Questions tagged [celestial-mechanics]

Use this tag for questions about the branch of astronomy dealing with motions of celestial objects.

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Relationship between lifetime of tech civilisations and time taken for a roundtrip between two such civilisations

Here is a statement of the question, which is taken from Chaisson and McMillan's Astronomy: A Beginner's Guide to the Universe. Assume that the number of (technological) civilisations in the Milky Way ...
Mike Wills's user avatar
1 vote
1 answer
242 views

κ̄₀ for Mercury—Formula (BIS—Different definition of the term/angle)

Following my other question about a specific “hidden” formula in Ptolemy’s model for Mercury, I am now looking for yet another “hidden” formula, this time the one used to find $\bar\kappa_0$ so that $...
Pierre Paquette's user avatar
4 votes
1 answer
324 views

κ₀ for Mercury—Formula

I refer here to Ptolemy’s epicycle-and-deferent model of the Solar System, specifically that of Mercury (see drawing). In this model, Mercury (not shown) revolves on an epicycle of center C, which ...
Pierre Paquette's user avatar
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24 views

Maximum possible rate of change of length-of-day?

What is the maximum possible rate of change of length-of-day (in units of minutes per day), on Earth? It is well known (and easy to observe) that it's larger near the equinoxes, and at higher ...
Rob Wall's user avatar
2 votes
0 answers
44 views

One body problem: position as function of time

Consider the typical one body problem (e.g. earth-sun system), where the orbit is elliptical. It is known that there is no "closed-form" formula for the position of the point (earth) as a ...
Plemath's user avatar
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Are there conditions under which it is practicable to use Vesc = sqrt(2gr) instead of Vesc = sqrt(2GM/r)?

Given that ...
elvexo's user avatar
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1 vote
1 answer
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By how much would the length of the solar day change if Earth's rotation were suddenly to reverse direction?

The question in the title has to some extent been answered here: https://worldbuilding.stackexchange.com/questions/79619/does-earths-direction-of-rotation-affect-day-length?newreg=...
Mike Wills's user avatar
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1 answer
263 views

How to derive the angular velocity $\omega$ of a local ENU frame relative to the non-rotating Earth

I recently started studying a book about Orbital Mechanics. Its title is "Orbital Mechanics for Engineering students", 4th edition. I am currently at section 1.7 (page 30) trying to find the ...
Alexis's user avatar
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Derivation of Spatial Delaunay Elements

I am trying to calculate the Spatial Delaunay Elements with actions $(L, G, H)$ and angles $(l, g, h)$ from polar coordinate $\left( \rho, \theta, \phi\right)$ and momentum $\left( P, \Theta, \Phi\...
pink1243's user avatar
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Dominated splitting in dynamical system

I am studying paper '' A (SHORT) SURVEY ON DOMINATED SPLITTING '' by MART´IN SAMBARINO It is mentiend that on page 2 '' Nevertheless, since the action of Df on the invariant subbundles might not have ...
user500089's user avatar
2 votes
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how to solve single nonlinear algebraic equation in two variables?

(I am not a mathematician; I am having physics background.) How to solve a single nonlinear algebraic equation in two variables, $x$ and $y$? (I know that - if there are two variables, one needs two ...
atom's user avatar
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How to find the set of solutions to this non-linear, underdetermined equation?

The Equation I have this equation: $$\frac{1}{2} \left( x^2+y^2 \right) + \frac{2(1-m)}{r_1} + \frac{2m}{r_2} = C,$$ where $m$ and $C$ are constants and $x,y$ are two variables and where $$r_1 = \sqrt ...
atom's user avatar
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6 votes
2 answers
269 views

Geometric interpretation of integral $-\int\frac{1}{\sqrt{a+2bx-hx^{2}}}dx=\frac{1}{\sqrt{h}}\arccos\frac{b-hx}{\sqrt{b^{2}+ah}}$

The following formula is given as "the familiar arc-cosine form" by Joos, in his Theoretical Physics. The German language original has $e$ in place of $h$. $$-\int\frac{1}{\sqrt{a+2bx-hx^{2}...
Steven Thomas Hatton's user avatar
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Do Kepler's laws (really) solve the Kepler problem? And should the terms of the central force problem be treated as dependent variables?

See the bottom for an added definition and discussion. Context Note: I've never had a course in differential equations, so my question may demonstrate gross ignorance. When the distinction is clear, I ...
Steven Thomas Hatton's user avatar
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1 answer
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Finding area of a circle from circular motion

We can find the change in area by $$dA=1/2 |r⃗ ×dr⃗| $$ $$ dA = 1/2 r dr \sin(θ) = 1/2 r dr $$ since velocity vector is orthogonal to r $$ dA = 1/2 r vdt $$ T =2πr/v $$A = 1/2\int_t^T r vdt $$ couldn'...
mark's user avatar
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1 answer
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Deriving a second-order system of differential equations that describes the motion of a planet (in cartesian coordinates)

I got a question that’s probably very basic but I just can’t figure it out. I want to derive a system of second order differential equations that describes the motion of a planet in (x,y)-coordinates (...
Avina's user avatar
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1 vote
0 answers
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Connected component in stability theory

I am studying Chapter 3, section 2.14 of the book Stability Theory by Lyapunov Direct Method by Laloy. Given the equations of motion $\dot{q} = \frac{\partial H}{\partial p}(q, p)$ and $\dot{p} = -\...
yumika's user avatar
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4 votes
1 answer
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Trying to model the Flyby anomaly

Thinking about it, the following non-linear ordinary differential equation crossed my mind: $$ \frac{d^2 r}{dt^2} - \frac{1}{2} H \frac{dr}{dt} + \frac{\mu}{r^2} = 0 $$ Apparently, I've been trying to ...
Han de Bruijn's user avatar
-3 votes
1 answer
456 views

How can I calculate the right obit period of planets with python?

Here is my python code which can plot the 2D orbit of Earth, Mars and Mercury with Sun. However, I cannot calculate the proper orbit of Earth and others . attempt: I use the $r_\max$, $r_\min$ of ...
john chong's user avatar
0 votes
1 answer
119 views

How to calculate the new position at time t of bodies with variable acceleration?

Intro For an N-body simulation of calestial bodies I need to calculate on the one hand the accelerations of the calestial bodies based on the received gravitational forces [done] and on the other ...
Dawid's user avatar
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3 votes
2 answers
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Find the eccentricity of the elliptical orbit when the central force is changed from one focus to another

A body describing an ellipse of eccentricity e under the action of a force directed to focus when at the nearer apse , the centre of force is transferred to the other focus . Prove that eccentricity ...
Bertie's user avatar
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1 vote
1 answer
171 views

Gravity Differential Equation

Is it possible to solve this system of differential equations in terms of $G$? $$r''=\frac{-Gr}{\sqrt{\left(r^2+(1-s)^2\right)^3}},\ s''=\frac{G(1-s)}{\sqrt{\left(r^2+(1-s)^2\right)^3}}$$ with initial ...
moqui's user avatar
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0 answers
70 views

In the calculation of precession of Mercury, why we can evaluate a definite integral using a contour integral

In the calculation of Mercury using perturbation, we have to evaluate the integral: $$ \delta \phi =-\frac{GMm}{2a^3}\sqrt{\frac{m}{2E}}\frac{\partial}{\partial J}\int^{r_{max}}_{r_{min}}\frac{r^3 dr}{...
Link's user avatar
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1 vote
1 answer
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How do I apply the Stereographic projection to generate the Star Finder templates at different latitudes?

I am trying to create my own Star Finder based on the templates from the 2120-D model. I started with a semi-sphere expressed as circles of latitude and semi circles of longitude, rotated along the X ...
PolAndre's user avatar
1 vote
0 answers
57 views

Can quadrics be applied to the n-body problem?

Gravitational orbits within the 2-body problem can be visualized as conics on the surface of a double cone. Is it reasonable to imagine that 3-body systems can be visualized as quadrics on the surface ...
James Burns's user avatar
0 votes
1 answer
28 views

Estimate the coverage shift of a satellite given its distance to the original satellite

I am doing some satellite-related research and would like to verify whether an assumption I am making about the satellite geometry is reasonable or not. Essentially, a LEO (low-earth-orbit) satellite ...
Danny's user avatar
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2 votes
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Solve differential equation $\ddot{x} = -a \frac{x}{|x|^3}$ [duplicate]

Came across this in a physics problem. Let $x$ be a plane vector dependent on time $t$. Let $\ddot{x}$ be the second order derivative of $x$ against $t$. Let $|x|$ be the Euclidean norm of vector $x$ ...
zvi's user avatar
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2 votes
0 answers
92 views

Orbital resonances - expansion of disturbing function

I want to study the orbital resonance type $3:1$ between an asteroid and Jupiter. For this purpose, I found the expansion of the disturbing function in Celletti A., Stability and Chaos in Celestial ...
Augustin's user avatar
0 votes
2 answers
458 views

Least distance of a particle under central force

A particle starts at great distance with velocity V. Let p be length of perpendicular from the center of a star on the tangent to the initial path of particle. Show that the least distance of the path ...
Sharmi C's user avatar
  • 419
1 vote
2 answers
73 views

A problem in mechanics

This is an apparently simple problem that I could not solve. The source is a colleague who interviewed for the European space agency. You are standing in a field with a chronometer and a camera. ...
AndreA's user avatar
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0 votes
1 answer
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A set couple ODE's [duplicate]

$$\frac{d^2x}{{\rm dt}^2}=-GM\frac{x}{{(x^2+y^2)}^\frac{3}{2}}$$ $$\frac{d^2y}{{\rm dt}^2}=-GM\frac{y}{{(x^2+y^2)}^\frac{3}{2}}$$ How can I solve those equations, I tried using wolfram alpha but it ...
alper akyuz's user avatar
2 votes
1 answer
315 views

Developing Kepler's first law from the two-body problem

I'm supposed to develop Kepler's first law from the two-body problem. I've the information that I'd get a differential equation which can't be solved by elementary function but however the orbitals ...
Matti Hokkanen's user avatar
0 votes
0 answers
21 views

Interpretation about the exchange of energy among oscillators

I'm studying Hamiltonian system and in particular the role of frequencies in these systems. What I want to understand is about the physic interpretation of some definition. Considerer a Hamiltonian ...
Giovanni Febbraro's user avatar
1 vote
0 answers
36 views

Proving the existence of functions of $C^k$ in variable change

The problem states as follow: Given a parametrization $\alpha:I\rightarrow\mathbb{R}^3$ of class $C^k$ $k\geq1$, $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ verifying $$\alpha_1(t)^2+\alpha_2(t)^2$$ prove ...
Yábir Garcia's user avatar
4 votes
1 answer
128 views

Why isn't a jagged orbit ever observed in the two-body problem?

The two-body problem deals with two planets revolving around a common center relative to one another. Why doesn't the model ever exhibit a jagged orbit and it is always smoothly elliptical? Is there ...
develarist's user avatar
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1 vote
0 answers
143 views

Two-body problem, but with one of the bodies fixed

The two-body problem discusses the orbit of two celestial objects (planets) around each other, often with one of them having heavier mass than the other, which has smaller mass. If the heavier mass is ...
develarist's user avatar
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0 votes
1 answer
30 views

Arbitrary $n-\text{masses}$ for $n-\text{points}$ along the same cirunference create a $n-\text{regular}$ polygon, for $n=3, 4, 5.$

Suppose three points $(q_1,q_2,q_3)$ of abritrary masses $(m_1,m_2,m_3)$ are on a circunference with center at the origin, creating a triangle (non necessarily equilateral). Suppose the the barycenter ...
Davshock's user avatar
  • 127
3 votes
2 answers
231 views

Interval of convergence of Lagrange's infinite series

I am reading a book on Orbital Mechanics for Engineering Students by Howard D. Curtis. In that book it was mentioned (in page 119) that there is no closed form solution for $E$ as a function of the ...
sai saandeep's user avatar
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2 votes
0 answers
133 views

what is the math for gravity causing different kinds of orbit ( ellipse, parabola ,straight line ) for an object in different cases?

case-1: the motion or you can say the orbit of a projectile is parabolic because gravity acts on it. case-2: the motion or the orbit of an object ...
AL vees's user avatar
  • 163
-1 votes
2 answers
46 views

Constant circular motion: understanding $\underline{e_{\theta}}=\frac{d(\underline{e_r})}{d\theta}$

Context: 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion. This may be trivial, but can someone tell me what's wrong with the following reasoning? $$\underline{e_r}=\...
mjc's user avatar
  • 2,249
0 votes
1 answer
90 views

The flow generated by integral of motion sends orbits of Hamiltonian into orbit of Hamiltonian?

I read these two statement in the notes of my teacher that seem to me opposing. Let $H$ an Hamiltonian and let $\Phi$ an integral of motion of $H$, so that $\Phi$ keeps constant value along the orbit ...
Giovanni Febbraro's user avatar
0 votes
1 answer
148 views

On the gradient of gravitational potential energy

This question is about a mistake in working. Consider the gravitational two-body problem, where two particles $\mathbf{r}_1, \mathbf{r}_2 $ with masses $m_1, m_2$ are attracted to each other under the ...
user avatar
1 vote
1 answer
50 views

A quite obvious result on one-variable differential functions

It states as follows: If $f\in C^1((a,\infty))$ is such that its limits exists, that is $ \hat{f}=\lim_{t \to \infty}f(t)$, then there exists a sequence $\lbrace t_n\rbrace$ with $t_n \longrightarrow ...
juan zaragoza's user avatar
1 vote
0 answers
46 views

How off is my understanding of Lambert's theorem to solve Lambert's Problem?

Math Overflow-ites! Hopefully, this question fits here better than places like Space and Physics. And sorry if I ramble a little - I'm rather new to this all. My understanding of Lambert's theorem is ...
kingofthenerdz3's user avatar
2 votes
1 answer
111 views

Periodic Behavior of a Two-Body Problem?

Given the following equations and parameters: does the orbit repeat? If not, why? and what would make it repeat? Any clarification on what makes an orbit periodic would be greatly appreciated! I ...
Clark's user avatar
  • 514
1 vote
2 answers
931 views

A very simple question on motion in a circle.

Question A spacecraft of mass m orbits Earth at a radius R and speed $v_0$ as shown below. An aerospace engineer decides it should orbit at a radius of $\frac{2R}{3}$ instead. The mass of Earth is M. ...
SFR's user avatar
  • 445
2 votes
1 answer
70 views

Normal distance

Recently, I saw a paper "G. H. Darwin, Periodic Orbits" and I don't undertand the concept of "normal displacement $\delta$p". "... Now suppose that x, y are the coordinates of a point on an orbit, ...
José Psicodélico's user avatar
0 votes
1 answer
112 views

Equation of Motion of a particle

I've tried this question over and over, and I'm getting nowhere. I've even tried looking for a solution to help make sense of how to get there, but I've had no luck. Can anyone help me please? A ...
T JM's user avatar
  • 13
0 votes
0 answers
238 views

Two Body Problem with differential equations?

I have been learning up on the two body problem that uses differential equations to solve for a equation of a satellite relative to the planet. The two sources (linked below) both use different ...
Dylan 's user avatar
2 votes
0 answers
74 views

Different kinds of stability that apply to planar periodic orbits, and what do they mean?

This is a question about terminology related to orbit stability. I had wanted to ask about stability of orbits described in the paper Three Classes of Newtonian Three-Body Planar Periodic Orbits ...
uhoh's user avatar
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