Up..Up.. into the sky

Considering variation in air pressure (neglecting thermodynamic and humidity variations) and acceleration due to gravity with altitude as the only parameters, the air density as a function of altitude can be given as:- $\rho \ = \ \rho(h)$ where $h$ is the altitude above mean sea level.

Given the following constants:-

• $\rho_{0} \ = \ 1.25 \frac{kg}{m^3}$ = Air density at sea level [ $\rho(0)$ ]

• $R_{⨁} \ = \ 6400 km$ = Radius of Earth

• $g_{0} \ = \ 10 \frac{m}{s^2}$ = Acceleration due to gravity at earth's surface

• $B \ = \ 100 kPa$ = Bulk modulus of air (assumed constant),

Find $\rho(R_{⨁})$ in terms of these.

If your answer is $\sqrt{\frac{a}{b}}$ where $a,b \in \mathbb N$ , $gcd(a,b) = 1$ ; enter ( $b-a$ ).

Bonus: Include humidity, temperature and other parameters.

Original :).

Also, do post a solution as I am still looking for an optimal solution to this problem.