Helioseismology
Key points
- Oscillations contain information about the conditions in the solar interior.
- Duvall's law: a collapse of the p-modes with different ns to one curve, the general dispersion relation for all the p-modes together. It may be inverted (analytically) to learn about the sound-speed profile in the interior.
- Forward problem: a theoretical computation of the helioseismic observables (frequencies, travel times, etc.) from the given solar model. The model here includes also non-homogeneous deviations, such as velocity anomalies or hot/cold spots. The forward problem is usually a solution of the integral equation, where the model and the wave-field description is given.
- Inverse problem: the inverse of the integral relation from the above. In general it cannot be done. There are methods that allow to invert this relation under a set of assumptions/approximations. Such as when the model perturbations are small. Then the integral relation can be changed so that it simplifies to integration of the physical perturbations weighted by the sensitivity kernel over the volume. The sensitivity kernels are Frechét derivatives of the observable functional with respect to the perturbation function. They are the functions of the background model only (!Important: this is valid only in the linear regime!). Then the modelling is easier because the kernels may be pre-computed once and for all the only read-in when the computation of the forward or inverse problem need to be done.
- In the linear regime, the actions of various perturbers add up. Yet, they may have a different significance/magnitude, so in some cases they may be neglected. This decision depends on the amplitude of the sensitivity kernels and the expected amplitude of the perturbers. For instance locally, the flows are usually stronger perturbers, sound-speed perturbations are a weak perturber. On the other hand, when dealing with large-scale regions, the contributions of the flow may average out, wherease the weak sound-speed perturbations do not vanish and become significant.
- Results of the inversions are usually unstable (matrices to be inverted are usually ill-posed), thus regularisation must be part of the inverse problem. This, however, bring a non-uniqueness of the solution (it is a function of the trade-off parameter). The solutions may be "manipulated" (exaggerating!).
- Inversions are described not only by the inverted values, but also by their uncertainties and by the smoothing (averaging) kernel.
- Ring-diagram and time--distance are the methods of local (=limited field of view, spatially resolved) helioseismology. The power-spectrum analysis is the method of global (=full Sun/full star, spatially unresolved) seismology.