Dynamics of the 3/1 planetary mean-motion resonance: an application to the HD60532 b-c planetary system

Abstract

In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance. The case where the outer planet is more massive than its inner companion is considered. We show that the resonant dynamics can be described using two fundamental parameters, the total angular momentum and the spacing parameter. The topology of the Hamiltonian function describing the resonant behaviour is investigated on a large domain of the phase space without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The families of the Apsidal Corotation Resonances (ACR) parameterized by the planetary mass ratio are obtained and their stability is analyzed. The main dynamical features in the domains around the ACR are also investigated in detail by means of spectral analysis techniques, which allow us to detect the regions of different regimes of motion of resonant systems. The construction of dynamical maps for various values of the total angular momentum shows the evolution of domains of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems.

Appendix

The explicit expressions for the coefficients of the analytical first-order expansion of the Hamiltonian (10) of the 3/1 MMR:

$$\begin{aligned} A= & {} - \dfrac{M_1}{2 J_1^3}+\dfrac{3}{2}\dfrac{M_2}{2 J_2^3}, \qquad \qquad B= - \dfrac{3}{8}\left( \dfrac{1}{2}\right) ^2\dfrac{M_1}{J_1^4}, \nonumber \\ C= & {} -c_2 \dfrac{M_3}{J_1 J_2^2}+\dfrac{3}{2}c_1 \dfrac{M_1}{J_2^3}, \qquad D =- c_3 \dfrac{M_3}{J_2^3}+\dfrac{3}{2}c_1 \dfrac{M_3}{J_2^3},\nonumber \\ E= & {} -c_4\dfrac{M_3}{J_2^2 \sqrt{J_1 J_2}}, \qquad \qquad \, F_1 = -\dfrac{1}{2}c_5\dfrac{M_3}{J_2^2 J_1},\\ F_2= & {} -\dfrac{1}{2}c_6\dfrac{M_3}{J_2^3},\qquad \qquad \qquad F_3 = \dfrac{1}{2}c_7\dfrac{M_3}{J_2^2\sqrt{J_1 J_2}},\nonumber \end{aligned}$$

(12)

where $M_1 = \mu _1^2 m_1^{\prime 3}$, $M_2 = \mu _2^2 m_2^{\prime ^3}$ and $M_3 = m_1 M_2/m_0$, while $c_i$ are the Laplace coefficients:

$$\begin{aligned} c_0= & {} -\dfrac{1}{2}\alpha b_{\frac{3}{2}}^{\left( 1\right) }\left( \alpha \right) ,\nonumber \\ c_1= & {} \dfrac{1}{2}b_{\frac{1}{2}}^{\left( 0\right) }\left( \alpha \right) ,\nonumber \\ c_2= & {} \dfrac{1}{8}\left[ 2\alpha D_\alpha +\alpha ^2 D_\alpha ^2\right] b_{\frac{1}{2}}^{\left( 0\right) },\\ c_4= & {} \dfrac{1}{4}\left[ 2-2\alpha D_\alpha -\alpha ^2 D_\alpha ^2\right] b_{\frac{1}{2}}^{\left( 1\right) },\nonumber \\ c_5= & {} \dfrac{1}{8}\left[ 21+10\alpha D_\alpha +\alpha ^2 D_\alpha ^2\right] b_{1/2}^{\left( 3\right) },\nonumber \\ c_6= & {} \dfrac{1}{4}\left[ -20-10\alpha D_\alpha -\alpha ^2 D_\alpha ^2\right] b_{1/2}^{\left( 2\right) },\nonumber \\ c_7= & {} \dfrac{1}{4}\left[ 17+10\alpha D_\alpha +\alpha ^2 D_\alpha ^2\right] b_{1/2}^{\left( 1\right) }-\dfrac{27}{8}\alpha ,\nonumber \end{aligned}$$

(13)

where $D_\alpha ^n$ are n-th order derivative in $\alpha =a_1/a_2$. The coefficients $b_i$ are obtained by the series:

$$\begin{aligned} \dfrac{1}{2}b_s^{\left( j\right) }\left( \alpha \right)= & {} \dfrac{s\left( s+1\right) \cdots \left( s+j-1\right) }{1\cdot 2\cdot 3\cdots j}\alpha ^j\\&\times \left[ 1+\dfrac{s\left( s+j\right) }{1\left( j+1\right) }\alpha ^2+\dfrac{s\left( s+1\right) \left( s+j\right) \left( s+j+1\right) }{1\cdot 2\left( j+1\right) \left( j+2\right) }\alpha ^4+ \cdots \right] .\nonumber \end{aligned}$$

(14)

when $j=0$,

$$\begin{aligned} \dfrac{s\left( s+1\right) \cdots \left( s+j-1\right) }{1\cdot 2\cdot 3\cdots j}\alpha ^j= 1, \end{aligned}$$

(15)

and the series is convergent for $\alpha <1$.