# 4.3.5.2: Real Gas in Varying Gravity

- Page ID
- 674

The regular assumption of constant compressibility, \(Z\), is employed. It has to remember when this assumption isn't accurate enough, numerical integration is a possible solution. Thus, equation (111) is transformed into

\[
\int_{P_{b}}^{P} \dfrac{dP}{P} =

- \dfrac {G}{Z\,R\,T} \int_{r_{b}} ^r \dfrac {dr}{r^2}

\label{static:eq:gravityGchangeBZ}

\]

With the same process as before for ideal gas case, one can obtain

\[
\dfrac{\rho}{\rho_{b}} =

\dfrac{P}{P_{b}} =

\text{ e} ^{ -\dfrac{G}{Z\,R\,T}

\dfrac{r-r_{b}}{r\,r_{b}}

}

\label{static:eq:gravityGchangeFZ}

\]

Equation (113) demonstrates that the pressure is reduced with the distance. It can be observed 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 Taylor series for equation (113) is

\[
\dfrac{\rho}{\rho_{b}} =

\dfrac{P}{P_{b}} =

\overbrace{1 -

\dfrac{G\,\left( r-r_b\right) }{Z\,R\,T} }^{\text{standard}} -

\overbrace{\dfrac{\left( 2\,G\,{Z\,R\,T}+ \\

{G}^{2}\,r_b\right) \,{\left( r-r_b\right)

}^{2}}{2\,r_b\,{(Z\,R\,T)}^{2}}

+\cdots}^{\text{correction factor}}

\label{static:eq:gravityGchangeFZexpended}

\]

## Contributors and Attributions

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.