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Fig. 4

image

From panel a) to panel c), three PSS at ω/N0,max = [ 0.66676,0.50200,0.05522 ] in a non-rotating polytropic model of star. The fL2(kr,r) = 0 curves shown are the imprint on the PSS of the invariant tori of the integrable gravity ray dynamics. They are drawn here for L = 1 / 2 + with taking integer values from 1 to 30 corresponding to degrees of spherical harmonics. The value of L of a given tori can be easily estimated from the intersection with the blue vertical dashed line, since the ordinate of the intersection point is approximately equal to 4L. The EBK quantisation enables us to specify the tori where positive interferences of rays produce modes. For example, the (n,ℓ) = (3,3) gravity mode on panel b) and the (n,ℓ) = (50,3) gravity mode on panel c) are constructed on the tori highlighted in green. The red curve on panel a) is a separatrix also called a homoclinic orbit because it connects the hyperbolic point at (kr,r/R) = (0,0.7968) to itself. As explained in Sect. 4.3.1, it is at the origin of a large chaotic phase-space region observed in the rotating case.

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