Matter ghost matter astrophysics.6:
Spiral structure.(p4)
Returning to the first order terms,
we have :
(15)
In polar coordinates :
(16)
The third order terms vanish.
(17)
i.e :
(18)
The 2d distribution function is
:
(19)
And the axis of the velocity ellipse
follow:
(20)
Then, introducing the number of
density n(
)
we get :
(21)
and :
(22)
In the twin fold F* we also take
an Eddington-type solution.
(23)
(24)
(25)
(26)
From reference [1] we know that
the Poisson equation is :
(27)
where
is
the gravitational potential. 
is the mass density in the first fold and 
the mass-density in the second fold. The final differential equation, for
this axially symmetric system, is :
(28)
Introduce :
(29)
where Vo
and Vo* and characteristic
velocities. Introduce the following adimensional quantities :
(30)
Let us write the axis of the velocity
ellipses as :
(31)
Then we get the Poisson differential
equation, refering to a non-rotating axisymmetric system, written in terms
of adimensional parameters 
, 
, 
, 
(32)
-
runs
the importance of the twin structure (characteristic mass-ratio). -
is
the ratio of the thermal velocities in the two adjacent folds F
and F*.
- and
refer
to the characteristic lengths (equivalent to the Jeans
length) in the two populations.
The mass densities, written in
adimensional form, obey :
(33)
Initial conditions, for numerical computation, will be given for 
= 0 . Then
:
(34)
Strictly talking, this is not
physical, for the -motions are basicly neglected, but 2d simulation are not
physical too. We build this material in order to pilot numerical 2d simulations,
searching, as a starting point, steady-state conditions.
