The Navarro-Frenk-White profile looks as follows [10]

Here and hereinafter―the density of dark matter in the center of galaxy,―the size of its halo,― the current radius. Assume that following relationship exists between them. Therefore, the profile function can be expanded in Taylor series with respect to above introduced small parameter. With the first order on accuracy we get

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 [11]

where―constant of Newtonian potential for a gravitating mass. Marking and it’s clear that expressions (5) and (8) can be written down in the form of coupled elliptical trajectory

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 [12] [13] ). Observations have shown that these curves cross at the critical point and the corresponding mean velocities at it approximately

equals. (Really, for the dwarf galaxy Messier 32 the observable velocities interval is [14] ; for the dwarf galaxy Leo I― [15] ). Because of that we’ll use these values for estimation of unknown dark matter halo’s parameters.

Here it’s necessary to point out that later on we’ll consider distances no larger than. At these distances curved line in the gravitation field of dark matter approximately looks like the straight line type of Note that such relation also follows from the equality centrifugal force and gravitational force of dark matter that relates to the first term in potential energy (3).

Put that angles of trajectories are coincide also, i.e. Hence, the following equality takes place

Since the total energy is larger than kinetic one and it, in its turn, is greater than potential energy , we can assume that Then expression (11) simplifies―

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,. After substitution the above mentioned numerical values for the position of probe body and its velocity, we obtain the following estimate of the central density of dark matter’s halos

It is interesting to compare this result with the previously obtained similar values. For example, in [16] . However, in [17] it was shown that the central part of dark matter density should not be larger than Furthermore, in [3] [18] [19] it is shown that the central density of dark matter . Analysis of these estimates gives that the most acceptable central density of the dark matter’s halo, probably, lays within the interval.

Note that Navarro-Frank-White profile is completely identified. Nevertheless, in literature are known profiles of dark matter that contain from one [20] up to several [4] free parameters. In our article we’ll consider Einasto profile [20]

with one unknown parameter. Here as usual―the density of dark matter in the center of galaxy,―the size of its halo,―the current radius. Based on the previously adopted relationship between size of the halo and the current radius, the exponential function of Einasto profile may be expanded in Taylor series. Restricting ourselves by two terms only we get

In doing this the new additional condition we have used here―the smallness of the power parameter. Then Einasto profile in the approximate form is written down as

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 is

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. Basing on our previous article [21] , it is possible to estimate and this 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 [3] it follows that from [18] ―from [19] ―from [21] ―(see also (15)). As for the central density of dark matter’s halo in Einasto profile, we write down it as

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

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 and above given values of X, we find: And finally, for the Galaxy Leo I when we have: (At calculations we took into account that for such galaxy).

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

Note that in article [1] the corresponding interval is closed to (31) and equals Also important to emphasize that in [22] the next interval of parameter have been argued