Differential Photometry
Spacebased Photometric Performance

Reliable detection with a >8 sigma statistical significance with >3 transits (four sigma, single transit) of terrestrial planets requires relative photometry of 2x10-5 on time scales of 2 to 16 hours. The brightness change due to a transit is proportional to the the ratio of the planet's area to that of the star. When observing stars the size of the Sun, the decrease in brightness for giant planets, such as Jupiter, Saturn or the planet orbiting HD209458, etc., is approximately 1%. For planets like Uranus and Neptune it is about 0.1%. For Earth-size planets it is about 0.01%. Obstacles to achieving the necessary precision are:

  • Variations in seeing and extinction (for ground based observing)
  • Photon shot noise
  • Intrinsic variability of the target stars and
  • Instrument instabilities.

Variations in seeing and extinction are eliminated by using a spaceborne photometer. Shot noise is reduced to an acceptable level using Kepler's moderate-aperture photometer. The Sun's brightness variations on time scales of a few hours are small compared to the necessary precision and, by extension, so are those of most stars of similar spectral type and age. The principal challenge in attaining adequate photometric precision therefore lies in using a stable method of photometry. This is done using the techniques of differential ensemble photometry.

The key factors used to achieve the required differential performance for the Kepler Mission are:

  • Differential spatial photometry: The brightness of each target star is normalized to the average of all nearby stars, providing common-mode rejection in the measuring system.
  • Differential temporal photometry: Transit durations are a few hours to less than a day. Brightnesses are compared to data just shortly before and after the test interval, so there is no need for long term stability.
  • Decorrelation of image motion: Motion due to the image drifting over time scales long compared to a transit produce highly correlated amplitude variations that can be removed.
  • Optimal weighting of pixels: The individual pixels that comprise each star image can be weighted to maximize the SNR.
  • Keeping each star image on the same pixels for three months: Eliminates effects of inter- and intra-pixel quantum-efficiency variations.
  • Operating the CCDs near full well capacity: Read noise and dark current are negligible.
  • Selection of an Earth-trailing heliocentric orbit: Stable thermal environment and negligible scattered light background.
Proper Photometer Design

In a well-designed experiment, the total of all controllable noise sources should be similar to that of the uncontrollable noise. For Kepler, stellar variability is the limiting uncontrollable noise source. By design the shot and instrument noises are of the same magnitude. Given Kepler's aperture, bandpass and optical efficiency, the number of detected photoelectrons, Ne, from a G2 dwarf star of magnitude mv is given by:

Ne = 7.8x108x 10-0.4 (mv-12) e-/hr

For a typical target star with mv=12 and an integration time of 6.5 hours, Ne=5x109, giving a relative Poisson noise of 1.4x10-5. Instrumental noise is suppressed because the photometric measurements are differential with respect to nearby stars and differential with respect to time. By measuring the ratio of the brightness of each star to the average of its neighbors on the same CCD detector, the Kepler photometry is largely immune to temperature variations, drifting amplifier gains and zero-point offsets, as well as changes in the focus, alignment and transmission of the optical system. This technique of "ensemble normalization" or "common-mode rejection" was applied to ground-based photometry of the cluster M67 by Gilliland et al. 1993. They found that a precision of 8x10-4 could be attained at transit time scales, limited by the (often 50%) variations in seeing and transmission imposed by the Earth's atmosphere. Absolute, not differential, HST photometry of the transiting planet HD209458b has also achieved near-Poisson-limited precision (6x10-5 in ten min) with the largest non-random errors resulting from rapidly changing environmental conditions related to HST's low-Earth orbit.

Several noise sources are not addressed by differential photometry, notably noise arising from spacecraft LOS motion and from some kinds of variation of the stellar Point Spread Function (PSF). These occur because neighboring stars fall at different fractional-pixel displacements relative to the CCD's pixel grid. Also, different stars sit on differing distributions of stray light from neighboring and background stars. If the position and PSF variations are suitably small, this kind of noise can be nearly eliminated by decorrelating the time series of relative brightness against the accurately measured star positions and PSF parameters. The Kepler technology demonstration clearly shows that by applying these methods to realistic laboratory measurements, an instrument precision of <1x10-5 was maintained for weeks at a time well at the same time detecting transit signals equivalent to Earth-size planets.

In summary, the noise sources limiting Kepler's photometry are known and understood. Methods for suppressing them have been proven in circumstances ranging from laboratory demonstrations to ground-based observations at large telescopes to spaceborne photometry. Applied to intrinsically stable time series data from Kepler's benign Earth-trailing heliocentric orbit, these same methods can perform even better. No new technology is required for Kepler to produce the photometric precision necessary for this mission.

Technology Demonstration

An end-to-end laboratory simulation was conducted to demonstrate that the technology to do differential ensemble photometry at the precision required to detect Earth-size planets was ready. The simulation reproduced the important features of a spaceborne system including the on-orbit noise sources that limit system performance. When software was used that compensated for both jitter and drift motions of the images, the measured precision routinely produced the required value.

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