3.1 Solar-rotation stereoscopy

First we are describing here stereoscopic and tomographic 3D reconstruction techniques that use only one single viewpoint (at Earth), while the aspect angle change is produced by the solar rotation.

3.1.1 Static solar-rotation stereosocopy

For quasi-stationary structures in the solar corona, the solar rotation with a synodic period of Tsys = 26.24 days provides a natural change in aspect angle that can be exploited for stereoscopic measurements. If an image of the Sun is aligned with the solar rotation axis in y-direction and the heliographic latitude of the Sun center is small (b0 ≈ 0), the solar rotation rate introduces a displacement Δx of a source at heliographic longitude l1 and latitude b1 in east-west direction as

Δx12 = x(t2) − x (t1) = (R ⊙ + h)cos(b1)[sin (l1 + ωsyn(t2 − t1)) − sin(l1)] , (1 )
where l1 is measured relative to the longitude of the observer and ω = 2π∕T is the differential rotation rate that slightly depends on the latitude b = b1,
2 4 ωsid(b) = A + B sin (b) + C sin (b) , (2 )
with the coefficients A = 14.71 ± 0.05 deg/day, B = –2.4 ± 0.2 deg/day, and C = –1.8 ± 0.3 deg/day, yielding an equatorial (sidereal) rotation period of Tsid = 24.47 days, which has to be multiplied with a factor of Tsyn∕Tsid = 26.24/24.47 = 1.0723 for the synodic coordinate system. The height dependence of the displacement Δx (t) in Eq. (1View Equation) yields a radial altitude measurement h above the solar surface with a radius of R ⊙ = 696, 000 km, since all other parameters (x1,x2,l1,b1) can be measured at two different times t1 and t2, which gives a unique value for the altitude h, supposed that the altitude of the source does not change during the observing interval. The lower time limit of a stereoscopic measurement is given by the desired accuracy of altitude measurement Δh and the spatial resolution Δx of the instrument, because the stereoscopic parallax effect has to exceed the spatial resolution of the instrument, which is given by
TsynΔx-- t ≈ 2 π Δh . (3 )
Say, for a spatial resolution of ′′ Δx = 1 and an accuracy of ′′ Δh = 3000 km ≈ 4 we need t ≈ 1.0 day, which obviously exceeds most evolutionary time scales of coronal structures (typically from minutes to hours).

The first solar-rotation based stereoscopic height measurements are described in Berton and Sakurai (1985Jump To The Next Citation Point), who measured the 3D coordinates of a set of coronal loops identified in Skylab XUV images from 1973, measuring the stereoscopic parallax effect during 1 – 2 days. They estimated the measuring error to ≈ 0.004 R ⊙ ≈ 3000 km and reported the 3D coordinates of a large inter-active region loop with a height of ≈ 0.15 R⊙ = 100,000 km, for which the loop plane inclination angle of 𝜃 ≈ 25∘ could be determined.

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Figure 2: The principle of solar-rotation stereoscopy is illustrated with this example of radio brightness maps observed with the Very Large Array (VLA) at two different days (middle panels), where partial maps covering sources A or B (from day 1) are rotated to the locations A’ and B’ expected at day 2 as a function of the distance R + h ⊙ from Sun center (top), which are then cross-correlated with the actually observed maps A” and B” at day 2 (bottom), which yields a direct altitude measurement hA and hB of the source centroids (from Aschwanden and Bastian, 1994bJump To The Next Citation Point).

The same method of solar-rotation based stereoscopic parallax measurement has been applied to radio maps of active regions at a wavelength of λ = 20 cm (Figure 2View Image), observed with the Very Large Array (VLA) over the course of several days (Aschwanden et al., 1992Jump To The Next Citation Point; Aschwanden and Bastian, 1994aJump To The Next Citation Point,bJump To The Next Citation Point; Aschwanden, 1995Jump To The Next Citation Point). While the VLA has a spatial resolution of ≈ 5′′ at this frequency ν ≈ 1.5 GHz, the accuracy of the source positions could be optimized to ′′ ≈ 0.1 by cross-correlating partial source maps, which yields sub-resolution accuracy for the locations of the source centroids. Radio emission from active regions at ν ≈ 1.5 GHz is dominated by free-free emission and, hence, the 3D centroid position corresponds to the altitude where the source becomes optically thick.

Sunspots are also relatively stable over a time interval of a day and, thus, the iso-Gauss surfaces in the lower corona. Radio mapping of sunspots at microwave frequencies of ν ≈ 10 –15 GHz is dominated by gyro-resonance emission, which originates from dome-like iso-Gauss surfaces above sunspots, typically at harmonics of s = 2,3,4 of the fundamental gyrofrequency of the magnetic field. Measuring the stereoscopic parallax of the dominant gyroresonance source above a sunspot over the course of 4 days with the Owens Valley Radio Observatory (OVRO), the altitudes of the gyroresonant layers could be determine for a set of 7 frequencies in the range of ν ≈ 10 –15 GHz, which were found in a height range of h ≈ 3000– 12,000 km, and allowed to constrain the local magnetic field with a potential field model and to identify the correct harmonics at every frequency (Aschwanden et al., 1995Jump To The Next Citation Point). Of course, stationarity of the magnetic field over time scales of 4 days can only be expected for stable large sunspots, while small-scale magnetic fields change during much shorter time scales in active regions.

3.1.2 Dynamic solar-rotation stereosocopy

Coronal loops are generally not stable over a time period of a day, because the heating rate appears to be discontinuous and intermittent and cooling by conductive and radiative loss occurs on time scales of less than an hour. This means that loops at the same location observed a day apart are not identical, but continuously replaced by plasma upflows and downflows. However, what stays more permanent is the magnetic field, say over time scales of a few days, especially the large-scale dipolar fields that represent the main magnetic field structure of an active region, produced by the leading sunspot and the trailing region with opposite magnetic polarity. The property of the quasi-stationarity of the magnetic field can thus be used to constrain the 3D geometry of near-cospatial loops at different times, regardless how often the plasma is flushed through the magnetic conduits. A technique that uses the stable 3D geometry of the magnetic field and is not sensitive to the fast-paced plasma dynamics has been developed for the 3D reconstruction of a set of dipolar loops that make up an active region, called dynamic stereoscopy (Figure 3View Image), which was applied to a few days of SOHO/EUV images (Aschwanden et al., 1999Jump To The Next Citation Point, 2000Jump To The Next Citation Point). This method allows us to determine the approximate 3D geometry of coronal loops under the assumption of planarity, which requires only one free parameter (the inclination angle of the loop plane) to determine the 3D coordinates of loops from the observed 2D projections [x(s),y(s)] as a function of the loop length coordinate s.

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Figure 3: The principle of dynamic solar-rotation stereoscopy is illustrated for an example of two adjacent loops (top). The only free parameter in the 3D reconstruction is the loop inclination plane, which yields different 2D projections at 3 consecutive days (middle). The best-fitting inclination angle can be determined by optimizing the (parallel) co-alignment between the observed and model loop stripe as a function of the inclination angle (bottom) (from Aschwanden et al., 1999Jump To The Next Citation Point).

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