Using the procedure explained before with the data of paper by Huanget al. [4] it is got in Table 1 the parameter a, which is a 3% lower than the one got by Sofue. See Table 2.

Table 1. Parameter a using the data from Huang et al. (2016) [4] for the MW.

In this table and hereafter, the letter E means the power of 10. e.g. 1.0E+10 = 1.0·10¹⁰ and 1.5E−1= 1.5·10⁻¹; Notice that the maximum of the relative difference regarding the average a² is about 5%.

The data of rotation curve, see [4], table 3 page 2633, have only velocity errors. In average the relative error of velocity is 10% so consequently by the formula (2.2) the relative error associated to a² is 20%.

In Table 2 the rotation curve data (radius and velocity) have been taken from table A2, page 15 of Sofue (2020) [5]. In the last column the parameter a is calculated for each point of the rotation curve and finally its average value a and a² is calculated.

Table 2. Parameter a using the Sofue (2020) data for the MW.

Notice that the maximum of the relative difference regarding the average a² is about 5%.

The data of rotation curve, see [5], page 15, have an average error of velocities about 6% or bigger. The author does not give error of radius, so by the formula (2.2) the relative error of a² is bigger than 12%.

The average a² got using the Sofue data is a 3% bigger than the one got by Huang. See Table 1. Therefore if it is considered the experimental errors both values of parameter a² may be considered as equivalents.

In the framework of NFW method, R₂₀₀ is the radius of a sphere whose mean density of DM is 200 times bigger than the critic density of Universe

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

and M₂₀₀ is the DM mass enclosed by the sphere with the radius R₂₀₀.

Considering the spherical volume formula, it is simple to get the following relations between both concepts.

The NFW profile for the DM density is defined by two parameters: the scale radius and the characteristic density.

It is supposed a spherical symmetry and its formula it is

being ρ 0 a characteristic density, x = r/R₀ a dimensionless magnitude related with radius by R₀, which is called the scale radius.

By integration it is obtained the formula for the Dark matter enclosed by a sphere with radius r.

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

Two important concepts for NFW profiles are M₂₀₀ and R₂₀₀ both referred to DM only.

is called the concentration parameter and R 200 = R 0 ⋅ c

As M 200 R 200 3 = 100 H 2 G see formula (3.4) then M 200 c 3 R 0 3 = 100 H 2 G and 4 ⋅ π ρ 0 ⋅ R 0 3 ⋅ f ( c ) c 3 R 0 3 = 100 H 2 G so

That it is the equation for the concentration parameter. This equation is quite easy to solve numerically, and it is clear that c depend on the characteristic density only.

With this parameter c, it is easily calculated M₂₀₀ and R₂₀₀.

For example, shown in Table 3 are the NFW parameters published by Sofue [5].

Table 3.The NFW parameters for MW by Sofue (2020).

Using the characteristic density it is simple 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 (3.7) K_NFW= 3.4·10¹¹ M_☉ then using (3.9) M₂₀₀ = 6.498·10¹¹ M_☉.

It is easy to check that the density of the sphere with the previous data M₂₀₀ and R₂₀₀ is virtually equal to 200 ρ C .

Since the direct mass formula has only one parameter, knowing the total mass associated with a specific radius is enough to calculate the parameter a².

So from this formula M TOTAL ( < R 200 ) = a 2 ⋅ R 200 G it is got

In the following epigraphs the parameter a² will be calculated using the data provided by three authors.

4.1.1. Calculation of Parameter a² Using the Karukes Virial Data

For example, in Table 4, the virial data for MW published by Karukes et al. (2020) [6] are shown in Table 3, page 15 of his paper.

The first column shows the DM enclosed into the sphere with radius R₂₀₀ and the second one shows the total mass (DM plus baryonic) and consequently its difference gives the total baryonic mass according this author.

Using the value M_200-TOTAL and R₂₀₀ into the formula (4.2) it is got a² = 1.54·10²¹ m^5/2/s² that it is virtually equal to the value got for this parameter in the chapter 2, using the rotation curve data method with the data provided by Huang or Sofue.

Using M_200-DM and R₂₀₀ it is right to check that the average density is virtually 200 ρ C .

Table 4. The virial data for M.W. according Karukes et al. [6].

In Table 5, the parameter a² is calculated using a different set of data pairs, see [6] page 25. The only value to be excluded is that corresponding to 28 kpc, as it deviates significantly from the remaining data points. This discrepancy can be attributed to the fact that the galactic halo is considered to begin at approximately 30 kpc. At radii smaller than 30 kpc, the contribution of baryonic matter is non-negligible, and therefore the direct mass estimation formula is not applicable.

The direct mass formula is so simple because its domain is the halo region, where the baryonic matter density is negligible compared to the dark matter density.

As it is shown in Table 5, into the halo region the parameter a² is virtually constant.

Table 5. Total Mass and its radius according to [6].

4.1.2. Calculation of Parameter a² Using the Sofue Virial Data

In Table 6 are collected the NFW parameters of MW from the paper [5] table 3, page 12. The baryonic mass may be calculated from the figure 3 (for the mass of bulge) and the table 3 for the mass of disc. Notice how different is the baryonic mass regarding the value calculated by Karukes. See Table 4.

Table 6. NFW parameters for M.W. according to [5].

With the same data of Table 6, in the epigraph 3.3 was calculated the virial mass M_200-DM = 6.5·10¹¹ M_☉ and R₂₀₀ = 178.85 kpc.

So adding the baryonic mass it is got M_200-TOTAL = M_200-DM + M_BA=8.3·10¹¹ M_☉ and by the formula (4.2) it is calculated easily the parameter a² = 1.484·10²¹ m^2.5/s². In Table 7 are summarized the results.

Table 7. Parameter a² using the Sofue virial data.

4.1.3. Calculation of Parameter a² Using the Huang Virial Data

In Table 8 are the NFW parameters provided by this author and the baryonic mass of the MW. These data have been got from table 4 at the page 2636 from Huang et al. [4].

Table 8.The NFW parameters for M.W. according Huang (2016).

The equation for the concentration factor is c 3 f ( c ) = 4 π G ρ 0 100 ⋅ H 2 = 1335.50 whose solution is c = 13.2046 and f(c) =1.7240.

Therefore R₂₀₀ = 13.2046·14.39 = 190.014 kpc and M₂₀₀ = K_NFW·f(c) = 7.81·10¹¹ M_☉ where K_NFW is the characteristic mass of NFW profile, K_NFW= 4.531·10¹¹ M_☉.

As M₂₀₀is the DM enclosed by the sphere with radius R₂₀₀, to calculate the total mass it is added the baryonic mass, so M_200-TOTAL= (7.81+0.81)·10¹¹ M_☉ = 8.62·10¹¹ M_☉.

Finally it is possible to calculate the parameter a² using M_200-TOTAL andR₂₀₀ obtaining by the formula (4.2) a² = 1.4949·10²¹ m^2.5/s² that it is very close to the one got in the table 1, a² = 1.4942·10²¹ m^2.5/s².

In Table 9 are summarized the results.

Table 9. Parameter a² using the virial data by Huang et al. (Over density 200).

In the paper [4], the authors give a different virial data because they have considered an over density = 95 times the critic density of the Universe instead 200. In Table 10 are shown the virial data provided by the authors. See the table 4 in the page 2636. The procedure to calculate the parameter a² is the same.

Table 10. Parameter a² using the virial data by Huang et al. (Over density 95).

This value of the parameter a² is only 1.8% lower than the value one got by the Huang rotation curve data, see Table 1.

6.1.1. Calculation of Parameter a² Using the Rotation Curve Data of Sofue

In the chapter 2 of paper [1], it is introduced the rotation curve data published by [7] Sofue, Y. (2015) and in the chapter 8 of paper [1] it is calculated the parameter a using such rotation curve. Namely a² = 2.235·10²¹ m^5/2/s² is the value obtained by the rotation curve data of M31.

Unfortunately, the author does not provide the numerical data for the rotation curve, and the associated errors cannot be reliably inferred from the graph alone. As a result, it is not possible to conduct an error analysis for this method when calculating the parameter a².

6.1.2. Virial Data and Calculation of Parameter a² by the Direct Mass Formula

In the table 2 of paper [7], it is introduced the NFW parameters shown in Table 12.

Table 12. The NFW parameters for M31 according to Sofue [7].

Now it will be calculated the virial data by the NFW method. The equation for the concentration factor is c 3 f ( c ) = 4 π G ρ 0 100 ⋅ H 2 = 246.048 whose solution is c = 6.579 and f(c) =1.1573.

Therefore R₂₀₀ = 6.579·34.6 = 227.63 kpc and M₂₀₀ = K_NFW · f(c) = 1.342·10¹² M_☉ where K_NFW is the characteristic mass of the NFW DM profile, K_NFW = 1.16·10¹² M_☉.

As M₂₀₀is the DM enclosed by the sphere with radius R₂₀₀, to calculate the total mass, and the baryonic mass is added, so M_200-TOTAL = (1.34 + 0.16)·10¹² M_☉ = 1.50·10¹² M_☉.

Finally, it is possible to calculate the parameter a² using M_200-TOTAL and R₂₀₀ getting the value a² = 2.377·10²¹ m^2.5/s² that it is only 6% bigger than the value one got by the Sofue rotation curve data, whose value is a² = 2.235·10²¹ m^5/2/s².

In this case, the error analysis is more complex because the total mass depends on the parameter c, which is obtained by solving a transcendental equation involving c³. The characteristic density D₀ carries an uncertainty of approximately 10%. Consequently, the total mass must have an error significantly larger than 10%, which in turn propagates to the parameter a². Despite this, the relative difference in a² between the two methods is only 6%. Therefore, it can be concluded that both methods are consistent, i.e., the NFW profile is compatible with a velocity decay law in the halo region of the form v = a·r^−0.25, mirroring what was found in the chapter 5 for the MW galaxy.

These authors provide both the rotation curve data and the virial data, each with known uncertainties. The error analysis in this case is somewhat lengthy and is presented in the following sections.

6.2.1. Calculation of Parameter a² Using the Rotation Curve Data of Zhang et al.

On page 9 of Zhang et al. [8], the rotation curve data are tabulated. In Chapter 5 of Abarca [1], it is shown that for galactocentric radii exceeding 40 kpc, the baryonic mass density becomes negligible in comparison to the dark matter (DM) density. Consequently, the parameter a² is evaluated starting at 40 kpc. Nevertheless, the velocity measurements at 46 kpc and 52 kpc reported by Zhang et al. [8] are anomalously low and have therefore been excluded from the analysis.

In Table 13, the remaining rotation curve data are presented. Their corresponding a² values are calculated, along with the average a².

Table 13. Calculus of a² using the rotation curve of M31 data from Zhang et al. [8].

The average a² is a 19% lower than the one calculated by the Sofue rotation curve data (epigraph 6.1.1).

Error analysis ofa²based on rotation curve data errors

From the formula of parameter a² = v²·r^0.5, it is easy to calculate its relative error d(a²)/a² = 2dv/v + (1/2)·dr/r where dv/v is the relative error of the velocity and dr/r is the relative error of radius.

In Table 14 are shown the rotation curve data and its errors provided by Zhang et al.

Table 15 presents the rotation curve data along with their associated errors, expressed in SI units. Table 15 also includes the relative error of parameter a² for each data point, with an average value of 43%.

Considering this relative error, the value of a² obtained by Zhang (1.8·10²¹) is reasonably consistent with that reported by Sofue (2.2·10²¹), since their relative difference is only 20%.

Table 14. Rotation curve data from Zhanget al. [8].

Table 15. R.C. with its data errors into SI units and relative error of a².

6.2.2. Calculation of Parameter a² Using the Direct Mass Formula

In the graphic of page 8 of paper [8], it is plotted the rotation curves associated to the disc and the bulge mass, so by a simple calculus it is got the baryonic mass of M31, M_BA = 10.5·10¹⁰ M_☉. In addition, in the page 11 are given the virial data associated to DM purely, see Table 16 below.

Table 16. Virial data and baryonic mass of M31 according to Zhang et al. [8].

As it is shown in the virial data, Table 16, the radius error is about 11%, the up error of mass is 45% and the low error one is 31%.

Using such data it is got M_200-TOTAL = 1.245·10¹² M_☉ and as in the previous epigraphs, the parameter a² it is calculated by the equation (4.2), a² = 2.006·10²¹m^2.5/s². This value is only a 10% bigger regarding the value got by the rotation curve method a² = 1.8·10²¹ m^2.5/s² (see epigraph 6.2.1).

Error analysis ofa²based on the virial data errors

As it was shown in the chapter 5, the relative error of the parameter a² got by the formula (4.2) is d(a²)/a² = dm/m + (1/2)·dr/(r) and considering the relative error of the virial data (see Table 16) is obtained a relative error of the parameter a² shown in Table 17.

Table 17. Relative error of parameter a².

6.2.3. Equivalence of a² Obtained by the Two Different Methods Using the Zhang’s Data

In Table 18 are summarized the results obtained in the two previous epigraphs.

Table 18. The a² obtained by the two different methods and its errors.

If we consider the relative difference in the parameter a² between the two results—approximately 10%—alongside its own relative uncertainties (around 40%), it is reasonable to conclude that the two results are equivalent.

Once again, the values of a² obtained through the two different methods reinforce the central thesis of this work: the NFW dark matter profile is consistent with the velocity decay law v = a·r^−0.25in the halo region of M31, mirroring the conclusion obtained in Chapter 5 for the Milky Way galaxy.

The principal difficulty in obtaining adequate rotation curves for these galaxies arises from their large distances. As a consequence, the velocity measurements are affected by larger uncertainties, and the radial extent over which the rotation curves are reliably measured is more limited.

The DMbQG framework requires only a single parameter, a, to define the direct mass formula. Therefore, knowledge of a single point on the rotation curve is, in principle, sufficient to determine the value of a. However, this point must be located at a sufficiently large radius, where most of the baryonic mass is enclosed within the corresponding sphere.

In the work of Vijayakumar et al. [9], three different rotation curves are presented, see Figure 1, in which it is possible to distinguish between the contribution of the stellar disc and that associated with the neutral hydrogen (H I) gas. When the gas rotation curve exhibits a declining behavior similar to that of the stellar disc, it can be inferred that the majority of the galactic baryonic mass is enclosed within the outer radius.

Figure 1 shows three different rotation curves, each graphic have three curves, the total velocity traced by white dots, the velocity associated to the stellar disc traced by the yellow line and the velocity associated to the gas HI, traced by the green line.

It is clear that NGC 3521 is the only galaxy whose gas rotation curve associated to the gas displays a “Keplerian-like” decline at the largest radii. Consequently, it is the only rotation curve from which data can be reliably extracted to determine the parameter a within the framework of the DMbQG theory because it is assumed that the majority of the baryonic mass is enclosed within the outer radius.

Figure 1. Three different rotation curves clipped from paper by Vijayakumar et al. [9].

In this section, the NFW method is compared with the Direct Mass approach. It is ultimately concluded that both methods are equivalent when the range of measurement uncertainties associated with the rotation curve point at 40 kpc is taken into account.

The data, see Table 19, and Figure 2 of NGC 3521 comes from Vijayakumar et al. [9].

Table 19. NFW parameters for the NGC 3521 galaxy.

Figure 2 shows the merged rotation curves and the joint stellar (in yellow)-gas (in green)-dark matter fitting results (in magenta), with the stellar component rescaled according to the maximal disk assumption.

Figure 2. R.C. of NGC 3521 galaxy clipped from paper [9].

Baryonic mass calculus

In the table 1 of Vijayakumar et al. [9], the authors inform about the stellar mass of NGC 3521 M_* = 10¹¹ M_☉. In addition they inform that its distance is 13.2 Mpc.

As the green line in the graphic represents the rotation curve associated to the gas and at the 40 kpc its slope is similar to the keplerian rotation curve associated to the stars, the yellow line, then it is possible to estimate the total mass of the gas by the dynamical mass formula.

At 40 kpc the estimated velocity of the gas is 61.7 km/s so the M_GAS = 3.5·10¹⁰ M_☉ and consequently the estimated total baryonic mass for NGC 3521 is M_BA = 1.35·10¹¹ M_☉

Virial data calculus by the NFW method

By the equation c 3 f ( c ) = 4 π G ρ 0 100 ⋅ H 2 = 349 being f ( c ) = ln ( 1 + c ) − c / ( 1 + c ) it is possible to calculate the concentration parameter c.

So the equation c³/f(c) = 349 that solved numerically gives the solution c = 7.626 and consequently R₂₀₀ = c·R_S = 175.4 kpc; V₂₀₀ =10·H·R₂₀₀ = 122.8 km/s and M 200 = V 200 2 ⋅ R 200 / G = 6.1 × 10 11   M ☉ . Adding the baryonic mass it is obtained the value M_200-TOTAL = 7.45·10¹¹ M_☉ enclosed within the virial radius R₂₀₀ = 175.4 kpc

Comparison Virial total mass by NFW versus Direct mass for NGC 3521

By Figure 2, it is estimated the velocity of rotation curve for the total mass at 40 kpc equal to 168.9 km/s and by the formula a² = V²·R^0.5 it is obtained the value a² = 1·10²¹ m^5/2/s² in the framework of DMbQG theory.

Through the direct mass, formula (4.1) it is calculated the total mass associated to the radius R₂₀₀ = 175.4 kpc obtaining the value M_DIRECT (<R₂₀₀) = 5.54·10¹¹ M_☉

Table 20 compares the virial total mass by the NFW method and the Direct mass.

Table 20. Comparison Virial total mass by NFW versus Direct mass.

This relative difference between the virial total masses is compatible with the thesis of this work because this galaxy is quite far away and the measures of rotation curve have high errors. The reader can check how big the error bars are in Figure 2, especially the ones associated to radius bigger than 35 kpc.

Therefore one more time this result backs the thesis of this work i.e. the NFW method is compatible with a decaying velocity into the halo region according the law v = a·v^−0.25, which is the central hypothesis of DMbQG theory for the NGC 3521 galaxy.

Calculus of theR_200-TOTALand theM_200-TOTALin the framework of DMbQG

In the paper by Abarca, M. [3], the formulas are derived:

M 200 -TOTAL = a 12 / 5 G ⋅ ( 10 ⋅ H ) 2 / 5 and R 200 -TOTAL = [ a 2 100 ⋅ H 2 ] 2 / 5 (see the formulas 3.3 and 3.4) so knowing the parameter a² it is right to calculate such galactic parameters. See Table 21.

Table 21. Virial data for NGC 3521 galaxy by the parameter a².

In the paper of Sorgho et al. [10] there are published two rotation curves where it is discriminated the rotation curves associated to the stellar disc and the gas. It has been selected NGC 3621 because its distance is only 6.6 Mpc whereas the other one is 13.6 Mpc far away. Obviously, the more near the more trustable the measures are.

The data of Table 22 and Figure 3 of NGC 3621 comes from Sorgho, A. et al. [10].

Table 22. NFW parameters for the galaxy NGC 3621 [10].

Baryonic mass calculus

As the green line in the graphic represents the rotation curve associated to the gas and at the 47 kpc its slope is similar to the keplerian rotation curve associated to the stars disk, the blue line, then it is possible to estimate the total mass of the gas by the dynamical mass formula.

Figure 3. Rotation curve of NGC 3621 galaxy.

At 47 kpc the estimated velocity of the gas is 30.8 km/s so the M_GAS = 1·10¹⁰ M_☉.

At 47 kpc the estimated velocity of the stars curve is 40 km/s so M_*= 1.75·10¹⁰ M_☉ and consequently the total baryonic mass for NGC 3621 is M_BA = 2.75·10¹⁰ M_☉.

Virial data calculus by the NFW method

As R₂₀₀ = 162.7 kpc and V₂₀₀ =10·H·R₂₀₀ = 113.9 km/s then M₂₀₀ = 4.9·10¹¹ M_☉ Adding the baryonic mass it is obtained the value M_200-TOTAL = 5.17·10¹¹ M_☉ enclosed within the virial radius R₂₀₀ = 162.7 kpc.

Comparison Virial total mass by NFW versus Direct mass for NGC 3621

As at the maximum radius, 47 kpc, the rotation curves of stars and the gas are keplerian, then may be considered that the majority of baryonic mass is enclosed within the radius 47 kpc and therefore it is possible to calculate the parameter a² using the point of the rotation curve at 47 kpc.

By Figure 3 it is estimated the velocity of rotation curve at 47 kpc equal to 154 km/s and by the formula a² = R^0.5·V² it is obtained the value a² = 9·10²⁰ m^5/2/s².

Through the direct mass, formula (4.1) it is calculated the total mass associated to the radius R₂₀₀ = 162.7 kpc obtaining the value M_DIRECT (<R₂₀₀) = 4.8·10¹¹ M_☉

Table 23 compares the virial total mass by the NFW method and the Direct mass.

Table 23. Comparison Virial total mass by NFW versus Direct mass.

Therefore one more time this result backs the thesis of this work i.e. the NFW method is compatible with a decaying velocity into the halo region according the law v = a·v^−0.25, which is the central hypothesis of DMbQG theory for the NGC 3621 galaxy.

Relation formula between the NFW parameters and the parameter a in NGC 3621

In the epigraph (7.3) of the paper by Abarca, M. [3], it is shown that a 2 ≈ 2 π G ρ 0 ⋅ R 0 5 / 2 is a very good relation between the parameters of NFW method and the parameter a² for the MW and M31 galaxies.

Here it will be shown that this relation is quite close for the galaxy NGC 3621 as well.

Through the data R₂₀₀ = 162.7 kpc and c = 5.3 it was calculated R₀ = 30.7 kpc and M₂₀₀ = 4.9·10¹¹ M_☉ and using these data, through the formula of virial mass M 200 -DM = M DM ( < R 200 ) = K NFW ⋅ f ( c ) being K NFW = 4 π ρ 0 R 0 3 and f ( c ) = 0.99928 it is possible to calculate ρ 0 = 9.13 × 10 − 23 kg / m 3 .

So the expression 2 ⋅ π ⋅ G ⋅ ρ 0 ⋅ R 0 2.5 = 1.057 × 10 21   m 2.5 ⋅ s − 2 and this value is quite close to the parameter a² = 9·10²⁰ m^5/2 /s² obtained in the previous paragraph. Namely the relative difference is below 15%.

In conclusion, it can be stated that for these galaxies far from the Local Group, the virial mass calculated using the NFW method is consistent with a power-law decay, with an exponent of −0.25, for the rotation curve associated with the total mass in the halo region of these galaxies.