The effects of deformation inertia (kinetic energy) in the orbital and spin evolution of close-in bodies

Abstract

The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boué et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is analyzed, and equations for the small inertia regime are proposed. The analysis of Love numbers shows that, independently of the rheology, deformation inertia can be neglected if the tide-forcing frequency is much smaller than the frequency of small oscillations of an ideal body made of a perfect (inviscid) fluid with the same inertial and gravitational properties of the original body. Finally, numerical integration of the full set of equations, which couples tide, spin and orbit, is used to evaluate the effect of inertia on the overall motion. The results are consistent with those obtained from the Love number analysis. The conclusion is that, from the point of view of orbital evolution of celestial bodies, deformation inertia can be safely neglected. (Exceptions may occur when a higher-order harmonic of the tide forcing has a high amplitude.)

Appendix: The small inertia limit for the Maxwell oscillator

In this appendix, we apply three steps of averaging to the Maxwell oscillator in Eq. (3) that can be written as (see Eq. 49):

$$\begin{aligned} \tau _1\tau _0^2\dddot{x}+\tau _0^2\ddot{x}+ \tau _2\dot{x} +x=\frac{F(t)}{\gamma }+\tau _1 \frac{\dot{F}(t)}{\gamma } \end{aligned}$$

(61)

or

$$\begin{aligned} \frac{\mu \eta }{\gamma \alpha }\dddot{x}+\frac{\mu }{\gamma }\ddot{x}+ \eta \frac{\alpha +\gamma }{\alpha \gamma }\dot{x} +x=\frac{F(t)}{\gamma }+\frac{\eta }{\alpha } \frac{\dot{F}(t)}{\gamma }. \end{aligned}$$

Let $t=s\tau _4$ define a nondimensional time s, where $\tau _4=\sqrt{\mu /(\alpha +\gamma )}$, and $x^\prime (s)=\frac{\hbox {d}}{\hbox {d}s}x(s)$. Then in the nondimensional time scale the equation for the Maxwell oscillator becomes

$$\begin{aligned} x^{\prime \prime \prime }+x^\prime +\sqrt{\varepsilon \frac{\tau _1}{\tau _2}} \left( \frac{\tau _2}{\tau _1}x^{\prime \prime }+x-H(t)\right) =0, \end{aligned}$$

where

$$\begin{aligned} H(t)=\frac{F(t)}{\gamma }+\tau _1 \frac{\dot{F}(t)}{\gamma },\quad \sqrt{\varepsilon }=\frac{\tau _0}{\tau _2}, \quad \tau _1=\frac{\eta }{\alpha },\quad \tau _2=\eta \frac{\gamma +\alpha }{\gamma \alpha }. \end{aligned}$$

Using the definitions

$$\begin{aligned} x^\prime =u,\quad x^{\prime \prime }=v=u^\prime ,\quad w=x+v, \end{aligned}$$

and $\tau _3=\eta /\gamma$, the third-order equation can be written as

$$\begin{aligned} \begin{aligned} u^\prime =&v\\ v^\prime =&-u-\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\left( \frac{\alpha }{\gamma }v+w-H(t)\right) \\ w^\prime =&-\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\left( \frac{\alpha }{\gamma }v+w-H(t)\right) \\ t^\prime =&\sqrt{\varepsilon \tau _1\tau _2}. \end{aligned} \end{aligned}$$

This equation can be explicitly solved when $\varepsilon =0$. The solution motivates the change of variables $(u,v,w,t)\rightarrow (p,q,w,t)$ where

$$\begin{aligned} \left( \begin{matrix} p \\ q\end{matrix}\right) = \left( \begin{matrix} \cos s &{} - \sin s \\ \sin s &{} \cos s\end{matrix}\right) \left( \begin{matrix} u \\ v\end{matrix}\right) . \end{aligned}$$

In the new variables, the equation becomes

$$\begin{aligned} \begin{aligned} p^\prime =&\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\sin (s) \left( \frac{\alpha }{\gamma }v+w-H(t)\right) \\ q^\prime =&-\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\cos (s)\left( \frac{\alpha }{\gamma }v+w-H(t)\right) \\ w^\prime =&-\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\left( \frac{\alpha }{\gamma }v+w-H(t)\right) \\ t^\prime =&\sqrt{\varepsilon \tau _1\tau _2}, \quad \text {where}\\ v(s)=&-\sin (s)p+\cos (s)q. \end{aligned} \end{aligned}$$

Using the definitions

$$\begin{aligned} z=\left( \begin{matrix} p \\ q\end{matrix}\right) ,\quad A(s)=\frac{1}{4}\left( \begin{matrix} \sin 2s &{} - \cos 2s \\ -\cos 2s &{} -\sin 2s\end{matrix}\right) , \quad \xi (s)=\left( \begin{matrix} \cos s \\ \sin s\end{matrix}\right) ,\quad \varGamma =\frac{\gamma }{\alpha }(-w+H(t)), \end{aligned}$$

this equation can be written as

$$\begin{aligned} \begin{aligned} z^\prime =&\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}}\left( -\frac{z}{2}+A^\prime z+\varGamma \xi ^\prime \right) \\ w^\prime =&\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}}\left( -\xi ^\prime \cdot z+\varGamma \right) \\ t^\prime =&\sqrt{\varepsilon \tau _1\tau _2}, \quad \text {where}\\ \frac{\hbox {d}}{\hbox {d}s}A(s)=&A^\prime (s)=\frac{1}{2}\left( \begin{matrix} \cos 2s &{} \sin 2s \\ \sin 2s &{} -\cos 2s\end{matrix}\right) , \quad \frac{\hbox {d}}{\hbox {d}s}\xi (s)= \xi ^\prime (s)=\left( \begin{matrix} -\sin s \\ \cos s\end{matrix}\right) . \end{aligned} \end{aligned}$$

The following change of variables $(z,w,t)\rightarrow (y,w,t)$ was obtained after two steps of averaging Hale (1980),

$$\begin{aligned} \begin{aligned}&z=y+(Ay+\varGamma \xi )\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}}+(cA^\prime y+\varLambda \xi ^\prime )\varepsilon \frac{\tau ^2_3}{\tau _1\tau _2},\quad \text {where}\\&c=\frac{1}{8}-\frac{\gamma }{2\alpha },\quad \varLambda =\left( -\varGamma +\tau _2\frac{\gamma }{\alpha }\dot{H}\right) \frac{\gamma }{\alpha }. \end{aligned} \end{aligned}$$

(62)

Applying this change of variables, using $\varGamma ^\prime =\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}(\xi ^\prime \cdot z+\frac{\alpha }{\gamma }\varLambda )$ and $\varLambda ^\prime =\{-\varGamma ^\prime +\sqrt{\varepsilon \frac{\tau _1}{\tau _2}}\tau _2^2\frac{\gamma }{\alpha }\ddot{H}\}\frac{\gamma }{\alpha }$, and averaging the resulting equations we obtain, after a long computation,

$$\begin{aligned} \begin{aligned} y^\prime&=\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}} \left( -\left\{ \frac{1}{2}+\varepsilon \frac{\tau _3}{\tau _2}c \right\} y+ \sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}} \left( \frac{1}{8}+\frac{\gamma }{2\alpha }\right) \left( \begin{matrix} 0 &{} -1 \\ 1 &{} 0\end{matrix}\right) y\right) +\mathscr {O}(\varepsilon ^2)\\ w^\prime&=\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}} \left( \varGamma -\varepsilon \frac{\tau _3^2}{\tau _1\tau _2} \varLambda \right) +\mathscr {O}(\varepsilon ^2). \end{aligned} \end{aligned}$$

In the original time t, and neglecting the correction term, these equations become

$$\begin{aligned} \begin{aligned} \tau _2 \dot{y}&=\frac{\tau _3}{\tau _1}\left( -\left\{ \frac{1}{2}+ \varepsilon \frac{\tau _3}{\tau _2}c \right\} y+\sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}} \left( \frac{1}{8}+\frac{\gamma }{2\alpha }\right) \left( \begin{matrix} 0 &{} -1 \\ 1 &{} 0\end{matrix}\right) y\right) \\ \tau _2 \dot{w}&=\frac{\tau _3}{\tau _1} \left( \varGamma -\varepsilon \frac{\tau _3^2}{\tau _1\tau _2} \varLambda \right) . \end{aligned} \end{aligned}$$

The second of these equations implies the first equation in (27).

Notice that $|y(t)|\rightarrow 0$ as $t\rightarrow \infty$, so after some transient where |y(t)| decay as $\exp [-t\tau _3/(2\tau _1\tau _2)]$ Eq. (62) implies

$$\begin{aligned} z=\varGamma \xi \sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}}+ \varLambda \xi ^\prime \varepsilon \frac{\tau _3^2}{\tau _1\tau _2}. \end{aligned}$$

This and the relation $x=w-v=w+\sin (s)p-\cos (s) q=w-\xi ^\prime (s)\cdot z$ imply

$$\begin{aligned} x=w-\xi ^\prime \cdot z=w-\varGamma \underbrace{\xi ^\prime \cdot \xi }_{=0} \sqrt{\varepsilon }\frac{\tau _3}{\sqrt{\tau _1\tau _2}}- \varLambda \underbrace{\xi ^\prime \cdot \xi ^\prime }_{=1} \varepsilon \frac{\tau _3^2}{\tau _1\tau _2} =w-\varepsilon \frac{\tau _3^2}{\tau _1\tau _2}\varLambda . \end{aligned}$$

This equation plus the definition of $\varLambda$ in Eq. (62) implies the second equation in (27).