MODERN COSMOLOGY

Simple lens models 395

14.3.3 Singular isothermal sphere

A simple model for describing the matter distribution in a galaxy is to assume that
the stars forming the galaxy behave like the particles in an ideal gas, confined by
the total gravitational potential, which we assume to have spherical shape. The
equation of state of the ‘particles’ (stars) has the form

p=

ρkBT

m

, (14.76)

whereρandmare the matter density and the mass of a star, respectively. In the
equilibrium case the temperatureTis defined via the one-dimensional dispersion
velocityσvof the stars as obtained from

mσv^2 =kBT. (14.77)

In principle the temperature could depend on the radius; however, in the simplest
model, of the isothermal spherical model, one assumes that the temperature is
constant and hence alsoσv. The equation for hydrostatic equilibrium is given by

p′

ρ

=−

GM(r)

r^2

, (14.78)

with

M′(r)= 4 πr^2 ρ, (14.79)

whereM(r)is the mass inside the sphere of radiusr. A solution of the previous
equations is

ρ(r)=

σv^2

2 πG

1

r^2

. (14.80)

This mass distribution is calledsingular isothermal sphere(it is indeed singular
forr →0). Sinceρ(r) ∼r^2 ,M(r)∼r, the velocity of the stars in the
gravitational field of an isothermal sphere is given by

v^2 rot(r)=

GM(r)

r

= 2 σv^2 , (14.81)

which is constant. Such a mass distribution can (at least in a qualitative way)
describe the flat rotation curves of the galaxies, as measured beyond a certain
galactic radius. Thus the dark matter in the halo can, in a first approximation, be
described by a singular isothermal sphere model.
The projected mass density on the lens plane perpendicular to the line of
sight is:

&(ξ)=

σv^2

2 G

1

ξ

, (14.82)