spiral structure 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 follows: (20)
Then, introducing the number density n(ρ), we obtain: (21)
and: (22)
In the twin fold F*, we also adopt a solution of the Eddington type. (23)
(24)
(25)
(26)
According to reference [1], we know that the Poisson equation is written as: (27)
where φ is the gravitational potential. ρ₁ is the mass density in the first fold and ρ₂ is the mass density in the second fold. The final differential equation for this axially symmetric system is: (28)
Introduce: (29)
where V₀ and V₀* are characteristic velocities. Introduce the following dimensionless quantities: (30)
Let us write the axis of the velocity ellipses as: (31)
We then obtain the Poisson differential equation, referring to a non-rotating axisymmetric system, expressed in terms of dimensionless parameters ρ, ξ, η, and ζ: (32)
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η represents the importance of the twin structure (characteristic mass ratio).
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ξ is the ratio of the thermal velocities in the two adjacent folds F and F*.
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ρ and ζ refer to the characteristic lengths (equivalent to the Jeans length) in the two populations.
The mass densities, written in dimensionless form, obey: (33)
Initial conditions, for numerical computation, will be given for η = 0. Then: (34)
Strictly speaking, this is not physical, since the -motions are essentially neglected, but 2D simulations are not physical either. We construct this material in order to drive numerical 2D simulations, seeking, as a starting point, steady-state conditions.
