Calculation of gravitational potential

According to the analysis of Section 3.4, we can write

$$\rho(r,\theta)=\sum_{n=0,\infty}\rho_n(r)\,P_n(\cos\theta),$$ (D.8)

where

$$\rho_n(r)=(n+1/2)\int_0^\pi \rho(a)\,P_n(\cos\theta)\,\sin\theta\,d\theta.$$ (D.9)

Here, use has been made of Equation (D.5). Now, to first order in $\vert\epsilon\vert$, Equation (D.6) can be inverted to give

$$a = r\left[1+\frac{2}{3}\,\epsilon(r)\,P_2(\cos\theta)\right].$$ (D.10)

Hence, to the same order, Equation (D.9) gives

$$\rho_n(r)= (n+1/2)\int_0^\pi\left[\rho(r)+\frac{2}{3}\,r\,\frac{d... ...}{dr}\,\epsilon(r)\,P_2(\cos\theta)\right]P_n(\cos\theta)\,\sin\theta\,d\theta.$$ (D.11)

Making use of Equation (3.42), we deduce that, to first order in $\vert\epsilon\vert$,

$$\rho_0(r) =\rho(r),$$ (D.12)

$$\rho_2(r) =\frac{2}{3}\,r\,\frac{d\rho}{dr}\,\epsilon(r),$$ (D.13)

with all of the other $\rho_n(r)$ zero.

The analysis of Section 3.4, combined with the previous two equations, also implies that

$${\mit\Phi}(r,\theta)={\mit\Phi}_0(r)+ {\mit\Phi}_2(r)\,P_2(\cos\theta),$$ (D.14)

where

$${\mit\Phi}_0(r)=-4\pi\,G\left[\frac{1}{r}\int_0^r\rho(r')\,r'^{\,2}\,dr'+\int_r^\infty \rho(r')\,r'\,dr'\right],$$ (D.15)

and

$${\mit\Phi}_2(r)=-\frac{8\pi\,G}{15}\left[\frac{1}{r^{\,3}}\int_0^... ...r'^{\,5}\,dr'+r^{\,2}\int_r^\infty \frac{d\rho}{dr'}\,\epsilon(r')\,dr'\right].$$ (D.16)

Now, to first order in $\vert\epsilon\vert$, we can write

$$r(a,\theta)= a + \delta(a,\theta),$$ (D.17)

where

$$\delta(a,\theta) = -\frac{2}{3}\,a\,\epsilon(a)\,P_2(\cos\theta).$$ (D.18)

Substitution of Equation (D.17) into Equations (D.15) and (D.16), followed by an expansion to first order in $\vert\epsilon\vert$, yields

$${\mit\Phi}(a,\theta)= {\mit\Phi}_0'(a)+{\mit\Phi}_2'(a)\,P_2(\cos\theta),$$ (D.19)

where

$${\mit\Phi}_0'(a)= -4\pi\,G\left[\frac{1}{a}\int_0^a\rho(a')\,a'^{\,2}\,da'+\int_a^\infty \rho(a')\,a'\,da'\right],$$ (D.20)

and

$${\mit\Phi}_2'(a) =-\frac{8\pi\,G}{3}\left\{\frac{\epsilon(a)}{a}\int_0^a \rho(a')\... ...2}\,da'-\frac{1}{5\,a^{\,3}}\int_0^a \rho(a')\,d[\epsilon(a')\,a'^{\,5}]\right.$$

$$\left.\phantom{=}-\frac{a^{\,2}}{5}\int_a^\infty \rho(a')\,d[\epsilon(a')]\right\}.$$ (D.21)

Here, we have integrated the last two terms in curly brackets by parts.