3.2 Solar-rotation tomography

There are three wavelength regimes where solar-rotation tomography has been applied, but each one requires a different treatment of the radiation: tomography of (i) optically thin emission in EUV and soft X-rays, (ii) Thompson-scattered white light, or (iii) radio wavelengths. These tomographic 3D reconstruction techniques are using only one single viewpoint (at Earth), while the aspect angle change is produced by the solar rotation.

3.2.1 EUV and Soft X-rays

Let us consider first the simplest method of optically thin EUV or soft X-ray emission. The brightness of the solar corona observed in EUV or soft X-rays is produced by optically thin free-free emission, which can be characterized by an emission measure EM that is proportional to the squared electron density ne and column depth along the line-of-sight at a given temperature T (assuming that the plasma is in local thermal dynamic equilibrium, which is not always the case for loops undergoing rapid heating),

∫ EM ∝ n2e(z,T )dz , (4 )
while the intensity or flux F λ(x, y) measured at a wavelength λ represents the temperature integral of the differential emission measure dEM βˆ•dT ∝ n2dz βˆ•dT e multiplied with the filter response function R λ(T),
∫ dEM (T ) F λ = ---------R λ(T)dT , (5 ) dT
which defines the general inversion problem for the 3D density distribution ne (x, y,z), for a given temperature filter λ,
F (X ,y ,πœƒ ) = ∑ A n2 (X ,y ,Z ) + σ (X ,y ,πœƒ ) , (6 ) λ i j k l λ e i j l λ i j k
with A λ being the conversion factor of the squared density n2e to the observed brightness F λ (as defined by Eqs. (4View Equation) and (5View Equation)), and σλ is the data noise. The absolute coordinate system with origin in the Sun center is given in coordinates (xi,yj,zk) with the x-axis co-aligned to the solar rotation direction, while (X ,y ) i j is the image coordinate system rotated around the solar axis (y-axis) by the line-of-sight angle πœƒk according to,
( ) ( ) ( ) | Xi | | cos πœƒk − sin πœƒk| | xi| ( Zl ) = ( sin πœƒk cos πœƒk) ( zl) . (7 )
Thus, if we have an image with a size of N 2 pixels and want to reconstruct the 3D density distribution n (x, y,z) e with N 3 voxels, the problem is under-constrained. Using the variation of the aspect angle, for instance by taking multiple images (Nk) over a time interval with significant solar rotation (of the aspect angle πœƒk), will increase the number of constraints to N 2Nk, as long as the reconstructed volume is static, but a unique solution would theoretically require Nk = N images, which is not feasible in practice. The inversion of the 3D density distribution is therefore always under-constrained and requires special inversion techniques with additional constraints. One way to reduce the degrees of freedom is to reduce the solution volume by using a spherical coordinate system aligned with the solar surface and by restricting the number or radial voxels to a small number that covers only about the lowest density scale height, see for instance Frazin et al. (2009bJump To The Next Citation Point) (Figure 15View Image, bottom left).
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Figure 4: The principle of the backprojection method used in medical tomography is visualized for the case of two 1-D projections observed from two arbitrary directions (angle πœƒ), from which the unknown true 2-D brightness distribution is reconstructed (from Davila and Thompson, 1992Jump To The Next Citation Point).

The simplest tomographic reconstruction method is the backprojection method, which yields a probability distribution based on the linear addition of projections from different directions (Figure 4View Image). Another (under-constrained) inversion method that has been used for solar tomography is the robust, regularized, positive estimation scheme (Frazin, 2000Jump To The Next Citation Point; Frazin and Janzen, 2002Jump To The Next Citation Point). The combination of differential emission measure analysis (Eq. (4View Equation)) and solar rotation tomography allows in principle to reconstruct the average density ne (x,y,z) and temperature Te(x,y, z) in each voxel (Frazin et al., 2005b; Frazin and Kamalabadi, 2005bJump To The Next Citation Point). Besides the problem of under-constrained inversion, the time variability is an additional challenge, which could be overcome with Kalman filtering (Frazin et al., 2005aJump To The Next Citation Point). The first attempt to reconstruct the 3D density distribution in the solar corona by means of solar-rotation tomography was done using a 2-week’s dataset of soft X-ray images from Yohkoh (Hurlburt et al., 1994).

3.2.2 White-light

Let us consider the second wavelength regime, i.e., white light, which undergoes Thompson scattering in the solar corona, which is sensitive to the geometry of the distribution of scattering particles and the direction to the observer. The scattering cross-section depends on the angle χ between the line-of-sight and the radial direction through the scattering electron as (Jackson, 1962),

dσ 1 2 2 --- = --re(1 + cos χ), (8 ) dω 2
where dσβˆ•dω is the differential cross-section in units of [2 −1 cm sr] and 2 2 −13 re = e βˆ•mec = 2.82 × 10 cm is the classical electron radius. By integrating over all solid angles we obtain the total cross-section for perpendicular scattering, the so-called Thompson cross-section for electrons,
8π σT = ---r2e = 6.65 × 10−25 cm2 . (9 ) 3
The total scattered radiation I(x, y) can then be calculated by integrating over the source locations of the photons (the photosphere) and the scattering electrons (with a 3D distribution ne(x, y,z)) along the line-of-sight z, as a function of the scattering angle χ (x,y,z) with respect to the observers line-of-sight, which was first calculated by Minnaert (1930Jump To The Next Citation Point), van de Hulst (1950Jump To The Next Citation Point), and Billings (1966Jump To The Next Citation Point). For a recent review see Howard and Tappin (2009aJump To The Next Citation Point). The degree of polarization p, which is observed in the polarized brightness (pB) component of white-light images, is defined as,
p = IT-−-IR- = -IP-, (10 ) IT + IR Itot
where I T and I R represent the tangential and radial terms of the total scattered radiation, as given in Minnaert (1930), van de Hulst (1950), Billings (1966), or Howard and Tappin (2009a). Many coronagraphs (such as those on STEREO) have the capability to measure linear polarization in three orientations, from which the total Itot and polarized brightness IP can be derived. The total scattered radiation is proportional to the electron density ne(x,y, z) of the scattering corona, in contrast to the square-dependence of the observed brightness on the density (Eq. (4View Equation)) for free-free emission in soft X-rays and EUV.

Since the differential cross section dσ βˆ•dω (Eq. (8View Equation)) varies only by a factor of two with angle χ, the plane-of-the-sky or plane-of-max-scattering approximations are very poor. It is essential for any tomography approach which aims to reconstruct an extended density distribution to treat the observations as the result of an extended integral along the line-of-sight. The plane of maximum scattering has often been approximated with the plane-of-sky in the past, which is appropriate for locations near the solar limb, but needs to be corrected with the actual plane of maximum scattering, for sources at large distances from the Sun, indicated with the impact radius d in Figure 5View Image (Vourlidas and Howard, 2006Jump To The Next Citation Point). This correction becomes relevant for tomographic 3D reconstruction of CMEs that propagate far away from the Sun. Using a time series of images in polarized brightness IP taken over a time interval with significant solar rotation (which varies the line-of-sight angles) allows then to deduce the 3D density distribution n (x,y,z) e in a coronal volume.

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Figure 5: Generalized Thompson scattering geometry: The spherical Thompson surface represents the source locations of the maximum (90°) scattering angle with respect to an observer at distance R. The white-light brightness has to be integrated over all scattering positions P along a given line-of-sight (from Vourlidas and Howard, 2006).

The first tomographic reconstruction of the 3D density distribution of the solar corona, based on a time series of coronagraph images from Skylab in white light has been accomplished by Altschuler (1979Jump To The Next Citation Point). Tomographic inversion of coronagraph images from Mark III K-coronameter on Hawaii and from LASCO C-1 were conducted by Zidowitz et al. (1996), Zidowitz (1997), and Zidowitz (1999Jump To The Next Citation Point). Tomography of the solar corona in an altitude range of a few solar radii has been systematically investigated by regularization inversion methods (e.g., Frazin, 2000; Frazin and Kamalabadi, 2005b), and applied to LASCO C-2 datasets (Frazin and Janzen, 2002; Morgan et al., 2009Jump To The Next Citation Point; Morgan and Habbal, 2010Jump To The Next Citation Point), to Mauna Loa Solar Observatory Mark-IV coronameter data (Butala et al., 2005), to STEREO COR-1 datasets (Kramar et al., 2009Jump To The Next Citation Point; Barbey et al., 2011), and by separating contributions of the K- and F-corona (Frazin and Kamalabadi, 2005a).

3.2.3 Radio wavelengths

Solar radio emission has many different emission mechanisms, which moreover have complicated properties depending on the frequency (or wavelength), polarization, opacity, and magnetic field (for an overview see, e.g., Chapter 15 in Aschwanden, 2005Jump To The Next Citation Point, and references therein). For 3D tomography it matters a lot whether radio emission is observed in an optically thin or thick regime. If an optically thin feature is observed (e.g., free-free emission at decimetric frequencies), the radio brightness has to be calculated from a line-of-sight integral with varying opacity, similar as for EUV or soft X-ray wavelengths. In contrast, if optically thick emission is observed (e.g., gyroresonance emission in microwaves), the radio brightness originates from a localized source surface that can be treated like an opaque body in the 3D reconstruction.

Let us consider the case of free-free (bremsstrahlung) emission, which has a free-free absorption coefficient αν(z) for thermal electrons that depends on the ambient ion density ni(z), electron density n (z) e, and temperature T (z) e, at position z along a given observer’s line-of-sight, and radio frequency ν as,

n (z)∑ Z2n (z) αff(z,ν ) ≈ 9.786 × 10 −3-e-----i--i-i--- ln Λ , (11 ) ν2T 3βˆ•2(z)
where ln Λ(z) ≈ 20 is the Coulomb integral, which yields the free-free opacity τ (z, ν) ff as a function of position z by integrating over the column depth range ′ z = [− ∞, z],
∫ z ′ ′ ′ τff(z,ν ) = −∞ αff[Te(z ),ne(z),ν] dz . (12 )
From the free-free opacity we obtain the radio brightness temperature TB(ν ) at the observer’s frequency ν with a further integration of the opacity along the line-of-sight,
∫ 0 TB (ν) = Te(z)exp −τff(z,ν) αff(z,ν) dz. (13 ) −∞
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Figure 6: Quiet-Sun brightness temperature spectrum for an isothermal corona with T = 1.0 MK e (solid line) or Te = 5.0 MK (dashed line) with a base density of 9 −3 n0 = 10 cm and gravitational stratification (with a density scale height of λ ≈ 50 (Te βˆ•1 MK ) Mm) (from Aschwanden et al., 2004Jump To The Next Citation Point).

An example of a quiet-Sun radio brightness spectrum TB (ν) is given in Figure 6View Image for different coronal temperatures in hydrostatic equilibrium. The observed quantity is the flux density I (ν), which can be calculated from the brightness temperature TB(ν ) with the Rayleigh–Jeans approximation at radio wavelengths, integrated over the solid angle ΩS of the radio source,

2ν2kB-∫ I(ν) = c2 TBd ΩS . (14 )
These expressions describe how the radio brightness observed at a particular frequency ν depends on the 3D density ne(x, y,z) and temperature distribution Te(x,y, z) of an observed source (e.g., an active region), and this way describes the inversion problem that has to be solved to obtain the 3D density distribution ne(x, y,z). One way to obtain a large number of observational constraints, is to synthesize radio images at many frequencies ν, a method that is called frequency tomography (Aschwanden et al., 1992; Aschwanden, 1995Jump To The Next Citation Point), which was pioneered with the multi-frequency imaging radio interferometer RATAN-600 with up to 36 frequencies in the ν = 0.1 –5.0 GHz range (Bogod and Grebinskij, 1997; Gelfreikh, 1998; Grebinskij et al., 2000), and has been simulated for a proposed Frequency-Agile Solar Radiotelescope (FASR) (Aschwanden et al., 2004). However, the frequency tomography method allows only to invert the electron density ne(ν) and temperature Te(ν) as a function of the radio frequency ν, while an additional opacity model ν (z) is required to map the radio frequency into a geometric coordinate z for each line-of-sight. Such additional information on the absolute height h (ν ) = z(ν)cosπœƒ (at an angle πœƒ from Sun center) can be obtained from solar-rotation stereoscopy (see Section 3.1).
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Figure 7: Triangulation of point-like 1D (top), curvi-linear 2D (middle), and voluminous 3D (bottom) structures. The triangulation and back-projection of point-like and curvi-linear structures is unique for two stereoscopic viewpoints, while two projections of a voluminous structure do not unambiguously define the 3D surface or volume (from Inhester, 2006Jump To The Next Citation Point).

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