Tidal evolution for any rheological model using a vectorial approach expressed in Hansen coefficients

Abstract

We revisit the two-body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of motion are derived from the tidal potential developed in Fourier series expressed in terms of Keplerian elliptical elements, so that the variation of dissipation with amplitude and frequency can be examined. However, this method introduces multiple index summations and some orbital elements depend on the chosen frame, which is prone to confusion and errors. Here, we develop the quadrupole tidal potential solely in a series of Hansen coefficients, which are widely used in celestial mechanics and depend just on the eccentricity. We derive the secular equations of motion in a vectorial formalism, which is frame independent and valid for any rheological model. We provide expressions for a single average over the mean anomaly and for an additional average over the argument of the pericentre. These equations are suitable to model the long-term evolution of a large variety of systems and configurations, from planet–satellite to stellar binaries. We also compute the tidal energy released inside the body for an arbitrary configuration of the system.

Appendices

Hansen coefficients relations

From the definition of the Hansen coefficients (Eq. (48)), a number of recurrence relations can be obtained. In this work, we use the following ones (e.g. Challe and Laclaverie 1969; Giacaglia 1976):

$$\begin{aligned} X_k^{\ell ,-m}= & {} X_{-k}^{\ell ,m} , \end{aligned}$$

(154)

$$\begin{aligned} (1-e^2) \, X_k^{\ell ,m}= & {} X_k^{\ell +1,m} + \frac{e}{2} \left[ X_k^{\ell +1,m-1} + X_k^{\ell +1,m+1} \right] , \end{aligned}$$

(155)

$$\begin{aligned} \sqrt{1-e^2}\, k \, X_k^{\ell ,m}= & {} m (1-e^2) X_k^{\ell -2,m} + \frac{e \, \ell }{2 } \left[ X_k^{\ell -1,m-1} - X_k^{\ell -1,m+1} \right] \nonumber \\= & {} m \, X_k^{\ell -1,m} + \frac{e}{2} \left[ (m+\ell ) \, X_k^{\ell -1,m-1} + (m-\ell ) \, X_k^{\ell -1,m+1} \right] . \end{aligned}$$

(156)

The first relation (Eq. (154)) allows us to obtain the coefficients $X_k^{-3,-1}$ and $X_k^{-3,-2}$ from the coefficients $X_k^{-3,1}$ and $X_k^{-3,2}$, respectively (Table 1). It also allows us to obtain any other coefficient with $m<0$ from $X_k^{\ell ,m}$ with $m>0$. The second relation (Eq. (155)) with $\ell = -4$ provides

$$\begin{aligned} \begin{aligned} m=0 \quad \Rightarrow \quad (1-e^2) \, X_k^{-4,0}&= \frac{e}{2} \, X_k^{-3,-1} + X_k^{-3,0} + \frac{e}{2} \, X_k^{-3,1} , \\ m=1 \quad \Rightarrow \quad (1-e^2) \, X_k^{-4,1}&= \frac{e}{2} \, X_k^{-3,0} + X_k^{-3,1} + \frac{e}{2} \, X_k^{-3,2} , \\ m=2 \quad \Rightarrow \quad (1-e^2) \, X_k^{-4,2}&= \frac{e}{2} \, X_k^{-3,1} + X_k^{-3,2} + \frac{e}{2} \, X_k^{-3,3} . \end{aligned} \end{aligned}$$

(157)

Finally, from the last relation (Eq. (156)) with $\ell = -3$ we get

$$\begin{aligned} \begin{aligned} m=0 \quad \Rightarrow \quad \sqrt{1-e^2} \, k \, X_k^{-3,0}&= \frac{3}{2} e \, ( X_k^{-4,1} - \, X_k^{-4,-1}) , \\ m=1 \quad \Rightarrow \quad \sqrt{1-e^2} \, k \, X_k^{-3,1}&= 2 e \, X_k^{-4,2} + X_k^{-4,1} - e \, X_k^{-4,0} , \\ m=2 \quad \Rightarrow \quad \sqrt{1-e^2} \, k \, X_k^{-3,2}&= \frac{5}{2} e \, X_k^{-4,3} + 2 \, X_k^{-4,2} - \frac{1}{2} e \, X_k^{-4,1} . \end{aligned} \end{aligned}$$

(158)

Using these sets of relations, it is possible to express all Hansen coefficients appearing in this work solely as functions of $X_k^{-3,0}$, $X_k^{-3,1}$, and $X_k^{-3,2}$, using the following sequence:

$$\begin{aligned} \begin{aligned} X_k^{-4,3}&= \frac{1}{5e}\bigg [ e \, X_k^{-4,1} - 4 \, X_k^{-4,2} + 2 \sqrt{1-e^2} \, k \, X_k^{-3,2} \bigg ] , \\ X_k^{-3,3}&= \frac{1}{e} \bigg [ 2 (1-e^2) \, X_k^{-4,2} - 2 X_k^{-3,2} - e \, X_k^{-3,1} \bigg ] , \\ X_k^{-4,2}&= \frac{1}{2e} \bigg [ e \, X_k^{-4,0} - X_k^{-4,1} + \sqrt{1-e^2} \, k \, X_k^{-3,1} \bigg ] , \\ X_k^{-4,1}&= \frac{1}{1-e^2}\bigg [ \frac{e}{2} \, X_k^{-3,2} +X_k^{-3,1}+\frac{e}{2} \, X_k^{-3,0} \bigg ] , \\ X_k^{-4,0}&= \frac{1}{1-e^2} \bigg [ \frac{e}{2} \, X_k^{-3,-1} +X_k^{-3,0} +\frac{e}{2} \, X_k^{-3,1} \bigg ] . \end{aligned} \end{aligned}$$

(159)

Hansen coefficients combinations

We let

$$\begin{aligned} F_{+}= \left( \frac{r}{a} \right) ^\ell \mathrm {e}^{\mathrm {i}m \upsilon } , \quad \mathrm {and} \quad F_{-}= \left( \frac{r}{a} \right) ^\ell \mathrm {e}^{- \mathrm {i}n \upsilon } , \end{aligned}$$

(160)

with derivatives, respectively,

$$\begin{aligned} F_{+}'= & {} - \mathrm {i}\frac{d F_{+}}{d M} = m \sqrt{1-e^2}\left( \frac{r}{a} \right) ^{\ell -2} \mathrm {e}^{\mathrm {i}m v} - e \, \ell \left( \frac{r}{a} \right) ^{\ell -1} \frac{\mathrm {e}^{\mathrm {i}(m+1) \upsilon } - \mathrm {e}^{\mathrm {i}(m-1) \upsilon }}{2 \sqrt{1-e^2}} , \end{aligned}$$

(161)

$$\begin{aligned} F_{-}'= & {} \mathrm {i}\frac{d F_{-}}{d M} = n \sqrt{1-e^2}\left( \frac{r}{a} \right) ^{\ell -2} \mathrm {e}^{- \mathrm {i}n \upsilon } - e \, \ell \left( \frac{r}{a} \right) ^{\ell -1} \frac{\mathrm {e}^{-\mathrm {i}(n+1) \upsilon } - \mathrm {e}^{-\mathrm {i}(n-1) \upsilon }}{2 \sqrt{1-e^2}} . \end{aligned}$$

(162)

Using the definition of the Hansen coefficients (Eq. (48)), we have

$$\begin{aligned} \left\langle F_{+}F_{-}\right\rangle _M= & {} \sum _{k=-\infty }^{+\infty }X_k^{\ell ,m} X_k^{\ell ,n} = \left\langle \left( \frac{r}{a} \right) ^{2\ell } \mathrm {e}^{\mathrm {i}(m-n) \upsilon } \right\rangle _M = X_0^{2\ell ,m-n} , \end{aligned}$$

(163)

$$\begin{aligned} \left\langle F_{+}' F_{-}\right\rangle _M= & {} \sum _{k=-\infty }^{+\infty }k \, X_k^{\ell ,m} X_k^{\ell ,n} \nonumber \\= & {} \left\langle m \sqrt{1-e^2}\left( \frac{r}{a} \right) ^{2\ell -2} \mathrm {e}^{\mathrm {i}(m-n) \upsilon } - \frac{e \, \ell }{2 \sqrt{1-e^2}}\right. \nonumber \\&\left. \times \left( \frac{r}{a} \right) ^{2\ell -1} \left( \mathrm {e}^{\mathrm {i}(m-n+1) \upsilon } - \mathrm {e}^{\mathrm {i}(m-n-1) \upsilon } \right) \right\rangle _M \nonumber \\= & {} m \sqrt{1-e^2}X_0^{2\ell -2,m-n} - \frac{e \, \ell }{2 \sqrt{1-e^2}} \left( X_0^{2\ell -1,m-n+1} - X_0^{2\ell -1,m-n-1} \right) \nonumber \\= & {} \frac{m+n}{2} \sqrt{1-e^2}X_0^{2\ell -2,m-n} , \end{aligned}$$

(164)

$$\begin{aligned} \left\langle F_{+}' F_{-}' \right\rangle _M= & {} \sum _{k=-\infty }^{+\infty }k^2 \, X_k^{\ell ,m} X_k^{\ell ,n} \nonumber \\= & {} \left\langle \frac{\ell ^2 \, e^2}{4 (1-e^2)} \left( \frac{r}{a} \right) ^{2\ell -2} \left( 2 \mathrm {e}^{\mathrm {i}(m-n) \upsilon } - \mathrm {e}^{\mathrm {i}(m-n+2) \upsilon } - \mathrm {e}^{\mathrm {i}(m-n-2) \upsilon } \right) \right\rangle _M \nonumber \\&+ \left\langle (m-n) \, \ell \, \frac{e}{2} \left( \frac{r}{a} \right) ^{2\ell -3} \left( \mathrm {e}^{\mathrm {i}(m-n+1) \upsilon } - \mathrm {e}^{\mathrm {i}(m-n-1) \upsilon } \right) \nonumber \right. \\&\left. + m n \, (1-e^2) \left( \frac{r}{a} \right) ^{2\ell -4} \mathrm {e}^{\mathrm {i}(m-n) \upsilon } \right\rangle _M \nonumber \\= & {} \frac{\ell ^2 \, e^2}{4 (1-e^2)} \left( 2 X_0^{2l-2,m-n} - X_0^{2l-2,m-n+2} - X_0^{2l-2,m-n-2} \right) \nonumber \\&+ (m-n) \, \ell \, \frac{e}{2} \left( X_0^{2l-3,m-n+1} - X_0^{2l-3,m-n-1} \right) + m n \, (1-e^2) X_0^{2l-4,m-n} \nonumber \\= & {} \ell ^2 \left( 2 X_0^{2l-3,m-n} - X_0^{2l-2,m-n} \right) \nonumber \\&+ \left( \frac{\ell (m-n)^2}{2\ell -2} + m n - \ell ^2 \right) (1-e^2) X_0^{2l-4,m-n} ,\nonumber \\ \end{aligned}$$

(165)

where to simplify expression (164) we used equation (156), and to simplify expression (165) we used equations (155) and (156) together with (e.g. Giacaglia 1976):

$$\begin{aligned} (1-e^2)^2 X_k^{\ell ,m}= & {} \left( 1+\frac{e^2}{2}\right) X_k^{\ell +2,m} + e \left[ X_k^{\ell +2,m+1} + X_k^{\ell +2,m-1} \right] \nonumber \\&\quad + \frac{e^2}{4} \left[ X_k^{\ell +2,m+2} + X_k^{\ell +2,m-2} \right] . \end{aligned}$$

(166)