HW 5 Q6 (William Matzko)
6. (a)Usually, the temperature on a planet decreases with altitude. However, in inversion layers, this is not the case. In an inversion layer, temperature increases with altitude. Such a thing is important in the atmosphere of exoplanets because it can give insight into what that planet's atmosphere could be composed of. These inversion layers are usually composed of molecules that absorb UV radiation, so finding these inversion layers on exoplanets is desirable if we want to try and find life on them.
(b) We know that P = P_0 exp(-z/H), where P is the pressure, P_0 is the pressure at the surface level, z is the altitude above surface level and H is the scale height. We are given that, on Earth z = 8.8 km and H = 8 km. This corresponds to a pressure of ~0.333*P_0. On a modified planet, to find the z at which we have the same pressure, we first must compute the scale height. We can express the scale height equation as H = k*T/(u*m*g), where k is the Boltzmann constant, T is the temperature of the planet, um is average mass of gas particle, and g is the acceleration due to gravity. k is just a constant, and we can just assume that um is unchanged on this new planet (that is, its atmospheric composition is the same as Earth's, which makes um = (29)(1.67E-27 Kg). We then need to find g. We are given the new planet's radius and mass, so it is a trivial matter to compute the planet's surface gravity: g = GM/r^2. We can just plug in the new values for M and r (of course, assuming that we can neglect height variations in g). We find that g = 34 m/s roughly. Next, we must find an expression for T. We are given in the problem that a = 0.2 AU, so since the planet orbits a solar analog, we cannot just use the same temperature we would on Earth. We must therefore find a way to relate the semimajor axis, a, to the temperature of the planet. I managed to find a formula that does just this, with some additional decorations: T_p = T_eff * ((1-A)R^2/(4a^2))^(1/4), where A is the albedo of the planet, R is the radius of the host star, a is the semimajor axis, T_eff is the effective temperature of the host star, and T_p is the temperature of the planet. See "Compositions of Hot Super-Earth Atmospheres: Exploring Kepler Candidates" by Miguel et al., 2011, Astrophysical Journal for this equation. Plugging in our sun's effective temperature of 5700 K, Earth's albedo of 0.3, and our sun's radius, we find that T_p = 562 K. We now have all the information we need to calculate the scale height. Plugging all this in gives a new scale height of ~4.7 Km on this planet. We want to find the altitude at which we have the same pressure (0.333 P_0), so we simply solve exp(-z/H) = 0.333. Doing so gives us z ~ 5 Km.
(b) We know that P = P_0 exp(-z/H), where P is the pressure, P_0 is the pressure at the surface level, z is the altitude above surface level and H is the scale height. We are given that, on Earth z = 8.8 km and H = 8 km. This corresponds to a pressure of ~0.333*P_0. On a modified planet, to find the z at which we have the same pressure, we first must compute the scale height. We can express the scale height equation as H = k*T/(u*m*g), where k is the Boltzmann constant, T is the temperature of the planet, um is average mass of gas particle, and g is the acceleration due to gravity. k is just a constant, and we can just assume that um is unchanged on this new planet (that is, its atmospheric composition is the same as Earth's, which makes um = (29)(1.67E-27 Kg). We then need to find g. We are given the new planet's radius and mass, so it is a trivial matter to compute the planet's surface gravity: g = GM/r^2. We can just plug in the new values for M and r (of course, assuming that we can neglect height variations in g). We find that g = 34 m/s roughly. Next, we must find an expression for T. We are given in the problem that a = 0.2 AU, so since the planet orbits a solar analog, we cannot just use the same temperature we would on Earth. We must therefore find a way to relate the semimajor axis, a, to the temperature of the planet. I managed to find a formula that does just this, with some additional decorations: T_p = T_eff * ((1-A)R^2/(4a^2))^(1/4), where A is the albedo of the planet, R is the radius of the host star, a is the semimajor axis, T_eff is the effective temperature of the host star, and T_p is the temperature of the planet. See "Compositions of Hot Super-Earth Atmospheres: Exploring Kepler Candidates" by Miguel et al., 2011, Astrophysical Journal for this equation. Plugging in our sun's effective temperature of 5700 K, Earth's albedo of 0.3, and our sun's radius, we find that T_p = 562 K. We now have all the information we need to calculate the scale height. Plugging all this in gives a new scale height of ~4.7 Km on this planet. We want to find the altitude at which we have the same pressure (0.333 P_0), so we simply solve exp(-z/H) = 0.333. Doing so gives us z ~ 5 Km.
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