Three-Body Expansion
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Three-Body Expansion Documentation

This is a joint work with U. Locatelli and A. Giorgilli.

Introduction

The expansion of the Hamiltonian of the three-body is a common topic in Celestial Mechanics.

The program threebody_plane.c (see main) computes the expansion of the Hamiltonian of the three-body in Poincare' canonical heliocentric variables.

For sake of completeness, we report here a brief description of the main issues. All the details can be found in any books about Celestial Mechanics (see the References).

Orbital elements

The orbital elements describe uniquely a specific orbit. These elements are the ones generally considered in classical two-body systems, where a Kepler orbit is used.

The traditional orbital elements are the six Keplerian elements, after J. Kepler and his laws of planetary motion,

The mean anomaly, $l$, is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time,

\[ l = nt = \sqrt{\frac{ G( m_0 + m ) } {a^3}} t\,. \]

The relation between the mean anomaly and the true anomaly is not trivial. To transform an expansion in the true anomaly, into one in the mean anomaly (needed later), we use an algorithm by J. Henrard (see [1]).

The idea is to trasnform a function

\[ F(f,e) = \sum_{i\geq0} \frac{F_i^0}{i!}e^i\,, \]

into

\[ G(l,e) = F(f(l,e),e) = \sum_{i\geq0} \frac{F_0^i}{i!}e^i\,. \]

The algorithm by Henrard uses the Lie transforms method, and the generator is

\[ W(f,e) = \frac{\partial f}{\partial e} = \frac{4\sin(f) + \sin(2f)e}{2(1-e^2)} = \sum_{i\geq0} \frac{W_i}{i!}e^i\,. \]

The transformed function is computed by the recursive formula

\[ F_{i-j}^{j} = F_{i-j+1}^{j-1} + \sum_{k=0}^{i-j} {i-j \choose k} \langle\nabla F_{i-j-k}^{j-1},W_{k}\rangle\,. \]

Poincare' canonical heliocentric variables (planar case)

\begin{align*} \Lambda &= \beta\sqrt{\mu a}\,,& \xi &= \phantom{-}\sqrt{2\Lambda}\sqrt{1-\sqrt{1-e^2}}\cos(\omega)\,,\\ \lambda &= l+\omega\,,& \eta &= -\sqrt{2\Lambda}\sqrt{1-\sqrt{1-e^2}}\sin(\omega)\,, \end{align*}

where $\mu=G(m_0+m_1)$ and $\beta=\frac{m_0 m}{m_0+m}$.

It also useful to define the quantity

\[ |X| = \sqrt{2} \sqrt{1-\sqrt{1-e^2}}\,, \]

that is related to the modulus of the secular Poincare' canonical heliocentric variables

\[ \xi^2+\eta^2 = \Lambda |X|^2\,. \]

The Hamiltonian

The Hamiltonian of the three-body problem can be written as

\[ H = H_0 + H_1\,, \]

where $H_0$ is the unperturbed part and $H_1$ is the perturbation. Moreover, the perturbation can be written as the sum of the potential energy, $U_1$, and the kinetic energy, $T_1$,

\begin{align*} H_0&=-\sum_{i=1}^{2}\frac{\mu_i^2 \beta_i^3}{2\Lambda_i^2}\,,\\ U_1&=-G\sum_{0<i<j<2}\frac{m_i m_j}{\Delta_{ij}}\,,\\ T_1&=\sum_{0<i<j<2}\frac{{\bf \widetilde{r}}_i \cdot {\bf \widetilde{r}}_j}{m_0}\,. \end{align*}

The integrable part

\[ H_0=-\sum_{i=1}^3 \frac{\mu_i^2 \beta_i^3}{2 \Lambda_i^2}\,, \]

The expansion is simple,

\begin{align*} -\frac{\mu^2 \beta^3}{2 \Lambda^2}&=-\frac{\mu^2 \beta^3}{2\left(L+\Lambda^*\right)^2}= -\frac{\mu^2 \beta^3}{2 {\Lambda^*}^2} \frac{1}{\left( 1+\frac{L}{\Lambda^*} \right)^2}\\ &=-\frac{m_0 m}{2 a^*}\sum_{k=0}^{\infty} \left(-\frac{1}{\Lambda^*}\right)^k (k+1) L^k\,. \end{align*}

The perturbation function

\[ U_1=-G\sum_{0<i<j<2}\frac{m_i m_j}{\Delta_{ij}}\,. \]

For small values of the eccentricity and inclinations, we can introduce the small quantity

\[ \Xi=\|r_1-r_2\|^2-\left( a_1^2 + a_2^2 - 2 a_1 a_2 \cos(\lambda_1-\lambda_2) \right)\,. \]

We can now write

\[ U_1=\frac{G m_1 m_2}{a_2^*} \sum_{k=0}^{\infty}\frac{(-1)^k (\frac{1}{2})_k}{(1)_k} \left(\frac{\Xi}{{a_2^*}^2}\right)^k \left(\frac{a_2^*}{a_2}\right)^{2k+1} D^{-(2k+1)}\,, \]

where $D^{-(2k+1)}$ is defined by

\[ D^2=1+\left(\frac{a_1}{a_2}\right)^2-2\left(\frac{a_1}{a_2}\right)\cos(\lambda_1-\lambda_2)\,. \]

and the symbols $(a)_k$ are the Pochhammer symbols, defined as

\[ (a)_0 = 1,\qquad (a)_k = (a+k-1) (a)_{k-1}\,. \]

This expansion is easy to compute, except for the terms $D^{-(2k+1)}$. To complete the scheme we use the expansion in Laplace coefficients,

\[ D^{-(2k+1)} = \sum_{j=-\infty}^{\infty} \frac{1}{2} b_{k+\frac{1}{2}}^{(|j|)}(\alpha)\cos(j\lambda_1-j\lambda_2)\,. \]

The Laplace coefficients can be computed as

\begin{align*} \frac{1}{2} b_{i+\frac{1}{2}}^{(j)}(\alpha) &= \frac{(i+\frac{1}{2})_{j}}{(1)_{j}} \alpha^{j} F(i+\frac{1}{2},i+\frac{1}{2}+j,j+1;\alpha^2)\,. \end{align*}

The hypergeometric function can be computed as

\[ F(a,b,c;x) = \sum_{i\geq0} \frac{(a)_i(b)_i}{(c)_i} \frac{x^i}{(1)_i}\,. \]

and moreover we have the following relation

\begin{align*} F(a,b,c;x+y) =& \sum_{i\geq0} \frac{(a)_i(b)_i}{(c)_i} \frac{y^i}{(1)_i} F(a+i,b+i,c+i;x)\,. \end{align*}

The kinetic part

\begin{align*} T_1=-\frac{\beta_1 n_1 a_1 \beta_2 n_2 a_2}{m_0 \sqrt{1-e_1^2} \sqrt{1-e_2^2}}\ &\Big[ \left(\sin(f_1+\omega_1) + e_1\sin(\omega_1)\right) \left(\sin(f_2+\omega_2) + e_2\sin(\omega_2)\right)\\ &+\left(\cos(f_1+\omega_1) + e_1\cos(\omega_1)\right) \left(\cos(f_2+\omega_2) + e_2\cos(\omega_2)\right)\\ &-\left(\cos(f_1+\omega_1) + e_1\cos(\omega_1)\right) \left(\cos(f_2+\omega_2) + e_2\cos(\omega_2)\right)\left(1-\cos(J)\right) \Big]\,. \end{align*}

Important expansion

Binomial expansion

\[ (1 + x)^\alpha = \sum_{k=0}^{\infty} {\alpha \choose k} x^k \]

Trigonometric expansion

\[ \cos(nx)+i \, \sin(nx)= (\cos(x)+i\sin(x))^n \,, \]

from which we have

\begin{align*} \cos(nx)&= \Re\Big((\cos(x)+i\sin(x))^n \Big)\\ &=\sum_{\substack {k=0\\k {\rm\ even}}}^{n} \binom{n}{k}(-1)^{k/2}\cos^{n-k}(x)\,\sin^k(x)\\ \\ \sin(nx)&= \Im\Big((\cos(x)+i\sin(x))^n \Big)\\ &=\sum_{\substack {k=0\\k {\rm\ odd}}}^{n} \binom{n}{k}(-1)^{(k-1)/2}\cos^{n-k}(x)\,\sin^k(x)\\ \end{align*}

References