Detecting Transiting Planets
Simulated Transit Events
Detecting transit and reflected light signals
are classical signal-detection problems for deterministic signals
in colored noise (Van Trees 1968). Essentially, the optimal detector
whitens the observations and then correlates the whitened data
with the signal resulting from passing a transit through the
same whitening filter. An essential component of signal detection
is the characterization of the observational noise, including
stellar variability. The result is a time series of test statistics
representing the likelihood that a transit was occurring at each
point in time. Folding this time series at the orbital period
of a planet coherently adds the test statistics for the bin containing
the transit. The test statistics add incoherently if the wrong
period or bin is chosen. The detection threshold is set so that
no more than one false alarm occurs for the entire mission. Thus,
the result is still meaningful even if terrestrial planets are
infrequent or the mission had to be de-scoped. The detection
rate depends on the SNR of a single event, the number of observed
events and the detection threshold. The adaptive, nonparametric
matched filter algorithm suggested by Kay (1999) is well suited
to Kepler's needs, particularly when modified as per Jenkins
The number of independent tests depends on
the signal to be detected and the range of values each signal
parameter can have. In the case of transits, there are two basic
parameters: period and phase (time to first transit). For orbital
periods of less than 2 years, there are about 5x106 tests per star required, for a total of 5x1011 tests. In order to have no more than one false alarm
for the entire experiment, a detection threshold of 7 sigma is
required, assuming Gaussian statistics, which is important for
being able to address a null result. We note that the measurement
noise in the Tech Demo was well characterized as being Gaussian.
For the point design of a 4 sigma single event SNR the combined
SNR of a set of four transits is 8 sigma, yielding a detection
efficiency of 84%.
The figure below shows SNRs for four
transits of an Earth-size planet about an mv=12 G2 dwarf star as a function of transit duration
during periods of low, high and mean stellar variability. Here,
low, high and mean stellar variability correspond to data collected
by ACRIM 1 in 1985 and 1986, in the last half of 1988 and in
1989 and over the entire data set, respectively. For transits
between 3 and 5 hours, 1 Re
planets are detected at least 84% of the time on average. If
the transits are longer than 5 hours, 97% are detected on average,
and at least 84% are detected even if all transits occur during
high stellar activity. Planets of 1.3 Re are detected virtually always out to mv=12, and about 88% of the time for those orbiting
mv=14 G-dwarf stars. Planets
yielding average SNRs below the threshold are also detected,
but at a lower rate. For example, planets with SNRs of 7 sigma
and 6.5 sigma are detected 50% and 30% of the time, respectively.
Note that a central Earth-analog passage lasts
13 hours. Since the chord length of a circle falls as the cosine
with distance from the center >50% of transiting Sun-Earth-analogs
have durations >11.3 hours, >70% are >9.2 hours. Quoting
results for 6.5-hour periods represents a conservative estimate
of the single event SNR.
Signal to Noise Ratio as a Function of Transit Duration for Various Stellar Variabilities
The three curves give the SNR for 4 combined transits about an mv=12
solar-like star at times of low, medium and high stellar variability.
Detecting Giant-Inner Planets
A simulation was performed to demonstrate
the detectability of the transit signal for an Earth-size planet
in the presence of all the noise that is expected to be in the
stellar noise was generated to match the solar noise spectrum obtained from
the ACRIM 1 instrument aboard the SMM spacecraft to form a 4 year data sequence.
- Shot noise and pointing noise were added to the synthetic stellar noise to model the case for an mv=12 G2V (solar-like) star.
- A set of four 12 hour Earth-size transits (R=1.0 Re) was added to the noise sequence.
A matched filter algorithm was applied to
the data set to search for planets with periods between 100 and
400 days. The peak event of 8.5 sigma indicated a planet with
the correct period of 365 days, as shown in the figure below.
The probability of this event occurring by chance is less than
1x10-17 for Gaussian noise, demonstrating the significance
of the detection. The additional blue spikes result from folding
signals composed of multiple pulses. The red points are the result
for the same data set without the transits incorporated.
Probability of Occurrence by Chance for Simulated Data.
The strong minimum at 365 days in blue indicates the presence
of a planet with a high confidence level. The red points are
the result for the same data set without transits, with all events
attributable to noise.
The figure below shows the sections of the
data used in the above simulation. The individual transits are
shown (left-hand scale), along with the result of folding and
adding the data at the correct period derived with the detection
algorithm (right-hand scale).
Transit Detection Simulations
The four sections of a light curve containing the transits of an Earth-size planet
(1.0 Re) are folded at the correct period, with the sum shown
in red. The presence of the transit is unmistakable.
For the case of detecting giant-inner planets
by phase modulation of reflected light, the number of tests depends
only on the length of observation and on the largest planetary
period to be considered. Stellar variability limits this search
space to periods less than about 7 days as shown in the figure
below. Thus, there are about 1,278 tests per star, for a total
of 1.3x108 tests (assuming
the solar rotation period is typical of all stars in the sample).
Jovian-size planets are detectable with periods less than 6 days
and Uranus-size planets with periods less than 2.5 days. Because
the signal amplitude depends on the planetary albedo, planets
with low albedos are more difficult to detect than that shown
Detectability of Reflected Light from Short-Period Giant Planets
The blue, red and green curves represent the total noise
and include expected stellar variability, shot noise, CCD noise
and pointing noise appropriate for mv=10, 12, and 14 stars, respectively.
The blue spike at 4.2 days is the reflected light signature of
a 51 Peg-type planet with an albedo of 0.5 (assumed to match
Jupiter) in orbit about a mv=10
star. At other periods, the strength of this spike would vary,
as given by the black dotted reflected light envelope. Since
this envelope exceeds the measurement noise curves for periods
less than about 7 days, giant planets with periods up to 7 days