Pixel Response Function 36

The system Pixel Response Function (PRF) is the image property of most interest to the Kepler data analysis, as it describes the observed appearance of point sources. The PRF is a combination of the optical PSF, the jitter PSD, the module defocus, the CCD response function, and the electronic impulse response. It is a function of the source MORC location (§4.3), and the intrapixel location of the source centroid. In principle, the PRF will also be a function of stellar spectra type, since there is chromatic aberration in the Schmidt and wavelength dependence in the system optical throughput and CCD properties. However, the broadband emission from stars of any color washes out most of the fine structure calculated from the Code V model of the Kepler optics using monochromatic light at different wavelengths (Figure 14). The PRFs discussed in this Section are the measured PRFs, which means that each is the average of several stars in the Kepler bandpass. While the Code V model shows that there are real differences between the different stellar colors in detail, these differences are not enough to threaten target management, and they were not part of observation planning or data analysis. Since Kepler is a photometer, not an imager, the most compact PSF is not necessarily the best. The Kepler PSF shows significant fine structure, and concentrating too much light in the brightest pixel can actually degrade the photometry by increasing sensitivity to image motion, and degrade DIA (Differential Image Analysis) by degrading centroid accuracy. On the other hand, the SNR is decreased by making the PRF too broad, since a sum over more zodiacal light, background stars (and their noise), and pixels (with their read noise) will be needed to capture the flux. This noise increase is significant since, on average, each pixel has the equivalent of one 19th magnitude star, in addition to the zodiacal light. So both brightest pixel flux fraction (BPFF) and 95% encircled energy diameter (EE95) are used to bound the acceptable PRF, which is evaluated for the spectrum of a G2V star. The BPFF should be <60%

(Gilliland, 2004), while EE95 is required to be < 7.0 pixels.

The Kepler telescope was focus adjusted during Commissioning, by 40 µm in piston and a smaller amount in tilt, to minimize the RMS offset from best focus. A set of bright, nonsaturating, relatively isolated stars were defined as Short Cadence targets, which will be referred to as the “EE Targets” since EE is the performance requirement which BATC was verifying with these observations. The resulting image set substantially exceeds the encircled energy requirement. In addition, focusing resulted in a more symmetric variation of PRF across the focal plane, which reduces discontinuities in photometric aperture properties between quarterly rolls for a given target.

Figure 15: Example square-root scaled pixel-centered Pixel Response Functions calculated by the PRF model based on Commissioning data. The color bar indicates the square-root scaled value. PRFs are calculated near the center of the channel at row 535.0, column 550.0. Unscaled images are normalized to give a flux of unity integrated over the PRF. Left: Channel 9.2, smallest EE95; Middle: Channel 13.2, median EE95; Right: Channel 10.4, largest EE95. Images are in channel coordinate system, with the + ordinate increasing rows, and the + abscissa increasing columns.

Subsequent to focusing, the PRF was measured, using a much larger set of stars, by again moving the telescope in sub-pixel intervals. The representation of the PRF derived from the full data set, and delivered to the SOC and MAST, was used to compute pixel-centered PRFs at row 535.0 and column 550.0 as shown in Figure 15. Images from the EE activity, selected to have the centroid closest to a pixel

center are shown in the Y’ Z’ FPA coordinate system in Figure 16. The EE95 derived from the delivered PRF are shown in Figure 17. These images and 95% EE diameters are meant to give the observer a qualitative idea of how the PRF varies across the focal plane; for precise target aperture definition and photometric time series analysis, users should download the PRF model from MAST (KAM §2.3.5.17).

Figure 16: Pixel-centered images across the FPA after adjusting the focus by 40 µm during

Commissioning. Each EE Target is normalized to the brightest pixel in the target aperture, rotated and translated to the common FPA co-ordinate system, and linearly scaled to the color bar to the right. Black lines on color bar represent 10% intervals. Some channels did not have targets, or the targets were deemed unsatisfactory. The EE targets are magnified by 50x compared to the spacing between them in this image. Relatively dim stars near targets and smear correction artifacts are also visible.

Figure 17: Observed 95% EE diameter after focusing the Kepler telescope (see Bryson et al. 2011 for discussion), arranged in the FPA co-ordinate system. Each box is labeled by channel, followed by the 95% EE diameter in pixels. Green indicates a 95% EE diameter well within the 7.0 pixel requirement, shading to red in those few cases where the requirement is approached or slightly exceeded. The white corners indicate FGS channels.

A preliminary analysis of the percent flux in the center pixel of the pixel-centered images shown in Figure 16, grouped by module like the EE results shown in Figure 17, gives results from 20% to 62%, with a median value of 45%. It therefore appears that the focusing activity made the EE diameters smaller, without an excessive concentration of light in the brightest pixels. Table 14 shows 95% EE and central pixel brightness results for all channels.