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.

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

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.

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 $a^2=\frac{G\cdot M_{\text{VIRIAL}}}{\sqrt{R_{\text{VIRIAL}}}}$ it is right to get the parameter $a^2$ depending on M₂₀₀ only

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

$M_{200-TOTAL}=\frac{a^{12/5}}{G\cdot {\left(10\cdot H\right)}^{2/5}}$ (3.4)

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

$R_{200-TOTAL}={\left[\frac{a^2}{100\cdot H^2}\right]}^{2/5}$ (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 .

As $\frac{M_{200}}{R_{200}^3}=\frac{100H^2}{G}$ then $\frac{M_{200}}{c^3{R}_0^3}=\frac{100H^2}{G}$ and $\frac{4\cdot \pi {\rho }_0\cdot R_0^3\cdot f\left(c\right)}{c^3{R}_0^3}=\frac{100H^2}{G}$ so

$\frac{c^3}{f\left(c\right)}=\frac{4\pi G{\rho }_0}{100\cdot H^2}$ (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.

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) $K_{\text{NFW}}=3.4\times {10}^{11}{M}_{\Theta }$ then using (4.5) $M_{200\text{-DM}}=6.498\times {10}^{11}{M}_{\Theta }$.

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

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

${\rho }_0=\frac{100\cdot H^2{c}^3}{f\left(c\right)\cdot 4\cdot \pi \cdot G}$ (4.8)

in addition R₀ = R₂₀₀/c. This way, knowing ${\rho }_0$ 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. $M_{\text{BA-MW}}=6\times {10}^{10}{M}_{\Theta }$ according to this author.

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

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 $M_{200\text{-DM}}=M_{\text{DM}}\left(<R_{200}\right)=K_{\text{NFW}}\cdot f\left(c\right)$ then $K_{\text{NFW}}=4.057\times {10}^{11}{M}_{\Theta }$.

As $K_{\text{NFW}}=4\pi {\rho }_0{R}_0^3$ then ${\rho }_0=2.086\times {10}^{-21}\text{kg}\cdot {\text{m}}^{-3}$.

According to [9] (E. Karukes, 2020), the data of Table 4 about masses means that the baryonic mass enclosed by R₂₀₀ is $6\times {10}^{10}{M}_{\Theta }$. However for [2] (Sofue, 2015) the baryonic mass for the MW is $1.3\times {10}^{11}{M}_{\Theta }$ 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.

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 $200{\rho }_C$, 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 $200{\rho }_C$ as it was expected. In Table 9 it will be shown that the mean density of M_200-TOTAL into R_200-TOTAL is $200{\rho }_C$, 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.

$M_{\text{TOTAL}}\left(<r\right)=M_{BA}+M_{\text{DM}}\left(<r\right)=M_{BA}+K_{\text{NFW}}\cdot f\left(x\right)$

If it is defined

$f_{BA}=M_{BA}/K_{\text{NFW}}$ (4.9)

then

$M_{\text{TOTAL}}\left(<r\right)=\left[f_{BA}+f\left(x\right)\right]K_{\text{NFW}}$ (4.10)

As the sphere whose mean density is $200{\rho }_C$ verify

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

And defining a new parameter t

$t=R_{200\text{-TOTAL}}/R_0$ (4.11)

It is got $\frac{\left[f_{BA}+f\left(t\right)\right]\cdot 4\pi \cdot {\rho }_0\cdot R_0^3}{R_0^3\cdot t^3}=\frac{100H^2}{G}$ that leads to the expression $\frac{t^3}{f_{BA}+f\left(t\right)}=\frac{4\pi G\cdot {\rho }_0}{100\cdot H^2}$ that by (4.7) leads to

$\frac{t^3}{f_{BA}+f\left(t\right)}=\frac{c^3}{f\left(c\right)}$ (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.

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

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

So $R_{200\text{-TOTAL}}=t\cdot R_0=191.745\text{kpc}$ and as $f\left(t\right)=1.9732$ the $M_{200\text{-TOTAL}}=\left[0.382+f\left(t\right)\right]K_{\text{NFW}}=8.0077\times {10}^{11}{M}_{\Theta }$.

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

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

So $R_{200\text{-TOTAL}}=t\cdot R_0=198.273\text{kpc}$.

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

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.

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

$M_{200\text{-TOTAL}}=\frac{a^{12/5}}{G\cdot {\left(10\cdot H\right)}^{2/5}}$ (3.4) and $R_{200\text{-TOTAL}}={\left[\frac{a^2}{100\cdot H^2}\right]}^{2/5}$ (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 $200{\rho }_C$, which differs from unity in thousands for the three values.

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:

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

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%).

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 $a^2$.

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.

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

In addition $f\left(x\right)=\ln \left(1+x\right)-x/\left(1+x\right)$ (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

$M_{\text{TOTAL}}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}$ 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:

$M_{\text{TOTAL}}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}=\frac{a^2\cdot \sqrt{R_0}}{G}\cdot \sqrt{x}$

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:

$M_{\text{TOTAL}}\left(<r\right)=K_T\cdot \sqrt{x}$ (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 $K_T=2.11\times {10}^{11}{M}_{\Theta }$.

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

$M_{\text{TOTAL}}^{\text{DIRECT}}\left(<r\right)=f_T\cdot K_{\text{NFW}}\cdot \sqrt{x}$ (5.3)

This way this function may be compared with

$M_{\text{TOTAL}}^{\text{NFW}}\left(<r\right)=\left[f_{BA}+f\left(x\right)\right]K_{\text{NFW}}$ 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:

$F{M}_{\text{TOTAL}}^{\text{DIRECT}}\left(<r\right)=f_T\cdot \sqrt{x}$ (5.4)

and

$F{M}_{\text{TOTAL}}^{\text{NFW}}\left(<r\right)=\left[f_{BA}+f\left(x\right)\right]$ (5.5)

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.

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

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

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

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).

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 × ${10}^{11}{M}_{\Theta }$.

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 $K_T=2.03584\times {10}^{11}{M}_{\Theta }$ 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 $x\in \left[3,20\right]$.

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.

$F{M}_{\text{TOTAL}}^{\text{DIRECT}}\left(<r\right)=f_T\cdot \sqrt{x}$ Formula (5.4)

And

$F{M}_{\text{TOTAL}}^{\text{NFW}}\left(<r\right)=\left[f_{BA}+f\left(x\right)\right]$ 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%.

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

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.

In the framework of DMbQG theory was got the formulas

$M_{200\text{-TOTAL}}=\frac{a^{12/5}}{G\cdot {\left(10\cdot H\right)}^{2/5}}$ (3.4) and $R_{200\text{-TOTAL}}={\left[\frac{a^2}{100\cdot H^2}\right]}^{2/5}$ (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 $200{\rho }_C$ is almost perfect for both.

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:

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

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 .

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:

$F{M}_{\text{TOTAL}}^{\text{NFW}}\left(<r\right)=\left[f_{BA}+f\left(x\right)\right]$ 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:

$F{M}_{\text{TOTAL}}^{\text{DIRECT}}\left(<r\right)=f_T\cdot \sqrt{x}$ 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 $K_T=5.5\times {10}^{11}{M}_{\Theta }$ 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%.

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

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.

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.

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 ${\rho }_0$ 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.

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).

$f_T\cdot \sqrt{x}=f_{BA}+\ln \left(1+x\right)-\frac{x}{1+x}$ (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 ${\rho }_0$ 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 $a^2$.

$\frac{a^2\cdot \sqrt{R_0}}{G}\cdot \sqrt{x_2}=\left[f_{BA}+f\left(x_2\right)\right]\cdot 4\pi {\rho }_0\cdot R_0^3$ (7.2)

And clearing up the parameter

$a^2=1.1919608\frac{4\pi G{\rho }_0\cdot R_0^{5/2}}{\sqrt{x_2}}\approx 2.007\cdot \text{π}G{\rho }_0\cdot R_0^{5/2}$ (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)

${\rho }_0=\frac{a^2\sqrt{x_2}}{4\pi G{R}_0^{5/2}\cdot \left[f_{BA}+f\left(x_2\right)\right]}$ (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 ${\rho }_0=1.3918977\times {10}^{-22}$ kg/m³. With that density and R₀ = 30 kpc it is got rightly $K_{\text{NFW}}=6.9725\times {10}^{11}{M}_{\Theta }$

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.

The Formula (4.7) $\frac{c^3}{f\left(c\right)}=\frac{4\pi G{\rho }_0}{100\cdot H^2}=226.80343$ allows to calculate the new

parameter c knowing the new density. This equation is right to solve numerically and its solution is c = 6.35435 and f(c) = 1.1312659.

By (4.6) R₂₀₀ = R₀·c so R₂₀₀ = 30·c = 190.63 kpc and

By (4.5) $M_{200\text{-DM}}=K_{\text{NFW}}\cdot f\left(c\right)=7.8878\times {10}^{11}{M}_{\Theta }$

In Table 25 are compared the new virial mass and radius with the Karukes ones.

In the last row are shown the ratios: the mean density of the sphere R₂₀₀ versus $200{\rho }_C$.

To illustrate how stable the virial mass and radius are regarding the parameter R₀, now will be remake the virial data using R₀ = 20 kpc.

Using Formula (7.4) now ${\rho }_0=3.835619\times {10}^{-22}$ and K_NFW = 5.69303 × ${10}^{11}{M}_{\Theta }$.

Now the Formula (4.7) $\frac{c^3}{f\left(c\right)}=\frac{4\pi G{\rho }_0}{100\cdot H^2}=624.9968$ whose solution is c = 9.709302 and f(c) = 1.4644895 and finally:

By (4.6) R₂₀₀ = R₀·c so R₂₀₀ = 20·c = 194.186 kpc.

By (4.5) $M_{200-DM}=K_{NFW}\cdot f\left(c\right)=8.3374\times {10}^{11}{M}_{\Theta }$ and the ratio $\overline{\text{Den}}/200{\rho }_C$ = 1.0000006.

These results show that the election of parameter R₀ is quite flexible in the procedure used in this section.

In Table 17 are shown the NFW parameters published by [2] Sofue and in the Chapter 6 were calculated some parameters associated: K_NFW and f_BA, the baryonic fraction.

By other side in Table 1 is shown the parameter $a^2$ associated to the direct mass for M31, published in [1] Abarca. In the Epigraph 6.4 is calculated the parameter K_T and f_T both used to link the direct mass with the NFW total mass.

In Table 26 are collected all the parameters used in this epigraph.

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 (7.5).

$f_T\cdot \sqrt{x}=f_{BA}+\ln \left(1+x\right)-\frac{x}{1+x}$ (7.5)

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

$X_1=0.091984$, $X_2=3.96548$ and $X_3=30.826$

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 4 .

The value X₂ is used to search a relation between parameter $a^2$ and parameters ${\rho }_0$ 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 = 34.6 kpc then X₂ is equivalent to 137 kpc.

Firstly will be equated the direct mass Formula (5.2) and the NFW total mass Formula (4.10) at the point x = x₂ getting the Equation (7.6) in order to calculate $a^2$.

$\frac{a^2\cdot \sqrt{R_0}}{G}\cdot \sqrt{x_2}=\left[f_{BA}+f\left(x_2\right)\right]\cdot 4\pi {\rho }_0\cdot R_0^3$ (7.6)

And clearing up the parameter

$a^2=0.9439\frac{4\pi G{\rho }_0\cdot R_0^{5/2}}{\sqrt{x_2}}\approx 1.896\cdot \pi G{\rho }_0\cdot R_0^{5/2}$ (7.7)

As x₂ = 3.96548, f_BA + f(x₂) = 0.9439 and using the other values of parameters of Table 26 is got $a^2$ = 2.235 × 10²¹ that matches with the value of parameter $a^2$ of M31, see Table 1 .

This way, using the parameters of NFW total mass formula may be got the parameter $a^2$ associated to 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 (7.6) is cleared up the characteristic density, see Formula (7.8)

${\rho }_0=\frac{a^2\sqrt{x_2}}{4\pi G{R}_0^{5/2}\cdot \left[f_{BA}+f\left(x_2\right)\right]}$ (7.8)

The problem is that the scale radius R₀ is a parameter belonging to NFW DM formula. Namely its value in Table 26 is R₀ = 34.6 kpc. However, in the framework of DMbQG will be considered R₀ = 40 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 it will be shown that the relative difference versus the Sofue data, see the Chapter 6, is negligible.

In the Formula (7.8) are used the values $a^2$ = 2.235 × 10²¹, f_BA + f(x₂) = 0.9439, x₂ = 3.96548 and the novelty is to consider R₀ = 40 kpc and then it is got ${\rho }_0=$ $1.0506\times {10}^{-22}$ kg/m³ Using this value got for density and R₀ = 40 kpc it is got rightly $K_{\text{NFW}}=1.2475\times {10}^{12}{M}_{\Theta }$

It is clear that the values, density and K_NFW are lightly different to the same parameters shown in Table 26 . 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 at the beginning of Chapter 6 because the new parameter c now is lightly lower.

The Formula (4.7) $\frac{c^3}{f\left(c\right)}=\frac{4\pi G{\rho }_0}{100\cdot H^2}=171.19$ allows to calculate the new parameter c knowing the new density, this equation is right to solve numerically and its solution is c = 5.63 and f(c) = 1.0424.

By (4.6) R₂₀₀ = R₀·c so R₂₀₀ = 40·c = 225.2 kpc and

By (4.5) $M_{200\text{-DM}}=K_{\text{NFW}}\cdot f\left(c\right)=1.3\times {10}^{12}{M}_{\Theta }$.

In Table 27 are compared the new virial mass and radius with the Sofue ones.

In the last row are shown the ratios between the mean density of the sphere R₂₀₀ versus $200{\rho }_C$. Notice how the new virial values give an excellent approximation to $200{\rho }_C$.

To illustrate how stable the virial mass and radius are regarding the parameter R₀, now will be remake the virial data using R₀ = 30 kpc instead of 40 kpc.

Using the Formula (7.8) now with R₀ = 30 kpc it is right to get ${\rho }_0=2.1566568$ $\times {10}^{-22}$ kg/m³ and $K_{\text{NFW}}=1.08035\times {10}^{12}{M}_{\Theta }$

The Formula (4.7) becomes $\frac{c^3}{f\left(c\right)}=\frac{4\pi G{\rho }_0}{100\cdot H^2}=351.4175$ and the new solution for the parameter c is c = 7.647964 and f(c) = 1.272958.

By (4.6) R₂₀₀ = R₀·c so R₂₀₀ = 30·c = 229.439 kpc and

By (4.5) $M_{200\text{-DM}}=K_{\text{NFW}}\cdot f\left(c\right)=1.375\times {10}^{12}{M}_{\Theta }$

The ratio $\overline{\text{Den}}/200{\rho }_C=1.0000027$ which is an excellent result.

As it was expected the new virial data are very similar to the ones shown in Table 27 , so it has been demonstrated that the virial data are very stable regarding the R₀ considered in the procedure followed to calculate the virial mass and radius using the parameter $a^2$ as initial data.

In the previous study made for MW the results were even more spectacular because that time the value for R₀ was 300% bigger and the new virial mass and radius have a relative difference of 1% and 4.5% respectively. See Table 25 .

The Formulas (7.3) for MW galaxy and (7.7) for M31 galaxy may be approximated with accuracy by the formula

$a^2\approx 2\pi G{\rho }_0\cdot R_0^{5/2}$ (7.9)

This is a simple formula to link the direct mass parameter with the NFW parameters.

In order to check such formula will be used the data from three different authors.

The first one is [10] Ou, Necib et al. In this paper is published the MW virial data and the gNFW parameters for the rotation curve ( Table 28 ).

Through the normalization mass M₀ may be got the characteristic density ${\rho }_0=1.78\times {10}^{-21}\text{kg}/{\text{m}}^3$.

Knowing the scale radius R₀ and ${\rho }_0$, using the Formula (7.9) it is possible to calculate rightly the parameter $a^2$ = 1.38 × 10²¹ m^5/2/s² and using such value into the Formulas (3.4) and (3.5) it is got rightly the $M_{200\text{-TOTAL}}=8\times {10}^{11}{M}_{\Theta }$ and R_200-TOTAL = 191.6 kpc. Both values match fully with the values published in [10] if it is considered the range of errors.

These virial data results show that (7.9) is a good approximation for the parameter $a^2$.

A second test may be done with the MW data published by [11] (Labini 2024) ( Table 29 ).

By the Formula (7.9) it is calculated the parameter $a^2$ = 1.15 × 10²¹ m^5/2/s² and by the Formulas (3.4) and (3.5) it is got rightly the $M_{200\text{-TOTAL}}=6.4\times {10}^{11}{M}_{\Theta }$ and R_200-TOTAL = 178 kpc. The matching with the virial mass is almost perfect, which is a good test for the Formula (7.9).

A third test will be made with the M31 data published by [12] (Zhan 2024) ( Table 30 ).

Using R_VIR, by (4.6) it is got the scale radius R₀ = 25.26 kpc, by (4.8) and the parameter c it is got the characteristic density

${\rho }_0=2.939\times {10}^{-22}\text{kg}/{\text{m}}^3$

Now by (7.9) it is estimated the parameter $a^2$ = 2.09 × 10²¹ m^5/2/s².

Finally by (3.4) and (3.5) it is got $M_{200\text{-TOTAL}}=1.3\times {10}^{12}{M}_{\Theta }$ and R_200-TOTAL = 226 kpc.

Both values match fully with the virial data given by the author if it is considered the range of errors.

Therefore one more time it is tested the Formula (7.9) as a very good approximation for the parameter $a^2$.