Ecole doctorale de Physique de la région Parisienne (ED107)

108 Inertial modes in slowly rotating stars : An evolutionary description
Fermi temperature, which is roughly 10 10 K. Furthermore, in the Newtonian nonlinear
hydrodynamics of a not too young slowly rotating NS, it is not worth trying to use a very
sophisticated equation of state (EOS). We shall therefore adopt a barotropic EOS. With
those assumptions, we have
P = P [n], (4.1)
where P is the pressure and n the mass density, and the Poisson equation for the gravitational
potential U is
∆ U = 4π GN n (4.2)
The Newtonian Navier-Stokes equations (written in the inertial frame for a rigidly
rotating NS) are
(∂t + Ω ∂ϕ)
W + W · ∇
W + 2 Ω ∧ W + ∇H = (4.3)
1 ∇ ∇η · W
+ n
∇ ∧ W ∧ ∇η + η∆ W − W ∆η + ζ
η
∇ + ∇ · W
+ 3
A.
Here we take (r, ϑ, ϕ) for a spherical system of coordinates in the inertial frame. In
this system, ϑ = 0 coincides with the direction of the rotation axis of the NS, which is
parallel to the angular velocity : Ω = Ω ez. W is the velocity in the corotating frame, i.e.
the part that is added to the velocity of the rigidly rotating background when a mode is
present. Note that both velocities can be of the same order in the nonlinear case. This
is the reason why we shall not use the term Eulerian perturbation for W . Otherwise,
H = 1 dP dn is the enthalpy, η and ζ are, respectively, the dynamical shear and bulk
n dn
viscosity coefficients, and finally A contains any external accelerations (or force per unit of
mass). The modifications that are needed in the case of the linear study with differential
rotation or in the relativistic linear case are, respectively, discussed in Sections 4.5 and 4.6.
In what follows, A is mainly the effective gravitational acceleration, i.e. the gradient
of the difference between the centrifugal potential 1
2 Ω2 ρ 2 and the gravitational potential
U, where ρ is the distance from the rotation axis. In the linear regime, we will sometimes
introduce an RR acceleration [cf. Blanchet (1993) and (1997)] of the form
where
and
ARRj = 4
c 2 vi ∂jΥi ,
Υi = 4 G
45
Skm[t] = ɛpq(k
c 5 ɛijkx j x m S km(5) [t] (4.4)
d 3 x n xm )x p v q ,
with (km) = 1
2 (km + mk), vi being the full velocity and the superscript (5) the fifth time
derivative, which can not be easily calculated in a numerical work [see Rezzolla et al.
(1999)]. Note that this formula is valid only if written with Cartesian components of the