Solar-like oscillations

An echelle diagram for the Sun, using data for low-angular-degree modes from the Birmingham Solar Oscillations Network (BiSON).[7][8] Modes of the same angular degree ${\displaystyle \ell }$ form roughly vertical lines at high frequencies, as expected from the asymptotic behaviour of the mode frequencies.

The peak of the oscillation power roughly corresponds to lower frequencies and radial orders for larger stars. For the Sun, the highest amplitude modes occur around a frequency of 3 mHz with order ${\displaystyle n_{\mathrm {max} }\approx 20}$, and no mixed modes are observed. For more massive and more evolved stars, the modes are of lower radial order and overall lower frequencies. Mixed modes can be seen in the evolved stars. In principle, such mixed modes may also be present in main-sequence stars but they are at too low frequency to be excited to observable amplitudes. High-order pressure modes of a given angular degree ${\displaystyle \ell }$ are expected to be roughly evenly-spaced in frequency, with a characteristic spacing known as the large separation ${\displaystyle \Delta \nu }$.[9] This motivates the echelle diagram, in which the mode frequencies are plotted as a function of the frequency modulo the large separation, and modes of a particular angular degree form roughly vertical ridges.

The frequency of maximum oscillation power is accepted[10] to vary roughly with the acoustic cut-off frequency, above which waves can propagate in the stellar atmosphere, and thus are not trapped and do not contribute to standing modes. This gives

${\displaystyle \nu _{\mathrm {max} }\propto {\frac {g}{\sqrt {T_{\mathrm {eff} }}}}}$

Similarly, the large frequency separation ${\displaystyle \Delta \nu }$ is known to be roughly proportional to the square root of the density:

${\displaystyle \Delta \nu \propto {\sqrt {\frac {M}{R^{3}}}}}$

When combined with an estimate of the effective temperature, this allows one to solve directly for the mass and radius of the star, basing the constants of proportionality on the known values for the Sun. These are known as the scaling relations:

${\displaystyle M\propto {\frac {\nu _{\mathrm {max} }^{3}}{\Delta \nu ^{4}}}T_{\mathrm {eff} }^{3/2}}$

${\displaystyle R\propto {\frac {\nu _{\mathrm {max} }}{\Delta \nu ^{2}}}T_{\mathrm {eff} }^{1/2}}$

Equivalently, if one knows the star's luminosity, then the temperature can be replaced via the blackbody luminosity relationship ${\displaystyle L\propto R^{2}T_{\mathrm {eff} }^{4}}$, which gives

${\displaystyle M\propto {\frac {\nu _{\mathrm {max} }^{12/5}}{\Delta \nu ^{14/5}}}L^{3/10}}$

${\displaystyle R\propto {\frac {\nu _{\mathrm {max} }^{4/5}}{\Delta \nu ^{8/5}}}L^{1/10}}$

• Asteroseismology
• Helioseismology
• Variable stars

• Lecture Notes on Stellar Oscillations published by J. Christensen-Dalsgaard (Aarhus University, Denmark)