For transiting planets, derive the relations between their observed properties and their physical parameters, and then apply them, as follows. a. show that the impact parameter, b, of a planetary transit (the mid-transit angle on the sky between the centers of the star and the planet, in units of the star's angular radius), is related to the orbital inclination angle i, the stellar radius r., and the separation a, as cosi = br./a. further, show that the duration fdur of the transit (defined here as the interval between the times when the center ofthe planet crosses the stellar limb) is (eq. 6.7) /1- b tdur a where r is the orbital period. b. show that, when observing a population of stars of radius r. hosting planets of radius r, in circular orbits of radius a, with randomly oriented planetary orbits, the probability of observing a system that undergoes periodic transits is (eq. 6.5) pran = (r. +rp) /a. hint: calculate, from a vantage point at the center of a star, what is the solid angle on the sky that is covered by observers who will see a transit by the planet. transits are in the range between those that cross the star's equator to those where the planet's limb grazes the star's limb. dividing this solid angle by the full sky's 4t gives the probability. c. kepler-93, whose light curve is shown in fig. 6.2, is a star of mass m 0.91mo, radius r. 0.92ro, and effective temperature te 5700 k. by "eyeballing" the figure, find the observed parameters of the transit (orbital period, transit depth, transit duration; you may want to use a ruler), and use them to find the physical parameters of the system: separation a, planet radius r, and orbital inclination i. note that the