**Authors:** Manuel Abarca Hernandez

In this work has been calculated two new DM density profiles inside halo region of M31 galaxy and it has been demonstrated that both ones are mathematically equivalents. Its radius dominion is only the halo region because it is needed that baryonic matter density has to be negligible. The first profile is called direct DM density because it is got directly from rotation curve and represents DM density depending on radius The second one, called Bernoulli because it is got from a Bernoulli differential equation, represents DM density depending on local gravitational field according a power law. The power of E is called B. Hypothesis which is the basis to get Bernoulli profile stated that DM is generated locally by the own gravitational field according this formula. DM density = A• E^B where A& B are coefficients and E is gravitational intensity of field. Briefly will be explained method followed to develop this paper. Rotation curve data come from [5] Sofue,Y.2015. Thanks this remarkable rotation curve, the regression curve of velocity depending on radius has a correlation coefficient bigger than 0.96 and data range from 40 kpc up to 300 kpc. In fourth chapter it is got the function of DM density depending on radius, called direct DM density. In fifth chapter it is demonstrated that function direct DM density is mathematically equivalent to the function DM density depending on E. Namely a power of E whose exponent is B= 1.6682 In sixth chapter it is got that for radius bigger than 40 kpc the ratio baryonic density versus DM density is under 1% so it is reasonable to consider negligible baryonic matter density in order to simplify calculus. In seventh chapter it is got a Bernoulli differential equation for field and is solved. In eighth chapter it is made dimensional analysis for magnitudes Density, field E and universal constants G, h and c. It is demonstrated that it is needed a formula with two Pi monomials. It is found that B = 5/3 is the value coherent with Buckingham theorem and differs only two thousandth regarding B=1.6682 which was got by regression analysis. In ninth chapter are recalculated parameters a,b and A as a consequence of being B=5/3 instead B=1.6682. Thanks this change the formulas are now dimensionally right. In the tenth chapter has been developed the called reduced Bernoulli formulas, whose main importance is that radius is changed by a dimensionless variable X= radius/ Rs. Being Rs a parameter with length dimension. In the eleventh chapter the reduced formulas are applied to calculate total masses in M31 at different radii. In the twelfth chapter, it is introduced Sofue data for Milky Way and it is applied dark gravitation theory to calculate the total mass at different radius. Results match perfectly with Sofue results. The importance of this new version of paper is that by Milky Way data has been found parameter B and b coherent with Buckingham theorem. In addition has been calculated a new aOPTIMUS by a careful selection of data set in Milky Way. Also it has been written reduced Bernoulli formulas and it is calculated total masses of Milky Way. In the thirteenth chapter it has been introduced the concept of extended halo, which is a big difference regarding NFW theory. According dark gravitation theory it is possible to extend the halo up to 770 kpc when is studied the interaction between M31 and MW, whereas the halo has to be extended up to distance of nearby cluster when it is studied the interaction between Local Group and other nearby clusters. Furthermore the extended halo may explain successfully the fact that proportion of baryonic matter versus D.M. is lower at cluster scale. In the fourteenth chapter the theory is applied successfully to Solar System, it is got that the Dark matter generated by the Sun inside the Solar System is equivalent to 74.6 times the mass of the Earth, although a big big fraction of this DM is generated inside the orbit defined by Mercury, because in this region the field is far bigger than field far away the Sun and according the Dark Gravitation theory DM density depends on field with the power 5/3.In this version parameters C and D have been calculated with data from Mercury and Venus, whereas in previous version were calculated with data from Venus and Neptune. This version is more trustworthy because the influence of giant planets is a minimum.

**Comments:** 37 Pages. Dark Gravitation Theory works perfectly in Solar System

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[v1] 2019-09-27 11:31:03

[v2] 2019-10-01 17:06:30

[v3] 2019-10-13 16:52:54

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[v5] 2019-11-01 04:56:14

[v6] 2019-12-31 16:53:36

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