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 that is proportional to the squared electron density and column depth along the line-of-sight at a given temperature (assuming that the plasma is in local thermal dynamic equilibrium, which is not always the case for loops undergoing rapid heating),x-axis co-aligned to the solar rotation direction, while is the image coordinate system rotated around the solar axis (y-axis) by the line-of-sight angle according to, et al. (2009b) (Figure 15, bottom left).
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 4). Another (under-constrained) inversion method that has been used for solar tomography is the robust, regularized, positive estimation scheme (Frazin, 2000; Frazin and Janzen, 2002). The combination of differential emission measure analysis (Eq. (4)) and solar rotation tomography allows in principle to reconstruct the average density and temperature in each voxel (Frazin et al., 2005b; Frazin and Kamalabadi, 2005b). Besides the problem of under-constrained inversion, the time variability is an additional challenge, which could be overcome with Kalman filtering (Frazin et al., 2005a). 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).
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 between the line-of-sight and the radial direction through the scattering electron as (Jackson, 1962),Thompson cross-section for electrons, polarized brightness (pB) component of white-light images, is defined as,
Since the differential cross section (Eq. (8)) varies only by a factor of two with angle , 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 in Figure 5 (Vourlidas and Howard, 2006). 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 taken over a time interval with significant solar rotation (which varies the line-of-sight angles) allows then to deduce the 3D density distribution in a coronal volume.
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 (1979). 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 (1999). 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., 2009; Morgan and Habbal, 2010), to Mauna Loa Solar Observatory Mark-IV coronameter data (Butala et al., 2005), to STEREO COR-1 datasets (Kramar et al., 2009; Barbey et al., 2011), and by separating contributions of the K- and F-corona (Frazin and Kamalabadi, 2005a).
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, 2005, 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 for thermal electrons that depends on the ambient ion density , electron density , and temperature , at position along a given observer’s line-of-sight, and radio frequency as,free-free opacity as a function of position by integrating over the column depth range , radio brightness temperature at the observer’s frequency with a further integration of the opacity along the line-of-sight,
An example of a quiet-Sun radio brightness spectrum is given in Figure 6 for different coronal temperatures in hydrostatic equilibrium. The observed quantity is the flux density , which can be calculated from the brightness temperature with the Rayleigh–Jeans approximation at radio wavelengths, integrated over the solid angle of the radio source,frequency tomography (Aschwanden et al., 1992; Aschwanden, 1995), which was pioneered with the multi-frequency imaging radio interferometer RATAN-600 with up to 36 frequencies in the 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 and temperature as a function of the radio frequency , while an additional opacity model is required to map the radio frequency into a geometric coordinate for each line-of-sight. Such additional information on the absolute height (at an angle from Sun center) can be obtained from solar-rotation stereoscopy (see Section 3.1).
Living Rev. Solar Phys. 8, (2011), 5
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