# Questions tagged [celestial-mechanics]

Celestial Mechanics is the branch of astronomy devoted to the study of the motion of the celestial bodies on the basis of the laws of gravitation.

104 questions with no upvoted or accepted answers
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### Regularization: What is so special about the Coulomb/Newtonian and harmonic potential?

I wanted to know if the procedure for regularization of the Coulomb potential outlined in Celletti (2003): Basics of regularization theory could be generalized to arbitrary polynomial potentials. So ...
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### Why does planetary spin affect the stability of orbits?

Reading about the Hill Sphere I notice that "the region of stability for retrograde orbits at a large distance from the primary, is larger than the region for prograde orbits at a large distance from ...
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### Confusion about Post-Newtonian orbital motion (Damour-Deruelle)

In their famous paper in 1985 (link), Damour&Deruelle describe the orbital motion for a binary system taking into account first-order post-Newtonian corrections (1PN). The solution is given in ...
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### Why Kepler problem is equivalent to a free particle on 4 dimensional sphere?

In trying to understand Laplace-Runge-Lenz vector, I read in Wikipedia that the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four dimensional hypersphere....
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### When are two object guaranteed to keep getting further and further away?

In a two-body problem, it is known (if I understand correctly) that if the specific orbital energy of the system is $\varepsilon \geq0$, then the objects must eventually escape each other. My question ...
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### Relativistic Corrections to orbital elements

Given a massive compact object $M$ and a smaller object $m$ orbiting it in an elliptical orbit where $M \gg m$, Newtonian gravity describes the orbital elements $(a, e, i, \omega, \Omega, T)$, such as ...
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### Is there a rigorous proof regarding the non-linear stability of the $L_4$ and $L_5$ Lagrange points?

I have found that many proofs regarding the stability of the $L_4$ and $L_5$ Lagrange points are based on linear approximations of the equations of motion near these points. However, from a dynamical ...
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### Harmonic and subharmonic orbits in central fields

Using Newton's theorem of revolving orbits one can easily obtain orbits for central forces containing inverse cube terms, such as $$F(r)=F_0(r)+\frac{(1-k^2)|B|}{r^3},$$ from known orbits for $F_0$. ...
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### Where I can find obliquity direction?

I can find axial tilt of planets easily, but that doesn't specify the direction of that tilt, i.e. planet's rotation axis may be anywhere in circle defined on a sphere by axial tilt value. And I can't ...
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### Find semi-minor axis of elliptical binary orbit given semi-major axis and mass ratio

I'm working on a simulation of binary stars. I'm using Kepler's Laws to solve for the angular velocity given the semi-major axis $a$ and the masses $m_1$ and $m_2$ of the two objects in an elliptical ...
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### Keplerian orbits are intersections of planes with the the future light-cone in Minkowski 4-space

Near the end of this essay, Baez comments: In 2011, Guowu Meng pointed out a remarkable fact to me. Orbits of the Kepler problem are precisely the intersections of 2-dimensional planes with this cone ...
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### A question about almost stationary orbits

For an orbit around the earth with period equal to a sidereal day with eccentricity $e=0$ and inclination $i=0$, the path on earth would be a single point on the equator. I have done some simulations ...
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### Orbital period of a satellite in Kerbal Space Program

Let me start off by saying I am really bad at maths and physics, but lately I have been trying to calculate the orbital period of a satellite I have put in orbit around a planet called Kerbin in a ...
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### Calculating stable orbital resonance and other factors into orbits

Firstly here is a brief summary of what I trying to accomplish. From a predetermined total solar system mass , star count in that system, per-star classification and mass I am building a system of ...
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### Is there any free open source planetary orbit simulator software?

I need a free open source planetary orbit simulator software that should be able to calculate orbit stability for a couple simple planetary systems. Basically, I need to calculate orbit durations for ...
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### Coexistence at Lagrange points

I was wondering how precise the location of an object in a Lagrange point needs to be to maintain stability, since it seems that several natural objects (asteroids) exists together in some of these ...
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### Reduced Three-body problem

I know that three-body problem in celestial mechanics can't be solved analytically in general. But suppose that the 3rd body is much smaller than the others, so it does not perturb their orbits. This ...
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### How to calculate orbital eccentricity from the ratio of satellites' velocities?

I have a problem with working out how to calculate the orbital eccentricity from the ratio of two satellites' linear velocities. We know that two satellites encircles the Earth in two elliptical ...
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### For two planets in a 2:1 mean motion resonance , where will their periastron and apastron points be?

I want to animate the changing orbits of planets when they enter a mean motion resonance. Using a 2:1 resonance, I want to show a low-mass inner planet and a low-mass outer planet being tugged by a ...
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### Law for simulation of solar system

I heard that from simulations we know that the solar system is stable for the next 400-10000 years. And I am wondering, if you want to simulate the solar system, which equations do you take? Normally ...
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### Tidal forces in the early solar system

I'm reading a book called "Gravity from the ground up" by Bernard Schutz. I don't understand this section from Investigation 13.3 on page 159, which discusses the formation of the solar ...
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### Orbital obliquity of the Earth

Consider that we have all the orbital parameters that characterize the Earth. How would one calculate the orbital obliquity of Earth? One could argue that since the rotation of Earth doesn't change ...
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### Latitude of Impact of a satellite

I'm currently studying orbital mechanics, and am tasked with finding the latitude of impact of a suborbital satellite, given its location and velocity in geocentric coordinates at some time. Assume ...
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### How can I obtain the position of the Earth on the orbit for given date?

I would like to be able calculate (or at least download some time-series data of) the position of the Earth on it's orbit at given date. For my purposes, it would be sufficient to account for the ...
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I am confused about the derivation of the action $S(x,\mathbf{J})$ for a harmonic oscillator as given at page 219 in "Galactic Dynamics", J.Binney-S. Tremaine, 2nd Ed. 2008. The part of the derivation ...