twin universe astrophysics and cosmology Ghost matter astrophysics.6. Spiral structure.(p3)
- How to define the initial conditions for a 2D numerical simulation.
Constructing a 2D Eddington-type solution for the coupled Poisson + Vlasov equations.
Non-uniform (elliptical) solutions of the Vlasov equation have been extensively studied for a long time in 3D. In what follows, we consider motion and positions in 2D, so that we need to construct 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 velocity distribution function. Equation (1) is written in dyadic tensor 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).
..
Bold letters represent vectors. The last term in equation (2) denotes the scalar product of two dyadic tensors (see reference [20]). Now we introduce a 2D elliptical Eddington-type solution:
(3)
where C is the residual, i.e., the thermal velocity. Under steady-state conditions, the Vlasov equation becomes:
(4)
Combining with the Vlasov solution, we obtain:
(5)
This is a third-order polynomial in the components 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 these, we obtain the following system:
(10)
Let:
(11)
Then:
(12)
The distribution function becomes:
(13)
where C is the radial component of the thermal velocity C, and Cp its azimuthal component. We then obtain:
(14)
In the classical (three-dimensional) Eddington solution, we had a velocity ellipsoid whose major axis pointed toward the center of the system. See Figure 6.
Fig. 6: Velocity ellipsoid corresponding to an Eddington-type solution.
In the present 2D elliptical Eddington-type solution, we obtain a velocity ellipse whose major axis is constant and points toward the center of the system. At the center, the velocity ellipse becomes a circle (2D Maxwell-Boltzmann velocity distribution). As will be shown later, its major axis (mean radial thermal velocity) remains constant with respect to radial distance r. Its transverse axis (mean azimuthal thermal velocity) tends to zero at infinity. See Figure 7.
Fig. 7: Evolution of the velocity ellipse in the 2D Eddington-type solution, as a function of distance from the center of the system.