thermodynamics - Atmospheric pressure, density and temperature variation with altitude

I'm trying to understand how one can calculate pressure, density and temperature of the atmosphere as a function of altitude.

My assumptions are mostly sourced from https://en.wikipedia.org/wiki/Lapse_rate and https://en.wikipedia.org/wiki/Barometric_formula#Derivation. However, on these pages, there seems to be a little vagueness regarding what parameters are being held constant so I shall write them out explicitly here with dependence on height $z$ where appropriate:

1) Air is an ideal gas so $P(z)M = \rho(z)RT(z)$.

2) The pressure is hydrostatic i.e. $dP(z) = -\rho(z) g dz$

3) There is some temperature lapse rate as a function of altitude and density of air $T(z) = f(z, \rho(z))$. This allows me to take into account radiation and convection. Now, the Wikipedia page (https://en.wikipedia.org/wiki/Lapse_rate) treats $\rho$ as a constant and then assumes that air behaves like an adiabatic gas when it expands due to heat to obtain a valid expression for T(z). That seems incorrect though as density clearly does change with altitude.

4) It's not clear if I can obtain $\rho(z)$ from some other consideration independently.

Are there good tricks/reasonable physical assumptions to solve and obtain all three variables as a function of $z$?

EDIT: The constant density assumption is what I'm having trouble with. Why should this be true and if not, what is the way to obtain it (at least to some first order where we ignore temperature lapse)?

Answer

The two basic equations are

$$\rho=\frac{PM}{RT}$$ and $$\frac{dP}{dz}=-\rho g$$ If we eliminate the (altitude-dependent) density from these equations, we obtain: $$\frac{dP}{dz}=-\frac{PM}{RT}g\tag{1}$$ For the troposphere, the equation for the adiabatic reversible expansion and compression of convected air parcels is: $$\frac{P}{P_0}=\left(\frac{\rho}{\rho_0}\right)^{\gamma}$$ where the subscript 0 revers to the values at ground level (z = 0). If we combine this equation with the ideal gas law, we obtain: $$\frac{T}{T_0}=\left(\frac{P}{P_0}\right)^{\frac{\gamma-1}{\gamma}}\tag{2}$$ We can now substitute Eqn.2 into Eqn. 1 to obtain a equation (strictly valid for the troposphere) that involves only the pressure P (i.e., the density and temperature have been eliminated): $$\frac{dP}{dz}=-\frac{MgP_0^{\frac{\gamma-1}{\gamma}}}{RT_0}P^{1/\gamma}$$ If we integrate this equation between z = 0 and arbitrary z, we obtain: $$\left(\frac{P}{P_0}\right)^{\frac{\gamma-1}{\gamma}}=1-\frac{(\gamma-1)}{\gamma}\frac{Mgz}{RT_0}\tag{3}$$ Combining Eqns. 2 and 3 then yields: $$T=T_0-\frac{(\gamma-1)}{\gamma}\frac{Mg}{R}z\tag{3}$$ Eqn. 3 suggests that, for the assumed adiabatic convective expansion and compression of air parcels in the troposphere, the tropospheric temperature should vary linearly with altitude z. The vertical temperature gradient predicted by this equation is called the "dry adiabatic lapse rate," and has a value of 9.8 C/km. The actual temperature gradient observed in the atmosphere is less than this, with a value of 6.5 C/km.