All Questions
Tagged with newtonian-gravity celestial-mechanics
373
questions
0
votes
1
answer
62
views
Has our knowledge of astrophysics and gravity reached the point where we can accurately calculate Lagrange points?
is it possible for us today given the knowledge we possess of gravity and our success with inserting satellites in to steady/ geosynchronous orbit and any knowledge we have on the relative size (and ...
- 107
2
votes
1
answer
74
views
Gravitational collapse - proof that energy dissipation is required?
As an undergraduate, I took a short course on astrophysics, where I encountered the Jeans mass. This is the critical mass for a spherical cloud of interstellar gas above which the cloud is predicted ...
- 317
0
votes
0
answers
31
views
Restricted 3-body: one large mass and two smaller ones
A restricted 3-body problem is usually understood as two large bodies and one much smaller one that doesn't affect the motion of the other two.
I am curious about a 3-body problem with one large body, ...
6
votes
7
answers
1k
views
What changes the velocity perpendicular to radius in an elliptical orbit?
I'm working currently on a problem that asks to justify that angular momentum and kinetic energy conserves for a planet in an elliptical orbit. Although I've been taught that angular momentum should ...
- 69
0
votes
1
answer
61
views
Find eccentricity of orbit given the velocity and the semi-major axis
Is it possible to calculate eccentricity of orbit knowing only the semi-major axis and the velocity of both celestial bodies? If not, what other additional information is required.
Does the fact that ...
- 38
2
votes
0
answers
37
views
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 ...
- 125
0
votes
1
answer
48
views
Speed of satellite in elliptical orbit [closed]
A satellite $S$ orbits a planet of mass $M$ in an elliptical orbit. At perihelion, $S$ has a tangential velocity of $v_1$ and is distance $r_1$ from the planet. At aphelion, $S$ has a velocity of $v_2$...
- 3
1
vote
1
answer
70
views
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 ...
- 11
0
votes
2
answers
62
views
Average Speed in one half of an elliptical orbit
I was wondering whether the average speed along one half of an elliptical orbit (say in a star planet system) had a closed form exact solution using Kepler's laws.
My approach was using the ...
2
votes
2
answers
124
views
Orbiting body around a star
Let us assume there's a body with mass $m$ and velocity $v$, at a distance $r$ from another body of mass $M$. The velocity vector is perpendicular to the radial vector. With these values, how do we ...
- 91
1
vote
1
answer
78
views
Two bodies orbiting around barycenter
There are two bodies with masses $m_1$ and $m_2$ orbiting around barycenter. Distances to both bodies from barycenter are $r_1$ and $r_2$ . First body has known velocity $v_1$ as on the picture :
My ...
- 11
0
votes
0
answers
25
views
How does moving the pericenter affect the apocenter?
A perfectly circular orbit of a constant height (distance from the center of mass of the orbited planet) around a perfectly spherical planet with smooth surface and no gravitational anomalies will ...
4
votes
1
answer
730
views
Do orbits with positive energy tend to infinity?
Consider any potential field $$V = V(x)$$ (not limited to gravitational potential field, but we only consider time-independent ones) in 3-d space that satisfies the following conditions:
The ...
- 43
0
votes
0
answers
30
views
Different transfer trajectories between same planets
I was working on desmos drawing transfer trajectories between the earth and moon. I managed to draw both the trajectories but i noticed something rather odd. The transfer orbits were different ...
- 91
0
votes
1
answer
48
views
Dynamics of Eliptical Orbit
For a circular orbit, we can resolve the forces acting on it along the radial and tangential axes. However, along which orthogonal axes should I resolve the forces acting on a body traveling in an ...
0
votes
2
answers
83
views
Newtonian mechanics doubt
While solving a particular classical mechanics problem , I was told that for a system of particles to be bound under their mutual forces, their initial energy (With Respect To the COM) must be less ...
- 294
0
votes
0
answers
70
views
How can one predict asteroid orbit, with the use of vector calculus?
If you are given, (or found) the position and velocity vectors of an asteroid how can one use this to predict its orbit?
2
votes
1
answer
224
views
Question about Lagrange's solution for the three-body problem
The original text in my textbook, written in short:
"By Newton's second law,
$$\ddot{\mathbf{x}}_1=-Gm_{2}\frac{\mathbf{x}_1-\mathbf{x}_2}{|\mathbf{x}_1-\mathbf{x}_2|^3}-Gm_{3}\frac{\mathbf{x}_1-...
- 23
25
votes
7
answers
6k
views
Why are there so many objects perfectly orbiting each other? Isn't it infinitely more likely that two random objects crash/fly apart?
If, in free space, I throw two objects towards each other, they can either miss each other and fly apart (if the velocity is enough and there's not enough gravitational attraction between them), or ...
- 1,090
0
votes
1
answer
138
views
Sun-Earth-Moon-Spacecraft four-body simulation using MATLAB - unexpected results [closed]
I'm trying to simulate this four-body system by directly integrating the system of equations
$$\ddot{\textbf{x}} = \sum_{i=1\:i\neq k}^4 G\frac{m_i}{|\textbf{x}_i-\textbf{x}_k|^3}(\textbf{x}_i-\textbf{...
- 3
2
votes
1
answer
138
views
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 ...
- 51
2
votes
2
answers
200
views
Derivation of Kepler's third law using Virial theorem
I am familiar with the long derivation of Kepler's third law using the equations of motion. One starts with
\begin{equation}
\dot{r}=\sqrt{\frac{2}{m}\big[E-V(r)\big]}
\end{equation}
and integrates to ...
- 4,827
1
vote
0
answers
46
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:
$E_Y = \frac{GM^2}{...
0
votes
0
answers
36
views
Why doesn't the centre of mass of the solar system move away from the sun? [duplicate]
Consider our solar system, in the frame of the sun (i.e. "the sun is stationary"), with a simplified 5 planets and nothing else. Suppose that for a brief moment, all of the planets aligned (...
- 535
1
vote
2
answers
47
views
Is it possible to determine if a planet can have a moon based on its mass and gravitational pull?
I'm curious, if based on what we know with Newton's law, can we determine if a random planet, knowing it's mass and gravitational pull, can hold a moon in it's orbit.
Or to phrase it another way, is ...
- 111
5
votes
3
answers
249
views
What am I doing wrong? An easy gravity problem
The following problem is giving me a headache:
Halley's comet follows an elliptical orbit around the sun. At perihelion, its distance from the sun ($r_P$) is $8.823 \cdot 10^{10}$ metres. At aphelion,...
- 427
0
votes
2
answers
65
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
1
vote
1
answer
64
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
39
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
667
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
1
vote
3
answers
802
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 ...
0
votes
1
answer
82
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
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 ...
- 3,256
-1
votes
1
answer
92
views
1
vote
1
answer
77
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
45
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
184
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 ...
- 751
0
votes
1
answer
62
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 ...
- 354
1
vote
2
answers
162
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
55
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 ...
0
votes
1
answer
78
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
327
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 ...
- 15
3
votes
3
answers
2k
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
0
answers
109
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
49
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 ...
- 1,302
0
votes
1
answer
68
views
Kepler's Two-Body Problem Choice of Sign
I have been going over the solution to Kepler's problem and there is a subtlety I am missing. The notes I am following (my own notes from a while ago in fact...) get to the following expression
$$
\...
1
vote
2
answers
616
views
General solution for the two-body problem [closed]
I am trying to compare numerical solutions of the two-body problem with the analytical one. But, for some reason, the analytical one doesn't seem to agree with the numerical one. For the numerical ...
- 11
0
votes
0
answers
43
views
Why are the orbit of planets usually ellipses? [duplicate]
There has been a similar question about planets' orbits being ellipses but the answer circulates around how the circle is a special type of orbit which doesn't really answer my question.
Elaborate ...
- 159
13
votes
3
answers
2k
views
Is it possible for a moon to have the same orbital period as its planet?
Is it possible for a planet to take just as long to orbit its star as a moon takes to orbit the planet? If we assume circular orbits, then
$\text{orbital period}\sim \sqrt{\frac{\text{radius}^{3}}{\...
- 233
1
vote
1
answer
49
views
Pair of binary stars orbiting each other
Suppose we have a pair of binary stars orbiting around each other in their mutual gravitational field. My question is, is the trajectory of the combined system would be an ellipse?
And, if it is an ...
