In Table 2 , the first and second column represent the data set of Table 1 , the third and fourth columns are the same data into the I.S. of units. The fifth column shows the velocity by the fitted function and the last column shown the relative difference between velocity data and velocity fitted by the power regression function.

In Table 3 are shown the parameters of the fitted function velocity versus radius, third and fourth columns, using a power regression function.

In Figure 2 are plotted the data set of Table 2 in grey colour, also it is represented the fitted function by a power regression whose parameters are shown in Table 3 .

The correlation coefficient is 0.96 which is a superb result especially when dominion measures are up to 303 kpc.

There is not any other galaxy with a rotation curve data set so wide.

According to the DMbQG theory, the galaxy halo is unbounded, but as the distance MW and M31 is 770 kpc, it is right to consider that inside the data set radius of M31, i.e. from 40 kpc up to 300 kpc the gravitational influence of Milky Way may be neglected as a first approximation, using a model of spherical symmetry for the gravitational field.

In the halo region, the rotation curve it is fitted by power regression with a high correlation coefficient according to formula $v=a\cdot r^b$. As in the Newtonian framework, $M_{DYNAMIC}\left(<r\right)=\frac{v^2\cdot R}{G}$ represents total mass enclosed by a sphere with radius r, by substitution of velocity results $M=\frac{v^2\cdot R}{G}=\frac{a^2\cdot r^{2b+1}}{G}$. Hereafter this formula will be called Direct Mass

$M_{DIRECT}\left(<r\right)=\frac{a^2\cdot r^{2b+1}}{G}$ (3.1)

because it has been got rightly from rotation curve.

If it is considered the halo region where baryonic matter is negligible regarding dark matter it is possible to calculate DM density by a simple derivative. In the next chapter will be show that for r > 40 kpc baryonic matter is negligible.

As density of D.M. is $D_{DM}=\frac{\text{d}m}{\text{d}V}$ where $\text{d}m=\frac{a^2\cdot \left(2b+1\right)\cdot r^{2b}\text{d}r}{G}$ and $\text{d}V=4\pi r^2\text{d}r$ then results $D_{DM}=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}\cdot r^{2b-2}$. Writing $L=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}$ results

$D_{DM}\left(r\right)=L\cdot r^{2b-2}$ (3.2)

This formula will be called Direct DM density. If b = −1/2 then the L parameter is zero and the DM density will be zero, which is the Keplerian rotation.

Parameters a & b from power regression of M31 rotation curve, see Table 3 , allow calculate easily the direct DM density.

Direct DM density for M31 halo 40 < r < 300 kpc (3.3)

$D_{DM}\left(r\right)=L\cdot r^{2b-2}$ kg/m³ being L = 1.1255E+30 and 2b − 2= −2.4964529.

It is important to highlight that this formula is only a statistical approximation of DM density able to explain the rotation curve, into the dominion 40 kpc up to 300 kpc.

As it is known the total gravitational field, considering spherical symmetry, is the centripetal acceleration, E = v²/R whose unit is m/s² (I.S.). Hereafter, the gravitational field will be represented by E.

The key to use the centripetal acceleration to calculate E is the dynamical equilibrium. It is supposed that celestial bodies are quite close to dynamical equilibrium, because the most of celestial bodies belong to a specific galactic system from its formation times.

By substitution of $v=a\cdot r^b$ into the formula $E=\frac{v^2}{r}$ it is got $E=\frac{a^2\cdot r^{2b}}{r}$, briefly

$E=a^2\cdot r^{2b-1}$ (4.1)

According to the hypothesis of dark matter by quantum vacuum

$D_{DM}=A\cdot E^B$ (4.2).

i.e. the DM density at a point depend on the gravitational field according to a power function. The parameters A & B have to be calculated. This hypothesis has been widely studied by the author in previous papers: [2] Abarca, M.2014. [6] Abarca, M. 2016. [7] Abarca, M.2019 and [8] Abarca, M. 2023. This hypothesis fulfils the physic meaning of D.M. Density formula in the halo region because it is supposed that such D.M. density is generated as a consequence of gravitational field propagation in the framework of a quantum gravitation theory.

The formula (3.2) is $D_{DM}=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}\cdot r^{2b-2}$, the direct DM density depend on a & b parameters which come from power regression formula for velocity. In the previous epigraph has been got the formula for $E=a^2\cdot r^{2b-1}$ (4.1) which depend on a & b as well.

As D_DM depend on parameters a, b similarly to gravitational field E, then it is possible to get the formulas for the variable change of D_DM. i.e. D_DM depending on radius, formula (3.2) is changed into D_DM depending on E, formula $D_{DM}=A\cdot E^B$ (4.2).

Through a simple mathematical treatment it is possible to get the coefficients A & B depending on a & b, specifically these formulas are:

$A=\frac{a^{\frac{2}{2b-1}}\cdot \left(2b+1\right)}{4\pi G}$ & $B=\frac{2b-2}{2b-1}$, (4.3)

According to parameters, a & b shown in Table 3 , in Table 4 are shown A & B coefficients for M31 galaxy in the halo region.

Conversely

$b=\frac{B-2}{2B-2}$ and $a={\left[\frac{4\pi GA\left(B-1\right)}{2B-3}\right]}^{\frac{2b-1}{2}}$ being $B\ne 1$ and $B\ne 3/2$, (4.4)

As conclusion, in this chapter has been demonstrated that a power law for the rotation velocity $v=a\cdot r^b$ is mathematically equivalent to a power law for DM density depending on E. $D_{DM}=A\cdot E^B$ on condition that D.M. is generated by the gravitation field.

This formula is the cornerstone for DMbQG theory because it is supposed that DM is generated locally according to an unknown quantum gravity mechanism. In other words, the propagation of gravitational field has this additional effect on the space as the gravitational wave goes by.

The formulas $D_{DM}=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}\cdot r^{2b-2}$ and $E=a^2\cdot r^{2b-1}$ have been got rightly from rotation curve. Therefore it can be considered more specific for each galaxy. However the formula $D_{DM}=A\cdot E^B$ is much more essential.

The basis of this theory is that such formula is right for different gravitational systems. Therefore A & B parameters have to be similar for different galaxies on condition that the galaxies are similar. In further chapters will be got that power B is exactly the same for M31 and Milky Way although parameter A will be a bit different.

However, there is an important fact to highlight. It is clear that A depend on a and b, both parameters are global parameters. As the gravitational interaction time between masses is proportional to distance, it is right to think that DM generated by a gravitational field has a bigger proportion as the system increase its size. For example, inside the Solar system it is clear that Newton and General Relativity Theory are able to explain with total accuracy every gravitational phenomenon without DM hypothesis. Therefore, it is right to conclude that DM is non negligible when gravitational interaction takes a longer time to link the matter. Namely, at galactic scale.

The formula (4.2) $D_{DM}=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}\cdot r^{2b-2}$ is a local formula because it has been got by differentiation. However E, which represents a local magnitude

$E=\frac{G\cdot M\left(<r\right)}{r^2}=\frac{a^2\cdot r^{2b}}{r}=a^2\cdot r^{2b-1}$ has been got through $v=a\cdot r^b$ whose parameters a & b were got by a regression process on the whole dominion of rotation speed curve. Therefore, the D_DM formula has a character more local than E formula because the former was got by a differentiation process whereas the later involves M(<r) which is the mass enclosed by the sphere of radius r.

In other words, the process of getting D_DM involves a derivative whereas the process to get E(r) involves M(<r) which is a global magnitude. This is a not suitable situation because the formula $D_{DM}=A\cdot E^B$ involves two local magnitudes. Therefore it is needed to develop a new process with a more local nature or character.

It is clear that a differential equation for E is the best method to study locally such magnitude.

As it is known in this formula $\stackrel{\to }{E}=-G\frac{M\left(r\right)}{r^2}\hat{r}$, M(r) represents mass enclosed by a sphere with radius r. If it is considered a region where does not exist any baryonic matter, such as any galactic halo, then the derivative of M(r) depend on dark matter density essentially and therefore

$M^{\prime }\left(r\right)=4\pi r^2{D}_{DM}\left(r\right)$, (6.1)

If $E=G\frac{M\left(r\right)}{r^2}$, vector modulus, is differentiated then it is got $E^{\prime }\left(r\right)=G\frac{M^{\prime }\left(r\right)\cdot r^2-2rM\left(r\right)}{r^4}$

If $M^{\prime }\left(r\right)=4\pi r^2{D}_{DM}\left(r\right)$ is replaced above then it is got $E^{\prime }\left(r\right)=4\pi G{D}_{DM}\left(r\right)-2G\frac{M\left(r\right)}{r^3}$ As $D_{DM}\left(r\right)=A\cdot E^B\left(r\right)$ it is right to get

$E^{\prime }\left(r\right)=4\pi \cdot G\cdot A\cdot E^B\left(r\right)-2\frac{E\left(r\right)}{r}$ (6.2)

which is a Bernoulli differential equation.

$E^{\prime }\left(r\right)=K\cdot E^B\left(r\right)-2\frac{E\left(r\right)}{r}$ Being

$K=4\pi \cdot G\cdot A$ (6.3)

Calling y to E, the differential equation is written in this simple way

$y`=K\cdot y^B-\frac{2\cdot y}{r}$ (6.4)

The Bernoulli differential Equation (6.4) may be converted into a differential linear equation with this variable change $u=y^{1-B}$ and this is the linear equation:

$\frac{u`}{1-B}+\frac{2u}{r}=K$ (6.5)

The homogenous equation is

$\frac{u`}{1-B}+\frac{2u}{r}=0$ (6.6)

and its general solution is

$u=C\cdot r^{2B-2}$ (6.7)

being C the integration constant.

If it is searched a particular solution for the complete differential Equation (6.5) with a simple linear function u = z·r then it is got that

$z=\frac{K\cdot \left(1-B\right)}{3-2B}$. (6.8)

Therefore the general solution for (6.5) is $u=C\cdot r^{2B-2}+z\cdot r$

When it is inverted the variable change it is got the general solution for (6.3)

And it is got

$E\left(r\right)={\left(C{r}^{2B-2}+\frac{K\left(1-B\right)\cdot r}{3-2B}\right)}^{\frac{1}{1-B}}\text{with}B\ne 1\text{and}B\ne 3/2$ (6.9)

where C is the parameter of initial condition of gravitational field at a specific radius.

Calling $\alpha =2B-2$, $\beta =\frac{1}{1-B}$ and $D=\frac{K\left(1-B\right)}{3-2B}$ then

$E\left(r\right)={\left(C{r}^{\alpha }+Dr\right)}^{\beta }$ (6.10)

Calculus of parameter C through initial conditions R₀ and E₀

Suppose R₀ and E₀ are the specific initial conditions for radius and E, then

$C=\frac{E_0^{1/\beta }-D\cdot R_0}{R_0^{\alpha }}$ (6.11)

Final comment

It is clear that the Bernoulli solution contains implicitly the fact that it is supposed there is not any baryonic matter inside the radius dominion and the only DM matter is added by $D_{DM}\left(r\right)=A\cdot E^B\left(r\right)$. Therefore this solution for field works only in the halo region and R₀ and E₀ could be the border radius of galactic disk where it is supposed begins the halo region and the baryonic density is negligible, although it is possible to select any other point belonging to the halo.

As it is supposed that DM density as power of E come from a quantum gravity theory, it is right to think that constant Plank h should be considered and universal constant of gravitation G as well.

So the elements for dimensional analysis are D, density of DM whose units are kg/m³, E gravitational field whose units are m/s², G and finally h.

Table 6 developed dimensional expression for these four elements D, E, G and h.

According to Buckingham theorem it is got the following formula for Density

$D=\frac{K}{\sqrt[7]{G^9\cdot h^2}}\cdot E^{\frac{10}{7}}$ Being K a dimensionless number which may be understood as a coupling constant between field E and DM density.

As it is shown in Table 4 , for M31 galaxy, the parameter B = 1.6682469

So the relative difference between B = 1.6682469 and 10/7 ≈ 1.428 is 16.7% which is not excessive, however in the following epigraph will be demonstrated the impossibility of this formula in the framework of the theory.

The formulas (4.3) are $A=\frac{a^{\frac{2}{2b-1}}\cdot \left(2b+1\right)}{4\pi G}$ and $B=\frac{2b-2}{2b-1}$. Being a, b the parameters got to fit rotation curve of velocities $v=a\cdot r^b$

As A has to be a positive quantity, see formula (4.2), then $2b+1>0$. As $2b+1=\frac{2B-3}{B-1}>0$ therefore mathematically $B\in \left(-\infty ,1\right)\cup \left(3/2,\infty \right)$.

If B = 3/2 then 2b + 1 = 0 and A = 0 so dark matter density is cero which is the Keplerian rotation case.

In every galactic rotation curve studied, B parameter has been bigger than 3/2. See Abarca papers quoted in Bibliographic references. Therefore the experimental data got in several galaxies match perfectly with mathematical findings made in this paragraph, namely for $B\in \left(3/2,\infty \right)$.

The main consequence of this mathematical analysis is that formula

$D=\frac{K}{\sqrt[7]{G^9\cdot h^2}}\cdot E^{\frac{10}{7}}$

got with only one pi monomial has to be rejected because B = 10/7 < 3/2.

As consequence of the previous theorem it is compulsory to explore a new formula with two pi monomials.

As this formula come from quantum gravitation theory it is right to include c, the velocity of light as the additional Universal constant required in this analysis. So the elements to make dimensional analysis are D, E, G, h and c ( Table 7 ).

According to Buckingham theorem, as matrix rank is three, there are two pi monomials. The first one was calculated in previous epigraph and the second one involves G, h, E and c. These are both pi monomials ${\pi }_1=D\cdot \sqrt[7]{G^9\cdot h^2}\cdot E^{-\frac{10}{7}}$ and ${\pi }_2=\frac{c}{\sqrt[7]{G\cdot h}}{E}^{-\frac{2}{7}}$. So according to the Buckingham theorem the formula for DM density through two pi monomials will be $D_{DM}=\frac{J}{\sqrt[7]{G^9\cdot h^2}}\cdot E^{\frac{10}{7}}\cdot f\left({\pi }_2\right)$ being J a dimensionless number and $f\left({\pi }_2\right)$ an unknown function, which cannot be calculated by the dimensional analysis theory.

This formula is physically more acceptable because it is got considering G, h and c as universal constant involved in formula of DM density. As the hypothesis of the theory estates that DM is generated through a quantum gravitation mechanism it is right to consider not only G and h but also c as it is supposed that gravitons are virtual particles whose velocity is c.

It is right to think that $f\left({\pi }_2\right)$ should be a power of ${\pi }_2$, because it is supposed that density of D.M. is a power of E. In other words, to select the function $f\left({\pi }_2\right)$ it is used the Ockham’s razor principle.

Taking in consideration the A & B parameters fitted to M31 halo galaxy, see Table 4 , it is right to consider 5/3 as the best simple number closer to the empirical value B = 1.6682

To achieve this goal the power for ${\pi }_2$ must be −5/6. This way, power of E in formula $D_{DM}=A\cdot E^B$ will be 5/3 and the formula of Density with two pi monomial $D_{DM}=\frac{J}{\sqrt[7]{G^9\cdot h^2}}\cdot E^{\frac{10}{7}}\cdot f\left({\pi }_2\right)$ becomes

$D_{DM}=\frac{M}{\sqrt[6]{G^7\cdot c^5\cdot h}}{E}^{\frac{5}{3}}$ (7.1)

being M a dimensionless number.

Table 3 comes from chapter 2 and represents regression curve of velocity depending on radius.

Due to the Buckingham theorem it is needed that b = −1/4. Therefore, it is needed to recalculate the parameter a in order to find a new couple of values a &b that fit perfectly to the experimental measures in M31 halo.

Recalculating A with the Minumun Square Method

When it is searched the parameter a, a method widely used is called the minimum squared method. So it is searched a new parameter a for the formula v = a·r^0.25 on condition that ${\displaystyle \underset{e}{\sum }{\left(v-v_e\right)}^2}$ has a minimum value. Where v represents the value fitted for velocity formula, r_e represents each radius measure and v_e represents its velocity associated. So using the data set in Table 2 , third and fourth columns, it is right to get the value for parameter a.

$a=\frac{{\displaystyle \underset{e}{\sum }Ve\cdot r_e^{-0.25}}}{{\displaystyle \underset{e}{\sum }{r}_e^{-0.5}}}=4.727513\times {10}^{10}{\text{m}}^{5/4}/\text{s}$

From now on it is possible to consider the dimension precise for parameter a, as this parameter is the coefficient for the formula of velocity v = a·r^–1/4 so its dimension will be m^5/4/s and for the parameter a² its dimension will be m^5/2/s².

This parameter a² will be used continuously in the following chapters.

With these new parameters recalculated, see Table 8 , it is going to get the new direct formulas for DM density, field E and total mass.

The formula (3.3) for DM Density depending on radius now becomes:

$D_{DM}\left(r\right)=L\cdot r^{2b-2}=L\cdot r^{\frac{-5}{2}}$ (8.3)

Being $L=\frac{a^2\cdot \left(2b+1\right)}{4\pi G}=\frac{a^2}{8\cdot \pi \cdot G}=1.33\times {10}^{30}\text{kg}/{\text{m}}^{1/2}$

The formula (4.1) for field E now becomes:

$E=a^2\cdot r^{2b-1}=a^2\cdot r^{\frac{-3}{2}}$ (8.4)

Being $a^2=2.235\times {10}^{21}{\text{m}}^{\text{5}/\text{2}}/{\text{s}}^{\text{2}}$

The formula (3.1) for direct mass now becomes:

$M_{DIRECT}\left(<r\right)=\frac{a^2\cdot r^{2b+1}}{G}=\frac{a^2\cdot \sqrt{r}}{G}$ (8.5)

In the chapter 6 was got the solution for field in the halo region, now thanks dimensional analysis it is possible to simply the Bernoulli formulas, using the Table 8 . Namely:

The formula (6.10) for field $E\left(r\right)={\left(C{r}^{\alpha }+Dr\right)}^{\beta }$ being $\alpha =2B-2=\frac{4}{3}$ and $\beta =\frac{1}{1-B}=\frac{-3}{2}$ becomes

$E\left(r\right)={\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-3}{2}}$ (8.6)

The formula (6.11) for the initial condition $C=\frac{E_0^{1/\beta }-D\cdot R_0}{R_0^{\alpha }}$ becomes

$C=\frac{E_0^{\frac{-2}{3}}-D\cdot R_0}{R_0^{\frac{4}{3}}}$ (8.7)

The parameter D becomes $D=\frac{4\cdot \pi \cdot G\cdot A\left(1-B\right)}{3-2B}=8\cdot \pi \cdot G\cdot A$ and as $A=\frac{a^{\frac{-4}{3}}}{8\pi G}$ (8.1) then

$D=a^{\frac{-4}{3}}=5.85\times {10}^{-15}{\text{m}}^{-5/3}\cdot {\text{s}}^{4/3}$ (8.8)

Bernoulli Solution for Dm Density in Halo Region

As the formula for DM density is $D_{DM}=A\cdot E^B$, formula (4.2), being B = 5/3 it is right to get the new formula for DM density.

$D_{BERNI}\left(r\right)=A\cdot E^{{}^{\frac{5}{3}}}=A{\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-5}{2}}=\frac{D}{8\pi G}{\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-5}{2}}$ (8.9)

Formula below expresses the dark matter contained inside a spherical corona defined by

R₁ and R₂ belonging at halo.

$\begin{array}{c}{M}_{BERNI}={\displaystyle \underset{R_1}{\overset{R_2}{\int }}4\pi \cdot r^2\cdot D_{BERNI}\text{d}r}={\displaystyle \underset{R_1}{\overset{R_2}{\int }}4\pi \cdot r^{2}A{E}^B\text{d}r}\\ =4\pi A{\displaystyle \underset{R_1}{\overset{R_2}{\int }}{r}^2{\left[C\cdot r^{4/3}+D\cdot r\right]}^{\frac{-5}{2}}\cdot \text{d}r}\end{array}$

The indefinite integral

$I=4\pi A\cdot {\displaystyle \int \frac{r^2}{{\left(C\cdot r^{4/3}+D\cdot r\right)}^{2.5}}}=\frac{8\pi A\sqrt{r}}{D\cdot {\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}=\frac{\sqrt{r}}{G\cdot {\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}$ (8.10)

as $\frac{8\pi A}{D}=\frac{1}{G}$

By the Barrow’s rule over the primitive (8.10) is got the DM contained into the spherical corona defined by R₂ and R₁.

$\underset{R_1}{\overset{R_2}{M_{BERNI}}}=M_{BERNI}\left(R_2\right)-M_{BERNI}\left(R_1\right)$ (8.11)

The name for this theorem has been chosen because the relation between field E and total mass M(<r) is the same that in Newton’s theory.

From Bernoulli field (8.6) $E_{BERNI}\left(r\right)={\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-3}{2}}=\frac{1}{r^{3/2}\cdot {\left(C\cdot r^{1/3}+D\right)}^{3/2}}$

Bernoulli mass formula (8.10) $M_{BERNI}\left(r\right)=\frac{\sqrt{r}}{G\cdot {\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}$ so $G\cdot M_{BER}\left(r\right)=\frac{\sqrt{r}}{{\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}$ and $\frac{G\cdot M_{BER}\left(r\right)}{r^2}=\frac{\sqrt{r}}{r^2{\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}=\frac{1}{r^{3/2}\cdot {\left(C\cdot r^{1/3}+D\right)}^{3/2}}=E\left(r\right)$

Therefore

$E_{BERNI}\left(r\right)=\frac{G\cdot M_{BER}\left(r\right)}{r^2}$ (8.12)

This identity shows how the DMbQG theory, adding an extra of mass depending on radius, is able to explain the DM in galaxies and clusters in the Newtonian framework.

Corollary

According to the Newtonian framework, the function of mass included in the formula of field E (8.12) means the total mass included inside the sphere with radius r. Therefore M_BER(r) must be renamed as M_BER(<r) = M_TOTAL(<r) where r ranges in the halo region i.e. M_BER(r) gives the total mass enclosed by the sphere with radius r.

In conclusion, the total mass enclosed by the sphere with radius r is given by the formula:

$M_{TOTAL}\left(<r\right)=\frac{\sqrt{r}}{G\cdot {\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}$ (8.13)

where r belong to the halo region. i.e. r > 40 kpc for M31 galaxy.

Put in brief, the formula (8.13) is a corollary of the identity (8.12)

Notice how in the formula the variable M(r) has been changed by M(<r) and the subscript Berni has been changed by Total.

This parameter, see formula (8.7), may be calculated by the data set of rotation curve in halo region, see Table 1 .

Theoretically any point belonging to the data set may be used as initial condition, and the value got must be the same.

E₀ is the gravitational field at R_O radius. Considering that data measures are in dynamical equilibrium, it is possible to estimate the field by $E_0=v_0^2/r_0$ using data in Table 1 .

Finally the parameter D = 5.85E−15 was calculated in epigraph 8.3 using the value B = 5/3 got by dimensional analysis.

It is clear that the data set are not in perfect dynamical equilibrium, but data selected are quite close to the dynamical equilibrium because the celestial bodies in the halo region are linked gravitationally to M31 since billions years ago.

In Table 9 it is calculated C₀ for every point of data set.

It is remarkable that all the values C₀ are close all of them and they are close to 0 as well.

Some values of parameters C₀ are negatives because these points are above the fitted curve, see Figure 2 , its points plotted are numbered from left to the right.

The points of Table 1 plotted in Figure 2 verify that the more close to the curve the point is, the smaller, in absolute value, the parameter C₀ is. The parameter C₀ corresponding to point 12 is the smaller. This close oscillation around the fitted curve suggests strongly that the data of Table 1 belong to celestial bodies very close to dynamical equilibrium.

This property of data set has inspired the following theorems of epigraph 8.7 and a surprising corollary in the 8.8 one.

Taking into account that by the dimensional analysis the exponent of rotation curve has to be –1/4 and the results got in previous epigraph about the values of parameter C, it is natural to consider that exist an ideal rotation curve whose power is –1/4 and the data set of Table 1 are very close to this curve. Notice that the fitted curve in Figure 2 has a power very close to –1/4, in fact differ 18 ten-thousandth only.

Definition. Hereafter, it will be named Buckingham halo curve to the points (r, v) r belonging to the halo region and its velocity is $v=a\cdot r^{-1/4}$ being a, the parameter associated to the galactic halo.

As the exponent –1/4 was got by the Buckingham theorem, it has been select such name for that curve.

In Table 10 , the columns first, second and fourth come from Table 2 , the third column is got with the formula $V_0=a\cdot R_0^{-1/4}$. The parameter a_M31 comes from Table 8 .

The fifth column shows the relative difference of velocities between the data and the ideal curve.

Velocity at radius 302.9 kpc has the lowest relative difference, 0.8% only.

In Figure 3 is plotted the ideal curve called Buckingham halo curve.

Direct Theorem: If it is supposed that a point belonging to Buckingham halo curve is in dynamical equilibrium and if it is selected such point as initial point to calculate C, formula (8.7), then such parameter is zero.

Proof: Suppose a point (R₀, V₀) belonging to Buckingham halo curve, then $V_0=a\cdot R_0^{-1/4}$ As dynamical equilibrium leads to $E_0=\frac{GM\left(<r\right)}{r^2}=\frac{V_0^2}{r}$ then $E_0=a^2\cdot R_0^{-3/2}$ and $E_0^{\frac{-2}{3}}=a^{-4/3}\cdot R_0=D\cdot R_0$ because , see (8.8). Therefore C = 0 because its numerator is zero, formula (8.7).

It is important to highlight that it is considered the hypothesis of dynamical equilibrium for the Buckingham halo curve. However this is a very plausible supposition by the reasons explained in this epigraph and the previous one.

Reverse Theorem

If it is selected a point (R₀, V₀) which is supposed to be in dynamical equilibrium and its parameter C = 0 then such point belong to Buckingham halo curve.

Proof: Suppose that $V_0=a\cdot R_0^b$ being the exponent b unknown.

If C = 0 then $E_0^{-2/3}=D\cdot R_0$ and as there is dynamical equilibrium $E_0=\frac{V_0^2}{r}$ then $E=\frac{a^2\cdot r^{2b}}{r}=a^2\cdot r^{2b-1}$ and $E_0^{-2/3}=a^{-4/3}\cdot R_0^{\frac{2-4b}{3}}$ so $a^{-4/3}\cdot R_0^{\frac{2-4b}{3}}=D\cdot R_0$ using the formula (8.8) it is got: $D\cdot R_0^{\frac{2-4b}{3}}=D\cdot R_0$ which leads to $R_0=R_0^{\frac{2-4b}{3}}$

So b = –1/4

Final comments

The data of rotation curve do not belong to Buckingham halo curve by two reasons:

The first one it is simple: measures have experimental errors. The second one is more subtle: the celestial bodies are not in perfect dynamical equilibrium. It is right to think that celestial bodies which belong to M31 gravitational system from its formation times, more than ten billions years ago, will be closer to dynamical equilibrium regarding other ones that were captivated by the gravitational field of M31 afterwards.

Watching the Figure 2 , it is clear that point 1 and point 3 at 40.5 kpc and 58.4 kpc are the points more distant regarding Buckingham halo curve. This important difference regarding dynamical equilibrium curve may be explained by the asymmetries of gravitational field during the history of dynamic evolution. Anyway it is undeniable that in general data are very close to Buckingham halo curve, see Table 10 .

The reader can check that in the paper [8] Abarca, M.2019 the DMbQG theory is essentially developed except the formula for total mass that is the Bernoulli mass instead of Direct mass. In practice, the relative differences between both formulas are totally negligible. However be able to demonstrate that the Direct mass is the Bernoulli formula when parameter C is zero means a better understanding of DM phenomenon. In addition allows continuing the development of the theory more easily, for example when the theory is extended to cluster of galaxies. See chapter 12.

Thanks the results got in the epigraphs 8.6 and 8.7 is natural to consider parameter C = 0 into the Bernoulli formulas.

For the Field E

When in formula $E\left(r\right)={\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-3}{2}}$ (8.6) C = 0 then it is got $E=a^2\cdot r^{\frac{-3}{2}}$ because D^-3/2 = a² which is precisely direct formula for E, (8.4).

For the D.M. Density

As D_DM = A·E^5/3 Using field got by Bernoulli solution it is right to get $D_{DM}\left(r\right)=A{\left(C{r}^{\frac{4}{3}}+Dr\right)}^{\frac{-5}{2}}$ (8.9) Being $A=\frac{D}{8\pi G}$ and $D=a^{\frac{-4}{3}}$ if C = 0 then formula becomes $D_{DM}\left(r\right)=A\cdot D^{\frac{-5}{2}}\cdot r^{\frac{-5}{2}}=L\cdot r^{\frac{-5}{2}}$ Being $L=\frac{a^2}{8\cdot \pi \cdot G}$ which is the direct DM density formula, (8.3).

For the Direct Mass Formula

If C = 0 then $M_{BERNI}\left(<r\right)=\frac{\sqrt{r}}{G\cdot {\left(C\cdot \sqrt[3]{r}+D\right)}^{\frac{3}{2}}}$ (8.13) becomes $M_{DIRECT}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}$ (8.5) because D^-3/2 = a²

As the Direct mass formula are a particular case of Bernoulli formulas when parameter C =0 and in the epigraph 8.5 the corollary of Newton’s theorem claims that the Bernoulli mass is the total mass, see formula (8.13) then it is concluded that the Direct mass is the total mass at a specific radius.

$M_{DIRECT}=M_{TOTAL}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}$ (8.14)

Final comment

According to findings got in this chapter it is clear that the DMbQG claims that the total mass is unbounded and grows up with the square root of radius.

In addition the total mass formula depends on the parameter a solely, instead two parameters C and D associated to Bernoulli formulas, which is a magnificent simplification of the theory.

Finally it is very important to highlight that from now on, the direct formulas have unbounded dominion as they are a particular case of Bernoulli formulas. Notice that before this epigraph the dominion of direct formulas was associated to the data, i.e. from 40 kpc up to 300 kpc.

As it is known, dynamical mass represents the total mass enclosed by a sphere with a radius r in order to produce a balanced rotation with a specific velocity at such radius, so it is right to consider dynamical mass as the total mass, baryonic and DM mass, enclosed at radius R. Ranging the radius in the interval of radius measured.

The formula of dynamical mass is $M_{DYN}\left(<r\right)=\frac{V^2\cdot r}{G}$ and $M_{DIRECT}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}$ being a²/G = 3.35 × 10³¹ (I.S. units)

Using the data of Table 2 , radius and velocities into the I.S. of units, it is got the results shown in Table 11 .

In Figure 4 are plotted both masses, Points over the line represent the direct masses and the other ones the dynamical masses.

The first and third points have the maximum difference regarding fitted curve whereas relative differences decreased as radius increased. Namely, relative differences are below 10% for radius bigger than 120 kpc and are below 2% for radius bigger than 210 kpc.

So direct mass is a very good approximation for dynamical mass enclosed at radius R, ranging the radius in the interval of radius measured.

In this epigraph will be shown that relative difference between both kinds of formulas is negligible through an unbounded dominion. Remember that in epigraph 8.8 was shown that the direct mass (8.5) is a particular case of Bernoulli mass (8.13)

In the corollary of Newton’s theorem, epigraph 8.5, was demonstrated that the Bernoulli mass is the total mass enclosed by a sphere with radius r. Now it will be shown that relative differences between Bernoulli and direct mass are quite small, even for extended haloes.

In the paper [9] Abarca, M.2023, epigraph 9.6 it is developed a method to calculate the optimal parameter C

In Table 12 are tabulated both functions and its relative difference. It is remarkable that even at 2 Mpc its difference is only 3.8%, despite the fact that its dominion has been extended 7 times.

Final comment

It is clear that the epigraphs 8.7 and 8.8 have stated that the direct mass is the most suitable formula to calculate the total mass in the halo region. Despite this fact this epigraph has been made to show that the difference between both kinds of masses is negligible. In addition, the reader interested, may read in [9] Abarca, M. 2023 the process to get the optimal parameter C for the Bernoulli mass formula.

As it is known, Bernoulli solution is only right into the halo region, where the baryonic mass is negligible.

To calculate baryonic volume density has been used the model provided by Sofue for the baryonic disc of MW.

The data below comes from [3] Sofue. They are the parameters for baryonic matter at disc in Milky Way.

Where $\Sigma \left(r\right)={\Sigma }_0\exp \left(-r/a_d\right)$ represents the superficial density at disc region. Similarly as it was done for M31 galaxy, to convert superficial baryonic density to volume density it is right to get the formula $D_{BARYONIC}^{VOLUME}=\frac{\Sigma \left(r\right)}{2r}$.

So $D_{BARYONIC}^{VOLUME}\left(30.5\text{kpc}\right)=1.34\times {10}^{-24}\text{kg}/{\text{m}}^{\text{3}}$, the formula of Direct Dark matter density was got in epigraph 8.8. $D_{DM}\left(r\right)=L\cdot r^{\frac{-5}{2}}$ Being L_MILKY-WAY = 9.1E+29, according with parameter a got in epigraph 11.3, see Table 14 .

For example D_DM (30.5 kpc) = 3.35E−23 kg/m³. So the ratio of both volume density (baryonic versus DM) at 30.5 kpc is 0.04.

In conclusion: it is right to consider negligible the baryonic density for radius bigger than 30.5 kpc, therefore it is possible to estate that halo dominion begins at 30.5 kpc for Milky Way.

The data of rotation curve of Milky Way, Table 13 , comes from [3] Sofue.2020, and there have been selected data with radius bigger than 30 kpc. i.e. its halo region.

This new set of Sofue data is very important for the theory of DMbQG theory because gives a rotation curve at halo region with a power for radius, whose exponent is very close to –1/4, which is the same for M31.

This fact backs strongly the hypothesis of this theory.

In his previous paper [1] Sofue, Y.2015, the author gave an extended dominion up to 300 kpc, However data with radius bigger than 100 kpc have too high velocity and the fitted power function did not fit properly with exponent –1/4.

The logical explanation about the “bad” behaviour of these data is to consider that the celestial bodies more far away than 100 kpc are not in dynamical equilibrium. Perhaps they came from the outskirts of MW and were captivated by MW gravitational field afterwards so it is right to consider that these data are far away to be in dynamical equilibrium, whereas celestial bodies below 100 kpc of radius are properly in dynamical equilibrium with Milky Way.

Anyway, the important data are those closer, because it is right to think that celestial objects with lower radius belong to MW halo from times of MW formation so these objects may have a better dynamic equilibrium with MW.

Figure 5 plotted the data of Table 13 and the statistical fitted function. According to the statistical power regression v = a·r^b being a = 3.68918E+10 and b = –0.248717 It is remarkable its high correlation coefficient.

In addition the exponent b is almost identical to the one associated to M31 galaxy and it is very close to –1/4

Using the method developed in chapter 8, it has been stated b = –1/4 so it is needed to recalculate the parameter a, using the formula for a optimal, see epigraph 8.1

$a_{OPTIMAL}=\frac{{\displaystyle \underset{e}{\sum }Ve\cdot r_e^{-0.25}}}{{\displaystyle \underset{e}{\sum }{r}_e^{-0.5}}}=3.90787373\times {10}^{10}\approx 3.9\text{E}10{\text{m}}^{5/4}/\text{s}$

So the optimal parameter a_M-W = 3.9E+10 m^5/4/s

Which is lightly bigger compared with the one associated to b = –0.248717, see the formula written in the Figure 5 .

According to the hypothesis of DMbQG theory, the parameter a is similar for similar galaxies, for example a_M31 = 4.7275E+10 m^5/4/s

However parameter b is the same, so it is right to consider that parameter b is constant for different types of galaxies.

Table 14 shows the most important parameters for MW at the halo region.

As it has been demonstrated in the chapter 8, the direct mass, formula (8.14) is the total mass enclosed by a sphere with a radius r.

In Table 15 , the total mass is tabulated form 30 kpc up to 2 Mpc.

$M_{TOTAL}\left(<r\right)=\frac{a^2\cdot \sqrt{r}}{G}$ Being $a^2/G=2.2885\times {10}^{31}\text{kg}/{\text{m}}^{1/2}$ (10.1)

In this section will be compared the results got by direct mass, formula (10.1) with the results published in the prestigious Journal of Cosmology and Astroparticle Physics by [10] E.V. Karukes et al. 2020 in the paper A robust estimate of the Milky Way mass from rotation curve data .

Results come from GAIA DR2 and others remarkable sources.

Table 16 made the comparison only with the four radiuses bigger than 30 kpc, as DMbQG theory only works in the halo region.

In the last column is shown the relative difference between both results of mass, being quite small indeed.

It is awesome how a simple theory which associates only one parameter a to the galactic halo, which has been calculated with a data set from 30 kpc up to 95 kpc, is able to give results so close with results got by GAIA DR2 which have been got with the highest current technology and processed through sophisticated software. Notice how the relative difference at 193 kpc is even lower that the ones at 74 kpc or 119 kpc.

Table 17 comes from [10] E.V. Karukes et al. 2020.

Notice that there is a perfect concordance, between direct mass and measures if it is considered the interval of errors.

Notice that although the parameter a was got using a data set ranging from 30 kpc up to 95 kpc the Direct mass formula gives accuracy calculus for mass not only at 193 kpc but up to 770 kpc and beyond as it will show in the next chapter.

In this epigraph will be compared the results published in [11] Jeff Shen.2022 with the results calculated by the Direct mass formula in Milky Way, formula (10.1).

Below is copied a fragment of the abstract of the paper [11] J. Shen. 2022, where it is possible to see two masses results at different radii.

Now it will be calculated the total mass at the same radius with the direct mass formula (10.1). In Table 18 , both kind of masses and the last column shows its relative difference are shown.

As it is shown the relative difference is small especially at 100 kpc. In addition both results match if it is considered the range of the errors measures.

It is important to remark that results by the direct mass formula at 216 kpc is calculated through the direct mass whose parameter a² was got using a data set whose dominion ranges between 30 kpc and 95 kpc.

In Table 19 , the parameter a² belonging to the four most important galaxies in the L.G are tabulated.

In order to estimate the total mass of Local Group only it will be considered M31, MWand their main galaxy satellites: M33 and LMC. The rest of galaxies have a mass negligible to estimate the total mass of Local Group. In fact M33 add only a 9% of total mass and LMC add only a 2.8%.

The values of parameters a² for the LMC and M33 have been got in the paper [9] Abarca, M.2023, chapter 14, using its rotation curves.

By the formula (8.14) of the Direct mass, it is right to get the table of masses at different distances using parameters a² associated to galaxies.

Calculus written in Table 20 are only an estimation, as the gravitational interaction between the four galaxies is quite complex. The masses calculated below are in $M_{\Theta }$ units.

In conclusion adding MW + LMC + M31 + M33 at 770 kpc the mass is $4.97\text{E}12M_{\Theta }$ that match fully with the stated in [4] Azadeh Fattahi, Julio F. Navarro. et al. 2020 and [1] Sofue, Y.2015.

As DMbQG theory stated an unbounded dominion of DM, it is possible to extend the radius. For example at 2 Mpc the total mass is $8\times {10}^{12}{M}_{\Theta }$.

These results are a magnificent success of DMbQG theory.

For example, in the Local Group of galaxies, the dynamical data according to [1] Sofue, Y. 2015, are 770 kpc for distance between M31 and MW and 170 km/s for its relative velocity, so using $M_{DYNAMIC}\left(<r\right)=\frac{v^2\cdot R}{G}$ it is got a M_DYNAMIC = 1.03E+43 kg. Supposing that there is dynamical equilibrium between MW and M31, see [12] A. Fattahi et al. 2020, we have the equation:

$M_{DYNAMIC}=M_{TOTAL}=\frac{a^2\cdot \sqrt{r}}{G}=1.03\times {10}^{43}\text{kg}$, that for a radius = 770 kpc leads to the value a² = 4.45E+21, which is very close to the parameter a² = 4.28E+21 shown in Table 19 . Even these values would be closer if it was considered other galaxies such as the Small Cloud of Magellan and the others dwarfs satellite galaxies of MW and M31. Anyway its relative difference is lower than 4%.

It is remarkable the fact that the parameters a² of M31 and MW were calculated with data in the halo whose radius range from 40 kpc up to 300 kpc in M31 case and range from 35 kpc up to 100 kpc in MW case, whereas calculus for parameter a² associated to the Local Group has been made with only one data radius 770 kpc and velocity 170 km/s. So, despite the fact that both methods are independent they give a value to the parameter a² whose relative difference is only 4%.

Clearly, this result is another validation for the DMbQG theory.

In Table 21 there are recent data for the Coma and Virgo cluster. With such data will be calculated its parameter a² with the formula (12.1).

Using the data [13] Kashibadze 2020, it is right to get the parameter a² = 3.65E+23 m^5/2/s² for the Virgo cluster, which is 17 Mpc far away from MW.

As it has been comment, this value is a lower bound, because always there will be a fraction of baryonic mass outside from the virial radius. Also, it is clear that parameter a² error depend on the measure errors for virial mass and radius.

Similarly using the data for the Coma cluster [14] Seong-A.2023, it is right to get the parameter a² = 1.22E+24 m^5/2/s²

According to [15] Chernin, A.D. in the current cosmologic model ΛCDM, dark energy has an effect equivalent to antigravity i.e. the mass associated to dark energy is negative and the dark energy have a constant density for all the Universe equal to ${\phi }_{DE}={\phi }_C\cdot {\Omega }_{DE}=6.444\times {10}^{-27}\text{kg}/{\text{m}}^{\text{3}}$ being ${\Omega }_{DE}=0.7$ and

${\rho }_C=\frac{3H^2}{8\pi G}=9.205\text{E}-27\text{kg}/{\text{m}}^{\text{3}}$ the critic density of the Universe.

As DE density is constant, the total DE mass is proportional to Radius with power 3, whereas DM mass grows with radius power 0.5 so it is right to get a radius where DM is counter balanced by DE.

According to [15] Chernin, A.D. The mass associated to DE is

$M_{DE}\left(<R\right)=-\frac{{\rho }_{DE}8\pi R^3}{3}$, (13.1)

or equivalently

$M_{DE}\left(<R\right)=-{\phi }_{DE}\frac{8\pi R^3}{3}=\frac{-H^2\cdot {\Omega }_{DE}}{G}\cdot R^3$ (13.2).

Notice that the author multiplies by two the volume of a sphere by reasons explained in his work.

[15] Chernin.2013 defines gravitating mass

M_G(<R) = M_DE(<R) + M_TOTAL(<R) (13.3)

where M_TOTAL is baryonic plus dark matter mass, and defines R_ZG, Radius at zero Gravity as the radius where M_DE(<R_ZG) + M_TOTAL(<R_ZG) = 0. i.e. where the gravitating mass is zero.

The definition of R_ZG leads rightly to the equation

$M_{TOTAL}\left(<R_{ZG}\right)={\phi }_{DE}\frac{8\pi R_{ZG}^3}{3}$, (13.4).

Using the formula for total mass (8.14), the expression (13.4) leads to

$\frac{a^2\cdot \sqrt{R_{ZG}}}{G}={\rho }_{DE}\cdot \frac{8\pi \cdot R_{ZG}^3}{3}$, (13.5),

where it is possible to clear up

$R_{ZG}={\left[\frac{3a^2}{8\pi G{\rho }_{DE}}\right]}^{2/5}$ (13.6)

and as

${\phi }_{DE}=\frac{3\cdot H^2}{8\pi G}{\Omega }_{DE}$ (13.7)

then by substitution results

$R_{ZG}={\left[\frac{a^2}{H^2\cdot {\Omega }_{DE}}\right]}^{2/5}$ (13.8)

This formula will be called R_ZG (parameter a²) because depend on the parameter a².

As the radius R_ZG is the distance to cluster centre where is zero the gravitating mass, it is right to consider R_ZG as the halo radius and its sphere defined as the halo cluster.

According to the parameter a² value got in chapter 11 for the Local group, $a_{L-G}^2=4.28\times {10}^{21}$, by (13.8) it is right to get R_ZG = 2.19 Mpc. So at that radius the gravitating mass is zero, in other words, for radius under 2,19 Mpc dark matter dominates and for bigger radius dark energy dominates and it is not possible for the Local Group to have any dwarf galaxy linked gravitationally beyond this radius.

According to the parameter a² value got in chapter 12 for the Coma cluster it is right to get R_ZG = 21 Mpc. In other words 21 Mpc is the radius of region where the DM of Coma Cluster dominates versus dark energy.

Similarly for the Virgo cluster the parameter a² = 3.65E+23 leads to R_ZG = 12.97 Mpc.

In Table 22 , the previous calculus is summarized.

With these three important clusters of galaxies, it has been illustrated how the total mass, formula (8.14), is counter balanced by the dark energy at mega parsecs scale, and precisely this Radius at zero gravity determines the regional size where the cluster has gravitational influence.