2.2 Properties of solar oscillations

The five-minute solar oscillations were first discovered by Leighton et al. (1962

) and interpreted as standing acoustic waves by Ulrich (1970 ) and Leibacher and Stein (1971 ). Deubner (1975 ) then confirmed that the power in the oscillations is concentrated at discrete frequencies for any given horizontal wavenumber, as predicted by Ulrich’s theory. The driving mechanism of solar oscillations is believed to be near-surface turbulent convection (Goldreich and Keeley, 1977 ). Solar and stellar oscillations are discussed in details by Cox (1980

), Gough (1993

), Unno et al. (1989

), and Christensen-Dalsgaard (2002 ). Particularly useful are the lecture notes of J. Christensen-Dalsgaard (Christensen-Dalsgaard, 2003 ).

The small oscillations of a sphere can be represented by a linear superposition of eigenmodes, each characterized by a set of three indices: the radial order $n$ the spherical harmonic degree $l$ and the azimuthal order $m$ . For instance, the radial displacement of a fluid element can be written as

where $r$ is the radius, $h$ and $f$ are spherical-polar coordinates (colatitude and longitude), and $t$ is time. The $m Yl$ are spherical harmonics, $anlm$ is a complex mode amplitude, and $qnl(r)$ is the radial eigenfunction of the mode with frequency $wnlm$ . By convention, $n$ corresponds to the number of nodes of the radial eigenfunction, $l$ indicates the total number of nodal lines on spheres, and $m$ tells how many of these nodal lines cross the equator. A spherically symmetric star would give rise to a spectrum of azimuthally degenerate frequencies. However, rotation and other perturbations lift the $(2l + 1)$ -fold $m$ -degeneracy of the frequency of nonradial mode ( $n,l$ ).

Figure 1:

$m$ -averaged power spectra of solar oscillations obtained from $512 min$ series of tracked Doppler images of size $o o 30 × 30$ (near disk center) observed by MDI (left) and TON (right), versus spherical harmonic degree $l$ and temporal frequency. The ridges are labelled by the value of the radial order $n$ . From González Hernández et al. (1998 ).

Figure 1

shows $m$ -averaged power spectra of solar oscillations derived from series of tracked Doppler images of size $o o 30 × 30$ (near disk center) observed simultaneously by the MDI and TON instruments. Each ridge in the power spectrum corresponds to a different radial order $n$ . The lowest frequency ridge ( $n = 0$ ) is for the fundamental (f) modes. The f modes are identified as surface gravity waves, with nearly the dispersion relation for deep water waves, $w2 = gk$ , where $w$ is the angular temporal frequency, $g = 274$ $-2 m s$ is the gravitational acceleration at the Sun’s surface, $k -~ l/Ro .$ is the horizontal wavenumber, and $Ro . = 696 Mm$ is the solar radius. The f modes propagate horizontally. All other ridges, denoted $pn$ , correspond to acoustic modes, or p modes. The restoring force for p modes is pressure. The ridge immediately above the f mode ridge is $p 1$ , the next one $p 2$ , and so forth. Low- $l$ and high- $n$ modes penetrate deeper inside the Sun. For frequencies above the acoustic cutoff frequency ( $5.3 mHz$ ), acoustic waves are not trapped inside the Sun. Acoustic modes with similar values of $wnlm/l$ propagate to similar depths inside the Sun. For degrees larger than about 150, wave damping becomes significant and modes are not resolved anymore (continuous ridges). Figure 2

is another beautiful example of a power spectrum of solar oscillations, obtained from data collected during the 1994 south pole campaign.

Figure 2:

Power spectrum of solar oscillations observed in brightness from the geographic south pole in 1994 ( $Ca II K1$ line, $6 ºA$ bandpass). Notice that ridges of power can be seen well beyond the acoustic cutoff frequency. Courtesy of T.L. Duvall.