The Solar X-ray Limb II

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Introduction

How big is the Sun? This basic astronomical question, no doubt asked by Assyrian or Mayan astronomers in their day, has not been answered very much better yet than in the 1891 paper of Ref. [1]. In principle RHESSI can now improve on this result substantially by using a radically different technique.

In RHESSI Nugget No. 135 we introduced an idea of Gordon Hurford's: to make use of RHESSI's unique capability to measure visibilities, the individual Fourier components of a source's spatial distribution. RHESSI can do this with extraordinary precision, depending mainly upon the total number of photon counts from a given source. The idea Hurford proposed in the previous Nugget was to take the particular visibility component exactly perpendicular to the limb (fringes exactly parallel to the limb), measure its phase precisely, and then by reference to the accurate RHESSI metrology, determine the exact height of the X-ray limb. We have carried this idea further and obtained some representative data, but have not succeeded yet in obtaining a final result. This Nugget thus gives a progress report, including simulations of the expected effect and observations of a representative event.

The Background

See the previous Nugget for more detail, but the essential point of the measurement is captured here in Figure 1. The Figure shows results of model calculations for a 2D Gaussian source cut in the middle by a sharp absorption edge as would be produced by an occultation. Here we are imagining a flare (but any source would do) behind the Sun, producing back-lighting and a sharply defined shadow resulting from the occultation. If RHESSI were very sensitive, a cosmic X-ray source in the far background would do nicely as it drifted into occultation behind the Sun, but in the short term we just plan to use solar flares. They are bright (a good thing) but highly variable and not really Gaussian, providing interesting possible problems to solve. We note that the visibilities combine linearly, and so the integration of many events may help to increase the signal-to-noise ratio.

Figure 1: Left, some model visibility amplitudes for a Gaussian model with diameter (FWHM) 12 arc s. The vertical lines show the spatial frequencies of RHESSI grids 1 and 3. The colored lines show different degrees of occultation (at 50%, the Gaussian is just cut in half). The right panel shows the variation of visibility amplitude with position angle for two different source sizes.

So, from the model point of view, we have a clear shot at measuring the absorption height of the solar limb - the physical radius of the Sun - by this technique.

Some Data and Some Problems

The X-ray sources produced by solar flares have finite dimensions, and so with a bit of trigonometry one can conclude that many flare sources should be partially occulted by the limb. The Nugget of April 1, 2014 shows images of some possible examples. For an image that looks like an occulted source, though, one really does not know about its spatial content for the purpose of limb determination: one needs to refer to the visibilities themselves to be sure. Accordingly we have searched for suitable flare events. The requirements are actually fairly stringent. As Figure 1 shows the occultation signal becomes stronger for greater occultations and finer collimator resolution. Thus it is only the visibility observed by RHESSI's highest-resolution collimator that really counts, and only a small fraction of the position angles it observes as RHESSI rotates.

Figure 2 shows an example that we have worked extensively with. The three panels show (left) a CLEAN image, revealing the source to be compact and possibly simple; and (right) the details of the observed visibilities. The visibility amplitudes plotted as a function of position angle (center) show a peak just at the angle expected for the limb orientation (fringes parallel to the limb); this is Collimator 1, and the scan also shows measureable visibilities at other position angles that must reveal fine structure at other orientations, intrinsic to the source rather than to the external occultation. The right panel just extends the same information in phasor form, showing the real and imaginary parts of the visibilities. The phase information encodes the height of the limb itself.

Figure 2: Actual data. Left, an image of a flare just behind the limb (SOL2002-04-04T15:29); middle, the dependence of visibility amplitude in Collimator 1 on position angle; right, the complex (phasor) representation of the data.

This all looks very proper, but our work on this has turned up a couple of problems. First, there is little data (none for this flare) from RHESSI Collimator 2, whose visibility would independently check that of Collimator 1. Second, the position angle of the limb invariably falls close to the "turning point" of the RHESSI modulation patterns, leading to visibility artifacts that are hard to control (see Nugget No. 8 for information about these patterns). For these reasons we have not published the results shown below, in Figure 3

Figure 3: Tentative results: left, the RHESSI limb height compared with various earlier estimates. The point labeled "Allen 1973" is that of Ref. [1]. Right, the visible limb profile normalized to its disk-center intensity, and a line showing a model the X-ray absorption that RHESSI measures. The red point (the data of Figure 2) miss the model by slightly more than one sigma, but the error bars are quite small.

In Figure 3 we are comparing the visible emission limb with our X-ray extinction limb. There are model dependences in relating these to the actual radius of the Sun, which lies some distance below the limb height. The X-ray model dependences are not so great, since the absorption (at higher energies, at least) comes mainly from [Thomson scattering] which directly reflects the mass distribution on the line of sight.

Conclusions

We have carried our work on the X-ray limb a step further than when the topic was first discussed, and now we have tentative results. They are very encouraging but not conclusive at present. Once we have understood the problems, we believe that we can significantly reduce the uncertainties in the measurement since the visibilities can be added linearly. We thus can use multiple observations, learning about systematics and refining the result. This may not happen soon but the data look quite promising. We note that a related alternative way of doing this physical measurement would be to observe solar occultations of cosmic X-ray sources, but this would require much better sensitivity and coverage that we have now or in the foreseeable future.

References

[1] "Der Sonnendurchmesser und der Venusdurchmesser nach den Beobachtungen an den Heliometern der deutschen Venus-Expeditionen"