**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. The first is called direct DM density because it is got directly from velocity as power regression of radius in rotation curve. In other words velocity of rotation curve depend on radius as a power function. In fact galactic rotation curve inside M31 halo has a power regression of velocity depending on radius with a correlation coefficient bigger than 0,95 inside halo region. The second one, called Bernoulli profile has been introduced by author in previous papers, [8] Abarca,M.2016, and others papers quoted in bibliography, where it has been used to study DM in several galaxies. It is called Bernoulli because it is got from a Bernoulli differential equation. Hypothesis which is the basis to get Bernoulli profile stated that DM is generated locally by the own gravitational field according a power law. DM density = A• E^B where A& B are coefficients and E is gravitational intensity of field. In addition A& B are similar for different galaxies on condition that galaxies are similar and giants, not dwarfs. To find reasons that author has to do so daring statement, reader can consult [1] Abarca,M.2014. Dark matter model by quantum vacuum. [8] Abarca,M.2016. Dark matter density on big galaxies depend on gravitational field as Universal law and other papers quoted in bibliography. Briefly will be explained method followed to develop this paper. Firstly are presented rotation curve and table with data points inside M31 halo. These data come from [5] Sofue,Y.2015. In addition it is got a power regression for rotation curve points in halo region whose function is v = a•r^b getting a correlation coefficient bigger than 0,95. In fourth chapter it is developed a mathematical method to get a new DM density depending on radius called direct DM density because it is got directly from power regression function got in chapter three. In fifth chapter it has been demonstrated that a power regression function for rotation curve is mathematically equivalent that DM density depend on gravitational field, as a power function i.e. DM density = A• E^B where A& B are cleared up depending on a & b (parameters of power regression of rotation curve). In sixth chapter it has been got that for radius bigger than 40 kpc then ratio baryonic density versus DM density is under 1% so it is reasonable to consider negligible baryonic density in order to develop theory introduced in this work. The seventh is a short chapter where is compared direct DM density got with NFW density profile fitted by Sofue in his paper. [5] Sofue, Y.2015. Relative differences between both density profiles are under 25% inside main part of halo dominion. In addition it is shown that NFW profile is bigger than direct DM through all dominion. In eighth chapter is got a Bernoulli differential equation for gravitational field. Hypothesis to state this equation is that DM density is a power function of gravitational field i.e. DM density = A• E^B. Solution for E allows to get Bernoulli density profile. In ninth chapter is demonstrated that Bernoulli profile is mathematically equivalent to direct DM profile. In tenth chapter has been got masses for M31 by direct DM and NFW profiles at different radius. Through whole dominion masses got by NFW are bigger than direct DM profile although relative difference between both profiles are under 25 %. In conclusion chapter will be pointed reason, which in author opinion, explain that NFW profile gives bigger values than direct DM profile.

**Comments:** 20 Pages. Direct DM density profile has been calculated directly from rotation curve of M31

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[v1] 2016-09-03 09:36:01

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