Identification of astrophysically interesting signals in the Kepler time
series requires an assessment of the noise characteristics of the photometer. Users
of Kepler data will need information about both the usual signal-to-noise issues, e.g.,
photon and read noise, background flux, and the temporal stability due to systematic and
episodic noise sources. Detection of weak transits against a bright stellar
background is the primary objective of the Kepler mission, therefore the photometric
stability of the instrument was of paramount importance during the design phase.
The Project is investing considerable effort to identify and account for systematic
noise sources in the Kepler time series. These efforts are
described in the Instrument Handbook, the Kepler Data Analysis Handbook (Fall 2010),
and a series of data release notes, distributed when large datasets are released to the
archive. Individual data release notes are found in the
folder at MAST. Users of Kepler data are strongly encouraged to read these detailed
Observations with the Kepler photometer are obtained in either a 1 minute ("short")
or 30-minute ("long") observation, termed a cadence. Both types of
observations may be
summed to improve the overall signal detection. A number of factors effect the
signal-to-noise ratio at both the bright and faint observing limits. A factor of
2 estimate of the photometric precision expected within either a single short or long
observation for isolated (uncrowded) FGK stars can be made using the expression
presented at left.
For cooler and hotter stars, these estimates may be subject to larger uncertanties,
reflecting the tranformation from gr magnitudes to the Kepler magnitudes.
The expected photocurrent of the source, expressed in
electrons per second, is provide by the first expression. f12
is the benchmark photoelectron current at the focal plane for a G2 V with a Kepler
magnitude (Kp) = 12.0, which is 1.74 × 105 e-/s. This value
is the best pre-launch prediction of the flux estimate, and may change as understanding
of the photometer performance increases.
An estimate of the precision is given in the
second expression, where tint is the per-frame integration time
= 6 seconds; nfr = the total number of frames in an observation;
nais the number of pixels in the photometric aperture;
Kp is the apparent magnitude in the Kepler bandpass;
NR is the read noise = 120 e−, and p
is an empirical constant, with a value between 1 and 2, designed to account for
all other noise sources. A current estimate for p is 1.2, and, as with
the benchmark photocurrent, may change with operational experience.
The number of frames in an observation must also account for the CCD readout time,
0.66 seconds. There are 270 frames per 30 minute observation (1 long cadence),
and 9 frames per 1-minute observation (1 short cadence).
The table at right provides an estimate of
this precision, P, expressed in parts per million. These values
were derived using a nominal aperture, na = 20
pixels, and integration times of 60 seconds = 1 short cadence, 1800
seconds = 1 long cadence and 5 hours (10 long cadences, 18000 seconds).
The derived values do not account for "noise" introduced by instrinsic
variability in the observed sources, which, in general, will differ for different
variability classes. Photometric precision estimates in the table should be
considered applicable to quiescent sources. While intrinsic variability
is a complication for transit detections; it is generally the property
under study by guest observes.
Note that 1 millimag ~ 1000 parts per million; 0.01 magnitudes ~
104 parts per million.
In practice, the target apertures, expressed in pixels, are adjusted for
source brightness. Apertures decline with increasing magnitude, to minimize
background and other noise sources when the expected signal will fall in a
just a few pixels. As the program matures, we will provide additional
estimates of the precision for a range of target apertures.
The expression above provides an estimate of the precision for stars fainter than
Kp ~ 12.0. For brighter sources the expression above becomes increasingly
inaccurate. Bright star photometry is dominated by systematic effects rather than
just photon and read noise. Users should utilize the table values with caution, and
consult the Instrument Handbook for further details concering the issues relevant to
bright source photometry.
P (ppm) SC
P (ppm) LC
P (ppm) 5 hr
Using flight data, we measured the observed precision
to compare to the pre-flight estimate. As noted above,
variations in the received signal are driven by photon statistics,
instrumental uncertainties and intrinsic source variations. Intrinsic
variability is present in many stars at unknown levels and will cause
estimates of the overall precision, measured in PPM, to be larger than
that ratio defined by the sum of photon statistics, instrumental and
spacecraft-induced limitations. Note that the estimate provided above
is calculated for a single observation, either a 9-frame short cadence or
a 270-frame long cadence. The onboard solid state recorder accumulates
all CCD reads in each cadence; the individual frames are not stored for
later downlink and analysis. Therefore we cannot directly measure
the precision within a single cadence observation by averaging the frames
within that cadence. Measurements
of the signal variations using a variety of timescales provide a
sense of the overall precision of Kepler light curve.
For this measure of the precision, we selected 23 cool dwarfs, all but the
faintest three within a temperature range 5700 < Teff < 6000, and
log g > 4.0, as listed by the Kepler Input Catalog. The choice of parameter
range was driven by the desire to select stars with minimal intrinsic
variations (ie., no giants), and to compare stars of similar physical
properties. These stars are nominally early-G dwarfs.
The table at right provides a measure of the observed precision, P,
expressed in parts per million for long cadence data. These values were
derived from the Q2 light curves after processing through the pixel
calibration (CAL), photometric analysis (PA), and pre-search data conditioning
(PDC) modules within the Kepler pipeline. The Kepler magnitude of each source is
tabulated, along with the observed count rate, and precision. Counts are the median
value measured for one long cadence as output by PDC, averaged over an
entire quarter, presented in electrons/cadence. The precision, expressed in
parts-per-million, is the variance of the counts, again averaged over an entire
quarter. These values are intended to provide a straighforward meaasure of the
noise in the long cadence light curves; in the future we will provide estimates
over a broader range of timescales.
The observed count rate per second, interpolated from the pipeline output
through the PDC step, at a Kepler magnitude = 12.0, = 1.72 × 105 e-/s.
This value agrees to within
1% with the value estimated from the pre-flight instrument characterization,
described above. Note that output from the pipeline Photometric Analysis (PA) module
will generally display a larger count rate. The PDC module corrects for light in the
source aperture arising from surrounding stars,based on a derived source-specific crowding
metric. Isolated sources will show the same count level, while crowded stars may have
a significant reduction in the observed source flux. PA produces a sum of counts within
the optimal aperture, with no correction for nearby sources.
The measured precision tracks the pre-flight estimates, with signifcant deviations at the
bright end. As noted above, the expression used for deriving the expected precision
break down at bright magnitudes, largely due to the larger apertures used for bright
sources (not reflected in the calculation above), pixel saturation and other systematic
effects. The values to the right were derived using data for most of a quarter, excluding
data gaps due to safe modes, loss-of-pointing, and the monthly downloads.
A few of the selected stars showed significant intrinsic variations during Q2. We
retained those stars in order to demonstrate the effect of variability on simple
measures of the precision. The source at Kp = 14.35 turned out to be an
eclipsing binary; the source at Kp = 15.69 showed a long-period change in the observed
count rate. The values displayed in the plot for these two source reflect a measure of
the precision excluding most of the observed large amplitude variability (the precision
for these two objects was estimated by dividing the light curve into ~25 3-day
segments, and deriving the median value of the measured variance).
Observed count rate, expressed as log photoelectrons electrons per
cadence (one 30-minute observation), as a function of Kepler magnitude
for the full brightness range of sources observed by Kepler during
Quarter 2. This dataset consisted of 23 solar analogues with 5700
< Teff < 6000, and log g > 4.0, as listed in the Kepler
The photometric precision, expressed in log parts per million,
(Log PPM) for the sources displayed to the left. Values are given in the
tables listed above. The points indicate an empircal measurement
over an entire quarter (excluding data gaps). The line indicates the precision
calculated from the expression given above, for a single 30 minute
observation, 1 long cadence. Note that the calculation used a single, 20-pixel
aperture for each integration, a choice generally applicable to stars in the
range 11 < Kp < 14. The divergence of the calculated value from empirical measures
at the bright end is due to the larger apertures used for brighter sources and the
onset of saturation in the center pixels for these sources.
The limiting factor for observations of
faint sources is set by source confusion, rather than the
photometric accuracy computed for isolated sources. Users wanting
to observe objects with Kepler magnitudes less than 17.0, should carefully
examine the field around their source, and estimate the
contamination from the PSFs of surrounding objects. Note that
an estimate of the crowding metric is provided for most sources
brighter than Kp = 17.0 in the Kepler Target Catalog, look under the
field labeled "Contamination" in the output of a target search. Contamination
is defined as (1 - crowding_value), where the crowding value was
derived when the Kepler Target Catalog was created. A value of 0
implies no contamination, 1 implies essentially all background,
i.e., complete contamination of the source by surrounding
objects. For fainter sources, observers can estimate the
contamination using imagery from the Digital Sky Survey,
supplemented with the Kepler Full Frame Images (FFIs) as the
latter become available.
Three views of a portion
of a Kepler CCD, scaled to illustrate source crowding at faint
magnitudes. White on the grey scale indicates that the source
flux exceed the maximum value shown.
For bright sources the observed signal in the central pixel(s) will
increase linearly until the well depth is reached. Beyond that level,
charge will bleed into adjacent pixels in the column containing that source.
However, even when the central pixel is saturated, the target aperture can extend
along the bleed column, preserving most or all of the signal from the source.
Guest observers may propose for bright stars, with the caveats that these
sources are expensive in pixel terms. A custom aperture may be required,
and the project cannot guarentee that a custom aperture can be created for the
An example of charge bleed in a Kepler image.
A single output channel is displayed after 270, 6-second integrations.
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