a column of a planet's atmosphere. The planet's atmosphere is a compressible ideal gas at rest that obeys the polytropic relation Po %3D 3/2 3/2 Po here p is pressure and pis density. Here, p, and e are the values of pressure and density, espectively, at the planet's surface. Take z (altitude) to be positive upward with z=0 at the urface, take R to be the gas constant for the planet's atmosphere, and take g to be the lownward acceleration due to gravity. a) Starting from hydrostatic balance and the polytropic relation above, derive an expression for the pressure field, p(z), in terms of the given parameters. Leave all parameters except the polytropic index as algebraic. b) Derive an expression for the density field, p(z), in terms of the given parameters. Leave all parameters except the polytropic index as algebraic. c) Derive an expression for the temperature field, T(z), in terms of the given parameters. Leave all parameters except the polytropic index as algebraic.