Dark Matter Dogma: A Study of 214 Galaxies

Abstract

The aim of this paper is to test the need for non-baryonic dark matter in the context of galactic rotation and the apparent difference between distributions of galactic mass and luminosity. We present a set of rotation curves and 3.6 μm surface brightness profiles for a diverse sample of 214 galaxies. Using rotation curves as the sole input into our Newtonian disk model, we compute non-parametric radial profiles of surface mass density. All profiles exhibit lower density than parametric models with dark halos and provide a superior fit with observed rotation curves. Assuming all dynamical mass is in main-sequence stars, we estimate radial distributions of characteristic star mass implied by the corresponding pairs of density and brightness profiles. We find that for 132 galaxies or 62% of the sample, the relation between density and brightness can be fully explained by a radially declining stellar mass gradient. Such idealized stellar population fitting can also largely address density and brightness distributions of the remaining 82 galaxies, but their periphery shows, on average, 14 M_⊙/pc² difference between total density and light-constrained stellar density. We discuss how this density gap can be interpreted, by considering a low-luminosity baryonic matter, observational uncertainties, and visibility cutoffs for red dwarf populations. Lastly, we report tight correlation between radial density and brightness trends, and the discovered flattening of surface brightness profiles—both being evidence against dark matter. Our findings make non-baryonic dark matter unnecessary in the context of galactic rotation.

1. Introduction

Elevated to the status of an axiomatic assumption, the concept of dark matter has settled firmly in modern astrophysics and cosmology. Galactic mass templates presuppose dark halos, with debates mostly focusing on their features such as “core-cusp problem” [1,2,3] rather than questioning their existence. Dark matter rush burgeons with state-of-the-art experiments [4,5,6,7] and theories on the nature of the alleged enigmatic substance [8,9,10]; but is there an impeccable basis for the excitement? Has empirical evidence of anomalous phenomena been established beyond a reasonable doubt, backed by a wealth of strong data and realistic assumptions? If so, then have alternative simpler explanations been exhausted? What if dark matter is regular baryons obscured by observational limitations and analytical habits? Considerations like these inspired us to write this paper.

Our focus here is on the apparent difference between expected mass distribution and luminosity in galaxies, which is one of the main reasons for the dark matter hypothesis. It has been widely reported that in most galaxies, light intensity declines faster than mass density1 away from the center [11,12,13,14]. Consequently, the implied mass-to-light (M/L) ratio is orders of magnitude higher at the periphery than at the center. There were many attempts to explain M/L gradient with various forms of non-luminous baryonic matter including brown dwarfs [15], black holes [16,17], and fractal cold hydrogen [18,19], to name a few. Yet, these ideas never fully addressed the problem, and dark matter in a form of unknown, non-baryonic substance became a commonly accepted explanation2. Most of its proponents theorize that dark matter accounts for several times [25,26,27,28] more mass than baryonic matter. Despite intensive search however, all attempts to detect non-baryonic dark matter so far failed.

In a 2014 paper [29], we presented a Newtonian thin disk model for deriving radial distributions of mass directly from galactic rotation curves (RC). Applied to a set of 47 galaxies, the model produced mass estimates lower than those in the literature. We have subsequently refined the model, expanded the galaxy sample to 214, and used 3.6 μm luminosity data as an independent constraint to identify differences between total density and light-constrained stellar density. We present our methods, data, and results in the corresponding sections of this paper. Our conclusion is that the need for dark matter was artificially inflated by the dominant analytical approach, and that dark matter is unnecessary to model galactic rotation and to explain the sliding M/L phenomenon.

2. Methods

2.1. General Approach

Our approach to M/L problem is straightforward—we compute radial distributions of surface mass density and compare them to respective surface brightness profiles. There are two aspects of our method, different from the traditional approach, that ensure a fair basis for exploring the M/L problem. First, rather than using popular parametric modular templates to infer density from brightness, we compute non-parametric dynamical density profiles directly and solely from RCs, using a thin disk model. Second, we avoid making apriori assumptions about a stellar M/L profile and speculative dark halo. Instead, we match two independent datasets—the computed density and the observed brightness profiles, deriving implied radial gradients of characteristic star mass that fit both density and light data, serving as a test for dark matter in galaxies. Below we explain the mechanics and discuss the merits of this approach.

2.2. Incumbent Component Models

Conventionally, mass distribution is modeled analytically, treating a galaxy as a set of conspiring standard components inferred from the shape of surface brightness profile. This is an ambiguous procedure [30], which suffers, in addition to reliance on surface brightness data, from two major deficiencies—the assumptions of uniform or rigid stellar M/L and presupposition of spherical dark halos. A combination of these unproven assumptions leads to a wide gap between underestimated baryonic matter and overestimated total mass, lending support for dark matter hypothesis.

2.3. Disk Mass Density Model

Considering the deficiencies of incumbent models, what would be a better method for computing galactic mass distribution? We adopt two key principles: (a) a thin disk is a better geometrical approximation for the overall galactic shape than a sphere; (b) galactic mass distribution must be computed solely from rotation data, using Newtonian gravitation and invariant mathematical solution.

We start with the basic idea shown in Figure 1a, where point A is any location at the zeroth azimuth of an ultrathin disk and point B is any other location in the disk.

The gravitational force $\overrightarrow{F}$ acting on mass at point A due to mass at point B has a centripetal (or centrifugal) component $\overrightarrow{F_C}$ with magnitude F cos φ, where φ is the angle between vectors $\overrightarrow{F}$ and $\overrightarrow{F_C}$. The total centripetal force acting on mass at point A is the sum of all centripetal and centrifugal gravitational pulls due to mass at all other points in the disk. Assuming uniform circular motion of material at point A, and integrating gravitation over all other points in the disk using polar coordinates (r, θ), we can write:

$$\frac{m_A{V}_A^2}{r_A}=G{m}_A{{\displaystyle \iint }}^{}{\mathsf{\Sigma }}_B\left(r_B,{\theta }_B\right)\frac{r_A-r_B\cos {\theta }_B}{{\left(r_A^2-2r_A{r}_B\cos {\theta }_B+r_B^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}_{B}d{\theta }_{B}d{r}_B$$

(1)

where $m_A$ and $V_A$ are the mass and the orbital speed at point A $\left(r_A,0°\right),$ G is the Newtonian universal constant of gravitation; ${\mathsf{\Sigma }}_B$ is the surface mass density at point B $\left(r_B,{\theta }_B\right)$; $\left(r_A-r_B\cos {\theta }_B\right)$/${\left(r_A^2-2r_A{r}_B\cos {\theta }_B+r_B^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ = cos φ; A ≠ B. The differential in polar coordinates includes factor $r_B$.

In an axisymmetric disk, ${\mathsf{\Sigma }}_B$ is independent of angle $\theta$ and varies only radially; we will denote the corresponding radius-specific mass as $m_r$. Canceling out $m_A$ from both sides of Equation (1), moving ${\mathsf{\Sigma }}_B$ outside the inner integral, and approximating integrals numerically, we get:

$$\frac{V_A^2}{r_A}=G{\displaystyle \sum }_r\left(m_r{\displaystyle \sum }_{\theta }\frac{r_A-r\cos \theta }{{\left(r_A^2-2r_{A}r\cos \theta +r^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)$$

(2)

Given a rotation curve V_A(r_A) comprised of N datapoints, Equation (2) can be specified for each of the corresponding N orbits. Assuming that galactic mass is distributed across the same N orbits, we get an N × N linear system with m_r as the only variable (N × 1 column vector, in matrix form). The linear system has a straightforward and unique solution for $m_r$ (see Appendix A for details).

Using this approach, we developed Disk Mass Density Model (DM2) for which details are provided in Appendix A. DM2 takes an observed galactic rotation curve as the sole input and converts it, using the above logic, into a surface mass density profile.

Considering that the output density profile is a non-parametric (the sought distribution of mass is not constrained by any predefined function) and invariant algebraic solution, DM2 ensures exact fits to input RCs (see Figure 1b).

Given a disk rather than a spherical setup, DM2 density estimates are approximately 20–40% lower than those derived from models with dark halos. Figure 2 presents the difference in terms of surface mass density and enclosed mass (<r).

2.4. Mass–Luminosity Analysis

To qualify the computed surface mass density in the context of dark matter investigations, we analyze density profiles in conjunction with corresponding observed surface brightness profiles.

For each radial point of a galactic disk, we divide the corresponding local density by local brightness to obtain the local mass-to-light (M/L) ratio. For each galaxy, a set of all M/L values along the disk radius comprises its radial M/L profile. The numerator in these M/L ratios represents all matter, including stars, substellar objects, stellar remnants, gas, etc. However, we deliberately and conservatively treat all mass as stars, thinking of total M/L profiles as if they were stellar M/L profiles.

The reason for such treatment of M/L profiles is two-fold. First, prosaically, there are no accurate observational data on the radial distribution of non-stellar material in galaxies: It is impossible to reliably adjust galactic total density by subtracting non-stellar density. Second, beneficially, we can make use of the empirical mass–luminosity relation (MLR) of main-sequence stars to infer, from each radial M/L value (local density/brightness ratio), a corresponding main-sequence star with a unique combination of mass and luminosity. We refer to such star as the characteristic star and compute characteristic star mass for each radial datapoint of each galaxy in our sample, using the following logic and procedures.

Considering that the empirical scaling relation between main-sequence star mass and luminosity is given by power-law function L = aM^b, where M and L are in units of solar luminosity (L_⊙) and solar mass (M_⊙), respectively, we can express star mass as a function of its mass/luminosity ratio:

M = (a M/L)^{1/(1 − b)}

(3)

In this form, given coefficients a and b4, the formula can be used to compute characteristic star mass $\overline{\mathrm{M}}$ for a locale with known surface mass density Σ(r) and surface brightness S(r):

$${\overline{\mathrm{M}}=(a\mathsf{\Sigma }/S)}^{1/(1-\mathrm{b})}$$

(4)

The application of this procedure to all pairs of Σ(r) and S(r) yields the galactic radial profile of characteristic star mass. We use this idealized stellar population fitting to check compliance of each Σ(r)/S(r) pair with the limits of main-sequence MLR5.

As the M/L ratio tends to increase in galaxies towards the periphery (causing characteristic star mass to decline radially), and considering that we implicitly include non-stellar matter in M/L analysis, the above procedure may encounter unphysical values of $\overline{\mathrm{M}}$ < 0.075 M_⊙ (below the minimal star mass limit). At these occurrences, the bottleneck is surface brightness S(r), and so we tentatively assume:

$${\overline{\mathrm{N}}}_{\min \mathrm{stars}}=S/L_{\min \mathrm{star}}$$

(5)

where ${\overline{\mathrm{N}}}_{\min \mathrm{stars}}$ is such count of smallest red dwarf stars that would match the value of the observed surface brightness if the local stellar population consisted of lowest-mass stars; and $L_{\min \mathrm{star}}$ is the luminosity of a minimal red dwarf estimated as L = a(0.075 M_⊙)^b, with a and b calibrated for the very tip of the Main Sequence as per Appendix B.

This theoretical count of minimal stars allows to calculate the excess surface mass density (also referred to here as excess density) which represents the difference between the total density and the light-constrained maximum stellar density for a given level of surface brightness:

$${\mathsf{\Sigma }}_{\mathrm{ex}}\left(r\right)=\mathsf{\Sigma }\left(r\right)-0.075{\mathrm{M}}_{\odot }\times {\overline{\mathrm{N}}}_{\min \mathrm{stars}}$$

(6)

We compute radial profiles of characteristic star mass and radial profiles of excess density for each galaxy in our sample and then aggregate individual galaxy findings and explore general sample-level trends, which we present in Section 4.

2.5. Further Analysis

It is not always possible to perform M/L analysis out to the edge of RC. There are cases where pairwise density/brightness matching halts around the radius where the brightness profile is truncated by the limiting magnitude of a telescope, with density profile carrying on towards galactic periphery alone. In such a situation, one can only study surface mass density. Accordingly, we assess dynamics between the peak value of excess density and the total density at the end of a rotation curve, both for individual galaxies and groups.

2.6. Conclusions Regarding Dark Matter

For each galaxy in our sample, as well as for sample-level “average galaxy”, we study the radial profiles of the two above-defined metrics, characteristic star mass and excess density. For galaxies where characteristic star mass always stays above the minimal star mass limit (and hence no excess density is observed), we conclude that no dark matter is required at all within the density/brightness observational radii to explain empirical observations of rotation and luminosity.

The rationale for such a conclusion is that the combination of total surface mass density and surface brightness at all radii can be explained, in principle, by fitting a main-sequence stellar population. Naturally, such stellar population will exhibit a radially declining star mass gradient. We believe this is a norm, rather than an oddity, and put forward a testable hypothesis that average star mass tends to decline along a galactocentric radius. Considerations about the declining radial star mass gradient and references to relevant empirical evidence are provided in Section 5.

For galaxies exhibiting non-zero excess density, we check whether these values are within plausible estimates of non-stellar baryonic mass and/or whether they can be explained by other ordinary factors such as visibility cutoffs and uncertainty of empirical data. We discuss and attempt to quantify some of these factors in Section 5. We consider that galaxies where average computed excess density is below an ordinarily explainable level (adopted at 20 M_⊙/pc²) require no dark matter.

2.7. Summary of the Analytical Framework

Here we summarize the logical sequence discussed above. We start with an observed galactic rotation curve which we convert into a radial density profile (using DM2 model based on two key assumptions: thin disk, Newtonian physics). We then divide each datapoint of the surface mass density profile by a corresponding datapoint of the observed surface brightness profile, to produce the radial M/L profile of the galaxy. Assuming that all galactic mass is in stars, we use empirical main-sequence stellar MLR to convert each datapoint of the radial M/L profile into a characteristic star mass, thus obtaining the radial distribution of characteristic star mass. If all datapoints of the characteristic star mass profile contain a physically plausible value (>0.075 M_⊙), we conclude that no dark matter is required within the density/brightness observational radius. If some of the characteristic star masses fall below the minimum mass limit, we compute the corresponding value of excess density. If all excess density points can be explained with ordinary factors, we conclude that no dark matter is required within the density/brightness observational radius. To the extent there are excess density datapoints which cannot be fully explained with ordinary factors, we conclude that further checks are needed. Figure 3 recaps the key points of our analytical framework in a flowchart format.

3. Data Sample

We assembled a dataset of 214 galaxies covering a broad range of sizes, heliocentric distances, morphologies, luminosities, and rotation velocities (see Appendix E). Figure 4 provides a snapshot of the sample along with correlations between these general properties. Consistent with other studies [35,36], our sample exhibits tight Tully–Fisher relation (correlation of peak rotational velocities with total galactic 3.6 μm luminosities is >0.86), as well as other well-known correlations.

To make galaxies of a different size comparable, we express their radii in units that are multiples of R₂₇—the radius at which 3.6 μm apparent surface brightness reaches 27 mag_AB/arcsec². R₂₇ marks the area within which Spitzer IRAC Channel 1 sensitivity stays relatively high, and is a useful common basis for comparing galaxies.

For each galaxy, our dataset includes two major components: A Spitzer IRAC channel 1 surface brightness profile (3.6 μm) and a rotation curve. Sixty-nine percent of brightness data were obtained from The Spitzer Survey of Stellar Structure in Galaxies (S4G) [37,38,39,40] and 31% from the SPARC database [35]. Appendix C details our surface brightness data processing, which included unit conversion from mag_AB/arcsec² to L_⊙,3.6/pc², data cleaning, and inclination adjustments. Figure 5 shows all 214 inclination-corrected surface brightness profiles superimposed.

Given the absence of extensive RC databases, we collected rotation data piecemeal from various sources ranging from RCs of individual galaxies to aggregated datasets. In particular, we greatly benefited from THINGS [36,41], GHASP [42,43], and SPARC databases, professor Yoshiaki Sofue’s papers [11,44,45], as well as Swaters et al. [46]. One hundred and fifty-six RCs in our sample came from a single source and were adopted “as is”. For the remaining 58 galaxies, where RCs were available from multiple sources, we created composite RCs following the procedure outlined in Appendix D. All RCs are superimposed in Figure 6.

Lastly, we used NASA Extragalactic Database (NED) and HyperLeda Database as a crosscheck for morphologies, inclinations, and heliocentric distances, and compiled a set of physical parameters of 724 main-sequence stars to calibrate empirical stellar mass–luminosity relation (MLR) function for Spitzer 3.6 μm passband (see Appendix B).

4. Results

4.1. Statistical Summary

For 132 or 62% of the galaxies in our sample, the entire M/L profile can be fully explained by a radially declining star mass gradient. Although M/L profiles of the remaining 82 galaxies cannot be completely addressed by an idealized stellar population fitting, the average excess density in these galaxies is merely 14 M_⊙/pc², hardly sufficient to invoke dark matter7 (Figure 7). Even a more conservative average of just peak excess densities is only 18 M_⊙/pc². Moreover, in most galaxies, these peaks are around r~1.1 R₂₇, declining thereafter. Many of these excess density values are at the cusp of Spitzer telescope visibility for red dwarf stellar populations and comparable with likely densities of gas and other low-luminosity baryonic matter.

Charts below (Figure 8) illustrate surface mass density profiles superimposed with surface brightness profiles for 214 galaxies, as well as basic mass–luminosity analytics, including brightness/density correlations, M/L profiles, and characteristic star mass profiles.

Figure 9 presents sample-level radial trends of the same metrics as shown in Figure 8. Individual profiles of 214 galaxies were cut into radial bins, each with a width of 0.01 R₂₇; and average values of density and brightness were computed for each bin (mean average of all datapoints falling into a bin). The resulting compressed profiles represent average density and brightness trends across our galaxy sample on the r/R₂₇ radial scale.

Before we proceed with the analysis of Figure 9 trends, we remind that the density and brightness data in our sample are independent data series. The former was computed using the DM2 model, exclusively from RCs (see source info in Appendix F), without any additional data inputs. The brightness is completely unrelated data from S4G and SPARC, and our galaxy sample is highly diversified in terms of morphologies, sizes, inclinations, etc.

Figure 9a shows sample-level radial trends in density and brightness. Whilst diverging initially to ~2 dex vertical gap, the two profiles flat out notably at r~1.1–1.3 R₂₇. This gearshift in density–brightness interplay is a likely reason for excess density ${\mathsf{\Sigma }}_{\mathrm{ex}}\left(r\right)$ peaking around the r~1.1 R₂₇ breakpoint in most galaxies. This existence of such inflection point, if confirmed on wider datasets with higher quality photometry, will have profound consequences for dark matter hypothesis, as the automatic assumption that light intensity necessarily continues to decline considerably faster than mass density at all radii, will be invalidated. We believe that the reason this breakpoint has not been reported earlier might be that large galaxy samples have not been scrutinized the way we propose in this paper, namely: Empirical-only (without extrapolations) light radially bin-averaged on r/R₂₇ scale for a sufficiently large and diversified set of galaxies. Unfortunately, the current limitations of instruments make post-R₂₇ surface brightness data scarce and barely reliable, but next-generation telescopes will be capable to definitively confirm or disprove this breakpoint and the flattening of surface brightness profile tails.

Figure 9b shows something even more interesting—an incredibly tight sample-level radial correlation between the 3.6 μm surface brightness and the dynamical surface mass density. Polynomial curve fits for brightness as a function of density achieve R-squared of 0.9998. If one increases the radial bin width when averaging density and brightness across the sample, and excludes the noisy post-R₂₇ data, the already amazing correlation between the two independent data series (average density and brightness profiles of an eclectic ensemble of 214 different galaxies) becomes even stronger.

To round off the sample-level results, we review the M/L profile and the characteristic star mass profile (Figure 9c,d, respectively). As expected, the M/L ratio is gradually increasing as a function of radius, but it remains compliant with empirical stellar mass-luminosity relation (Appendix B) until r~1 R₂₇, beyond which excess density becomes non-zero. The growth of the M/L ratio is very smooth. This picture is echoed in the radial profile of characteristic star mass, which gradually declines with radius, in accordance with our hypothesis that the average star mass tends to decline as one proceeds from the center towards galactic periphery. The profile has visible breakpoints, which can be compared with the breakpoints in main-sequence stellar MLR (see Appendix B). The last breakpoint (at r~1 R₂₇) marks the start of the “sub-stellar” zone where the characteristic star mass slips below 0.075 M_⊙ and excess density starts to show, reaching a~14 M_⊙/pc² peak before starting to descend. The gap between the total dynamical density and the excess density narrows to ~33% around r~1.5 R₂₇, with ${\mathsf{\Sigma }}_{\mathrm{ex}}\left(1.5{\mathrm{R}}_{27}\right)$~11.1 M_⊙/pc² and Σ$\left(1.5{\mathrm{R}}_{27}\right)$~16.7 M_⊙/pc²). By r~1.9 R₂₇, excess density declines to ~8.2 M_⊙/pc², which is 14% below the corresponding value of total density (~9.6 M_⊙/pc²).

All contemporary surface brightness-related metrics, including our estimates of excess density, must be taken with a grain of salt for r >> R₂₇ due to higher uncertainties in light data beyond the 27 mag_AB/arcsec² level of 3.6 μm surface brightness. Nevertheless, we think that the above measurements and computed values are good first approximations that can be used to evaluate overall M/L trends in galaxies.

4.2. Representative Galaxies

To offer a more granular view at the level of individual galaxies, Figure 10 displays six representative specimens out of 132 galaxies without any excess density as per our M/L analysis. These six galaxies have a broad range of sizes, morphologies, velocities, distances, and inclinations, but all are explainable with radially declining star mass hypothesis. NGC2998 and NGC4303 have both RCs and surface brightness profiles longer than R₂₇, which makes successful stellar fitting along the whole length of RC even more remarkable. Not only all six galaxies do not require any dark matter within observational radii where both density and brightness data are available, but with their low peripheral density, dark matter can be ruled out even if respective RCs are extended with future measurements.

To provide a balanced and proportional illustration of our sample, Figure 11 shows four representative galaxies out of 82, the radial M/L profiles of which cannot be fully addressed by stellar MLR. For the first two examples—NGC5907 and NGC4157—peak excess density is under 15 M_⊙/pc², never quite approaching the total density curves and declining to 3.9 and 5.4 M_⊙/pc², respectively. We consider these and similar galaxies as benign cases, with hardly enough excess density and low total peripheral density to warrant dark matter speculation. Sixty-seven of 82 galaxies are like this, with average ${\mathsf{\Sigma }}_{\mathrm{ex}}\left(r\right)$ < 20 M_⊙/pc² within observational radii. The third example, NGC0300, is one of the nearest galaxies in our sample at 2 kpc distance from the Sun and offers a peculiar case. Although its maximum ${\mathsf{\Sigma }}_{\mathrm{ex}}\left(r\right)$ is 28 M_⊙/pc², it traces total density downward slope to 10.9 M_⊙/pc² and then rapidly declines to 1.2 M_⊙/pc² matching a bounce in luminosity and reflecting sensitivity to the low light data, which could also be an error of measurement. Lastly, blue dwarf galaxy NGC2915 is our worst case with a maximum excess density of 65 M_⊙/pc² coming down to ≤10 M_⊙/pc² over a radial distance of ~2.6 R₂₇.

5. Discussion

The results presented in Section 4 rest on key assumptions that (a) all dynamical mass is in a disk, and that (b) stellar M/L varies with radius. Given this premise, the fact that for 132 galaxies or 62% of our sample the radial M/L profile can be explained solely by declining star mass gradient is a strong argument against dark matter.

We believe, furthermore, that most of the remaining cases can also be addressed without dark matter. Datapoints with non-zero excess density can be attributed to a combination of several factors—limits of light detection for red dwarf populations, the underestimated presence of low-luminosity baryonic mass, the uncertainty of distances, inclinations, rotation curves, and surface brightness data. Below we discuss these considerations and provide general observations on the dark matter hypothesis.

5.1. Limits of Detection of Red Dwarf Populations

The average S4G surface brightness profile is stated to trace isophotes at μ3.6(AB)(1σ) ∼27 mag/arcsec², which is stated to be equivalent to a stellar mass surface density of ∼1 M_⊙/pc² [37]. Such a low surface density threshold is misleading. The 1 M_⊙/pc² interpretation is correct only for the stellar population including Sun-like or heavier stars. If one assumes a population of red dwarfs of masses approaching 0.075 M_⊙ (minimal star mass corresponding to the hydrogen-burning threshold), then Spitzer IRAC channel 1 sensitivity drops to ≈C × 40 M_⊙/pc², where C is inclination correction (Figure 12). Such a red dwarf population with a density below C × 40 M_⊙/pc² will likely appear noisy or indistinguishable from the background in Spitzer 3.6 μm filter. Forty percent of 82 galaxies have excess density datapoints below the visibility threshold for 0.075 M_⊙ stars.

5.2. Low-Luminosity Baryonic Mass

Favoring a simpler approach in our mass–luminosity analysis, we ignored all low-luminosity baryonic mass such as gas, dust, brown dwarfs, etc. Estimates for these classes are often imprecise and, except for gas content, rarely provide radial distributions making it difficult to apply directly to the M/L problem.

Table 1. Summary of contemporary estimates for various classes of low-luminosity baryonic mass. Estimates for brown dwarfs (BD), white dwarfs (WD), neutron stars (NS), and black holes (BH) assuming surface mass density ~27 M_⊙/pc², an average density across our sample at R₂₇ (see Figure 11), and applying consistent global mass ratios to respective categories.

Category	Estimates	M⊙/pc2
Gas (HI)	<10 M⊙/pc2 at R25 [51]	~6.0
Gas (H2)	<1 M⊙/pc2 at R25 [51]	~0.6
Gas (He)	He mass fraction Yp = 0.233 [52]	~2.0
Dust	Dust-to-gas ratio ~2 × 10−5 [53]	~0.001
Brown Dwarfs (BD)	1011 in Milky Way; >0.03 M⊙ BD mass [54]	~0.7
White Dwarfs (WD)	232 WDs within 25 pc around Sun [55,56,57]; 0.6 M⊙ WD mass	~0.9
Neutron Stars (NS)	108–109 in Milky Way [58]; 1.4 M⊙ NS mass	~0.3
Black Holes (BH)	108 in Milky Way; 10 M⊙ BH mass [59]	~0.2
Total		~10.7

provides the summary of contemporary research, as well as our attempt to quantify likely densities for different categories of baryonic matter assuming a total surface mass density of 27 M_⊙/pc², the average density for our sample at r~R₂₇.

Acknowledging a high degree of uncertainty, we believe that, in total, low-luminosity baryons could account for ∼10 M_⊙/pc² as a first approximation estimate. This is probably a conservative number for several reasons. First, new findings, such as the discovery of massive HI medium enveloping M31 a thousand times larger than previously believed [47], consistently pushed these estimates upward, often by an order of magnitude, with the improvement of instruments and analytical techniques. Second, H₂ may be undetectable at lower abundances as there is not enough C and O present to give a signal [48]. Third, estimates for low-luminosity objects are based either on a limited sample set tied to particular locale (white dwarf census was just recently expanded from 20 to 40 pc around the Sun [49,50]) or on stellar evolution models backed by minimal empirical data (black holes, neutron stars). Fourth, brown dwarf frequencies are grounded in observations around the Sun and a few star-forming clusters, suggesting a ratio of 2–6 stars per brown dwarf. If our hypothesis of radially declining star mass gradient is valid, its logical extension is to expect a higher proportion of brown dwarfs in low-density galactic outskirts.

5.3. Uncertainty of Inclinations and Distances

Some of the variables critical for galaxy rotation and M/L problem admit a significant degree of uncertainty. Distances to galaxies, for example, are far from being resolved precisely. In our sample, 74 galaxies have the upper bound of distance estimates at least double of the lower bound number. Galactic inclinations also present several problems. Face-on galaxies are good for surface brightness data but unreliable for rotation curves. Vice versa, highly inclined galaxies are good for velocity measurements but questionable for surface brightness data. For some galaxies, inaccuracies in these key input parameters can drastically alter the results of their mass/luminosity analysis.

5.4. Uncertainty of RCs

RC measurements are uncertain, especially for warped and low-inclination disks. The methodology for averaging of approaching and receding side velocities is not standardized. Many RCs have very low resolution, with only a few datapoints. For multiple RCs, there are strong disagreements between authors. Appendix D shows an example of such RC disagreement.

5.5. Uncertainty of Surface Brightness Data

There is a significant variation in surface brightness estimates, typically caused by differences in equipment used and data processing methodology applied, especially in the case of warped disks and irregular galaxies. For some galaxies, there is up to an order of magnitude difference between S4G “fr” and “fx” series at some radii. As noted in Appendix C, in only 20% cases of the S4G-SPARC overlap set of light data, the total luminosity estimates by SPARC differ from those by S4G by less than 10%. In the most extreme case, the difference reaches 9 times. Another source of significant uncertainty is the choice of filters. We used 3.6 μm data, but some radiation sources shine much brighter in other passbands, giving higher L_⊙/pc² values. A truly unambiguous image of a galaxy is bolometric, but unfortunately, such data are not available for our sample.

5.6. Uncertainty of MLR Slope for Smallest Red Dwarfs

Given relatively few datapoints for the smallest red dwarfs at the very tip of the Main Sequence, we took a conservative approach and ignored a breakpoint in power-law MLR function around 0.1 M_⊙, below which the function changes slope. Were we to use just the seven lightest stars to compute the MLR function, the coefficients would have been a = 4.1 and b = 2.84, with R² = 0.98 (instead of the adopted a = 0.82, b = 2.17). The application of this alternative set of coefficients in our mass–light analysis would have further reduced average excess density as well as the number of “light-deficient” galaxies.

5.7. Light Extinction

We have not applied any extinction adjustment to surface brightness profiles in our sample. We believe the impact of internal extinction ranges from zero for face-on galaxies, to 0.25 mag for edge-on galaxies. This is probably a conservative estimate as even face-on galaxies are likely to have some internal light extinction. Moreover, light is lost both in the intergalactic medium and within Milky Way. Although the former is assumed to be almost transparent in Spitzer 3.6 μm, even minor light loss over intergalactic distances can be cumulatively significant. Overall, it seems probable that substantially higher extinction corrections are necessary to fully account for all light loss. A 0.25–0.5 mag correction would have reduced the number of “light deficiency” datapoints in our sample by 19–38%.

5.8. Plateau in Surface Brightness Profiles beyond R₂₇

As shown in Section 4, we discovered an inflection point around r~1.1 R₂₇ in our sample-level average surface brightness profile. Outside this radial point, the gap between density and brightness appears to be no longer expanding. This agrees well with and explains the fact that for most of the relatively “light-deficient” galaxies in our sample excess density peaks around r~1.1 R_27. Current technological limitations make it hard to confirm empirically with high confidence the flattening of light profile beyond the identified inflection point. However, this is, in principle, a testable hypothesis, which can be another strong argument against dark matter.

5.9. Evidence for Radially Declining Star Mass

The ideal empirical test for our hypothesis of radially declining star mass gradient would be direct observation of star distributions in different galaxies. Unfortunately, present knowledge of such distribution is limited even for the Milky Way. Galactic O-Star Spectroscopic Survey (GOSSS) [60] identifies the 590 brightest O-Star systems within our galaxy. Most of the identified O-Stars are located within central I and IV quadrants [61], directionally supporting our thesis, but the study accounts only for 1% of estimated total 50,000 O-type systems in the galaxy. According to the authors of GOSSS though, over 90% of the hidden O-stars are also in the quadrants I and IV. There is also a growing body of evidence of top-heavy initial mass function in the Galactic center [62,63] and more generally with increasing density [64].

Concentration of heavy stars in the center is expected given the minimal density and mass of maternal molecular cloud required for star formation. Theoretical reasons suggest a minimum density of (5 ± 2) × 10³ cm⁻³ [65] and cloud mass of 30 M_⊙ for the lowest star mass of 0.075 M_⊙ required by a hydrogen burning limit. This is consistent with the smallest known star-forming globules with the mass range of 5–50 M_⊙. The minimal mass of a maternal cloud required to form a heavy star is considerably higher. For an O-type star with mass >20 M_⊙ to form with 95% probability, the cloud should be approximately 7 × 10⁴ M_⊙ [66,67]. In a typical galactic periphery, where surface density drops below 50 M_⊙/pc², such cloud has to vacuum all mass within ~1 million pc³.

5.10. Evidence of Extended Stellar Disks Around Galaxies

Several recent papers provided independent evidence of extended stellar disks around galaxies. For example, an extremely faint, outer stellar disk is observed to 10 scale lengths in NGC 300 [68]. In our neighbor, M31 Andromeda, there is evidence for stellar outer disk substructure [69]. M33 is also suspected to have significant outer stellar disk [70]. In some galaxies, like NGC 4013, giant stellar tidal streams around the disk were confirmed [71]. With new-generation telescopes, we expect the evidence for extended stellar disks to increase dramatically.

5.11. Strong Correlations between Dynamics and Light

Even without modelling density and auditing every radial M/L datapoint for compliance with stellar MLR (89% of the all density/brightness datapoints in our sample do pass the test), it is strikingly apparent from simple correlation analysis of basic properties for our 214 galaxies, that galactic rotation is shaped by baryons. As mentioned in passing in Section 3, the Tully–Fisher relation is confirmed with 0.83 correlation between total luminosity and peak rotational speed.

The Tully–Fisher relation is not the only confirmation that baryonic mass defines galactic dynamics. In Section 4, we presented sample-level radial mass–luminosity trends. Unlike Tully–Fisher, we do not compress a galaxy into a pair of numbers to then compare a group of galaxies. Instead, we squeeze together radial profiles of a large group of galaxies and look at general patterns inside an averaged galaxy. This inner perspective allows to see things impossible with Tully–Fisher analytics, such as the correspondence between density and brightness at different radii. Our approach and Tully–Fisher complement each other and jointly offer a strong argument against the idea of dynamically dominant non-baryonic dark matter. As showed in this paper, sample-level average galactic density profile correlates extremely with galactic light (R² = 0.9998) at all radii8. Furthermore, the density is dynamically sufficient. Therefore, the pro-dark matter interpretation, which is usually applied to Tully–Fisher relation (that baryonic matter must be correlated with dark matter), does not stand in the case of our analysis of galactic mass-luminosity trends.

5.12. Lack of Evidence for Dark Matter in the Solar Neighborhood

It is unlikely for the Solar system to be in a dark matter-free bubble, if dark matter is as abundant in the universe as claimed. Considerable effort went into experiments attempting to detect weakly interacting massive particles [72,73], without any positive results. On the other hand, a study of surface mass density at the solar Galactocentric position between 1.5 and 4 kpc from the Galactic plane [74] accounted for visible mass only and compared expected and empirical kinematics of stellar objects. According to the authors, the visible mass strikingly matches the observations, presenting another empirical hurdle for the dark matter concept.

6. Conclusions

Using a diversified sample of 214 galaxies:

▪

We have demonstrated that all types of galactic rotation curves can be precisely tracked by a non-parametric disk model which requires lower surface mass density than estimated with parametric models with a dark matter halo.

▪

We have showed that in 132 galaxies, all radially corresponding density/brightness datapoint pairs within observational radii can be addressed solely by the assumption of radially declining average star mass9.

▪

We have discussed how the average excess density of 14 M_⊙/pc² in the remaining 82 galaxies can be explained without appealing to non-baryonic dark matter, but instead with a combination of factors such as visibility cutoffs for dim stellar populations, the quality of various estimates, incompleteness of light data, and the low-luminosity baryonic content.

▪

We have presented an empirically testable hypothesis that radially rising mass/luminosity profiles might be a natural outcome of radially declining average star mass in galaxies.

▪

We have reported a finding of 3.6 μm galactic surface brightness profiles flattening beyond the r~1.1 R₂₇ inflection point.

▪

Lastly, we have demonstrated the extremely tight correlation between sample-average radial profiles of dynamical mass density and surface brightness, with polynomial fit R² ≈ 1.

For an unbiased rational thinker abiding by the Occam’s razor principle, our analysis suggests that the widely popular and yet hopelessly theoretical dark matter must be rejected in favor of a simpler explanation. Perhaps dark matter is like Confucius’ black cat—hard to find in a dark room, especially if there is no cat.