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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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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What happens in the event that the cooling radius is shorter than the virial radius of a Cold Dark Matter Halo?
The cooling radius of a cold dark matter halo is defined to be the time at which the cooling time $t_{cool} = t_{free fall}$
where
$$t_{cool}=\frac{\rho \varepsilon }{\Lambda \left ( T \right )n_{H}...
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Good sources to get velocity/position vectors for all planets and moons in the Solar System for building an orbital simulator?
I’m building an $N$-body simulator, and I have everything ready to begin simulating. But my issue is is that I have no idea how to get all the starting positions and velocities for the celestial ...
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Three-body solutions with no syzygies?
Montgomery has shown in his paper The zero angular momentum, three-body problem: All but one solution has syzygies, here (PDF), that all negative-energy, zero-angular-momentum solutions of the three-...
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Explicit construction of action-angle variables for the two-fixed-centers problem
After studying action-angle variables and Eulers two-fixed-center problem in a course on mechanics and symplectic geometry, I understand that a two-fixed-center system is Liouville integrable and ...
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Horseshoe orbit cycle times
I asked on Worldbuilding originally, but was directed here instead
I've been working on a habitable moon system for a story, and I'm now trying to populate the other moons around the parent planet. ...
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Why do the orbit equations have to be symmetric about two axes even the orbit is not bounded?
In the book of Classical Mechanics by Goldstein, at page 88, it is given that:
$$
\frac{d^{2} u}{d t^{2}}+u=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right) .
$$
The preceding equation is such ...
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Determining velocity of moons
I have a question that I believe is relatively easy to answer, I am working on an $N$-body simulation of a fictional star system and am having trouble finding the velocity of moons so that they will ...
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Story about a mathematician, a dinner party, and the three-body problem
I remember dimly hearing a story, coincidentally also at a dinner party, and I was trying recently to track the details down with no success. I was hoping someone here might have also heard this story ...
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Isn't the Jacobi constant just the Lagrangian times 2?
At this wikipedia page the Jacobi constant is expressed as:
$$C_J=2\left(\frac{v^2}{2}-U\right)$$
where $U$ is the potential energy and $v$ is velocity. If kinetic energy $T$ is defined (as it ...
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Simple model of the solar system. Parameters? Accuracy?
I was thinking of making a simple 2D model of the solar system, with planets moving along ellipses like
$$x(t) = k_x \sin(t + k_t) (\sin(k_\phi) + \cos(k_\phi))$$
$$y(t) = k_y \cos(t + k_t) (cos(k_\...
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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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How to add kinetic energy to gravitational energy to obtain total energy?
Cambridge Pre-U 9792/03/M/J/22
Examination question: What is the total energy E of the binary star system?
Given: The kinetic energy of star X is $E_x = \frac {2GM^2}{9D}$
Working:
$E_Y = \frac{GM^2}{...
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Problem in an ellipse circumscribed on an auxiliary circle
I was reading the book "an introduction to the evolution of single and binary stars", by Mattew Benacquista, and I couldn't understand a specific step in topic 2.1 (Time-Depedent Orbits), ...
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Solution to two-body problem in orbital mechanics for $r(t)$ and $\theta(t)$, rather than $r(\theta)$?
I have written a simple numerical integration code to calculate the orbits of two planetary bodies orbiting a star, in order to calculate the transit-timing variation for one body due to the ...
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Does the Lagrange Point $L_3$ exist in practice in the multi-body solar system?
Lagrange Points ($L_1$ through $L_5$) in a restricted 3-body system are well documented. Traditionally body 1 (M1) is the central object with a mass much greater than the other two objects. M2 is ...
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How to calculate the new position at time $t$ of celestial bodies with variable acceleration?
Intro
For an $N$-body simulation of celestial bodies I need to calculate
on the one hand the accelerations of the celestial bodies based on the received gravitational forces [done]
and on the other ...
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What is the logic behind this vis viva equation based calculation?
"If the moon were shrunk to having a radius of one meter, without changing its distance (of the apogee) from the earth, what would be the slowest it could be slowed down to and still orbit the ...
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Kepler problem, How can I prove that the period of elliptic orbit is only dependent on energy?
We are also given the following hint:
Equate the energies and sectorial velocities at the apogee and perigee of an elliptic orbit to express the area of the ellipse in terms of E.
For some ...
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Is a satellite orbit around the Earth Lyapunov stable?
Presume there is a satellite orbiting the Earth in an orbit that follows a closed path around the planet (that is, escape orbits are not permitted here). As I understand it, there are two ...
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What happens after a collision?
2 different planets far away from each other, each have an extremely strong water canon. Those 2 canons are aimed to the sky, aimed at each other. They both shoot a massive load of water to each other,...
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Why don't the galaxies collide due to gravity?
Why doesn't the galaxies collide due to gravitational attraction and why
doesn't all the matter in the universe stick together due to gravitational attraction?
user250074
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Why are so few inclined circumbinary planets known?
For a research project I'm studying the orbits of circumbinary planets, most of these planets orbits are coplanar. However I was wondering if orbits with a high inclination could be stable. I made a ...
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Restricted 3-body problem - Rotating coordinate system
I'm trying to follow the formulation of the restricted 3-body problem in Taff's "Celestial Mechanics" book as I'm writing an essay on Lagrange points and am confused by the transformation into the ...
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The shape characteristics of gravitational wells given different masses and spread of objects
I am curious as to research that calculates the shape of gravitational wells, and their limits, and affect on time, for different masses and spread of mass. The actual question is st the end.
For ...
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Planar Precession Frequency of Orbit
What is the general relation between orbital precession $\Phi$, orbital frequency $\Omega$ and a radial perturbation frequency $\omega$?
For certain cases the answer is "clear", for example:
1) If $\...
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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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Sign of the action for the harmonic osccilator?
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 ...
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How do I use Kepler's Laws to propagate an orbit for a Two-Line Element (TLE) Set?
I need to propagate a large number of orbits (get a satellite's position at a certain time) from TLEs (two-line element sets) using Kepler's laws. Ordinarily I'd use a real propagator like SGP4/SDP4, ...
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