Performance Criteria
System Characterization
Data Processing and Analysis

To meet the goals and objectives of the Kepler Mission the spacecraft/photometer design criteria for an acceptable one-sigma total noise level has been established as 2x10^-5 or 0.25 Earth-area for an m_v=12 star in five hours of observing. (For the proposal the same noise and stellar flux is taken to be for 6.5 hours.) These parameters are listed at the bottom of Table 1. The total noise includes: photon-shot noise (1.4x10^-5), instrument noise (1x10^-5) and stellar variability (1x10^-5). The signal (light collected for a given star) is determined by the photometer collecting area and system efficiency. In line 1 of Table 1 the signal for various stellar magnitudes is scaled from a 12th magnitude star. The shot noise (Table 1, line 2) is the square root of the measured signal (line 1). The fractional instrument noise (all noise sources except photon-shot noise and stellar variability) is converted to an absolute noise and given in line 3. The shot and instrument noise are combined as a root-sum-square (RSS) and given in line 4. The fractional noise in line 5 is the ratio of line 4 to line 1. (The fractional noise is also calculated for 1-hour integrations, line 5a, which is what was measured in the 2-day tests.) Stellar variability (line 8) is a given for any particular star. For the Sun and presumably other solar-like stars it is less than 1x10^-5 at the relevant temporal frequencies and spectral bandpass. The total fractional noise (line 10) is obtained by an RSS combination of lines 5 and 8. The total noise relative to an Earth-size transit is line 9 divided by 8x10^-5. For a 12th magnitude star this is 0.25. The reciprocal of the total noise gives the number of sigma for a one-Earth transit (line 11).

The performance reported below is to be compared to the values given in bold in lines 5 or 5a. The better performance shown for 9th, 10th, and 11th magnitude stars is not required, since the objective of the Kepler Mission is to detect Earth-size planets. The required performance at these brightnesses is taken to be the same as for 12th magnitude stars, even though the better performance is realized in the tests.

Prior to establishing a Baseline performance without inclusion of all the confounding factors, a number of tests were performed to characterize the system.

Operating temperature
The CCD was operated at temperatures of -40°, -50° and -60°C and the dark current and its effects on the noise measured. Going from -40°C to -50°C reduced both the dark current and the system noise. Below -50°C there was no improvement in the system noise. At -50°C the dark current no longer dominates other noise sources. A temperature of -50°C was used for all of the test results reported here, unless otherwise indicated.

Focus and point spread function
The star images were intentionally not made into sharp unresolved points of light on the CCD. The point spread function was intentionally extended over many pixels for two reasons: 1) to reduce the sensitivity of the photometric precision to subpixel variations in quantum efficiency when there is motion and 2) to provide an adequately large integrated well depth to prevent saturation. An optimum focus was found wherein a 5x5 pixel aperture contained 80% of the energy and about a 3-pixel diameter contained 40% of the energy. This focus was used throughout the remainder of the tests reported here.

Thermal effects and uncorrected signal variation with motion
A limitation to the photometric precision of aperture photometry comes from the spatial stability of the Point Spread Function (PSF) relative to the aperture. Two effects take place as the PSF moves within the aperture: 1) the gains and losses of the PSF at the edges of the aperture are never equal at some level of precision and 2) the detector may also have significant spatial variations at the pixel or subpixel scale in quantum efficiency. There are two types of motion to consider: random motion that has a mean of zero over the long term and drifts that cause a change in the mean position over the long term. Long term is meant to refer to time scales on the order of expected transits. For transit times of two hours to 16 hours, this implies the mean motion needs to be stable on time scales of 40 minutes to two days. What is stable? If the total for instrument noise is to be less than 10^-5, then changes caused by either: a variation in the mean position due to random motion not averaging out or a drift of the mean position after any corrections, need to be less than or on the order of 0.5x10^-5.

Small long-term changes in the temperatures of the CCD and its internal mount and the wall and base of the CCD dewar were found to significantly affect the spatial stability of the images. In addition to an actively controlled thermal enclosure for the entire Testbed, proportional temperature control was instituted for these components and for the CCD electronics box. This permitted identification and mitigation of thermal effects of these components.

Tests were conducted to measure the effect of motion at the millipixel level on the amplitude of the signal. The relative amplitudes were found to be highly correlated with an amplitude variation of up to 10^-4 per one millipixel of motion, similar to what has been reported previously (Robinson, et al. 1995; Jenkins, et al. 1997). For motion when the PSF is nearly centered or moving tangent to the center there is very little variation. When the center of the PSF moves radially the amplitude increases while moving towards the center and decreases for motion away from the center. The amplitude of the effect increases with increasing distance from the center. The response in all cases can be explained as a result of the wings of the PSF moving in and out of the set of pixels chosen as the photometric aperture for a given star.

Calculations were performed to quantify the signal amplitude changes for motions at the one-millipixel level. The results reproduce the behavior in the lab and give comparable amplitude variations with motion measured in the lab, thereby confirming the interpretation of the effects. The distribution of energy near the peak of the PSF (uniform, spiky, etc.) contributes a constant amount and therefore is not relevant. The only variance is due to the wings of the PSF moving relative to the edges of the aperture. The integrations were performed numerically.

Thus long-term drifts in the mean stellar positions at the millipixel level are significant and undesirable. However, these variations being highly correlated, are correctable and the corrections have been incorporated in the data analysis program used for decorrelating normalized fluxes that is described below.

Aperture photometry
Aperture photometry is one choice for measuring the amplitude of the flux versus time. Point Spread Function (PSF) fitting would be another method, but is much more computationally intensive and requires a very good knowledge of the PSF. The size of the aperture chosen relative to the PSF along with the amplitude of the shot-noise relative to other noise sources affects the measured noise. Using the optimum PSF, the noise was measured as a function of the aperture size in pixels for each stellar magnitude and is given in Table 2. The one-sigma noise is given in units of an Earth-area transit. The maximum allowable noise is from Table 1 line 6.

It is evident that for brighter stars (m_v=9) the noise is minimized with the use of a larger aperture where the instrument noise is small relative to the shot-noise (compare lines 2 and 3 in Table 1). For the fainter stars, the outer pixels of a large aperture become dominated by instrument noise and contribute insignificantly to the measured value. The improvement realized by varying the aperture are, to first order, achieved in optimum pixel weighting, which is part of the analysis software (Jenkins et al., 2000) that reduces the photometric errors produced by small amplitude motions.

Table 2 Measured Noise (Earth-area)
The minimum noise is indicated in bold for the optimum aperture size in pixels versus stellar magnitude.

mv	Pix=5	=7	=9	=11	Max Allowed
9	0.19	0.11	0.09	0.08	0.22
11	0.15	0.11	0.10	0.11	0.22
12	0.22	0.16	0.18	0.18	0.22
13	0.36	0.33	0.33	0.34	0.42
14	0.57	0.52	0.66	0.78	0.87

The flow of the data processing and analysis is illustrated in Figure 1. The test conditions are defined external to the data system. The operation of the CCD and acquisition of the data are performed with Lowell Observatory Instrument System (LOIS) software. LOIS is an interface to the CCD controller (Taylor, 2000). To prevent saturation the CCD is readout every three seconds. During readout the pixels are binned 2x2 on the chip to simulate use of a CCD with larger pixels. There is no shutter in the system. Each image is co-added for either 3 or 15 minutes and then written to disk as a FITS file. The effect of cosmic rays can be simulated by using software to inject them into the individual three-second readouts. They can then either be left in the data or removed by a separate software module. At the end of the run the data are archived onto DLT tape.

Figure 1 Data Processing Flow Diagram

During the Kepler Mission the full CCD images are not telemetered to the ground. Rather the individual pixel data for each star being monitored are extracted on-board after each 15-min. co-add. The software module Space3 performs this function. It produces several files in the Testbed allowing for a choice of further processing to be performed. Simulated stellar variability can be added to any of these files as desired. Raw pixel data are used by the program optflux, which uses optimal pixel weighting, rather than equal weights to reduce the sensitivity to motion. The best results are obtained by pre-screening the pixels and including only those pixels that contribute significant statistical information to the flux calculation. This is a way to choose the appropriate photometric aperture based on the stellar brightness, the location of the PSF relative to the fixed pixel grid and account for noise and structure in the background (Jenkins, et al. 2000). The program optflux produces a flux file identical in format to the other flux files produced by Space3. A single flux file constitutes what is telemetered to the ground during the mission.

The program decorr removes any time-varying component in the relative fluxes that is highly correlated, thereby mitigating the effects of long-term drifts in the centroids. During the mission this process is performed on the ground. The result is light curves for each star for each point in time. The results of the analysis are summarized by either the program polynorm, which provides a tabular summary of the test that can be compared to the values in Table 1 or plotnoise, which provides a graphical summary (Figure 2, on the next page).