# 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.

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### Conserved Quantities in the Two-Body Problem

I'm having trouble matching the number of conserved quantities in the two-body problem with the number of differential equations. We know that, considering positions and velocities in the two-body ...
1 vote
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### Does the integral of the vis-viva equation have any meaning?

If orbital speed of an elliptical orbit can be described by $$v(r) =\sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)},$$ then what would the meaning of its ($\mathrm{d}r$) integral be?
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### How do we know when the earth completes an orbit?

Two bodies in space always orbit their center of mass. So the relative motion of the Sun and the Earth happen in the same line, save for the rotation of the Sun. So, how do we measure The time taken ...
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### How much energy is required to remove Earth from it's orbit and exit the solar system under perfect conditions?

Ok, this is my first question on this site. But it's one I've been thinking about for a while. Say through whatever means, we place a device capable of generating thrust/ kinetic energy on the surface ...
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### Runge-Lenz vector when the earth is neither at the aphelion nor at the perihelion

Assuming the earth is either at the perihelion or at the aphelion, it is easy to see the Runge-Lenz (RL) vector is directed along the line joining the perihelion and aphelion. Since the RL vector is a ...
1 vote
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### How exactly does the Moon stabilizes Earth axial tilt?

There are many references regarding the Moon stabilizing the tilt of the Earth's rotational axis. I'd like to see some support for that claim, more than non-sequitur handwaving "Moon causes tides,...
1 vote
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### Proof for equal eccentricity in a binary star system

What is the proof that the orbits of two stars orbiting around a common center of mass have equal eccentricities? You can use: $m_1r_1 = m_2r_2,$ then say that $r_1= a(1+e_1)$, $r_2=a(1-e_2)$ and from ...
1 vote
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### Moon, Earth and the Sun [closed]

How to prove that the geometric locus of the points where the attractive forces of the Sun and the Moon are of equal intensity is a sphere of radius $r = \frac{R \sqrt{Mm}}{M-m }$, where $M$ is ...
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### Calculating mass of object by its orbit

Can you calculate the mass of an object by its own orbit around another object? I found many ways to calculate the mass of an object by measuring the trajectories of other objects orbiting it, but ...
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### How to prove that the phase curve for the effective potential $V_{\rm eff}(r)$ in the Kepler problem have inflection points?

Just as in the plot for effective potential $V_{\rm eff}(r)$, we get inflection point at $r = 3P/2$, $P$ being the conic parameter and given by $c^2/µ$, how can we find the inflection points on its ...
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### Derive Sun's trajectory from movement of two planets in a 2D plane

Consider a solar system with 1 sun and two planets revolving around the sun in a 2D Euclidean space. While time continues, the sun moves forward, while the two planets revolve around the sun (move up ...
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### How do the planets stay in their orbit? [duplicate]

The Sun has a strong gravity. The planets also have gravity. So they attract each other. But then why dont they go and mix up with the Sun? If it is the orbit of the planets or a pre-existing motion ...
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### Are moons always smaller than the planets they orbit?

I'm not a physicist, asking for knowledge. Is there any moon orbiting a planet, but bigger than that planet? If not, is it mathematically possible for a bigger object to orbit around a smaller object ...
1 vote
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### Deriving energy for elliptical orbit

So I wanted to derive the total energy for an elliptical orbit, $E = -GmM/2a,$ and while I was doing it, I ran into this hurdle. So at the closest point to the focus, the orbiting object is at a ...
1 vote
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### Can a planetary system orbit two black holes instead of one sun?

A comic book depicts an exoplanetary system in another galaxy that orbits two black holes instead of a sun. Or it was a life-bearing exoplanet that was in proximity of two black holes as if they were ...
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### How to build up correct numerical model for osculating orbit?

The situation is that I want to make a numerical model of rstricted 3-body problem (heavy body on a fixed position, one massive body with known law of motion and one lightweight whose motion I want to ...
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### Applications of Hamiltonian formalism in classical or celestial mechanics

I am looking for a reference (or just a brief explanation) to applications of the Hamiltonian formalism to classical mechanics, e.g. to planetary motion. In all known to me textbooks on classical ...
1 vote