# Rotating Disk Galaxies without Dark Matter Based on Scientific Reasoning

## Abstract

**:**

## 1. Introduction

^{5}to 10

^{12}stars distributed in a flattened, roughly axisymmetric structure, rotating around a common axis in nearly circular orbits. Besides stars, the galactic “disk” is also known to contain an interstellar medium such as gases (mostly atomic and molecular hydrogen) as well as relatively small solid “dust particles”. The general behavior of stellar systems, including disk galaxies, has been believed to follow Newton’s laws of motion and Newton’s law of universal gravitation [1].

## 2. Astronomical Measurements

^{16}m).

^{15}m, the distance for light to travel in 1 year in vacuum). For example, the nearest star to our solar system is about 4.22 light-years away. The distance between the Sun and Earth is ~1.5 × 10

^{11}m, taking about 8.3 minutes for light from the Sun to reach us. Even our next-door neighbor, the Moon, is about 3.8 × 10

^{8}m away, a lot farther than most people would think. Only objects within our own solar system can our present spacecraft reach. For the most part, the cosmos is out of our reach, except light that travels throughout the universe can bring information about distant objects to us on the Earth.

## 3. Mass Determined by Newtonian Dynamics—“Gravitational Mass”

^{11}m) and the Earth’s orbital period, p ((≈ 3.15 × 10

^{7}s, i.e., 1 year) [14,15]. In other words, having centripetal acceleration of the Earth:

^{−11}m

^{3}kg

^{−1}s

^{−2}) is the gravitational constant and M

_{sun}the Sun’s mass, yields the value of M

_{sun}≈ 2.0 × 10

^{30}kg (= 1 solar mass M

_{⊙}). Here V denotes the Earth’s (or the test object’s) orbital velocity.

_{1}m

_{2}/ (m

_{1}+ m

_{2}) is orbiting about a fixed mass M = m

_{1}+ m

_{2}at a distance a = a

_{1}+ a

_{2}where the subscripts “1” and “2” denotes the masses and radii of the star “1” and star “2” [16]. In fact, the value of M

_{sun}determined from an equation of (1) = (2) is actually reduced from the solution for two-body Kepler’s problem to an extreme case when m

_{1}>> m

_{2}such that m $\to $ 0, a

_{1}$\to $ 0, and M $\to $ m

_{1}(= M

_{sun}or 1.0 M

_{⊙}). Thus, the value of M (= m

_{1}+ m

_{2}) in a binary star systems can be determined. With the known M, the values of m

_{1}and m

_{2}can be determined from the relationship of m

_{2}/ m

_{1}= a

_{1}/ a

_{2}= v

_{1}/ v

_{2}(where v

_{1}and v

_{2}are the orbital velocities of the two stars) derived from the two-body problem. In reality, the distances a

_{1}and a

_{2}are not easy to determine accurately. Instead, the values of v

_{1}and v

_{2}can be measured much more reliably based on the measured Doppler shifts, especially for the so-called “eclipsing binaries” with their orbit planes lying very close in the line of sight [14,15].

^{5}–10

^{12}) of stars, with an interstellar medium of gas and cosmic dust, among others, are distributed in an extensive space such as a thin disk of radius about 10 kpc (or 3.09 × 10

^{20}m). Simply adding the “point mass” fields of such a distributed stellar system with ~10

^{11}stars is impractical to compute the gravitational field in a typical galaxy, so it becomes a common practice, for most purposes, to model the gravitational field or potential “by smoothing the mass density in stars on a scale that is small compared to the size of the galaxy, but large compared to the mean distance between stars” [1], i.e., to treat the distributed mass system as a continuum, as a reasonable approximation.

_{g}denotes the galactic radius (or the galactocentric distance of the galaxy edge, taken as the cut-off radius in rotation curve beyond which the detectable signal diminishes), M

_{g}the total mass of the galaxy, and V

_{0}the characteristic rotation velocity as a representative value of the flat part of rotation curve. All the variables in (3) and (4) are made dimensionless by measuring length r, h in units of the galactic radius R

_{g}, mass density ρ in units of M

_{g}/ R

_{g}

^{3}, and rotation velocity V in units of V

_{0}, with the disk thickness h assumed to be constant.

_{d}), i.e., the computed ρ verses r in a log-linear plots appear to be nearly straight lines with negative slopes, for the most part when the abruptly varying ends at r = 0 and 1 are trimmed out [18,19,20,21]. Hence, the “gravitational mass”—determined from rotation curves--distributions in disk galaxies qualitatively agree with the measured radial distributions of surface brightness for a large number of disk galaxies [24,25,26], but the disk radial scale length, R

_{d}, determined from rotation curve based thin-disk model appears to be larger than that from fitting the brightness data (e.g., 4.5 kpc versus 2.5 kpc for Milky Way [20]). A straightforward interpretation of such a discrepancy would be an indication of increasing mass-to-light ratio with galactocentric distance, namely the (baryonic) matter becomes less luminous in regions further away from the galactic center. This is consistent with typical edge-on views of disk galaxies that often revealing a dark edge against a bright background central bulge (cf. the image of NGC 891 in Figure 1).

## 4. Mass Determined by Mass-to-Light Ratio—“Luminous Mass”

^{−8}W m

^{−2}K

^{−4}is the Stefan-Boltzmann constant for black-body radiation. However, it should be kept in mind that the emissions of galaxies may not exactly follow that of a blackbody with emissivity equal to 1, in the presence of dust and gases.

## 5. Galaxy Rotation Curves Described without Dark Matter

_{g}can be calculated from the predicted value of A as M

_{g}= V

_{0}

^{2}R

_{g}/ (G A). For example, the Milky Way total mass is determined as 1.41 × 10

^{11}M

_{⊙}from predicted A = 1.6365 with V

_{0}= 220 km/s and R

_{g}= 20.55 kpc [21], very close to the Milky Way star counts of about 100 billion [1,27]. The numerical approach for solving (5) and (7) can also account for the effect of a spherical bulge at galactic center with slight mathematical manipulation [21], with results illustrated in Figure 3 for the Milky Way. It has been shown that even for a bulge of mass as large as 7.57 × 10

^{10}M

_{⊙}, the Milky Way total mass would only change to 1.52 × 10

^{11}M

_{⊙}(i.e., a less than 8% increase [21]). The total mass of the Andromeda (NGC 224) can also be calculated as 2.76 × 10

^{11}M

_{⊙}from A = 1.6450 with V

_{0}= 250 km/s and R

_{g}= 31.25 kpc [21], about twice that of the Milky Way, as commonly being anticipated.

_{0}is taken as a representative value of the flat part of rotation curve [18,19,20,21]. For Mestel’s disk of constant rotation velocity with available analytical solution, the value of A is determined as π/2 = 1.5707063 [20]. A proportionality constant from a lognormal density distribution model results of 38 galaxies for M

_{g}versus V

_{max}

^{2}R

_{g}[28] indicates a value of A equal to 1.57063 if V

_{max}and V

_{0}happen to be the same as with Mestel’s disk. Comparing with the scalar virial theorem, M

_{g}= <V

^{2}> r

_{g}/ G with <V

^{2}> and r

_{g}denoting the mean-square speed of the system’s stars and the gravitational radius [1], this suggests that <V

^{2}> r

_{g}= V

_{0}

^{2}R

_{g}/ A, i.e., r

_{g}~ 0.59 R

_{g}if <V

^{2}> = V

_{0}

^{2}is assumed. Given the fact that <V

^{2}> also includes the velocity dispersion in addition to the rotational orbital part, it seems that r

_{g}= 0.5 R

_{g}(corresponding to <V

^{2}> ~ 1.18 V

_{0}

^{2}) could be a reasonable approximation for rough estimate of the total mass in a disk galaxy from the measured rotational velocity based on the virial theorem.

^{−3}in the galactic plane [35,36], but apparently without an independent method to validate. There are also hydrogen molecules (as molecular hydrogen) found in molecular clouds and in the Interstellar Medium (ISM), which appear to be literally dark when cold as majority of them are (e.g., around 10-20K [37,38]). The amount of “dark” hydrogen molecules could only be estimated by assuming a constant ratio from the luminosity of carbon monoxide, with unknown uncertainties, of course. Furthermore, condensed baryonic matter in the form of dust and debris is expected to have minimal effect on optical extinction and can easily avoid detection due to the small optical cross section [39]. The presence of those optically undetectable components (with Μ / L approaching to infinity) makes it certain that there must be more baryonic matter than what can be represented in terms of mass-to-light ratio. Indeed, the baryonic Tully-Fisher relation was shown to be optimally improved when the HI mass is multiplied by a factor of about 3 [40]. Therefore, estimating mass in a galaxy simply based on a constant mass-to-light ratio can be seriously flawed, though convenient. Its usage may provide some order-of-magnitude rough idea about the amount of mass, but should not be considered for serious comparison and verification in scientific analysis. In view of the technical difficulties in detecting astronomical matter and evaluating celestial mass, the existence of invisible baryonic matter that cannot be accounted for by a simple M/L is naturally expected following scientific logic.

^{12}M

_{⊙}from an assumed composition of a nucleus, bulge, disk, and a halo of a virial radius over 200 kpc [42]. Interestingly, the mass within 21.5 kpc (where the Gaia rotation curve terminates) was estimated about 2.1 × 10

^{11}M

_{⊙}[42], quite comparable to 1.52 × 10

^{11}M

_{⊙}or 1.41 × 10

^{11}M

_{⊙}with or without a central bulge as numerically determined from a measured rotation curve up to 20.55 kpc [21].

_{⊙}/pc

^{2}using a pure disk model or ~74 M

_{⊙}/pc

^{2}when a sizable bulge is included in the computation [21]. As a reference, the current textbook value of surface mass density for solar neighborhood is ~49 M

_{⊙}/pc

^{2}based on estimates from observations [1]. In view the fact that an axisymmetric disk model describes a surface mass density only meaningful in terms of averaging over the entire circular ring of radius ~8 kpc, while the local mass density may actually vary significantly along the ring (as shown in the photographic images), shouldn’t we consider the Newtonian dynamic model to be reasonably accurate? Moreover, a surface mass density of 100 M

_{⊙}/pc

^{2}in the Milky Way model [21] corresponds to equivalently ~20 hydrogen atoms or ~10 hydrogen molecules per cm

^{3}for an assumed disk thickness of 200 pc [9], extremely tenuous by the terrestrial standards and well within the reported range of estimated gas density in the Interstellar Medium [1,47]. If the typical density of cold molecular clouds to enable star formation ranges from 10

^{2}to 10

^{6}molecules per cm

^{3}[47], it is not difficult to realize the possible magnitude of variations in mass density just within a ring containing the solar neighborhood.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Photographic images of circular thin-disk galaxies with small, amorphous, centrally located bulge.

**Figure 2.**Definition sketch of the rotating thin-disk galaxy model, where the mass is assumed to distribute axisymmetrically in the circular disk of uniform thickness h with a variable density ρ and rotation velocity V as functions of the radial distance from galactic center r (but independent of the polar angle θ).

**Figure 3.**Profiles of the Milky Way rotation velocity V(r) and mass density ρ(r) for the disk portion and bulge portion as noted with the thick line as a reference from the pure disk model without a bulge (taken from Figure 7 of Ref. [21]). Noteworthy here is that the portion of mass density profile (shown with the thick line, as roughly a combined mass density profile from both disk and bulge) for r in the interval [0.1, 0.9] appears nearly linear in the semi-log plot (when when the abruptly varying ends around r = 0 and 1 are trimmed out), indicating an approximately exponential decline of mass density with galactocentric distance.

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Feng, J.Q. Rotating Disk Galaxies without Dark Matter Based on Scientific Reasoning. *Galaxies* **2020**, *8*, 9.
https://doi.org/10.3390/galaxies8010009

**AMA Style**

Feng JQ. Rotating Disk Galaxies without Dark Matter Based on Scientific Reasoning. *Galaxies*. 2020; 8(1):9.
https://doi.org/10.3390/galaxies8010009

**Chicago/Turabian Style**

Feng, James Q. 2020. "Rotating Disk Galaxies without Dark Matter Based on Scientific Reasoning" *Galaxies* 8, no. 1: 9.
https://doi.org/10.3390/galaxies8010009