Matter
ghost matter astrophysics.6.
Spiral structure.(p3)
- How to define the initial
conditions for a 2d numerical simulation.
Building a 2d Eddington-like solution of the couple Poisson + Vlasov equations
.
The non-uniform (elliptical) solutions
of the Vlasov equation have been intensively studied since a long time, in
3d. In the following we deal with 2d movements and locations, so that we have
to build the 2d self-consistent elliptical solution of the Vlasov equation.
Let us write the Vlasov equation
:
(1)
where :
(2)
f (x , y , u , v , t ) is the distribution function of the velocity. The equation (1)
is written using dyadics tensors notation, in terms of peculiar (residual
or thermal) velocity C = ( u , v ).
<V>
is the macroscopic velocity. m is the mass of a particle.
is the position vector ( x , y ).
..
The bold letters represent vectors.
The last term of the equation (2) represents the scalar product of two dyadic
tensors (see reference [20]). Now we introduce an elliptic Eddington-like
2d solution :
(3)
Where C
is the residual, the thermal velocity. In steady-state condition, the Vlasov
equation becomes :
(4)
Combining with the Vlasov solution,
we get :
(5)
This is a third order polynom
with respect to the component u and v of the thermal velocity
C . A solution arises :
(6)
Then :
(7)
From the third order terms, we
get :
(8)
From the second order terms
(9)
Combining, we get the following set :
(10)
Let :
(11)
Then :
(12)
The distribution function becomes
:
(13)
Where
C
is the radial component of the thermal velocity
C and Cp its
azimutal component. Then we get :
(14)
In the classical (three-dimensional) Eddington solution, we had an ellipsoïd
of velocities, whose major axis pointed towards the center of the system.
See figure 6.
Fig. 6 : Ellipsoid
of velocities corresponding
to an Eddington-type solution.
In the present 2d elliptic Eddindton-like
solution we get a vellocity ellipse, whose major axis is constant and points
towards the center of the system. At the center the velocity ellipse becomes
a circle (2d Maxwell-Boltzmann distribution of the velocity). As will be shown
further its major axis < Cv
(mean radial thermal velocity
) is constant versus the radial distance v
. Its transverse axis
(mean azimutal
thermal velocity) tends to zero at infinite. See figure 7.
Fig. 7 :The evolution of the velocity ellipse, in the 2d Eddington-like solution,
versus distance from the center
of the system.
