Asteroseismology

Asteroseismology or astroseismology is the study of oscillations in stars. Because a star's different oscillation modes are sensitive to different parts of the star, they inform astronomers about the internal structure of the star, which is otherwise not directly possible from overall properties like brightness and surface temperature. Asteroseismology is closely related to helioseismology, the study of stellar oscillations specifically in the Sun. Though both are based on the same underlying physics, more and qualitatively different information is available for the Sun because its surface can be resolved. Asteroseismology or astroseismology is the study of oscillations in stars. Because a star's different oscillation modes are sensitive to different parts of the star, they inform astronomers about the internal structure of the star, which is otherwise not directly possible from overall properties like brightness and surface temperature. Asteroseismology is closely related to helioseismology, the study of stellar oscillations specifically in the Sun. Though both are based on the same underlying physics, more and qualitatively different information is available for the Sun because its surface can be resolved. By linearly perturbing the equations defining the mechanical equilibrium of a star (i.e. mass conservation and hydrostatic equilibrium) and assuming that the perturbations are adiabatic, one can derive a system of four differential equations whose solutions give the frequency and structure of a star's modes of oscillation. The stellar structure is usually assumed to be spherically symmetric, so the horizontal (i.e. non-radial) component of the oscillations is described by spherical harmonics, indexed by an angular degree ℓ {displaystyle ell } and azimuthal order m {displaystyle m} . In non-rotating stars, modes with the same angular degree must all have the same frequency because there is no preferred axis. The angular degree indicates the number of nodal lines on the stellar surface, so for large values of ℓ {displaystyle ell } , the opposing sectors roughly cancel out, making it difficult to detect light variations. As a consequence, modes can only be detected up to an angular degree of about 3 in intensity and about 4 if observed in radial velocity. By additionally assuming that the perturbation to the gravitational potential is negligible (the Cowling approximation) and that the star's structure varies more slowly with radius than the oscillation mode, the equations can be reduced approximately to one second-order equation for the radial component of the displacement eigenfunction ξ r {displaystyle xi _{r}} , d 2 ξ r d r 2 = ω 2 c s 2 ( 1 − N 2 ω 2 ) ( S ℓ 2 ω 2 − 1 ) ξ r {displaystyle {frac {d^{2}xi _{r}}{dr^{2}}}={frac {omega ^{2}}{c_{s}^{2}}}left(1-{frac {N^{2}}{omega ^{2}}} ight)left({frac {S_{ell }^{2}}{omega ^{2}}}-1 ight)xi _{r}} where r {displaystyle r} is the radial co-ordinate in the star, ω {displaystyle omega } is the angular frequency of the oscillation mode, c s {displaystyle c_{s}} is the sound speed inside the star, N {displaystyle N} is the Brunt-Vaisala or buoyancy frequency and S ℓ {displaystyle S_{ell }} is the Lamb frequency.The last two are defined by N 2 = g ( 1 Γ 1 P d p d r − 1 ρ d ρ d r ) {displaystyle N^{2}=gleft({frac {1}{Gamma _{1}P}}{frac {dp}{dr}}-{frac {1}{ ho }}{frac {d ho }{dr}} ight)}