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 derived the following expressions that describe analytically various structural properties of these configurations.

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

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

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

$=$

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

See Also