Equivalence between Direct Mass and NFW-Total Mass Formula in MW and M31 GalaxiesC

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1. Introduction

Since 2014 up to 2024, I have published several papers studying DM in galactic halos, especially in M31 and Milky Way although also I have published some papers studying other galaxies and clusters.

This paper is focused on the equivalence of NFW mass formula and the Direct mass formula, which is the formula for the total mass in the halo region developed in the framework of DMbQG. So the reader has to have at least a general knowledge about this original theory to read this paper. The paper [1] (Abarca), is the best work about it, so the reader may consult such paper in order to understand the DMbQG theory.

The Chapter 3 is dedicated to introduced the Direct mass and some derived formulas, and the Chapter 4 is dedicated to introduced the NFW method and his extension to the total mass (baryonic plus DM). As the reader knows, the NFW is a density profile for DM only, so in this work has been necessary to integrate the baryonic matter into the NFW profile because the Direct mass function gives the total mass into the halo region.

As reader knows, M31 is the twin galaxy of Milky Way in the Local Group of galaxies. According to [2] (Sofue, Y. 2015), its baryonic masses are $M_{M 31} = 1.6 \times 10^{11} M_{Θ}$ and $M_{MILKY WAY} = 1.4 \times 10^{11} M_{Θ}$ .

The DM by Quantum Gravitation, DMbQG hereafter, theory was introduced in [3] (Abarca, M. 2014). Dark matter model by quantum vacuum). It considers that DM is generated by the own gravitational field according to an unknown quantum gravitational phenomenon.

In order to study purely the phenomenon it is needed to consider a radius dominion where it is supposed that baryonic matter is negligible. i.e. radius bigger than 30 kpc for MW and 40 kpc for M31, according to some calculus made about it.

This hypothesis has two main consequences: the first one is that the law of dark matter generation, in the halo region, has to be the same for all the galaxies. In the paper [1] (Abarca, M. 2024) is developed the theory using the rotation curve of M31 published by [2] (Sofue, Y. 2015) and the rotation curve of MW by [4] (Sofue, Y. 2020).

The second consequence is that the haloes are unlimited so the total dark matter goes up without limit. In the paper [1] (Abarca, M. 2024) is solved the divergence of the total mass, thanks to the Dark energy.

As I have mentioned before, this theory has been developed assuming the hypothesis that DM is a quantum gravitational effect. However, it is possible to remain into the Newtonian framework to develop the theory. In my opinion there are two factors to manage the DM conundrum with a quite simple theory.

The first one, that it is developed into the halo region, where baryonic matter is negligible. The second one, that the mechanics movements of celestial bodies are very slow regarding velocity of light, which is supposed to be the speed of gravitational bosons.

It is known that community of physics is researching a quantum gravitation theory since many years ago, but it does not exist yet, however I think that my works in this area support strongly that DM is a quantum gravitation phenomenon.

Use a simpler theory instead the general theory is a typical procedure in physics. For example the Kirchhoff’s laws are the consequence of Maxwell theory for direct current and remain valid for alternating current, introducing complex impedances, on condition that signals must have low frequencies.

So these reasons support the possibility to study a complex phenomenon as it is the DM with a theory mathematically simple in the framework of Newtonian mechanics.

In the paper [1] (Abarca, M. 2024) in the framework of DMbQG theory it is calculated by the Direct mass with unbounded dominion for radius the dynamical mass of the Local Group, that according to [5] (Azadeh Fattahi, Julio F. Navarro). 2020 is estimated to be $5 \times 10^{12} M_{Θ}$ . The result given by the direct mass considering the four main galaxies of the L. G. match perfectly with such estimation, whereas using the virial masses associated to MW, M31, M33 and LMC calculated by NFW the total amount of masses is scarcely $3 \times 10^{12} M_{Θ}$ . This calculus has been made without considering the dark energy because according some calculus made, into the L. G. the D. E. is important for radius bigger than 1 Mpc, so for the system MW and M31 the DE may be neglected.

The DMbQG theory has been developed successfully in cluster of galaxies in the paper [6] (Abarca, M. 2024), and there have been found a set of remarkable theoretical results tested in the L. G. and the Virgo cluster, that is the nearest big cluster and consequently the cluster where measures reach the maximum of accuracy. Namely some theoretical findings match perfectly with the results published by [7] (Kashibadze, Karachentsev, 2020) and by [8] (Karachentsev, I. D., Tully, R. B 2014).

Despite the fact that the DMbQG theory has been tested successfully in galaxies and clusters, the prove reach in this paper is valuable because the NFW is a trustable profile for DM, tested in thousands of galaxies, and although the DMbQG theory claims that DM has an unbounded halo region, it gives similar results in the halo region common for both theories i.e. up to the virial radius.

2. Virial Mass and Virial Radius in Galaxies and Clusters

In galaxies and clusters, it is a good estimation about virial radius and virial mass to consider R_vir = R₂₀₀ and M_vir = M₂₀₀. Where R₂₀₀ is the radius of a sphere whose mean density is 200 times bigger than the critic density of Universe

$ρ_{C} = \frac{3 H^{2}}{8 π G} = 9.2055 \times 10^{- 27} kg \cdot m^{- 3}$ (2.1)

(In this work it will be considered H = 70 km/s/Mpc)

and M₂₀₀ is the total mass enclosed by the radius R₂₀₀.

Considering the spherical volume formula, it is right to get the following relation between both concepts.

$R_{200}^{3} = \frac{G \cdot M_{200}}{100 \cdot H^{2}}$ (2.2)

$M_{200} = \frac{100 H^{2} R_{200}^{3}}{G}$ (2.3)

$\frac{M_{200}}{R_{200}^{3}} = \frac{100 H^{2}}{G}$ (2.4)

These parameters are common in cluster of galaxies as well. In the Chapter 2 of paper [6] (Abarca, M. 2024) is checked the above relation between R₂₀₀ and M₂₀₀ on a set of clusters.

In the Chapter 4 will be introduced the NFW density profile and the NFW mass function, both linked to DM, so in the framework of NFW the R₂₀₀ and the M₂₀₀ are connected with DM exclusively.

As the Direct mass is linked to the total mass (baryonic mass plus DM), in the Chapter 4 will be developed a procedure to integrate the baryonic mass in the NFW function mass, and this is the reason why R₂₀₀ and the M₂₀₀ are written with the subscript R_200-TOTAL and the M_200-TOTAL when they are linked to the total mass.

3. Virial Theorem as a Method to Get the Direct Mass Formula in Galaxies or Galaxy Clusters

In Chapter 8, of paper [1] (Abarca, M. 2024) was demonstrated that the direct mass formula

$M_{TOTAL} (< r) = \frac{a^{2} \cdot \sqrt{r}}{G}$ (3.1)

is the most suitable formula to calculate the total mass (baryonic and DM) enclosed by a sphere with a specific radius that ranges into the galactic halo.

The halo is the region where the density of baryonic matter is negligible versus the D. M. density. e.g. the halo for Milky Way may have a radius bigger than 30 kpc, or the halo for M31 may have a radius bigger than 40 kpc. See [1] (Abarca, M. 2024).

3.1. Parameter a² Formula Depending on Virial Radius and Virial Mass

Due to the fact that the Direct mass formula has one parameter only, is enough to know the mass associated to a specific radius to be able to calculate parameter $a^{2}$ .

According to DMbQG theory is possible to do an equation between M₂₀₀ (<R₂₀₀) = M_DIRECT (<R₂₀₀) i.e.

$M_{200} = M_{TOTAL} (< R_{200}) = \frac{a^{2} \cdot \sqrt{R_{200}}}{G}$

And clearing up

$a^{2} = \frac{G \cdot M_{200}}{\sqrt{R_{200}}}$ (3.2)

This formula is called parameter $a^{2}$ (M₂₀₀, R₂₀₀) because depend on both measures.

3.2. Parameter a² Formula Depending on Virial Mass Only

In Chapter 2 was got this formula $R_{200}^{3} = \frac{G \cdot M_{200}}{100 \cdot H^{2}}$ (2.2) as a good approximation between the virial mass and the virial radius. So using that formula and by substitution of the virial radius in $a^{2} = \frac{G \cdot M_{VIRIAL}}{\sqrt{R_{VIRIAL}}}$ it is right to get the parameter $a^{2}$ depending on M₂₀₀ only

$a^{2} = {(G \cdot M_{200})}^{5 / 6} \cdot {(10 \cdot H)}^{1 / 3}$ (3.3)

This formula will be called parameter $a^{2}$ (M₂₀₀) as depend on M₂₀₀ only. Conversely it is possible to clear up the virial mass from the previous formulas.

$M_{200 - T O T A L} = \frac{a^{12 / 5}}{G \cdot {(10 \cdot H)}^{2 / 5}}$ (3.4)

or using the Formula (2.3) and clearing up the virial radius then

$R_{200 - T O T A L} = {[\frac{a^{2}}{100 \cdot H^{2}}]}^{2 / 5}$ (3.5)

It is important to insist that the parameter $a^{2}$ is linked to the total mass and this is the reason why the radius and mass are written with the subscript 200-TOTAL.

In [1] (Abarca, 2024) using the rotation curve of M31 published by [2] (Sofue, 2015) was got the parameter $a$ = 4.727513 × 10¹⁰ m^5/4/s or $a^{2}$ = 2.235 × 10²¹ m^5/2/s² and using the rotation curve of [4] (Sofue, 2020) was got the same parameter for MW:

$a^{2}$ = 1.527 × 10²¹ m^5/2/s², so using the previous formulas are got the following values. See Table 1.

Table 1. M_200-TOTAL and R_200-TOTAL using the parameter $a^{2}$ .

Parameter $a^{2}$	M_200-TOTAL	R_200-TOTAL
m^5/2/s²	$M_{Θ}$	kpc
M31 2.235 × 10²¹	1.42 × 10¹²	232.15
MW 1.527 × 10²¹	9.02 × 10¹¹	199.33

4. The NFW Profile for Dark Matter Mass Density

The NFW profile for DM density in galaxies is

$ρ (r) = \frac{ρ_{0}}{x \cdot {(1 + x)}^{2}}$ (4.1)

being $ρ_{0}$ a characteristic density, x = r/R₀ a dimensionless magnitude related with radius by R₀, which is called scale radius.

By integration it is right to get the Dark matter enclosed by a sphere with radius r.

$M_{DM} (< r) = K_{NFW} \cdot f (x)$ (4.2)

being

$K_{NFW} = 4 π ρ_{0} R_{0}^{3}$ (4.3)

and

$f (x) = \ln (1 + x) - x / (1 + x)$ (4.4)

where x = r/R₀, being ln the natural logarithm.

Two important concepts for NFW profiles are M₂₀₀ and R₂₀₀ both referred to DM only i.e. the DM enclosed into a sphere with R₂₀₀ as radius whose mean

density is 200 times the critic density $ρ_{C} = \frac{3 H^{2}}{8 π G} = 9.2055 \times 10^{- 27} kg \cdot m^{- 3}$ .

$M_{200 -DM} = M_{DM} (< R_{200}) = K_{NFW} \cdot f (c)$ (4.5)

where

$c = R_{200} / R_{0}$ (4.6)

is called the concentration parameter and R₂₀₀ = R₀ × c.

4.1. Calculus of Concentration Parameter

As $\frac{M_{200}}{R_{200}^{3}} = \frac{100 H^{2}}{G}$ then $\frac{M_{200}}{c^{3} R_{0}^{3}} = \frac{100 H^{2}}{G}$ and $\frac{4 \cdot π ρ_{0} \cdot R_{0}^{3} \cdot f (c)}{c^{3} R_{0}^{3}} = \frac{100 H^{2}}{G}$ so

$\frac{c^{3}}{f (c)} = \frac{4 π G ρ_{0}}{100 \cdot H^{2}}$ (4.7)

This equation is quite easy to solve numerically, and it is clear that c depends on the characteristic density only.

With this parameter c, it is rightly calculated M_200-DM and R₂₀₀.

In Table 2 are shown the NFW parameters published by [4] (Sofue, 2020) for MW.

Table 2. The NFW parameters for M. W. according to Sofue (2020).

Characteristic density $ρ_{0}$	Scale radius $R_{0}$
$0.787 \pm 0.037 GeV \cdot {cm}^{- 3} = 1.403 \times 10^{- 21} kg \cdot m^{- 3}$	10.94 ± 1.05 kpc

Using the characteristic density it is right to get the equation c³/f(c) = 2286.125 that gives the value c = 16.348, and f(c) = 1.91.

So R₂₀₀ = R₀·c = 178.85 kpc.

Using (4.3) $K_{NFW} = 3.4 \times 10^{11} M_{Θ}$ then using (4.5) $M_{200 -DM} = 6.498 \times 10^{11} M_{Θ}$ .

In Table 3 is checked the density of the sphere with the radius R₂₀₀.

Table 3. Density of R₂₀₀ sphere versus $200 ρ_{C}$ .

Mean density M_200-DM versus R₂₀₀	$200 ρ_{C}$
$1.837 \times 10^{- 24} kg \cdot m^{- 3}$	$1.841 \times 10^{- 24} kg \cdot m^{- 3}$
Ratio Density of R₂₀₀ sphere versus $200 ρ_{C}$ = 0.997827

4.2. Determining the NFW Profile by R₂₀₀ and the Concentration Parameter c

Conversely, some authors give the NFW profile using three parameters M_200-DM, R₂₀₀ and c.

Using (4.7) and knowing the parameter c is possible to clear up the characteristic density

$ρ_{0} = \frac{100 \cdot H^{2} c^{3}}{f (c) \cdot 4 \cdot π \cdot G}$ (4.8)

in addition R₀ = R₂₀₀/c. This way, knowing $ρ_{0}$ and $R_{0}$ , it is defined the NFW profile.

Although M_200-DM is derived from the previous ones, as it is very important all the authors publish its value. Namely its value may be calculated by (2.3) or by (4.5).

For example, in Table 4, are shown the NFW parameters published by [9] (E. Karukes, 2020).

The value M_TOTAL-R200 represents the total mass enclosed by the sphere R₂₀₀ so by subtraction of M_TOTAL-R200 minus M_200-DM may be calculated the baryonic mass of MW i.e. $M_{BA-MW} = 6 \times 10^{10} M_{Θ}$ according to this author.

Table 4. The NFW parameters for M. W. according to Karukes (2020).

M_200-DM	M_TOTAL-R200	R₂₀₀	Concentration factor
$M_{Θ}$	$M_{Θ}$	kpc
${8.3}_{- 0.8}^{+ 1.2} \times 10^{11}$	${8.9}_{- 0.8}^{+ 1} \times 10^{11}$	$193_{- 6}^{+ 9}$	c = 19

Obviously using the M_TOTAL-R200 into the sphere R₂₀₀ does not verify that the mean density is $200 ρ_{C}$ as it is shown in Table 5.

Table 5. Comparing two different density mean with $200 ρ_{C}$ .

Density_MEAN

M_200-DM into R₂₀₀

Density_MEAN

M_TOTAL-R200 into R₂₀₀

200 ρ_{C}

1.867 \times 10^{- 24} kg \cdot m^{- 3}

2 \times 10^{- 24} kg \cdot m^{- 3}

1.841 \times 10^{- 24} kg \cdot m^{- 3}

Match well with

200 ρ_{C}

Does not match with

200 ρ_{C}

Using the data of Table 4 is got the two typical parameters of NFW density profile.

As R₀ = R₂₀₀/c then R₀ = 10.1578 kpc.

As c = 19 then f(c) = 2.04573.

As $M_{200 -DM} = M_{DM} (< R_{200}) = K_{NFW} \cdot f (c)$ then $K_{NFW} = 4.057 \times 10^{11} M_{Θ}$ .

As $K_{NFW} = 4 π ρ_{0} R_{0}^{3}$ then $ρ_{0} = 2.086 \times 10^{- 21} kg \cdot m^{- 3}$ .

According to [9] (E. Karukes, 2020), the data of Table 4 about masses means that the baryonic mass enclosed by R₂₀₀ is $6 \times 10^{10} M_{Θ}$ . However for [2] (Sofue, 2015) the baryonic mass for the MW is $1.3 \times 10^{11} M_{Θ}$ and others authors give different values. It is known that the measures for the baryonic mass of MW has a high imprecision. Similarly the relative differences about the virial masses and radius are not negligible, although both authors give results compatibles if it is considered the range of errors.

In Table 6 are summarized the virial data of Sofue and Karukes.

Table 6. DM virial data Sofue vs Karukes.

	M_200-DM	R_200-DM
	$M_{Θ}$	kpc
[4] Sofue 2020 data	6.498 × 10¹¹	178.85
[9] Karukes 2020 data	${8.3}_{- 0.8}^{+ 1.2} \times 10^{11}$	$193_{- 6}^{+ 9}$
Relative difference %	21%	7.8%

4.3. Calculus of Concentration Parameter t for the Total Mass

As in this paper it will be compared the R₂₀₀ for the total masses, in this epigraph it will be developed a method to calculate the R_200-TOTAL in the framework of the NFW profile density for DM. i.e. the radius of the sphere where the total mass has a mean density of $200 ρ_{C}$ , this total mass will be denoted by M_200-TOTAL. In the epigraph 4.2 was defined M_TOTAL-R200 and it was shown that its mean density into the R₂₀₀ sphere was bigger than $200 ρ_{C}$ as it was expected. In Table 9 it will be shown that the mean density of M_200-TOTAL into R_200-TOTAL is $200 ρ_{C}$ , as R_200-TOTAL is bigger than R₂₀₀. Below is developed a procedure to calculate the new R_200-TOTAL.

The total mass is the addition of baryonic plus the DM, as the baryonic mass is mainly concentrated into the bulge and disk of a galaxy, this amount of mass is a constant quantity into the halo dominion i.e.

$M_{TOTAL} (< r) = M_{B A} + M_{DM} (< r) = M_{B A} + K_{NFW} \cdot f (x)$

If it is defined

$f_{B A} = M_{B A} / K_{NFW}$ (4.9)

then

$M_{TOTAL} (< r) = [f_{B A} + f (x)] K_{NFW}$ (4.10)

As the sphere whose mean density is $200 ρ_{C}$ verify

$\frac{M_{200 -TOTAL}}{R_{200 -TOTAL}^{3}} = \frac{100 H^{2}}{G}$ Formula(2.4)

And defining a new parameter t

$t = R_{200 -TOTAL} / R_{0}$ (4.11)

It is got $\frac{[f_{B A} + f (t)] \cdot 4 π \cdot ρ_{0} \cdot R_{0}^{3}}{R_{0}^{3} \cdot t^{3}} = \frac{100 H^{2}}{G}$ that leads to the expression $\frac{t^{3}}{f_{B A} + f (t)} = \frac{4 π G \cdot ρ_{0}}{100 \cdot H^{2}}$ that by (4.7) leads to

$\frac{t^{3}}{f_{B A} + f (t)} = \frac{c^{3}}{f (c)}$ (4.12)

This equation allows calculating the concentration parameter t for the total mass.

The parameter t depends on the parameter c and the fraction f_BA.

In the following two epigraphs will be used this method with two different data set provided by two authors about the NFW profile in MW.

4.3.1. R_200-TOTAL and M_200-TOTAL by Parameter t Using NFW Sofue Data

In Table 7 are the [4] Sofue (2020) data to calculate the parameter t.

Table 7. NFW parameters according Sofue to calculate the parameter t.

$R_{0} = 10.94 \pm 1.05 kpc$ and $ρ_{0} = 1.403 \times 10^{- 21} kg \cdot m^{- 3}$
See epigraph 4.1 for calculus of: c, R₂₀₀, M_200-DM, K_NFW
c = 16.348	R₂₀₀ = 178.85 kpc
$K_{NFW} = 3.4 \times 10^{11} M_{Θ}$	$M_{200 -DM} = 6.498 \times 10^{11} M_{Θ}$
[2] Sofue (2015) $M_{B A} = 1.3 \times 10^{11} M_{Θ}$	f_BA = M_BA/K_NFW = 0.382

So $\frac{t^{3}}{f_{B A} + f (t)} = \frac{c^{3}}{f (c)}$ leads to $\frac{t^{3}}{0.382 + f (t)} = \frac{4369.12}{1.911} = 2286.156$ whose solution is $t = 17.527$ .

So $R_{200 -TOTAL} = t \cdot R_{0} = 191.745 kpc$ and as $f (t) = 1.9732$ the $M_{200 -TOTAL} = [0.382 + f (t)] K_{NFW} = 8.0077 \times 10^{11} M_{Θ}$ .

4.3.2. R_200-TOTAL and M_200-TOTAL by the Parameter t Using NFW Karukes Data

In Table 8 are the [9] (Karukes, 2020) data to calculate the parameter t.

Table 8. NFW parameters according to Karukes to calculate the parameter t.

$M_{200 -DM}$	$R_{200}$	Concentration factor c
${8.3}_{- 0.8}^{+ 1.2} \times 10^{11} M_{Θ}$	$193_{- 6}^{+ 9}$ kpc	c = 19
	R₀ = 193/19 = 10.158	f(c) = 2.045732274
$K_{NFW} = 4.0572269 \times 10^{11} M_{Θ}$	$M_{B A} = 6 \times 10^{10} M_{Θ}$	f_BA = M_BA/K_NFW = 0.147884

So $\frac{t^{3}}{f_{B A} + f (t)} = \frac{c^{3}}{f (c)}$ leads to $\frac{t^{3}}{0.1479 + f (t)} = 3352.886$ whose solution is $t = 19.5191$ .

So $R_{200 -TOTAL} = t \cdot R_{0} = 198.273 kpc$ .

and as $f (t) = 2.07$ the $M_{200 -TOTAL} = [0.1479 + f (t)] K_{NFW} = 8.998 \times 10^{11} M_{Θ}$ .

In Table 9 are summarized the total mass and the total radius data, according Sofue and Karukes data. Although the relative differences are not negligible both match if it is considered the range of error measures.

Table 9. Virial total mass and virial radius data. Sofue versus Karukes.

	M_200-TOTAL	R_200-TOTAL	Mean density vs $200 ρ_{C}$
	$M_{Θ}$	kpc	Dimensionless
Using Sofue data	8.0 × 10¹¹	191.7	0.99735
Using Karukes data	9.0 × 10¹¹	198.3	1.01368
Relative Diff. %	11%	3.3%

5. Testing the Equivalence between Direct Mass and NFW Total Mass in MW Halo

As the direct mass is referred to the total mass into the halo region, it is needed to extend the NFW for DM formula to the total mass, following the procedure developed in the Epigraph 4.3, in order to be able to compare both formulas.

In this chapter will be introduced a set of four tests to check the equivalence between the two formulas into the halo region up to the R_200-TOTAL radius.

Although the set of tests developed in this chapter is general, are used the MW data because our galaxy is the best well known with the most accuracy data.

5.1. Comparison between R_200-TOTAL and M_200-TOTAL Values Got by Direct Mass and NFW Total Mass. Test I

In [1] (Abarca, M. 2024) was got the direct mass (3.1) formula, that in the framework of DMbQG theory is the formula for the total mass depending on the radius into the halo and as a consequence of (3.1) were got the formulas

$M_{200 -TOTAL} = \frac{a^{12 / 5}}{G \cdot {(10 \cdot H)}^{2 / 5}}$ (3.4) and $R_{200 -TOTAL} = {[\frac{a^{2}}{100 \cdot H^{2}}]}^{2 / 5}$ (3.5) which depend on the parameter $a^{2}$ solely. See Table 1.

By other side, as in the epigraph 4.3 has been got the R_200-TOTAL and the M_200-TOTAL in the framework of NFW, now it is possible to compare both parameters got with the two different methods.

In Table 10 is shown the three different values for R_200-TOTAL and M_200-TOTAL, they match perfectly if it is considered the range of errors. Also it is checked the ratio mean density versus $200 ρ_{C}$ , which differs from unity in thousands for the three values.

Table 10. Comparison of virial total mass and virial radius for MW. Test I.

MW Galaxy	R_200-TOTAL	M_200-TOTAL	Density_MEAN/ $200 ρ_{C}$
MW Galaxy	kpc	$M_{Θ}$	Dimensionless
By parameter $a^{2}$	199.33	9.02 × 10¹¹	1.00026
Sofue NFW-total	191.745	8.0077 × 10¹¹	0.9976
Karukes NFW-total	198.273	8.998 × 10¹¹	1.0139

5.2. Calculus of Parameter a² Using M_200-TOTAL Got by NFW. Test II

As in the Epigraph 4 was calculated M_200-TOTAL using the NFW method, then it is possible to use such result to calculate the parameter $a^{2}$ by the formula:

$a^{2} = {(G \cdot M_{200})}^{5 / 6} \cdot {(10 \cdot H)}^{1 / 3}$ Formula(3.3)

Then this result may be compared with the parameter $a^{2}$ got in [1] (Abarca, M. 2024) in the framework of DMbQG theory that for MW is 1.527 × 10²¹ m^5/2∙s⁻² that is the reference value. See Table 11.

Table 11. Comparing the parameter $a^{2}$ . Test II.

	M_200-TOTAL	Parameter $a^{2}$	Relative diff.
	$M_{Θ}$	m^5/2∙s⁻²	%
Sofue NFW	8.0077 × 10¹¹	1.383 × 10²¹	9.4 (good)
Karukes NFW	8.998 × 10¹¹	1.524 × 10²¹	0.2 (excellent)
$a^{2}$ as reference. See Table 1		1.527 × 10²¹

The comparison between the reference parameter $a^{2}$ and the one got by Sofue NFW data is good (9.4%), but the comparison with the one got by Karukes data is excellent (0.2%).

5.3. Calculus of Parameter a² Using M_200-TOTAL and R_200-TOTAL Got by NFW. Test III

In this test the parameter $a^{2}$ is got by the formula $a^{2} = \frac{G \cdot M_{200}}{\sqrt{R_{200}}}$ Equation (3.2),

using M_200-TOTAL and R_200-TOTAL got by NFW method. See Table 12, in the second and third columns are summarized the data got in the epigraph 4.3. The fourth column shows the three different values of parameter $a^{2}$ .

Table 12. Comparing the parameter $a^{2}$ . Test III.

MW	R_200-TOTAL	M_200-TOTAL	Parameter $a^{2}$	Relative diff.
MW	kpc	$M_{Θ}$	m^5/2s⁻²	%
Sofue NFW	191.745	8.0077 × 10¹¹	1.3824 × 10²¹	9.4
Karukes NFW	198.273	8.998 × 10¹¹	1.5276 × 10²¹	0.03
$a^{2}$ as reference. See Table 1			1.527 × 10²¹

The result of this test is very similar to the test II because in the framework of DMbQG the Formulas (3.3) and (3.2) are mathematically equivalents.

5.4. Comparison of Direct Mass Formula with the NFW-Total Mass Formula into the Halo Region up to R_200-TOTAL. Test IV

The NFW mass formula extended to the total mass was developed in the Chapter 4, being $M_{TOTAL} (< r) = [f_{B A} + f (x)] K_{NFW}$ (4.10), being K_NFW the constant defined by (4.3), and f_BA as the fraction of baryonic matter, $f_{B A} = M_{B A} / K_{NFW}$ (4.9).

In addition $f (x) = \ln (1 + x) - x / (1 + x)$ (4.4) where x = r/R₀ is the dimensionless variable associated to the variable radius.

The direct mass formula for the total mass in the framework of DMbQG theory is

$M_{TOTAL} (< r) = \frac{a^{2} \cdot \sqrt{r}}{G}$ Formula(3.1)

In order to compare both formulas it is needed to use the same dimensionless variable x. So the Equation (3.1) will be changed in this way:

$M_{TOTAL} (< r) = \frac{a^{2} \cdot \sqrt{r}}{G} = \frac{a^{2} \cdot \sqrt{R_{0}}}{G} \cdot \sqrt{x}$

where x = r/R₀ being R₀ the NFW scale radius.

Defining

$K_{T} = \frac{a^{2} \cdot \sqrt{R_{0}}}{G}$ (5.1)

then the direct mass is:

$M_{TOTAL} (< r) = K_{T} \cdot \sqrt{x}$ (5.2)

For example, for the MW galaxy using R₀ = 10.94 kpc, see Table 2, and parameter $a^{2}$ = 1.527 × 10²¹ m^5/2∙s⁻² then $K_{T} = 2.11 \times 10^{11} M_{Θ}$ .

In order to compare the direct mass (5.2) with the NFW total mass (4.10) it is defined f_T = K_T/K_NFW and so

$M_{TOTAL}^{DIRECT} (< r) = f_{T} \cdot K_{NFW} \cdot \sqrt{x}$ (5.3)

This way this function may be compared with

$M_{TOTAL}^{NFW} (< r) = [f_{B A} + f (x)] K_{NFW}$ Formula(4.10)

and If it is cancelled the common factor K_NFW, with mass dimension, both functions become as dimensionless functions factors, written as FM:

$F M_{TOTAL}^{DIRECT} (< r) = f_{T} \cdot \sqrt{x}$ (5.4)

and

$F M_{TOTAL}^{NFW} (< r) = [f_{B A} + f (x)]$ (5.5)

5.4.1. Comparison of Direct Mass Formula with the NFW-Total Mass Formula Using the Sofue Data

For example using the MW Sofue data, see Table 7, may be defined (5.4) and (5.5). The parameter $a^{2}$ comes from Table 1. In Table 13 are summarized all the parameters for the both functions.

Table 13. Parameters for the both function factor FM.

	f_BA	f_T	R_200-TOTAL	R₀ kpc
MW	0.382	0.62	192 kpc	10.94

As the dominion for radius is the halo region from 30 kpc up to 200 kpc, the dominion for the variable x is $x \in [3, 18]$ being x dimensionless.

In Table 14 are tabulated both dimensionless function factor mass and it is shown its relative difference.

Table 14. Direct mass factor function versus NFW-total mass factor function.

Radius kpc	Variable X	Dimensionless Factor Direct mass	Dimensionless factor NFW-total mass	Relative Diff. %
32.82	3	1.07387	1.01829	5.18
43.76	4	1.24000	1.19144	3.92
54.7	5	1.38636	1.34043	3.31
65.64	6	1.51868	1.47077	3.16
76.58	7	1.64037	1.58644	3.29
87.52	8	1.75362	1.69034	3.61
98.46	9	1.86000	1.78459	4.05
109.4	10	1.96061	1.87080	4.58
120.34	11	2.05631	1.95024	5.16
131.28	12	2.14774	2.02387	5.77
142.22	13	2.23544	2.09249	6.39
153.16	14	2.31983	2.15672	7.03
164.1	15	2.40125	2.21709	7.67
175.04	16	2.48000	2.27404	8.30
185.98	17	2.55633	2.32793	8.93
196.92	18	2.63044	2.37907	9.56

Although the relative difference increases continuously, remains below 10% in the whole dominion. See Figure 1.

Figure 1. Function factor of direct mass vs function factor of NFW-total mass. Sofue data.

This relative difference is acceptable because the f_BA is a value with a very high imprecision. In the following epigraph will be used f_BA 0.1479 given by [9] (E. Karukes).

5.4.2. Comparison of Direct Mass Formula with the NFW-Total Mass Formula Using the Karukes Data

In order to compare both formulas are needed the new parameters provided by this author, see Table 8, R₀ = 10.158 kpc, f_BA = 0.147884 and K_NFW = 4.0572269 × $10^{11} M_{Θ}$ .

In addition it is needed the factor f_T = K_T/K_NFW where $K_{T} = \frac{a^{2} \cdot \sqrt{R_{0}}}{G}$ .

As the parameter $a^{2}$ is got by the rotation curve into the halo region and this author did not publish such curve, it is need to use the parameter calculated with [4] Sofue data, see Table 1 parameter $a^{2}$ = 1.527 × 10²¹ so $K_{T} = 2.03584 \times 10^{11} M_{Θ}$ and f_T = 0.50178.

As the dominion for radius is the halo region from 30 kpc up to 198 kpc, the dominion for the variable x is $x \in [3, 20]$ .

Comparing Table 13 and Table 15 it is clear that parameters f_T are lightly different but parameters f_BA are very different because according Karukes the baryonic mass of MW is lower than a half the value considered by Sofue.

Table 15. Parameters for the both function factor FM using Karukes data.

	f_BA	f_T	R₀	R_200-TOTAL
MW	0.147884	0.50178	10.158 kpc	198 kpc

$F M_{TOTAL}^{DIRECT} (< r) = f_{T} \cdot \sqrt{x}$ Formula(5.4)

And

$F M_{TOTAL}^{NFW} (< r) = [f_{B A} + f (x)]$ Formula(5.5)

In Table 16 are tabulated both dimensionless mass function factors and its relative difference. Excepting the value for radius 30.5 kpc the other one’s values have a relative difference below 5%, however with the Sofue data a half of data have its relative difference under 5% and the other half data range between 5% and 10%.

Table 16. Direct mass factor function vs NFW-total mass factor function. Karukes.

Radius kpc	Variable X	Dimensionless Factor Direct mass	Dimensionless Factor NFW-total mass	Relative Diff. %
30.474	3	0.8657	0.7842	9.413
40.632	4	0.9996	0.9573	4.228
50.79	5	1.1176	1.1063	1.008
60.948	6	1.2243	1.2367	−1.014
71.106	7	1.3223	1.3523	−2.268
81.264	8	1.4136	1.4562	−3.013
91.422	9	1.4994	1.5505	−3.407
101.58	10	1.5805	1.6367	−3.556
111.738	11	1.6576	1.7161	−3.529
121.896	12	1.7314	1.7898	−3.374
132.054	13	1.8021	1.8584	−3.126
142.212	14	1.8701	1.9226	−2.809
152.37	15	1.9357	1.9830	−2.442
162.528	16	1.9992	2.0399	−2.038
172.686	17	2.0607	2.0938	−1.606
182.844	18	2.1205	2.1450	−1.155
193.002	19	2.1786	2.1936	−0.691
203.16	20	2.2352	2.2400	−0.218

The cause about this discrepancy is that the baryonic mater in MW has a high level of imprecision. E.g. for Karukes $M_{B A} = 6 \times 10^{10} M_{Θ}$ and for Sofue M_BA = 1.3 × $10^{11} M_{Θ}$ .

However, in my opinion both examples demonstrate the main thesis of this paper:

The direct mass formula is equivalent to NFW formula extended to the total mass, into the halo region of MW from 30 kpc up to 200 kpc.

In Figure 2 is shown how close both functions are.

Figure 2. Function factor of direct mass vs function factor of NFW-total mass. Karukes.

6. Testing the Equivalence between Direct Mass and NFW Total Mass in M31 Halo

In this chapter it will made the same four tests made to MW in the previous chapters but to M31 galaxy. It will be used the NFW DM density profile published by [3] (Sofue, Y. 2015) and the direct mass formula published in [1] (Abarca, M. 2024).

In Table 17 are shown the [2] (Sofue, 2015) data for the M31 NFW DM profile.

Table 17. NFW parameters and baryonic mass for M31. Sofue data.

NFW parameters	R₀ Radius scale	$ρ_{0}$ Characteristic density
Measures of parameters	34.6 ± 2.1 kpc	$1.51 \pm 0.16 \times 10^{- 22}$ kg∙m⁻³
M31 Baryonic mass	$1.6 \times 10^{11} M_{Θ}$

Using the method developed in the epigraph 4.3 now it will be calculated R_200-_TOTAL and M_200-TOTAL for M31.

So from (4.3) formula $K_{NFW} = 1.16 \times 10^{12} M_{Θ}$ and $f_{B A} = M_{B A} / K_{NFW} = 0.14$ .

From Formula (4.7) it is right to get $\frac{c^{3}}{f (c)} = 246.048$ the equation to calculate

numerically the concentration parameter c whose solution is c = 6.579 and f(c) = 1.15734 so R_200-DM = R₀·c = 227.66 Kpc and from the Formula (4.5) it is got M_200-DM = 1.34 × $10^{12} M_{Θ}$ .

The Equation (4.12) allows to calculate the concentration parameter t for the total mass, so $\frac{t^{3}}{f_{BA} + f (t)} = \frac{c^{3}}{f (c)}$ becomes $\frac{t^{3}}{0.14 + f (t)} = 246.048$ whose numerical solution is t = 6.89639.

And so R_200-TOTAL = t × R₀ = 238.6 kpc, in addition 0.14 + f(t) = 1.333.

Finally by (4.10) $M_{200 -TOTAL} = [0.14 + f (t)] K_{NFW} = 1.546 \times 10^{12} M_{Θ}$ .

These two parameters R_200-TOTAL and M_200-TOTAL are the adequate parameters because are linked to the total mass and they verify that the mean density in the R_200-_TOTAL radius sphere is $200 ρ_{C}$ . In Table 18 is checked this property almost with mathematical accuracy.

Table 18. Comparison of virial total mass and virial radius for M31. Test I.

	R_200-TOTAL	M_200-TOTAL $M_{Θ}$	Mean dens/ $200 ρ_{C}$
NFW-total	238.6 kpc	1.546 × 10¹²	0.999595
By parameter $a^{2}$	232.15	1.4245 × 10¹²	0.999959
Relative diff. %	2.7%	7.8%

6.1. Comparison between R_200-TOTAL and M_200-TOTAL Values Got by Direct Mass and NFW Total Mass. Test I

In the framework of DMbQG theory was got the formulas

$M_{200 -TOTAL} = \frac{a^{12 / 5}}{G \cdot {(10 \cdot H)}^{2 / 5}}$ (3.4) and $R_{200 -TOTAL} = {[\frac{a^{2}}{100 \cdot H^{2}}]}^{2 / 5}$ (3.5) so using

the parameter $a^{2}$ for M31 = 2.235 × 10²¹ m^5/2∙s⁻², see Table 1, it is possible to calculate rightly both values. In Table 18 are summarized the value of R_200-TOTAL and M_200-TOTAL got by the two different methods and also it is shown its relative differences, which are very low.

In Table 18 are checked the mean density of the R_200-TOTAL radius sphere got by the two different methods, and the matching versus $200 ρ_{C}$ is almost perfect for both.

6.2. Calculus of Parameter a² Using M_200-TOTAL Got by NFW. Test II

Now using the value calculated for M_200-TOTAL using the NFW-total procedure, it is possible to use such result to calculate the parameter $a^{2}$ by the formula:

$a^{2} = {(G \cdot M_{200})}^{5 / 6} \cdot {(10 \cdot H)}^{1 / 3}$ Formula(3.3)

Then this result may be compared with the parameter $a^{2}$ got in [1] (Abarca, M. 2024) in the framework of DMbQG theory.

In Table 19 are compared both results of parameter $a^{2}$ with an excellent result.

Table 19. Comparing the parameter $a^{2}$ for M31. Test II.

M_200-TOTAL by NFW total	Parameter $a^{2}$	Relative difference
$M_{Θ}$	m^5/2s⁻²	%
1.546 × 10¹²	2.393 × 10²¹	6.6
$a^{2}$ as reference. See Table 1	2.235 × 10²¹

6.3. Calculus of Parameter a² Using M_200-TOTAL and R_200-TOTAL Got by NFW. Test III

In this test the parameter $a^{2}$ is got by the formula $a^{2} = \frac{G \cdot M_{200}}{\sqrt{R_{200}}}$ (3.2) using M_200-TOTAL and R_200-TOTAL got by the NFW-total method.

The result of parameter $a^{2}$ in this test is the same that in the test II because the Formula (3.3) is mathematically equivalent to (3.2) in the framework of DMbQG theory. See Table 20.

Table 20. Comparing the parameter $a^{2}$ for M31. Test III.

NFW-total	R_200-TOTAL	M_200-TOTAL	$a^{2}$ Formula (3.2)	Relative diff.
	kpc	$M_{Θ}$	m^5/2s⁻²	%
	238.6 kpc	1.546 × 10¹²	2.393 × 10²¹	6.6
$a^{2}$ as reference. See Table 1			2.235 × 10²¹

6.4. Comparison of Direct Mass Formula with the NFW-Total Mass Formula into the Halo Region up to R_200-TOTAL. Test IV

As it was shown in the Epigraph 5.4 to compare both formulas of the masses is enough to compare the called dimensionless function factor of masses.

Table 21 is right to define the NFW function mass and his dimensionless function factor mass associated whose formula is:

Table 21. NFW parameters for the dimensionless function factor mass.

Using NFW-Sofue data

R₀

Radius scale

R_200-TOTAL

NFW-total

K_NFW

f_BA = M_BA/K_NFW

kpc

M_{Θ}

34.6

238.6

1.16 × 10¹²

0.14

$F M_{TOTAL}^{NFW} (< r) = [f_{B A} + f (x)]$ Being $x = r / R_{0}$ Formula(5.5)

The data of this table were calculated at the beginning of Chapter 6.

By other side, the dimensionless function factor direct mass formula is:

$F M_{TOTAL}^{DIRECT} (< r) = f_{T} \cdot \sqrt{x}$ Formula(5.4)

Being x = r/R₀ being f_T = K_T/K_NFW and $K_{T} = \frac{a^{2} \cdot \sqrt{R_{0}}}{G}$ .

Being $a^{2}$ = 2.235 × 10²¹ and R₀ = 34.6 kpc then $K_{T} = 5.5 \times 10^{11} M_{Θ}$ and f_T = 0.474.

With these parameters the dimensionless function factor of direct mass is defined.

As the radius dominion is from 40 kpc up to 240 kpc the variable x ranges from 1.2 up to 6.9.

In Table 22 are tabulated both functions into its dominion and its relative difference, that for x bigger than 2 its relative difference is under 15%.

Table 22. Direct mass factor function versus NFW-total mass factor function. M31.

Radius kpc	Variable X	Dimensionless Factor Direct mass	Dimensionless factor NFW-total mass	Relative diff. %
41.52	1.2	0.51924	0.38300	26.238
48.44	1.4	0.56084	0.43214	22.949
55.36	1.6	0.59957	0.48013	19.921
62.28	1.8	0.63594	0.52676	17.168
69.2	2	0.67034	0.57195	14.678
103.8	3	0.82099	0.77629	5.444
138.4	4	0.94800	0.94944	−0.152
173	5	1.05990	1.09843	−3.635
207.6	6	1.16106	1.22877	−5.832
242.2	7	1.25409	1.34444	−7.205

In Figure 3 it is shown how close both functions are into its dominion.

Figure 3. Function factor of direct mass vs function factor of NFW-total mass for M31.

In Table 23 both functions are tabulated from 40 kpc up to 1380 kpc to show how in this dominion so wide, the relative difference remain negligible.

Table 23. Direct mass factor function versus NFW-total mass factor function. M31.

Radius kpc	Variable X	Dimensionless Factor Direct mass	Dimensionless factor NFW-total mass	Relat. Diff %
41.52	1.2	0.5192	0.3830	26.24
51.9	1.5	0.5805	0.4563	21.40
69.2	2	0.6703	0.5719	14.68
86.5	2.5	0.7495	0.6785	9.47
138.4	4	0.9480	0.9494	−0.15
207.6	6	1.1611	1.2288	−5.83
276.8	8	1.3407	1.4483	−8.03
415.2	12	1.6420	1.7819	−8.52
588.2	17	1.9544	2.0859	−6.73
761.2	22	2.2233	2.3190	−4.31
899.6	26	2.4169	2.4729	−2.31
1038	30	2.5962	2.6062	−0.39
1211	35	2.8042	2.7513	1.89
1384	40	2.9978	2.8780	4.00

In Figure 4 is plotted Table 23 and it is shown how close both functions are into a dominion so wide.

According the DMbQG theory the DM grows with the square root of radius without limit. In [1] (Abarca, M. 2024), was demonstrated that the halo of the L. G. is about 2 Mpc, so it is right to study these functions in this wide dominion.

Figure 4. Function factor of direct mass vs function factor of NFW-total mass for M31.

7. Relation between NFW Mass Formula Parameters and the Direct Mass Parameter a²

In this epigraph will be developed a procedure to calculate the parameter $a^{2}$ associated to direct mass using the parameters associated to NFW total mass formula and reciprocally another procedure to calculate the parameters associated to NFW DM mass formula using the parameter $a^{2}$ . This study will be made in the MW and M31 galaxies.

7.1. Milky Way Case

In Table 4 are shown the NFW parameters published by [9] Karukes and in the Epigraph 4.2 were calculated some parameters associated: K_NFW, f_BA the baryonic fraction, R₀ the scale radius and $ρ_{0}$ the characteristic density.

By other side in Table 1 is shown the parameter $a^{2}$ associated to the direct mass for MW, published in [1] Abarca. In the epigraph 5.4.2 is calculated the parameter K_T and f_T both used to link the direct mass with the NFW total mass.

In Table 24 are collected all the parameters used in this epigraph.

Table 24. Parameters of MW linked to NFW total mass and Direct mass formulas.

NFW

K_{N F W} = 4.0572269 \times 10^{11} M_{Θ}

f_BA = 0.147884

R₀ = 10.158 kpc

ρ_{0} = 2.0864 \times 10^{- 21}

kg/m³ units.

Direct

a^{2}

= 1.527 × 10²¹ m^5/2s⁻²

f_T = 0.50178

K_{T} = 2.0358395 \times 10^{11} M_{Θ}

With the previous parameters is possible to do an equation of the dimensionless function factor linked to the direct mass and the NFW total mass. By equation the Formulas (5.4) and (5.5) are got the x values where both functions match mathematically, see Formula (7.1).

$f_{T} \cdot \sqrt{x} = f_{B A} + \ln (1 + x) - \frac{x}{1 + x}$ (7.1)

This equation is quite easy to solve numerically and its solutions are:

$X_{1} = 0.09126639$ , $X_{2} = 5.6428337$ and $X_{3} = 19.6232587$

It is clear that the direct mass and the NFW total mass are functions very similar throughout its dominion, as it was shown in Figure 2.

The value X₂ is used to search a relation between parameter $a^{2}$ and the parameters $ρ_{0}$ and R₀.

The value X₁ has been rejected because does not belong to the halo. In addition the value X₂ is placed in the intermediate region of NFW total mass function dominion. As R₀ = 10.16 kpc then X₂ is equivalent to 57 kpc.

Firstly will be equated the direct mass Formula (5.2) and NFW total mass Formula (4.10) at the point x = x₂ getting the Equation (7.2) in order to calculate the parameter $a^{2}$ .

$\frac{a^{2} \cdot \sqrt{R_{0}}}{G} \cdot \sqrt{x_{2}} = [f_{B A} + f (x_{2})] \cdot 4 π ρ_{0} \cdot R_{0}^{3}$ (7.2)

And clearing up the parameter

$a^{2} = 1.1919608 \frac{4 π G ρ_{0} \cdot R_{0}^{5 / 2}}{\sqrt{x_{2}}} \approx 2.007 \cdot π G ρ_{0} \cdot R_{0}^{5 / 2}$ (7.3)

As x₂ = 5.64283367, f_BA + f(x₂) = 1.1919608 and using the other values of parameters of Table 24 is got $a^{2}$ = 1.527 × 10²¹ that match mathematically with the value of parameter $a^{2}$ of MW, see Table 1.

This way, using the parameters of NFW total mass formula may be got the parameter $a^{2}$ associated to the direct mass formula.

The next challenge is the reciprocal problem which is not so easy because the NFW DM mass formula has two parameters whereas the direct mass has only one.

From Equation (7.2) is cleared up the characteristic density, see Formula (7.4)

$ρ_{0} = \frac{a^{2} \sqrt{x_{2}}}{4 π G R_{0}^{5 / 2} \cdot [f_{B A} + f (x_{2})]}$ (7.4)

The problem is that the scale radius R₀ is a parameter belonging to NFW DM formula. Namely its value in Table 24 is R₀ = 10.16 kpc. In the framework of DMbQG, now it will be considered R₀ = 30 kpc because in [1] Abarca was estimated that such radius may be the halo border where the baryonic mass density versus DM density is negligible.

In this epigraph will be calculated the virial mass M₂₀₀ and R₂₀₀ using such radius and will be shown that the relative difference versus Karukes data, see Table 8, is negligible despite the fact that the value considered now for R₀ is three times bigger i.e. 300% bigger.

In the Formula (7.4) is used the values $a^{2}$ = 1.527 × 10²¹, f_BA + f(x₂) = 1.1919608, x₂ = 5.6428337 and the novelty is to consider R₀ = 30 kpc and then it is got $ρ_{0} = 1.3918977 \times 10^{- 22}$ kg/m³. With that density and R₀ = 30 kpc it is got rightly $K_{NFW} = 6.9725 \times 10^{11} M_{Θ}$

It is clear that the values, density and K_NFW are very different to the same parameters shown in Table 24. However when it is calculated the new parameter c then the virial mass and the radius will be very similar to the same concepts calculated by Karukes, and showed in Table 8 because the new parameter c now will be about three times lower.

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