Compare the transit probabilities of a 3 Earth-mass planet in the habitable zone of a solar analogue and a 3 Earth-mass planet in the habitable zone of a 0.25 solar-mass red dwarf.
From problem 2, we know that the probability of transit is the radius of the star divided by the orbit distance of the exoplanet. Because of the relationship between a star's mass and radius, this scales with the ratio between the star's mass and the exoplanet's distance. We also know from the habitable zone problem that the distance of the habitable zone scales with the square of the star's mass. This means that the probability of transit scales inversely with the mass of the star. From this, we can determine that the exoplanet around the red dwarf is four times as likely to transit as the planet around the solar analogue.
Compare the transit depth and duration for the same two planets.
By comparing the area of an exoplanet and its star, we get that the transit depth scales with the planet's radius squared divided by the star's radius squared. Since the star's radius scales with its mass, we get that the depth scales with the planet's radius squared divided by the star's mass squared. This means that the depth of the planet transiting around the red dwarf is 4 times greater than that of the planet around the solar analogue.
Using our formula for radial velocity, we can get that the transit time scales with the radius of the star, the cube root of the orbit period, and the inverse cube root of the star's mass. Using the handy-dandy relation between the star's mass and radius again, we get that the transit time scales with P^(1/3) * M^(2/3). From Kepler's third law, P^2 scales with a^3. This scales with M^6 given that both planets are in the habitable zone. This means that P scales with M^3, and the transit time t scales with M^(5/3), and that the transit time is 10 times longer for the planet around the solar analogue.
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