## Astrophysics   ## A D.M. Formula for Milky Way and M31 Halo Got by Buckingham Theorem –V5

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  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. Also it has been introduced a new expression for DM density and mass called reduced formulas because the previous formulas has been rewritten by a dimensionless variable similarly to NFW or Burkert formulas. Furthermore thanks these new expression is more easy to calculate the exact values of density and masses. In tenth chapter are calculated some different type of DM masses in M31 to be compared with DM calculated by Sofue. There is a good agreement between both results. In the eleventh chapter through the Hubble law it is demonstrated that galaxies in the ancient universe were smaller than at present and as a consequence it is got that proportion of DM versus baryonic matter was lower in the ancient universe. Results got are in agreement with current observational evidences. In the twelfth chapter, it is introduced Sofue data for Milky Way and it is applied DM theory by gravitational field to calculate the DM contained inside the halo at different radius. Results match perfectly with Sofue results. The importance of this chapter is based on the fact that the parameters of M31 has been used to do the calculus in Milky Way. These results back strongly the fact that DM generated by field theory is a general theory to explain DM nature. In the thirteenth chapter it has been introduced the concept of extended halo and thanks to this concept the total mass calculated of Milky Way and M31 is equal to 4*10 ^12 Msun , which is 10 ^12 Msun heavier than total mass with standard halos, so there is only a lack of 10 ^12 Msun to get the dynamical mass of Local Group instead of 2*10 ^12 Msun.

Comments: 31 Pages. There is anew expression for DM density and Mass called Reduced formulas.

### Submission history

[v1] 2019-09-27 11:31:03
[v2] 2019-10-01 17:06:30
[v3] 2019-10-13 16:52:54
[v4] 2019-10-17 17:12:38
[v5] 2019-11-01 04:56:14