Stellar Variability

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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