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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Can the Sun orbit the Earth?
You know how the earth orbits the sun because of gravity?...well if the sun were the right distance from the earth and it had the perfect tangential velocity...could it orbit the earth?
In other words....
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First Law of Motion, Gravitational acceleration and Law of Conservation of Energy [duplicate]
When an object leaves a celestial body (e.g. planet or star) with it's escape velocity, according to first law of motion, it will continue travel away from the body with the same speed (without having ...
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How to transform coordinates from heliocentric to a view from earth?
I'm trying to find and model the apparent motion of the retrograde motion of Mars as viewed on Earth, incorporating Lagrangian or Hamiltonian mechanics.
For the lagrangian I have:
$L = \frac{1}{2}m_{...
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Does the Earth orbit an ellipse or the barycentre of the Solar system?
I'm taking physics lessons in university and my professor described the Earth's orbit as en ellipse, where one of the focal points is the Sun (1st Kepler's law).
However, this doesn't make much sense ...
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Orbit with crash
Let's assume I shoot an object from a high tower horizontal to the earth's surface. As far as I understand, depending on the velocity I will get different types of orbits. With decreasing velocity I ...
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Tidal locking and heat dissipation
I need to know that how does one measure (approximate) the energy loss of a planet or satellite during the whole tidal locking process?
Where do I find this topic to study?
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How to calculate the total energy of a planet at rest?
Let's just say we have the Earth-Moon system isolated in space, with Earth at rest, and the moon orbiting it.
How can we calculate the total energy of Earth in such a case (Kinetic energy would be ...
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What are the sources of error in the formula I derived to estimate sunrise and sunset times, and how can I improve its accuracy? [closed]
I derived the following formula to estimate sunrise and sunset times:
$$ t=\pm\cos^{-1}\frac{\sin\theta-\sin L_{loc}\sin L_{sun}}{\cos L_{loc}\cos L_{sun}} $$
$L_{loc}$ is the local latitude
$L_{sun}$...
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Deriving Laplace-Runge-Lenz Vector
consider a particle in a central potential, i.e. the potential $V$ only depends on the distance $ r = \| \vec{x} \|$ to the origin. The equation of motion thus reads
$$ m\ddot{\vec{x}}=-\frac{\partial}...
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Kepler's Second Law: conservation of the energy or of the angular momentum?
Reading from example the old question, Angular Momentum and Kepler's Second Law
Considering that now I not remember the proof because starting from the angular moment conservation of $L$ we have ...
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Proof that derives Newton's law of gravity [duplicate]
I was wondering if it is possible to derive Newton's law of universal gravitation, without utilizing Kepler's 3rd law. I have found many proofs that use this type of proof. I read somewhere that ...
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Derivation of Newton's law of gravitation [duplicate]
How did Newton get $F=\frac{Gm_1m_2}{r^2}$?
What is intuition behind it?
What kind of experiment or thought experiment can I do to derive this?
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Estimating of the size of the Milky Way from the Sun's motion
I am trying to run a simple first-principles calculation to estimate the diameter of the Milky Way from what we know about the motion of the Sun around its center. In particular, from various online ...
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In the earth-asteroid two-body problem, is it possible for the asteroid to be captured into an elliptical orbit around earth?
Consider an asteroid of mass $m$ rushing towards the earth. What is the mathematical condition under which the asteroid will be captured into an orbit around the earth (given that the asteroid and the ...
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What about Newton's potential prevents singularities in the two-body problem?
The solutions to the gravitational two-body problem can be described by conic sections.
In particular, two abstract point masses don't collapse into each other (that is, do not get arbitrarily close ...
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How does $\frac{mv^2}{r} = \frac{GMm}{r^2}$ work if both planets attract each other?
According to this and this, both posts describe the fact that the earth and the moon are attracting each other due to gravitation.
The gravitation force is $\frac{GMm}{r^2}$.
However, I am not able to ...
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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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Why is the predictability of the solar system in the Lyapunov timescale limited to 5 million years?
Is this due to a mathematical problem that is not solved? Or could this be due to our current amount of information regarding mass and other such factors in our system?
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Orbit around irregularly shaped asteroids
I'm curious how would one calculate the shape of the orbit around irregular objects (let's call them asteroids).
How do you tackle this problem? How do you write down basic equations?
In classical ...
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Can Binary Stars Escape Each Other?
updated 8/27/2020
While the recession of our Moon from the Earth may slow and even stop,
(see When will the Moon reach escape velocity?)
binary star systems will
(1) never stop experiencing mutual ...
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Escape velocity questions
This topic really screwed with my head in school and my teacher could never answer my questions so I thought I would ask here. I will break it down with mathematics and then logic.
Let's assume we ...
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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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Converting elliptic orbits to circular ones [closed]
My physics teacher has told me that if we have an elliptic orbit under consideration, we can consider it equivalent to a circular orbit of radius equal to that of the semi-major axis of the ellipse.
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Confusion on Gravitation/Orbital Mechanics
In polar coordinates, we can write:
$$\frac{d^2\vec{r}}{dt^2}=\frac{-GM}{r^2}\hat{r}=(2\dot{r}\dot{\theta}+r\ddot{\theta})\hat{\theta}+(\ddot{r}-r\dot{\theta}^2)\hat{r}$$
$$\frac{-GM}{r^2}=\ddot{r}-r\...
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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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Time taken to collide [duplicate]
Two point masses m1 and m2, separated initially by distance d, move towards each other under mutual gravitational force.
Find the time they take to collide?
The main problem I'm having is to solve the ...
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Why aren't orbits transferred at the perigee?
In example 10.6 titled "Satellite Orbit Transfer 1" of Kleppner and Kolenkow the author says
The most energy-efficient way to put a satellite into circular orbit
is to launch it into an ...
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Why center of mass is at the focus for elliptical orbits?
I know there are similar questions with answers like this one in physics stackexchange but before I read these I had a completely different argument with a different conclusion which I thought was ...
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Understanding the third Kepler law [closed]
Consider a system of two masse $M$ and $m$ with $m<<M$ and the mass $m$ orbiting around $M$. Then $m$ describes a elliptic orbit with period $P$ and the third Kepler law states that:
$$\frac{P^2}...
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Why do we use specific orbital energy and specific angular momentum?
In the derivation for vis-viva equation
$$v^2 = GM \left({ 2 \over r} - {1 \over a}\right),$$
here they use specific quantities such as specific energy and specific angular momentum but I don't ...
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What's the actual path of the planets?
I read about Kepler's Laws and in one of them he mentions that the path of a planet is an ellipse, with the sun as one of its foci (I'm narrowing down this to only our solar system).
However though I'...
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Are the orbits in my simulation correct? [closed]
I'm creating a simulation of our solar system.
I used data from https://nssdc.gsfc.nasa.gov/planetary/factsheet/ and I supplied the planets with perihelion and aphelion values. But I'm afraid the ...
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Satellite path dependence on velocity
How does a satellite planetary path depend on its speed and other factors.
I want to know more about parabolic, elliptic and hyperbolic paths of satellite.
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Orbit eccentricity and initial velocity
I was thinking about orbital velocities, and came across this question (Velocity of satellites greater than required velocity). Does the answer to this question imply that for planets going round the ...
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How do scientists calculate the orbital period of a planet?
It is known that you can calculate the distance to a planet using parallax, but how do scientists calculate the orbital period of a planet?
(Assuming they don't know the distance and can't use Kepler'...
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Relation of speeds of Earth at extremities of orbit through force
I came across this article in my book where they had related the velocities of earth at the aphelion and at the perihelion . Their approach was
$1)$ conservation of angular momentum at the desired ...
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If Saturn's rings cannot coalesce into a moon because of tidal forces, then how are shepherd moons able to exist?
From Wikipedia:
In celestial mechanics, the Roche limit, also called Roche radius, is the distance within which a celestial body, held together only by its own force of gravity, will disintegrate due ...
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Help understanding why Newtonian mechanics doesn't allow planets to follow Rosetta orbits
The argument I am trying to understand is the following:
For an orbit to be closed, the angle between successive perihelions (points where the planet at its closest to the sun) must be a rational ...
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Why does a satellite moving around earth in circular motion its speed is constant?
I do know the satellite's tangential velocity is always perpendicular to gravity, therefore its speed must remain unchanged, however, I have some confusion.
If I separate both the tangential velocity ...
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Proof for elliptical orbits
Its mentioned in several books that a satellite launched with a velocity less than the escape velocity and other than the critical velocity will follow an elliptical orbit. However I can't find a ...
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Could an asteroid capture an object and bring it into orbit as the asteroid moves through space?
I'm a scifi writer and have very limited knowledge of astrophysics and I'm working on a story where my character lands on a moving asteroid.
In the original draft of the story, my character is on an ...
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How do I obtain the trajectory of a particle in the plane using Newton's law of gravitation? [closed]
Problem:
Suppose we have a fixed planet $A$ exerting gravitational force and a particle $B$:
I want to obtain the function that gives the trajectory of $B$ to $A$. If $B$ has no velocity and no ...
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The Precession of the Perihelion of Mercury 43'' / century or 12000 km / century difference to Newton's Laws
I was reading Foundations of Astronomy by M Seeds and was wondering about the precession of Mercury.
There he says it is 43''/century or about 29 km per revolution past the position predicted by ...
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What determines if an object will stay in a planet's orbit? [duplicate]
Say you threw an object 10 AU from a planet at a certain speed, would this object stay orbiting around the planet or would it shoot off into space? And does it depend of the speed the object when it ...
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Why total mechanical energy does not 'seem' to be conserved in a typical Earth-Sun system?
Considering mass of the Sun and Earth to be $M$ and $m$ respectively, we have the potential energy of the system to be equal to $\frac{-GMm}{r}$ when defined zero at infinite separation, $r$ is the ...
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Under which conditions do two moving bodies start orbiting each other around their center of mass?
If two bodies are close, both will get attracted to each other and collide. Under what conditions will the two bodies start revolving around their common center of mass? I understand that such bodies ...
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For which planets Kepler has formulated his laws? [closed]
Kepler has formulated his three laws of planetary motion around the Sun.
Did he formulate the laws for all 9 planets or only for a part of them? Say I do not know if all the planets have been ...
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Would Earth always face Sun the same way if Earth didn't rotate around it's axis?
Imagine for a second that Earth no longer rotates around its axis. It still revolves around the Sun.
However, would the Earth:
face the same way towards the Sun as it goes around it (permanent time ...
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How can these two definitions of the eccentric anomaly give the same answer?
The eccentric anomaly,$E$, is defined via the equation,
$$ r = a(1-e \cos E) \tag{1} $$
for radius $r$, semi-major axis $a$ and eccentricity $e$.
It is ALSO defined in terms of the Kepler Equation ...
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Vis viva equation - finding how change in velocity changes the semi-major axis distances - math trick [closed]
I'm reading about how to deorbit. I stumped upon a trick here. In the question, we wish to find how a chance in orbital speed changes the semi-major axis distances. So in the answer we start with the ...
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