2.1 Introduction

The raw data of helioseismology consist of measurements of the photospheric Doppler velocity – or in some cases intensity in a particular wavelength band – taken at a cadence of about one minute and generally collected with as little interruption as possible over periods of months or years; the measurements can be either imaged or integrated (“Sun as a Star”). An overview of the observation techniques can be found in Hill et al. (1991a). Figure 2View Image shows a typical single Doppler velocity image of the Sun, and Figure 3View Image a portion of an l = 0 time series, derived by averaging the velocity over the visible disk for each successive image in a set of observations. The five-minute period and the rich beat structure are clearly visible in the time series. For an example of an integrated-sunlight spectrum from a long series of observations, see Figure 15View Image.
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Figure 2: A single Doppler velocity image of the Sun from one GONG [Global Oscillation Network Group] instrument (left), and the difference between that image and one taken a minute earlier (right), with red corresponding to motion away from, and blue to motion towards, the observer. The shading across the first image comes from the solar rotation.
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Figure 3: A segment of an l = 0 time series of Doppler velocity observations, showing the dominant five-minute period and the rich beat structure.

As was first discovered by Deubner (1975), the velocity or intensity variations at the solar surface have a spectrum in k − ω or l − ν space that reveals their origin in acoustic modes propagating in a cavity bounded above by the solar surface and below by the wavelength-dependent depth at which the waves are refracted back towards the surface. These “p modes” can be classified by their radial order n, spherical harmonic degree l, and azimuthal order m; as discussed, for example, in Section 2.2 of Birch and Gizon (2005), the radial displacement of a fluid element at time t, latitude 𝜃 and longitude ϕ can be written in the form

l δr(r,𝜃,ϕ,t) = ∑ a ξ l(r)Ym (𝜃,ϕ)eiωnlmt, (1 ) nlm n l m= −l
where ξnlm is the radial eigenfunction of the mode with frequency ωnlm and Ylm(𝜃,ϕ ) is a spherical harmonic. As seen in Figure 4View Image, the power in the spectrum falls along distinct “ridges” in the l − ν plane, each ridge corresponding to one radial order. The modes making up the n = 0 ridge are the so-called f modes, which are surface gravity waves. The p modes, so called because their restoring force is pressure, are excited at the surface and have their largest amplitudes there. Another class of modes, the g modes with gravity as the restoring force, excited in the core and with amplitudes vanishing at the surface, are hypothesized to exist but have so far not been definitely observed (Section 5.9).
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Figure 4: Typical l − ν spectrum from one day of GONG observations (image courtesy NSO/GONG).

The longer the horizontal wavelength – and the lower the degree – the more deeply the modes penetrate, with the radial l = 0 mode going all the way to the core of the Sun (but providing no rotational information), while modes with l ≥ 200 or so penetrate only a few megameters below the surface and are generally too short-lived to form global standing waves; these are the modes used for local helioseismology. The lower turning point radius, rt, is a monotonic function of the phase speed ν∕L, where ∘-------- L = l(l + 1) ≈ l + 1∕2, as shown in Figure 5View Image. The varying penetration depth with degree, as illustrated in Figure 6View Image, makes it possible to deduce the rotation and other properties of the solar interior profile as a function of depth.

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Figure 5: Lower turning point of acoustic modes as a function of phase speed ν∕L, based on Model S of Christensen-Dalsgaard et al. (1996).

The lowest-degree modes are observed in integrated sunlight, but for the purposes of measuring the interior rotation profile we are mostly concerned with what are termed medium-degree (l ≤ 300) modes, which can be observed with imaging instruments of relatively modest (≈ 10 arcsec) resolution. The power in the modes peaks at about 3 mHz, or a period of 5 minutes; useful measurements can be made for modes between about 1.5 and 5 mHz, with the frequency determination becoming more challenging at the extremes due to signal-to-noise issues and, at the high-frequency end, to the increasing breadth of the peaks.

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Figure 6: Locations of modes in the l,ν plane for a typical MDI mode set. The colored shading represents the radial regions in which the modes have their lower turning points; the core, r ≤ 0.2 R⊙, the radiative interior, 0.2 ≤ r∕R ⊙ ≤ 0.65, the tachocline, 0.65 ≤ r∕R ⊙ ≤ 0.75, the bulk of the convection zone, 0.75 ≤ r∕R ⊙ ≤ 0.95, and the near-surface shear layer, r∕R ⊙ ≥ 0.95; the dashed line on the lower right corresponds to r∕R = 0.99 ⊙.

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