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Astron. Astrophys. 337, 945-954 (1998)

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3. The coma model

Given the shortcomings described above, we developed an improved model, which we now describe in detail. The first step is to construct an anisotropic model of the coma using the observed spatial brightness profiles. The noise in the outer part of the coma, where the observed signal became weaker, was reduced by applying an inverse gradient filter, which preserves the slope of the brightness distribution. We then performed a polar transform of the images centered on the nucleus (the pixel having the largest signal) with an angular resolution of [FORMULA] (i.e., 360 profiles were created). Each of the 360 lines in the polar image may be represented by a power law [FORMULA], [FORMULA]. The [FORMULA] parameters, [FORMULA] and [FORMULA], were determined by least-squares fits to the observed profiles in a restricted interval that contained "uncontaminated" coma. Specifically, we used the region defined by [FORMULA] km to fit the coma parameters, as the contribution to the signal by the nucleus was negligible there and the gradient was essentially unaffected by convolution with the PSF. Fig. 2 shows an example of the angular variations of [FORMULA] and [FORMULA] and illustrates both the strong anisotropy of the coma and the clear deviation from the canonical law [FORMULA]. We note that the fit to the coma is excellent, as indicated by the small residuals displayed in the figure.

[FIGURE] Fig. 2. The angular variations of the parameters [FORMULA] (top) and [FORMULA] (bottom) for the third observation are plotted. The residuals from the fit (stick plots) and the adopted smoothed values for the parameters are also displayed (dashed-line).

The full coma model is now constructed assuming that the above parametric representation holds also for [FORMULA] km. Small-scale variations in the fitted parameters were filtered out (see Fig. 2), as they may be artifacts of our coordinate transformation and, in any case, are irrelevant for determining the size of the nucleus. The model images of the coma were generated on an 8 times finer grid than the original WFPC2 pixel. At this new scale, in which the sub-pixel projects to a linear distance at the comet of [FORMULA] km, the effect of the finite size of the nucleus can be introduced. Our preliminary analysis of Sect. 2 indicated that the nucleus had an effective radius of 2.5 km, so we assumed that a [FORMULA] sub-pixel region defined the location of the nucleus. For all pixels other than the ones containing the nucleus, polar coordinates were computed from the cartesian coordinates and the appropriate parametric laws were used to calculate the coma signal in each sub-pixel. The four sub-pixels containing the nucleus were not set to zero but to one-half the average value of the surrounding pixels, since half the line-of-sight is blocked by the nucleus. While the actual coma contribution to the sub-pixels containing the nucleus may differ somewhat from our prescription, the derived nuclear magnitude is only weakly dependent on the exact choice because the sub-pixels are so much smaller than the actual WFPC2 pixel (i.e., there is a significant dilution effect). Furthermore, convolution with the PSF washes out fine details, as illustrated in Fig. 3. We note that the center of the nucleus does not necessarily coincide with the opto-center of the coma, as expected for an anisotropic coma. This effect was dramatically illustrated in the high resolution images of comet 1P/Halley obtained by the GIOTTO/HMC camera (Keller et al. 1994); in both cases (1P/Halley and 19P/Borrelly) the brightness distribution is skewed in the solar direction due to the enhanced dust emission from the sub-solar region.

[FIGURE] Fig. 3. Profiles of the coma model generated on the resampled grid before (solid line) and after (dashed-line) convolution with the PSF. This case corresponds to the third observation.

Finally, each coma model was convolved with a PSF that was also sampled on the finer grid, using the TinyTIM software discussed in Sect. 2.

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998