1. Measuring the Wilson effect: observations and modeling with RHESSI H. Jabran Zahid M. D. Fivian H. S. Hudson

2. Abstract Originally observed over 200 years ago, the Wilson effect is interpreted as a physical depression in the surface of the photosphere in a sunspot umbra. The magnitude of the depression may be several hundred km (~ 1 arc sec). The SAS (solar aspect system) on-board the RHESSI spacecraft gives us solar radius measurements of unprecedented precision, with statistical errors below 20 mas. Here we introduce a method to disentangle photometric effects from the physical displacement of the limb due to the Wilson effect. We propose to use this method to measure the depression with measurement errors on the order ~10 km.

3. Direct Observation of Wilson Depression 1,000 x RHESSI limb shape measurement at the passage of a sunspot, with radius values magnified by 1000 for clarity. The depression is clearly seen above the noise which is a combination of statistical and photospheric effects.

4. We make the following simplifying assumptions: the photosphere and umbra have blackbody continuum spectra at our wavelength, 670 nm, and they have identical limb- darkening function (LDF; we use the RHESSI observations to characterize this). At this wavelength the blackbody intensity ratio is 0.24 for an assumed umbral temperature. We also assume an idealized Gaussian spatial profile with  = 20 arcsec, as shown in Figure 2. These assumptions can be relaxed in more sophisticated analyses. Model Assumptions

5. Figure 2 Gaussian model Limb darkening Product

6. By assuming a sunspot to be characterized by the same limb darkening function (LDF) as the photosphere, we can readily model its photometric deficit. As a first approximation, we model the sunspot intensity profile in space as a Gaussian (Figure 2, top). The photometric deficit (Figure 2, bottom) resulting from the sunspot can be calculated by taking our normalized LDF (Figure 2, middle) and multiplying it by our estimated photometric ratio and the sunspot intensity profile. In Figure 2 we have characterized the deficit at the center of the disk where there is little limb-darkening, but the derivative of the LDF increases rapidly as we approach the limb, and here the optical effects of sunspot darkening are much greater. Spatial Model

7. Sunspot Transit Figure 3 shows our model sunspot as it traverses the disk. In the last panel, the spot is just crossing the limb. Note that when the spot axis reaches the exact limb, there is confusion of brightness and position. Figure 3

8. Figure 4 shows our model of the LDF (black) along with our measured point spread function (red) and limb profile (green). The limb profile is the convolution of the LDF with the measured point spread of our system. Our measured data corresponds to this curve. This LDF, point spread and limb profile are taken as our reference curves. See Poster 022.22. Figure 4

9. The dashed black curve shows our reference LDF at the limb. The red curve shows the sunspot intensity deficit with the same LDF just as the axis of the spot crosses the limb. This corresponds to the fifth panel of the sunspot transit plot (Figure 3). Figure 5

10. The two dashed black curves are our reference LDF and limb profile. The two red curves represent a LDF with a photometric deficit resulting from a sunspot as it traverses the limb and the resulting profile calculated by convolving the dashed LDF with our measured point spread function. Note that the curve appears to have both a depression in intensity as well as a shift of the limb position. Figure 6

11. The two dashed black curves are our reference curves. The two blue curves represent a LDF with a depressed radius and the resulting profile calculated by convolving the dashed limb darkening function with our measured point spread function. To first order, the blue dashed limb profile is the same as the solid black limb profile shifted by 1 arc second. Figure 7

12. The two dashed black curves are our reference curves. The two green curves represent a LDF with a photometric deficit from a sunspot along with a depressed radius due to the Wilson depression and a resulting profile calculated by convolving the dashed LDF with our measured point spread function. Figure 8

13. Figure 6 and 7 illustrate the individual contributions of the photometric deficit and spatial position (Wilson depression) of the sunspot respectively. Figure 8 illustrates the combination of these two effects. At the level of this model, these two effects are independent in the sense that we can decompose our observations of sunspots at the limb and estimate the photometric deficit and limb altitude separately. Figure 9 shows how this is done. Method of Decomposition

14. Figure 9

15. Figure 9 shows three profiles. The solid black is our measured limb profile,showing the four RHESSI limb pixels, and the dashed blue is our measured limb profile from a LDF with depressed radius. The dashed green curve is the measured limb profile with a LDF that has a photometric deficit and a depressed radius. The orthogonal arrows show how, to first order, the two effects that sunspots have on our measured data are decomposable. By properly parametrizing our sunspot model we can account for the photometric deficit. Based on this, we should be able to fit for the magnitude of the Wilson depression, which in our data will appear as a shift characterized by the horizontal displacement of the dashed green profile relative to our reference profile.

16. Conclusions The RHESSI limb observations clearly show sunspot effects a the limb With a simple model we can decompose the observations and assess the Wilson depression directly We hope to compare our observations with direct Hinode imaging