ParabolicDensity/Spheres/Structure

Parabolic Density Distribution

Part I:   Gravitational Potential

Part II:   Spherical Structures

Part III:   Axisymmetric Equilibrium Structures

Part IV:   Triaxial Equilibrium Structures (Exploration)

Spherically Symmetric Equilibrium Structure

In an article titled, "Radial Oscillations of a Stellar Model," 📚 C. Prasad (1949, MNRAS, Vol 109, pp. 103 - 107) investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,

$\rho (r)={\rho }_c[1-(\frac{r}{R}{)}^2],$

where, ${\rho }_c$ is the central density and $R$ is the radius of the star.

Radial Profiles

In a related discussion we have derived the following expressions that describe analytically various structural properties of this equilibrium configuration.

$M_r(r)$	$=$	$\frac{4\pi {\rho }_c{r}^3}{3}[1-\frac{3}{5}(\frac{r}{R}{)}^2];$
$g_0(r)\equiv \frac{G{M}_r(r)}{r^2}$	$=$	$\frac{4\pi G{\rho }_{c}r}{3}[1-\frac{3}{5}(\frac{r}{R}{)}^2];$
${\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$	$=$	$\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{8R}\{-15+10(\frac{r}{R}{)}^2-3\left(\frac{r}{R}{)}^4\right\};$
$P(r)$	$=$	$\frac{4\pi G{\rho }_c^2{R}^2}{15}[1-(\frac{r}{R}{)}^2{]}^2[1-\frac{1}{2}(\frac{r}{R}{)}^2];$
$H(r)$	$=$	$\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{8R}[7-10(\frac{r}{R}{)}^2+3\left(\frac{r}{R}{)}^4\right].$

Note that the total mass is obtained by setting $r=R$ in the expression for $M_r(r)$, namely,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$\frac{4\pi {\rho }_c{R}^3}{3}\left[\frac{2}{5}\right]=\frac{8\pi {\rho }_c{R}^3}{15}$       $\Rightarrow$       $2\pi {\rho }_c=\frac{15M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4R^3}.$

Effective Barotropic Relations

By replacing $r/R$ with $\rho /{\rho }_c$, we obtain analytic expression for, respectively, the pressure-density and enthalpy-density (effective barotropic) relations that are relevant in this parabolic configuration. Specifically,

$\frac{\rho }{{\rho }_c}$	$=$	$[1-(\frac{r}{R}{)}^2]$
$\Rightarrow (\frac{r}{R}{)}^2$	$=$	$[1-(\frac{\rho }{{\rho }_c}\left)\right]$
$\Rightarrow \frac{P(\rho )}{P_c}$	$=$	$\{1-[1-\left(\frac{\rho }{{\rho }_c}\right)\left]{\}}^2\right\{1-\frac{1}{2}[1-(\frac{\rho }{{\rho }_c}\left)\right]\}$
	$=$	$\frac{1}{2}\left(\frac{\rho }{{\rho }_c}{)}^2\right[1+\left(\frac{\rho }{{\rho }_c}\right)],$

where,

$P_c$

$\equiv$

$\frac{4\pi G{\rho }_c^2{R}^2}{15}.$

And,

$\frac{H(\rho )}{H_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$	$=$	$7-10[1-(\frac{\rho }{{\rho }_c}\left)\right]+3[1-(\frac{\rho }{{\rho }_c}){]}^2$
	$=$	$4\left(\frac{\rho }{{\rho }_c}\right)+3(\frac{\rho }{{\rho }_c}{)}^2,$

where,

$H_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$

$\equiv$

$\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{8R}.$

See Also