# 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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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### The magnetic force between the earth and the sun

There is a magnetic field around the earth and a stronger one around the sun. I guess there should be a magnetic force between the sun and earth. Now, shouldn't we take the magnetic force into account ...
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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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### 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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### 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 $v_2$...
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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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### How to find the length of a travel between planets using the Hohmann Transfer?

I am trying to figure out how long it would take to get from one planet to another. This is for a worldbuilding project of mine. I would put my question on the Worldbuilding Stack Exchange but I ...
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### 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 ...
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### Kepler orbit from mass, period, and eccentricity [closed]

I want to calculate the Keplerian orbital elements in the central force case. Given the mass M of the "sun" in kg, the eccentricity $e$ of the orbit, and the period $T$ in seconds, I believe ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### Proving Kepler's second law of planetary motion using conservation of angular momentum: What about gravity from other planets?

I'm reading An Introduction to Mechanics by Kleppner and Kolenkow. In the chapter on angular momentum, a (beautiful!) example is given by discussing Kepler's second law of planetary motion. The law ...
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### Is there a way to use the distances of the two opposite apsides to determine the eccentricity of an orbit?

Is there a way to use the distances of the two opposite apsides to determine the eccentricity of an orbit? The ratio between the distances (i.e. perihelion & aphelion) seem like they'd have a ...
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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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### 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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### 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 ...
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### 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 ...
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### Estimate Saturn's mass [closed]

How can you estimate Saturn's mass using data from Cassini's final moments in September 2017 (apoapsis on September 12 at 1:27 a.m. EDT Saturn time at a distance of about $1.3*10^6$ km from Saturn, ...
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### 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 ...
1 vote
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### How does tugging objects across long distances work? [duplicate]

Let's say there's an object on Mars, I stand on Earth, and hold a non-elastic, unbreakable rope (or an iron bar or something) tied to the object on Mars. And I try to tug it. The tug cannot travel ...
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### Mars' orbital period

The orbital period of Mars, is, as anyone can find at Wikipedia, $T=686.98$ d, and the semi-major axis of its orbit is $a=2.2794\cdot10^{11}$ m. This gives $T=2\pi\sqrt{\dfrac{a^3}{GM}}=686.84\text{d}$...
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### 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 ...
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### 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?
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### Measurement of the velocity of a celestial body by means of (relativistic and classical) gravitational effects on clocks

Imagine a planet with the same properties as Earth, this time moving in an elliptical orbit around a black hole of a large number of solar masses. Also imagine that the surface of this planet is as ...
1 vote
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### Why is $G*M_{sun}=4\pi^2$ when using AU/year units?

So, when using AU/year units, it turns out that 3rd Kepler Law: $\frac{r^3}{T^2}=\frac{G*M_{sun}}{4\pi^2}=1$, meaning $G*M_{sun}=4\pi^2$, any easy explanation for this? Cheers.
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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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### Do the line of Apsides and Line of Nodes of the Lunar orbit ever coincide (or change their relative position and does this explain Annular eclipses?

In the context of solar eclipses, my understanding is that an annular eclipse occurs when specific conditions are met, involving the relative positions of the Earth, the Moon, and the Sun. To be more ...
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### 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 $$\dot{r}=\sqrt{\frac{2}{m}\big[E-V(r)\big]}$$ and integrates to ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### Conserved Quantities in Kepler Problem?

In our classical mechanics class, professor said that Kepler's problem is a kind of Integrable System such that the number of conserved quantities would be equal to the number of degrees of freedom. ...
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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}{...