Technical Comments

Comment on “A Large Excess in Apparent Solar Oblateness Due to Surface Magnetism”

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Science  29 May 2009:
Vol. 324, Issue 5931, pp. 1143
DOI: 10.1126/science.1170358

Abstract

Fivian et al. (Reports, 24 October 2008, p. 560) analyzed data from the Reuven Ramaty High-Energy Solar Spectroscopic Imager satellite and reported that the Sun is more oblate than previous measurements have suggested. We argue that their threshold-based analysis yields a biased measure of the solar limb shape geometry.

Space observations can yield precise information on the shape of the Sun, which is an important tool for understanding the solar cycle and the solar interior. Fivian et al. (1) recently argued that the shape of the Sun could be accurately obtained from optical observations from the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) using a technique that locates the edge of the Sun where the limb-darkening function (LDF) reaches a threshold value. Unfortunately, this technique ignores the change in functional form of the LDF at different limb angles. If we describe a solar distortion as a shift in the LDF radially by an amount we label with the function β(θ), then brightness changes can be described by an independent function, say α(θ), that scales the LDF at each latitude (or limb position angle). Another spacecraft experiment, using observations from the Michaelson Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory, demonstrated (2) that the LDF does indeed shift and change shape with solar limb position as L(r,θ)=[1+α(θ)]L¯(rβ(θ)). In this case, a threshold-only–based limb definition, as in (1), mixes the intrinsic solar limb shape and intensity variations. Thus, even a spherical Sun (with β = 0 identically) that has real brightness variations will exhibit a spurious apparent shape, β′(θ), of the form β′(θ) = αθdlnL¯dr=1.5α(θ). Here, the radial derivative of the logarithm of the average LDF was evaluated near the LDF intensity threshold used in (1) to obtain the dimensional constant 1.5 arc seconds.

Fivian et al. used extreme ultraviolet observations to obtain a spatial mask for deleting RHESSI data points from the analysis to try to isolate the effects of brightness contamination from their shape determination. Their mask was a proxy for the near-limb magnetic field, just as the limb brightness due to faculae is a well-known proxy for magnetic fields. They found a plateau in their analysis for the solar oblateness after 80% of the brightest pixels were deleted and then hypothesized that this small subset of the data determined the true solar shape. In contrast, the MDI data (2) determine both the brightness (α) and limb position (β) with no data masking, so that with these results one can directly compute how the apparent oblateness is biased by a threshold-only limb definition. From the MDI analysis, we can easily identify the bright pixel data and simulate the RHESSI analysis by measuring how the apparent oblateness varies with the severity of the masking. Figure 1 shows the difference between the solar oblateness derived from the full LDF analysis of (2) and the RHESSI-style analysis as we mask away successively fainter data until less than 40% of the limb pixels are used. The plot is based on results from solar minimum when there was very little photospheric B-field flux. Still, even after most of the 1997 MDI data were masked (as the RHESSI data were), the brightness-effect oblateness was significantly biased to 0.6 milli–arc sec above the value derived from the full analysis in (2). This bias must increase with the degree of solar activity and is likely to be several times larger at the time of the 2004 RHESSI observations.

Fig. 1

MDI observations obtained near solar minimum in 1997 were used to derive a threshold-based solar oblateness. The solid curve shows the bias in the true oblateness (2) introduced by successively deleting fainter pixels from the analysis. The horizontal axis shows the fraction of bright limb data removed from the calculation, and the plot illustrates how even after 60% of the data are deleted and a plateau in the apparent oblateness is achieved, the derived solar shape remains biased by such a threshold-based limb definition.

We have shown that discarding most of a threshold-LDF measurement using a technique that deletes data with diminishing brightness does not recover the true limb shape. Furthermore, the geometric solar oblateness is not described by the plateau in this apparent shape variation. We contend that the Fivian et al. RHESSI analysis confuses the Sun’s oblateness with brightness contributions and that data masking does not correct this systematic error. In contrast with the Fivian et al. results (1), earlier satellite observations (2) did account for these limb brightness effects.

References and Notes

  1. This research was funded by the NASA Solar and Heliospheric Observatory/MDI grant NNX07AK36G to Stanford University and the NASA Solar Dynamics Observatory/Helioseismic Magnetic Imager contract NAS5-02139 to Stanford University and subcontract to the University of Hawaii.
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