A planet's atmosphere contains a perfect gas with polytropic equa- tion, p = Ap, where p is the pressure, p is the density, y is the polytropic index and A is a constant. Assuming that the acceleration due to gravity, g, is practically constant with height: a) i) show that the expression of the ratio of the pressure, p (at a vertical height z from the surface in terms of the surface pressure), to Po (the surface height), and the surface temperature To is: ²₁ = (₁ - (^² = ²¹); (2) Р 1 Po where y is the adiabatic constant and the gas constant is R. g RT. -Z ii) Use the equation in 1) a) i) to find the numerical value of the temperature lapse dT/dz, given that g = 9.8 m/s², R = 287 Nm/kg K, and y 1.23. =

A planet's atmosphere contains a perfect gas with polytropic equa- tion, p = Ap, where p is the pressure, p is the density, y is the polytropic index and A is a constant. Assuming that the acceleration due to gravity, g, is practically constant with height: a) i) show that the expression of the ratio of the pressure, p (at a vertical height z from the surface in terms of the surface pressure), to Po (the surface height), and the surface temperature To is: ²₁ = (₁ - (^² = ²¹); (2) Р 1 Po where y is the adiabatic constant and the gas constant is R. g RT. -Z ii) Use the equation in 1) a) i) to find the numerical value of the temperature lapse dT/dz, given that g = 9.8 m/s², R = 287 Nm/kg K, and y 1.23. =