Ecole doctorale de Physique de la région Parisienne (ED107)

138 Inertial modes in slowly rotating stars : An evolutionary description
Note that in this equation, δU 0 is not a dynamical variable but is determined according
to the constraint that U µ is a 4-velocity (U µ Uµ = −1). Furthermore, δU r , δU ϑ and δU ϕ
are not contravariant components of a 4-vector but convenient variables that are the
components of a 3-vector on the orthonormal basis associated with the spherical system
of coordinates for the flat 3-space. It enables us to write the motion equations in a way
very similar to the Newtonian EE. Indeed, writing the 3-velocity on the orthonormal basis
associated with the spherical coordinates as
V =
and defining on the same basis
⎧
⎨
−→
ϖ[t, r, ϑ, φ] =
⎩
with A[r] = (Ω − N ϕ a[r]
d ln[ N[r]
[r])
]
d ln[r]
coordinate, we have
δU r
δU ϑ
δU ϕ
(Ω − N ϕ [r]) cos[ϑ]
− sin[ϑ] (Ω − N ϕ [r] + A[r] )
0
− r N ϕ′ [r]
2
(4.30)
(4.31)
where ′ is the derivative versus the radial
(∂t + Ω ∂ϕ) V + 2 ϖ ∧ V + ∇h = 0. (4.32)
This equation is very close to the Newtonian EE, the main difference being that the
3-vector that appears instead of Ω is now depending on the coordinates (as in the case of
differential rotation) but no longer parallel to the rotation axis.
Concerning the baryonic number conservation, writing it in a Newtonian like way, we
have in the slow rotation limit
2 N[r]
(∂t + Ω ∂ϕ) ñ + div ñ
a[r]
−→
V = 0 (4.33)
where ñ is defined as ñ = n N 2 a, n being the baryonic number density. The natural generalization
of the anelastic approximation is then
div ñ −→
V = 0. (4.34)
As we are using the strong Cowling approximation, the background star appears in
the equations of motion only as “external” (from the point of view of the mode) data.
For any relativistic calculation, what we do is to calculate the background configuration
using the already existing code illustrated in Bonazzola et al. (1993) and then to use the
resulting lapse, shift, conformal factor and their derivatives with respect to the radial
coordinate in our equations.