dark matter halo - universe in problems

Dark Matter Halo

Problem 1Estimate the local density of the dark halo in the vicinity of the Earth, assuming that its density decreases as .

For and assuming that the dark matter dominates in the halo, one obtains

Problem 2Build the model of the spherically symmetric dark halo density corresponding to the observed galactic rotation curves.

The simplest halo model is the isothermic spherically symmetric one. The radial profile of the density in the model is restricted only by theobserved rotation curves. This restriction leads to the following requirements for radial dependence of the density:a) it must provide linear growth of the rotation curves at small distances.b) it must follow as for large distances, thus providing flat rotation curves.The conditions are satisfied by the following function

= C/g r2solution [hide]

= 4 (r) dr = 4C = .Mg Rg0 g r2 Rg g Mg4Rgr2 , 10kpc, r 6.7kpcMg 1011M Rg

0.2 .g DM 1025g/cm3 GeV/cm3

solution [hide]

1/r2

+2 2

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Dark Matter Halo - Universe in Problems 3/5/15

http://universeinproblems.com/index.php/Dark_Matter_Halo 1 / 5

where is the local halo density in vicinity of the Sun (if it concerns the dark halo in Milky Way) at and is the coreradius, inside which the density grows (with decreasing ) not faster than and goes to constant, thus providing the linear growth of the

rotation curves) at small

Problem 3In frames of the halo model considered in the previous problem determine the local dark matter density basing on the givenrotation velocities of satellite galaxies at the outer border of the halo and in some point .

Let some satellite galaxy orbits on the distance from the center of the main galaxy of mass with velocity , then

As

then substitution of the expression for (see problem) one obtains

so it follows that

and

From the other hand the core radius can be determined from the relation

(r) = 0 +r2c r20

+r2c r2= ( )0 r0 r = r0 rc

r 1/r2r.

0 v(r )v r0solution [hide]

r M(r) v(r)= G .v2r

M(r)r2

M(r) = 4 ( )d , r0r 2 r r

(r)(r) = 4G ( )d = 4G ( + )[1 ( )arctg( )],v2 1r

r

0r 2 r r 0 r2c r20 rcr

rr0

= (r ) = 4G ( + )v2 v2 0 r2c r20

= .0 v2

4G( + )r2c r20rc

( ) ( ) ( )2

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Thus the local density and the core radius can be determined as soon as the rotation velocities and are measured.

Problem 4For the halo model considered in problem about halo model obtain the dependencies and in terms of and . Plot thedependencies and .

For one obtains . The dependencies and are plotted on Figure.

( )arctg( ) = 1 .rcr0 r0rc ( )v2 r0v2

v( )r0 v

(r) v(r) 0 v(r) v(r)solution [hide]

(r) = ; v(r) =v24G1+r2c r2 v 1 arctan

rcr

rrc

= 220km/s, = 2.6kpc, = 8kpcv rc r0 5 0.30 1025g/cm3 GeV/cm3 (r)

v(r)



11 17.jpg

Problem 5Many clusters are sources of X-ray radiation. It is emitted by the hot intergalactic gas filling the cluster volume. Assuming that the hotgas ( ) is in equilibrium in the cluster with linear size and core radius , estimatethe mass of the cluster.

Equation of hydrostatic equilibrium reads:

kT 10keV R = 2.5Mpc = 0.25Mpcrcsolution [hide]

= ,1dpdr

GM(r)r2



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where is pressure, and is density of the ideal gas with the state equation , where is the concentration of the gas . Let us assume the isothermic temperature distribution then

and it follows that

Using results of the problem

one obtains

Category: Dark Matter

p p = nkT nn = /mp T = const,

= ,kTmpddr

GM(r)r2

M(r) = .rkTGmpd lnd ln r

(r) = ,v24G( + r)r2c

M(r) = 2 .rkTGmpr2+ rr2c 10

16M



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