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.
703
questions
0
votes
0
answers
64
views
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
2
answers
83
views
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?
- 111
4
votes
2
answers
227
views
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 ...
2
votes
3
answers
118
views
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 ...
-1
votes
1
answer
29
views
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 ...
- 11.1k
1
vote
0
answers
39
views
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}{...
0
votes
0
answers
42
views
Earth's motion in the universe
In the context of working with atomic clocks I have to obtain the orientation of the Earth for every given Julian date. I am trying to obtain a 3d vector summing up all of earth's motions in the ...
16
votes
6
answers
4k
views
If another planet was opposite Earth, would we be able to observe it?
Imagine another Earth-sized planet, in the exact same orbit as Earth, but 180 degrees out-of-phase. In this arrangement, at all times, you would be able to draw a single straight line through space ...
- 331
4
votes
1
answer
114
views
How are Lie series used as canonical transformations in perturbation theory?
I have a few questions on how to use Lie series as a canonical transformation, which are widely used in perturbation theory (celestial mechanics).
I know that these series are related to a Taylor ...
4
votes
0
answers
68
views
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....
- 405
3
votes
3
answers
112
views
Why do the planets tend to spin in the same direction as they orbit the center sun?
I mean, why do the spin angular momentum and the orbit angular momentum of a planet tend to have the same direction?
As we all know, a planetesimal $m$ orbiting a sun with mass $M_{sun}$ at $r$ will ...
- 309
0
votes
0
answers
36
views
Why doesn't the centre of mass of the solar system move away from the sun? [duplicate]
Consider our solar system, in the frame of the sun (i.e. "the sun is stationary"), with a simplified 5 planets and nothing else. Suppose that for a brief moment, all of the planets aligned (...
- 477
0
votes
2
answers
37
views
Is it possible to determine if a planet can have a moon based on its mass and gravitational pull?
I'm curious, if based on what we know with Newton's law, can we determine if a random planet, knowing it's mass and gravitational pull, can hold a moon in it's orbit.
Or to phrase it another way, is ...
- 1
0
votes
1
answer
90
views
Number of Geostationary Orbits
It is stated that there is only one geostationary orbit whose height can be calculated using:-
$H = [\frac{GM_ET^2}{4π^2}]^{\frac{1}{3}} - R$
But there can be more than one geostationary orbits if I ...
- 836
0
votes
1
answer
56
views
An application of spherical trigonometry to the celestial sphere
In the book 'An Introduction to Modern Astrophysics' (Carroll and Ostlie, 2017), the first chapter presents an application of spherical trigonometry. At a certain point, the authors derive an equation ...
- 3
5
votes
3
answers
211
views
What am I doing wrong? An easy gravity problem
The following problem is giving me a headache:
Halley's comet follows an elliptical orbit around the sun. At perihelion, its distance from the sun ($r_P$) is $8.823 \cdot 10^{10}$ metres. At aphelion,...
- 401
0
votes
2
answers
69
views
How does Kepler's Second Law show that a planet further from the sun will move slower?
This is probably a very stupid question. We are told that due to Kepler's Second Law, which according to this very straightforward explanation:
"Kepler's second law of planetary motion describes ...
- 48
0
votes
2
answers
50
views
Attraction between two objects in the universe. The resulting number of forces between them
Right now I am studying Newton's Law of Universal Gravitation and I already learned his Third Law. It is said that there is an action-reaction pair between the falling apple and the Earth which ...
- 3
0
votes
0
answers
8
views
Calculating the Local Lunar Sky at a specific point in lunar coordinates?
How can one calculate the local sky observable from any given point on the Moon, expressed in lunar coordinates? Specifically, I would like to understand the methodology and equations involved in ...
0
votes
0
answers
35
views
Component free computation of vector poisson brackets
How can the Poisson bracket {H, A} be computed directly without components.
H is the Hamiltonian for the inverse square force, $H=\frac{p^2}{m} - \frac{k}{|r|}$ , and A is the integration constant ...
- 75
1
vote
1
answer
52
views
Dependence of areal velocity on distance between sun and planet
We know velocity of a planet in an elliptical orbit is given by:
$$v^2 = GM * (\frac{2}{r} - \frac{1}{a})$$
in an elliptical orbit. [Here r is distance between particle and sun] source
We also know, ...
0
votes
1
answer
35
views
Angular Momentum or Gravitation question [closed]
Question: If the earth suddenly shrinks to $\frac{1}{64}$th of its original volume keeping mass same, the period of rotation of earth becomes $\frac{24}{x}$ hours, what is $x$?
So basically, why can’...
- 1
6
votes
4
answers
605
views
WHY can't we use conservation of energy to find speed of the earth around the sun?
I was trying to calculate the velocity of the earth around the suns orbit using the conservation of mechanical energy hence:
$$\frac{GMm}{R} = \frac{mv^2}{2}$$
$$\sqrt{\frac{2GM}{R}} = v$$
why is the ...
- 111
0
votes
0
answers
32
views
How much does mass center of satellite oscillates about elliptic orbit as its spinning around the Earth?
I would like to know how much the real orbit of satellite can differ to the elliptic one.
Some perturbation of mass center position of the satellite must occur - Earth gravity accelaration changes as ...
- 1
1
vote
3
answers
411
views
Angular Momentum vs. Force Due to Gravity
I'm getting my feet wet with orbital mechanics and have a very basic question. Kepler's 2nd Law shows that 2 objects in an elliptical orbit sweep out equal areas in equal time, implying objects ...
-2
votes
2
answers
47
views
Sling-Shooting Around a Star to Catch a Solar Flare
How much thrust could a solar sail receive from sling shooting around the Sun and riding the wave of a solar flare during the escape?
I found this link https://ww4.fmovies.co/film/the-ark-season-1-...
- 453
0
votes
1
answer
76
views
How to calculate Earth's speed due to moon induced orbit? [closed]
In https://stackoverflow.com/q/75297814/
the answer for the problem was that the earth like the moon had a speed due to the moon induced orbit. I don't understand how this was calculated? I have ...
- 1
1
vote
0
answers
36
views
How to calculate the distance of Mercury's perihelion shift in meters? [closed]
Since the shift in mercury's perihelion shift is 2.88E-5 degrees and mercury's altitude above the Sun at its perihelion is 46001200000 meters, can I use the formula for an isosceles triangle to ...
- 83
1
vote
1
answer
43
views
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), ...
0
votes
0
answers
71
views
How to obtain Mercury Precession equation $\frac{d^2u}{d\theta^2}+u=a$?
Let $r=R(t)$ and $\theta=\Theta(t)$ describe the orbit of the planet. Define $$u(\theta)\equiv\frac{\bar r}{R[\Theta^{-1}(\theta)]}$$ where $\bar r=5.83\times 10^{12}$ cm be the mean value of the ...
- 3,076
-1
votes
1
answer
70
views
1
vote
1
answer
61
views
Kepler third law for circular orbits [closed]
This question may be uber trivial, but it has been stuck in my head for a while.
Kepler's third law states that the period of the orbit $T$ is related to the semi-major axis $a$ though
\begin{equation}...
- 2,302
1
vote
1
answer
39
views
Resolving varying $ω$ in Kepler's Law's Proof
I'm having trouble understanding where $d^2r/dt^2$ comes from and what it stands for. What force is this? I'm not able to find any FBD's on google that mention any other force besides gravitational ...
- 11
0
votes
1
answer
99
views
Why Saturn's rings lie in the same plane? [duplicate]
This is Saturn.
In every picture of Saturn, we can see that her rings are arranged in a perfect 2-d plane.
So the question I ask is simple, why are they arranged so? Why aren't the rings arranged in ...
- 715
0
votes
1
answer
59
views
What is the condition for a body to revolve around another body? [duplicate]
For a given system consisting of two bodies, when will one body orbit another body as given by Kepler's law? Sometimes the body just gets attracted linearly and sometimes it orbits the other body ...
- 344
0
votes
1
answer
96
views
Keplers' law of motion's proof
We know from Kepler's laws of planetary motion that the radius vector from the sun to a planet sweeps equal areas in equal times, and in a book of Feynman, he proves it by letting $\vec{A}$ = area, $\...
- 5
6
votes
1
answer
176
views
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,...
- 1,909
1
vote
2
answers
71
views
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
0
answers
50
views
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 ...
4
votes
2
answers
642
views
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 ...
- 141
0
votes
0
answers
70
views
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 ...
0
votes
1
answer
58
views
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 ...
- 103
0
votes
1
answer
109
views
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 ...
- 7
0
votes
3
answers
202
views
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 ...
- 113
1
vote
3
answers
475
views
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
1
answer
99
views
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 ...
- 131
0
votes
0
answers
37
views
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 ...
- 1
0
votes
0
answers
91
views
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
0
answers
87
views
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 ...
0
votes
0
answers
46
views
How does this trick work in solving the 2-body central force equation of orbit?
I am working on understanding the derivation of Kepler orbits via section 8.5 of John R. Taylor's classical mechanics textbook, and one small detail has been tripping me up.
The equation being solved ...
- 1,250