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Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

Whitworth's (1981) Isothermal Free-Energy Surface
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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, Quarterly Journal of Pure and Applied Mathematics, 14, 1 - 22) showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

~\rho

~=

~  \rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .

SUMMARY — copied from accompanying, Trial #2 Discussion

After studying the relevant sections of both EFE and BT87 — this is an example of a heterogeneous 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{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{6}  \biggl(3A_{11}x^4 +  3A_{22}y^4 + 3A_{33}z^4  \biggr) 
\, ,

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}

[ 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}) \, .


Derivation

Other references to Ferrers Potential:

Following §2.3.2 (beginning on p. 60) of BT87, let's examine inhomogeneous configurations whose isodensity surfaces (including the surface, itself) are defined by triaxial ellipsoids on which the Cartesian coordinates ~(x_1, x_2, x_3) satisfy the condition that,

~m^2

~\equiv

~a_1^2 \sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \, ,

[ EFE, Chapter 3, §20, p. 50, Eq. (75) ]
[ BT87, §2.3.2, p. 61, Eq. (2-97) ]

be constant. More specifically, let's consider the case (related to the so-called Ferrers potentials) in which the configuration's density distribution is given by the expression,

~\rho(m^2)

~=

~\rho_c \biggl[1 - \frac{m^2}{a_1^2}\biggr]^n

~=

~
\rho_c \biggl[1 - 
\sum_{i=1}^{3} \frac{x_i^2}{a_i^2} \biggr]^n

~=

~
\rho_c \biggl[1 - \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\biggr) \biggr]^n 
\, .

NOTE:     In our accompanying discussion of compressible analogues of Riemann S-type ellipsoids, we have discovered that — at least in the context of infinitesimally thin, nonaxisymmetric disks — this heterogeneous density profile can be nicely paired with an analytically expressible stream function, at least for the case where the integer exponent is, n = 1.

According to Theorem 13 of EFE — see his Chapter 3, §20 (p. 53) — the potential at any point inside a triaxial ellipsoid with this specific density distribution is given by the expression,

~\Phi_\mathrm{grav}(\bold{x})

~=

~
- \frac{\pi G \rho_c a_1 a_2 a_3}{(n+1)}  \int_0^\infty \frac{ du}{\Delta } Q^{n+1}  \, ,

[ EFE, Chapter 3, §20, p. 53, Eq. (101) ]

where, ~\Delta has the same definition as above, and,

~Q

~\equiv

~
1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \, .

For purposes of illustration, in what follows we will assume that, ~a_1 > a_2 > a_3.

The Case Where n = 0

When ~n = 0, we have a uniform-density configuration, and the "interior" potential will be given by the expression,

~\Phi_\mathrm{grav}(\bold{x})

~=

~
- \pi G \rho_c a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

~=

~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{x^2}{ a_1^2 + u } \biggr)
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{y^2}{ a_2^2 + u } \biggr) 
- \int_0^\infty \frac{ du}{\Delta } \biggl( \frac{z^2}{ a_3^2 + u }  \biggr)
\biggr\}

 

~=

~
- \pi G \rho_c a_1 a_2 a_3 \biggl\{  
\int_0^\infty \frac{ du}{\Delta } 
~ - ~x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u) } 
~ - ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u) }  
~ - ~ \int_0^\infty \frac{ du}{\Delta (a_3^2 + u) } 
\biggr\}

 

~=

~
-\pi G \rho_c \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr] \, .

As a check, let's see if this scalar potential satisfies the differential form of the

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

Given that,

~\sum_{\ell = 1}^3 A_\ell

~=

~2 \, ,

[ EFE, §21, Eq. (108) ]

we find,

~\nabla^2\Phi_\mathrm{grav} = \biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr]\Phi_\mathrm{grav}

~=

~
+ 2\pi G \rho_c (A_1 + A_2 + A_3) = 4\pi G\rho_c \, .

Q.E.D.

The Case Where n = 1

When ~n = 1, we have a specific heterogeneous density configuration, and the "interior" potential will be given by the expression,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} a_1 a_2 a_3  \int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]^2

 

~=

~
\frac{1}{2} a_1 a_2 a_3  \biggl\{  
\int_0^\infty \frac{ du}{\Delta } \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~- ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

 

~
~- ~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~ - ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .

The first definite-integral expression inside the curly braces is, to within a leading factor of ~\tfrac{1}{2}, identical to the entire expression for the normalized potential that was derived in the case where n = 0. That is, we can write,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{1}{2} a_1 a_2 a_3  \biggl\{ ~ x^2 \int_0^\infty \frac{ du}{\Delta (a_1^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

 

 

~
~+~y^2 \int_0^\infty \frac{ du}{\Delta (a_2^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
~ + ~z^2 \int_0^\infty \frac{ du}{\Delta (a_3^2 + u)} \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]
\biggr\} \, .

Then, from §22, p. 56 of EFE, we see that,

~a_1 a_2 a_3 \int_0^\infty \frac{ du}{\Delta (a_i^2 + u)}  \biggl[ 1 - \sum_{\ell = 1}^3 \frac{x_\ell^2}{ a_\ell^2 + u }  \biggr]

~=

~
\biggl( A_i - \sum_{\ell=1}^3 A_{i\ell} x_\ell^2 \biggr) \, .

[ EFE, Chapter 3, §22, p. 53, Eq. (125) ]

Applying this result to each of the other three definite integrals gives us,

~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)}

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2}  \biggl( A_1 - \sum_{\ell=1}^3 A_{1\ell} x_\ell^2 \biggr)
~- \frac{y^2}{2}  \biggl( A_2 - \sum_{\ell=1}^3 A_{2\ell} x_\ell^2 \biggr)
~- \frac{z^2}{2}  \biggl( A_3 - \sum_{\ell=1}^3 A_{3\ell} x_\ell^2 \biggr) \, .

 

~=

~
\frac{1}{2} \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr]
~- \frac{x^2}{2}  \biggl[ A_1 - \biggl( A_{11}x^2 + A_{12}y^2 + A_{13}z^2 \biggr) \biggr]

 

 

~
~- \frac{y^2}{2}  \biggl[ A_2 - \biggl( A_{21}x^2 + A_{22}y^2 + A_{23}z^2 \biggr) \biggr]
~- \frac{z^2}{2}  \biggl[ A_3 - \biggl( A_{31}x^2 + A_{32}y^2 + A_{33}z^2 \biggr) \biggr]

 

~=

~
\frac{1}{2} I_\mathrm{BT} a_1^2 
- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\, ,

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}

[ EFE, §21, Eq. (109) ]


and we have made use of the symmetry relation, ~A_{ij} = A_{ji}. Again, as a check, let's see if this scalar potential satisfies the differential form of the

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

We find,

~\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr]

~=

~
\frac{1}{2}\biggl[\frac{\partial^2}{\partial x^2}  + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\biggr] 
\biggl[- \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) 
~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr)
~+ \frac{1}{2}  \biggl(A_{11}x^4 +  A_{22}y^4 + A_{33}z^4  \biggr) 
\biggr]

 

~=

~
\frac{\partial}{\partial x}   \biggl[- A_1 x  ~+ A_{12} x y^2 + A_{13} x z^2 ~+ A_{11}x^3  \biggr]
+\frac{\partial}{\partial y}  \biggl[- A_2 y   ~+  A_{12} x^2y  + A_{23} y z^2~+ A_{22}y^3  \biggr]
+ \frac{\partial}{\partial z}\biggl[- A_3 z  ~+ A_{13} x^2z + A_{23} y^2z~+ A_{33}z^3  \biggr]

 

~=

~
\biggl[- A_1  + A_{12} y^2 + A_{13} z^2 ~+ 3A_{11}x^2  \biggr]
+ \biggl[- A_2 +  A_{12} x^2  + A_{23} z^2~+ 3A_{22}y^2  \biggr]
+ \biggl[- A_3  + A_{13} x^2 + A_{23} y^2~+ 3A_{33}z^2  \biggr]

 

~=

~ - (A_1 + A_2 + A_3) + x^2(3A_{11} + A_{12} + A_{13}) + y^2( 3A_{22} + A_{12} + A_{23}) + z^2( 3A_{33} + A_{13} + A_{23})\, .

In addition to recognizing, as stated above, that ~(A_1 + A_2 + A_3) = 2, and making explicit use of the relation,

~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}

~=

~\frac{2}{a_i} \, ,

this last expression can be simplified to discover that,

~\nabla^2 \biggl[ \frac{\Phi_\mathrm{grav}}{-2\pi G \rho_c} \biggr]

~=

~ - (2) + \frac{2x^2}{a_1^2} + \frac{2y^2}{a_2^2} + \frac{2z^2}{a_3^2}

~\Rightarrow ~~~ \nabla^2 \Phi_\mathrm{grav}

~=

~ 4\pi G \rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .

This does indeed demonstrate that the derived gravitational potential is consistent with our selected mass distribution in the case where n = 1, namely,

~\rho

~=

~  \rho_c \biggl[ 1 -  \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .

Q.E.D.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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