Decomposition of Tide Generating Potential

Let $r$ , $\theta$ , $\varphi$ be right-handed spherical coordinates in a non-rotating reference frame whose origin lies at the center of the planet, and whose symmetry axis coincides with the planetary rotation axis. Thus, the vector position of a general point is

$${\bf r} = r\,\sin\theta\,\cos\varphi\,{\bf e}_r + r\,\sin\theta\,\sin\varphi\,{\bf e}_\theta + r\,\cos\theta\,{\bf e}_\varphi,$$ (12.15)

where ${\bf e}_r=\nabla r/\vert\nabla r\vert$ , et cetera. Let the coordinates of the moon's center be $r'$ , $\theta'$ , $\varphi'$ . It follows that

$${\bf r}' = r'\,\sin\theta'\,\cos\varphi'\,{\bf e}_r + r'\,\sin\theta'\,\sin\varphi'\,{\bf e}_\theta + r'\,\cos\theta'\,{\bf e}_\varphi.$$ (12.16)

Hence, from Equation (12.5),

$$\cos\gamma = \cos\theta\,\cos\theta'+\sin\theta\,\sin\theta'\,\cos(\varphi-\varphi').$$ (12.17)

Now, according to the spherical harmonic addition theorem (Arfken 1985),

$$P_n(\cos\gamma) = P_n(\cos\theta)\,P_n(\cos\theta')$$ (12.18)

$$\phantom{=} + 2\sum_{m=1,n}\frac{(n-m)!}{(n+m)!}\,P_n^{\,m}(\cos\theta)\,P_n^{\,m}(\cos\theta')\,\cos[m\,(\varphi-\varphi')],$$ (12.19)

which implies that

$$P_2(\cos\gamma) = P_2^{\,0}(\cos\theta)\,P_2^{\,0}(\cos\theta')+ \frac{1}{3}\,P_2^{\,1}(\cos\theta)\,P_2^{\,1}(\cos\theta')\,\cos(\varphi-\varphi')$$

$$\phantom{=}+ \frac{1}{12}\,P_2^{\,2}(\cos\theta)\,P_2^{\,2}(\cos\theta')\,\cos\left[2\,(\varphi-\varphi')\right].$$ (12.20)

Here,

$$P_n^{\,m}(x)=(-1)^m\,(1-x^{\,2})^{m/2}\,\frac{d^{\,m}[P_n(x)]}{dx^{\,m}},$$ (12.21)

for $n\geq 0$ and $m\leq n$ , denotes an associated Legendre function (Abramowitz and Stegun 1965). In particular,

$$P_2^{\,0}(x) = \frac{1}{2}\,(3\,x^{\,2}-1),$$ (12.22)

$$P_2^{\,1}(x) = -3\,x\,(1-x^{\,2})^{1/2},$$ (12.23)

$$P_2^{\,2}(x) = 3\,(1-x^{\,2}).$$ (12.24)

Note that $P_n^{\,0}(x)=P_n(x)$ .

Let

$${\cal R}_n^{\,(-m)}(r,\theta,\varphi) = \left(\frac{r}{a}\right)^n P_n^{\,m}(\cos\theta)\,\cos(m\,\varphi),$$ (12.25)

$${\cal R}_n^{\,(0)}(r,\theta) = \left(\frac{r}{a}\right)^n P_n^{\,0}(\cos\theta),$$ (12.26)

$${\cal R}_n^{\,(+m)}(r,\theta,\varphi) = \left(\frac{r}{a}\right)^n P_n^{\,m}(\cos\theta)\,\sin(m\,\varphi),$$ (12.27)

where $n\geq m>0$ . According to Equations (12.14), (12.20), and (12.25)-(12.27),

$${\mit\Phi}_{\rm tide}(r,\theta,\varphi) = - \frac{G\,m'\,a^{\,2}}{r'^{\,3}}\left[P_2^{\,0}(\cos\theta')\,{\cal R}_2^{\,(0)}(r,\theta)\right.$$

$$\phantom{ - \frac{G\,m'\,a^{2}}{r'^{\,3}}} +\frac{1}{3}\,P_2^{\,1}\,(\cos\theta')\,\cos\varphi'\,{\cal R}_2^{\,(-1)}(r,\theta,\varphi)$$

$$\phantom{ - \frac{G\,m'\,a^{2}}{r'^{\,3}}}+\frac{1}{3}\,P_2^{\,1}\,(\cos\theta')\,\sin\varphi'\,{\cal R}_2^{\,(+1)}(r,\theta,\varphi)$$

$$\phantom{ - \frac{G\,m'\,a^{2}}{r'^{\,3}}} +\frac{1}{12}\,P_2^{\,2}\,(\cos\theta')\,\cos(2\,\varphi')\,{\cal R}_2^{\,(-2)}(r,\theta,\varphi)$$

$$\phantom{ - \frac{G\,m'\,a^{2}}{r'^{\,3}}}\left.+ \frac{1}{12}\,P... ...(\cos\theta')\,\sin(2\,\varphi')\,{\cal R}_2^{\,(+2)}(r,\theta,\varphi)\right].$$ (12.28)

Here, we have neglected the unimportant constant term in Equation (12.14).