Spacebased Photometric Performance
Proper Photometer Design
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.
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
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.
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