The Spin–Orbit Resonances of the Solar System: A Mathematical Treatment Matching Physical Data

Abstract

In the mathematical framework of a restricted, slightly dissipative spin–orbit model, we prove the existence of periodic orbits for astronomical parameter values corresponding to all satellites of the Solar System observed in exact spin–orbit resonance.

Appendices

Appendix 1: Proof of Lemma 2.2

Proof

We first prove (14). Up to a rescaling we can prove (14) assuming $\Vert v'\Vert _{C^0}=1$. Assume by contradiction that

$$\begin{aligned} \Vert v\Vert _{C^0}=:c>\pi /2. \end{aligned}$$

Note that it is obvious that $c\le \pi$, since v has zero average and, therefore, must vanish at some point. Since |v| is a continuous periodic function it attains a maximum at some point; up to a translation we can assume that |v| attains its maximum in $-c$. In that case, multiplying by $-1$, we can also assume that $-c$ is a minimum, namely

$$\begin{aligned} \Vert v\Vert _{C^0}=c=-v(-c). \end{aligned}$$

Since $\Vert v'\Vert _{C^0}=1$, we get

$$\begin{aligned} v(t)\le -c +|t+c|, \quad \forall t\in [-2c,0] \end{aligned}$$

and, therefore,

$$\begin{aligned} v(0)\le 0,\ \ v(-2c)\le 0,\quad \int \limits _{-2c}^0 v\le -c^2. \end{aligned}$$

(46)

Since $\Vert v'\Vert _{C^0}=1$, we also get

$$\begin{aligned} v(t)\le \pi -c -|t-\pi +c|,\quad \forall \, t\in [0,2\pi -2c]. \end{aligned}$$

Then,

$$\begin{aligned} \int \limits _0^{2\pi -2c}v\le (\pi -c)^2. \end{aligned}$$

Combining with the last inequality in (46), we get

$$\begin{aligned} \int \limits _{-2c}^{2\pi -2c}v\le (\pi -c)^2-c^2=\pi (\pi -2c)<0, \end{aligned}$$

which contradicts the fact that $v$ has zero average, proving (14).

We now prove (15). Up to a rescaling we can prove (15) assuming $\Vert v''\Vert _{C^0}=1$. Assume by contradiction that

$$\begin{aligned} \Vert v\Vert _{C^0}=:c>\pi ^2/8. \end{aligned}$$

(47)

Up to a translation we can assume that $|v|$ attains maximum at $0$. In that case, multiplying by $-1$, we can also assume that $-c$ is a minimum, namely

$$\begin{aligned} \Vert v\Vert _{C^0}=c=-v(0). \end{aligned}$$

Since $\Vert v''\Vert _{C^0}=1$, we get

$$\begin{aligned} v(t)\le -c +t^2/2, \quad \forall t\in \mathbb R. \end{aligned}$$

Since v has zero average must exist $t_1 < 0 < t_2$

$$\begin{aligned} s.t. v(t_1) = v(t_2) = 0, v(t) < 0 \ \forall \ t \in (t_1, t_2), \end{aligned}$$

(48)

Moreover,

$$\begin{aligned} \hbox {and} \ \ t_1 \le - \sqrt{2 c}, t_2 \ge \sqrt{2 c}, t_2 - t_1 < 2 \pi . \end{aligned}$$

Since $v$ has zero average and is $2\pi$-periodic,

$$\begin{aligned} \int \limits _{t_2}^{2\pi +t_1} v= -\int \limits _{t_1}^{t_2}v \ge \frac{2}{3} (2c)^{3/2}. \end{aligned}$$

(49)

Set

$$\begin{aligned} a:=\pi + (t_1-t_2)/2 \end{aligned}$$

and note that

$$\begin{aligned} 0<a\le \pi -\sqrt{2c}<\pi /2, \quad a^2<2c \end{aligned}$$

(50)

by (48) and (47). Set

$$\begin{aligned} u(t):=v\left( t+\pi +(t_1+t_2)/2\right) . \end{aligned}$$

Note that $u\in \mathbb B\cap C^2$ and, by (48),

$$\begin{aligned} \Vert u\Vert _{C^0}=c,\quad \Vert u''\Vert _{C^0}=1,\quad \ u(-a)=u(a)=0,\quad \int \limits _{-a}^a u = \int \limits _{t_2}^{2\pi +t_1} v \mathop {\ge }\limits ^{(49)}\frac{2}{3} (2c)^{3/2}. \end{aligned}$$

Consider now the even function

$$\begin{aligned} w(t):=\frac{1}{2}(u(t)+u(-t)). \end{aligned}$$

Note that $w\in \mathbb B\cap C^2$ and

$$\begin{aligned} \Vert w\Vert _{C^0}\le c,\ \ \Vert w''\Vert _{C^0}\le 1,\ \ w(-a)=0,\ \ \int \limits _{-a}^0 w =\frac{1}{2} \int \limits _{-a}^{a} u \ge \frac{1}{3} (2c)^{3/2}. \end{aligned}$$

(51)

Set

$$\begin{aligned} z(t):=c-\frac{c}{a^2}t^2. \end{aligned}$$

We claim that

$$\begin{aligned} z(t)\ge w(t),\quad \forall \, -a\le t\le 0. \end{aligned}$$

(52)

Then,

$$\begin{aligned} \int \limits _{-a}^0 w\le \int \limits _{-a}^0 z= \frac{2}{3} ca \mathop {<}\limits ^{(50)} \frac{1}{3} (2c)^{3/2} \mathop {\le }\limits ^{(51)}\int \limits _{-a}^0 w, \end{aligned}$$

which is a contradiction.

Let us prove the claim in (52). Note that $z(-a)=w(-a)=0$. Assume by contradiction that there exists $\bar{t}\in [-a,0)$ such that

$$\begin{aligned} z(\bar{t})=w(\bar{t}),\quad z(t)\ge w(t),\quad \forall \, t\in [-a,\bar{t}],\quad z'(\bar{t})\le w'(\bar{t}). \end{aligned}$$

Then, since $\Vert w''\Vert _{C^0}\le 1$,

$$\begin{aligned} w(t)&\ge w(\bar{t}) +w'(\bar{t})(t-\bar{t}) -\frac{1}{2} (t-\bar{t})^2\\&\mathop {>}\limits ^{(50)} z(\bar{t}) +z'(\bar{t})(t-\bar{t}) -\frac{c}{a^2} (t-\bar{t})^2 =z(t) ,\quad \forall \, t\in (\bar{t},0]. \end{aligned}$$

Then,

$$\begin{aligned} w(0)>z(0)=c, \end{aligned}$$

which contradicts the first inequality in (51). This completes the proof of (15). $\square$

Appendix 2: Fourier Coefficients of the Newtonian Potential

Properties of the Fourier coefficients $\alpha _j$ of the Newtonian potential $f$, including Eq. (32), have been discussed, e.g., in Appendix 1 of¹⁷ Biasco and Chierchia (2009).

Here we provide a simple formula for the Fourier coefficients $\alpha _j$ of the Newtonian potential $f$ in (3) [compare (d) of §1, and (31)–(32)]; namely we prove that

$$\begin{aligned} \alpha _j= -\frac{1}{4 \pi } \int \limits _0^{2 \pi } \frac{1}{\rho ^2(w^2+1)^2} \left[ (w^4-6w^2+1) c_j (u) -4w(w^2-1)s_j (u) \right] \mathrm{d}u,\quad \end{aligned}$$

(53)

where $w=w(u;\mathbf{e}) :=\sqrt{\frac{1+\mathbf{e}}{1-\mathbf{e}}} \tan \frac{u}{2}$, $\rho =1-\mathbf{e}\cos u$, and

$$\begin{aligned} c_j(u):=\cos (ju-j\mathbf{e}\sin u),\quad s_j(u):=\sin (ju-j\mathbf{e}\sin u). \end{aligned}$$

Proof

If $z=\arctan w$, then

$$\begin{aligned} e^{2iz}=\frac{i- w }{w+i}=-\frac{(w-i)^2}{w^2+1}, \end{aligned}$$

(54)

so that if $w_\mathbf{e}(t):=w(u_{\mathbf{e}}(t),\mathbf{e})$ one has $f_\mathbf{e}= 2 \arctan w_\mathbf{e}$ and

$$\begin{aligned} G_\mathbf{e}&= -\frac{1}{2\rho _\mathbf{e}^3} \frac{(w_\mathbf{e}-i)^2}{(w_\mathbf{e}+i)^2} = -\frac{1}{2\rho _\mathbf{e}^3} \frac{(w_\mathbf{e}-i)^4}{(w_\mathbf{e}^2+1)^2}\\&= -\frac{1}{2\rho _\mathbf{e}^3} \frac{1}{(w_\mathbf{e}^2+1)^2} \left( w_\mathbf{e}^4-6w_\mathbf{e}^2+1 -4iw_\mathbf{e}(w_\mathbf{e}^2-1) \right) . \nonumber \end{aligned}$$

(55)

By parity properties, it is easy to see that the $G_j$’s are real, namely $G_j=\bar{G}_j$, so that

$$\begin{aligned} \alpha _j&= G_j = \frac{1}{2 \pi } \int \limits _0^{2 \pi } G(t) e^{-ijt} \, \mathrm{d}t = -\frac{1}{4 \pi } \int \limits _0^{2 \pi } \frac{e^{i2f_\mathbf{e}(t) - ijt}}{\rho _\mathbf{e}(t)^3} \, \mathrm{d}t \\&= -\frac{1}{4 \pi } \int \limits _0^{2 \pi } \frac{1}{\rho _\mathbf{e}^3(w_\mathbf{e}^2+1)^2} \left[ (w_\mathbf{e}^4-6w_\mathbf{e}^2+1)\cos (jt) -4w_\mathbf{e}(w_\mathbf{e}^2-1)\sin (jt) \right] \mathrm{d}t. \end{aligned}$$

Making the change of variable given by the Kepler equation (5), i.e., integrating from $t$ to $u=u_\mathbf{e}$ and setting $u_\mathbf{e}(t)' = \frac{1}{\rho _\mathbf{e}(t)}$, one gets (53). $\square$

Appendix 3: Small Bodies

In the Solar System, besides the 18 moons listed in Table 1 and Mercury, there are five other minor bodies with mean radius smaller than 100 km observed in 1:1 spin–orbit resonance around their planet: Phobos and Deimos (Mars), Amalthea (Jupiter), and Janus and Epimetheus (Saturn), as listed in Table 4.

Table 4 Physical data of minor bodies in 1:1 spin–orbit resonance

Besides being small, such bodies have also a quite irregular shape and only Janus and Epimetheus have good equatorial symmetry.¹⁸ Indeed, for these two small moons (and only for them among the minor bodies), our theorem holds as shown by the data reported in Table 5.¹⁹

Table 5 Check of the hypotheses of Proposition 1 for the small satellites in spin–orbit resonance