An isothermal atmosphere layer with temperature $T$ starts at altitude $h_1,$ where the pressure is $p_1.$ In this problem, use a non$-$constant gravitational acceleration,

$g(h)=g_0(\frac{R_e}{R_e+h}{)}^2,$

where $g_0$ is the acceleration at $h=0,$ and $R_e$ is the radius of the Earth. Assume an ideal gas, $p=RT.$

a$)$ Determine the pressure $p_2$ at another altitude $h_2>h_1$ in the same layer.

b$)$ Show that when $h_1,h_2{R}_e,$ your expression in the previous part reduces to the expression in Section $3\mathrm{.}2\mathrm{.}4$ of the notes.

c$)$ Assume that the entire atmosphere is isothermal at $T=250K.$ Write a code that computes and plots the pressure for altitudes $0h100km,$ using both a constant and variable gravitational acceleration. Use a $log$ scale for the pressure $(semilogx),g_0=9\mathrm{.}80665\frac{m}{s^2},R=287\mathrm{.}053\frac{J}{k}g.K,p_0=101325\frac{N}{m^2},$ and $R_e=6371km.$ Include the plot in your submission and discuss the differences.

Exponentiating both sides gives

$p_2=p_1{e}^{-\frac{g}{RT}(h_2-h_1)}$

So the pressure in an isothermal layer decreases exponentially with altitude. Note that for point $2,$ we can use any other location in the layer, not necessarily the top $-$ the above equation then gives us the pressure at the chosen point.