- #1

- 102

- 11

**Summary::**i) Set up a differential equation that describes how the pressure ##p## varies with the distance

r from the center of the planet. Hint: You can base your reasoning on static

equilibrium and Archimedes' principle.

ii)Calculate how the atmospheric pressure p and the density of the atmosphere ##ρ## depend on r.

Assume that pressure and density only depend on r.

Consider a planet with a thin atmosphere. The planet is assumed to have radius ##r_0## and mass M.

The gravitational field outside the planet is given by: $$\vec g = - \frac {GM} {r^2} \vec e_r$$ where G is Newton's gravitational constant. The atmosphere near the surface of the planet can be considered as a stationary linearly compressible fluid, which means that a relationship ##ρ = ρ_0 + α(p - p_0)## applies between the density of the fluid ##ρ## and the pressure ##p##. Here ##ρ_0## and ##p_0## are the density of the atmosphere respective pressure at the planetary surface.

i) Set up a differential equation that describes how the pressure ##p## varies with the distance

r from the center of the planet. Hint: You can base your reasoning on static

equilibrium and Archimedes' principle.

ii)Calculate how the atmospheric pressure p and the density of the atmosphere ##ρ## depend on r.

Assume that pressure and density only depend on r.

I am not quite sure how to start on i) since equilibrium in the radial direction is given by ##-m\frac {GM} {r^2} + ρVg = 0## and ##ρV = m## the previous expression is just zero and that doesn't give me anything

Thanks in advance!