Asymptotic Period Spacing in Red Giants

What you are seeing: a vertical cross-section of a red giant with the buoyancy-trapped $\ell = 1$ g-mode oscillating inside its core cavity. The cavity is where $N(r) > 0$ (radiative buoyancy region); a convective core or envelope shuts it off. The panels below show the Brunt-Vaisala profile $N(r)$ that sets the cavity, and the resulting comb of g-mode periods spaced by $\Pi_1$ . Switching between RGB and red-clump profiles changes the buoyancy integral, which changes $\Pi_1$ from $\sim 80$ s to $\sim 250$ s

Figure 1. Asymptotic g-mode period spacing in a red giant. Top: a cross-section of the star with the WKB displacement eigenfunction xi_n(r) P_l(cos theta) cos(omega t) (red and blue opposite phase); the dashed circles bound the g-mode cavity. Middle: the buoyancy frequency N(r) that sets the cavity, with the radial nodes of mode n ticked. Bottom: the comb of g-mode periods P_n = (n+1/2) Pi_l, evenly spaced by Pi_1. Method: WKB on the buoyancy integral. Reference: Aerts, Christensen-Dalsgaard and Kurtz (2010), Ch. 3.4; Bedding et al. (2011).

profileRGB

ℓ1

mode order n14

animation speed1

WHAT TO TRY

Switch from RGB to red clump: the convective core punches a hole in the buoyancy cavity, the integral drops, and Pi_1 jumps from about 80 s to about 250 s. This is how seismology tells an inert-core red giant from a He-burning one.
Raise the mode order n: the standing wave gains nodes (count the ticks under N(r)), and the highlighted comb tooth steps along by exactly Pi_1 each time.
Switch l from 1 to 2: the comb spacing shrinks by sqrt(3), the asymptotic Pi_l = Pi_0/sqrt(l(l+1)).
Watch the cross-section: the displacement oscillates fastest where N(r) is largest, because that is where the local g-mode wavelength is shortest.