## Dynamics of resonant planets: The resonances 2:1, 3:2 and 5:2

Transactions of IAA RAS, issue 8, 47–50 (2002)

**Keywords**:
Very Long Baseline Interferometry (VLBI), celestial mechanics, resonant planets, resonances

### Abstract

This paper considers the planar problem of two planets orbiting a star with periods close to a commensurability. The problem is stated in Hamiltonian form, in heliocentric canonical variables, and is reduced to two degrees of freedom through the elimination of all non{critical terms involving the mean longitudes. In the adopted approximation, the reduced Hamiltonian includes the main resonant and secular terms up to fourth order in the eccentricities. In the case of first-order resonances (Callegari Jr. et al, 2002) the second-order critical terms are also kept in the model. Numerical tests have shown good agreement between the solutions of the model and the solutions of the correspondent exact equations of a two-planet system, at least as far as the eccentricities are kept small. In the domains studied here, the eccentricity of the planets generally remains below 0.05 in the resonances of higher α, reaching 0.1 only in those of lower (α is the ratio of the semimajor axes of the 2 planets.)

### Citation

`N. Callegari Jr., T. A. Michtchenko, S. Ferraz-Mello. Dynamics of resonant planets: The resonances 2:1, 3:2 and 5:2 // Transactions of IAA RAS. — 2002. — Issue 8. — P. 47–50.`

```
@article{jr.2002,
abstract = {This paper considers the planar problem of two planets orbiting a star with periods close to a commensurability. The problem is stated in Hamiltonian form, in heliocentric canonical variables, and is reduced to two degrees of freedom through the elimination of all non{critical terms involving the mean longitudes. In the adopted approximation, the reduced Hamiltonian includes the main resonant and secular terms up to fourth order in the eccentricities. In the case of first-order
resonances (Callegari Jr. et al, 2002) the second-order critical terms are also kept in the model. Numerical tests have shown good agreement between the solutions of the model and the solutions of the correspondent exact equations of a two-planet system, at least as far as the eccentricities are kept small. In the domains studied here, the eccentricity of the planets generally remains below 0.05 in the resonances of higher α, reaching 0.1 only in those of lower (α is the ratio of the semimajor axes of the 2 planets.)},
author = {N.~Callegari Jr. and T.~A. Michtchenko and S. Ferraz-Mello},
issue = {8},
journal = {Transactions of IAA RAS},
keyword = {Very Long Baseline Interferometry (VLBI), celestial mechanics, resonant planets, resonances},
pages = {47--50},
title = {Dynamics of resonant planets: The resonances 2:1, 3:2 and 5:2},
url = {http://iaaras.ru/en/library/paper/317/},
year = {2002}
}
```

```
TY - JOUR
TI - Dynamics of resonant planets: The resonances 2:1, 3:2 and 5:2
AU - Jr., N. Callegari
AU - Michtchenko, T. A.
AU - Ferraz-Mello, S.
PY - 2002
T2 - Transactions of IAA RAS
IS - 8
SP - 47
AB - This paper considers the planar problem of two planets orbiting a
star with periods close to a commensurability. The problem is stated
in Hamiltonian form, in heliocentric canonical variables, and is
reduced to two degrees of freedom through the elimination of all
non{critical terms involving the mean longitudes. In the adopted
approximation, the reduced Hamiltonian includes the main resonant and
secular terms up to fourth order in the eccentricities. In the case
of first-order resonances (Callegari Jr. et al, 2002) the second-
order critical terms are also kept in the model. Numerical tests have
shown good agreement between the solutions of the model and the
solutions of the correspondent exact equations of a two-planet
system, at least as far as the eccentricities are kept small. In the
domains studied here, the eccentricity of the planets generally
remains below 0.05 in the resonances of higher α, reaching 0.1 only
in those of lower (α is the ratio of the semimajor axes of the 2
planets.)
UR - http://iaaras.ru/en/library/paper/317/
ER -
```