Stellar Oscillation Modes
A star does not just pulse in and out; it rings in patterns. Each non-radial mode splits into an angular part, the spherical harmonic $Y_l^m(\theta, \phi)$ that tiles the surface into expanding and contracting cells, and a radial part, the eigenfunction $\xi_r(r)$ that sets how many shells the wave crosses on the way in. The surface here shows the live displacement $Y_l^m(\theta,\phi)\cos(\omega t)$ in red (outward) and blue (inward) with its nodal lines, the still curves separating the cells: $l - |m|$ circles of latitude and $2|m|$ meridians. The diagnostic shows the radial eigenfunction of a real $n_{\mathrm{poly}}=3$ polytrope, computed here, whose number of interior nodes is the radial order $n$. So the three integers split the work cleanly: $l$ and $m$ shape the surface, $n$ counts the nodes in depth. Acoustic (p-mode) frequencies come from the asymptotic JWKB quantisation of the sound-speed cavity, scaled to a solar-like large separation $\Delta\nu = 135\ \mu$Hz.
WHAT TO TRY
- Hold n and l fixed and sweep m from -l to l: the surface pattern reorganises from rings (m=0) to a checkerboard of meridional cells (|m|=l), while the radial eigenfunction below never changes. m is purely an orientation.
- Raise n: count the extra nodes appearing in the eigenfunction below. Each new node is one more shell where the gas reverses direction, and the frequency climbs by the large separation.
- Raise l at fixed n: the p-mode turning point moves outward, the cavity shrinks toward the surface, and the eigenfunction's inner edge retreats. High-degree modes never reach the core.
- Set l=0: the turning point sits at the centre, the mode fills the whole star, and the surface breathes uniformly (a radial mode).