1. A simple model for a gas giant planet assumes a polytropic equation of state, P...

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1. A simple model for a gas giant planet assumes a "polytropic" equation of state, P = Kp, (1) where K is a parameter with units of kg m's-2. This model yields the density profile sin(7 r/R) (Tr/R) p(r) = Pe (2) where pe is the mass density at the center (r = 0) and R= VTK/(2G) is the planet's outer radius. (a) The mass in a shell of radius r and thickness dr is dM = 4rr p(r)dr. Integrate dM from r 0 to r = R to find the model's total mass M. (Hint: fx sin(kr)dx = sin(kx)/k2 – x cos(kr)/k.) (b) Given Jupiter's radius Ry = 7.1 x 107 m and mass Mj = 1.90 x 1027 kg, what are the corresponding values for K and p.? (Note: these values "fit" the model to Jupiter.) (c) Using the values of K and pe you obtained, what central pressure Pe does this model predict? 1. A simple model for a gas giant planet assumes a "polytropic" equation of state, P = Kp, (1) where K is a parameter with units of kg m's-2. This model yields the density profile sin(7 r/R) (Tr/R) p(r) = Pe (2) where pe is the mass density at the center (r = 0) and R= VTK/(2G) is the planet's outer radius. (a) The mass in a shell of radius r and thickness dr is dM = 4rr p(r)dr. Integrate dM from r 0 to r = R to find the model's total mass M. (Hint: fx sin(kr)dx = sin(kx)/k2 – x cos(kr)/k.) (b) Given Jupiter's radius Ry = 7.1 x 107 m and mass Mj = 1.90 x 1027 kg, what are the corresponding values for K and p.? (Note: these values "fit" the model to Jupiter.) (c) Using the values of K and pe you obtained, what central pressure Pe does this model predict?