Magnetic processes in late-type stars produce
brightness variations that dominate the power spectrum at frequencies
corresponding to the stellar rotational period. Even for the
Sun - a star of low rotation rate and relatively evenly distributed
active regions (in longitude) - variability is concentrated at
time scales comparable to the rotational period. Fortunately,
the time scales of interest to planet detection are considerably
shorter.
Solar Variability
We have quantified solar variability at the
requisite time scales using observations from the Active Cavity
Radiometer for Irradiance Monitoring (ACRIM 1) onboard the Solar
Maximum Mission (SMM) satellite. This instrument measured the
total solar flux over a 4.5-year period (Willson et al. 1981)
from 1985 (near solar minimum) into 1989 (near solar maximum).
The figure below illustrates the power spectrum
of the Sun from the SMM data taken between 1985 and 1989, roughly
during solar maximum. The power in the data on time scales similar
to that for planetary transits is 10,000 times less than that
for the rotation period of the Sun. The variability depends on
the square root of the power, so it is 100 times less than the
noise level associated with the rotation period. The noise level
at the rotation period is about 1 x 10-3, so the noise on the time scale of a planetary transit
is about 1 x 10-5. Brightness
variations with durations greater than 16 hours have little affect
on the detectability of planetary transits. These data are believed
to show 30% more variability than what would be expected had
the UV been excluded as it is for the Kepler Mission.
Power Spectrum of the Sun Near Solar Maximum
These results are validated by comparing them
with measurements made in 1996 by VIRGO aboard the Solar and
Heliospheric Observatory (SOHO) at the same solar cycle phase
(Froehlich et al. 1997).
The figure below illustrates the noise environment
for detecting transits and reflected light signatures based on
these data. There is little power due to solar variability at
time scales comparable to transits. Most power in the measurement
noise occurs at frequencies less than 1 mHz (10 days), corresponding
to the rotation of sunspot groups and solar-cycle scale variations
(Froehlich 1987). However, at frequencies of 10 to 100 mHz (3
to 30 hours), the power spectrum is dominated by convection-induced
processes such as granulation and super granulation (Rabello
Soares et al. 1997, Andersen et al. 2000). Recent studies by
R. Radick and T. Brown (2000) suggest this noise is caused by
plage structures being carried by rotation across the limb of
the star. The shallow slope beyond this region is attributed
to the combined effects of gravity oscillations and the non-solar
noise injected into the data set. The differences in the high
frequency levels of the measurement noise are due to the varying
amounts of shot noise at the different stellar magnitudes represented.
We conclude that the detection of terrestrial planets around
a solar-like star is feasible with the Kepler photometric
system.
Estimated Power Spectral Density
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 are detectable.
Expected Stellar Variability
Not all stars behave like the Sun. Ground-based
photometric surveys of solar-like stars find that the solar irradiance
is a factor of 2 to 3 times more stable than the sample stars
of similar spectral type and activity index (Radick et al. 1998).
Hence, convection depends only on stellar mass while magnetic
processes are highly sensitive to a star's rotation rate. Convective
variability (which dominates on the time scales of interest)
should be more similar among dwarf stars of a given spectral
class than magnetic variability. Assuming that convection induced
variations are similar for all late-type stars, then detection
of Earth-size transits is feasible as long as the rotational
periods are sufficiently long.
We can estimate what percentage of solar-type
stars in a magnitude limited survey in the galactic plane are
slow rotators by making use of the rotation-activity and activity-age
relations. Studies of open cluster stars show that the large
spread in rotational velocities found among young, late-type
stars largely disappears by 700 Myrs (the age of the Hyades)
(Radick et al. 1987). The distribution of rotational periods
among the fastest Hyades stars is 1 to 2 weeks and slows down
as they age. By 1 to 2 Gyrs, rotation is sufficiently small to
render magnetic-induced variability innocuous to terrestrial
planet detection in all stars. Measurements of the star formation
history in the galactic plane based on CaII H&K indices of
a large sample of solar-type stars suggest that 75% of F, G,
and K dwarfs are older than 1 to 2 G yrs (Rocha-Pinto and Maciel
1998, Rocha-Pinto et al 2000). For the younger, more active stars,
we obtain an excellent history of their behavior from our data
that can be used to filter out magnetic variations.
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