The Precise Solar Shape and Its Variability

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Science  28 Sep 2012:
Vol. 337, Issue 6102, pp. 1638-1640
DOI: 10.1126/science.1223231


The precise shape of the Sun has not been convincingly determined, despite half a century of modern photoelectric observations. The expected deviation of the solar-limb shape from a perfect circle is very small, but such asphericity is sensitive to the Sun’s otherwise invisible interior conditions, as well as the solar atmosphere. We use evidence from a long-running experiment based in space to show that, when analyzed with sufficiently high spatial resolution, the Sun’s oblate shape is distinctly constant and almost completely unaffected by the solar-cycle variability seen on its surface. The solar oblateness is significantly lower than theoretical expectations by an amount that could be explained by a slower differential rotation in the outer few percent of the Sun.

The magnitude of the solar oblateness (expressible as a pole-equator radius difference or a Legendre polynomial shape coefficient) is a fundamental quantity and is potentially a constraint on theories of gravity (1). The solar-limb shape responds to the solar gravitational potential, the Sun’s interior rotation, and magnetic and fluid-flow stresses at the visible photosphere. Modern observations over the past half-century have concentrated on determining the Sun’s shape, or solar oblateness, but various instruments on the ground and space have not yielded consistent measurements (Fig. 1) (18). The discrepant measurements have been interpreted as evidence that the Sun’s shape varies with the solar cycle (3, 7, 9) or that it may be dominated by surface magnetic contamination (8).

Fig. 1

Summary of modern photoelectric solar oblateness measurements. The difference in apparent equatorial and polar radii (Δr) versus the mean observation date is plotted. The dashed lines show the 1σ limits of the precise value obtained from the HMI series of measurements.

The Helioseismic and Magnetic Imager (HMI) aboard NASA’s Solar Dynamics Observer (SDO) is a satellite instrument unique in its capability to observe the solar limb. The HMI benefits from the experience of NASA’s Solar and Heliospheric Observatory and Michelson Doppler Imager (SOHO/MDI) experiment, which provided previous sensitive, space-based solar-limb–shape measurements (10). HMI has 16 times as many detector pixels as MDI and uses precise thermal control of the instrument and its optics. Because the SDO spacecraft and HMI undergo semiannual roll calibrations, the HMI can yield very sensitive measurements of the solar-limb–shape variability. From its geosynchronous orbit, the HMI generates more than 15,000 solar images per day and is immune from the optical distortion effects of the terrestrial atmosphere. Solar measurements from the past two years are particularly important for understanding how magnetism may affect the Sun’s shape, either through global solar-cycle changes or by superficial effects in the magnetized outer solar atmosphere. During this time, the Sun has evolved from its extended magnetic minimum, when it exhibited periods with no observable sunspots or solar activity, to its present approach toward maximum, where daily sunspot numbers have exceeded 90.

Observations of the solar disk from the ground suffer from terrestrial atmospheric blurring and distortion noise at the arc second scale. Over the 0.5° angular diameter of the Sun’s disk, this noise is incoherent around the limb. In contrast, from space the dominant limb-shape noise comes from spacecraft pointing and optical distortion. Because these errors are either coherent across the image or temporally slowly varying, they can be removed to yield a precise solar shape that may be orders of magnitude more accurate than that obtained from ground-based data. For the SOHO/MDI satellite, these corrections were made by combining multiple images while the satellite rotated with respect to the Sun-pointing direction (7, 10). This allowed the solar distortion, which is ~10 milli–arc sec, to be separated from nonaxisymmetric optical distortions, which could be more than 20 times larger. Previous MDI measurements showed that the limb position and brightness may be accurately extracted from image pixels by evaluating the radial and angular dependence of the limb-darkening function (LDF), which we empirically determine from each image. The derived image-mean LDF, Γ(r), is the template and reference for measuring local limb position and brightness variations. It is critical for obtaining the solar-limb distortion function. We determined this function with the use of least-squares techniques (11), which took as input the HMI 4-by-4–K pixel images with an image scale of 0.5 arc sec per pixel (12). The brightness and shape were then obtained in 180 (2° wide) θ bins.

Previous MDI measurements (7, 10) were obtained from relatively defocused, 2–arc sec per pixel sampled data and suffered from blurring caused by instrumental distortion. The instrument point-spread function (and effective angular resolution) was also temporally more variable than HMI because of limited temperature control. The results presented here are more accurate because of HMI’s improved stability and because our analysis corrects for distortion before evaluating the template LDF, Γ(r) (Fig. 2). The spatial resolution increased between the MDI and HMI experiments from ~2.5 to better than 1 arc sec.

Fig. 2

Solar limb spatial resolution improvement with HMI. The solid curve shows the normalized HMI LDF, Γ(r), whereas the dashed curve shows the MDI LDF. The steeper slope of the HMI function is due to the considerably higher spatial resolution of HMI, which cleanly resolves the steep edge in the LDF.

True solar and instrument contributions to the apparent limb shape were distinguished by rolling the SDO spacecraft in 16 or 32 steps of 22.5° or 11.25°, respectively. The solar signal rotated through the data as SDO rolled while the instrument noise was fixed to the satellite reference frame. Full disk images were obtained at several narrow (7.6 pm) wavelength bands and in various Stokes polarization states at each SDO roll angle. This analysis used the far-red HMI continuum pass-band and the circularly polarized limb data, but the results were insensitive to the optical polarization. Six SDO/HMI 7-hour roll sequences have been obtained. HMI provided nearly simultaneous measurements from two separate charge-coupled device cameras that observed different optical polarization states (12), but notably, the two resulting local limb position and brightness data sets were identical to within ~3 × 10−4 of a pixel and 2 × 10−5 of the mean limb brightness (13). The dominant HMI limb noise came from the intrinsic solar-limb roughness and brightness variability with uncorrected instrument noise that was one to two orders of magnitude smaller.

Our high-resolution analysis effectively separates (and decorrelates) local LDF brightness from the measured limb displacement in all bins that are not “bright.” The low-resolution MDI analysis (7, 10) shows that brighter limb regions move the limb inward at all signal levels. With HMI, only outlier data bins brighter than ~0.5% affect the limb position. This excess brightness is associated with magnetic regions and a real change in the solar atmosphere known as the Wilson depression (14). In HMI data, the Wilson depression appears as an inward shift of the solar limb of ~0.02 pixels or 10 milli–arc sec in bins with a brightness excess of 0.5%. With the greater sensitivity of HMI, smaller limb brightness values show no correlation between brightness and limb position, unlike results from MDI (15). Simply removing these bright data bins from the global shape analysis eliminates magnetic contamination of the limb shape to the level of our noise threshold of less than 0.5 milli–arc sec in the oblateness. Depending on the solar activity, including these bins can bias the oblateness by as much as a few milli–arc seconds. Less than 5% of the HMI data were deleted in this way. This finding contrasts with the Reuven Ramaty High-Energy Solar Spectroscopic Imager oblateness procedure performed by Fivian et al. (8), which removed ~80% of their measurements from analysis.

We divided the brightness-corrected limb data into 12 angular bins from which we evaluated the mean and standard deviation of the limb position. We then obtained the results for six spacecraft rolls, collected over a 2-year period, from a weighted fit to a quadrupole (oblateness, C2) and fourth-order Legendre polynomial (a hexadecapole, C4) (Fig. 3). The final temporal variations of the oblateness and hexadecapole solar-shape terms are shown in Fig. 4, along with the corresponding sunspot numbers. The implied difference between the equatorial and polar solar radii due to these shape terms is –3/2 of the oblateness and –5/8 of the hexadecapole amplitude. Predicted dimensionless shape coefficients (C2 and C4 were normalized by the solar radius) have been calculated from the solar interior-rotation profile (16) and can be compared with these data by dividing the Legendre polynomial amplitudes by a nominal solar radius of 960 arc sec.

Fig. 3

Binned continuum wavelength, circular polarization, and limb-shape data from roll 1 (table S1). The dashed curve shows the oblateness and hexadecapole fit to the HMI shape versus angle. Error bars indicate statistical 1σ errors from these roll data.

Fig. 4

Oblateness (asterisks) and hexadecapole (dashes) coefficients versus time. The dashed and dotted lines show the mean and standard error, respectively, of the weighted average of these shape amplitudes. Triangles denote the 3-day mean sunspot numbers (right y axis) centered at the time of the HMI measurements. Error bars indicate 1σ statistical errors derived from the measurements.

The HMI oblateness coefficient is –4.80 ± 0.33 milli–arc sec (corresponding to an equator-pole radius difference of 7.20 ± 0.49 milli–arc sec) and rules out the possibility of a substantial solar-cycle oblateness variation of more than 0.1 milli–arc sec during the rising phase (17). The oblateness is apparently constant, despite the large solar-cycle change in surface magnetism, and confirms that this analysis is not sensitive to spurious facular limb-shape contamination. Earlier oblateness measurements have different systematic errors and have yielded marginally larger values. Fivian et al. (8), in particular, come close and also have a potentially long time series of measurements to compare with HMI [but see also (18)].

It is, perhaps, surprising that the solar oblateness does not vary with the global solar 11-year cycle, as some of the earlier measurements have suggested (3, 7, 9). We have seen that lower–spatial resolution LDF measurements can introduce a weak correlation between the apparent limb position and brightness (7, 8). Because the limb brightness is intrinsically variable due to faculae, this may affect the apparent solar shape. Our results show that the apparent shape variability is not due to a changing global oblateness. Solar-shape results are now also consistent with a constant solar radius, as measured recently (19), and these results support the helioseismic inference that the solar magnetic cycle does not substantially perturb the interior rotation (by more than ~2%) and stratification at low solar latitudes or, on average, in the outer 70% of the solar interior (20).

The dimensionless oblateness coefficient is C2 = –5.00 ± 0.34 × 10−6, which is significantly lower in magnitude than the value –5.87 × 10−6 implied by helioseismic rotation data (16). Though this oblateness value is consistent with the early MDI measurement (10), it is now almost 3σ lower than the helioseismic oblateness prediction (16). This difference cannot be accounted for by decreasing the core rotation (for example, in the inner 20% of a solar radius), which is not well constrained by helioseismology. On the other hand, the difference in values could be explained by a 3 to 10% perturbation in the local rotation rate in the outer few percent of the Sun (16).

Finally, the Sun’s mean hexadecapole shape amplitude is small (–0.1 ± 0.4 milli–arc sec) but shows a hint of variability (21). This value is marginally correlated with the sunspot cycle with an amplitude of 2.1 ± 2 milli–arc sec. The hexadecapole shape is also sensitive to the internal solar differential rotation, but if due only to rotation, it would require large changes (on the order of 50%) in the outer parts of the Sun (16) that are not consistent with the constant helioseismic rotation (20) and the constant oblateness. In contrast, solar-cycle changes in near-surface flows or magnetic stresses localized near mid-latitudes could affect C4 and not the oblateness.

Supplementary Materials

Supplementary Text

Figs. S1 to S3

Table S1


References and Notes

  1. We let L(r,θ) be the observed LDF function from a binned satellite image. From this, we used the circular average mean LDF represented by Γ(r) to solve for a brightness function α(θ) and the limb shape β(θ). The binned LDF function was then expressed as L(r,θ) = [α(θ) + 1]Γ[r – β(θ)], where α and β represent the mean limb brightness change and position around the limb. We then linearized this equation and solved it as a least-squares problem to find α and β. We obtained the function Γ(r) from the binned intensity of limb pixels, whereas we iterated the solution for β(θ) so that Γ(r) was adjusted at each iteration by correcting the limb-pixel binning by shifting pixels by the local β(θ) from the previous iteration. This was done for each of the typically 13,000 images obtained during an SDO spacecraft roll. After two iterations, the solution was stable to better than 5%.
  2. Figure S1 shows that the analysis recovers the limb shape, independent of any limb brightness variations. Figure S2 shows that independent simultaneous HMI solar-limb shape and brightness measurements agree on all angular scales and that the limb position and brightness measurements are dominated by solar atmosphere inhomogeneity and its global asphericity.
  3. Figure S3 shows how the limb brightness and position are correlated and how the brightness measurements [α, see (11)] can be used to flag localized magnetic limb contamination of the limb shape. The shape analysis is broadly insensitive to the brightness threshold, with no significant change in the derived global oblateness, even with large changes in the assumed brightness threshold.
  4. The χ2statistic for these 5 degrees of freedom and a constant to describe the Fig. 4 C2 data are both equal to 2.8. This indicates no statistical basis for a nonconstant C2 at better than the 99.9% confidence level.
  5. The χ2 statistic for describing the hexadecapole amplitude as a constant was 9.4. This was marginally inconsistent (at 95% level) with a constant. Linear regression of the hexadecapole measurements against the sunspot number time series suggested a marginally significant hexadecapole solar-cycle variation with an amplitude of 2.1 ± 2 milli–arc sec.
  6. Acknowledgments: The raw and astrometric data used for this analysis are available via the HMI public archives ( The development of the HMI astrometry pipeline was supported by NASA and a grant to Stanford Univ. and the Univ. of Hawaii (NNX09AI90G). The HMI experiment aboard the SDO satellite was funded, in part, by a NASA contract to Stanford (NAS5-02139). J.R.K. was supported by a senior Humboldt prize while some of this work was done at the Kiepenheuer Institut fur Sonnenphysik. M.E. was partially supported by Instituto Nacional de Estudos do Espaço (CNPq), CNPq grant 303873/2010-8, and Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior grant 0873/11-0. We thank H. Hudson for comments on this manuscript.
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