هيكل لولبي

tspiral structure 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.