14. (14 points) While hiding in the dense atmosphere of the planet Lacos, the USS Nittany Lion experiences a fierce ion storm that interacts with the radioactive Schneideronium atmosphere and thrusts the ship and its inhabitants into a parallel universe. Everything is identical to our universe, except that three fundamental items have changed:
• The surface area of a sphere, when used to calculate fluxes or surface areas of planets, is
$A = \frac{2\pi r^3}{r_0}$
$r_0 = 7.23 \times 10^9 cm$
(The cross sectional area of a sphere remains $\pi r^2$, as in our universe.)
• The surface brightness of a blackbody (erg s$^{-1}$ cm$^{-2}$) is
S.B. $= \sigma_0 T^3$
$\sigma_0 = 0.00732 erg s^{-1} cm^{-2} K^{-3}$
• The pressure produced by an ideal gas is
$P = n k_0 T^3$
$k_0 = 7.03 \times 10^{-19} erg K^{-3}$
In this Universe, what is the scale height of Earth's atmosphere? (Assume an isothermal, constant gravity atmosphere, and that the sun's luminosity in this universe is $3.828 \times 10^{33} erg s^{-1}$.)