# 4.3.5.1: Ideal Gas in Varying Gravity

In physics, it was explained that the gravity is a function of the distance from the center of the plant/body. Assuming that the pressure is affected by this gravity/body force. The gravity force is reversely proportional to $$r^2$$. The gravity force can be assumed that for infinity, $$r \rightarrow \infty$$ the pressure is about zero. Again, equation (11) can be used (semi one directional situation) when $$r$$ is used as direction and thus
$\dfrac{\partial P }{\partial r} = - \rho \dfrac {G}{r^2} \label{static:eq:gravityGchange} \tag{110}$
where $$G$$ denotes the general gravity constant. The regular method of separation is employed to obtain
$\int_{P_{b}}^{P} \dfrac{dP}{P} = - \dfrac {G}{RT} \int_{r_{b}} ^r \dfrac {dr}{r^2} \label{static:eq:gravityGchangeB} \tag{111}$
where the subscript $$b$$ denotes the conditions at the body surface. The integration of equation (111) results in
$\ln \dfrac{P}{P_{b}} = - \dfrac {G}{RT} \left( \dfrac{1}{r_{b}}-\dfrac{1}{r}\right) \label{static:eq:gravityGchangeA} \tag{112}$
Or in a simplified form as
$\dfrac{\rho}{\rho_{b}} = \dfrac{P}{P_{b}} = \text{ e} ^{ -\dfrac{G}{RT} \dfrac{r-r_{b}}{r\,r_{b}} } \label{static:eq:gravityGchangeF} \tag{113}$
Equation (113) demonstrates that the pressure is reduced with the distance. It can be noticed that for $$r \rightarrow r_{b}$$ the pressure is approaching $$P \rightarrow P_{b}$$. This equation confirms that the density in outer space is zero $$\rho(\infty) = 0$$. As before, equation (113) can be expanded in Taylor series as
$\dfrac{\rho}{\rho_{b}} = \dfrac{P}{P_{b}} = \overbrace{1 - \dfrac{G\,\left( r-r_b\right) }{R\,T} }^{\text{standard}} - \overbrace{\dfrac{\left( 2\,G\,{R\,T}+ {G}^{2}\,r_b\right) \,{\left( r-r_b\right) }^{2}}{2\,r_b\,{(R\,T)}^{2}} +\cdots}^{\text{correction factor}} \label{static:eq:gravityGchangeFexpended} \tag{114}$
Notice that $$G$$ isn't our beloved and familiar $$g$$ and also that $$G\,r_b/RT$$ is a dimensionless number (later in the Chapter 9 a discution about the definition of the dimensionless number and its meaning).

### Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.