ParabolicDensity/GravPot

Parabolic Density Distribution

Part I:   Gravitational Potential

Part II:   Spherical Structures

Part III:   Axisymmetric Equilibrium Structures

Part IV:   Triaxial Equilibrium Structures (Exploration)

Gravitational Potential

In an accompanying chapter titled, Properties of Homogeneous Ellipsoids (1), we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids. In that discussion, we largely followed the derivations of [EFE]. In the latter part of the nineteenth-century, 📚 N. M. Ferrers (1877, Quart. J. Pure Appl. Math., Vol. 14, pp. 1 - 22) showed that very similar analytic expressions can be derived for ellipsoids that have certain inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

$\rho$

=

${\rho }_c[1-(\frac{x^2}{a_1^2}+\frac{y^2}{a_2^2}+\frac{z^2}{a_3^2}\left)\right],$

that is, configurations with parabolic density distributions. Much of our presentation, here, is drawn from our separate, detailed description of what we will refer to as Ferrers potential.

Uniform-Density Reminders

We begin by reminding the reader that, for a uniform-density configuration, the "interior" potential will be given by the expression,

${\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})$

=

$-\pi G{\rho }_c[I_{\mathrm{B}\mathrm{T}}{a}_1^2-(A_1{x}^2+A_2{y}^2+A_3{z}^2\left)\right].$

As we have shown in a separate presentation, if the three principal axes of the configuration are unequal in length and related to one another such that $a_1>a_2>a_3$, the appropriate expressions for the four leading coefficients are,

$A_1$	$=$	$\frac{2a_2{a}_3}{a_1^2}\left[\frac{F(\theta ,k)-E(\theta ,k)}{k^2{\sin }^3\theta }\right];$
$A_2$	$=$	$\frac{2a_2{a}_3}{a_1^2}\left[\frac{E(\theta ,k)-(1-k^2)F(\theta ,k)-(a_3/a_2)k^2\sin \theta }{k^2(1-k^2){\sin }^3\theta }\right];$
$A_3$	$=$	$\frac{2a_2{a}_3}{a_1^2}\left[\frac{(a_2/a_3)\sin \theta -E(\theta ,k)}{(1-k^2){\sin }^3\theta }\right];$
$I_{\mathrm{B}\mathrm{T}}$	$=$	$\frac{2a_2{a}_3}{a_1^2}\left[\frac{F(\theta ,k)}{\sin \theta }\right].$

[EFE], Chapter 3, Eqs. (33), (34) & (35)

As can readily be demonstrated, this scalar potential satisfies the differential form of the

Poisson Equation

${\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho$

As we have also demonstrated, if the longest axis, $a_1$, and the intermediate axis, $a_2$, of the ellipsoid are equal to one another, then an equatorial cross-section of the object presents a circle of radius $a_1$ and the object is referred to as an oblate spheroid. For homogeneous oblate spheroids, evaluation of the integrals defining $A_i$ and $I_{\mathrm{B}\mathrm{T}}$ gives,

$A_1$	$=$	$\frac{1}{e^2}[\frac{{\sin }^{-1}e}{e}-(1-e^2{)}^{1/2}](1-e^2{)}^{1/2};$
$A_2$	$=$	$A_1;$
$A_3$	$=$	$\frac{2}{e^2}[(1-e^2{)}^{-1/2}-\frac{{\sin }^{-1}e}{e}](1-e^2{)}^{1/2};$
$I_{\mathrm{B}\mathrm{T}}$	$=$	$2A_1+A_3(1-e^2)=2(1-e^2{)}^{1/2}[\frac{{\sin }^{-1}e}{e}],$

[EFE], Chapter 3, Eq. (36)
[T78], §4.5, Eqs. (48) & (49)

where the eccentricity,

$e\equiv [1-(\frac{a_3}{a_1}{)}^2{]}^{1/2}.$

Note the following, separately derived limits:

Table 1:  Limiting Values
	$e\to 0$	$\frac{a_3}{a_1}\to 0$
$\frac{{\sin }^{-1}e}{e}$	$1+\frac{e^2}{6}+{𝒪}\left(e^4\right)$	$\frac{\pi }{2}-\left(\frac{a_3}{a_1}\right)+\frac{\pi }{4}(\frac{a_3}{a_1}{)}^2-{𝒪}(\frac{a_3^3}{a_1^3})$
$A_1=A_2$	$\frac{2}{3}[1-\frac{e^2}{5}-{𝒪}(e^4\left)\right]$	$\frac{\pi }{2}\left(\frac{a_3}{a_1}\right)-2(\frac{a_3}{a_1}{)}^2+{𝒪}(\frac{a_3^3}{a_1^3})$
$A_3$	$\frac{2}{3}[1+\frac{2e^2}{5}+{𝒪}(e^4\left)\right]$	$2-\pi \left(\frac{a_3}{a_1}\right)+4(\frac{a_3}{a_1}{)}^2-{𝒪}(\frac{a_3^3}{a_1^3})$
$I_{\mathrm{B}\mathrm{T}}$	$2$	$0$

Hence, for a uniform-density sphere $(e=0)$,

${\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})$	=	$-\pi G{\rho }_c[I_{\mathrm{B}\mathrm{T}}{a}_1^2-(A_1{x}^2+A_2{y}^2+A_3{z}^2\left)\right]$
	=	$-\pi G{\rho }_c[2a_1^2-\frac{2}{3}(x^2+y^2+z^2\left)\right]$
	=	$-2\pi G{\rho }_c{a}_1^2[1-\frac{1}{3}(\frac{r}{a_1}{)}^2]$
	=	$-\frac{3GM}{2a_1}[1-\frac{1}{3}(\frac{r}{a_1}{)}^2].$
J. B. Tatum (2021) Celestial Mechanics class notes (UVic), §5.8.9, p. 36, Eq. (5.8.23)

This matches the expression for the gravitational potential inside (and on the surface) of a uniform-density sphere, as we have derived in an accompanying chapter.

Parabolic Density Distribution

SUMMARY — copied from accompanying, Trial #2 Discussion

After studying 📚 N. M. Ferrers (1877, Quart. J. Pure Appl. Math., Vol. 14, pp. 1 - 22) and the relevant sections of both [EFE] and [BT87], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription. As is discussed in a separate chapter, the potential that it generates is sometimes referred to as a Ferrers potential, for the exponent, n = 1.

In our accompanying discussion we find that,

$\frac{{\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})}{(-\pi G{\rho }_c)}$

$=$

$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}{a}_1^2-(A_1{x}^2+A_2{y}^2+A_3{z}^2)+(A_{12}{x}^2{y}^2+A_{13}{x}^2{z}^2+A_{23}{y}^2{z}^2)+\frac{1}{6}(3A_{11}{x}^4+3A_{22}{y}^4+3A_{33}{z}^4),$

where,

for $i\ne j$ $A_{ij}$ $\equiv$ $-\frac{A_i-A_j}{(a_i^2-a_j^2)}$ [ EFE, §21, Eq. (107) ]

for $i=j$ $2A_{ii}+\sum _{\ell =1}^3{A}_{i\ell }$ $=$ $\frac{2}{a_i^2}$ [ EFE, §21, Eq. (109) ]

More specifically, in the three cases where the indices, $i=j$,

$3A_{11}$	$=$	$\frac{2}{a_1^2}-(A_{12}+A_{13}),$
$3A_{22}$	$=$	$\frac{2}{a_2^2}-(A_{21}+A_{23}),$
$3A_{33}$	$=$	$\frac{2}{a_3^2}-(A_{31}+A_{32}).$

In the case of an axisymmetric $(a_m=a_{\ell })$, but nearly spherical configuration,

$A_1=A_2$	=	$\frac{2}{3}[1-\frac{e^2}{5}-{𝒪}(e^4)];$
$A_3$	=	$\frac{2}{3}[1+\frac{2e^2}{5}+{𝒪}(e^4)].$

Hence,

$A_{13}=A_{23}$	$=$	$-\frac{A_1-A_3}{a_1^2{e}^2}$
	$=$	$-\frac{2}{3a_1^2{e}^2}\left[\right(1-\frac{e^2}{5})-(1+\frac{2e^2}{5})+{𝒪}(e^4)]$
	$=$	$\frac{2}{3a_1^2}[\frac{3}{5}+{𝒪}(e^2)]$
	$\approx$	$\frac{2}{5a_1^2}.$

These results match our separate derivations in the case of a sphere. Specifically,

$A_1=A_2=A_3$

$=$

$\frac{2}{3};$

and,

$A_{11}=A_{12}=A_{13}=A_{22}=A_{23}=A_{33}$

$=$

$\frac{2}{5a_1^2}.$

Hence, for a sphere with a parabolic density distribution, we find,

$\frac{{\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})}{(-\pi G{\rho }_c)}$

$=$

$a_1^2-\frac{2}{3}{r}^2+\frac{2}{5a_1^2}(x^2{y}^2+x^2{z}^2+y^2{z}^2)+\frac{1}{5a_1^2}(x^4+y^4+z^4).$

This matches the gravitational potential derived for a parabolic density distribution using spherical coordinates.

See Also