# Thermodynamic Constraints on the Non-Baryonic Dark Matter Gas Composing Galactic Halos

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Classical Thermodynamics with and without Radiative Transfer

_{BB}(ν,T), where ν is frequency), allowing for the material property of emissivity (Section 2.3). In 1893, Wein showed that the wavelength (λ = c/ν) of the blackbody peak is inversely proportional to T:

^{−1}was experimentally determined, c = lightspeed, k

_{B}is Boltzman’s constant, h = Planck’s constant., and w

_{3}is near 2.821439 (see below). Wein’s derivation and spectral measurements ended the debate regarding the equivalence of heat and light [39].

_{3}. From Marr and Wilkin [45], the average frequency, <ν> is constrained by:

_{3}.

_{j}, and the sum of these integers (N = ΣNj) should also be an integer. In fact, Planck’s derivation of I

_{BB}in 1900 did not invoke energy quantization [46]. Planck portrayed his 1901 formulation, which added constants, as a mathematical construction [47]. Quantization of photons is of course associated with processes such as transitions between electronic or vibrational states, but from the above does not describe the thermal state of a body. Ref. [33] provides details and also discusses the connection of blackbody emissions with thermodynamic laws.

#### 2.2. Kinetic Theories of Gas, Elastic and Inelastic

_{int}) is negligible compared to the lifetime (τ

_{c}) between collisions; (5) collisions are perfectly elastic.

#### 2.2.1. Gas Temperature Depends on Kinetic Energy

^{2}. Hence, the VT for this bound state can be expressed as 2 <K.E.> = sF, where F is the force governing the translational motions and <K.E.> is the average kinetic energy specifically of these translations. This is true for indivisible particles or for monatomic gas atoms, which lack internal molecular rotations or vibrations. The force need not be conservative, since isothermal conditions require constant energy inside the sphere. Equating pressure (P) with F divided by area introduces volume (V) into the VT:

_{gc}is the gas constant. Combining the above, while recognizing that all three translations involve the same average velocity u, yields:

_{gc}. However, average velocities and temperature should be linked by some proportionality constant.

#### 2.2.2. Inelastic Collisions and Blackbody Emissions

_{B}T is about 90% of the K.E. of two atoms colliding (3k

_{B}T). This proportion seems large, but gas atoms are far apart relative to their size and collisions involving any given atom are infrequent. Importantly, for a gas to be isothermal, it must have as much thermal energy coming in (the fire in Figure 2b) as leaving (the photons in Figure 2b). In summary, with particles being small compared to the volume of the gas body, collisions are infrequent, so the loss during all collisions at any given time is large per collision, but small compared to the total energy available. Flux differs fundamentally from energy.

#### 2.2.3. Cross Sections, Interactions during Collisions, and Transport Properties

_{heat}= thermal diffusivity, D

_{mass}= mass diffusivity, and υ = kinematic viscosity) requires estimating both interaction times and cross sections during collisions. Classically, only head-on collisions as in Figure 2 were considered, where cross sections were equated to 2× particle area [32]. Classic EKTG predicts equal values for all three transport properties, which is not correct. Glancing collisions can also occur and will affect viscous drag, but this cross section only involves the area of one particle. Assuming that interaction times are the same in both types of collisions [33] leads to relative sizes for the transport properties in IKTG as being:

^{1.5}(Figure 3b). This discrepancy and both diffusivities having a 1/m dependence, as in IKTG [33] (p. 176; Figure 5.12), point to problems accompanying the use of mean free paths and u between collisions to determine τ.

#### 2.3. Interactions of Matter with Light (Heat)

#### 2.4. Simplifications Associated with Astronomical Scales and Monatomic Baryonic Gases

_{2}exists, which has more complicated energetics associated with its molecular vibrations and rotations. Importantly, the proportionality between velocity and temperature was originally derived by considering gas to be made of particles without any prescribed properties. Real gas behaves in this manner because translational K.E. is the bulk of the thermal energy. On this basis, we approximate baryon gas in the IGM to consist of H or He atoms while considering NBDM in halos as being indivisible particles. Motions of the electrons around H or He nuclei are not considered for simplicity and because this behavior is not part of classical macroscopic models. The kinetic energy of these three gases being predominantly translational permits the use of Equation (5).

## 3. Thermodynamic Behavior of Non-Baryonic Dark Matter Halos

#### 3.1. NBDM Halos Are a Type of Gas

_{H}is the mass of the halo inside r = s and m is the mass of baryons rotating at that equatorial radius. Halos occupy a volume and thus also have a density.

^{−4}to 10

^{6}atoms per cubic centimeter for its various components, which is much more rarified than the density associated with dispersing our Sun to Neptune’s orbit, as the latter would yield a gas of 10

^{12}H atoms per cm

^{3}or 10

^{−8}kg per m

^{3}. A cloud equivalent to the dispersed Sun would also be denser than astronomical environments such as molecular cloud cores, yet would be 10

^{8}times more rarified than Earth’s atmosphere. Thus, NBDM halos are a very dilute gas.

#### 3.1.1. NBDM Gas Contains Particles

#### 3.1.2. Motions and Forces Inside an NBDM Halo

#### 3.2. Gravitation and Collisions

#### 3.2.1. Gravitational Attraction of NBDM to Baryons Requires Collisions

^{3}[34]), the scale length of this region (~10

^{17}m for a small galaxy [63]) requires that the number of such collisions around a spiral galaxy be large. Since the mean free path between collisions is roughly particle separation distances (see [32] or Section 2.2), then roughly 10

^{13}collisions should occur along a line-of-sight. Implications of baryon–NBDM particle collisions on halo energetics and stability are discussed below.

#### 3.2.2. Limitations on the Size of NBDM Particles in View of Their Collisions

^{−10}nm) being four orders of magnitude smaller than the size of the H nucleus (~10

^{−6}nm) supports approximating the particles as point masses. Glancing collisions being possible suggests that point masses are required to minimize and possibly to avoid any interaction of NBDM with baryons.

#### 3.3. Thermal Consequences of Pure NBDM Halos Not Interacting with Photons

#### 3.3.1. Implications on Halo Gas Volume

#### 3.3.2. Implications on Galaxy Structure

#### 3.3.3. Can the Particles Be Very, Very Tiny Rather Than Being Point Masses?

^{−10}m wavelengths were recently explored [5]). Irrespective of the exact volume of a photon, it is thus possible that the volume occupied by NBDM particles would not be zero, but would be tiny. A high density relative to baryonic gas is still expected, and consequently NBDM should be preferentially located towards the center of a galaxy.

#### 3.3.4. Summary on the Thermodynamics of a Pure NBDM Gas

#### 3.4. Thermal Consequences of Baryons Colliding with NBDM Particles

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Rubin, V.C.; Ford, W.K. Rotation of the Andromeda nebula from a spectroscopic survey of emission regions. Astrophys. J.
**1970**, 159, 379. [Google Scholar] [CrossRef] - Faber, S.M.; Gallagher, J.S. Masses and mass-to-light ratios of galaxies. Ann. Rev. Astron. Astrophys.
**1979**, 17, 135–187. [Google Scholar] [CrossRef] - Trimble, V. Existence and nature of dark matter in the universe. Ann. Rev. Astron. Astrophys.
**1987**, 25, 425–472. [Google Scholar] [CrossRef] - Geist, K. Wimps and machos. In Encyclopedia of Astronomy and Astrophysics; Murdin, P., Ed.; Institute of Physics Publishing: Bristol, UK, 2006. [Google Scholar] [CrossRef]
- Ackermann, M.; Albert, A.; Anderson, B.; Baldini, L.; Ballet, J.; Barbiellini, G.; Bastieri, D.; Bechtol, K.; Bellazzini, R.; Bissaldi, E.; et al. Dark matter constraints from observations of 25 Milky Way satellite galaxies with the Fermi Large Area Telescope. Phys. Rev. D
**2014**, 89, 042001. [Google Scholar] [CrossRef][Green Version] - De Vega, H.J.; Salucci, P.; Sanchez, N.G. Observational rotation curves and density profiles versus the Thomas-Fermi galaxy structure theory. Mon. Not. R. Astron. Soc.
**2014**, 442, 2717–2727. [Google Scholar] [CrossRef][Green Version] - Gaitskill, R.J. Direct detection of dark matter. Annu. Rev. Nucl. Part. Sci.
**2004**, 54, 315–359. [Google Scholar] [CrossRef][Green Version] - Dark Matter. Available online: https://en.wikipedia.org/wiki/Dark_matter (accessed on 26 April 2020).
- Ade, P.A.R.; Aghanim, N.; Armitage-Caplan, C.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; et al. Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys.
**2014**, 571, A16. [Google Scholar] - Feng, J.L. Dark matter candidates from particle physics and methods of detection. Annu. Rev. Astron. Astrophys.
**2010**, 48, 495–545. [Google Scholar] [CrossRef][Green Version] - Giagu, S. WIMP dark matter searches with the ATLAS detector at the LHC. Front. Phys.
**2019**, 7. [Google Scholar] [CrossRef] - Nagano, K.; Fujita, T.; Michimura, Y.; Obata, I. Axion dark matter search with interferometric gravitational wave detectors. Phys. Rev. Lett.
**2019**, 123. [Google Scholar] [CrossRef][Green Version] - Garcia-Ruiz, I.; Sancisi, R.; Kuijken, K. Neutral hydrogen and optical observations of edge-on galaxies: Hunting for warps. Astron. Astrophys.
**2002**, 394, 769–789. [Google Scholar] [CrossRef][Green Version] - Wiegert, T.; Irwin, J.; Miskolczi, A.; Schmidt, P.; Carolina Mora, S.; Damas-Segovia, A.; Stein, Y.; English, J.; Rand, R.J.; Santistevan, I. CHANG-ES IV. Radio continuum emission of 35 edge-on galaxies observed with the Karl G. Jansky very large array in D configuration—Data release 1. Astronom. J.
**2015**, 150, 81. [Google Scholar] [CrossRef] - CHANG-ES Continuum Halos in Nearby Galaxies- and EVLA Survey. Available online: http://www.queensu.ca/changes (accessed on 26 January 2020).
- Burbidge, G. On the masses and relative velocities of galaxies. Astrophys. J.
**1975**, 196, L7–L10. [Google Scholar] [CrossRef] - Bottema, R.; Pestaña, J.L.G. The distribution of dark and luminous matter inferred from extended rotation curves. Mon. Not. R. Astron. Soc.
**2015**, 448, 2566–2593. [Google Scholar] [CrossRef][Green Version] - Hofmeister, A.M.; Criss, R.E. Debated Models for Galactic Rotation Curves: A Review and Mathematical Assessment. Galaxies
**2020**, 8, 47. [Google Scholar] [CrossRef] - Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J.
**1983**, 270, 365–370. [Google Scholar] [CrossRef] - Brownstein, J.R.; Moffat, J.W. Galaxy rotation curves without nonbaryonic dark matter. Astrophys. J.
**2006**, 636, 721–741. [Google Scholar] [CrossRef][Green Version] - Lin, H.-N.; Li, M.-H.; Li, X.; Chang, Z. Galaxy rotation curves in the Grumiller’s modified gravity. Mon. Not. R. Astron. Soc.
**2013**, 430, 450–458. [Google Scholar] [CrossRef] - Feng, J.Q.; Gallo, C.F. Mass distribution in rotating thin-disk galaxies according to Newtonian dynamics. Galaxies
**2014**, 2, 199–222. [Google Scholar] [CrossRef] - Pavlovich, K.; Pavlovich, A.; Sipols, A. Newtonian explanation of galaxy rotation curves based on distribution of baryonic matter. arXiv
**2014**, arXiv:1406.2401P. [Google Scholar] - Marr, J.H. Galaxy rotation curves with lognormal density distribution. Mon. Not. R. Astron. Soc.
**2015**, 448, 3229. [Google Scholar] [CrossRef][Green Version] - McGaugh, S.S. A tale of two paradigms, the mutual incommensurability of LCDM and MOND. Can. J. Phys.
**2015**, 93, 250–259. [Google Scholar] [CrossRef][Green Version] - Hofmeister, A.M.; Criss, R.E. The physics of galactic spin. Can. J. Phys.
**2017**, 95, 156–166. [Google Scholar] [CrossRef][Green Version] - Hofmeister, A.M.; Criss, R.E. Implications of Geometry and the Theorem of Gauss on Newtonian Gravitational Systems and a Caveat Regarding Poisson’s Equation. Galaxies
**2017**, 5, 89. [Google Scholar] [CrossRef][Green Version] - De Swart, J.G.; Bertone, G.; van Dongen, J. How dark matter came to matter. Nat. Astron.
**2017**, 1, 0059. [Google Scholar] [CrossRef][Green Version] - Suleiman, R. A Model of Dark Matter and Dark Energy Based on Relativizing Newton’s Physics. World J. Condens. Matter Phys.
**2018**, 8, 130–155. [Google Scholar] [CrossRef][Green Version] - Zemansky, M.W.; Dittman, R.H. Heat and Thermodynamics, 6th ed.; McGraw-Hill: New York, NY, USA, 1981. [Google Scholar]
- Tolman, R.C. Relativity, Thermodynmaics and Cosmology; Oxford University Press: Oxford, UK, 1934. [Google Scholar]
- Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill Book Company: St. Louis, MO, USA, 1965. [Google Scholar]
- Hofmeister, A.M. Measurements, Mechanisms, and Models of Heat Transport; Elsevier: Amsterdam, The Netherlands, 2019; Chapters 1, 2, 5 and 8. [Google Scholar]
- Fang, T.; Buote, D.A.; Humphrey, P.J.; Canizares, C.R.; Zappacosta, L.; Maiolino, R.; Tagliaferri, G.; Gastaldello, F. Confirmation of X-ray absorption by warm-hot intergalactic medium in the sculptor wall. Astrophys. J.
**2010**, 714, 1715–1724. [Google Scholar] [CrossRef] - Sofue, Y.; Rubin, V.C. Rotation curves of spiral galaxies. Ann. Rev. Astron. Astrophys.
**2001**, 39, 137–174. [Google Scholar] [CrossRef][Green Version] - Truesdell, C. The Tragicomical History of Thermodynamics; Springer: New York, NY, USA, 1980. [Google Scholar]
- Fegley, B., Jr. Practical Chemical Thermodynamics for Geoscientists; Academic Press/Elsevier: Waltham, MA, USA, 2015. [Google Scholar]
- Norton, J.D. The impossible process: Thermodynamic reversibility. Stud. Hist. Philos. Mod. Phys.
**2016**, 55, 43–61. [Google Scholar] [CrossRef][Green Version] - Barr, E.S. Historical survey of the early development of the infrared spectral region. Am. J. Phys.
**1960**, 28, 42–54. [Google Scholar] [CrossRef] - Purrington, R.D. Physics in the Nineteenth Century; Rutgers University Press: New Brunswick, NJ, USA, 1997. [Google Scholar]
- McGucken, W. Nineteenth-Century Spectroscopy; The Johns Hopkins Press: Baltimore, MD, USA; London, UK, 1969. [Google Scholar]
- Pippard, A.B. The Elements of Classical Thermodynamics; Cambridge University Press: London, UK, 1974. [Google Scholar]
- Williams, B.W. A specific mathematical form for Wien’s displacement law as νmax/T = constant. J. Chem. Educ.
**2014**, 91, 623. [Google Scholar] [CrossRef] - Valluri, S.R.; Corless, R.M.; Jeffrey, D.J. Some applications of the Lambert W function to physics. Can. J. Phys.
**2000**, 78, 823–831. [Google Scholar] - Marr, J.M.; Wilkin, F.P. A better presentation of Planck’s radiation law. Am. J. Phys.
**2012**, 80, 339–405. [Google Scholar] [CrossRef][Green Version] - Kangro, H. Early History of Planck’s Radiation Law; Taylor and Francis: London, UK, 1976. [Google Scholar]
- Kragh, H. Max Planck: The reluctant revolutionary. Phys. World
**2000**, 13, 31–35. [Google Scholar] [CrossRef] - Bergin, E.A.; Tafalla, M. Cold dark clouds: The initial conditions for star formation. Ann. Rev. Astron. Astrophys.
**2007**, 45, 339–396. [Google Scholar] [CrossRef][Green Version] - Trusler, J.P.M. Kinetic Theory of Gases. Available online: http://www.thermopedia.com/content/907/ (accessed on 9 September 2018).
- Brouard, M.; Chadwick, H.; Gordon, S.D.S.; Hornung, B.; Nichols, B.; Aoiz, F.J. Stereodynamics in NO(X)1+Ar inelastic collisions. J. Chem. Phys.
**2016**, 144, 224301. [Google Scholar] [CrossRef] [PubMed] - Clausius, R. On a mechanical theorem applicable to heat. Phil. Mag.
**1870**, 40, 122–127. [Google Scholar] [CrossRef] - Hofmeister, A.M.; Criss, R.E. Spatial and symmetry constraints as the basis of the virial theorem and astrophysical implications. Can. J. Phys.
**2016**, 94, 380–388. [Google Scholar] [CrossRef] - Berberan-Santos, M.N.; Bodunov, E.N.; Polliani, L. The van der Waals equation: Analytical and approximate solutions. J. Math. Chem.
**2008**, 43, 1437–1457. [Google Scholar] [CrossRef][Green Version] - Kestin, J.; Knierrim, K.; Mason, E.A.; Najafi, B.; Ro, S.T.; Waldman, M. Equilibrium and transport properties of the noble gases and their mixtures at low density. J. Phys. Chem. Ref. Data
**1984**, 13, 229–303. [Google Scholar] [CrossRef] - Suárez-Iglesias, O.; Medina, I.; Sanz, M.; Pizarro, C.; Bueno, J.L. Self-diffusion in molecular fluids and noble gases: Available data. J. Chem. Eng. Data
**2015**, 60, 2757–2817. [Google Scholar] [CrossRef] - Bates, J.B. Infrared emission spectroscopy. Fourier Transform. IR Spect.
**1978**, 1, 99–142. [Google Scholar] - Kellogg, O.D. Foundations of Potential Theory; Dover Publications: New York, NY, USA, 1953. [Google Scholar]
- Hofmeister, A.M.; Criss, R.E.; Criss, E.M. Verified solutions for the gravitational attraction to an oblate spheroid: Implications for planet mass and satellite orbits. Planet. Space Sci.
**2018**, 152, 68–81. [Google Scholar] [CrossRef] - Brewster, M.Q. Thermal Radiative Transfer and Properties; John Wiley & Sons: New York, NY, USA, 1992. [Google Scholar]
- Siegel, R.; Howell, J.R. Thermal Radiation Heat Transfer; McGraw-Hill: New York, NY, USA, 1972. [Google Scholar]
- Wiegert, T.; English, J. Kinematic classification of non-interacting spiral galaxies. New Astron.
**2014**, 26, 40–61. [Google Scholar] [CrossRef][Green Version] - Ferrière, K. The interstellar environment of our galaxy. Rev. Mod. Phys.
**2011**, 73, 1031. [Google Scholar] [CrossRef] - NASA/IPAC Extragalactic Database. Available online: https://ned.ipac.caltech.edu/ (accessed on 10 January 2020).
- Müller, I. Entropy: A subtle concept in thermodynamics. In Entropy; Greven, A., Keller, G., Warnecke, G., Eds.; Princeton University Press: Princeton, NJ, USA, 2003; pp. 17–36. [Google Scholar]
- Electromagnetic Spectrum. Available online: https://en.wikipedia.org/wiki/Electromagnetic_spectrum (accessed on 30 April 2020).

**Figure 1.**Postulated and measured distributions of baryonic and non-baryonic types of matter in spiral galaxies: (

**a**) schematic of a spiral galaxy with a purported halo of non-baryonic dark matter (NBDM), as used in Newtonian Orbit Models (NOMs). Components in this perspective view are labeled. For simplicity, the disk represents stars along with detectable H atoms. Other figure parts show cross sections; (

**b**) contours of H atoms surrounding an edge-on spiral galaxy (NGC 4010, also known as UCC 6964) detected using a 21-cm line. The black lens shape shows the extent of the visibly detected galaxy. Panel (b) is modified after Figure 13 (panel 11) of Reference [13], with permission. This distribution of H atoms is analogous to an atmosphere around a star or planet; (

**c**,

**d**) radio contours of two additional edge-on galaxies in the C- and L-bands, with superimposed optical images, modified after Wigert et al. [14]. Both panels are publically available from CHANG-ES [15].

**Figure 2.**Schematics of inelastic collisions in a gas of finite-size atoms: (

**a**) progression of an inelastic collision (viewed in the center of mass reference frame) with time. Black = nuclei. Grey = shells of electrons, forming a shielding cloud, which deforms during the collision, thereby consuming some of the energy of the collision. Squiggle arrow = thermal photon, emitted as the electron cloud adjusts to the reduction in force. Hence, each collision results in a tiny reduction in the thermal energy of the gas; (

**b**) steady-state conditions in a body of gas. Applied heat (the fire) establishes T and a distribution of molecular velocities. The velocity distribution produces a thermal photon gas with its energy distribution linked to the velocities. Thermal photons escape, providing blackbody emissions, with this same distribution. To maintain constant T of the substance, heat must be continually supplied. Thus, isothermal conditions require that the heat input = heat output with a negligible thermal gradient. Reprinted from Figure 5.2 a,d of Reference [33], with permission from Elsevier.

**Figure 3.**Data on transport properties. Circles and solid line = thermal diffusivity. Squares and dotted line = mass self-diffusivity. Diamond and dashed line = kinematic viscosity. Least squares fits are shown: (

**a**) comparison of physical properties at ambient condition to EKTG of hard spheres (Equation (7)). From data compiled in [33] (Table 5.2). For clarity, fits are shown for noble gas only; (

**b**) dependence of measured transport properties on temperature. Data on He from Kestin et al. [54]. Data on H

_{2}from the compilation of Suárez-Iglesias et al. [55] and various sources, listed in [33] (Table 5.2). For clarity, fits for D

_{mass}are omitted.

**Figure 4.**Newtonian attraction to a highly flattened constant density oblate spheroid (black shape) with a 1:10 axial ratio similar to those of spiral galaxies. One quadrant is shown: (

**a**) contour plot of the Newtonian gravitational potential exterior to the oblate, shown in units of GMm/a; (

**b**) lines of exterior force. Reprinted from Figure 7e,f in Reference [58], with permission from Elsevier.

**Figure 5.**Schematics of collisions in a gas of NBDM: (

**a**) section of the halo that is spinning (grey arrow) around the galaxy. The atom (spot) is immense compared to the particles (thin black arrows), offering a large cross section; (

**b**) collision of particle with an H atom. The nucleus is the draw, which has >10

^{4}× more mass. Distortions of the nucleus or of the electron cloud during the collision result in emission of a photon as the particle and atom each rebound (dotted arrows).

**Figure 6.**Schematic showing thermodynamic consequences of the postulated behavior of NBDM in galactic halos. Black arrows trace implications, where subhorizontal lines indicate considerations being applied simultaneously and heavy arrows emphasize key deductions. White boxes list postulated behavior of NBDM in halos; pale ovals indicate consequences for the nature of NBDM; various grey shapes indicate additional consequences; very dark grey boxes indicate violations of thermodynamic laws.

Postulated Properties of Non-Baryons | Key Consequences ^{1} |
---|---|

Gravitationally interacts with ordinary matter | NBDM has mass; collisions must occur |

Negligibly interacts with photons | Nearly point masses; elastic collisions |

Distributed in a large halo around galaxies | A dilute assembly of particles (a gas) |

^{1}Details are in Section 3.

Law No. | Classical Statement ^{1} |
---|---|

0th | Equilibrium between systems is communicable |

1st | Energy is conserved if heat is accounted for |

2nd | Flow of heat from a colder to a hotter body cannot occur as a sole result |

3rd | Absolute zero is unattainable by processes involving finite steps |

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Hofmeister, A.M. Thermodynamic Constraints on the Non-Baryonic Dark Matter Gas Composing Galactic Halos. *Galaxies* **2020**, *8*, 77.
https://doi.org/10.3390/galaxies8040077

**AMA Style**

Hofmeister AM. Thermodynamic Constraints on the Non-Baryonic Dark Matter Gas Composing Galactic Halos. *Galaxies*. 2020; 8(4):77.
https://doi.org/10.3390/galaxies8040077

**Chicago/Turabian Style**

Hofmeister, Anne M. 2020. "Thermodynamic Constraints on the Non-Baryonic Dark Matter Gas Composing Galactic Halos" *Galaxies* 8, no. 4: 77.
https://doi.org/10.3390/galaxies8040077