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Questions tagged [celestial-mechanics]

Use this tag for questions about the branch of astronomy dealing with motions of celestial objects.

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Two Body Problem with differential equations?

I have been learning up on the two body problem that uses differential equations to solve for a equation of a satellite relative to the planet. The two sources (linked below) both use different ...
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Different kinds of stability that apply to planar periodic orbits, and what do they mean?

This is a question about terminology related to orbit stability. I had wanted to ask about stability of orbits described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits ...
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Symmetry of a planar system and product of inertia

I read the following sentence here: "The products of inertia occupy the off-diagonal positions and measure the asymmetry of the mass distribution with respect to the planes of the inertia frame of ...
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Moon and relative inclination between planes

I had read that the moon was inclined relative to the sun at 5 degrees from it and with respect to earths equatorial it plane; the ecliptic would have inclination 23.5 degrees with longitude of ...
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Object Intercept in Space question

I have a formula, which given a time $t$, returns the $x$ and $y$ position of a planet, essentially it fakes a planet orbiting a star. I have a ship that is stationary with respect to the planet, ...
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Derivation of Vis-Viva Equation/Total Orbital Energy

I'm studying for my Astrodynamics course, however 1 step in the derivation of the Vis-Viva equation I don't understand. The way the equation is derived is by taking the dot-product of the trajectory ...
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Reparametrization of ellipse with constant trajectory “speed”. [duplicate]

One can derive a parametrization for ellipse in polar coordinates (origo at one of the focal points) $$\varphi(t) = ct$$ $$r(\varphi) = \frac {k+1}{k+\cos(\varphi)} $$ where for width of ellipse $w$: ...
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Seeking an example of potential.

Consider a compact Lie group (the dimension of the group is greater than 1) and a representation $R^n$ of the group G. I am seeking an example of potential $p: R^n \rightarrow R$ which is invariant ...
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Meaningful interpretation of an integral

Consider the following integral, and it's closed forms: $$\displaystyle \int_{\mathbb{R}} \frac{\tan^{-1}\left(\frac{\sqrt{x^2+a^2}}{b}\right) \, \text{d}x}{(x^2+b^2) \left( \frac{\sqrt{x^2+a^2}}{b} \...
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56 views

Reference request for Arnold Diffusion

I'm trying to understand Arnold diffusion from its original paper: Instability of dynamical systems with several degrees of freedom by V.I. Arnold. Is there any book where this topic is detailed or ...
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Double Fourier Series (external satellite resonance)

I'm working through a paper "Dynamics of Planetary Rings" by Goldreich and Tremaine (http://www.annualreviews.org/doi/pdf/10.1146/annurev.aa.20.090182.001341). I'm working through p.22 about expanding ...
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Poincare's last geometric theorem

Which problem in celestial mechanics led Poincare to his conjecture about number of fixed points of area preserving maps of the annulus?
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circular movement of 3 bodies

Im trying to find the $(x,y)$ coordinate of 3 bodies which move in circle - for example (sun > earth > moon). Assume sun is always at (0,0). Earth orbit around the sun. Moon orbit around earth. (no ...
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Satellite elevation angles negative

Until recently, the code I used to calculate the elevation and azimuth of a satellite from one particular site appeared to work. Then I used a different file from a site, not in the northern but in ...
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A celestial topology?

I recently asked for natural topologies on the set of lines in $\mathbb R^2$. Now I'm aiming for a similar question on the set $S_p$ of conic sections in $\mathbb R^2$ sharing the same focus $p$ (but ...
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101 views

Mcgehee transformation, conversion to polar coordinates and blowing up the singularity

I am looking for any reference on the above topics as I am struggling to convert the below to polar coordinates in phase space: The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{...
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Distance to the celestial horizon

Calculating the distance to the horizon, defined as the point at which a ship will vanish from sight because it's blocked by the curvature of the earth, is fairly simple. But how about something a ...
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Closed form of planetary radial motion as time function

What function/ functions express radial motion of planet by means of non-linear ODE $$ \ddot r - \frac{A}{r^3} +\frac{B}{r ^2}=0 $$ (The Kepler/Newton constants are: $\,B= a^3 \omega^2\, ; A=B p \,; ...
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321 views

northmost latitude

From the book Astronomy, principles and practice. I cannot solve the second part. Assuming the Earth to be a sphere of radius 6378 km calculate the great circle distance in kilometers between London (...
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Getting started on Celestial Mechanics

I am searching for a math-accurate book on this subject, in particular for this topics: $n$-body problem, getting more detailed when $n=2$. Efeméride calculation. Orbit determination. Perturbation ...
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multyplication of 2 vectors forming a matrix - meaning

I am trying understand an algorithm used to determine orientations. Knowing a cross product of 2 vectors gives you a third vector which is orthogonal. What does the multiplication of a 3x1 and 1x3 ...
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Keplerian orbits and closest approaches to Earth.

This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers: Let's consider a small astronomical object following an ideal elliptic ...
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planetary motion: Particle describes an ellipse as a central orbit about a focus

A particle describes an ellipse as a central orbit about a focus. Show that the velocity at the end of the minor axis is the geometric mean between the greatest and least velocities. My attempt: I ...
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Overlap of Planets in Elliptical Orbit

I'm investigating further into my orbital overlap problem. I've already looked into the overlap ($0°$ angle between the two orbits) of two planets in a circular orbit around the sun. I'm now trying ...
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Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...