Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [celestial-mechanics]

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

Celestial mechanics is the branch of astronomy dealing with motions of celestial objects.

Historically, celestial mechanics uses laws of gravity to give the position of a natural celestial object or satellite in the sky at a given time.

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Proof of a definite integral whose result is the difference of arithmetic and geometric means

This might have been already asked in this site but I can't find it. So here's the integral: $$\int_{r_\text{min}}^{r_\text{max}} \sqrt{\left(1-\frac{r_\text{min}}{r}\right)\left(\frac{r_\text{max}}{r}-1\right)}~dr=\pi…
Sanjana
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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 Keplerian orbit around the sun. For concreteness I'm…
hmakholm left over Monica
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Geometric interpretation of integral $-\int\frac{1}{\sqrt{a+2bx-hx^{2}}}dx=\frac{1}{\sqrt{h}}\arccos\frac{b-hx}{\sqrt{b^{2}+ah}}$

The following formula is given as "the familiar arc-cosine form" by Joos, in his Theoretical Physics. The German language original has $e$ in place of…
Steven Thomas Hatton
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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 not necessary having the same major axis). The…
Lehs
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κ₀ for Mercury—Formula

I refer here to Ptolemy’s epicycle-and-deferent model of the Solar System, specifically that of Mercury (see drawing). In this model, Mercury (not shown) revolves on an epicycle of center C, which itself turns on an eccentric circle (later called…
Pierre Paquette
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Trying to model the Flyby anomaly

Thinking about it, the following non-linear ordinary differential equation crossed my mind: $$ \frac{d^2 r}{dt^2} - \frac{1}{2} H \frac{dr}{dt} + \frac{\mu}{r^2} = 0 $$ Apparently, I've been trying to modify and simplify an equation for Orbital…
Han de Bruijn
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Why isn't a jagged orbit ever observed in the two-body problem?

The two-body problem deals with two planets revolving around a common center relative to one another. Why doesn't the model ever exhibit a jagged orbit and it is always smoothly elliptical? Is there something about the math actively enforcing the…
develarist
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Interval of convergence of Lagrange's infinite series

I am reading a book on Orbital Mechanics for Engineering Students by Howard D. Curtis. In that book it was mentioned (in page 119) that there is no closed form solution for $E$ as a function of the eccentricity $e$ in the equation $E-e\sin E=M_e$…
sai saandeep
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One body problem: position as function of time

Consider the typical one body problem (e.g. earth-sun system), where the orbit is elliptical. It is known that there is no "closed-form" formula for the position of the point (earth) as a function of time. This is because the calculation of the…
Plemath
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Find the eccentricity of the elliptical orbit when the central force is changed from one focus to another

A body describing an ellipse of eccentricity e under the action of a force directed to focus when at the nearer apse , the centre of force is transferred to the other focus . Prove that eccentricity of the new orbit is…
Bertie
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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 an AI for this game that needs to compute the…
Elliott
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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 \,; $ There is no need to remind.. this differential…
Narasimham
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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 (51◦ 30 N, 0◦) and New York (40◦ 45 N, 74◦…
ted
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Calculating Partial Sector of an Ellipse

I was trying to calculate and find out the area of a sector of an ellipse when the angle of the sector is drawn from one of the focii and one arm is taken as the x-axis. I found the following question asked a decade ago: How to calculate ellipse…
Math_Maven
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how to solve single nonlinear algebraic equation in two variables?

(I am not a mathematician; I am having physics background.) How to solve a single nonlinear algebraic equation in two variables, $x$ and $y$? (I know that - if there are two variables, one needs two different equations to solve them.) Equation is…
atom
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