Asteroseismology

In Asteroseismology, the oscillations of a star are observed to study its interior. In this entry we present two example for such inferences. The oscillation spectrum of stars is often harmonic, which means that peaks in the spectrum occur at regular intervals in the frequency $\Delta \nu$. The only relevant time scale in the problem is the time it takes a sound wave to travel from the center of the star to the surface, and from dimensional analysis

$\Delta \nu \propto \left( \int_0^R \frac{dr}{c_s} \right)^{-1} \propto \sqrt{G \bar{\rho}}$

where $\bar{\rho} \propto M/R^3$ is the mean density.

The series of harmonics will continue in higher and higher frequencies, until the wavelength becomes comparable with the scale height

$\nu_{\max} \propto \frac{c_s}{\lambda} \propto \frac{c_s}{H} \propto \frac{c_s}{c_s^2/g} \propto \frac{g}{c_s}$

The speed of sound at the surface can usually be estimated from the temperature and the composition of the atmosphere. So from frequency we can infer the gravitational acceleration on the stellar surface $g = G M / R^2$. When the wavelength becomes smaller than the scale height, the waves will "leak" out of the star rather then being reflected and amplified by constructive interference.