Interplay of noise from stellar variability Poisson variations and the instrument

As described earlier, it is important to find planets with a size sufficient to produce 8a detections with three or more transits. Equation (6.1) presents a simplified relationship of the SNR as a function of ratio of the area of the planet to area of the star (Ap/A*), number of transits (Ntran), and the noise due to stellar variability (v), Poisson noise in the stellar flux (F), and instrument noise (i):

where i, v, and F are in electrons counted over a sample period equal to the transit duration. It should be noted that i represents many separate noise sources (Koch 2002), and that noise variance due to stellar variability is a function of the transit duration (see Figure 3). An examination of Equation (6.1) shows that for very bright stars where the shot noise and instrument noise are small, the SNR is dominated by the stellar variability. The opposite is true for dim stars where 1/F is large and the SNR is dominated by the shot noise in the stellar flux. Table 1 presents calculated noise values for Kepler. For all entries in this table, a stellar variability of 10 ppm is assumed. Note that at 14th magnitude, the effect of stellar variability and instrument noise are negligible, and the photometric precision is dominated by the shot noise. It is also clear that for stars 14th magnitude and dimmer, detectable planets must be somewhat larger than the diameter of the Earth if they are to be reliably detected orbiting G2 dwarfs. Planets much smaller than the Earth are readily found if they orbit smaller stars, are in short-period orbits, or orbit the 70% of stars that are expected to have lower variability than the Sun.

Visual magnitude Stellar signal (photo electrons) Stellar shot noise (ppm) Instrument noise (ppm) Solar variability (ppm)

Combined differential photometric precision (ppm) Relative signal for Earth transit across the Sun (ppm) SNR for four transits

Minimum detectable planet radius (Earth = 1) at 8a

0.75

0.98

Table 1. Comparison of SNR and minimum size planet that produces an 8a detection for a planet transiting a G2 dwarf

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