Dark matter halo

Last updated
Simulated dark matter halo from a cosmological N-body simulation Dark matter halo.png
Simulated dark matter halo from a cosmological N-body simulation

In modern models of physical cosmology, a dark matter halo is a basic unit of cosmological structure. It is a hypothetical region that has decoupled from cosmic expansion and contains gravitationally bound matter. [1] A single dark matter halo may contain multiple virialized clumps of dark matter bound together by gravity, known as subhalos. [1] Modern cosmological models, such as ΛCDM, propose that dark matter halos and subhalos may contain galaxies. [1] [2] The dark matter halo of a galaxy envelops the galactic disc and extends well beyond the edge of the visible galaxy. Thought to consist of dark matter, halos have not been observed directly. Their existence is inferred through observations of their effects on the motions of stars and gas in galaxies and gravitational lensing. [3] Dark matter halos play a key role in current models of galaxy formation and evolution. Theories that attempt to explain the nature of dark matter halos with varying degrees of success include cold dark matter (CDM), warm dark matter, and massive compact halo objects (MACHOs). [4] [5] [6] [7]

Contents

Galaxy rotation curve for the Milky Way. Vertical axis is speed of rotation about the galactic center. Horizontal axis is distance from the galactic center. The sun is marked with a yellow ball. The observed curve of speed of rotation is blue. The predicted curve based upon stellar mass and gas in the Milky Way is red. Scatter in observations roughly indicated by gray bars. The difference is due to dark matter or perhaps a modification of the law of gravity. Rotation curve (Milky Way).svg
Galaxy rotation curve for the Milky Way. Vertical axis is speed of rotation about the galactic center. Horizontal axis is distance from the galactic center. The sun is marked with a yellow ball. The observed curve of speed of rotation is blue. The predicted curve based upon stellar mass and gas in the Milky Way is red. Scatter in observations roughly indicated by gray bars. The difference is due to dark matter or perhaps a modification of the law of gravity.

Rotation curves as evidence of a dark matter halo

The presence of dark matter (DM) in the halo is inferred from its gravitational effect on a spiral galaxy's rotation curve. Without large amounts of mass throughout the (roughly spherical) halo, the rotational velocity of the galaxy would decrease at large distances from the galactic center, just as the orbital speeds of the outer planets decrease with distance from the Sun. However, observations of spiral galaxies, particularly radio observations of line emission from neutral atomic hydrogen (known, in astronomical parlance, as 21 cm Hydrogen line, H one, and H I line), show that the rotation curve of most spiral galaxies flattens out, meaning that rotational velocities do not decrease with distance from the galactic center. [11] The absence of any visible matter to account for these observations implies either that unobserved (dark) matter, first proposed by Ken Freeman in 1970, exist, or that the theory of motion under gravity (general relativity) is incomplete. Freeman noticed that the expected decline in velocity was not present in NGC 300 nor M33, and considered an undetected mass to explain it. The DM Hypothesis has been reinforced by several studies. [12] [13] [14] [15]

Formation and structure of dark matter halos

The formation of dark matter halos is believed to have played a major role in the early formation of galaxies. During initial galactic formation, the temperature of the baryonic matter should have still been much too high for it to form gravitationally self-bound objects, thus requiring the prior formation of dark matter structure to add additional gravitational interactions. The current hypothesis for this is based on cold dark matter (CDM) and its formation into structure early in the universe.

The hypothesis for CDM structure formation begins with density perturbations in the Universe that grow linearly until they reach a critical density, after which they would stop expanding and collapse to form gravitationally bound dark matter halos. The spherical collapse framework analytically models the formation and growth of such halos. These halos would continue to grow in mass (and size), either through accretion of material from their immediate neighborhood, or by merging with other halos. Numerical simulations of CDM structure formation have been found to proceed as follows: A small volume with small perturbations initially expands with the expansion of the Universe. As time proceeds, small-scale perturbations grow and collapse to form small halos. At a later stage, these small halos merge to form a single virialized dark matter halo with an ellipsoidal shape, which reveals some substructure in the form of dark matter sub-halos. [2]

The use of CDM overcomes issues associated with the normal baryonic matter because it removes most of the thermal and radiative pressures that were preventing the collapse of the baryonic matter. The fact that the dark matter is cold compared to the baryonic matter allows the DM to form these initial, gravitationally bound clumps. Once these subhalos formed, their gravitational interaction with baryonic matter is enough to overcome the thermal energy, and allow it to collapse into the first stars and galaxies. Simulations of this early galaxy formation matches the structure observed by galactic surveys as well as observation of the Cosmic Microwave Background. [16]

Density profiles

A commonly used model for galactic dark matter halos is the pseudo-isothermal halo: [17]

where denotes the finite central density and the core radius. This provides a good fit to most rotation curve data. However, it cannot be a complete description, as the enclosed mass fails to converge to a finite value as the radius tends to infinity. The isothermal model is, at best, an approximation. Many effects may cause deviations from the profile predicted by this simple model. For example, (i) collapse may never reach an equilibrium state in the outer region of a dark matter halo, (ii) non-radial motion may be important, and (iii) mergers associated with the (hierarchical) formation of a halo may render the spherical-collapse model invalid. [18]

Numerical simulations of structure formation in an expanding universe lead to the empirical NFW (Navarro–Frenk–White) profile: [19]

where is a scale radius, is a characteristic (dimensionless) density, and = is the critical density for closure. The NFW profile is called 'universal' because it works for a large variety of halo masses, spanning four orders of magnitude, from individual galaxies to the halos of galaxy clusters. This profile has a finite gravitational potential even though the integrated mass still diverges logarithmically. It has become conventional to refer to the mass of a halo at a fiducial point that encloses an overdensity 200 times greater than the critical density of the universe, though mathematically the profile extends beyond this notational point. It was later deduced that the density profile depends on the environment, with the NFW appropriate only for isolated halos. [20] NFW halos generally provide a worse description of galaxy data than does the pseudo-isothermal profile, leading to the cuspy halo problem.

Higher resolution computer simulations are better described by the Einasto profile: [21]

where r is the spatial (i.e., not projected) radius. The term is a function of n such that is the density at the radius that defines a volume containing half of the total mass. While the addition of a third parameter provides a slightly improved description of the results from numerical simulations, it is not observationally distinguishable from the 2 parameter NFW halo, [22] and does nothing to alleviate the cuspy halo problem.

Shape

The collapse of overdensities in the cosmic density field is generally aspherical. So, there is no reason to expect the resulting halos to be spherical. Even the earliest simulations of structure formation in a CDM universe emphasized that the halos are substantially flattened. [23] Subsequent work has shown that halo equidensity surfaces can be described by ellipsoids characterized by the lengths of their axes. [24]

Because of uncertainties in both the data and the model predictions, it is still unclear whether the halo shapes inferred from observations are consistent with the predictions of ΛCDM cosmology.

Halo substructure

Up until the end of the 1990s, numerical simulations of halo formation revealed little substructure. With increasing computing power and better algorithms, it became possible to use greater numbers of particles and obtain better resolution. Substantial amounts of substructure are now expected. [25] [26] [27] When a small halo merges with a significantly larger halo it becomes a subhalo orbiting within the potential well of its host. As it orbits, it is subjected to strong tidal forces from the host, which cause it to lose mass. In addition the orbit itself evolves as the subhalo is subjected to dynamical friction which causes it to lose energy and angular momentum to the dark matter particles of its host. Whether a subhalo survives as a self-bound entity depends on its mass, density profile, and its orbit. [18]

Angular momentum

As originally pointed out by Hoyle [28] and first demonstrated using numerical simulations by Efstathiou & Jones, [29] asymmetric collapse in an expanding universe produces objects with significant angular momentum.

Numerical simulations have shown that the spin parameter distribution for halos formed by dissipation-less hierarchical clustering is well fit by a log-normal distribution, the median and width of which depend only weakly on halo mass, redshift, and cosmology: [30]

with and . At all halo masses, there is a marked tendency for halos with higher spin to be in denser regions and thus to be more strongly clustered. [31]

Milky Way dark matter halo

The visible disk of the Milky Way Galaxy is thought to be embedded in a much larger, roughly spherical halo of dark matter. The dark matter density drops off with distance from the galactic center. It is now believed that about 95% of the galaxy is composed of dark matter, a type of matter that does not seem to interact with the rest of the galaxy's matter and energy in any way except through gravity. The luminous matter makes up approximately 9×1010 solar masses. The dark matter halo is likely to include around 6×1011 to 3×1012 solar masses of dark matter. [32] [33] A 2014 Jeans analysis of stellar motions calculated the dark matter density (at the sun's distance from the galactic centre) = 0.0088 (+0.0024 −0.0018) solar masses/parsec^3. [33]

See also

Related Research Articles

<span class="mw-page-title-main">Accelerating expansion of the universe</span> Cosmological phenomenon

Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time. The accelerated expansion of the universe was discovered in 1998 by two independent projects, the Supernova Cosmology Project and the High-Z Supernova Search Team, which used distant type Ia supernovae to measure the acceleration. The idea was that as type Ia supernovae have almost the same intrinsic brightness, and since objects that are farther away appear dimmer, the observed brightness of these supernovae can be used to measure the distance to them. The distance can then be compared to the supernovae's cosmological redshift, which measures how much the universe has expanded since the supernova occurred; the Hubble law established that the farther away that an object is, the faster it is receding. The unexpected result was that objects in the universe are moving away from one another at an accelerating rate. Cosmologists at the time expected that recession velocity would always be decelerating, due to the gravitational attraction of the matter in the universe. Three members of these two groups have subsequently been awarded Nobel Prizes for their discovery. Confirmatory evidence has been found in baryon acoustic oscillations, and in analyses of the clustering of galaxies.

<span class="mw-page-title-main">Hubble's law</span> Observation in physical cosmology

Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible light spectrum. The discovery of Hubble's law is attributed to Edwin Hubble's work published in 1929.

<span class="mw-page-title-main">Galaxy rotation curve</span> Observed discrepancy in galactic angular momenta

The rotation curve of a disc galaxy is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre. It is typically rendered graphically as a plot, and the data observed from each side of a spiral galaxy are generally asymmetric, so that data from each side are averaged to create the curve. A significant discrepancy exists between the experimental curves observed, and a curve derived by applying gravity theory to the matter observed in a galaxy. Theories involving dark matter are the main postulated solutions to account for the variance.

<span class="mw-page-title-main">Friedmann–Lemaître–Robertson–Walker metric</span> Metric based on the exact solution of Einsteins field equations of general relativity

The Friedmann–Lemaître–Robertson–Walker metric is a metric based on an exact solution of the Einstein field equations of general relativity. The metric describes a homogeneous, isotropic, expanding universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker – are variously grouped as Friedmann, Friedmann–Robertson–Walker (FRW), Robertson–Walker (RW), or Friedmann–Lemaître (FL). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.

In astrophysics, dynamical friction or Chandrasekhar friction, sometimes called gravitational drag, is loss of momentum and kinetic energy of moving bodies through gravitational interactions with surrounding matter in space. It was first discussed in detail by Subrahmanyan Chandrasekhar in 1943.

A galactic halo is an extended, roughly spherical component of a galaxy which extends beyond the main, visible component. Several distinct components of a galaxy comprise its halo:

The cuspy halo problem is a discrepancy between the inferred dark matter density profiles of low-mass galaxies and the density profiles predicted by cosmological N-body simulations. Nearly all simulations form dark matter halos which have "cuspy" dark matter distributions, with density increasing steeply at small radii, while the rotation curves of most observed dwarf galaxies suggest that they have flat central dark matter density profiles ("cores").

<span class="mw-page-title-main">Lambda-CDM model</span> Model of Big Bang cosmology

The Lambda-CDM, Lambda cold dark matter, or ΛCDM model is a mathematical model of the Big Bang theory with three major components:

  1. a cosmological constant, denoted by lambda (Λ), associated with dark energy
  2. the postulated cold dark matter, denoted by CDM
  3. ordinary matter
<span class="mw-page-title-main">Friedmann equations</span> Equations in physical cosmology

The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.

<span class="mw-page-title-main">Structure formation</span> Formation of galaxies, galaxy clusters and larger structures from small early density fluctuations

In physical cosmology, structure formation describes the creation of galaxies, galaxy clusters, and larger structures starting from small fluctuations in mass density resulting from processes that created matter. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.8 billion years ago. However, looking at the night sky today, structures on all scales can be seen, from stars and planets to galaxies. On even larger scales, galaxy clusters and sheet-like structures of galaxies are separated by enormous voids containing few galaxies. Structure formation models gravitational instability of small ripples in mass density to predict these shapes, confirming the consistency of the physical model.

<span class="mw-page-title-main">Satellite galaxy</span> Galaxy that orbits a larger galaxy due to gravitational attraction

A satellite galaxy is a smaller companion galaxy that travels on bound orbits within the gravitational potential of a more massive and luminous host galaxy. Satellite galaxies and their constituents are bound to their host galaxy, in the same way that planets within our own solar system are gravitationally bound to the Sun. While most satellite galaxies are dwarf galaxies, satellite galaxies of large galaxy clusters can be much more massive. The Milky Way is orbited by about fifty satellite galaxies, the largest of which is the Large Magellanic Cloud.

<span class="mw-page-title-main">Deceleration parameter</span>

The deceleration parameter in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by: where is the scale factor of the universe and the dots indicate derivatives by proper time. The expansion of the universe is said to be "accelerating" if , and in this case the deceleration parameter will be negative. The minus sign and name "deceleration parameter" are historical; at the time of definition was expected to be negative, so a minus sign was inserted in the definition to make positive in that case. Since the evidence for the accelerating universe in the 1998–2003 era, it is now believed that is positive therefore the present-day value is negative. In general varies with cosmic time, except in a few special cosmological models; the present-day value is denoted .

<span class="mw-page-title-main">Lyman-alpha emitter</span>

A Lyman-alpha emitter (LAE) is a type of distant galaxy that emits Lyman-alpha radiation from neutral hydrogen.

The Navarro–Frenk–White (NFW) profile is a spatial mass distribution of dark matter fitted to dark matter halos identified in N-body simulations by Julio Navarro, Carlos Frenk and Simon White. The NFW profile is one of the most commonly used model profiles for dark matter halos.

<span class="mw-page-title-main">Weak gravitational lensing</span>

While the presence of any mass bends the path of light passing near it, this effect rarely produces the giant arcs and multiple images associated with strong gravitational lensing. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the deflection is impossible to detect in a single background source. However, even in these cases, the presence of the foreground mass can be detected, by way of a systematic alignment of background sources around the lensing mass. Weak gravitational lensing is thus an intrinsically statistical measurement, but it provides a way to measure the masses of astronomical objects without requiring assumptions about their composition or dynamical state.

<span class="mw-page-title-main">Baryon acoustic oscillations</span> Fluctuations in the density of the normal matter of the universe

In cosmology, baryon acoustic oscillations (BAO) are fluctuations in the density of the visible baryonic matter of the universe, caused by acoustic density waves in the primordial plasma of the early universe. In the same way that supernovae provide a "standard candle" for astronomical observations, BAO matter clustering provides a "standard ruler" for length scale in cosmology. The length of this standard ruler is given by the maximum distance the acoustic waves could travel in the primordial plasma before the plasma cooled to the point where it became neutral atoms, which stopped the expansion of the plasma density waves, "freezing" them into place. The length of this standard ruler can be measured by looking at the large scale structure of matter using astronomical surveys. BAO measurements help cosmologists understand more about the nature of dark energy by constraining cosmological parameters.

The Press–Schechter formalism is a mathematical model for predicting the number of objects of a certain mass within a given volume of the Universe. It was described in an academic paper by William H. Press and Paul Schechter in 1974.

In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within the virial radius of a gravitationally bound system, a radius within which the system obeys the virial theorem. The virial radius is determined using a "top-hat" model. A spherical "top hat" density perturbation destined to become a galaxy begins to expand, but the expansion is halted and reversed due to the mass collapsing under gravity until the sphere reaches equilibrium – it is said to be virialized. Within this radius, the sphere obeys the virial theorem which says that the average kinetic energy is equal to minus one half times the average potential energy, , and this radius defines the virial radius.

<span class="mw-page-title-main">Jeans equations</span> System of differential equations

The Jeans equations are a set of partial differential equations that describe the motion of a collection of stars in a gravitational field. The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system for systems without collision. They are analogous to the Euler equations for fluid flow and may be derived from the collisionless Boltzmann equation. The Jeans equations can come in a variety of different forms, depending on the structure of what is being modelled. Most utilization of these equations has been found in simulations with large number of gravitationally bound objects.

In cosmology, the missing baryon problem is an observed discrepancy between the amount of baryonic matter detected from shortly after the Big Bang and from more recent epochs. Observations of the cosmic microwave background and Big Bang nucleosynthesis studies have set constraints on the abundance of baryons in the early universe, finding that baryonic matter accounts for approximately 4.8% of the energy contents of the Universe. At the same time, a census of baryons in the recent observable universe has found that observed baryonic matter accounts for less than half of that amount. This discrepancy is commonly known as the missing baryon problem. The missing baryon problem is different from the dark matter problem, which is non-baryonic in nature.

References

  1. 1 2 3 Wechsler, Risa; Tinker, Jeremy (September 2018). "The Connection between Galaxies and their Dark Matter Halos". Annual Review of Astronomy and Astrophysics. 56: 435–487. arXiv: 1804.03097 . Bibcode:2018ARA&A..56..435W. doi:10.1146/annurev-astro-081817-051756. S2CID   119072496.
  2. 1 2 Mo, Houjun; van den Bosch, Frank; White, Simon (2010). Galaxy Formation and Evolution. Cambridge University Press. pp. 97–98. ISBN   978-0-521-85793-2.
  3. Khullar, Gourav (4 November 2016). "The Bullet Cluster – A Smoking Gun for Dark Matter!". Astrobites. Retrieved 30 May 2019.
  4. Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M. (May 1996). "The Structure of Cold Dark Matter Halos". The Astrophysical Journal. 462: 563–575. arXiv: astro-ph/9508025 . Bibcode:1996ApJ...462..563N. doi:10.1086/177173. S2CID   119007675.
  5. Lovell, Mark R.; Frenk, Carlos S.; Eke, Vincent R.; Jenkins, Adrian; Gao, Liang; Theuns, Tom (21 March 2014). "The properties of warm dark matter haloes". Monthly Notices of the Royal Astronomical Society. 439 (1): 300–317. arXiv: 1308.1399 . doi: 10.1093/mnras/stt2431 . S2CID   55639399.
  6. Alcock, C (10 October 2000). "The MACHO Project: Microlensing Results from 5.7 Years of Large Magellanic Cloud Observations". The Astrophysical Journal. 542 (1): 281–307. arXiv: astro-ph/0001272 . Bibcode:2000ApJ...542..281A. doi:10.1086/309512. S2CID   15077430.
  7. Alcock, C (20 September 2000). "Binary Microlensing Events from the MACHO Project". The Astrophysical Journal. 541 (1): 270–297. arXiv: astro-ph/9907369 . Bibcode:2000ApJ...541..270A. doi:10.1086/309393. S2CID   119498357.
  8. Peter Schneider (2006). Extragalactic Astronomy and Cosmology. Springer. p. 4, Figure 1.4. ISBN   978-3-540-33174-2.
  9. Theo Koupelis; Karl F Kuhn (2007). In Quest of the Universe . Jones & Bartlett Publishers. p. 492; Figure 16–13. ISBN   978-0-7637-4387-1. Milky Way rotation curve.
  10. Mark H. Jones; Robert J. Lambourne; David John Adams (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 21; Figure 1.13. ISBN   978-0-521-54623-2.
  11. Bosma, A. (1978), Phy. D. Thesis, Univ. of Groningen
  12. Freeman, K.C. (1970). "On the disks of spiral and S0 galaxies". Astrophys. J. 160: 881. Bibcode:1970ApJ...160..811F. doi: 10.1086/150474 .
  13. Rubin, V. C.; Ford, W. K.; Thonnard, N. (1980). "Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)". Astrophys. J. 238: 471. Bibcode:1980ApJ...238..471R. doi: 10.1086/158003 .
  14. Bregman, K. (1987), Ph. Thesis, Univ. Groningen
  15. Broeils, A. H. (1992). "The mass distribution of the dwarf spiral NGC 1560". Astron. Astrophys. J. 256: 19. Bibcode:1992A&A...256...19B.
  16. V Springel; SDM White; A Jenkins; CS Frenk; N Yoshida; L Gao; J Navarro; R Thacker; D Croton; J Helly; JA Peacock; S Cole; P Thomas; H Couchman; A Evrard; J Colberg; F Pearce (2005). "Simulations of the formation, evolution and clustering of galaxies and quasars". Nature. 435 (7042): 629–636. arXiv: astro-ph/0504097 . Bibcode:2005Natur.435..629S. doi:10.1038/nature03597. PMID   15931216. S2CID   4383030.
  17. Gunn, J. and Gott, J.R. (1972), Astrophys. J. 176.1
  18. 1 2 Mo, Houjun; van den Bosch, Frank; White, Simon (2010). Galaxy Formation and Evolution. Cambridge University Press. ISBN   978-0-521-85793-2.
  19. Navarro, J. et al. (1997), A Universal Density Profile from Hierarchical Clustering
  20. Avila-Reese, V., Firmani, C. and Hernandez, X. (1998), Astrophys. J. 505, 37.
  21. Merritt, D. et al. (2006), Empirical Models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models
  22. McGaugh, S. et al. (2007), The Rotation Velocity Attributable to Dark Matter at Intermediate Radii in Disk Galaxies
  23. Davis, M., Efstathiou, G., Frenk, C. S., White, S. D. M. (1985), ApJ. 292, 371
  24. Franx, M., Illingworth, G., de Zeeuw, T. (1991), ApJ., 383, 112
  25. Klypin, A., Gotlöber, S., Kravtsov, A. V., Khokhlov, A. M. (1999), ApJ., 516,530
  26. Diemand, J., Kuhlen, M., Madau, P. (2007), ApJ, 667, 859
  27. Springel, V.; Wang, J.; Vogelsberger, M.; Ludlow, A.; Jenkins, A.; Helmi, A.; Navarro, J. F.; Frenk, C. S.; White, S. D. M. (2008). "The Aquarius Project: the subhaloes of galactic haloes". MNRAS. 391 (4): 1685–1711. arXiv: 0809.0898 . Bibcode:2008MNRAS.391.1685S. doi: 10.1111/j.1365-2966.2008.14066.x . S2CID   119289331.
  28. Hoyle, F. (1949), Problems of Cosmical Aerodynamics, Central Air Documents Office, Dayton.
  29. Efstathiou, G., Jones, B. J. T. (1979), MNRAS, 186, 133
  30. Maccio, A. V., Dutton, A. A., van den Bosch, F. C., et al. (2007), MNRAS, 378, 55
  31. Gao, L., White, S. D. M. (2007), MNRAS, 377, L5
  32. Battaglia, Giuseppina; Helmi, Amina; Morrison, Heather; Harding, Paul; Olszewski, Edward W.; Mateo, Mario; Freeman, Kenneth C.; Norris, John; Shectman, Stephen A. (2005). "The radial velocity dispersion profile of the Galactic halo: constraining the density profile of the dark halo of the Milky Way". Monthly Notices of the Royal Astronomical Society. 364 (2): 433–442. arXiv: astro-ph/0506102 . Bibcode:2005MNRAS.364..433B. doi: 10.1111/j.1365-2966.2005.09367.x . S2CID   15562509.
  33. 1 2 Kafle, P.R.; Sharma, S.; Lewis, G.F.; Bland-Hawthorn, J. (2014). "On the Shoulders of Giants: Properties of the Stellar Halo and the Milky Way Mass Distribution". The Astrophysical Journal. 794 (1): 17. arXiv: 1408.1787 . Bibcode:2014ApJ...794...59K. doi:10.1088/0004-637X/794/1/59. S2CID   119040135.

Further reading