Potential due to uniform ring

Consider a uniform ring of mass $M$, radius $a$, and negligible cross-sectional area, centered on the origin, and lying in the $x$-$y$ plane. Let us consider the gravitational potential ${\mit\Phi}(r)$ generated by such a ring in the $x$-$y$ plane (which corresponds to $\theta = 90^\circ$). It follows, from Section 3.4, that for $r>a$,

$${\mit\Phi}(r) = - \frac{G\,M}{a}\sum_{n=0,\infty} [P_n(0)]^2\left(\frac{a}{r}\right)^{n+1}.$$ (3.75)

However, $P_0(0)=1$, $P_1(0) = 0$, $P_2(0)=-1/2$, $P_3(0)=0$, $P_4(0)=3/8$, $P_5(0)=0$, $P_6(0)=-5/16$, $P(7)=0$, and $P_8(0)=35/128$ (Abramowitz and Stegun 1965b). Hence,

$${\mit\Phi}(r) = - \frac{G\,M}{r}\left[1 + \frac{1}{4}\left(\frac{... ...rac{a}{r}\right)^6+ \frac{1225}{16384}\left(\frac{a}{r}\right)^8+\cdots\right].$$ (3.76)

Likewise, for $r<a$,

$${\mit\Phi}(r) = - \frac{G\,M}{a}\sum_{n=0,\infty} [P_n(0)]^2\left(\frac{r}{a}\right)^{n},$$ (3.77)

giving

$${\mit\Phi}(r) = - \frac{G\,M}{a}\left[1 + \frac{1}{4}\left(\frac{... ...rac{r}{a}\right)^6+ \frac{1225}{16384}\left(\frac{r}{a}\right)^8+\cdots\right].$$ (3.78)