Three parameters describe the characteristics of a planetary transit:
- the period of recurrence of the transit;
- the duration of the transit.and
- the fractional change in brightness of the star.
The Planet's Orbital Period
The period, P, is used to determine the semi-major axis, a, given that the stellar mass, M*, is known from the spectral type of the star. Kepler's Third Law is used to calculate the semi-major axis:
The Transit Duration
For transits across the center of the star the transit duration is given by:
where d* is the stellar diameter in solar diameters,
M* is the stellar mass in solar masses and
a is the orbital semi-major axis in AU.
The duration does not tell us anything physically about the planet. However, like the transit depth, the duration must by consistent for all the transits of a given planet-star combination. If not, then one suspects that there are multiple planets in the system that are being detected or some other non-transit phenomena are taking place.
The Transit Depth, i.e., the Planet's Size
The fractional change in brightness or transit depth is equal to the ratio of the area of the planet to the area of the star. This measurement is used to calculate the size of the planet given that the size of the star is known from its spectral type. The spectral type can be determined from ground-based observations of the star. A more precise determination of stellar mass can be made using asteroseismology, the study of star oscillations.
The table below lists the duration and transit depth (equal to the ratio of the planet area to the area of the Sun) for several cases. Single transits of giant planets have depths on the order of 1%. These transits are so large that follow-up verification can be done from the ground.
Transit Properties of Solar System Objects
M* = star mass (Sun = 1)
d* = dia. of star
Our instrument is designed to produce an SNR=8 sigma for an Earth-size planet orbiting an mv=12 solar-like star with 4 near-grazing transits having a duration of 6.5 hrs. The signal-to-noise ratio (SNR) varies as (nt)1/2, where t is the transit duration and n is the number of transits which equals the mission life divided by the orbital period. Thus, for a given star and using Kepler's third law, the SNR relative to that at 1 AU, SNR1AU, increases with decreasing a as a-1/2:
Transits can only be detected if the planetary orbit is near the line-of-sight (LOS) between the observer and the star. This requires that the planet's orbital pole be within an angle of d*/a (part 1 of the figure below) measured from the center of the star and perpendicular to the LOS, where d* is the stellar diameter (=0.0093 AU for the Sun) and a is the planet's orbital radius.
This is possible for all 2π angles about the LOS, i.e., for a total of 4π d*/2a steradians of pole positions on the celestial sphere (part 2 of figure).
Thus the geometric probability for seeing a transit for any random planetary orbit is simply d*/2a (part 3 of figure) (Borucki and Summers, 1984, Koch and Borucki, 1996).
For the Earth and Venus this is 0.47% and 0.65% respectively (see above Table). Because grazing transits are not easily detected, those with a duration less than half of a central transit are ignored. Since a chord equal to half the diameter is at a distance of 0.866 of the radius from the center of a circle, the usable transits account for 86.6% of the total. If other planetary systems are similar to our solar system in that they also contain two Earth-size planets in inner orbits, and since the orbits are not co-planar to within 2d*/D, the probabilities can be added. Thus, approximately 0.011 x 0.866 = 1% of the solar-like stars with planets should show Earth-size transits.
Probability for Detection of Multiple Planets Per System
Current models for planetary system formation assume that the planets are formed out of a common nebula with the star and that the orbital planes should have small relative inclinations. For the solar system, these inclinations are on the order of a few degrees (see table above). Similarly, the inclinations are also small for the inner moons of Jupiter, Saturn and Neptune. If one were to view the system near either node of the intersection of the orbital planes of two planets, then both planets would be observed. For small relative inclinations of the planes, (φ1 + φ2) < d*/a, both planets would always be observed, and for (φ1 + φ2)/ ≥ d*/a the probability for seeing a second planet in the system is given by (Koch and Borucki, 1996):
For the Venus-Earth combination, there is a 12% chance of seeing both planets. Thus, there appears to be a significant chance that multiple-planetary systems can be seen. This result should lead to a further refinement of the models that describe both the frequency of planet formation as well as the co-planarity of their orbits.