Questions tagged [celestial-mechanics]

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

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Formula for Stereographic Projection of Celestial Sphere centered at north point of horizon ,not at North Pole.

I'm currently making planetarium on web just for fun, and have problem with finding projection formula. Several search results shows formulas for projection of upper celestial sphere(centered at north ...
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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 ...
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what is the math for gravity causing different kinds of orbit ( ellipse, parabola ,straight line ) for an object in different cases?

case-1: the motion or you can say the orbit of a projectile is parabolic because gravity acts on it. case-2: the motion or the orbit of an object ...
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Constant circular motion: understanding $\underline{e_{\theta}}=\frac{d(\underline{e_r})}{d\theta}$

Context: 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion. This may be trivial, but can someone tell me what's wrong with the following reasoning? $$\underline{e_r}=\...
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The flow generated by integral of motion sends orbits of Hamiltonian into orbit of Hamiltonian?

I read these two statement in the notes of my teacher that seem to me opposing. Let $H$ an Hamiltonian and let $\Phi$ an integral of motion of $H$, so that $\Phi$ keeps constant value along the orbit ...
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Three-body problem: computing the gradient of gravitational potential energy in vector form

Consider the gravitational three-body problem, where three particles $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3 $ with masses $m_1, m_2, m_3$ are attracted to each other under the inverse square law. $$...
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A quite obvious result on one-variable differential functions

It states as follows: If $f\in C^1((a,\infty))$ is such that its limits exists, that is $ \hat{f}=\lim_{t \to \infty}f(t)$, then there exists a sequence $\lbrace t_n\rbrace$ with $t_n \longrightarrow ...
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How off is my understanding of Lambert's theorem to solve Lambert's Problem?

Math Overflow-ites! Hopefully, this question fits here better than places like Space and Physics. And sorry if I ramble a little - I'm rather new to this all. My understanding of Lambert's theorem is ...
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“Push a point” along a curve at a linear rate (Or for some distance) / “Linearizing” a function.

To begin off, I'd like to clarify I am a high schooler; I have no formal education in integral calculus or professional mathematics, but I'm exuberant to learn more! This is also my first question so ...
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33 views

Periodic Behavior of a Two-Body Problem?

Given the following equations and parameters: does the orbit repeat? If not, why? and what would make it repeat? Any clarification on what makes an orbit periodic would be greatly appreciated! I ...
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About solving complex recurrence relations

A simple approach is to use Newton's classical mechanics to find a free orbital function, rather than using Kepler's law, and the following recurrence relations problem arises. How should I solve it?
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A very simple question on motion in a circle.

Question A spacecraft of mass m orbits Earth at a radius R and speed $v_0$ as shown below. An aerospace engineer decides it should orbit at a radius of $\frac{2R}{3}$ instead. The mass of Earth is M. ...
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Normal distance

Recently, I saw a paper "G. H. Darwin, Periodic Orbits" and I don't undertand the concept of "normal displacement $\delta$p". "... Now suppose that x, y are the coordinates of a point on an orbit, ...
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Equation of Motion of a particle

I've tried this question over and over, and I'm getting nowhere. I've even tried looking for a solution to help make sense of how to get there, but I've had no luck. Can anyone help me please? A ...
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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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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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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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63 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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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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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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138 views

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 ...