Исследование динамической эволюции компактной планетной системы KEPLER-51

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Abstract

В работе исследуется динамическая эволюция компактной трехпланетной системы Kepler-51. Анализируются возможные резонансные состояния системы и проводится поиск потенциальных цепочек резонансов средних движений. С помощью программного комплекса Posidonius исследуется динамическая эволюция системы на интервале времени 100 млн лет с учетом приливного взаимодействия. Также для различных начальных значений эксцентриситетов, наклонов, аргументов перицентров и долгот восходящих узлов орбит проводится моделирование динамической эволюции планетной системы в рамках численно-аналитической теории движения. Показано, что компактная планетная система Kepler-51 не является резонансной. При начальных условиях, соответствующих массам и элементам орбит планет, определенным из наблюдений с учетом их погрешностей, эволюция системы является устойчивой и регулярной на исследуемом интервале 100 млн лет.

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About the authors

Э. Д. Кузнецов

Уральский федеральный университет

Author for correspondence.
Email: eduard.kuznetsov@urfu.ru
Russian Federation, Екатеринбург

А. С. Перминов

Уральский федеральный университет

Email: perminov12@yandex.ru
Russian Federation, Екатеринбург

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Supplementary files

Supplementary Files
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2. Fig. 1. Distribution of low-order resonance regions (up to and including the fourth order) in the Kepler-51 system, taking into account errors in determining the masses of the planets and the eccentricities of their orbits.

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3. Fig. 2. Evolution of the orbital elements of the planets (1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d) over an interval of 100 million years for different values ​​of the stellar mass M: (a) – major semi-axes at , (b) – major semi-axes at , (c) – eccentricities at , (d) – eccentricities at .

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4. Fig. 3. Evolution of the relationships between the orbital periods of the planets (blue dots): (a) Kepler-51 c and Kepler-51 b; (b) Kepler-51 d and Kepler-51 c. The red dotted lines correspond to the resonances of the mean motions.

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5. Fig. 4. Evolution of the Kepler-51 system in the region of three-body resonances – chains of two-body resonances of mean motions. The red lines correspond to the position of three-body resonances. The system evolves from left to right.

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6. Fig. 5. Evolution of the mean orbital elements of the planets over an interval of 100 million years for zero initial values ​​of orbital inclinations and longitudes of the ascending nodes: 1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d.

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7. Fig. 6. Evolution of the mean elements of planetary orbits over an interval of 100 thousand years for zero initial values ​​of orbital inclinations and longitudes of ascending nodes: 1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d.

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8. Fig. 7. Evolution of the average eccentricities of the planetary orbits (1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d) over an interval of 100 million years for the corresponding initial values ​​of the osculating elements of the orbits: I1 = 0.9°, I2 =I3 = 0°, Ω2 = Ω3 = 0°, arguments of pericenters are nominal, orbital eccentricities are maximum, Ω1 varies in increments of 30°.

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9. Fig. 8. Evolution of the average orbital inclinations of the planets (1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d) over an interval of 100 million years for the corresponding initial values ​​of the osculating elements of the orbits: I1 = 0.9°, I2 =I3 = 0°, Ω2 = Ω3 = 0°, arguments of pericenters are nominal, orbital eccentricities are maximum, Ω1 varies in 30° increments.

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10. Fig. 9. Evolution of the average eccentricities of the planetary orbits (1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d) over an interval of 100 million years for the corresponding initial values ​​of the osculating elements of the orbits: I2 = 0.6°, I1 =I3 = 0°, Ω1 = Ω3 = 0°, arguments of pericenters are nominal, orbital eccentricities are maximum, Ω2 varies in 30° increments.

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11. Fig. 10. Evolution of the average orbital inclinations of the planets (1 – Kepler-51 b, 2 – Kepler-51 c, 3 – Kepler-51 d) over an interval of 100 million years for the corresponding initial values ​​of the osculating elements of the orbits: I2 = 0.6°, I1 =I3 = 0°, Ω1 = Ω3 = 0°, arguments of pericenters are nominal, orbital eccentricities are maximum, Ω2 varies in 30° increments.

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12. Fig. 11. Maps of minimum (left column) and maximum (right column) values ​​of planetary orbital eccentricities over a time interval of 100 million years (NU: orbital eccentricities are nominal, I1 = 0.9°, I2 = 0.6°, I3 = 0°, g1 = 252.1, g2 = 322.9°, g3 = 263.5°): (a) – minimum achievable values ​​of orbital eccentricity of planet Kepler-51 b; (b) – maximum achievable values ​​of orbital eccentricity of planet Kepler-51 b; (c) – minimum achievable values ​​of orbital eccentricity of planet Kepler-51 c; (d) – maximum achievable values ​​of orbital eccentricity of planet Kepler-51 c; (d) – minimum achievable values ​​of the orbital eccentricity of the planet Kepler-51 d; (e) – maximum achievable values ​​of the orbital eccentricity of the planet Kepler-51 d.

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13. Fig. 12. Maps of the minimum (left column) and maximum (right column) values ​​of the planetary orbital inclinations over a time interval of 100 million years (NU: orbital eccentricities are nominal, I1 = 0.9°, I2 = 0.6°, I3 = 0°, g1 = 252.1, g2 = 322.9°, g3 = 263.5°): (a) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 b; (b) – maximum achievable values ​​of the orbital inclination of the planet Kepler-51 b; (c) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 c; (d) – maximum achievable values ​​of the orbital inclination of the planet Kepler-51 c; (d) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 d; (e) – the maximum achievable values ​​of the orbital inclination of the planet Kepler-51 d.

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14. Fig. 13. Maps of minimum (left column) and maximum (right column) values ​​of planetary orbital eccentricities over a time interval of 100 million years (NU: orbital eccentricities are nominal, I1 = 0.9°, I2 = 0.6°, I3 = 0°, g1 = 252.1, g2 = 269.4°, g3 = 316.9°): (a) – minimum achievable values ​​of orbital eccentricity of planet Kepler-51 b; (b) – maximum achievable values ​​of orbital eccentricity of planet Kepler-51 b; (c) – minimum achievable values ​​of orbital eccentricity of planet Kepler-51 c; (d) – maximum achievable values ​​of orbital eccentricity of planet Kepler-51 c; (d) – minimum achievable values ​​of the orbital eccentricity of the planet Kepler-51 d; (e) – maximum achievable values ​​of the orbital eccentricity of the planet Kepler-51 d.

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15. Fig. 14. Maps of the minimum (left column) and maximum (right column) values ​​of the planetary orbital inclinations over a time interval of 100 million years (NU: orbital eccentricities are nominal, I1 = 0.9°, I2 = 0.6°, I3 = 0°, g1 = 252.1, g2 = 269.4°, g3 = 316.9°): (a) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 b; (b) – maximum achievable values ​​of the orbital inclination of the planet Kepler-51 b; (c) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 c; (d) – maximum achievable values ​​of the orbital inclination of the planet Kepler-51 c; (d) – minimum achievable values ​​of the orbital inclination of the planet Kepler-51 d; (e) – the maximum achievable values ​​of the orbital inclination of the planet Kepler-51 d.

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