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Article devoted to searching the parameters of dark matter halos on the base of dwarf galaxies’ dynamics (Messier 32 and Leo I). For doing this, we propose the new approach founded on construction the coupled elliptical trajectory for a probe body in the gravitational fields of Newtonian potential and potential of dark matter’s halo. This allows more accuracy estimate its central density for the Navarro-Frenk-White profile and free parameter for the Einasto profile . Our result is in good correlation with results of other authors that are got by different numerical methods.

One of the urgent problems of modern cosmology is searching the dark matter properties. Dark matter―an unusual type of cosmic substance, which is in the overall energy balance of the Universe approximately equals 27% [

It should be noted that distribution of dark matter in the halo of galaxies is not uniform―it concentrates in their centers and decreases to peripheries. The corresponding distribution function of dark matter (profile) is usually founded on numerical methods that are modeling the dynamics of stars in galaxies. Today a number of profiles are known [

Finding numerical values of these quantities is one of unsolved cosmological problems that from our viewpoint, can be partially solved by examining the dynamics of dwarf galaxies. Note that influence of dark energy and dark matter on galaxies’ dynamics successfully described even in the framework of Newtonian approach [

Now consider the following dynamic model―in the gravitational field of a massive galaxy, surrounded by a halo of dark matter, the probe body moves. The prototype of this model, for example, is the model of dwarf galaxy Messier 32 motion in the Andromeda galaxy or the motion of dwarf galaxy Leo I in the Milky Way. As for the distribution function of dark matter we’ll choose two most known of them―Navarro-Frenk-White profile and Einasto profile.

The Navarro-Frenk-White profile looks as follows [

Here and hereinafter

Then the potential energy of the dark matter of the field can be written down as

where coefficients in standard designations are

Later on we’ll use first term in (3) only because such potential energy leads to a closed trajectory that corresponds to the real observations of dwarf galaxies’ movement. Consequently, based on the conservation laws of energy and momentum for the test body dynamics, its trajectory can be write down in the standard form

From expressions (5)-(7) sees that it describes the elliptical trajectory that is similar to standard trajectory for one body in Newtonian mechanics [

where

Now require that trajectories (5) and (8) coincide each other at the real movement of probe body. To substantiate this declaration we use the graphs of rotation curves for the Newtonian movement and for the movement of body in the gravitation field of dark matter. It is known that for the first case such curve is the hyperbola, while for the second case―the quasi-logarithmic line (for example, see [

equals

Here it’s necessary to point out that later on we’ll consider distances no larger than

Put that angles of trajectories are coincide also, i.e.

Since the total energy is larger than kinetic one and it, in its turn, is greater than potential energy

Now from (12) can be found the value of an unknown factor in the first term of expression (3)

Comparison (4) and (13) allows get the expression of central density of dark matter’s halos

For numerical estimations assume that the probe mass m in (14) equals to one. Thus roughly put

It is interesting to compare this result with the previously obtained similar values. For example, in [

Note that Navarro-Frank-White profile is completely identified. Nevertheless, in literature are known profiles of dark matter that contain from one [

with one unknown parameter

In doing this the new additional condition we have used here―the smallness of the power parameter

where

The potential energy of dark matter field in this case is as follows

where

Now for further calculations we use only the second term in (22), because it leads to a closed trajectory as before. Therefore, using the conservation laws of energy and momentum, its trajectory can be written as

From the expression (24) sees that it describes an elliptical trajectory that is similar to the standard trajectory in Newtonian mechanics. Repeating the previous arguments about the procedure of analyzing trajectories and their shapes, we find the relation

Hence, the expression of an unknown coefficient

From the comparison of expressions (21), (26) and usage (19) we get

Important to notice that expression (27), besides dynamic characteristics of a particle E and M, contains the current radius r and unknown parameter

We write down the expression of central density of dark matter’s halo in the form

Here the coefficient X has various numerical values which are known in literature. In fact, from [

where Y is unknown coefficient. Substituting the necessary numerical values into (27) we find―for the dwarf galaxy Messier 32

From comparison of (27), (28) and (29) follows the expression of unknown parameter

Using the found values X and Y we can obtain this parameter. As for the galaxy Messier 32

From the saying above we conclude―free parameter in Einasto profile must satisfy the following interval

Note that in article [

In the article, the new method for search characteristics of dark matter’s halo is proposed. Searching the Navarro-Frenk-White profile shows that the central part of density of dark matter’s halos must satisfy the following

magnitude

Authors of [

For describing this broken line in terms of Einasto profile (and not only) they used results of the survey THINGS were have been obtained highest quality rotational curves for 34 nearby spiral and irregular galaxies. Naturally that these curves depend not only the dark matter, but on some baryonic parts (baryonic substrate, relativistic gas, etc.) and their quantitative relation also. Therefore authors got the cited above range of Einasto parameter

In the framework of N-body simulations―“Aquarius” project with number of particles is 4.4 billions―au- thors [

Really, in [

From this examination, it’s clear the difference between ours models―in our case model based on the dwarf galaxies Newtonian dynamics and allowed get the improved estimate of dark matter core. Moreover, it’s very simple and allowed get the very plausible interval for the Einasto parameter. In contrary, the model in [

Two above mentioned other articles [

Therefore all of these models must be regard as additional each other which give rather satisfactory description of dark matter halo on different cosmological scales and for different space objects.

L. M. Chechin,T. K. Konysbayev, (2016) Searching the Parameters of Dark Matter Halos on the Basis of Dwarf Galaxies’ Dynamics. Journal of Modern Physics,07,982-988. doi: 10.4236/jmp.2016.79089