7.2. STARS 425
- For the stars with rotation~Ω, the corresponding eigenvalue equations with the Coriolis
force are
(7.2.43)
Pr[
∆P−
f
r^2+
√
Re
r^2~kT]
− 2 ~Ω×P=βP,∆ ̃T+
√
Re
r^2Pr=βT,divP= 0 ,with the boundary conditions (7.2.37).
In (Ma and Wang,2013b) we showed that there is a lower boundΩ 0 such that asΩ>Ω 0
the first eigenvalueβ 1 of (7.2.43) is complex in the critical state (7.2.25). Therefore, for the
stars with the bigger rotation their atmosphere convections in the first phase transition are
time periodic.
7.2.5 Dynamics of stars with variable radii
For stars with varying sizes and for supernovae, their radiiexpand and shrink periodically.
Therefore, the metric in the interior of such stars is as follows:
ds^2 =−ψc^2 dt^2 +R^2 (t)[
αdr^2 +r^2 (dθ^2 +sin^2 θdφ^2 )]
,
whereψ=ψ(r,t),α=α(r,t), andR(t)is the scalar factor representing the star radius. For
convenience, we denote
ψ=eu(r,t), α=ev(r,t), R^2 (t) =ek(t), 0 ≤r≤ 1.Then the metric is rewritten as
(7.2.44) ds^2 =−euc^2 dt^2 +ek
[
evdr^2 +r^2 (dθ^2 +sin^2 θdφ^2 )]
.
The stars with variable radii are essentially in radial motion. Hence, the horizontal mo-
mentum(Pθ,Pφ)is assumed to be zero:
(7.2.45) (Pθ,Pφ) = 0.
In the following we develop dynamic models for astronomicalobjects with variable sizes.
1.Gravitational field equations.We recall the gravitational field equations (Ma and Wang,
2014e):
(7.2.46) Rμ ν=−
8 πG
c^4(Tμ ν−1
2
gμ νT)−(Dμ νφ−1
2
gμ νΦ).The nonzero components of the metric (7.2.44) are
g 00 =−eu, g 11 =ek+v, g 22 =ekr^2 , g 33 =ekr^2 sin^2 θ,