In this Study we plane to calculate the gravitational potential generated by a ring bent into an elliptical mass distribution. In a first part we suppose that the mass distribution is homogeneous, after we take the case in which the mass distribution is inhomogeneous or anisotropic. The aim of this study is to generalise that of circular case and work out the behaviour of a test particle in the vicinity of that distribution. The rings around asteroids are recently discovered. This study will have an impact concerning the behaviour of a test particle moving around these distributions.

Keywords: potential–elliptic functions–gravitation, static case, dynamical case, anisotropic distribution, mass

In an earlier work Najid et al.,¹ we studied the gravitational potential generated by an inhomogeneous asteroid with a parabolic mass distribution. We established the close form of this potential. It allows us to study both, the static and dynamical case in which a test particle is near the distribution.

In another work Najid et al.,² we studied the case of a circular ring with an anisotropic distribution of mass. The mass distribution depends on the direction of observation. We manage to work out the problem in terms of elliptic functions.

In this work, we plane to generalise the case of ring by scrutinizing a more realistic case. In general, around all structures the rings are elliptical, and even with inhomogeneous or anisotropic mass distribution. In the literature we find studies like ellipsoids, material points and a segment of double material were used.^3,4 Fred et al.,⁵ 1982established the expressions for both the potential and the field of a disk. They suggested a formula of a ring. Anisotropic and inhomgeneous distributions are our new ideas.

We will express the potential dV in a point P(x,y) generated by the infinitesimal point M of mass dm in the plane of the ellipse (Figure 1)

Figure 1 Gravitational potential generated by elliptical distribution at plane.

with

$FM=r$ and $FP=\rho$ .
$\left(r,\theta \right)$ are the polar coordinates of M.
$\left(\rho ,\phi \right)$ are the polar coordinates of P.
$\overrightarrow{MP}=\overrightarrow{FP}-\overrightarrow{FM}$ .
$dm=\lambda rd\theta$ , $\lambda$ is the linear mass of the ellipse.

We deduce

$dV=-G\frac{\lambda rd\theta }{\sqrt{r^2+{\rho }^2-2\rho rcos\left(\theta -\phi \right)}}$

If we take the origin at the focus F, we are getting :

$r=\frac{a\left(1-e^2\right)}{1+ecos\theta }$

Where a is the semi axis of the ellipse and e is the eccentricity

$V=-Ga\left(1-e^2\right){\displaystyle \underset{\theta =0}{\overset{\theta =2\pi }{\int }}\frac{\lambda \left(\theta \right)d\theta }{\sqrt{a^2{\left(1-e^2\right)}^2+{\rho }^2{\left(1+ecos\theta \right)}^2-2\rho a\left(1-e^2\right)\left(1+ecos\theta \right)\text{cos}\left(\text{θ}-\text{φ}\right)}}}$

In this case r is given by (Figure 2)

$r=\frac{b}{\sqrt{1-e^{2}co{s}^2\theta }}$

Figure 2 Gravitational potential generated by elliptical distribution at origin.

b is the semi–minor axis of the ellipse.
In this case
$OM=r$ and $OP=\rho$
$\left(r,\theta \right)$ are the polar coordinates of M.
$\left(\rho ,\phi \right)$ are the polar coordinates of P
$\overrightarrow{MP}=\overrightarrow{OP}-\overrightarrow{OM}$

In this case we get :

$dV=-Gb\frac{\lambda \left(\theta \right)d\theta }{\sqrt{b^2+{\rho }^2\left(1-e^{2}co{s}^2\theta \right)-2b\rho \text{cos}\left(\theta -\phi \right)\sqrt{1-e^{2}co{s}^2\theta }}}$

Simplified solution–Particular case.2 In the case of a circular ring of radius a and $\lambda \left(\theta \right)={\lambda }_0\left(1+dco{s}^2\frac{\theta }{2}\right)$ . Where ${\lambda }_0$ and d are constants, in the plane xy the potential is given by:

$V=-\frac{4G{\lambda }_{0}a}{p{k}^2}\left[\left(k^2+d\right)K\left(k\right)-dE\left(k\right)\right]$

p:The largest distance between P and the ring.
K(k):The complete elliptic integral of first kind
E(k ): The complete elliuptic integral of second kind (Handbook of Mathematics)

$k:k^2=1-\frac{q^2}{p^2}$ .

q:The smallest distance between P and the ring.

The calculation of V depends on the expression of$\lambda \left(\theta \right)$ . According to this we expect to have an elliptical integral, in a close form. With V we study the behaviour of a test particle in the vicinity of the distribution.⁶

None.

Author declares there is no conflict of interest.

1. Najid NE, Elourabi EH, Zegoumou M. Potential generated by a massive inhomogeneous straight segment. Research in Astronomy and Astrophysics. 2011;1(3):345–352.
2. Najid NE, Zegoumou M, Elourabi EH. Dynamical behavior in the vicinity of a circular anisotropic ring. The Open Astronomy Journal. 2012;5:54–60.
3. Bartczak P, Breiter S. Double Material Segment as the Model of Irregular Bodies. Celestial Mechanics and Dynamical Astronomy. 2003;86(2):131–141.
4. Bartczak P, Breiter S, Jusiel P. Ellipsoids, material points and material segments. Celestial Mechanics and Dynamical Astronomy. 2006;96(1):31–48.
5. Krogh FT, NG EW, Snyder WV. The gravitational field of a disk. Celestial Mechanics. 1982;26(4):395–405.
6. Abramowitz M, Stegun IA. Handbook of Mathematical Functions. USA: National Bureau of Standards Applied Mathematics Series; 1965. p. 1–470.