Helioseismology

Helioseismology infers the Sun's structure and dynamics from its stellar oscillations. Spatial projection of resolved surface observations onto spherical harmonics identifies angular degree and azimuthal order; a temporal Fourier transform measures mode frequencies. Stellar acoustic modes constrain the adiabatic sound speed, while rotational splitting constrains stellar rotation. Their finite wavelength, surface reflection phases and finite observational coverage limit inversion resolution.

Slow stellar rotation breaks the azimuthal degeneracy of a stellar oscillation. First-order rotational frequency shifts are weighted integrals of the angular velocity with kernels computed from the unperturbed mode. Multiple radial orders and angular degrees provide overlapping constraints on internal rotation. Surface rotation alone cannot replace these internal measurements, and a finite kernel set does not give pointwise uniqueness.

For a high-order stellar acoustic mode, neglecting buoyancy and representing surface reflection by a phase, WKB quantization gives $F(w)={\int }_{r_t}^R\sqrt{1-c^2/(w^2{r}^2)}dr/c$, where $w=\omega /L$ and $L=\sqrt{\ell (\ell +1)}$. Thus the scaled frequencies depend mainly on $\omega /L$, rather than independently on degree and radial order. A frequency-dependent surface phase can be fitted when a constant $\alpha$ is inadequate.

Assume $a=c/r$ decreases outward, tends to zero at $R$, and define $h(a)=-d\ln r/da$. Differentiation of the Duvall law gives $F^{\prime }(w)={\int }_0^{w}h(b)b/[w^2\sqrt{w^2-b^2}]db$. Interchanging integrals and using ${\int }_b^{a}dw/[w\sqrt{w^2-b^2}\sqrt{a^2-w^2}]=\pi /(2ab)$ proves the displayed Abel transform inversion. For a finite outer reference value $w_s$, both lower limits become $w_s$, after removing the known outer-layer contribution to the travel time. Inversion requires a monotone branch and amplifies errors in differentiated frequency data.