Oscillations in centrifugally flattened polytrope

1
Oscillations in centrifugally flattened polytrope
F. Lignières and M. Rieutord
Observatoire Midi-Pyrénées

• Non-pertubative model
• Polytropic and uniformly rotating model of stars
• The effect of the centrifugal force on this
  equilibrium model is fully adressed
• At the moment, we neglect the effect of Coriolis
  and centrifugal forces on the oscillating motions

2
The formalism

• Projection on
• Because of symmetries, only modes with the same
  azimuthal number m and the same parity of degree
  are coupled

where is the surface and
3
The ODE system
The coupling terms
4
The numerical method

• Two different methods
• QZ algorithm
• Incomplete Arnoldi-Chebyshev
• Numerical parameters lmax, nr

Test
The formalism and the numerics have been tested
in the case of a uniform density ellipsoid
(separable)
5
Axisymmetric modes (m0), l 0 to 7, n1 to 8 or
10
w
(GM/R3p)1/2
e
6
Spherical Harmonic Spectrum of the modes
l2, n5
l6, n4
e 0
e0.1
e 0.15
w
e
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Departure from the order 1 perturbation in e
l0
l1
D w/ w
l2
e
(D w/w) x 200 Hz gtgt 0.1 m Hz
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Large and small separation
e 0
e 0.1
e 0.15
D w
n
n
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Amplitude of the mode at the surface
l4, n4
e 0
e 0
l0, n1
e 0.05
e 0.05
e 0.1
e 0.1
e 0.15
e 0.15
Pole
Pole
Equator
Equator
10
l0,n1 at e0.15
r r1/2v
11
l2,n5 at e0.15
r r1/2v
12
l4,n4 at e0.15
r r1/2v
13
CONCLUSION
In the frequency and flatness range considered

• Modes keep memory of their zero rotation degree
  number up to large flatness
• Differences with order one pertubation in
  flatness are significant
• The large separation persists the small
  separation dont
• The mode amplitude is amplified at the equator