ParabolicDensity/Axisymmetric/Structure

Parabolic Density Distribution

Part I:   Gravitational Potential

Part II:   Spherical Structures

Part III:   Axisymmetric Equilibrium Structures

Part IV:   Triaxial Equilibrium Structures (Exploration)

Axisymmetric (Oblate) Equilibrium Structures

Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

$\rho$

=

${\rho }_c[1-(\frac{x^2+y^2}{a_{\ell }^2}+\frac{z^2}{a_s^2}\left)\right],$

that is, axisymmetric ($a_m=a_{\ell }$, i.e., oblate) 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.

Gravitational Potential

As we have detailed in an accompanying discussion, for an oblate-spheroidal configuration — that is, when $a_s<a_m=a_{\ell }$ — the gravitational potential may be obtained from the expression,

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

$=$

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

where, in the present context, we can rewrite this expression as,

$\frac{{\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})}{(-\pi G{\rho }_c)}$	$=$	$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}{a}_{\ell }^2-[A_{\ell }(x^2+y^2)+A_s{z}^2]+[A_{\ell \ell }{x}^2{y}^2+A_{\ell s}{x}^2{z}^2+A_{\ell s}{y}^2{z}^2]+\frac{1}{6}[3A_{\ell \ell }{x}^4+3A_{\ell \ell }{y}^4+3A_{ss}{z}^4]$
	$=$	$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}{a}_{\ell }^2-[A_{\ell }{\varpi }^2+A_s{z}^2]+[A_{\ell \ell }{x}^2{y}^2+A_{\ell s}{\varpi }^2{z}^2]+\frac{1}{2}[A_{\ell \ell }(x^4+y^4)+A_{ss}{z}^4]$
	$=$	$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}{a}_{\ell }^2-[A_{\ell }{\varpi }^2+A_s{z}^2]+\frac{A_{\ell \ell }}{2}[(x^2+y^2{)}^2]+\frac{1}{2}[A_{ss}{z}^4]+[A_{\ell s}{\varpi }^2{z}^2]$
	$=$	$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}{a}_{\ell }^2-[A_{\ell }{\varpi }^2+A_s{z}^2]+\frac{A_{\ell \ell }}{2}\left[{\varpi }^4\right]+\frac{1}{2}\left[A_{ss}{z}^4\right]+\left[A_{\ell s}{\varpi }^2{z}^2\right]$
$\Rightarrow \frac{{\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})}{(-\pi G{\rho }_c{a}_{\ell }^2)}$	$=$	$\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}-\left[A_{\ell }\right(\frac{{\varpi }^2}{a_{\ell }^2})+A_s(\frac{z^2}{a_{\ell }^2}\left)\right]+\frac{1}{2}\left[A_{\ell \ell }{a}_{\ell }^2\right(\frac{{\varpi }^4}{a_{\ell }^4})+A_{ss}{a}_{\ell }^2(\frac{z^4}{a_{\ell }^4})+2A_{\ell s}{a}_{\ell }^2(\frac{{\varpi }^2{z}^2}{a_{\ell }^4}\left)\right].$

Index Symbol Expressions

The expression for the zeroth-order normalization term $(I_{BT})$, and the relevant pair of 1^st-order index symbol expressions are:

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

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

where the eccentricity,

$e\equiv [1-(\frac{a_s}{a_{\ell }}{)}^2{]}^{1/2}.$

The relevant 2^nd-order index symbol expressions are:

$a_{\ell }^2{A}_{\ell \ell }$	$=$	$\frac{1}{4e^4}\{-(3+2e^2)(1-e^2)+3(1-e^2{)}^{1/2}\left[\frac{{\sin }^{-1}e}{e}\right]\};$
$\frac{3}{2}{a}_{\ell }^2{A}_{ss}$	$=$	$\frac{(4e^2-3)}{e^4(1-e^2)}+\frac{3(1-e^2{)}^{1/2}}{e^4}\left[\frac{{\sin }^{-1}e}{e}\right];$
$a_{\ell }^2{A}_{\ell s}$	$=$	$\frac{1}{e^4}\{(3-e^2)-3(1-e^2{)}^{1/2}\left[\frac{{\sin }^{-1}e}{e}\right]\}.$

We can crosscheck this last expression by drawing on a shortcut expression,

$A_{\ell s}$	$=$	$-\frac{A_{\ell }-A_s}{(a_{\ell }^2-a_s^2)}$
$\Rightarrow a_{\ell }^2{A}_{\ell s}$	$=$	$\frac{1}{e^2}\{A_s-A_{\ell }\}$
	$=$	$\frac{1}{e^2}\left\{\frac{2}{e^2}\right[(1-e^2{)}^{-1/2}-\frac{{\sin }^{-1}e}{e}](1-e^2{)}^{1/2}-\frac{1}{e^2}[\frac{{\sin }^{-1}e}{e}-(1-e^2{)}^{1/2}](1-e^2{)}^{1/2}\}$
	$=$	$\frac{1}{e^4}\left\{\right[2-2(1-e^2{)}^{1/2}\frac{{\sin }^{-1}e}{e}]-[(1-e^2{)}^{1/2}\frac{{\sin }^{-1}e}{e}-(1-e^2)\left]\right\}$
	$=$	$\frac{1}{e^4}\{(3-e^2)-3(1-e^2{)}^{1/2}\frac{{\sin }^{-1}e}{e}\}.$

Meridional Plane Equi-Potential Contours

Here, we follow closely our separate discussion of equipotential surfaces for Maclaurin Spheroids, assuming no rotation.

Configuration Surface

In the meridional $(\varpi ,z)$ plane, the surface of this oblate-spheroidal configuration — identified by the thick, solid-black curve below, in Figure 1 — is defined by the expression,

$\frac{\rho }{{\rho }_c}$	$=$	$1-[\frac{{\varpi }^2}{a_{\ell }^2}+\frac{z^2}{a_s^2}]=0$
$\Rightarrow \frac{{\varpi }^2}{a_{\ell }^2}+\frac{z^2}{a_s^2}$	$=$	$1$
$\Rightarrow z^2$	$=$	$a_s^2[1-\frac{{\varpi }^2}{a_{\ell }^2}]=a_{\ell }^2(1-e^2)[1-\frac{{\varpi }^2}{a_{\ell }^2}]$
$\Rightarrow \frac{z}{a_{\ell }}$	$=$	$\pm (1-e^2{)}^{1/2}[1-\frac{{\varpi }^2}{a_{\ell }^2}{]}^{1/2},$	for $0\le \frac{|\varpi |}{a_{\ell }}\le 1.$

Expression for Gravitational Potential

Throughout the interior of this configuration, each associated ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ = constant, equipotential surface is defined by the expression,

${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}\equiv \frac{{\mathrm{\Phi }}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}(\mathbf{𝐱})}{(\pi G{\rho }_c{a}_{\ell }^2)}+\frac{1}{2}{I}_{\mathrm{B}\mathrm{T}}$

$=$

$\left[A_{\ell }\right(\frac{{\varpi }^2}{a_{\ell }^2})+A_s(\frac{z^2}{a_{\ell }^2}\left)\right]-\frac{1}{2}\left[A_{\ell \ell }{a}_{\ell }^2\right(\frac{{\varpi }^4}{a_{\ell }^4})+A_{ss}{a}_{\ell }^2(\frac{z^4}{a_{\ell }^4})+2A_{\ell s}{a}_{\ell }^2(\frac{{\varpi }^2{z}^2}{a_{\ell }^4}\left)\right].$

Letting,

$\zeta \equiv \frac{z^2}{a_{\ell }^2}$,

we can rewrite this expression for ${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}$ as,

${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}$	$=$	$A_{\ell }\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)+A_s\zeta -\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2\left(\frac{{\varpi }^4}{a_{\ell }^4}\right)-\frac{1}{2}{A}_{ss}{a}_{\ell }^2{\zeta }^2-A_{\ell s}{a}_{\ell }^2\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)\zeta$
	$=$	$-\frac{1}{2}{A}_{ss}{a}_{\ell }^2{\zeta }^2+[A_s-A_{\ell s}{a}_{\ell }^2(\frac{{\varpi }^2}{a_{\ell }^2}\left)\right]\zeta +A_{\ell }\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)-\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2\left(\frac{{\varpi }^4}{a_{\ell }^4}\right).$

Potential at the Pole

At the pole, $(\varpi ,z)=(0,a_s)$. Hence,

${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}{|}_{\mathrm{m}\mathrm{i}\mathrm{d}}$	$=$	$-\frac{1}{2}{A}_{ss}{a}_{\ell }^2(\frac{a_s^2}{a_{\ell }^2}{)}^2+[A_s-A_{\ell s}{a}_{\ell }^2{\xcancel{\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)}}^0\left]\right(\frac{a_s^2}{a_{\ell }^2})+A_{\ell }{\xcancel{\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)}}^0-\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2{\xcancel{\left(\frac{{\varpi }^4}{a_{\ell }^4}\right)}}^0$
	$=$	$A_s\left(\frac{a_s^2}{a_{\ell }^2}\right)-\frac{1}{2}{A}_{ss}{a}_{\ell }^2(\frac{a_s^2}{a_{\ell }^2}{)}^2.$

General Determination of Vertical Coordinate (ζ)

Given values of the three parameters, $e$, $\varpi$, and ${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}$, this last expression can be viewed as a quadratic equation for $\zeta$. Specifically, $0$ $=$ $\alpha {\zeta }^2+\beta \zeta +\gamma ,$ where, $\alpha$ $\equiv$ $\frac{1}{2}{A}_{ss}{a}_{\ell }^2$   $=$ $\frac{1}{3}\{\frac{(4e^2-3)}{e^4(1-e^2)}+\frac{3(1-e^2{)}^{1/2}}{e^4}[\frac{{\sin }^{-1}e}{e}\left]\right\},$ $\beta$ $\equiv$ $A_{\ell s}{a}_{\ell }^2\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)-A_s$   $=$ $\frac{1}{e^4}\{(3-e^2)-3(1-e^2{)}^{1/2}\frac{{\sin }^{-1}e}{e}\left\}\right(\frac{{\varpi }^2}{a_{\ell }^2})-\frac{2}{e^2}[(1-e^2{)}^{-1/2}-\frac{{\sin }^{-1}e}{e}](1-e^2{)}^{1/2},$ $\gamma$ $\equiv$ ${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}+\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2\left(\frac{{\varpi }^4}{a_{\ell }^4}\right)-A_{\ell }\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)$   $=$ ${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}+\frac{1}{8e^4}\{-(3+2e^2)(1-e^2)+3(1-e^2{)}^{1/2}\left[\frac{{\sin }^{-1}e}{e}\right]\left\}\right(\frac{{\varpi }^4}{a_{\ell }^4})-\frac{1}{e^2}[\frac{{\sin }^{-1}e}{e}-(1-e^2{)}^{1/2}](1-e^2{)}^{1/2}(\frac{{\varpi }^2}{a_{\ell }^2}).$ The solution of this quadratic equation gives, $\zeta$ $=$ $\frac{1}{2\alpha }\{-\beta \pm [{\beta }^2-4\alpha \gamma {]}^{1/2}\}.$ Given that in this physical system, $\zeta =z^2/a_{\ell }^2$ must be positive, we must choose the superior root. We conclude therefore that, $\frac{z^2}{a_{\ell }^2}$ $=$ $\frac{1}{2\alpha }\left\{\right[{\beta }^2-4\alpha \gamma {]}^{1/2}-\beta \}.$ But check this statement because it appears that $\beta$ will sometimes be negative.

Here we present a quantitatively accurate depiction of the shape of the (Ferrers) gravitational potential that arises from oblate-spheroidal configurations having a parabolic density distribution. We closely follow the discussion of equi-gravitational potential contours that arise in (uniform-density) Maclaurin spheroids. In order to facilitate comparison with Maclaurin spheroids, we will focus on a model with …

$\frac{a_s}{a_{\ell }}=0.582724,$	$e=0.81267,$
$A_{\ell }=A_m=0.51589042,$	$A_s=0.96821916,$	$I_{\mathrm{B}\mathrm{T}}=1.360556,$
$a_{\ell }^2{A}_{\ell \ell }=0.3287756,$	$a_{\ell }^2{A}_{ss}=1.5066848,$	$a_{\ell }^2{A}_{\ell s}=0.6848975.$

[NOTE:   Along the Maclaurin spheroid sequence, this is the eccentricity that marks bifurcation to the Jacobi ellipsoid sequence — see the first model listed in Table IV (p. 103) of [EFE] and/or see Tables 1 and 2 of our discussion of the Jacobi ellipsoid sequence. It is unlikely that this same eccentricity has a comparably special physical relevance along the sequence of spheroids having parabolic density distributions.]

The largest value of the gravitational potential that will arise inside (actually, on the surface) of the configuration is at $(\varpi ,z)=(1,0)$. That is, when,

${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}{|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$

$=$

$A_{\ell }-\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2=0.3515026.$

So we will plot various equipotential surfaces having, $0<{\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}<{\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}{|}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, recognizing that they will each cut through the equatorial plane $(z=0)$ at the radial coordinate given by,

${\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}$	$=$	$-\frac{1}{2}{A}_{ss}{a}_{\ell }^2{\xcancel{{\zeta }^2}}^0+[A_s-A_{\ell s}{a}_{\ell }^2(\frac{{\varpi }^2}{a_{\ell }^2}\left)\right]{\xcancel{\zeta }}^0+A_{\ell }\left(\frac{{\varpi }^2}{a_{\ell }^2}\right)-\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2\left(\frac{{\varpi }^4}{a_{\ell }^4}\right)$
$\Rightarrow 0$	$=$	$\frac{1}{2}{A}_{\ell \ell }{a}_{\ell }^2{\chi }^2-A_{\ell }\chi +{\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}},$

where,

$\chi \equiv \frac{{\varpi }^2}{a_{\ell }^2}.$

The solution to this quadratic equation gives,

$\chi$	$=$	$\frac{1}{A_{\ell \ell }{a}_{\ell }^2}\{A_{\ell }\pm [A_{\ell }^2-2A_{\ell \ell }{a}_{\ell }^2{\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}{]}^{1/2}\}$
	$=$	$\frac{A_{\ell }}{A_{\ell \ell }{a}_{\ell }^2}\{1\pm [1-\frac{2A_{\ell \ell }{a}_{\ell }^2{\varphi }_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}}}{A_{\ell }^2}{]}^{1/2}\}$

See Also