)(-{\gamma }_{\star }+\alpha -3)$. Let us define $R_3$ as ${\gamma }_{\star }(R_3)=3$⁴ since $\alpha (R_3)\ll 3$ from the observed ${\text{σ}}_{\mathrm{l}\mathrm{o}\mathrm{s}}(r)$ profiles, then, at $R=R_3$, we have, independently of the value of the anisotropy:
我们可以通过在半光半径 $R_{1/2}$ 测量该半径处的视线速度色散 ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(R_{1/2})$ 来测量包围在其中的总质量（Wolf 等人 2010 年）。由于 ${\text{σ}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2={\text{σ}}_r^2+{\text{σ}}_{\theta }^2+{\text{σ}}_{\varphi }^2=(3-2\beta ){\text{σ}}_r^2$ 我们可以将 Jeans 方程写为 $GM(r)r^{-1}={\text{σ}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2(r)+{\text{σ}}_r^2(r)(-{\gamma }_{\star }+\alpha -3)$ 。让我们定义 $R_3$ 为 ${\gamma }_{\star }(R_3)=3$ ⁴ 自 $\alpha (R_3)\ll 3$ 从观测到的 ${\text{σ}}_{\mathrm{l}\mathrm{o}\mathrm{s}}(r)$ 轮廓中，然后，在 $R=R_3$ 处，我们独立于各向同性值：

$$\begin{array}{r}M(R_{1/2})\approx 3G^{-1}{\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}^2(R_{1/2})R_{1/2}.\end{array}$$

(20)

APOSTLE cosmological hydro dynamical simulations have tested the validity and accuracy of this mass estimator and found that the resulting measurements are, at most, biased by 20% (Campbell et al. 2017).
APOSTLE 宇宙学热力学模拟已经测试了这种质量估计器的有效性和准确性，并发现由此得出的测量结果最多偏差 20%（Campbell 等人 2017 年）。

4.8 Tracer mass estimator

Given a number of N of tracers in dynamical pressure supported equilibrium with no systematic rotation and moving with l.o.s. velocities within a dark halo of mass profile M(r), the TME is expressed as
假设有 N 个示踪剂处于动力压力支持的平衡状态，没有系统旋转，并且以质量分布 M(r)的暗物质晕内以 l.o.s.速度运动，TME 表示为

$$\begin{array}{r}M(r_{\mathrm{o}\mathrm{u}\mathrm{t}})=\frac{C}{GN}\sum _{i=1}^N{V}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.,\mathrm{i}}^2{R}_i^{ϵ}.\end{array}$$

(21)

The prefactor C depends on (i) the slope $ϵ$ of the gravitational potential, assumed to be: $\text{Φ}(r)\propto \frac{v_0^2}{ϵ}{\left(\frac{a}{r}\right)}^{ϵ};v_0^2\log \left(\frac{a}{r}\right)(ϵ=0)$. (ii) The “slope” ${\gamma }_{\star }$ of the de-projected density profile of the tracers (${\rho }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}}(r)\propto r^{-{\gamma }_{\star }}$). (iii) The orbital anisotropy $\beta$ of the tracers.
前置因子 C 取决于（i）引力势的斜率 $ϵ$ ，假定为： $\text{Φ}(r)\propto \frac{v_0^2}{ϵ}{\left(\frac{a}{r}\right)}^{ϵ};v_0^2\log \left(\frac{a}{r}\right)(ϵ=0)$ 。 （ii）追踪器的去投影密度轮廓的“斜率” ${\gamma }_{\star }$ （ ${\rho }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}}(r)\propto r^{-{\gamma }_{\star }}$ ）。 （iii）追踪器的轨道各向异性 $\beta$ 。

We then have $C=\frac{(ϵ+{\gamma }_{\star }-2\beta )}{I_{ϵ,\beta }}{r}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{1-ϵ}$ with $r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ the distance of the outermost tracer and $I_{ϵ,\beta }=\frac{{\pi }^{1/2}\Gamma (ϵ/2+1)}{4\Gamma ((ϵ/2+5)/2)}[ϵ+3-\beta (ϵ+2)]$, where $\Gamma$ is the Gamma function (Watkins et al. 2010; An and Evans 2011).
然后我们有 $C=\frac{(ϵ+{\gamma }_{\star }-2\beta )}{I_{ϵ,\beta }}{r}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{1-ϵ}$ 与 $r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ 外层示踪剂的距离和 $I_{ϵ,\beta }=\frac{{\pi }^{1/2}\Gamma (ϵ/2+1)}{4\Gamma ((ϵ/2+5)/2)}[ϵ+3-\beta (ϵ+2)]$ ，其中 $\Gamma$ 是 Gamma 函数（Watkins 等人 2010 年；An 和 Evans 2011 年）。

The mass estimator in Eq. (21) performs very well, especially in the case in which the tracers are in random orbits, so that $\beta =0$ and for ellipticals where we have $\alpha =0\pm 0.1$. In these cases, the uncertainties on the two latter quantities do not bias the mass estimate.
方程（21）中的质量估计器表现非常好，特别是在追踪器处于随机轨道的情况下，即 $\beta =0$ 和对于椭圆体，我们有 $\alpha =0\pm 0.1$ 。在这些情况下，对后两个量的不确定性不会对质量估计造成偏差。

4.9 Weak lensing

We briefly recall here that weak gravitational lensing is a powerful tool for probing the dark matter distribution in galaxies (Schneider 1996; Hoekstra and Jain 2008; Munshi et al. 2008; Bartelmann and Maturi 2016). It is known that observed images of distant galaxies are coherently deformed by weak lensing effects caused by foreground matter distributions. These distortions enable the measurement of the mean mass profiles of foreground lensing galaxy through the stacking of the background shear fields (Zu and Mandelbaum 2015). To determine halo mass, we measure the excess surface mass density $\text{Δ}\text{Σ}(R)=\overline{\text{Σ}(<R)}-\overline{\text{Σ}(R)}$, which is the difference between the projected average surface mass within a circle of radius R and the surface density at that radius. The tangential shear ${\gamma }_t$ is directly related to the above quantities through $\text{Δ}\text{Σ}(R)={\text{Σ}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}⟨{\gamma }_t(R)⟩$, where ${\text{Σ}}_{\mathrm{c}}$ is the critical surface density defining the Einstein radius of the lens
我们在此简要回顾了弱引力透镜是探测星系中暗物质分布的有力工具（Schneider 1996; Hoekstra 和 Jain 2008; Munshi 等人 2008; Bartelmann 和 Maturi 2016）。已知远处星系的观测图像会受到前景物质分布引起的弱透镜效应的一致变形。这些扭曲使得通过叠加背景剪切场来测量前景透镜星系的平均质量轮廓成为可能（Zu 和 Mandelbaum 2015）。为了确定暗物质晕的质量，我们测量超出表面质量密度 $\text{Δ}\text{Σ}(R)=\overline{\text{Σ}(<R)}-\overline{\text{Σ}(R)}$ ，即在半径 R 的圆内的平均表面质量与该半径处的表面密度之间的差异。切向剪切 ${\gamma }_t$ 与上述量直接相关，通过 $\text{Δ}\text{Σ}(R)={\text{Σ}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}⟨{\gamma }_t(R)⟩$ ，其中 ${\text{Σ}}_{\mathrm{c}}$ 是定义镜头爱因斯坦半径的临界表面密度

$$\begin{array}{r}{\text{Σ}}_{\mathrm{c}}=\frac{c^2}{4\pi G}\frac{D_{\mathrm{s}}}{D_{\mathrm{l}}{D}_{\mathrm{l}\mathrm{s}}},\end{array}$$

(22)

where $D_{\mathrm{s}}$, $D_{\mathrm{l}}$, and $D_{\mathrm{l}\mathrm{s}}$ are the distances to the source, to the lens and the lens-source one, respectively. The lens equation relates ${\gamma }_t$ with the distribution of matter in the lensing galaxy:
其中 $D_{\mathrm{s}}$ ， $D_{\mathrm{l}}$ 和 $D_{\mathrm{l}\mathrm{s}}$ 分别是到源的距离，到透镜的距离和透镜-源之间的距离。透镜方程将 ${\gamma }_t$ 与透镜星系中物质分布联系起来：

$$\begin{array}{r}{\gamma }_t(R)=(\overline{\text{Σ}}(R)-\text{Σ}(R))/{\text{Σ}}_{\mathrm{c}},\end{array}$$

(23)

where $\text{Σ}(R)=2{\int }_0^{\mathrm{\infty }}\rho (R,z)\mathrm{d}z$ is the projected mass density of the object distorting the galaxy image, at projected radius R and $\overline{\text{Σ}}(R)=\frac{2}{R^2}{\int }_0^{R}x\text{Σ}(x)\mathrm{d}x$ is the mean projected mass density interior to the radius R.
其中 $\text{Σ}(R)=2{\int }_0^{\mathrm{\infty }}\rho (R,z)\mathrm{d}z$ 是扭曲星系图像的物体的投影质量密度，在投影半径 R 处， $\overline{\text{Σ}}(R)=\frac{2}{R^2}{\int }_0^{R}x\text{Σ}(x)\mathrm{d}x$ 是半径 R 内的平均投影质量密度。

4.10 Strong lensing

Gravitational lensing occurring in very aligned galaxy–galaxy–observer structures magnifies and distorts the images of a distant galaxy providing us with relevant information on the mass structure of the intervening galaxy so as of the background source (see Treu 2010).
在非常对齐的星系-星系-观测者结构中发生的引力透镜效应会放大和扭曲远处星系的图像，为我们提供有关介入星系质量结构和背景源的相关信息（参见 Treu 2010）。

The lens system is axially symmetric, and the radial coordinate r is related to cylindrical polar coordinates by $r=\sqrt{{\xi }^2+z^2}$, where $\xi$ is the impact parameter measured from the center of the lens. The mean surface density inside the radius $\xi$ is
镜头系统是轴对称的，径向坐标 r 与柱坐标之间的关系由 $r=\sqrt{{\xi }^2+z^2}$ 确定，其中 $\xi$ 是从镜头中心测量的冲击参数。半径 $\xi$ 内的平均表面密度是

$$\begin{array}{r}\overline{\text{Σ}}(\xi )=\frac{1}{\pi {\xi }^2}{\int }_0^{\xi }2\pi {\xi }^{\prime }\text{Σ}({\xi }^{\prime })\mathrm{d}{\xi }^{\prime }.\end{array}$$

(24)

The presence of an Einstein ring of radius $R_{\mathrm{E}}$, at projected galactocentric distance $\xi$ (see Fig. 6), allows us to obtain the projected total mass inside $\xi$:
在投影的银河中心距离 $\xi$ 处存在一个爱因斯坦环，半径为 $R_{\mathrm{E}}$ （见图 6），这使我们能够获得 $\xi$ 内的投影总质量：

$$\begin{array}{r}(M_{\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{o}}(\xi )+M_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{s}}(\xi ))=\pi R_{\mathrm{E}}^2{\text{Σ}}_{\mathrm{c}}.\end{array}$$

(25)

Fig. 6

Einstein ring (artist’s concept). This extraordinary GR effect provides us with the value of the projected mass of the galaxy lens inside $R_{\mathrm{E}}$
爱因斯坦环（艺术家概念）。这种非凡的广义相对论效应为我们提供了在 $R_{\mathrm{E}}$ 内的星系透镜的投影质量值。

4.11 X-ray emission and hydrostatic equilibrium
4.11 X 射线发射和静力平衡

Isolated ellipticals have an X-ray emitting halo of regular morphology, that extends out to very large radii. The gravitating mass inside a radius r, M(r) can be derived from their X-ray flux if the emitting gas is in hydrostatic equilibrium. From its density and the temperature profiles we obtain the total mass profile (Fabricant et al. 1984; Ettori and Fabian 2006:
孤立的椭圆星系具有规则形态的 X 射线发射晕，延伸至非常大的半径。如果发射气体处于静力平衡状态，可以从其 X 射线通量推导出半径 r 内的引力质量 M(r)。通过其密度和温度分布，我们可以得到总质量分布（Fabricant 等人，1984 年；Ettori 和 Fabian，2006 年）：

$$\begin{array}{r}M(<r)=\frac{k{T}_{\mathrm{g}}(r)r}{G\mu m_{\mathrm{p}}}(\frac{\mathrm{d}\log {\rho }_{\mathrm{g}}(r)}{\mathrm{d}\log r}+\frac{\mathrm{d}\log T_{\mathrm{g}}(r)}{\mathrm{d}\log r}),\end{array}$$

(26)

where $T_{\mathrm{g}}$ is the (measured) ionised gas temperature, ${\rho }_{\mathrm{g}}$ the gas density, k is the Boltzmann’s constant, $\mu =0.62$ is the mean molecular weight and $m_{\mathrm{p}}$ is the mass of the proton.
其中 $T_{\mathrm{g}}$ 是（测得的）电离气体温度， ${\rho }_{\mathrm{g}}$ 是气体密度，k 是玻尔兹曼常数， $\mu =0.62$ 是平均分子量， $m_{\mathrm{p}}$ 是质子质量。

5 The mass of the stellar component in galaxies
星系中恒星成分的质量

We can assume that the stellar mass surface density ${\text{Σ}}_{\star }(r)$ is proportional to the luminosity surface density, which in galaxies is well measured by CCD infrared photometry. Radial variations of the $M_{\star }/L$ ratio exist and often are astrophysically relevant, but rarely they play a role in the determination of the mass profile of galaxies.
我们可以假设恒星质量面密度 ${\text{Σ}}_{\star }(r)$ 与亮度面密度成正比，在星系中，这通常通过 CCD 红外光度测量得到。径向 $M_{\star }/L$ 比例的变化存在且通常在天体物理学中很重要，但很少在确定星系质量分布中起作用。

The total galaxy luminosity is related to its stellar content and hence, the direct approach to derive the galactic mass in stars by modelling their spectral energy distribution in terms of age, metallicity, initial mass function of the stellar component. This modelling, pioneered by Tinsley (1981), is performed by the well-known stellar population synthesis technique. The SED of a galaxy, selected colour indices and absorption lines are all reproduced by a theoretical models calculated under different assumptions regarding the above physical quantities. In practice, the exercise is not straightforward because degeneracies among age, metallicities, IMF and dust content, to name some, do arise and different combinations of the former quantities yield to very similar SEDs.
总星系亮度与其恒星内容相关，因此，通过对其光谱能量分布进行建模，以年龄、金属丰度、恒星成分的初始质量函数为参数，直接推导出星系中恒星的质量的方法。这种由 Tinsley（1981 年）开创的建模是通过众所周知的恒星种群合成技术进行的。星系的 SED、选定的颜色指数和吸收线都可以通过在不同假设下计算的理论模型来复制。实际上，这个过程并不简单，因为年龄、金属丰度、IMF 和尘埃含量等之间存在退化，前述物理量的不同组合会产生非常相似的 SED。

Bell and de Jong (2001) found rather simple relationships between mass-to-light ratios and certain colour indices. In detail, they investigated a suite of spectrophotometric spiral galaxy evolution models that assumed a Salpeter Initial Mass Function, an exponentially declining star formation rate and a current age of 12 Gyr and found that the stellar mass to light ratios correlate tightly with galaxy colours (see also Bell et al. 2003).
Bell 和 de Jong（2001 年）发现质量与光比率与某些颜色指数之间存在相对简单的关系。详细地说，他们研究了一套假设 Salpeter 初始质量函数、指数衰减的恒星形成率和当前年龄为 12 亿年的光谱光度螺旋星系演化模型，并发现恒星质量与光比率与星系颜色密切相关（另见 Bell 等人 2003 年）。

The important stellar mass-to-light ratios in the Spitzer $3.6\text{μ}\mathrm{m}$ band (${{\text{Υ}}_{\star }}^{3.6\text{μ}\mathrm{m}}$) and in the K-band (${{\text{Υ}}_{\star }}^K$) have also been derived by constructing stellar population synthesis models, with various sets of metallicity and star-formation histories (see Oh 2008; de Blok et al. 2008).
通过构建恒星种群合成模型，使用各种金属丰度和星形成历史，还可以推导出 Spitzer $3.6\text{μ}\mathrm{m}$ 波段（ ${{\text{Υ}}_{\star }}^{3.6\text{μ}\mathrm{m}}$ ）和 K 波段（ ${{\text{Υ}}_{\star }}^K$ ）中的重要恒星质量与光比率（见 Oh 2008; de Blok 等人 2008）。

We have

$$\begin{array}{r}\log ({{\text{Υ}}_{\star }}^K)=1.43\times (J-K)-1.38{\text{Υ}}_{\star }^{3.6\text{μ}\mathrm{m}}=0.92{\text{Υ}}_{\star }^K-\mathrm{0.05.}\end{array}$$

(27)

The values of the galaxy stellar masses as derived from their SEDs have been compared with those obtained by other methods. Grillo et al. (2009) investigated a sample of ellipticals with Einstein rings from which they derived the total projected mass (dominated by the stellar component) and, from the latter, the total mass of the spheroid. Then, by using the SDSS multicolour photometry they fitted the galaxy spectral energy distributions (SEDs) by means of composite stellar-population synthesis models of Bruzual and Charlot (2003) and Maraston (2013) and obtained the photometric mass of the stellar spheroid. The two different mass estimates agreed within 0.2 dex (see also Tiret et al. 2011).
通过与其他方法获得的值进行比较，从它们的 SED 中导出的星系恒星质量的值已经被比较过。Grillo 等人（2009 年）研究了具有爱因斯坦环的椭圆形样本，从中导出了总投影质量（由恒星成分主导）以及从后者导出了球状体的总质量。然后，通过使用 SDSS 多色光度法，他们利用 Bruzual 和 Charlot（2003 年）以及 Maraston（2013 年）的复合恒星种群合成模型拟合了星系的光谱能量分布（SED），并获得了球状体的光度质量。这两种不同的质量估算在 0.2 dex 内达成一致（另见 Tiret 等人 2011 年）。

Salucci et al. (2008) have estimated kinematically the disk mass from the rotation curve of 18 spirals of different luminosity and Hubble types and have compared them with the values obtained by fitting their SED with spectro-photometric models. They found $M_{\mathrm{p}\mathrm{h}\mathrm{o}}\propto M_{\mathrm{k}\mathrm{i}\mathrm{n}}^{1.0\pm 0.1}$ with a r.m.s. of 40% suggesting that photometric and kinematical estimate of the masses of the stellar galaxy disks are statistically consistent.
2008 年 Salucci 等人从 18 个不同亮度和哈勃类型的螺旋星系的旋转曲线中动力学估算了盘的质量，并将其与用光谱光度模型拟合其 SED 获得的值进行了比较。他们发现 $M_{\mathrm{p}\mathrm{h}\mathrm{o}}\propto M_{\mathrm{k}\mathrm{i}\mathrm{n}}^{1.0\pm 0.1}$ ，均方根为 40%，表明恒星星系盘的质量的光度和动力学估算在统计上是一致的。

We have to caution about one consequence of the found disagreement of about 0.15 dex among the dynamical and the spectro-photometric estimates. This value is small to affect existing colour stellar mass relationships, but it is large if we want to use it for mass modelling purposes. In fact, in spirals, for $R<R_D$, the dark and the luminous components of the circular velocity are of the same order of magnitude $V_h\simeq V_d(M_{\mathrm{D},\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}}/M_{\mathrm{D},\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}{)}^{-0.5}$ and, therefore, an uncertainty of $({10}^{0.15}-1)100\mathrm{\%}\sim 40\mathrm{\%}$ on the value of $M_{\mathrm{D},\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}$ jeopardizes the derivation of the DM velocity contribution and even more that of the subsequent DM halo density.
我们必须警惕在动力学和光谱光度估计之间发现的约 0.15 dex 的差异可能带来的后果。这个数值对现有的颜色恒星质量关系影响不大，但如果我们想将其用于质量建模目的，则这个数值就很大了。事实上，在螺旋星系中，对于 $R<R_D$ ，圆速度的暗部分和明亮部分的数量级相同 $V_h\simeq V_d(M_{\mathrm{D},\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}}/M_{\mathrm{D},\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}{)}^{-0.5}$ ，因此，对 $M_{\mathrm{D},\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}$ 数值的不确定性为 $({10}^{0.15}-1)100\mathrm{\%}\sim 40\mathrm{\%}$ 可能会危及导出暗物质速度贡献，甚至更多地危及随后的暗物质晕密度的推导。

For spiral galaxies there is a reliable method to estimate the disk mass which is immune from the latter uncertainty. We start from the gravitating mass inside $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$: $M_{\mathrm{g}}(R_{\mathrm{o}\mathrm{p}\mathrm{t}})\equiv G^{-1}{V}_{\mathrm{o}\mathrm{p}\mathrm{t}}^2{R}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ and $\mathrm{\nabla }$, the rotation curve logarithmic slope measured at $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$: $\mathrm{\nabla }\simeq 3.2(1-V(2.2R_D)/V(R_{\mathrm{o}\mathrm{p}\mathrm{t}}))$. From Persic and Salucci (1990) we have:
对于螺旋星系，有一种可靠的方法来估计免受后者不确定性影响的盘质量。我们从 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 内的引力质量开始： $M_{\mathrm{g}}(R_{\mathrm{o}\mathrm{p}\mathrm{t}})\equiv G^{-1}{V}_{\mathrm{o}\mathrm{p}\mathrm{t}}^2{R}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 和 $\mathrm{\nabla }$ ，在 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 测量的旋转曲线对数斜率： $\mathrm{\nabla }\simeq 3.2(1-V(2.2R_D)/V(R_{\mathrm{o}\mathrm{p}\mathrm{t}}))$ 。根据 Persic 和 Salucci（1990 年）我们有：

$$\begin{array}{r}{M}_D=(0.72-0.85\mathrm{\nabla })M_{\mathrm{g}}(R_{\mathrm{o}\mathrm{p}\mathrm{t}}),\end{array}$$

(28)

where the disk mass has uncertainty of 20%.
其中盘质量存在 20%的不确定性。

6 DM halo profiles

In this section, we will introduce the DM halo profiles that are presently adopted: the empirical ones and those emerging from specific theoretical scenarios, see Fig. 7. It is useful to remind that $M_h(r)=G^{-1}{V}_h^2(r)r={\int }_0^{r}4\pi r^2{\rho }_h(r)\mathrm{d}r$ with $V_h(r)$ the halo contribution to the circular velocity V(R).
在本节中，我们将介绍目前采用的 DM 暗物质晕轮廓：经验性的和从特定理论场景中出现的轮廓，参见图 7。提醒一下， $M_h(r)=G^{-1}{V}_h^2(r)r={\int }_0^{r}4\pi r^2{\rho }_h(r)\mathrm{d}r$ 与 $V_h(r)$ 是圆速度 V(R)的暗物质晕贡献。

6.1 BT-URC

The empirical DM halo density profile, adopted for the URC of Persic et al. (1996), takes the following form (see also Binney and Tremaine 2008):
采用 Persic 等人（1996 年）的 URC 的经验性 DM 暗物质晕密度分布轮廓采用以下形式（另见 Binney 和 Tremaine 2008）：

$$\begin{array}{r}{\rho }_{\mathrm{B}\mathrm{T}\text{-}\mathrm{U}\mathrm{R}\mathrm{C}}(r)=\frac{1}{G}\frac{v_0^2(r^2+3r_0^2)}{(r_0^2+r^2{)}^2},M_{\mathrm{B}\mathrm{T}\text{-}\mathrm{U}\mathrm{R}\mathrm{C}}(r)=\frac{1}{G}\frac{v_0^2{r}^3}{r_0^2+r^2},\end{array}$$

(29)

where $r_0$ and $v_0$ are the core radius and the asymptotic circular velocity of the halo, respectively.
其中 $r_0$ 和 $v_0$ 分别是暗物质晕的核半径和渐近圆速度。

6.2 Navarro–Frenk–White

In $\text{Λ}$CDM the structure of virialized DM halos, obtained by N-body simulations, have a universal spherically averaged density profile, ${\rho }_{\mathrm{N}\mathrm{F}\mathrm{W}}(r)$ (Navarro et al. 1997):
在 $\text{Λ}$ CDM 中，通过 N 体模拟获得的弛豫 DM 暗物质晕的结构具有普遍的球对称平均密度分布， ${\rho }_{\mathrm{N}\mathrm{F}\mathrm{W}}(r)$ （Navarro 等人，1997 年）：

$$\begin{array}{r}{\rho }_{\mathrm{N}\mathrm{F}\mathrm{W}}(r)=\frac{{\rho }_{\mathrm{s}}}{(r/r_{\mathrm{s}})(1+r/r_{\mathrm{s}}{)}^2},\end{array}$$

(30)

where ${\rho }_{\mathrm{s}}$ and $r_{\mathrm{s}}$ are strongly correlated: $r_{\mathrm{s}}\simeq 8.8{\left(\frac{M_{\mathrm{v}\mathrm{i}\mathrm{r}}}{{10}^{11}{M}_{\odot }}\right)}^{0.46}\mathrm{k}\mathrm{p}\mathrm{c}$ (e.g., Wechsler et al. 2006). We define $X\equiv r/R_{\mathrm{v}\mathrm{i}\mathrm{r}}$, the concentration parameter $c\equiv r_{\mathrm{s}}/R_{\mathrm{v}\mathrm{i}\mathrm{e}}$ is a weak function of mass (Klypin et al. 2011):
其中 ${\rho }_{\mathrm{s}}$ 和 $r_{\mathrm{s}}$ 强相关: $r_{\mathrm{s}}\simeq 8.8{\left(\frac{M_{\mathrm{v}\mathrm{i}\mathrm{r}}}{{10}^{11}{M}_{\odot }}\right)}^{0.46}\mathrm{k}\mathrm{p}\mathrm{c}$ (例如，Wechsler 等人 2006 年)。我们定义 $X\equiv r/R_{\mathrm{v}\mathrm{i}\mathrm{r}}$ ，浓度参数 $c\equiv r_{\mathrm{s}}/R_{\mathrm{v}\mathrm{i}\mathrm{e}}$ 是质量的一个弱函数(Klypin 等人 2011 年)：

$$\begin{array}{r}c=9.35{\left(\frac{M_{\mathrm{v}\mathrm{i}\mathrm{e}}}{{10}^{12}{M}_{\odot }}\right)}^{-0.13}\end{array}$$

(31)

but a very important quantity in determining the density shape at intermediate radii. The circular velocity for an NFW dark matter halo is given by
但在确定中间半径处密度形状方面是一个非常重要的量。NFW 暗物质晕的圆速度由下式给出

$$\begin{array}{r}{V}_{\mathrm{N}\mathrm{F}\mathrm{W}}(X)=V_{\mathrm{v}\mathrm{i}\mathrm{e}}^2\frac{1}{X}\frac{\ln (1+cX)-\frac{cX}{1+cX}}{\ln (1+c)-\frac{c}{1+c}},\end{array}$$

(32)

with $M_{\mathrm{v}\mathrm{i}\mathrm{r}}=1004/3\pi {\rho }_{\mathrm{c}}{R}_{\mathrm{v}\mathrm{i}\mathrm{r}}^3$ and ${\rho }_{\mathrm{c}}=1.0\times {10}^{-29}\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$.
与 $M_{\mathrm{v}\mathrm{i}\mathrm{r}}=1004/3\pi {\rho }_{\mathrm{c}}{R}_{\mathrm{v}\mathrm{i}\mathrm{r}}^3$ 和 ${\rho }_{\mathrm{c}}=1.0\times {10}^{-29}\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$ 。

6.3 Burkert-URC

The Burkert empirical profile (Burkert 1995; Salucci and Burkert 2000) well reproduces, in cooperation with the velocity components of the stellar and gaseous disks, the individual circular velocities of spirals, dwarf disks and low surface brightness systems. Furthermore, this profile is at the basis of the universal rotation curve of the above systems. The density profile reads as
Burkert 经验轮廓(Burkert 1995; Salucci 和 Burkert 2000)很好地再现了与恒星和气态盘的速度分量合作的螺旋、矮星盘和低表面亮度系统的个体圆速度。此外，这个轮廓是上述系统的通用旋转曲线的基础。密度轮廓如下

$$\begin{array}{r}{\rho }_{\mathrm{B}\text{-}\mathrm{U}\mathrm{R}\mathrm{C}}(r)=\frac{{\rho }_0{r}_0^3}{(r+r_0)(r^2+r_0^2)},\end{array}$$

(33)

$r_0$ and ${\rho }_0$ are the core radius and central density, respectively. The velocity profile is:
$r_0$ 和 ${\rho }_0$ 分别是核心半径和中心密度。速度轮廓为：

$$\begin{array}{r}{V}_{\mathrm{B}\text{-}\mathrm{U}\mathrm{R}\mathrm{C}}^2(r)=\frac{G}{r}2\pi {\rho }_0{r}_0^3[\ln (1+r/r_0)]+\frac{1}{2}\ln (1+r^2/r_0^2)-{\tan }^{-1}(r/r_0).\end{array}$$

(34)

This profile represents the (empirical) family of cored distributions (see Fig. 7). To discriminate among them the correct one is, currently, very difficult. It would require a large number of accurate measurements of RCs at inner radii $r<r_0$.
这个轮廓代表了核心分布的(经验)家族(见图 7)。要区分它们中的正确轮廓目前非常困难。这将需要大量准确测量内半径处 RC 的数据 $r<r_0$ 。

6.4 Pseudo-isothermal profile

The PI halo profile ${\rho }_{\mathrm{P}\mathrm{I}}(r)=\frac{{\rho }_0{r}_0^2}{((r^2+r_0^2)}$ is an alternative cored distribution to Eq. (34). This density profile implies that $V_{\mathrm{P}\mathrm{I}}(r)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ for $r\gg R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, which disagrees with the RC profiles at very outer radii that show a decline with radius (Salucci et al. 2007).
PI 晕轮廓 ${\rho }_{\mathrm{P}\mathrm{I}}(r)=\frac{{\rho }_0{r}_0^2}{((r^2+r_0^2)}$ 是 Eq 的另一种核心分布。(34)。这个密度轮廓意味着 $V_{\mathrm{P}\mathrm{I}}(r)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ 对于 $r\gg R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ，这与在非常外围半径处显示随半径下降的 RC 轮廓不符(Salucci 等人 2007 年)。

6.5 Fermionic halos

In this scenario there is a strong degeneracy limit for which the DM particles’ velocity dispersion ${\text{σ}}_{\mathrm{D}\mathrm{M},min}^2(\rho )$ has the minimal value. This represents the most compact configuration for a self-gravitating fermionic halo (see, e.g., Di Paolo et al. 2018). The density profiles of such fully degenerate halos are universal, depending only on the mass of the configuration:
在这种情况下，存在一个强度限制，使得 DM 粒子的速度离散 ${\text{σ}}_{\mathrm{D}\mathrm{M},min}^2(\rho )$ 具有最小值。这代表了自引力费米子暗物质晕的最紧凑配置（例如，参见 Di Paolo 等人 2018 年）。这些完全简并暗物质晕的密度配置文件是通用的，仅取决于配置的质量：

$$\begin{array}{r}\rho (x)={\rho }_0{\cos }^3\left[\frac{25}{88}\pi x\right],x=r/R_h,\end{array}$$

(35)

where ${\rho }_0$ is the central DM halo density. This profile is quite peculiar and recognizable in the RCs.
其中 ${\rho }_0$ 是中心 DM 暗物质晕密度。这个配置文件在旋转曲线中非常独特且易识别。

Fig. 7

DM halos density profiles. NFW (green), Burkert-URC (blue), fully degenerate fermionic particles (violet), Pseudo Isothermal (yellow) and Binney-URC (red)
DM 暗物质晕密度配置文件。NFW（绿色），Burkert-URC（蓝色），完全简并费米子粒子（紫色），伪等温（黄色）和 Binney-URC（红色）

6.6 Zhao halos

The following density profile (Zhao 1996):
下面的密度配置文件（Zhao 1996）：

$$\begin{array}{r}\rho (r)=\frac{{\rho }_0}{(\frac{r}{R_0}{)}^{\gamma }{(1+{\left(\frac{r}{R_o}\right)}^{\alpha })}^{\frac{\beta +\gamma }{\alpha }}},\end{array}$$

(36)

where ${\rho }_0$ is the central density and $R_0$ the “core radius”, that, initially, was not proposed for the DM halo density, is defined by the set of parameters: $\alpha$, $\beta$, $\gamma$. The case (1, 3, $\gamma$) is sometimes used as a “cored-NFW” profile. This is incorrect because both in the Burkert and in the NFW profiles, the inner regions are not related with the outermost regions, as, instead occurs in the Zhao model. Moreover, with the latter, we pass from the two free parameters of most of the halo models in the ballpark, to the five of Eq. (36). This seems in disagreement with observations in spirals, ellipticals and spheroidals that suggest that DM halos are one (two)-parameters family.
其中 ${\rho }_0$ 是中心密度， $R_0$ 是“核半径”，最初并非为 DM 暗物质晕密度提出，由参数集定义： $\alpha$ ， $\beta$ ， $\gamma$ 。有时使用情况（1，3， $\gamma$ ）作为“有核 NFW”配置文件。这是不正确的，因为在 Burkert 和 NFW 配置文件中，内部区域与外部区域无关，而在 Zhao 模型中却是如此。此外，通过后者，我们从大多数暗物质晕模型的两个自由参数转变为方程（36）的五个参数。这似乎与螺旋、椭圆和球状星系的观测结果不符，这些观测结果表明 DM 暗物质晕是一个（两个）参数家族。

6.7 Transformed halos

We want to draw the attention on the profiles which are the outcome of the primordial NFW halos after that these have experienced the effects that it is called baryonic feedback (e.g., Di Cintio et al. 2014). They seem in agreement with those observed around galaxies. However, the collisionless DM paradigm requires that such kind of transformation has occurred in every galaxy of any luminosity and Hubble type and to reach this goal seems extremely difficult. On the other side, the effect of the baryonic feedback to DM halos has to be investigated, no matter what the nature of DM is. In conclusion, a review on this crucial complex and still on its infancy issue must be a goal future work.
我们想要引起对那些是原始 NFW 暗暗物质晕的结果的注意，之后这些暗暗物质晕经历了所谓的重子反馈的影响（例如，Di Cintio 等人，2014 年）。它们似乎与围绕星系观察到的那些一致。然而，无碰撞 DM 范式要求这种转变发生在任何亮度和哈勃类型的每个星系中，要达到这个目标似乎极为困难。另一方面，必须调查重子反馈对 DM 暗暗物质晕的影响，无论 DM 的性质如何。总之，对这个关键的复杂问题进行审查，尽管仍处于初期阶段，但必须成为未来工作的目标。

7 Kinematics of galaxy systems
星系系统的运动学

A main channel to obtain the DM properties in galaxies is through their kinematics (rotation curves and dispersion velocities). The analysis could regard individual objects or stacked data of a sample of objects.
通过它们的动力学（旋转曲线和离散速度）获得星系中 DM 属性的主要渠道。分析可以涉及个体对象或对象样本的堆叠数据。

7.1 The Tully–Fisher and the baryonic Tully–Fisher
7.1 Tully–Fisher 和重子 Tully–Fisher

Tully and Fisher (1977) discovered that, in spirals, the neutral hydrogen 21-cm FHWM linewidths $w_{50}$, related, in a disk system, to the maximal rotational velocities $V_{max}$ by: $\log V_{max}\simeq -0.3+\log w_{50}-\log \sin i$, with i the inclination of the galaxy with respect to the l.o.s., correlate with the galaxy magnitudes M
Tully 和 Fisher（1977 年）发现，在螺旋星系中，中性氢 21 厘米 FHWM 线宽 $w_{50}$ ，与盘系统中的最大旋转速度 $V_{max}$ 相关，由以下关系： $\log V_{max}\simeq -0.3+\log w_{50}-\log \sin i$ ，其中 i 是星系相对于 l.o.s.的倾斜角，与星系的星等 M 相关

$$\begin{array}{r}M=a\log \left(\frac{w_{50}}{\sin i}\right)+b,\end{array}$$

(37)

where a is the slope of the relationship and b the zero-point.
其中 a 是关系的斜率，b 是零点。

With the availability of a large number of extended RCs, the relation evolved: a radius proportional to the disk length-scale $R_D$ (e.g. $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ or $R_{max}=2.2R_D$) emerged as the reference radius; moreover, the circular velocity at this reference radius substituted the linewidth w.
随着大量扩展的 RCs 的可用性，关系发展了：与盘长度尺度 $R_D$ （例如 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 或 $R_{max}=2.2R_D$ ）成比例的半径出现为参考半径；此外，在这个参考半径处的圆速度替代了线宽 w。

It is easy to realize that Eq. (37) just reflects the equilibrium configuration of rotating disks embedded in dark halos (Strauss and Willick 1995) and that the magnitude M in the relation is the prior for the stellar disk mass. However, it is worth going deeper: in fact, despite that in each spiral the disk and the dark components contribute in different proportions to the value of $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})=(V_d(R_{\mathrm{o}\mathrm{p}\mathrm{t}}{)}^2+V_h(R_{\mathrm{o}\mathrm{p}\mathrm{t}}{)}^2{)}^{1/2}$, one finds that $V_d(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ correlates better with magnitudes than $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ (Salucci et al. 1993). This finding can be understood in that the latter relationship couples two attributes that pertain exclusively to the stellar disk: its mass, measured kinematically and its luminosity.
易于意识到方程（37）只是反映了嵌入暗暗物质晕的旋转盘的平衡构型（Strauss 和 Willick，1995 年），而关系中的量 M 是恒星盘质量的先验。然而，值得深入探讨：事实上，尽管在每个螺旋中，盘和暗组分对 $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})=(V_d(R_{\mathrm{o}\mathrm{p}\mathrm{t}}{)}^2+V_h(R_{\mathrm{o}\mathrm{p}\mathrm{t}}{)}^2{)}^{1/2}$ 的值贡献的比例不同，但发现 $V_d(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ 与星等的相关性比 $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ 更好（Salucci 等人，1993 年）。这一发现可以理解为后者关系了两个仅属于恒星盘的属性：其质量，通过动力学测量，和其亮度。

The physical meaning of the TF relation as a link between circular velocities and stellar masses has been shown by means of 729 kinematically and morphologically different galaxies belonging to the SAMI Galaxy Survey sample (Bloom et al. 2017). It has been found:
TF 关系的物理意义是将圆速度和恒星质量联系起来，通过对属于 SAMI 星系调查样本的 729 个运动学和形态学不同的星系进行展示（Bloom 等人，2017 年）。已经发现：

$$\begin{array}{r}\log V_{2.2}=(0.26\pm 0.017)\log (M_{\star }/M_{\odot })-(0.5\pm 0.13),\end{array}$$

(38)

with $V_{2.2}\equiv V(2.2R_D)$. Such relationship results in very good agreement with the correspondent one we can derive from the URC (Salucci et al. 2007): $\log V_{2.2}=(0.263\pm 0.005)\log (M_{\star }/M_{\odot })-(0.57\pm 0.05)$.
与 $V_{2.2}\equiv V(2.2R_D)$ 。 这种关系与我们可以从 URC 推导出的对应关系非常一致 (Salucci 等人 2007)： $\log V_{2.2}=(0.263\pm 0.005)\log (M_{\star }/M_{\odot })-(0.57\pm 0.05)$ 。

A recent work (Ponomareva et al. 2018) has investigated the statistical properties of the Tully–Fisher relation for a sample of 32 galaxies with accurately measured distances and with (1) panchromatic photometry in 12 bands: from far ultra-violet to $4.5\text{μ}\mathrm{m}$, and (2) spatially resolved HI kinematics. For this sample they adopted, in turn, the following reference velocities: the linewidth $W_{50}$, the maximum velocity $V_{max}$ and $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ the average value of the RC in the range (2–$5)R_D$. With these quantities they constructed 36 correlations, each of them involving one magnitude and one kinematical parameter. They found that the slope of the relationships strongly depends on the band considered and that the tightest correlation occurs between the $3.6\text{μ}\mathrm{m}$ photometric band magnitude $M_{3.6\text{μ}\mathrm{m}}$ and $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ (see Fig. 8):
最近的一项研究 (Ponomareva 等人 2018) 调查了对精确测量距离并具有 (1) 12 个波段的全波段光度测量：从远紫外线到 $4.5\text{μ}\mathrm{m}$ ，以及 (2) 空间分辨 HI 动力学的 32 个星系样本的图利-费舍尔关系的统计特性。 对于这个样本，他们依次采用了以下参考速度：线宽 $W_{50}$ ，最大速度 $V_{max}$ 和 $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ RC 在范围内的平均值 (2- $5)R_D$ 。他们构建了 36 个相关性，每个相关性涉及一个幅度和一个动力学参数。 他们发现关系的斜率强烈取决于考虑的波段，并且最紧密的相关性发生在 $3.6\text{μ}\mathrm{m}$ 光度波段幅度 $M_{3.6\text{μ}\mathrm{m}}$ 和 $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ 之间 (见图 8)：

$$\begin{array}{r}{M}_{3.6}=(9.5\pm 0.3)\log V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}+(3.3\pm 1.7)\end{array}$$

(39)

in good agreement with the value of $8.6\pm 0.1$ found by Yegorova and Salucci (2007) for the slope of the I magnitude of the radial Tully–Fisher relationship at $R=1.2R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ that becomes $9.6\pm 0.3$ when translated in the $3.6\text{μ}\mathrm{m}$ band.
与 Yegorova 和 Salucci (2007) 发现的 I 幅度斜率的值 $8.6\pm 0.1$ 相当一致，即在 $R=1.2R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 处的径向图利-费舍尔关系，当转换为 $3.6\text{μ}\mathrm{m}$ 波段时变为 $9.6\pm 0.3$

Fig. 8

The slope and the scatter of the TF relation by adopting different reference velocities and different systems of magnitude
通过采用不同的参考速度和不同的幅度系统，TF 关系的斜率和散布

Fig. 9

(Top) The stellar mass (left) and baryonic (right) Tully–Fisher relations. (Bottom) The determination of the BTF. Images reproduced with permission from (top) McGaugh (2005), and (bottom) from Lelli et al. (2016a), copyright by AAS
(顶部) 恒星质量 (左) 和重子质量 (右) 图利-费舍尔关系。 (底部) BTF 的确定。 图像经许可复制自 (顶部) McGaugh (2005)，以及 (底部) Lelli 等人 (2016a)，AAS 版权所有

7.2 The baryonic Tully–Fisher

McGaugh et al. (2000) found a fundamental relationship by correlating the baryonic mass (i.e., the sum of the stellar and the (HI + He) gas mass) with the reference rotation velocity $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$. This Baryonic Tully–Fisher (BTF) relation has been thorough fully studied and confirmed by several works: (e.g., Bell and de Jong 2001; Verheijen 2001; Gurovich et al. 2004). A decisive step forward in understanding it came from McGaugh (2005), who investigated a sample of galaxies with extended 21-cm rotation curves spanning the range $20\mathrm{k}\mathrm{m}/\mathrm{s}<V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}<300\mathrm{k}\mathrm{m}/\mathrm{s}$. By using a grid of stellar population models they estimated the values of the stellar disks masses to which they added those of the HI disks derived by the observed 21-cm HI fluxes. They found:
McGaugh 等人（2000 年）通过将重子质量（即恒星和（HI + He）气体质量之和）与参考旋转速度 $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ 进行相关分析，发现了一个基本关系。这种重子 Tully-Fisher（BTF）关系已经被多项研究彻底研究和确认：（例如，Bell 和 de Jong 2001 年；Verheijen 2001 年；Gurovich 等人 2004 年）。对于理解它迈出了重要的一步，这一步来自 McGaugh（2005 年），他调查了一组延伸到 $20\mathrm{k}\mathrm{m}/\mathrm{s}<V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}<300\mathrm{k}\mathrm{m}/\mathrm{s}$ 范围的 21 厘米旋转曲线的星系样本。通过使用一系列恒星群体模型，他们估计了恒星盘的质量值，然后加上通过观测到的 21 厘米 HI 流量推导出的 HI 盘的质量。他们发现：

$$\begin{array}{r}{M}_{\mathrm{b}\mathrm{a}\mathrm{r}}=A{V}_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}^4;A=50M_{\odot }\mathrm{k}{\mathrm{m}}^{-4}{\mathrm{s}}^{-4}\end{array}$$

(40)

(see Fig. 9). Notice that, by including the HI mass in the galaxy baryonic mass, the BTF becomes log-log linear and has less intrinsic scatter.
（见图 9）。请注意，通过将 HI 质量包括在星系重子质量中，BTF 变为对数对数线性，并且具有更少的固有散射。

Lelli et al. (2016a) investigated the BTF relationship with a sample of 118 disc galaxies (spirals and irregulars) with data of the highest quality: extended HI high-quality rotation curves tracing the total mass distribution and Spitzer photometry at $3.6\text{μ}\mathrm{m}$ tracing the stellar mass distribution. They assumed the stellar mass-to-light ratio ($M_{\star }/L_{3.6\text{μ}\mathrm{m}}$) to be constant among spirals and found that the scatter, slope, and normalization of the relation vary with the adopted $M_{\star }/L_{3.6\text{μ}\mathrm{m}}$ value, though the intrinsic scatter is always modest: $\le 0.1$ dex. The BTF relationship gets minimized for $M_{\star }/L_{3.6\text{μ}\mathrm{m}}>0.5$. This result, in conjunction with the RC profiles of the galaxies in the sample, implies maximal discs in the high-surface-brightness.⁵
Lelli 等人（2016a）研究了一个包含 118 个盘状星系（螺旋和不规则）的样本的 BTF 关系，这些星系具有最高质量的数据：扩展的 HI 高质量旋转曲线追踪总质量分布和 Spitzer 光度追踪星体质量分布。他们假设螺旋星系中的恒星质量与光比（）是恒定的，并发现关系的散射、斜率和归一化随所采用的值变化，尽管固有散射始终适度：dex。BTF 关系对最小化。这一结果与样本中星系的 RC 剖面相结合，意味着高表面亮度中的最大盘。

The BTF relationship slope comes close to 4.0, see Fig. 9(bottom) and the residuals show no correlation with the galaxy structural parameters (radius or surface brightness). The above relationship seems to play an important cosmological role; however, the value of its slope strongly depends on the vagueness in the definition of the reference velocity $V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ (Brook et al. 2016). The DM enters in this relation principally through the value of the dark/ total matter fraction at $R_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$: this indicates that the BTF is related more to the disk formation process than to the DM nature.
BTF 关系斜率接近 4.0，参见图 9（底部），残差与星系结构参数（半径或表面亮度）没有相关性。上述关系似乎在宇宙学中起着重要作用；然而，其斜率的值强烈取决于对参考速度的定义的模糊性（Brook 等人 2016）。DM 主要通过在这个关系中进入暗/总物质分数的值：这表明 BTF 更与盘形成过程相关，而不是与 DM 性质相关。

7.3 The universal rotation curve and the radial Tully–Fisher
7.3 通用旋转曲线和径向 Tully–Fisher

We can represent all the rotation curves of spirals by means of the universal rotation curve (URC), pioneered in Rubin et al. (1980), expressed in Persic and Salucci (1991) and set in Persic et al. (1996) and in Salucci et al. (2007). By adopting the normalized radial coordinate $x\equiv r/R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, the RCs of spirals are very well described by a universal profile, function of x and of $\lambda$, where $\lambda$ is one, at choice, among $M_I$, the I magnitude, $M_D$, the disk mass and $M_{\mathrm{v}\mathrm{i}\mathrm{r}}$, the halo virial mass (Salucci et al. 2007).
我们可以通过通用旋转曲线（URC）来表示所有螺旋星系的旋转曲线，这是在 Rubin 等人（1980 年）中首创的，由 Persic 和 Salucci（1991 年）表达，并在 Persic 等人（1996 年）和 Salucci 等人（2007 年）中设定。通过采用归一化径向坐标，螺旋星系的 RCs 可以非常好地由一个通用剖面描述，该剖面是 x 和的函数，其中是选择之一，包括 I 星等级，盘质量和暗物质 virial 质量（Salucci 等人 2007）。

The universal magnitude-dependent profile is evident in the 11 coadded rotation curves $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ (Fig. 6 of Persic et al. 1996 and top of Fig. 11), built from the individual RCs of a sample of 967 spirals with luminosities spanning their whole I-band range: $-16.3<M_I<-23.4$. I-band surface photometry measurements provided these objects with their stellar disk length scales $R_D$ (Persic and Salucci 1995).⁶
通用的幅度相关剖面在 11 个叠加旋转曲线中明显可见（Persic 等人 1996 年的图 6 和图 11 顶部），这些曲线是从一组包含 967 个螺旋星系的个体 RCs 构建而成，这些星系的亮度跨越它们整个 I 波段范围：I 波段表面光度测量为这些物体提供了它们的恒星盘长度尺度（Persic 和 Salucci 1995）。

The coadded RCs are built in a three-step way: (1) We start with a large sample of galaxies with RC and suitable photometry (in the case of Persic et al. 1996: 967 objects and suitable I-band measurements). The whole (I) magnitude range is divided into 11 successive bins centred at $M_I$, as listed in Table 1 of Persic et al. (1996). (2) The RC of each galaxy of the sample is assigned to its corresponding I magnitude bin, normalized by its $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ value and then expressed in terms of its normalized radial coordinate x. (3) The double-normalized RCs $V(x)/V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ curves are coadded in 11 magnitude bins and in 20 radial bins of length 0.1 and then averaged to get: $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)/V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(1,M_I)$, the points with errorbars in Fig. 11. The 11 values of $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(1,M_I)$ are given in Table 1 of Persic et al. (1996). The RCs are usually increasing or decreasing. Simplifying, they increase when they are dark matter dominated or always for $r<R_D$ and decrease for $r>2R_D$ when they are disk dominated.⁷ The recent finding of RCs of six massive star-forming galaxies that, outside $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, decrease with radius (Genzel et al. 2017) has been considered very surprising. Rightly, it has been proposed that this trend arises because this high-redshift galaxy population was strongly baryon dominated. However, while the importance of such objects in the cosmological context is obvious, there is a presence, also in the local Universe, of many baryon-dominated decreasing RCs. This was first drawn to the attention by Persic and Salucci (1991) and, moreover, it is inbuilt in the URC.
合并的 RC 是通过三个步骤构建的：（1）我们从具有 RC 和适当光度的大样本星系开始（在 Persic 等人 1996 年的情况下：967 个对象和适当的 I 波段测量）。整个（I）星等范围被分成 11 个连续的区间，中心位于 $M_I$ ，如 Persic 等人（1996 年）的表 1 所列。 （2）样本中每个星系的 RC 被分配到相应的 I 星等区间，通过其 $V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ 值归一化，然后用其归一化的径向坐标 x 表示。 （3）双归一化的 RC 曲线 $V(x)/V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 在 11 个星等区间和 20 个长度为 0.1 的径向区间中合并，然后平均得到： $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)/V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(1,M_I)$ ，图 11 中带有误差线的点。 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(1,M_I)$ 的 11 个值在 Persic 等人（1996 年）的表 1 中给出。 RC 通常是增加或减少的。简单地说，当它们被暗物质主导或始终为 $r<R_D$ 时，它们会增加，当它们被盘组分主导时，它们会减少 $r>2R_D$ 。最近发现的六个大质量星形成星系的 RC，除 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 外，随半径减小（Genzel 等人 2017 年）被认为非常令人惊讶。正确地说，这种趋势是因为这些高红移星系群体在很大程度上由重子组成。然而，尽管这些对象在宇宙学背景下的重要性是显而易见的，但在当地宇宙中也存在许多重子主导的减小 RC。这是由 Persic 和 Salucci（1991 年）首次引起注意的，并且它是 URC 内在的。

The URC is the analytical function devised to fit the stacked/coadded RCs $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$. In principle, it could be any suitable empirical function of ($x,M_I$); the idea of Persic et al. (1996) was to choose, as fitting function, the sum in quadrature of the velocity components to the circular velocity. Namely, the Freeman stellar disk with one free parameter, its mass $M_D$ and the dark halo with an assumed profile and two free parameters, the central density ${\rho }_0$ and the core radius $r_0$. Then, the data $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ are fitted by the $V_{\mathrm{U}\mathrm{R}\mathrm{C}}$ universal function:
URC 是设计用来拟合堆叠/合并 RC 的解析函数 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ 。原则上，它可以是任何适当的经验函数（ $x,M_I$ ）；Persic 等人（1996 年）的想法是选择作为拟合函数的速度分量到圆速度的平方和。即，Freeman 恒星盘带有一个自由参数，其质量 $M_D$ 和暗暗晕带有一个假定的轮廓和两个自由参数，中心密度 ${\rho }_0$ 和核半径 $r_0$ 。然后，数据 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ 被 $V_{\mathrm{U}\mathrm{R}\mathrm{C}}$ 通用函数拟合：

$$\begin{array}{r}{V}_{\mathrm{U}\mathrm{R}\mathrm{C}}^2(x,M_I)\equiv V_{\mathrm{U}\mathrm{R}\mathrm{C}\mathrm{d}}^2(x;M_D(M_I))+V_{\mathrm{U}\mathrm{R}\mathrm{C}\mathrm{h}}^2(x;{\rho }_0(M_I),r_0(M_I))\end{array}$$

(41)

The first component of the RHS is the standard Freeman disk of Eq. (8); the second is the B-URC halo of Eq. (34). In dwarf galaxies, a HI term must be included (Karukes and Salucci 2017).
RHS 的第一个分量是方程（8）的标准 Freeman 盘；第二个是方程（34）的 B-URC 暗晕。在矮星系中，必须包括一个 HI 项（Karukes 和 Salucci 2017）。

The excellent fit (see Fig. 11) has led us to the validation of the URC idea: there exists a universal function of (normalized) radius and luminosity that well fits the RC of any spiral galaxy (see Salucci et al. 2007).⁸
出色的拟合（见图 11）使我们验证了 URC 理念：存在一个通用的（归一化的）半径和亮度函数，很好地拟合了任何螺旋星系的 RC（见 Salucci 等人 2007 年）。 ⁸

Fig. 10

The radial TF. The variation of the slopes $a_i$ with $r_i$ is very evident. Image reproduced with permission from Yegorova and Salucci (2007), copyright by the authors
径向 TF。 斜率的变化 $a_i$ 与 $r_i$ 非常明显。 图像经授权从 Yegorova 和 Salucci（2007）复制，版权由作者所有。

The radial Tully–Fisher is a relationship on the URC surface, orthogonal to the various RCs (Yegorova and Salucci 2007; see Fig. 10 top). At different galactocentric distances, measured in units of the optical size, $r_i\equiv i{R}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ($i=0.2,0.3,\dots ,1$), a family of independent Tully–Fisher-like relationships emerges:
径向 Tully-Fisher 关系是 URC 表面上的一种关系，与各种 RCs（Yegorova 和 Salucci 2007；见图 10 顶部）正交。在以光学尺寸为单位测量的不同星系中心距离 $r_i\equiv i{R}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ （ $i=0.2,0.3,\dots ,1$ ）处，出现了一系列独立的类 Tully-Fisher 关系：

$$\begin{array}{r}{M}_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}=b_i+a_i\log V(r_i),\end{array}$$

(42)

with $M_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}$ the magnitude in a specific band, often the (R, I)-bands. The RTF has a very small r.m.s. scatter, at any radius smaller than that of the classical TF. It also shows a large systematic variation of the slopes $a_i$ with $r_i$ that range, across the disk, between $-4$ and $-8$. This variation, in cooperation with the smallness of the scatter, indicates that the fractional amount of dark matter inside the optical radius is luminosity-dependent (Yegorova and Salucci 2007).
在特定波段中，通常是（R，I）波段，具有 $M_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}}$ 的幅度。RTF 具有非常小的均方根散射，在任何小于经典 TF 半径的半径处。它还显示出在盘面上的斜率 $a_i$ 与 $r_i$ 之间的大系统变化，范围在 $-4$ 和 $-8$ 之间。这种变化，与散射的小幅度一起，表明光学半径内的暗物质的分数量取决于亮度（Yegorova 和 Salucci 2007）。

It is important to stress that, given a sample of RCs, the RTF relationship provides us with an independent method of deriving (if it exists) the underlying coadded RCs and, in turn, the relative URC. Yegorova and Salucci (2007), in fact, have shown that samples with a similar $a_i$ vs $r_i$ relationship have also similar $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e})$. This has been applied to the large samples of Courteau (1997) and Vogt et al. (2004a, b) with the result of finding the same RTF discovered in the Persic et al. (1996) sample (see Fig. 8 of Yegorova and Salucci 2007) and, then, finding very similar coadded RCs.
强调一点很重要，给定一组 RC，RTF 关系为我们提供了一个独立的方法来推导（如果存在）基础的合并 RC，进而推导相对的 URC。事实上，Yegorova 和 Salucci（2007 年）已经表明，具有类似 $a_i$ vs $r_i$ 关系的样本也具有类似的 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e})$ 。这已经应用于 Courteau（1997 年）和 Vogt 等人（2004a、b）的大样本，结果发现与 Persic 等人（1996 年）样本中发现的相同 RTF（参见 Yegorova 和 Salucci 2007 年的图 8），然后发现非常相似的合并 RC。

8 The dark matter distribution in disk systems
8 盘系统中的暗物质分布

The general pattern is the following: spirals show a reference radius $R_T(L_I)$ whose size ranges from 1 to $3R_D$ according to the galaxy luminosity (see Fig. 8 of Persic et al. 1996 and Palunas and Williams 2000); inside $R_T(L_I)$ the ordinary baryonic matter fully accounts for the RC, while, for $R>R_T(L_I)$, is instead unable to justify the profile and the amplitude of the RC.
一般模式如下：螺旋显示一个参考半径 $R_T(L_I)$ ，其大小范围从 1 到 $3R_D$ ，根据星系的亮度（参见 Persic 等人 1996 年和 Palunas 和 Williams 2000 年的图 8）；在 $R_T(L_I)$ 内，普通重子物质完全解释了 RC，而对于 $R>R_T(L_I)$ ，则无法解释 RC 的轮廓和振幅。

8.1 Dark matter from stacked RCs
8.1 堆叠 RCs 中的暗物质

Very extended individual RCs and virial velocities $V_{\mathrm{v}\mathrm{i}\mathrm{r}}\equiv (G{M}_{\mathrm{v}\mathrm{i}\mathrm{r}}/R_{\mathrm{v}\mathrm{i}\mathrm{r}}{)}^{1/2}$ obtained in Shankar et al. (2006), further support the URC paradigm and help determining the universal velocity function out to the virial radius (Salucci et al. 2007). It is important to stress that the $V_{\mathrm{U}\mathrm{R}\mathrm{C}}$ function (and the relative mass model) has, in principle, three free parameters: the disk mass and two quantities related to the DM distribution (the halo central density ${\rho }_0$ and the core radius $r_0$). These are obtained by best fitting the $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$) and found to be correlated among themselves and with the luminosity. So, the RCs and the related gravitational potential of spirals belong to a family ruled by 1-parameter that we can choose among many possibilities, e.g., the halo mass, which is a combination of ${\rho }_0$ and $r_0$ and it ranges in spirals as: $3\times {10}^{10}{M}_{\odot }\le M_{\mathrm{v}\mathrm{i}\mathrm{r}}\le 3\times {10}^{13}{M}_{\odot }$ (Fig. 11).
在 Shankar 等人（2006）中获得的非常广泛的单个 RCs 和维里速度 $V_{\mathrm{v}\mathrm{i}\mathrm{r}}\equiv (G{M}_{\mathrm{v}\mathrm{i}\mathrm{r}}/R_{\mathrm{v}\mathrm{i}\mathrm{r}}{)}^{1/2}$ 进一步支持 URC 范式，并有助于确定通向维里半径的通用速度函数（Salucci 等人 2007）。 强调 $V_{\mathrm{U}\mathrm{R}\mathrm{C}}$ 函数（以及相关质量模型）原则上有三个自由参数：盘质量和与 DM 分布相关的两个量（暗物质晕中心密度 ${\rho }_0$ 和核半径 $r_0$ ）。这些参数通过最佳拟合 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ ）获得，并发现它们彼此之间以及与亮度相关。因此，螺旋星系的 RCs 和相关的引力势属于由我们可以从许多可能性中选择的 1 个参数控制的家族，例如，暗物质晕质量，它是 ${\rho }_0$ 和 $r_0$ 的组合，并在螺旋星系中范围为： $3\times {10}^{10}{M}_{\odot }\le M_{\mathrm{v}\mathrm{i}\mathrm{r}}\le 3\times {10}^{13}{M}_{\odot }$ （图 11）。

Fig. 11

(Top) The URC best-fit models of the coadded RCs (points with errorbars) (Persic et al. 1996). The following are shown: the bin magnitude $M_I$, the disk/halo contributions (dotted/dashed lines) and the resulting URC (solid line). (Bottom left) The 4-D relationship among the central DM density, its core radius in units of $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, the DM fraction at $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ and the galaxy I-luminosity (proportional to the area of the circles). (Bottom right) The URCs from Persic et al. (1996), (yellow) and from Catinella et al. (2006) (blue). Legenda: $x\equiv R/R_D$, $y\equiv \log (M_{\mathrm{v}\mathrm{i}\mathrm{r}}/({10}^{11}{M}_{\odot }))$, $z\equiv V(x)/V(3.2)$. The differences between the two URCs are also indicated
(顶部) 合并的 RCs 的 URC 最佳拟合模型（带有误差线）（Persic 等人 1996）。显示如下内容：bin magnitude $M_I$ ，盘/暗物质晕贡献（虚线/虚线）和得到的 URC（实线）。 （左下）中心 DM 密度、其单位为 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 的核半径、 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 处的 DM 分数和星系 I-亮度之间的 4-D 关系（与圆圈的面积成比例）。 （右下）来自 Persic 等人（1996）的 URCs（黄色）和来自 Catinella 等人（2006）的 URCs（蓝色）。图例： $x\equiv R/R_D$ ， $y\equiv \log (M_{\mathrm{v}\mathrm{i}\mathrm{r}}/({10}^{11}{M}_{\odot }))$ ， $z\equiv V(x)/V(3.2)$ 。还指出了两个 URCs 之间的差异

8.2 Dark matter from individual RCs
8.2 个体 RC 的暗物质

The study of individual RCs is very similar to that of the stacked ones as regards to their mass modelling, but it is complementary to it with respect to the data analysis. Moreover, in the core–cusp issue, the individual RCs have a special role: stacked RCs of spirals, as seen in the previous section, point unambiguously to a cored distribution, but cannot indicate to us whether this is a sort of average property of the entire population of spirals or a property of any single object. Only the analysis of fair number of individual RCs of systems of different luminosity and Hubble types can answer this.
对个体 RC 的研究与堆叠 RC 的研究在质量建模方面非常相似，但在数据分析方面是互补的。此外，在核心-尖峰问题上，个体 RC 具有特殊的作用：正如在前一节中所看到的，螺旋星系的堆叠 RC 明确指向核心分布，但不能告诉我们这是否是整个螺旋星系群体的一种平均特性，还是任何单个对象的特性。只有对不同亮度和哈勃类型系统的相当数量的个体 RC 进行分析才能回答这个问题。

It is worth pointing out that, in the first 15 years since the DM discovery from the profiles of the RCs, the latter have always been reproduced by models including a Freeman disk, a bulge and a dark halo with the cored Pseudo Isothermal distribution (e.g., Carignan and Freeman 1985; van Albada et al. 1985). It is well known that in the current $\text{Λ}$CDM cosmological scenario the dark matter halos have a very specific and universal cusped density distribution (Navarro et al. 1997). A debate has arisen on the level of the observational support for such profile (de Blok et al. 2001; Salucci 2001; Gentile et al. 2004; Simon 2005; Spekkens et al. 2005; Kuzio de Naray et al. 2008; de Blok et al. 2008; Oh et al. 2011; Adams et al. 2014 to name a few, reviews on this issue: Bullock and Boylan-Kolchin 2017; de Blok 2010).⁹
值得指出的是，在从 RC 的轮廓中发现 DM 以来的头 15 年里，后者一直被包括 Freeman 盘、一个球状体和一个具有核心伪等温分布的暗暗物质晕模型所复制（例如，Carignan 和 Freeman 1985; van Albada 等人 1985）。众所周知，在当前的 CDM 宇宙学场景中，暗物质晕具有非常特定和普遍的尖峰密度分布（Navarro 等人 1997）。关于对这种轮廓的观测支持水平的辩论已经出现（de Blok 等人 2001; Salucci 2001; Gentile 等人 2004; Simon 2005; Spekkens 等人 2005; Kuzio de Naray 等人 2008; de Blok 等人 2008; Oh 等人 2011; Adams 等人 2014 等等，关于这个问题的评论：Bullock 和 Boylan-Kolchin 2017; de Blok 2010）。

It is important to remark that the DM cores could come ab initio from the structural properties of the (exotic?) DM particles or been created, over all the Hubble time, by dynamical processes occurring inside the galaxies.
重要的是要指出，DM 核心可能起源于（奇异的？）DM 粒子的结构特性，或者是在整个哈勃时间内由发生在星系内的动力学过程创造的。

Fig. 12

The relationship between the size of the DM core radius $R_{\mathrm{C}}$ and the value of the central dark matter density ${\rho }_0$. Image reproduced with permission from Martinsson et al. (2013), copyright by ESO
暗物质核半径 $R_{\mathrm{C}}$ 与中心暗物质密度值 ${\rho }_0$ 之间的关系。图像经 Martinsson 等人（2013）许可复制，ESO 版权所有

Martinsson et al. (2013) devised and applied to a sample of 30 spirals, a method to decompose the rotation curves in its dark and luminous components. The method exploits the vertical velocity dispersions of the disk stars ${\text{σ}}_z$ (see Sect. 6.3). By reminding that $R_{max}\equiv 2.2R_D$ is the radius where the disk velocity component has its maximum, they found: $(V_d(R_{max})/V(R_{max}){)}^2=0.57\pm 0.07$, with a dependence on galaxy luminosity: in their velocity models, at $R_{max}$, the disk component prevails over the dark component in the biggest spirals, while, it is very sub-dominant in the smallest ones.
Martinsson 等人（2013）设计并应用于 30 个螺旋体样本，一种将旋转曲线分解为其暗和发光组分的方法。该方法利用了盘星的垂直速度离散 ${\text{σ}}_z$ （见第 6.3 节）。通过提醒 $R_{max}\equiv 2.2R_D$ 是盘速度分量达到最大值的半径，他们发现： $(V_d(R_{max})/V(R_{max}){)}^2=0.57\pm 0.07$ ，与星系亮度有关：在他们的速度模型中，在 $R_{max}$ ，盘组分在最大的螺旋体中占优势，而在最小的螺旋体中非常次要。

They also modeled the dark matter halos with either a PI or a NFW profile and found the former distribution performing something better and showing a tight ${\rho }_0$ vs. $r_0$ relationship, very similar to that found in spirals by means of a different analysis (see Fig. 12).
他们还用 PI 或 NFW 配置对暗暗物质晕进行建模，并发现前者的分布表现得更好，显示出一个紧密的 ${\rho }_0$ vs. $r_0$ 关系，与通过不同分析方法在螺旋体中发现的非常相似（见图 12）。

A recent study of NGC5005 (Richards et al. 2015) can be considered as a test case investigation of the mass distribution in spirals obtained by means of multi-messenger observations. These included images taken at $3.6\text{μ}\mathrm{m}$ from the Spitzer Space Telescope, B and R broadband and H$\alpha$ narrowband observations. Very Large Array (VLA) radio synthesis observations of neutral hydrogen provided the HI surface density and the kinematics. Spectroscopic integral field unit observations at WIYN 3.5-m telescope provided the ionized gas kinematics in the inner region. The surface brightness has been carefully decomposed in its disk and bulge component. The modelling of the composite high-resolution rotation curve clearly favours a PI DM halo, with core radius of $2.5\pm 0.1\mathrm{k}\mathrm{p}\mathrm{c}$, over the corresponding NFW configuration.
最近对 NGC5005（Richards 等人 2015）的研究可以被视为通过多信使观测手段获得的螺旋体质量分布的测试案例调查。这些包括从斯皮策空间望远镜拍摄的图像，B 和 R 宽带以及 H 窄带观测。非常大阵列（VLA）射电合成观测提供了中性氢的 HI 表面密度和动力学。WIYN 3.5 米望远镜的光谱积分场单元观测提供了内部区域的电离气体动力学。表面亮度已经被仔细分解为其盘和球状组分。复合高分辨率旋转曲线的建模清楚地支持 PI DM 暗暗物质晕，其核半径为 $2.5\pm 0.1\mathrm{k}\mathrm{p}\mathrm{c}$ ，而不是相应的 NFW 配置。

Fig. 13

Maximum disc best-fits (solid lines) to the RCs (dots with errorbars). Also shown the contribution of gas, disc, bulge, and PI dark halo (dotted, short dashed, long dashed, dash-dot lines). Image reproduced with permission from Bottema and Pestaña (2015), copyright by the authors
最大盘符合最佳拟合（实线）到 RCs（带误差条的点）。还显示了气体、盘、球状和 PI 暗暗物质晕的贡献（点状、短虚线、长虚线、虚点线）。图像经 Bottema 和 Pestaña（2015）许可复制，作者版权所有

Bottema and Pestaña (2015) obtained high-resolution kinematics for sample of 12 galaxies, whose luminosities are distributed regularly over a range spanning several orders of magnitude. They found that models with maximum disks, cored DM halos and a unique value of the mass-to-light ratio, i.e., $M_D/L_R=1.0$, fit very well all the RCs, see Fig. 13. NFW DM halos, independently of the baryonic distribution, cannot fit the RCs of the least massive galaxies of the sample, while, for the most massive ones, the best fitting values of the structural parameters of the NFW +stellar/HI disks models, namely the halo concentration and mass and the mass-to-light ratio of the stellar disk, take often non-physical values.
Bottema 和 Pestaña（2015）获得了 12 个星系样本的高分辨率动力学学，这些星系的亮度在跨越几个数量级的范围内均匀分布。他们发现，具有最大盘、中心核 DM 暗物质晕和质量与光比的唯一值，即 $M_D/L_R=1.0$ ，非常好地拟合了所有 RCs，见图 13。NFW 暗物质晕，无论有无重子分布，都无法拟合样本中最小星系的 RCs，而对于最大星系，NFW + 恒星/HI 盘模型的结构参数的最佳拟合值，即暗物质晕浓度和质量以及恒星盘的质量与光比，往往采用非物理值。

The Spitzer Photometry and Accurate Rotation Curves sample includes 175 nearby galaxies with surface photometry at $3.6\text{μ}\mathrm{m}$ and high-quality rotation curves. This sample spans a broad range of morphologies (S0 to Irr), luminosities ($\sim 5$ dex) and surface brightness ($\sim 4$ dex). These data have been used by Lelli et al. (2016b) to build the mass models of the galaxies. They adopted the specific value of 0.5 for the stellar mass-to-light ratio in the $3.6\text{μ}\mathrm{m}$-band as suggested by stellar population models and found that $V_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{y}}/V$ varies with luminosity and surface brightness: the stellar disks in high-mass, high-surface-brightness galaxies are nearly maximal, while in low-mass, low-surface-brightness galaxies they are very submaximal. Moreover, in these galaxies, the cored DM halo + (high mass) stellar disk model, generally, reproduces the sample RCs very well, differently from the cuspy halo + (low-mass) stellar disk model that often shows a bad fit and/or non-physical values for the parameters of the mass model.
Spitzer Photometry and Accurate Rotation Curves 样本包括 175 个附近星系，具有 $3.6\text{μ}\mathrm{m}$ 处的表面光度和高质量的旋转曲线。该样本涵盖了广泛的形态（S0 到 Irr）、亮度（ $\sim 5$ dex）和表面亮度（ $\sim 4$ dex）范围。这些数据已被 Lelli 等人（2016b）用于构建星系的质量模型。他们采用了在 $3.6\text{μ}\mathrm{m}$ 波段中由恒星群模型建议的 0.5 的特定值作为恒星质量与光比，发现 $V_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{y}}/V$ 随亮度和表面亮度变化：高质量、高表面亮度星系中的恒星盘几乎是极大的，而低质量、低表面亮度星系中它们是非常次极大的。此外，在这些星系中，具有中心核 DM 暗物质晕 +（高质量）恒星盘模型通常非常好地重现了样本 RCs，与常常显示拟合不良和/或质量模型参数的非物理值的尖峰暗物质晕 +（低质量）恒星盘模型不同。

The mass distribution of 121 nearby objects with high-quality optical rotation curves has been recently derived from the Fabry–Pérot kinematical GHASP survey of spirals and irregular galaxies (Korsaga et al. 2018). These galaxies cover all morphological types of spirals and have an infrared $3.6\text{μ}\mathrm{m}$ emission measurements, good tracers of the old stellar population. Combining the kinematical and the surface brightness data they obtained the mass models once they assumed a specific DM halo density profile. They considered the PI cored profile and the Navarro–Frenk–White cuspy profile. The value of the $M_D/L_{3.6}$ for the stellar disc was obtained for each objects in two different ways: (1) from the stellar evolutionary models and the WISE ${\text{W}}_1$–${\text{W}}_2$ colours, (2) from fitting the RC. Both approaches found that: (i) the rotation curves of most galaxies are better fitted with a cored rather than with a cuspy profile, (ii) there are luminosity/Hubble type dependent relationships between the parameters of the DM and those of the luminous matter. In detail, in the PI halos framework they found that core radius $\propto$ (central DM halo density)${}^{-1}$, in very good agreement with Kormendy and Freeman (2004) and Donato et al. (2009). In the NFW framework they found a very strong dependence of the concentration on the halo virial mass, in disagreement with the outcome of N-body simulations (e.g., Klypin et al. 2011).
最近从螺旋和不规则星系的 Fabry–Pérot 运动学 GHASP 调查中获得了 121 个附近对象的质量分布，这些对象具有高质量的光学旋转曲线（Korsaga 等人 2018）。这些星系涵盖了所有螺旋体的形态类型，并具有红外 $3.6\text{μ}\mathrm{m}$ 发射测量，是老年恒星群的良好示踪物。结合运动学和表面亮度数据，他们在假设特定 DM 晕密度剖面后获得了质量模型。他们考虑了 PI 中空心剖面和 Navarro–Frenk–White 尖峰剖面。对于每个对象，从两种不同方式获得了恒星盘的 $M_D/L_{3.6}$ 值：（1）从恒星演化模型和 WISE ${\text{W}}_1$ - ${\text{W}}_2$ 颜色中，（2）从拟合 RC 中。两种方法都发现：（i）大多数星系的旋转曲线更适合于中空而不是尖峰剖面，（ii）DM 参数与发光物质参数之间存在与亮度/哈勃类型相关的关系。详细来说，在 PI 晕框架中，他们发现核半径 $\propto$ （中心 DM 晕密度） ${}^{-1}$ ，与 Kormendy 和 Freeman（2004）以及 Donato 等人（2009）非常一致。在 NFW 框架中，他们发现浓度对晕维里质量有很强的依赖性，与 N 体模拟的结果不符（例如，Klypin 等人 2011）。

8.2.1 The galaxy

The investigation of DM distribution in our Galaxy is clearly important under many aspects, although it is made difficult by our location inside it. The stellar component can be modelled as a Freeman exponential thin disk of length scale $R_D=(2.5\pm 0.2)\mathrm{k}\mathrm{p}\mathrm{c}$ (e.g., Jurić et al. 2008).
在许多方面，研究我们银河系中 DM 分布显然是重要的，尽管由于我们位于其中而变得困难。恒星成分可以被建模为长度尺度为 $R_D=(2.5\pm 0.2)\mathrm{k}\mathrm{p}\mathrm{c}$ 的 Freeman 指数薄盘（例如，Jurić等人 2008）。

Very precise measurements of position and proper motion of maser sources (Honma et al. 2012) provide us with a reliable solar galactocentric distance of $R_{\odot }=8.29\pm 0.16\mathrm{k}\mathrm{p}\mathrm{c}$ and a circular speed, at $R_{\odot }$, of $V(R_{\odot })=(239\pm 5)\mathrm{k}\mathrm{m}/\mathrm{s}$. Adopting these values, for $R<R_{\odot }$, we can transform the available HI disk terminal velocities $V_T$ into circular velocities V(R): $V(R/R_{\odot })=V_T(R/R_{\odot })+\frac{R}{R_{\odot }}{V}_{\odot }$ (see McMillan 2011; Nesti and Salucci 2013 and references inside). For $R>R_{\odot }$ out to $\sim 100\mathrm{k}\mathrm{p}\mathrm{c}$, the MW circular motions are inferred from the kinematics of tracer stars in combination with the Jeans equation (Xue 2008; Brown et al. 2009).¹⁰ In Sofue (2017) the issue of the RC of the MW compared with those of spirals of similar luminosity is discussed.
非常精确的天线源位置和适当运动的测量（Honma 等人 2012）为我们提供了可靠的太阳银心距离 $R_{\odot }=8.29\pm 0.16\mathrm{k}\mathrm{p}\mathrm{c}$ 和圆速度，在 $R_{\odot }$ 处为 $V(R_{\odot })=(239\pm 5)\mathrm{k}\mathrm{m}/\mathrm{s}$ 。采用这些数值，对于 $R<R_{\odot }$ ，我们可以将可用的 HI 盘终端速度 $V_T$ 转换为圆速度 V(R)： $V(R/R_{\odot })=V_T(R/R_{\odot })+\frac{R}{R_{\odot }}{V}_{\odot }$ （参见 McMillan 2011；Nesti 和 Salucci 2013 以及内部参考文献）。对于 $R>R_{\odot }$ 延伸到 $\sim 100\mathrm{k}\mathrm{p}\mathrm{c}$ ，MW 的圆运动是从追踪星的运动学结合 Jeans 方程（Xue 2008；Brown 等人 2009）中推断出来的。在 Sofue（2017）中，讨论了 MW 的 RC 与类似亮度的螺旋体的 RC 之间的问题。

The mass model of the MW is that of any other spiral: it includes a central bulge, a stellar disk, an extended gaseous disk and all these components are embedded in a spherical dark halo (see Caldwell and Ostriker 1981; Catena and Ullio 2010; Nesti and Salucci 2013; Sofue 2013). As regards to the latter, in a number of studies, the available kinematics is not able to discriminate between the cored and a cusped DM halo profiles (e.g., Catena and Ullio 2010, 2012).
银河系的质量模型与其他螺旋星系相同：它包括一个中央球状体、一个恒星盘、一个扩展的气态盘，所有这些组件都嵌入在一个球形暗物质晕中（参见 Caldwell 和 Ostriker 1981；Catena 和 Ullio 2010；Nesti 和 Salucci 2013；Sofue 2013）。至于后者，在许多研究中，可用的运动学数据无法区分有核心和有尖峰的暗物质晕轮廓（例如，Catena 和 Ullio 2010，2012）。

Nesti and Salucci (2013) have alternatively assumed a B-URC and a NFW DM halo profile. They fitted the resulting velocity models to the available kinematical data: HI terminal velocities, circular velocities as recently estimated from maser star forming regions and velocity dispersions of stellar halo tracers in the outermost Galactic regions. They found, for the first model, the following best fit values: ${\rho }_0=4\times {10}^7{M}_{\odot }/{\mathrm{k}\mathrm{p}\mathrm{c}}^3$, $r_0=10\mathrm{k}\mathrm{p}\mathrm{c}$ and $M_D=6\times {10}^{10}{M}_{\odot }$, $M_{\mathrm{v}\mathrm{i}\mathrm{r}}=1.2\times {10}^{12}{M}_{\odot }$ that coincide with those of the URC with the same virial mass and optical radius. The mass model with NFW halo profile fits quite well the dynamical data; however, the resulting best fit value for the concentration parameter c is: $c=20\pm 2$, higher than the predicted value from only dark matter $\text{Λ}$CDM simulations. Similar findings were obtained also by Catena and Ullio (2010, 2012) and Deason et al. (2012).
Nesti 和 Salucci（2013 年）曾交替假设了一个 B-URC 和一个 NFW 暗物质晕轮廓。他们将得到的速度模型拟合到可用的运动学数据中：HI 终端速度，最近从激光器星形成区域估算的圆速度以及外部银河区域恒星晕迹的速度离散度。对于第一个模型，他们发现以下最佳拟合值： ${\rho }_0=4\times {10}^7{M}_{\odot }/{\mathrm{k}\mathrm{p}\mathrm{c}}^3$ ， $r_0=10\mathrm{k}\mathrm{p}\mathrm{c}$ 和 $M_D=6\times {10}^{10}{M}_{\odot }$ ， $M_{\mathrm{v}\mathrm{i}\mathrm{r}}=1.2\times {10}^{12}{M}_{\odot }$ ，与具有相同维里质量和光学半径的 URC 的值相符。具有 NFW 暗物质晕轮廓的质量模型相当好地拟合了动力学数据；然而，浓度参数 c 的最佳拟合值为： $c=20\pm 2$ ，高于仅来自暗物质的预测值 $\text{Λ}$ CDM 模拟。Catena 和 Ullio（2010 年，2012 年）以及 Deason 等人（2012 年）也得出了类似的发现。

Fig. 14

Rotational velocities in the Milky Way derived from gas and stellar kinematics (blue, orange) and masers measurements (black). Notice measurements with huge uncertainty. Image reproduced with permission from Pato and Iocco (2017), copyright by the authors
银河系中由气体和恒星动力学（蓝色，橙色）以及激光器测量（黑色）导出的旋转速度。请注意具有巨大不确定性的测量。图像经 Pato 和 Iocco（2017）许可复制，版权归作者所有

8.3 Low surface brightness galaxies
8.3 低表面亮度星系

There is a limited number of recent studies on the RCs of LSB galaxies, although some of these objects appear in well-studied samples of disk systems discussed in the previous sections. In LSB the 21-cm HI line provides us with the main observational channel probing the gravitational field: radio telescopes only now reach sufficient spatial resolution and sensitivity to map small and faint objects like LSB.¹¹
最近关于 LSB 星系旋转曲线的研究数量有限，尽管其中一些物体出现在先前章节讨论的盘系统的研究样本中。在 LSB 中，21 厘米 HI 线为我们提供了主要的观测通道，探测引力场：无线电望远镜现在才具有足够的空间分辨率和灵敏度来绘制 LSB 等小而暗淡的物体。 ¹¹

Di Paolo and Salucci (2018) applied to LSBs the concept of the stacked analysis of RCs that in spirals has led us to the URC. They investigated, in a sample of 72 objects with available rotation curves and infrared photometry, the distribution of the baryonic and the dark matter components. The galaxies were divided into five velocity bins according to their increasing values of $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$. Noticeably, when we plot them in physical units: $\log V(\log r)$, they show a great diversity: objects with a same maximum velocity possesses very different RC profiles, see Fig. 15. Instead, when we adopt the specifically normalized units: $x\equiv r/R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ and $v(x)=V(x)/V(1)$, the rotation curves $\log v(\log x)$ of each velocity bin are all alike, see Fig. 15, probing, as in spirals, the idea that by stacking and by coadding diverse RCs, we get a 3D universal profile, i.e., a surface function of x and of one galaxy structural quantity, e.g., $\log V_{\mathrm{o}\mathrm{p}\mathrm{t}}$. The diversity in the RCs is caused by the presence of another structural parameter in the mass distribution that the stacking processes and the double-normalization neutralize. From the double-normalized velocities, five coadded RCs have been built: $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,V_{\mathrm{o}\mathrm{p}\mathrm{t}})$. They are very well fit by the spirals URC velocity profile $V_{\mathrm{U}\mathrm{R}\mathrm{C}}(x;{\rho }_0,r_0,M_D)$ (see 41) see Figs. 5–6 of Di Paolo and Salucci (2018).
Di Paolo 和 Salucci（2018 年）将螺旋星系中导致 URC 的堆叠分析概念应用于 LSBs。他们研究了一个包含 72 个物体的样本，这些物体具有可用的旋转曲线和红外光度测量数据，研究了重子和暗物质成分的分布。根据它们递增的 $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 值，这些星系被分成五个速度桶。值得注意的是，当我们以物理单位绘制它们时： $\log V(\log r)$ ，它们显示出很大的多样性：具有相同最大速度的物体具有非常不同的旋转曲线剖面，见图 15。相反，当我们采用特定的标准化单位： $x\equiv r/R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 和 $v(x)=V(x)/V(1)$ 时，每个速度桶的旋转曲线 $\log v(\log x)$ 都是相似的，见图 15，这验证了在螺旋星系中的观点，即通过堆叠和合并不同的旋转曲线，我们得到一个 3D 通用剖面，即一个关于 x 和一个星系结构量的表面函数，例如 $\log V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 。旋转曲线的多样性是由质量分布中另一个结构参数的存在引起的，堆叠过程和双标准化可以中和这种多样性。从双标准化速度中，已建立了五个合并的旋转曲线： $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,V_{\mathrm{o}\mathrm{p}\mathrm{t}})$ 。它们非常适合螺旋 URC 速度剖面 $V_{\mathrm{U}\mathrm{R}\mathrm{C}}(x;{\rho }_0,r_0,M_D)$ （见 41），参见 Di Paolo 和 Salucci（2018 年）的图 5-6。

Fig. 15

The rotation curves of the LSBs sample of Di Paolo and Salucci (2018) in physical (black) and normalized units (red)
Di Paolo 和 Salucci（2018 年）的 LSBs 样本的旋转曲线，以物理单位（黑色）和标准化单位（红色）表示

The resulting URC of LSB galaxies (Fig. 18 of Di Paolo and Salucci 2018) implies that the B-URC halo parameters ${\rho }_0$ and $r_0$ connect with $R_D$ and $M_D$ in a way similar to that found in spirals ((see Fig. 16) Di Paolo and Salucci 2018). Moreover, also in these objects we find: ${\rho }_0{r}_0\sim 100M_{\odot }p{c}^{-2}$ (see Fig. 27).
LSB 星系的 URC 结果（Di Paolo 和 Salucci 2018 年的图 18）意味着 B-URC 暗晕参数 ${\rho }_0$ 和 $r_0$ 与螺旋星系中发现的方式类似地连接到 $R_D$ 和 $M_D$ （参见图 16，Di Paolo 和 Salucci 2018）。此外，在这些物体中我们还发现： ${\rho }_0{r}_0\sim 100M_{\odot }p{c}^{-2}$ （见图 27）。

Fig. 16

The 4D relationship in Fig. 11 (bottom, left) for low-surface-brightness galaxies (Di Paolo and Salucci 2018). Legenda: $R_{\mathrm{c}}\equiv r_0$
图 11 中低表面亮度星系的 4D 关系（底部左侧）（Di Paolo 和 Salucci 2018）。图例： $R_{\mathrm{c}}\equiv r_0$

Fig. 17

The slope $\alpha$ of the DM density: ${\rho }_{\mathrm{D}\mathrm{M}}\propto (r/R_{\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}{)}^{\alpha }$ with $R_{\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}$ the innermost radius with velocity measurement. Also shown are the predictions for halos of mass ${10}^{10}{M}_{\odot }$ with a pseudo-isothermal (ISO) or a cusped (NFW) halo profile. Image reproduced with permission from Oh et al. (2015), copyright by AAS
DM 密度的斜率 $\alpha$ ： ${\rho }_{\mathrm{D}\mathrm{M}}\propto (r/R_{\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}{)}^{\alpha }$ 与 $R_{\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}$ 内部半径和速度测量有关。还显示了质量为 ${10}^{10}{M}_{\odot }$ 的暗物质晕具有伪等温（ISO）或尖峰（NFW）晕轮廓的预测。图像经 Oh 等人（2015）许可复制，AAS 版权所有。

Remarkably, in LSBs, the URC, expressed in normalized radial units, has two independent parameters: one, as in spirals, is the stellar disk or the halo mass, the second is the compactness, either of the dark halo or of the luminous disk; in fact, a tight correlation between these two quantities emerges (without a plausible physical explanation) (see Fig. 28).
在低表面亮度星系中，以归一化径向单位表示的 URC 具有两个独立参数：一个是恒星盘或暗物质晕的质量，另一个是紧凑性，无论是暗物质晕还是发光盘；事实上，这两个量之间存在着密切的相关性（没有合理的物理解释）（见图 28）。

8.4 Dwarf disks

Oh et al. (2015) have investigated 26 high-resolution rotation curves of dwarf (irregular) disk (dd) galaxies from LITTLE THINGS sample, a high-resolution VLA HI survey of nearby dwarf galaxies. The rotation curves were decomposed into their baryonic and DM contributions in a very accurate way: in these objects, the first component is much less important than the second. Generally, the RCs of dds are found to increase with radius out to several disk length scales. Furthermore, the logarithmic inner slopes $\alpha$ of their DM halo densities are very high: $⟨\alpha ⟩=-0.32\pm 0.24$, in disagreement with the prediction of cusp-like NFW halos $⟨\alpha {⟩}_{\mathrm{N}\mathrm{F}\mathrm{W}}<-1$ (see Fig. 17). This result is confirmed also by the full mass modelling when it is possible to accurately perform it.
Oh 等人（2015）调查了 LITTLE THINGS 样本中 26 个高分辨率侏儒（不规则）盘（dd）星系的旋转曲线。这些旋转曲线被非常准确地分解为它们的重子和暗物质贡献：在这些对象中，第一个组分远不及第二个重要。通常，dd 的 RCs 被发现随着半径增加而增加到几个盘长度尺度。此外，他们的 DM 暗物质密度的对数内斜率 $\alpha$ 非常高： $⟨\alpha ⟩=-0.32\pm 0.24$ ，与 cusp-like NFW 暗物质晕的预测不符 $⟨\alpha {⟩}_{\mathrm{N}\mathrm{F}\mathrm{W}}<-1$ （见图 17）。当能够准确执行完整质量建模时，这一结果也得到了确认。

Karukes and Salucci (2017) investigated a sample of 36 objects with good-quality rotation curve drawn from the Local Volume Sample. They found that, although several objects have a RC suitable for individual mass modelling, on the whole, the stacked analysis yields very important results. They found that, despite variations in luminosities of $\sim 2$ dex and, above all, despite a great diversity in their rotation curves profiles V(R), when radii and velocities are normalized by ($R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$) the RCs look all alike (see Fig. 18) and lead to what can be considered as the low-mass continuation of $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$, the coadded RCs of spiral galaxies. This finding addresses the “diversity problem” (Oman et al. 2015); it confirms that dwarf disk galaxies, with the same maximum circular velocity, exhibit large differences in their inner RC profiles and then, in their inferred DM densities. However, this pattern disappears when the relevant quantities are expressed in normalized units (see Fig. 18). The reason is that these galaxies have a large scatter in the luminosity vs. size relationship (see Karukes and Salucci 2017) which, exactly as in LSB, gets neutralised by the normalization procedure performed while building the $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}$. Of course the issue itself does not disappear, but it actually thickens and manifests itself as arisen from the strong correlation between the distribution of dark and luminous dark matter and from the presence in these objects of an additional structural quantity: the compactness $C_{\star }$ (see later) belonging to the luminous world, but independent, by construction, of the galaxy luminosity (see Karukes and Salucci 2017).
Karukes 和 Salucci（2017）调查了一组来自 Local Volume Sample 的 36 个具有良好质量的旋转曲线对象。他们发现，尽管有几个对象具有适合个体质量建模的 RC，但总体而言，堆叠分析产生了非常重要的结果。他们发现，尽管 $\sim 2$ dex 的亮度变化，尤其是尽管它们的旋转曲线轮廓 V（R）存在很大的多样性，当半径和速度通过（ $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ， $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ）进行标准化时，RCs 看起来都一样（见图 18），并导致可以被视为螺旋星系的低质量延续的 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(x,M_I)$ 的合并 RCs。这一发现解决了“多样性问题”（Oman 等人，2015）；它证实了侏儒盘星系，具有相同的最大圆速度，展现出内部 RC 轮廓的巨大差异，然后，在它们推断的 DM 密度中。然而，当相关数量以标准化单位表示时（见图 18），这种模式消失了。原因是这些星系在亮度与大小关系中存在很大的散射（见 Karukes 和 Salucci 2017），这种散射正如在 LSB 中一样，通过构建 $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}$ 时执行的标准化程序中和了。当然，问题本身并没有消失，但实际上变得更加复杂，并表现为暗物质和明亮暗物质分布之间的强相关性以及这些对象中存在的额外结构数量：紧凑度 $C_{\star }$ （稍后见）属于明亮世界，但独立于星系亮度的构造（见 Karukes 和 Salucci 2017）。

Fig. 18

dds. The 36 RCs in physical units (left) and in two-normalized units (right). In Karukes and Salucci (2017) one finds the $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ vs. $L_K$ relationship whose scatter is responsible for the evident diversity of the various RC profiles when they are expressed in physical units. Image reproduced with permission from Karukes and Salucci (2017), copyright by the authors
dds。物理单位中的 36 个 RCs（左）和两个标准化单位中的 RCs（右）。在 Karukes 和 Salucci（2017）中，人们发现了 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ vs. $L_K$ 关系，其散射负责各种 RC 轮廓的明显多样性，当它们以物理单位表示时。图像经 Karukes 和 Salucci（2017）许可复制，作者版权所有

Let us stress that, differently from spirals and LSBs, we need just one $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ to represent all dds double-normalized RCs, laying in the range $10\mathrm{k}\mathrm{m}/\mathrm{s}<V_{\mathrm{o}\mathrm{p}\mathrm{t}}<80\mathrm{k}\mathrm{m}/\mathrm{s}$: in fact, all their (double normalized) velocity profiles are almost identical. The velocity modelling starts from the coadded RC $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(R/R_{\mathrm{o}\mathrm{p}\mathrm{t}},⟨V_{\mathrm{o}\mathrm{p}\mathrm{t}}⟩)$, with $⟨V_{\mathrm{o}\mathrm{p}\mathrm{t}}⟩=40\mathrm{k}\mathrm{m}/\mathrm{s}$. As in spirals and LSBs, these data are fitted by the (dd) URC model that includes an exponential Freeman disc, a B-URC DM halo and a gaseous disk. The fit is very successful, unlike that relative to the NFW halo + stellar and gaseous disks velocity model (Karukes and Salucci 2017).
让我们强调，与螺旋和低表面亮度星系不同，我们只需要一个 $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 来表示所有双标准化 RCs，其范围为 $10\mathrm{k}\mathrm{m}/\mathrm{s}<V_{\mathrm{o}\mathrm{p}\mathrm{t}}<80\mathrm{k}\mathrm{m}/\mathrm{s}$ ：实际上，它们所有的（双标准化）速度剖面几乎相同。速度建模始于合并的 RC $V_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{d}}(R/R_{\mathrm{o}\mathrm{p}\mathrm{t}},⟨V_{\mathrm{o}\mathrm{p}\mathrm{t}}⟩)$ ，具有 $⟨V_{\mathrm{o}\mathrm{p}\mathrm{t}}⟩=40\mathrm{k}\mathrm{m}/\mathrm{s}$ 。与螺旋和低表面亮度星系一样，这些数据通过包括指数 Freeman 盘、B-URC DM 暗物质晕和气态盘的（dd）URC 模型进行拟合。拟合非常成功，不像相对于 NFW 暗物质晕+恒星和气态盘速度模型的拟合（Karukes 和 Salucci 2017）。

These systems are strongly dominated by dark matter halos with cored density profile. The core sizes are proportional to the corresponding disk length scales: $r_0=3R_D$, continuing the relationship found in spirals and extending it 2 dex down in galaxy luminosity (Karukes and Salucci 2017). Also, all the other dark and luminous structural properties of the dark and luminous matter, including the stellar/DM compactness $C_{\star }$ and $C_{\mathrm{D}\mathrm{M}}$, result amazingly correlated (Karukes and Salucci 2017).
这些系统受到具有中心密度轮廓的暗物质晕的强烈支配。核心尺寸与相应的盘长度尺度成正比： $r_0=3R_D$ ，延续了在螺旋星系中发现的关系，并将其向下延伸 2 dex 到星系亮度（Karukes 和 Salucci 2017）。此外，所有其他暗物质和发光物质的结构性质，包括恒星/DM 紧凑性 $C_{\star }$ 和 $C_{\mathrm{D}\mathrm{M}}$ ，结果惊人地相关（Karukes 和 Salucci 2017）。

All structural relationships established in normal spirals extend down to “dd” galaxies, the relevant aspect being that also those that connect the dark and the luminous world continue, unchanged, in objects where the dark matter is, by far, the dominant component.
在正常螺旋星系中建立的所有结构关系都延伸到“dd”星系，相关的方面是，连接暗物质和发光世界的那些关系在对象中也继续保持不变，其中暗物质远远是主导成分。

9 The distribution of matter in spheroids
9 球状体中物质的分布

Spheroidal galaxies include the biggest and the smallest galaxies of the Universe. The investigation of their dark matter component is rather complicated. With respect to spirals, the bulk of stars in ellipticals is much more compact and then it probes much inner and more luminous matter-dominated galactic regions than the stellar and HI disks do in spirals. However, the halos of ellipticals are filled with objects, like planetary nebulae and globular clusters that can be good tracers of the gravitational potential, in spite of their limited number and totally unknown dynamical state.
球状星系包括宇宙中最大和最小的星系。对其暗物质成分的研究相当复杂。相对于螺旋星系，椭圆星系中的大部分恒星比较紧凑，因此它探测到比螺旋星系中的恒星和 HI 盘更内部和更亮的物质为主导的星系区域。然而，椭圆星系的暗物质晕充满了天体，如行星状星云和球状星团，它们可以是重力势的良好示踪者，尽管它们的数量有限且完全未知其动力学状态。

9.1 The fundamental plane in ellipticals
9.1 椭圆星系中的基本平面

The luminous regions of ellipticals show a 3D relationship, known as the fundamental plane, which is usually written as
椭圆星系的发光区域显示出一种三维关系，称为基本平面，通常写为

$$\begin{array}{r}\log \frac{R_{\mathrm{e}}}{\mathrm{k}\mathrm{p}\mathrm{c}}=a\log \frac{\text{σ}}{\mathrm{k}\mathrm{m}/\mathrm{s}}-\frac{b}{2.5}\frac{{\mu }_{\mathrm{e}}}{\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{s}}+c,\end{array}$$

(43)

where $R_{\mathrm{e}}$ is the effective radius, $\text{σ}$ is the central velocity dispersion (corrected to an aperture of $R_{\mathrm{e}}/8$). ${\mu }_{\mathrm{e}}$ and $\log I_{\mathrm{e}}$ are the surface brightness and surface luminosity within $R_{\mathrm{e}}$. It is worth reminding that for virialized stable objects, all with the same surface profile $I(r/R_{\mathrm{e}})$ and small amount of dark matter inside $R_{\mathrm{e}}$, one expects: $R_{\mathrm{e}}={\text{σ}}_0^a/I_{\mathrm{e}}^b$, with $a=2$ and $b=1$. It is well known that the FP has different parameters (Djorgovski and Davis 1987; Dressler et al. 1987; Jorgensen et al. 1996), e.g., $\log R_{\mathrm{e}}=1.24\log {\text{σ}}_0-0.82\log ⟨I{⟩}_{\mathrm{e}}$ with scatter 0.07 dex in $\log R_{\mathrm{e}}$. As a recent example, Hyde and Bernardi (2009) used a sample of about 50,000 early-type galaxies based on the SDSS-DR4/6, photometric and spectroscopic parameters and obtained $a=1.3\pm 0.05,b=0.3\pm 0.05$ with r.m.s. of 0.1 dex (see Fig. 19).
其中 $R_{\mathrm{e}}$ 是有效半径， $\text{σ}$ 是中心速度离散度（校正到 $R_{\mathrm{e}}/8$ 的孔径）。 ${\mu }_{\mathrm{e}}$ 和 $\log I_{\mathrm{e}}$ 是表面亮度和表面光度在 $R_{\mathrm{e}}$ 内。值得提醒的是，对于弥散稳定的物体，所有具有相同表面轮廓 $I(r/R_{\mathrm{e}})$ 和内部少量暗物质 $R_{\mathrm{e}}$ ，人们期望： $R_{\mathrm{e}}={\text{σ}}_0^a/I_{\mathrm{e}}^b$ ，具有 $a=2$ 和 $b=1$ 。众所周知，FP 具有不同的参数（Djorgovski 和 Davis 1987; Dressler 等人 1987; Jorgensen 等人 1996），例如， $\log R_{\mathrm{e}}=1.24\log {\text{σ}}_0-0.82\log ⟨I{⟩}_{\mathrm{e}}$ ，在 $\log R_{\mathrm{e}}$ 中散射 0.07 dex。作为最近的例子，Hyde 和 Bernardi（2009）使用了约 50,000 个早期星系的样本，基于 SDSS-DR4/6，光度和光谱参数，并获得了 $a=1.3\pm 0.05,b=0.3\pm 0.05$ ，r.m.s. 为 0.1 dex（见图 19）。

Fig. 19

The fundamental plane of ellipticals from Hyde and Bernardi (2009) in the coordinate system $I_{\mathrm{e}},R_{\mathrm{e}},\text{σ}(1/8R_{\mathrm{e}})$. Image reproduced with permission from Magoulas et al. (2012), copyright by the authors
Hyde 和 Bernardi（2009 年）提出的椭圆星系的基本平面在坐标系 $I_{\mathrm{e}},R_{\mathrm{e}},\text{σ}(1/8R_{\mathrm{e}})$ 中。图像由 Magoulas 等人（2012 年）获得许可后复制，作者版权所有

Moreover, Magoulas et al. (2012) investigated the near-infrared FP in $\sim {10}^4$ early-type galaxies (ETG) included in the 6dF Galaxy Survey (6dFGS). They fitted the distribution of central velocity dispersions, near-infrared surface brightness and half-light radii with a three-dimensional Gaussian model that provided an excellent match to the observed properties.
此外，Magoulas 等人（2012 年）研究了包括在 6dF 星系调查（6dFGS）中的 $\sim {10}^4$ 早期星系（ETG）的近红外 FP。他们用三维高斯模型拟合了中心速度离散度、近红外表面亮度和半光半径的分布，这与观测到的性质非常匹配。

The resulting FP reads as follows: $R_{\mathrm{e}}\propto {\text{σ}}^{1.52\pm 0.03}{I}_{\mathrm{e}}^{-0.89\pm 0.01}$, with a r.m.s. of 23%. The deviation of the FP with respect to the theoretical predictions, called the tilt of the FP, has been thought to be due a combination of several effects (Bernardi et al. 2003; Bolton et al. 2008; Hyde and Bernardi 2009; Graves and Faber 2010; Zaritsky 2012 for a review). However, from recent independent and accurate measurements of the total mass inside $R_{\mathrm{e}}$ by means of stellar dynamics (Cappellari et al. 2006; Thomas et al. 2011) and strong lensing (Bolton et al. 2007; Auger et al. 2010), it is clear that variations among ETGs of the stellar mass to light ratio M / L are the cause of the tilt. This has clearly emerged in Cappellari et al. (2013): they started with the FP which reads as
由于 FP 的结果如下： $R_{\mathrm{e}}\propto {\text{σ}}^{1.52\pm 0.03}{I}_{\mathrm{e}}^{-0.89\pm 0.01}$ ，均方根误差为 23%。FP 相对于理论预测的偏差，称为 FP 的倾斜，被认为是由几种效应的组合造成的（Bernardi 等人，2003 年；Bolton 等人，2008 年；Hyde 和 Bernardi，2009 年；Graves 和 Faber，2010 年；Zaritsky，2012 年进行了回顾）。然而，通过最近对星际动力学（Cappellari 等人，2006 年；Thomas 等人，2011 年）和强引力透镜（Bolton 等人，2007 年；Auger 等人，2010 年）进行的独立和准确的总质量测量，清楚地表明椭圆星系中恒星质量与光比 M / L 的变化是倾斜的原因。这在 Cappellari 等人（2013 年）中明显地体现出来：他们从 FP 开始，如下所示

$$\begin{array}{r}\log \left(\frac{L}{L_{\odot ,r}}\right)=a+b\log \left(\frac{{\text{σ}}_{\mathrm{e}}}{130\mathrm{k}\mathrm{m}\mathrm{s}}\right)+c\log \left(\frac{R_{\mathrm{e}}}{2\mathrm{k}\mathrm{p}\mathrm{c}}\right),\end{array}$$

(44)

${\text{σ}}_{\mathrm{e}}$ and $R_{\mathrm{e}}$ are normalized to the median values found in sample under study. The resulting values of the parameters are the following: $b=1.25\pm 0.04;c=0.96\pm 0.03$ and the r.m.s. scatter is 0.1 dex; when the galaxy luminosity is replaced by the dynamical mass $L\times (M/L)\mathrm{d}\mathrm{y}\mathrm{n}$, obtained by self-consistent JAM modelling (see Sect. 5.5) a smaller r.m.s. it is found and the parameters: $b=1.93\pm 0.03,c=0.96\pm 0.02$ acquire the virial values. This confirms that a major part of the scatter of the FP is actually due to variations in the M / Ls values.
${\text{σ}}_{\mathrm{e}}$ 和 $R_{\mathrm{e}}$ 被归一化为研究样本中找到的中位值。参数的结果值如下： $b=1.25\pm 0.04;c=0.96\pm 0.03$ ，均方根散布为 0.1 dex；当星系的亮度被动力学质量 $L\times (M/L)\mathrm{d}\mathrm{y}\mathrm{n}$ 替代时，通过自洽的 JAM 建模（见第 5.5 节）获得一个更小的均方根散布，并且参数： $b=1.93\pm 0.03,c=0.96\pm 0.02$ 获得维里值。这证实了 FP 的大部分散布实际上是由于 M / Ls 值的变化引起的。

Therefore, the fundamental plane of ETGs expresses the properties of the virialized stellar spheroids and, differently from the Tully–Fisher in spirals, is not directly related to the properties of DM distribution (inside $R_{\mathrm{e}}$). Finally, this result lends support to the idea, valid in spirals, that the dynamically measured mass is more accurate prior of luminous mass of a galaxy than the luminosity itself.
因此，ETGs 的基本平面表达了星系的维里化恒星球体的特性，并且与螺旋星系中的 Tully-Fisher 不同，它与暗物质分布的特性没有直接关联（在 $R_{\mathrm{e}}$ 内）。最后，这个结果支持了在螺旋星系中有效的观点，即动力学测量的质量比星系的亮度本身更准确。

9.2 The dark matter distribution in ellipticals
9.2 椭圆体中的暗物质分布

The derivation of the distribution of dark and luminous mass in ellipticals is far more difficult than in disk systems. The kinematics is more uncertain and the tracers of the gravitational field often do not cover sufficiently well the crucial region between $1/3R_{\mathrm{e}}$ and $3R_{\mathrm{e}}$ where the system becomes from stellar dominated to DM dominated.
在椭圆体中推导暗质和明亮质量的分布比在盘系统中困难得多。动力学更加不确定，重力场的追踪者通常不足够覆盖从 $1/3R_{\mathrm{e}}$ 到 $3R_{\mathrm{e}}$ 的关键区域，系统在那里从受恒星主导到受暗物质主导。

The main issues under investigation are as follows: (a) an universal power law slope of the total density profile: ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}\propto r^{-2}$ and (b) large variations of the $M_{\star }/L$ ratio with mass and other quantities. As regards to the first issue, let us stress that the above density law in ellipticals and the case $V(R)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ in spirals are different configurations (see Eq. 19). As regards to the second, at fixed galaxy luminosity, the stellar mass-to-light ratios vary in ellipticals much more than in spirals.
研究的主要问题如下：（a）总密度剖面的普遍幂律斜率： ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}\propto r^{-2}$ 和（b）质量和其他数量的 $M_{\star }/L$ 比率的大变化。关于第一个问题，让我们强调椭圆体中的上述密度定律和螺旋体中的 $V(R)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ 情况是不同的配置（见方程式 19）。关于第二个问题，在固定星系亮度时，椭圆体中的恒星质量与光比在很大程度上变化，远远超过螺旋体。

As regard to investigations in early-type galaxies (ETG)s one has to report the several different approaches devised to obtain their mass distribution. However, it is fair to stress that it is difficult to make a synthesis of the results obtained so far, being the situation still in full development.
关于早期型星系（ETG）的研究，必须报告为获得它们的质量分布而设计的几种不同方法。然而，公平地强调迄今为止获得的结果很难进行综合，情况仍在充分发展中。

Data from the Sloan Lens Advanced Camera for Surveys (SLACS) project (Bolton et al. 2006) provided us with the total matter density profiles for a sample of 73 ETGs with strong lenses and large stellar masses $(M_{\star }>{10}^{11}{M}_{\odot })$ (Auger et al. 2010). For each galaxy the relevant quantities are the Einstein radius $R_{\mathrm{E}}$, its relative enclosed mass, the stellar mass, and ${\text{σ}}_{\mathrm{E}}$ the velocity dispersions at $R_{\mathrm{E}}$. An isotropic mass model was assumed and they found the following: $({\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma })$ with $⟨\gamma ⟩=2.08\pm 0.03$ and with a scatter among galaxies of ${\text{σ}}_{\gamma }=0.16$.
来自 Sloan Lens Advanced Camera for Surveys（SLACS）项目（Bolton 等人，2006 年）的数据为我们提供了 73 个 ETGs 样本的总物质密度剖面，具有强透镜和大恒星质量 $(M_{\star }>{10}^{11}{M}_{\odot })$ （Auger 等人，2010 年）。对于每个星系，相关数量是爱因斯坦半径 $R_{\mathrm{E}}$ ，其相对封闭质量，恒星质量，以及 ${\text{σ}}_{\mathrm{E}}$ 在 $R_{\mathrm{E}}$ 处的速度离散度。假设各向同性质量模型，他们发现以下结果： $({\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma })$ ，带有 $⟨\gamma ⟩=2.08\pm 0.03$ ，并且在星系之间有 ${\text{σ}}_{\gamma }=0.16$ 的分散。

Cappellari et al. (2012) determined the total density profile for a sample of 14 ETGs fast-rotators (stellar masses $10.2<\log M_{\star }/M_{\odot }<11.7$). SLUGGS and ATLAS observations provided the 2D stellar kinematics out to about to $4R_{\mathrm{e}}$, reaching the region dominated by dark matter and poorly investigated before. They built axisymmetric dynamical models based on the Jeans equations solved with a spatially varying anisotropy $\beta$ and a general density profile for the dark matter halo. The resulting total density profiles were found to follow, from $R_{\mathrm{e}}/10$ to $4R_{\mathrm{e}}$, the power law: ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma }$ with $⟨\gamma ⟩=2.19\pm 0.03$. This extension of the above power law relationship to regions well outside $R_{1/2}\simeq R_{\mathrm{e}}$ is far than trivial and likely hides a connection between the dark halo and the stellar spheroid.
Cappellari 等人（2012 年）确定了 14 个 ETGs 快速旋转体的总密度剖面（恒星质量 $10.2<\log M_{\star }/M_{\odot }<11.7$ ）。SLUGGS 和 ATLAS 观测提供了 2D 恒星动力学，延伸到约 $4R_{\mathrm{e}}$ ，达到了暗物质主导且以前研究不足的区域。他们基于用空间变化各向同性 $\beta$ 和暗物质晕的一般密度剖面解决的 Jeans 方程构建了轴对称动力学模型。发现的总密度剖面从 $R_{\mathrm{e}}/10$ 到 $4R_{\mathrm{e}}$ 遵循幂律： ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma }$ ，带有 $⟨\gamma ⟩=2.19\pm 0.03$ 。将上述幂律关系扩展到 $R_{1/2}\simeq R_{\mathrm{e}}$ 之外的区域远非简单，可能隐藏着暗晕与恒星球体之间的联系。

Tortora et al. (2014) have investigated the central regions ($r<R_{\mathrm{e}}$) of ETGs using strong lensing data from SPIDER and kinematics and photometric data from ${\text{ATLAS}}^{3\mathrm{D}}$. The analysis extends the range of galaxy stellar mass ($M_{\star }$) probed by gravitational lensing down to $\sim {10}^{10}{M}_{\odot }$. Each galaxy was modelled by two components (dark matter halo + stellar spheroid). The following DM halo profiles were considered: NFW, NFW-contracted, and Burkert. The mass-to-light ($M_{\star }/L$) was normalized to the Chabrier IMF as $M_{\star }/L={\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}(M_{\star }/L{)}_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}}$ with ${\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}$, a free parameter describing the systematically variations of IMF among galaxies. They found that, generally: (1) ${\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}$ increases with galaxy size and mass. (2) $\alpha (R_{\mathrm{e}}/2)=\mathrm{d}\log M/\mathrm{d}\log r-3$ in the most massive ($M_{\star }\sim {10}^{11.5}{M}_{\odot }$) or largest ($R_{\mathrm{e}}\sim 15\mathrm{k}\mathrm{p}\mathrm{c}$) ETGs reaches the value of $-2$, while in low-mass ($M_{\star }\sim {10}^{10.2}{M}_{\odot }$) or very small ($R_{\mathrm{e}}\sim 0.5\mathrm{k}\mathrm{p}\mathrm{c}$) ETGs decreases to the value of $-2.5$. As regards to the DM distribution, the result of this work could not reach an explicit preference for a particular profile.
Tortora 等人（2014 年）使用 SPIDER 的强透镜数据以及 ${\text{ATLAS}}^{3\mathrm{D}}$ 的动力学和光度数据研究了 ETG 的中心区域（ $r<R_{\mathrm{e}}$ ）。分析将受引力透镜探测的星系恒星质量范围扩展到 $\sim {10}^{10}{M}_{\odot }$ 。每个星系由两个组件（暗物质晕+恒星椭球体）建模。考虑了以下 DM 晕轮廓：NFW、NFW-收缩和 Burkert。质量与光照（ $M_{\star }/L$ ）被归一化为 Chabrier IMF，如 $M_{\star }/L={\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}(M_{\star }/L{)}_{\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}}$ 与 ${\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}$ ，描述了星系之间 IMF 的系统变化。他们发现，一般来说：（1） ${\delta }_{\mathrm{I}\mathrm{M}\mathrm{F}}$ 随着星系大小和质量增加而增加。 （2） $\alpha (R_{\mathrm{e}}/2)=\mathrm{d}\log M/\mathrm{d}\log r-3$ 在最大（ $M_{\star }\sim {10}^{11.5}{M}_{\odot }$ ）或最大（ $R_{\mathrm{e}}\sim 15\mathrm{k}\mathrm{p}\mathrm{c}$ ）ETG 中达到 $-2$ 的值，而在低质量（ $M_{\star }\sim {10}^{10.2}{M}_{\odot }$ ）或非常小（ $R_{\mathrm{e}}\sim 0.5\mathrm{k}\mathrm{p}\mathrm{c}$ ）ETG 中下降到 $-2.5$ 的值。至于 DM 分布，这项工作的结果无法明确偏好于特定配置文��。

Chae (2014) investigated $\sim 2000$ nearly spherical Sloan Digital Sky Survey (SDSS) ETGs, at a mean redshift of $⟨z⟩=0.12$ and assembled mass models based on their aperture, velocity dispersions, and luminosity profiles measurements. A two-component mass model (i.e., stellar spheroid plus dark halo) successfully fitted, inside $R_{1/2}$, the SDSS aperture velocity dispersions. As result, they confirmed that, in the region $0.1R_{1/2}<R<R_{1/2}$, the total density (dark halo + stellar spheroid) exhibits a power-law behaviour: ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{\gamma }$ with $⟨\gamma ⟩=-2.15\pm 0.04$.
Chae（2014 年）研究了 $\sim 2000$ 几乎球形的 Sloan Digital Sky Survey（SDSS）ETG，平均红移为 $⟨z⟩=0.12$ 并根据其孔径、速度离散度和光度剖面测量组装了质量模型。两组分质量模型（即，恒星椭球体加暗暗晕）成功地拟合了 SDSS 孔径速度离散度内部。结果，他们确认，在区域 $0.1R_{1/2}<R<R_{1/2}$ ，总密度（暗暗晕+恒星椭球体）呈幂律行为： ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{\gamma }$ 与 $⟨\gamma ⟩=-2.15\pm 0.04$ 。

Oguri et al. (2014) investigated 161 strong gravitational lenses from SLACS and BELLS and a number of strongly lensed quasars. They derived the stellar mass $M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$ for each lensing galaxy by fitting the observed spectral energy distribution to a stellar population synthesis model with a Salpeter IMF (Bruzual and Charlot 2003). The measurement in these lens galaxies of the sizes of their Einstein rings $R_{\mathrm{E}}$ allowed them to build normalized total mass profiles for each object: $M_{\mathrm{t}\mathrm{o}\mathrm{t}}(<R_{\mathrm{E}})/M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$ and to normalize the projected radius R by the effective luminosity radius $R_{\mathrm{e}}$. Notice that this double-normalization is of the same kind of that performed in the dd galaxies (Karukes and Salucci 2017). They derived, from each Einstein ring, the relative scaled mass profile $M_{\mathrm{t}\mathrm{o}\mathrm{t}}(<R_{\mathrm{E}}/R_{\mathrm{e}})/M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$. These data were fitted by the model
Oguri 等人（2014 年）研究了来自 SLACS 和 BELLS 的 161 个强引力透镜以及一些强透镜类星体。他们通过将观测到的光谱能量分布拟合到具有 Salpeter IMF 的恒星群合成模型（Bruzual 和 Charlot 2003）中，为每个透镜星系推导出了恒星质量 $M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$ 。这些透镜星系的爱因斯坦环尺寸的测量 $R_{\mathrm{E}}$ 使他们能够为每个对象构建归一化的总质量剖面： $M_{\mathrm{t}\mathrm{o}\mathrm{t}}(<R_{\mathrm{E}})/M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$ 并通过有效亮度半径 $R_{\mathrm{e}}$ 归一化投影半径 R。请注意，这种双重归一化与 dd 星系中执行的归一化相同（Karukes 和 Salucci 2017）。他们从每个爱因斯坦环中推导出相对缩放质量剖面 $M_{\mathrm{t}\mathrm{o}\mathrm{t}}(<R_{\mathrm{E}}/R_{\mathrm{e}})/M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}$ 。这些数据通过模型拟合

$$\begin{array}{r}\frac{M_{\mathrm{t}\mathrm{o}\mathrm{t}}(<R)}{M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}}=A{\left(\frac{R}{R_{\mathrm{e}}}\right)}^{3+\gamma }.\end{array}$$

(45)

They found $\gamma =-2.11\pm 0.05$. Furthermore, they decomposed the total mass in its dark and luminous components: a power-law spherical DM dark halo and a Hernquist spheroid for which, with $y\equiv R/R_{\mathrm{e}}$: $M_{\mathrm{H}\mathrm{e}\mathrm{r}}(y)=M_{\mathrm{s}\mathrm{p}\mathrm{h}}{y}^2/({1.4}^2+y^2)$
他们发现 $\gamma =-2.11\pm 0.05$ 。此外，他们将总质量分解为其暗和明亮组件：一个幂律球形 DM 暗暗晕和一个 Hernquist 椭球体，其中，用 $y\equiv R/R_{\mathrm{e}}$ ： $M_{\mathrm{H}\mathrm{e}\mathrm{r}}(y)=M_{\mathrm{s}\mathrm{p}\mathrm{h}}{y}^2/({1.4}^2+y^2)$

$$\begin{array}{r}\frac{M_{\mathrm{D}\mathrm{M}}(<R)}{M_{\star }^{\mathrm{S}\mathrm{a}\mathrm{l}}}=A_{\mathrm{D}\mathrm{M}}{\left(\frac{R}{R_{\mathrm{e}}}\right)}^{3+{\gamma }_{\mathrm{D}\mathrm{M}}}.\end{array}$$

(46)

Quasar microlensing measurements break the IMF-stellar mass degeneracy, the DM fraction inside $R_{\mathrm{e}}$ results: $A_{\mathrm{D}\mathrm{M}}/A=0.2$ and ${\gamma }_{\mathrm{D}\mathrm{M}}=-{1.60}_{-0.13}^{+0.18}$ that implies that DM is distributed in a way shallower than the total matter, as it occurs in disk systems, see Fig. 20.
夸萨尔微引力测量打破了 IMF-恒星质量退化， $R_{\mathrm{e}}$ 内的 DM 分数结果： $A_{\mathrm{D}\mathrm{M}}/A=0.2$ 和 ${\gamma }_{\mathrm{D}\mathrm{M}}=-{1.60}_{-0.13}^{+0.18}$ 暗示 DM 分布方式比总物质更为浅，就像在盘系统中发生的那样，见图 20。

Fig. 20

Normalized mass of ETGs as function of its normalized radius. Also shown: the best-fit mass profile solid line and the stellar spheroid and the power law DM halo contributions green and red dotted lines. Image reproduced with permission from Oguri et al. (2014), copyright by the authors
作为其归一化半径函数的 ETGs 的归一化质量。还显示：最佳质量剖面实线和恒星球体以及幂律 DM 暗物质晕贡献绿色和红色点线。图像经 Oguri 等人（2014 年）许可复制，作者版权所有

Poci et al. (2017) (see also Cappellari et al. 2013), by modelling kinematical and photometric data of 258 early-type galaxies, belonging to the volume-limited ${\text{ATLAS}}^{3\mathrm{D}}$ survey, derived their density profiles and found the usual power law: ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)=r^{\gamma }$ with $\gamma =-2.2\pm 0.2$. Noticeably, however, they did find significant variations of $\gamma$ with ${\text{Σ}}_{\mathrm{e}}$ the surface brightness inside $R_{\mathrm{e}}$ and ${\text{σ}}_{\mathrm{e}}$, in some contrast with previous works (Fig. 21).
Poci 等人（2017 年）（另见 Cappellari 等人 2013 年），通过对属于限定体积 ${\text{ATLAS}}^{3\mathrm{D}}$ 调查的 258 个早期星系的动力学和光度数据进行建模，推导出它们的密度剖面，并发现通常的幂律： ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)=r^{\gamma }$ 与 $\gamma =-2.2\pm 0.2$ 。然而值得注意的是，他们确实发现了 $\gamma$ 与 ${\text{Σ}}_{\mathrm{e}}$ 内表面亮度的显著变化 $R_{\mathrm{e}}$ 和 ${\text{σ}}_{\mathrm{e}}$ ，与先前的研究形成对比（图 21）。

Fig. 21

The total density profile solid line and its stellar dotted line and DM dot-dashed line components for 16 galaxies from the ATLAS survey. Image reproduced with permission from Poci et al. (2017), copyright by the authors
来自 ATLAS 调查的 16 个星系的总密度剖面实线及其恒星点线和 DM 点划线组件。图像经 Poci 等人（2017 年）许可复制，作者版权所有

Serra et al. (2016) investigating a sample of 16 fast-rotator ETGs with HI disks extended out to $\sim 6R_{\mathrm{e}}$ established a tight linear relation between $V_{\mathrm{H}\mathrm{I}}$ the (flat) circular velocity measured from resolved HI observations in (external) DM dominated regions (i.e., for $R\gg R_{\mathrm{e}}$) and ${\text{σ}}_{\mathrm{e}}$. the velocity dispersion measured at $R_{\mathrm{e}}$, i.e., in a luminous matter dominated region:
Serra 等人（2016 年）研究了一组 16 个快速旋转的 ETGs 样本，其 HI 盘延伸到 $\sim 6R_{\mathrm{e}}$ ，建立了从解决的 HI 观测中测量的（外部）暗物质主导区域的（平坦）圆速度与 ${\text{σ}}_{\mathrm{e}}$ 之间的紧密线性关系。在 $R_{\mathrm{e}}$ 测量的速度离散度，即在明亮物质主导区域：

$$\begin{array}{r}{V}_{\mathrm{H}\mathrm{I}}=1.33{\text{σ}}_{\mathrm{e}},\end{array}$$

(47)

with an observed scatter of 12%. The tightness of the correlation suggests a strong coupling between luminous and dark matter, analogous to the situation in spirals, in LSBs and in dds. Equation (47) implies a decline in the effective circular velocities V(r) from $R_{\mathrm{e}}$ to the outer regions. Such drop is in excellent agreement with the results of Cappellari et al. (2015) and, remarkably, is similar to that observed in early-type spirals (Noordermeer et al. 2007) and in the most luminous late type spirals (Salucci et al. 2007). Assuming ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma }$, Eq. (47) implies $<\gamma >=2.18\pm 0.03$ across the sample, with a scatter of 0.11 around the average value (see Fig. 22).
观测到的散射为 12%。相关性的紧密性表明明亮物质和暗物质之间存在强耦合，类似于螺旋星系、LSBs 和 dds 中的情况。方程（47）暗示了有效圆速度 V(r)从 $R_{\mathrm{e}}$ 下降到外部区域。这种下降与 Cappellari 等人（2015 年）的结果非常一致，并且与早期螺旋星系（Noordermeer 等人 2007 年）和最明亮的后期螺旋星系（Salucci 等人 2007 年）中观察到的情况相似。假设 ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}(r)\propto r^{-\gamma }$ ，方程（47）意味着样本中的 $<\gamma >=2.18\pm 0.03$ ，围绕平均值散布 0.11（见图 22）。

Fig. 22

Radial profile of the normalized circular velocity for the sample of ellipticals in Serra et al. (2016). Data come from JAM models for $R<R_{\mathrm{e}}$ and from HI 21 cm for $R>R_{\mathrm{e}}$. Points and solid lines are coded according to the increasing $R_{\mathrm{H}\mathrm{I}}/R_{\mathrm{e}}$ ratio
Serra 等人（2016 年）研究了一组椭圆星系的径向归一化圆速度剖面。数据来自 JAM 模型 $R<R_{\mathrm{e}}$ 和 HI 21 厘米 $R>R_{\mathrm{e}}$ 。点和实线根据逐渐增加的 $R_{\mathrm{H}\mathrm{I}}/R_{\mathrm{e}}$ 比例进行编码

Alabi et al. (2018) (see also Alabi et al. 2016) used globular cluster kinematics data, primarily from the SLUGGS survey, to measure the dark matter fraction $f_{\mathrm{D}\mathrm{M}}(5R_{\mathrm{e}})$ and the average dark matter density ${\rho }_{\mathrm{D}\mathrm{M}}(5R_{\mathrm{e}}$ within $5R_{\mathrm{e}}$ for 32 nearby ETGs with stellar mass log $(M_{\star }/M_{\odot })$ ranging from 10.1 to 11.8. They found that $f_{\mathrm{D}\mathrm{M}}(R_{\mathrm{e}})\sim 0.6$ for galaxies with stellar mass lesser than $(M_{\star }/M_{\odot })\sim {10}^{11}$. At higher masses, a sudden large range of $f_{\mathrm{D}\mathrm{M}}(R_{\mathrm{e}})$ values emerges. This seems in contradiction with the total density power law ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}\propto r^{-2.1\pm 0.1}$ usually found in other determinations.
Alabi 等人（2018 年）（另见 Alabi 等人 2016 年）使用球状星团动力学数据，主要来自 SLUGGS 调查，测量了 32 个附近 ETGs 的暗物质分数 $f_{\mathrm{D}\mathrm{M}}(5R_{\mathrm{e}})$ 和平均暗物质密度 ${\rho }_{\mathrm{D}\mathrm{M}}(5R_{\mathrm{e}}$ 。这些星系的恒星质量对数为 10.1 至 11.8。他们发现对于恒星质量小于 $(M_{\star }/M_{\odot })\sim {10}^{11}$ 的星系， $f_{\mathrm{D}\mathrm{M}}(R_{\mathrm{e}})\sim 0.6$ 。在更高的质量下，出现了突然的大范围 $f_{\mathrm{D}\mathrm{M}}(R_{\mathrm{e}})$ 值。这似乎与通常在其他测定中找到的总密度幂律 ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{t}}\propto r^{-2.1\pm 0.1}$ 相矛盾。

Pulsoni et al. (2017) used planetary nebulae (PNe) as tracers of the gravitational field around ellipticals. They obtained two-dimensional velocity and velocity dispersion for 33 ETGs. The velocity fields were reconstructed from the measured PNe velocities. The data extend out from $3R_{\mathrm{e}}$ to $13R_{\mathrm{e}}$. The objects show a kinematic transition between the inner luminous matter dominated regions and the outer halo dominated ones. These transition radii, in units of $R_{\mathrm{e}}$, anti-correlate with stellar mass, differently from what occurs in spirals. The galaxies appear to have more diverse kinematic properties in their halos than in their central regions. It is noticeable the fact that 15% of the galaxies in the sample have steeply falling profiles implying that, inside $R_{\mathrm{e}}$, the fraction of dark matter is very negligible.
Pulsoni 等人（2017 年）使用行星状星云（PNe）作为椭圆星系周围引力场的示踪物。他们为 33 个 ETGs 获得了二维速度和速度离散度。速度场是从测量的 PNe 速度重建的。数据从 $3R_{\mathrm{e}}$ 延伸到 $13R_{\mathrm{e}}$ 。这些物体显示了内部明亮物质主导区域和外部暗物质主导区域之间的动力学过渡。这些过渡半径，以 $R_{\mathrm{e}}$ 为单位，与恒星质量呈反相关，与螺旋星系中发生的情况不同。这些星系在其晕中似乎具有比在其中心区域更多样化的动力学特性。值得注意的是，样本中有 15%的星系具有急剧下降的轮廓，这意味着在 $R_{\mathrm{e}}$ 内，暗物质的比例非常微不足道。

Fig. 23

The mass discrepancy–$\text{σ}$ relation, very likely created by systematical variations of the IMF among ETGs (see Cappellari 2016; Posacki et al. 2015)
质量差异- $\text{σ}$ 关系，很可能是由 ETGs 之间 IMF 的系统变化造成的（参见 Cappellari 2016; Posacki 等人 2015）

One important issue of the ETGs is the comparison between the M / Ls inferred from their dynamical or strong lensing modelling and those inferred from the fitting of their spectral energy distributions. Cappellari (2016) have investigated it with a large sample of objects. The values derived, see Fig. 23, indicate the existence of random variations of the IMF and variations with the galaxy dispersion velocity. Noticeably, the existence of a non universal initial mass function (IMF) is already present at intermediate redshift (Tortora et al. 2018).
ETGs 的一个重要问题是从它们的动力学或强引力透镜建模中推断的 M / Ls 与从其光谱能量分布拟合中推断的 M / Ls 之间的比较。 Cappellari（2016）用大量对象进行了调查。推导出的值，见图 23，表明 IMF 的随机变化和与星系色散速度的变化的存在。值得注意的是，非普遍初始质量函数（IMF）的存在已经在中间红移时期存在（Tortora 等人 2018）。

Evidences that ellipticals have variable IMF theme come also from their chemical evolution model reproducing the abundance patterns observed in the sample of the Sloan Digital Sky Survey Data Release 4 (De Masi et al. 2018). The model assumes ellipticals form by fast gas accretion, and suffer a strong burst of star formation followed by a galactic wind, which quenches star formation. The model, if assumes a fixed initial mass function (IMF) in all galaxies, fails in simultaneously reproducing the observed trends of chemistry with the galactic mass; only a varying IMF among ellipticals leads to an agreement between predictions and data.
证据表明，椭圆星系具有可变的 IMF 主题，这也来自于它们的化学演化模型，该模型重现了 Sloan Digital Sky Survey 数据发布 4 中观察到的丰度模式（De Masi 等人，2018 年）。该模型假设椭圆星系通过快速气体吸积形成，并经历了强烈的星形成爆发，随后是星系风，导致星形成停止。如果假设所有星系中的初始质量函数（IMF）固定不变，那么该模型无法同时重现化学与星系质量的观察趋势；只有椭圆星系中的 IMF 变化才能使预测与数据达成一致。

Fig. 24

The average density inside $2R_{\text{e}}$ in NGC 7113 and PGC 67207 blue hexagons as a function of their stellar spheroid mass $M_{\star }$ computed a dynamically (top) or b from the photometry (bottom). Also shown the values for ETGs in Coma Cluster (red filled) and in Abell 262 (red open). The lines show the corresponding spirals’ relationship. Image reproduced with permission from Corsini et al. (2017), copyright by the authors
NGC 7113 和 PGC 67207 蓝色六边形内部的平均密度作为它们的恒星球状体质量 $M_{\star }$ 的函数动态计算（顶部）或从光度计算（底部）。还显示了 Coma 星团中 ETGs（红色填充）和 Abell 262 中 ETGs（红色空心）的值。线条显示了相应螺旋星系的关系。图片经 Corsini 等人（2017 年）许可复制，作者版权所有。

Corsini et al. (2017) have investigated NGC 7113, and PGC 67207, two bright ETGs in low-density environments. These rare objects may help us disentangling in ellipticals what is of pertinence of the process of their formation and what is inherent to the properties of their dark matter halos. The surface-brightness distributions and their parameters were derived by $K_{\mathrm{S}}$-ugriz-band two-dimensional photometric decomposition. The line-of-sight stellar velocity distributions inside $R_{\mathrm{e}}$ were measured along several position angles. They assumed the BT-URC DM halo profile (see Eq. 29). The luminous and dark distributions were obtained from the orbit-based axisymmetric dynamical modelling (see Sect. 5.5). The fit model to the data is excellent and implies that these galaxies have a lower content of dark matter with respect to early-type galaxies living in high-density environments. Moreover, it is important to notice that their DM density inside $2R_{\mathrm{e}}$ is significantly higher than in similar mass spirals (see Fig. 24).
Corsini 等人（2017 年）调查了 NGC 7113 和 PGC 67207，这两个明亮的椭圆星系位于低密度环境中。这些罕见的天体可能有助于我们区分椭圆星系形成过程中的相关内容和它们暗物质暗晕属性之间的关系。表面亮度分布及其参数通过 $K_{\mathrm{S}}$ -ugriz 波段的二维光度分解得出。沿着几个位置角度测量了 $R_{\mathrm{e}}$ 内的视线恒星速度分布。他们假设了 BT-URC 暗物质暗晕剖面（见方程式 29）。明亮和暗分布是从基于轨道的轴对称动力学建模中获得的（见第 5.5 节）。对数据的拟合模型非常出色，表明这些星系的暗物质含量较低，相对于生活在高密度环境中的早型星系。此外，值得注意的是，它们内部的暗物质密度明显高于类似质量的螺旋星系（见图 24）。

Fig. 25

Line-of-sight dispersion velocities of the “classical” dSphs. A large r.m.s. is evident. Image reproduced with permission from Bonnivard (2015), copyright by the authors
“经典”dSphs 的视线色散速度。明显存在大的均方根。图像经 Bonnivard（2015）许可复制，作者版权所有。

9.3 DM in dwarf spheroidals

Dwarf spheroidal (dSph) galaxies are the smallest and least luminous galaxies in the Universe and provide unique hints on the nature of DM. They are old, in dynamical equilibrium and with no HI component. They contain a (small) number of stars, which provide us with tracers of the gravitational field. The very negligible baryonic content that they show does not affect their mass modelling (but see: Hammer et al. 2018) and it also indicates that this component may have never modified the primordial DM distribution (see Walker 2013) Then, by investigating these galaxies, we probe the original structure of the DM halos (see the review of Battaglia et al. 2013).
矮椭球（dSph）星系是宇宙中最小，最暗淡的星系，提供了关于 DM 性质的独特线索。它们古老，处于动态平衡状态，没有 HI 成分。它们包含（少量）星星，为我们提供了重力场的示踪者。它们所显示的非常微不足道的重子含量不会影响它们的质量建模（但请参见：Hammer 等人，2018），这也表明这个组分可能从未改变原始的 DM 分布（参见 Walker，2013）。然后，通过研究这些星系，我们探究 DM 暗物质晕的原始结构（参见 Battaglia 等人，2013）。

The stellar component for each dwarf spheroidal galaxy is modelled by means of a Plummer density profile with its scale radius $R_{\mathrm{e}}$, see Eq. (12). The main sample includes the eight larger dSphs of the Milky Way: Carina, Draco, Fornax, Leo I, Leo II, Sculptor, Sextans, and Ursa Minor. The determination of the DM mass profile M(r) requires the velocity dispersion profile along the line-of-sight $\text{σ}\mathrm{l}.\mathrm{o}.\mathrm{s}.(r)$ (see Fig. 25). The very limited number of these galaxies combined with the large range in the values of their physical quantities makes the stacked analysis approach impossible for investigating the dSphs mass distribution. There are three common methods that use available observations to infer the DM density profile in dSphs:
每个矮椭球星系的恒星成分都是通过使用其尺度半径 $R_{\mathrm{e}}$ 的普卢默密度分布模拟的，参见方程（12）。主要样本包括银河系的八个较大的 dSphs：Carina，Draco，Fornax，Leo I，Leo II，Sculptor，Sextans 和 Ursa Minor。确定 DM 质量分布 M(r)需要沿视线的速度色散分布 $\text{σ}\mathrm{l}.\mathrm{o}.\mathrm{s}.(r)$ （见图 25）。这些星系数量非常有限，加上它们的物理量值范围很大，使得堆叠分析方法无法用于研究 dSphs 的质量分布。有三种常见方法利用可用观测结果推断 dSphs 中的 DM 密度分布：

Jeans analysis In this approach one feeds Eq. (18) with the values of ${\nu }_{\star }(R)$, the stellar density profile, uses a large number of well determined dispersion velocities ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(r)$ (Walker et al. 2009a) and assumes a particular anisotropy profile (e.g., as in Bonnivard 2015). Then, through a Monte Carlo analysis, one obtains the free parameters of the DM density profile $\rho (r)$ and the anisotropy function $\beta$. There are views that this investigation, also when the tangential velocity dispersions are available, cannot resolve in these objects the cusp/core issue (Walker et al. 2009b; Strigari et al. 2008; Bonnivard 2015; Strigari et al. 2018). The degeneracy in the Jeans equation between the mass and the anisotropy profiles, combined with a kinematics of limited extension and quality, makes difficult to determine the density profile by means of this method.
在这种方法中，人们使用星密度分布的值 ${\nu }_{\star }(R)$ ，使用大量确定的色散速度 ${\text{σ}}_{\mathrm{l}.\mathrm{o}.\mathrm{s}.}(r)$ （Walker 等人，2009a），并假设特定的各向异性分布（例如，如 Bonnivard，2015）来馈送方程（18）。然后，通过蒙特卡洛分析，可以获得 DM 密度分布的自由参数 $\rho (r)$ 和各向异性函数 $\beta$ 。有观点认为，即使切向速度色散可用，这种研究也无法解决这些天体中的核心/核心问题（Walker 等人，2009b；Strigari 等人，2008；Bonnivard，2015；Strigari 等人，2018）。 Jeans 方程中质量和各向异性分布之间的退化，再加上有限范围和质量的动力学，使得通过这种方法确定密度分布变得困难。

Slope method Walker and Penarrubia (2011) first exploited the fact that in some dSphs there are multiple stellar populations, photometrically and chemo-dynamically distinct sub-components. They independently trace the (same) gravitational potential. Since $M(R_{\mathrm{e}})$, the mass contained within the effective radius $R_{\mathrm{e}}$ of each component, can be measured independently of their stellar orbital anisotropies, see Eq. (20) then, we can derive the quantity $\frac{\mathrm{d}\log M}{\mathrm{d}\log R}$ at different radii without adopting a DM halo profile. The method, applied to the dSph Fornax and Sculptor, for which two separate stellar sub-components have been disentangled, gives $\text{Δ}\log M/\text{Δ}\log r={2.61}_{-0.37}^{+0.43}$ and ${2.95}_{-0.39}^{+0.51}$, respectively, pointing to DM densities that keep an almost constant value within the central few-hundred parsecs of these objects. With the same method, Breddels et al. (2013) found that a NFW profile is only marginally allowed in Sculptor.
斜坡方法 Walker 和 Penarrubia（2011 年）首次利用了某些 dSphs 中存在多个恒星群体，光度和化学动力学上有明显差异的子组分的事实。它们独立地追踪（相同的）引力势。由于 $M(R_{\mathrm{e}})$ ，每个组分有效半径 $R_{\mathrm{e}}$ 内包含的质量可以独立测量，而不受其恒星轨道各向异性的影响，见方程（20），因此，我们可以在不采用 DM 暗物质晕轮廓的情况下，在不同半径处推导出数量 $\frac{\mathrm{d}\log M}{\mathrm{d}\log R}$ 。该方法应用于 Fornax 和 Sculptor dSph，这两个天体已经分离出两个独立的恒星子组分，分别给出 $\text{Δ}\log M/\text{Δ}\log r={2.61}_{-0.37}^{+0.43}$ 和 ${2.95}_{-0.39}^{+0.51}$ ，指向在这些天体的中心几百秒差距内保持几乎恒定值的 DM 密度。通过相同方法，Breddels 等人（2013 年）发现 NFW 轮廓在 Sculptor 中仅被边缘允许。

Fig. 26

Observed versus predicted dispersion velocities form different halo density profiles. Image reproduced with permission from Strigari et al. (2018), copyright by AAS
观察到的与预测的离散速度形成不同的晕密度轮廓。图像经 Strigari 等人（2018 年）许可复制，版权归 AAS 所有

This method has been carefully investigated by Strigari et al. (2018) in view of determining the level of its intrinsic bias, see Fig. 26 and finding improvements.
这种方法已经被 Strigari 等人（2018 年）仔细调查，以确定其固有偏差水平，参见图 26 并找到改进。

Schwarzschild modelling A promising method, based on distribution functions that depends on the action integrals, has been put forward by Pascale et al. (2018). This was applied to the Fornax galaxy, finding strong evidence for the presence of a cored density profile.
Schwarzschild 建模一种有前途的方法，基于依赖于作用积分的分布函数，由 Pascale 等人提出（2018 年）。这被应用于 Fornax 星系，发现了存在核心密度轮廓的强有力证据。

10 The LM/DM universal properties
10 明亮物质/暗物质的普遍性质

One could resume the state of the art of the issue of “DM in galaxies”, by stressing the unexpected scheme shown by the distributions of the dark and luminous matter in galaxies: halo masses, stellar component/baryonic masses, central densities, luminosities, DM density length scales, half-light radii, and galaxy morphologies are all engaged in a series of relationships, difficult to be understood in a physical sense. However, since the concurrent view argues that “galaxy formation is a complex phenomenon which could account for the apparently inexplicable observational scenario”, we stress that the above is far beyond a list of galaxy relationships, but a coherent pattern that can help us in the search of the unknown dark particle.
一个可以总结“星系中的暗物质”问题现状的方法是强调星系中暗物质和明亮物质的分布所展示的意外方案：暗物质和明亮物质的晕质量，恒星组件/重子质量，中心密度，亮度，DM 密度长度尺度，半光半径和星系形态都参与了一系列关系，难以在物理意义上理解。然而，由于共同观点认为“星系形成是一个复杂的现象，可以解释表面上无法解释的观测场景”，我们强调以上远不止是一系列星系关系，而是一个连贯的模式，可以帮助我们寻找未知的暗粒子。

In disk systems (dwarf disks, low surface brightness galaxies and spirals) when the values of their structural quantities are expressed in physical units, the stellar component forms a family ruled by three parameters: the disk length-scale $R_D$ and the magnitude (e.g., $M_I$) and the stellar disk concentration $C_{\star }$. In the same systems, also the dark component is represented by a family ruled by three parameters: the core radius $r_0$, the central density ${\rho }_0$ and $C_{\mathrm{D}\mathrm{M}}$ the DM concentration. The two families are closely and mysteriously related: the entanglement is so deep that it is difficult to understand which rules which.
在盘系统（矮星盘，低表面亮度星系和螺旋星系）中，当它们的结构量以物理单位表示时，恒星组件形成一个由三个参数规定的家族：盘长度尺度 $R_D$ 和幅度（例如， $M_I$ ）以及恒星盘浓度 $C_{\star }$ 。在相同的系统中，暗组件也由三个参数规定的家族代表：核心半径 $r_0$ ，中心密度 ${\rho }_0$ 和 $C_{\mathrm{D}\mathrm{M}}$ DM 浓度。这两个家族之间密切而神秘地相关：纠缠如此深刻，以至于很难理解哪个规则了哪个。

Remarkably, the situation much simplifies when we express the circular velocity V(r)¹² in the double-normalized form: $V(r/R_{\mathrm{o}\mathrm{p}\mathrm{t}})/V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ The profiles of the RCs emerge as a function of just one parameter, at choice among the above six, plus $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$, $M_{\mathrm{v}\mathrm{i}\mathrm{r}}$ and the angular momentum for unit mass j (see Lapi et al. 2018). Remarkably, this occurs independently on whether a galaxy is dark matter or luminous matter dominated for $R<R_{\mathrm{o}\mathrm{p}\mathrm{t}}$. The emerging evidence is that structural quantities deeply rooted in the luminous sector, like the disk length scales, tightly correlate with structural quantities deeply rooted in the dark sector, like the DM halo core radii.
显著的是，当我们以双标准化形式表达圆速度 V(r) ¹² 时，情况大大简化： $V(r/R_{\mathrm{o}\mathrm{p}\mathrm{t}})/V(R_{\mathrm{o}\mathrm{p}\mathrm{t}})$ RCs 的轮廓作为一个参数的函数出现，可以从上述六个中选择，再加上 $V_{\mathrm{o}\mathrm{p}\mathrm{t}}$ ， $M_{\mathrm{v}\mathrm{i}\mathrm{r}}$ 和单位质量角动量 j（见 Lapi 等人 2018 年）。值得注意的是，这发生在一个银河是暗物质还是明亮物质主导的情况下，对 $R<R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 都是独立的。出现的证据表明，根植于明亮部门的结构量，如盘长度尺度，与根植于暗部门的结构量，如 DM 暗物质核心半径，密切相关。

Let us conclude this section noticing that this scenario is, instead, still under investigation in spheroidal galaxies.
让我们总结这一节，注意到这种情况在球状星系中仍在调查中。

10.1 The cored distributions of dark matter halos around galaxies
10.1 星系周围暗物质晕的核心分布

The current situation is the following: (a) in disk systems of all morphologies and luminosities there is strong evidence that the DM halo density profile is very shallow out to the edge of the stellar distribution $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$, (b) in dwarf spheroidals and in ellipticals, also due to the intrinsic difficulty in these systems to disentangle the actual kinematics from the biased one, the situation is less clear, although, also in these objects, there are several claims of cored DM halo density profiles. In conclusion, the claim that DM around galaxies have a density distribution well represented by the cored B-URC profile is bald, but, I believe, correct.
目前的情况是：（a）在所有形态和亮度的盘系统中，有强有力的证据表明，DM 晕密度分布在星际分布的边缘 $R_{\mathrm{o}\mathrm{p}\mathrm{t}}$ 处非常浅，（b）在矮椭球体和椭圆体中，也由于这些系统内在的困难使得很难将实际动力学与有偏见的动力学分开，情况不太清楚，尽管在这些物体中也有几个声称有核心 DM 晕密度分布。总之，关于星系周围的 DM 具有由核心 B-URC 分布很好表示的密度分布的说法是光秃秃的，但我相信是正确的。

The most intriguing aspect of the DM in galaxies is not that they all possess a universal density profile, but that, this latter comes with a couple of very unexpected properties. The analysis of rotation curves, dispersion velocities, and weak-lensing data of large samples of dSphs, dwarf irregulars, spirals, and elliptical galaxies, found that the product of the DM core radius $r_0$ with the DM central density ${\rho }_0$ is nearly constant in galaxies, i.e., independent of their luminosity (Donato et al. 2009; see also Donato et al. 2004). This result, pioneered by Kormendy and Freeman (2004), is obtained in Donato et al. (2009) from the mass models derived from (1) about 1000 coadded RCs of spirals, (2) hundredths individual RCs of normal spirals of late and early types, (3) galaxy–galaxy weak lensing signals, (4) the inner kinematics of Local Group dwarf spheroidals, (5) the RCs of 36 dd and 72 LSBs (see Di Paolo and Salucci 2018). The relationship reads (see Fig. 27)
星系中暗物质最引人注目的方面并非它们都具有普遍的密度分布，而是，后者带有一些非常意外的特性。对大量 dSphs、矮不规则星系、螺旋星系和椭圆星系的旋转曲线、离散速度和弱引力透镜数据的分析发现，星系中暗物质核心半径 $r_0$ 与暗物质中心密度 ${\rho }_0$ 的乘积几乎在星系中保持恒定，即与它们的亮度无关（Donato 等人，2009 年；另请参见 Donato 等人，2004 年）。这一结果，由 Kormendy 和 Freeman（2004 年）率先提出，是由 Donato 等人（2009 年）从以下质量模型中获得的：（1）大约 1000 个螺旋星系的合并 RCs，（2）数百个晚期和早期类型正常螺旋星系的单个 RCs，（3）星系-星系弱引力透镜信号，（4）本地星系群矮椭球体的内部动力学，（5）36 个 dd 和 72 个 LSB 的 RCs（请参见 Di Paolo 和 Salucci，2018 年）。这种关系如下（见图 27）

$$\begin{array}{r}\log (r_0{\rho }_0)=2.15\pm 0.2,\end{array}$$

(48)

in units of $\log (M_{\odot }/\mathrm{p}{\mathrm{c}}^2)$.

Fig. 27

(Central DM halo density) $\times$ (halo core radius) as a function of a galaxy magnitude. Legenda: $r_{\mathrm{c}}\equiv r_0$. Data are from the URC of spirals (red circles), the scaling relation in Donato et al. (2009) (orange area), the Milky Way dSphs (purple triangles) Salucci et al. (2012), the dds (blue squares) Karukes and Salucci (2017). Also shown are the relationship by Burkert (2015): ${\rho }_0{r}_{\mathrm{c}}={75}_{-45}^{+85}{M}_{\odot }\mathrm{p}{\mathrm{c}}^{-2}$ (grey area) (see also Spano et al. 2008). Image reproduced with permission from Karukes and Salucci (2017), copyright by the authors
（中心 DM 暗物质密度） $\times$ （暗物质晕核心半径）作为星系星等的函数。图例： $r_{\mathrm{c}}\equiv r_0$ 。数据来自螺旋星系的 URC（红色圆圈），Donato 等人（2009 年）的比例关系（橙色区域），银河系 dSphs（紫色三角形）Salucci 等人（2012 年），dds（蓝色方块）Karukes 和 Salucci（2017 年）。还显示了 Burkert（2015 年）的关系： ${\rho }_0{r}_{\mathrm{c}}={75}_{-45}^{+85}{M}_{\odot }\mathrm{p}{\mathrm{c}}^{-2}$ （灰色区域）（另见 Spano 等人 2008 年）。图像经 Karukes 和 Salucci（2017）许可复制，作者版权

This relationship between the two structural quantities of the DM halos is found in galactic systems spanning over 14 magnitudes and it exploits mass profiles determined by several independent methods. In the same objects, the constancy of ${\rho }_0{r}_0$ is in sharp contrast with the systematic variations, by about 5 orders of magnitude, of all the other DM-related galaxy quantities, including the central DM density ${\rho }_0$ and many of the LM-related galaxy properties, as the magnitude.
DM 暗物质晕的这两个结构量之间的关系发现在跨越 14 个星等的星系系统中，并利用了由几种独立方法确定的质量剖面。在相同的物体中， ${\rho }_0{r}_0$ 的恒定性与所有其他与 DM 有关的星系量的系统变化形成鲜明对比，包括中心 DM 密度 ${\rho }_0$ 和许多与 LM 有关的星系属性，如星等，大约有 5 个数量级的变化。

At a higher level there is the correlation between the compactness of the stellar disks and that of the DM halos in dark matter dominated dds and LSBs (see Fig. 28 and the related caption). It is legitimate to interpret all this as an evidence of the dark and luminous worlds conjuring in galaxies.
在更高的层次上，恒星盘的紧凑性与暗物质主导的 dd 和 LSB 中 DM 暗物质晕的紧凑性之间存在相关性（见图 28 及相关说明）。将所有这些解释为暗物质和发光世界在星系中召唤的证据是合理的。

Fig. 28

The relationships between the compactness of the stellar disks and that of DM halos in dd and LSB galaxies (Karukes and Salucci 2017; Di Paolo and Salucci 2018). Let us set: $\mathbf{M}$ and $\mathbf{S}$ for a generic galaxy mass and size. We perform, in a sample of galaxies, the regression $\log \mathbf{S}=a+b\log \mathbf{M}$. For each galaxy i of the sample the compactness $\log {\mathbf{C}}_i$ is defined by $\log {\mathbf{C}}_i=-\log ({\mathbf{S}}_i)+a+b\log {\mathbf{M}}_i$. Here, for the luminous matter: $\mathbf{M}\equiv M_{\star }$, $\mathbf{S}\equiv R_D$ and ${\mathbf{C}}_i\equiv C_{\star }$ ; for the DM: $\mathbf{M}\equiv M_{\mathrm{v}\mathrm{i}\mathrm{r}}$, $\mathbf{S}\equiv r_0$ and ${\mathbf{C}}_i\equiv C_{\mathrm{D}\mathrm{M}}$
恒星盘的紧凑性与 dd 和 LSB 星系中 DM 暗物质晕的紧凑性之间的关系（Karukes 和 Salucci 2017; Di Paolo 和 Salucci 2018）。让我们设定： $\mathbf{M}$ 和 $\mathbf{S}$ 为一个通用星系的质量和大小。我们在一组星系中执行回归 $\log \mathbf{S}=a+b\log \mathbf{M}$ 。对于样本中的每个星系 i，紧凑性 $\log {\mathbf{C}}_i$ 由 $\log {\mathbf{C}}_i=-\log ({\mathbf{S}}_i)+a+b\log {\mathbf{M}}_i$ 定义。在这里，对于发光物质： $\mathbf{M}\equiv M_{\star }$ ， $\mathbf{S}\equiv R_D$ 和 ${\mathbf{C}}_i\equiv C_{\star }$ ；对于暗物质： $\mathbf{M}\equiv M_{\mathrm{v}\mathrm{i}\mathrm{r}}$ ， $\mathbf{S}\equiv r_0$ 和 ${\mathbf{C}}_i\equiv C_{\mathrm{D}\mathrm{M}}$

The relationship between the halo mass and the stellar mass located at its center is an important and well-investigated one. It is well known that the mass fraction $\frac{\mathrm{D}\mathrm{M}}{\mathrm{L}\mathrm{M}}$ as a function of the halo mass follows a characteristic U-shaped curve (Wolf et al. 2010; Moster et al. 2010) for which $M_{\mathrm{v}\mathrm{i}\mathrm{r}}/M_{\star }$ is minimized at the halo mass $M_{\mathrm{v}\mathrm{i}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\approx 3\times {10}^{11}{M}_{\odot }$ and rises at both lower and higher masses. According to the URC, the value of $M_{\mathrm{v}\mathrm{i}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}$ corresponds to $M_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\sim 1.2\times {10}^{10}{M}_{\odot }$ and to $L_{\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\sim 5\times {10}^9{L}_{\odot }$ in the $r\star$-band luminosity (see also Lapi et al. 2018). Outliers of this relationship do exist (Beasley et al. 2016); however, here we do not further enter this topic certainly related to the “galaxy formation process”.
暗物质晕质量与其中心处的恒星质量之间的关系是一个重要且深入研究的关系。众所周知，质量分数 $\frac{\mathrm{D}\mathrm{M}}{\mathrm{L}\mathrm{M}}$ 作为暗物质晕质量的函数遵循一个特征的 U 形曲线（Wolf 等人，2010 年；Moster 等人，2010 年），其中 $M_{\mathrm{v}\mathrm{i}\mathrm{r}}/M_{\star }$ 在暗物质晕质量 $M_{\mathrm{v}\mathrm{i}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\approx 3\times {10}^{11}{M}_{\odot }$ 处最小化，并在较低和较高质量处上升。根据 URC， $M_{\mathrm{v}\mathrm{i}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}$ 的值对应于 $M_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r},\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\sim 1.2\times {10}^{10}{M}_{\odot }$ 和 $L_{\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}}\sim 5\times {10}^9{L}_{\odot }$ 在 $r\star$ -波段亮度中（另请参阅 Lapi 等人，2018 年）。这种关系的异常值确实存在（Beasley 等人，2016）；然而，在这里，我们不进一步讨论这个明显与“星系形成过程”相关的主题。

Therefore, the empirical scenario includes six quantities that define a galaxy: three in the dark sector (halo mass and core radius and DM halo compactness) and three in the luminous sector (stellar/baryonic mass, half-light radius and stellar disk compactness). They all relate each other but, while some of these relationships lay in the heart of the DM mystery, others, instead, lay in the ball-park of the galaxy formation and evolution process.
因此，经验性情景包括定义星系的六个量：三个在暗部门（暗物质晕质量和核半径以及暗物质晕紧凑度）和三个在亮部门（恒星/重子质量、半光半径和恒星盘紧凑度）。它们彼此相关，但是，虽然其中一些关系潜藏在暗物质之谜的核心，另一些则潜藏在星系形成和演化过程的基本原理中。

10.2 The dark-luminous matter coupling 2.0
10.2 暗-亮物质耦合 2.0

In spirals, dwarf disks and LSBs there are extraordinary multiple connections between the dark and the luminous components. This occurs over many orders of magnitudes in halo masses and over the whole ranges of galaxies morphology and luminosity. The “standard” explanation relates to a dynamical evolution of the galaxies, in particular, of their DM halo densities, caused by powerful baryonic feedbacks. Although this scenario is far than being rejected, it seems, however, unable to cope with the intriguing wealth of correlations between quantities deep-rooted in opposite dark/luminous worlds that we have presented in this review. More in detail, while we cannot completely rule out the possibility that astrophysical phenomena can be responsible for the above intriguing scenario, on the other hand, what emerges in galaxies allow us to propose a shift of paradigm, according to which, the nature of dark matter is not given to us by convincing theoretical arguments, but must be searched in the various properties of the DM halos and stellar disks.
在螺旋、矮星盘和低表面亮度星系中，暗物质和亮物质之间存在着非凡的多重联系。这种联系发生在许多数量级的暗物质晕和整个星系形态和亮度范围内。“标准”解释与星系的动力学演化有关，特别是与它们的暗物质晕密度有关，这是由强大的重子反馈引起的。尽管这种情景远非被拒绝，但似乎无法解释我们在本综述中提出的暗/亮世界中根深蒂固的数量之间令人感兴趣的丰富相关性。更详细地说，虽然我们不能完全排除天体物理现象可能对上述有趣情景负责的可能性，另一方面，星系中出现的情况使我们提出了一种范式转变，根据这种转变，暗物质的性质并非由令人信服的理论论据给出，而必须在暗物质晕和恒星盘的各种属性中寻找。

In Salucci and Turini (2017), it is argued that these new ideas can be justified also by some direct hint: in spirals the DM pseudo-pressure ${\rho }_{\mathrm{D}\mathrm{M}}(r)V^2(r)$ reaches a maximum value always close to the core radius $r_0$ and this maximum takes the same value in all objects, no matter the galaxy mass. Moreover, at $r=r_0$ in all disk systems, the quantity $\rho (r){\rho }_{\star }(r)$ takes the same value. We notice that this density product is proportional to the interaction probability between the, the luminous and the dark matter. This is hardly a coincidence, in that, the quantity like $K_{\mathrm{S}\mathrm{A}}={\rho }_{\mathrm{D}\mathrm{M}}^2(r)$, which is proportional to the self interaction of the DM component, is largely varying in galaxies and among galaxies. One can speculate that the structure of the inner parts of the galaxies is driven by a direct interaction between dark and luminous components on timescales of the order of the age of the Universe. The DM central cusp, outcome of the proto-halo virialization, as time goes by, gets progressively eaten up/absorbed by the dominant luminous component. The interaction, then, flattens the density of DM and drops the pressure towards the center of the galaxy and it is likely to leave in inheritance the above galaxy relationships.
在 Salucci 和 Turini（2017）中，有人认为这些新观点也可以通过一些直接的线索来证明：在螺旋星系中，暗物质伪压力 ${\rho }_{\mathrm{D}\mathrm{M}}(r)V^2(r)$ 总是接近核半径 $r_0$ 达到最大值，并且这个最大值在所有物体中都取相同的值，无论星系质量如何。此外，在 $r=r_0$ 中的所有盘状系统中，数量 $\rho (r){\rho }_{\star }(r)$ 取相同的值。我们注意到，这个密度乘积与暗物质之间，光亮物质和暗物质之间的相互作用概率成正比。这几乎不可能是巧合，因为像 $K_{\mathrm{S}\mathrm{A}}={\rho }_{\mathrm{D}\mathrm{M}}^2(r)$ 这样的数量，它与暗物质成分的自相互作用成正比，在星系内部和星系之间变化很大。人们可以推测，星系内部结构受到暗物质和明亮成分之间直接相互作用的驱动，在宇宙年龄量级的时间尺度上。暗物质中心尖峰，作为原始晕的维里化结果，随着时间的推移，逐渐被主导的明亮成分吞噬/吸收。这种相互作用然后使暗物质的密度变平并将压力向星系中心降低，并且很可能会在上述星系关系中留下遗传。

11 Conclusions

On the fundamental issue of dark matter in galaxies there is a substantial difference between spheroidals and disk systems. Let us notice that also the latter statement shows that, although we are focused on DM halos, nonetheless, we must discuss galaxy morphology. And this has been the leitmotiv of this review: the DM component enters in aspects apparently of pertinence of the luminous matter and vice versa.
在星系中关于暗物质的基本问题上，球状星系和盘状系统之间存在实质性差异。让我们注意到，即使我们专注于暗物质晕，但我们仍然必须讨论星系形态。这一点一直是本综述的主题：暗物质成分涉及到明亮物质表面上相关的方面，反之亦然。

We have started to point out that the luminosity or a reference velocity is the tag that defines the dark and luminous mass distribution in galaxies. However, very recent results have proven that in spirals, “dd” and LSBs, the universal rotation curve, when expressed in physical units, needs two statistically independent controlling parameters: the luminosity and the compactness. It must be specified that we are not just flagging some empirical relationships: we have three structural properties of the stellar discs that enter in close relation with the three structural properties of the DM halos.
我们已经开始指出，在星系中，光度或参考速度是定义暗物质和明亮物质分布的标签。然而，最近的研究结果表明，在螺旋星系，“dd”和 LSBs 中，当以物理单位表示的普遍旋转曲线需要两个统计独立的控制参数：光度和紧凑度。必须指出的是，我们不仅仅是在标记一些经验关系：我们有三个与暗物质晕的三个结构特性密切相关的恒星盘的结构特性。

In elliptical galaxies, the situation is still very open. They also show regularities in their total mass distributions: its logarithmic derivative from $r=0$ to $r=R_{\mathrm{e}}$ and beyond is very near to 1, despite that in this region the galaxies pass from a totally LM dominated regime to one with a relevant fraction of dark matter. The fundamental plane of ellipticals and S0 entangles two quantities of the luminous world, the luminosity/stellar spheroidal mass, and the half-light ratio and a hybrid one: the dispersion velocity, which is rooted in both luminous and dark worlds. Universality in the distribution of matter in ellipticals has not been established yet. We believe that this is due to the insufficient quality and quantity of proper and useful probes of their gravitational potentials. We also have to notice that also for these systems there are evidences of cored DM distributions.
在椭圆星系中，情况仍然非常开放。它们在总质量分布中也显示出规律性：从 $r=0$ 到 $r=R_{\mathrm{e}}$ 及更远处的对数导数非常接近 1，尽管在这个区域内，星系从完全由 LM 主导的状态过渡到具有相当比例暗物质的状态。椭圆和 S0 的基本平面纠缠着两个光亮世界的量，即光度/恒星球状质量和半光比，以及一个混合的量：色散速度，它根植于光亮和暗暗物质世界。椭圆星系中物质的分布普遍性尚未建立。我们认为这是由于对其引力势的适当和有用的探测器的质量和数量不足。我们还必须注意，对于这些系统也有暗物质核心分布的证据。

Dwarf spheroidals, despite their limited number, are becoming always more crucial in the investigation of dark matter. Each of these dark spheres, lying at the lowest mass boundary of the cosmological structures harboring stars, is a wealth of information on the dark particle. Unfortunately, we can probe their gravitational field only very near to their centers, with tracers that provide data that are difficult to be unambiguously interpreted. It is worth saying, however, that also for this population of galaxies, there are evidences of cored DM halos with properties similar to those of the disk systems and ellipticals.
矮椭球体，尽管数量有限，正在变得越来越关键，用于研究暗物质。这些暗球中的每一个，位于宇宙结构中星体的最低质量边界上，都是有关暗粒子的丰富信息。不幸的是，我们只能在它们的中心附近探测到它们的引力场，使用的追踪器提供的数据很难被明确解释。然而，值得一提的是，对于这类星系群，也有暗物质核心的证据，其性质类似于盘系统和椭圆系统。

The non-gravitational nature of DM remains a mystery (Bertone 2010; de Swart et al. 2017). It seems impossible to explain the observational evidences gathered so far in a simple dark matter framework. In my opinion, they are portals to the new physics that seems to lurk behind the phenomenon called “dark matter”. I think that it will be important to recognize our prejudices and confront them head on, also if this means to end our fascination with the $\text{Λ}$CDM Weakly Interacting Massive Particles scenario.
暗物质的非引力性质仍然是一个谜（Bertone 2010; de Swart et al. 2017）。目前似乎不可能用简单的暗物质框架来解释迄今为止收集到的观测证据。在我看来，它们是通往似乎潜藏在所谓“暗物质”现象背后的新物理的门户。我认为重要的是要认识到我们的偏见并正视它们，即使这意味着结束我们对 $\text{Λ}$ CDM 弱相互作用大质量粒子方案的迷恋。

12 Future directions

As a consequence of the reverse-engineering approach to the mystery of the dark matter in galaxies that I advocate here, the future is the past. Namely, I argue that, in the observational properties of galaxies, there is much of the required information to solve the riddle. Unfortunately, we have recovered only a very small part of it, not because it is difficult or long to do, but because we were stuck in a different paradigm where, honestly, all this phenomenology is not so important.
作为我在这里提倡的对星系暗物质之谜进行逆向工程方法的结果，未来就是过去。换句话说，我认为，在星系的观测特性中，有许多解开谜团所需信息。不幸的是，我们只恢复了其中的一小部分，不是因为这很困难或很长时间，而是因为我们被困在一个不同的范式中，坦率地说，在那里，所有这些现象学并不那么重要。

However, the situation is extremely positive because, in the near future, from Gaia to SKA, we will be submerged by an enormous flux of information, coming from different messengers, on all aspects of galaxies, independently if one believes or not that this will lead to a solution of the old mystery of dark matter.
然而，情况非常积极，因为在不久的将来，从盖亚到 SKA，我们将被大量来自不同信使的信息淹没，涉及宇宙系的各个方面，无论一个人是否相信这将导致解决黑暗物质的古老谜团。