HW5 Q4 (William Matzko)

4. Here, we want to make use of the error propagation formula: σ = [((df/di)σ_i)^2 + ((df/dj)σ_j)^2 + . . .]^(1/2), where σ is the uncertainty of the quantity we want, f is the function we are differentiating, and i,j are the variables that have the associated uncertainty. Let's start with estimating the error in the radius of the planet, since that equation is easier to work with. The transit depth, g, is simply g = (r/R)^2, where r is the radius of the planet, and R is the radius of the star. We can solve this explicitly for r to give r = g^0.5 R. We assume the transit depth is perfectly known, so our only contributing uncertainty, σ_R, comes from the radius of the star. Filling in the error propagation formula, we just take the derivative of r with respect to R: dr/dR = g^0.5. We thus get an uncertainty of our planet to be σ_p = g^0.5 σ_R. If we have a 10% uncertainty in the radius of the star, then our uncertainty in the planet radius will be 0.1 * g^0.5. If our transit depth = 1, then a 10% uncertainty in our star's radius gives us a 10% uncertainty in the planet's radius. As the transit depth decreases, say to 0.5, then the planet has an uncertainty of ~7% in its radius. In a way, this goes against my intuition; I don't expect a smaller transit depth to produce a smaller uncertainty in the planet's radius. 

 We can use the same approach for estimating the uncertainty in the mass of the planet. We know that m_p sin(i) = k/(28.4 m/s) p^(-1/3) M^(2/3), where k is in m/s, p is in years, and M (the mass of the star) is in solar masses. To make things easier to work with, let's just say that the period of the planet is 1 year, let a = k/(28.4 m/s) and let m = m_p sin(i). Thus,
m = a M^(2/3). As above, we see that dm/dM = (2/3)aM^(-1/3). Since this is our only source of uncertainty in our planet's mass, we then have an uncertainty of σ_p = (2/3)aM^(-1/3) σ_M, where σ_M is now the uncertainty in the star's mass and σ_p is the uncertainty in the planet's mass. Again, we see that the uncertainty in the mass of the planet is proportional to the uncertainty in the mass of the star, times some constant values. If a = 1 and M = 1, then an uncertainty of 10% in the star's mass translates into ~6.7% planet mass uncertainty. Again, this sort of goes against my intuition. I expect that the minimum uncertainty on our planet should match the minimum uncertainty of our star (ie, if the star has a mass uncertainty of 10%, I expect the planet to have a mass uncertainty of at least 10%).

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