Helioseismology

Did You Know?

• The oscillations successfully utilized for seismology are essentially adiabatic, and their dynamics depend on pressure forces, matter with inertia density, and the adiabatic exponent.
• Helioseismology was born from analogy with geoseismology, but the Sun lacks a solid surface and cannot support shear waves.
• Because distant stars cannot be resolved, asteroseismology is restricted almost entirely to low degree modes, making inversion much more difficult.

Types of solar oscillation

The Sun's oscillations are understood as resonant vibrations within a self-gravitating fluid that is roughly spherical and in hydrostatic equilibrium. Each vibration pattern can be described as a radial function multiplied by a spherical harmonic, defined by three quantum numbers: the radial order, which counts the spherical shells where motion is zero; the angular degree, which counts the total nodal circles on a shell; and the azimuthal order, which counts how many of those circles run along lines of longitude. These oscillations fall into two types. Pressure modes, or p modes, are essentially standing sound waves driven by pressure rather than buoyancy. All solar oscillations used to study the interior are p modes, with frequencies from about 1 to 5 millihertz and angular degrees from zero up to a few hundred. Their energy density tends to be highest where sound speed is lowest, so they reveal more about the outer Sun than the core. Gravity modes, or g modes, are confined to convectively stable zones like the radiative interior or atmosphere, where buoyancy provides the restoring force. They fade out in the convection zone, making their surface signals extremely weak. No individual g mode has been definitively measured, though possible indirect detections have been reported and disputed.

Data analysis

To turn raw data into usable information, helioseismologists rely on the Fourier transform. Each oscillation mode behaves roughly like a damped harmonic oscillator, so its power across frequencies follows a Lorentz curve. Data with spatial resolution is first mapped onto spherical harmonics to create time series, which are then Fourier transformed; the resulting one-dimensional power spectra are usually merged into a two-dimensional spectrum. The low-frequency end is overwhelmed by granulation, which must be removed. Granulation shows up more strongly in intensity than in line-of-sight velocity, making velocity measurements the better choice for observatories. Local helioseismology uses several techniques. The Fourier–Hankel method was originally designed to detect wave absorption by sunspots. Ring-diagram analysis, which examines Doppler shifts in power spectra from 15° by 15° patches, reveals the speed and direction of horizontal flows beneath the surface. Time-distance helioseismology tracks travel times of waves between two surface points; any inhomogeneity along the wave path alters that travel time, and an inverse problem is solved to deduce the interior structure and motion.

Inversion

The Sun's oscillation modes represent a discrete set of observations sensitive to its continuous structure, allowing scientists to formulate inverse problems for interior structure and dynamics. Given a reference model, small differences between its mode frequencies and those of the Sun are weighted averages of structural differences, with weighting functions known as kernels. The first inversions used Duvall's law and later Duvall's law linearized about a reference model; these were supplemented by analyses linearizing the full set of stellar oscillation equations about a theoretical reference model, now a standard inversion method. Inversions demonstrated differences in solar models that were greatly reduced by implementing gravitational settling. Rotation breaks the degeneracy of modes with different azimuthal orders, producing rotational splittings that are weighted averages of angular velocity; different modes are sensitive to different parts of the Sun, enabling inference of the rotation rate throughout. Helioseismology has shown the Sun's rotation profile includes a rigidly-rotating radiative zone (though the inner core's rotation rate is not well known), a thin shear layer called the tachocline separating the rigidly-rotating interior from the differentially-rotating convective envelope, a convective envelope where rotation varies with depth and latitude, and a final shear layer just beneath the surface where rotation slows toward the surface.

History

Solar oscillations were first observed in the early 1960s as quasi-periodic intensity and line-of-sight velocity variations with a period of about 5 minutes. Scientists gradually realized the oscillations might be global modes and predicted clear ridges in two-dimensional power spectra; these ridges were confirmed in high-degree modes in the mid-1970s, and mode multiplets of different radial orders were distinguished in whole-disc observations. Jørgen Christensen-Dalsgaard and Douglas Gough suggested using individual mode frequencies to infer the Sun's interior structure, calibrating solar models against low-degree data and finding two similarly good fits. Tom Duvall and Jack Harvey connected the extreme data sets by measuring modes of intermediate degree, establishing the quantum numbers and favoring the higher model, suggesting the neutrino problem's resolution lay in nuclear or particle physics. New inversion methods in the 1980s allowed inference of sound speed and density profiles, corroborating that residual errors in solar structure were not the cause of the neutrino problem. Observations later showed oscillation mode frequencies vary with the Sun's magnetic activity cycle. Networks such as BiSON and GONG were assembled to overcome nighttime observation gaps, and long uninterrupted observations brought the field to maturity, summarized in a 1996 special issue of Science, coinciding with the start of normal operations of SoHO.