In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion.
In order to change the radius, additional energy is required, the total energy of mass m on a circular orbit is given by:
$$E_{total} = - \frac {GMm} {2R},$$
The change in energy between orbits ##R## and ##R_{0}## is:
$$\Delta E_{total} = \frac {GMm} {2} \cdot \left( \frac 1 R_{0} - \frac 1 R...
Obviously the mechanical energy of the total system remains the same.
But I'm having a hard time determining of the ME of the planet is constant or if it is changing.
Hi,
suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it).
Can its integral curves cross at any point ? Thanks.
Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...
Does the ISS remain oriented with the Earth? I.e. it rotates 360 degrees every 24 hours, or is it fixed with the stars?
If the latter, there should be a time when it's visible end-on.
I looked it up but did not find the deets.
I guess that what the gyros are for.
TL;DR Summary: I’m worldbuilding for a novel I’d like to write. I’d like my planet to have 3 moons, and I’m curious how to develop the planet and moons in a scientifically sound way.
I apologize in advance if any of this doesn’t make sense or if terminology is way off or misused. This stuff is...
As background, I grew up in a working class neighborhood where few parents went to school. In 1962, John Glenn came back from space. I was told this. I am not sure what the adults believed back them. "Space" was new. It would be 4 years before star trek, and even then,Space was labeled the...
I want to identify:
##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##.
##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##.
The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer.
In 1 how to define the...
I am really lost on how to deal with this. Since this is an elliptic orbit, the mechanical energy is negative. For the rocket to escape orbit, we have to get the mechanical energy to be equal to or greater than zero. I thought at first that it would escape in the perigee, since that's where the...
During one of my class lectures, we were shown a graph of star S2's orbit motion (shown at the bottom of this post) and was told to try and figure out how to find the mass of the central object using Kepler’s 3rd law in solar units, as shown below:
We were also given a hint to use the arcsec...
Schutz finds that the orbital period for a circular orbit in Schwarzschild is
$$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$
He gets this from
$$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$
Where previously he had ## \frac {d\phi}{d\tau} = \tilde L / r^2## and ## \frac {dt}{d\tau} =...
A satellite is orbiting a planet in a circular orbit. The planet's mass doubles instantly. What happens to the orbit of the satellite?
I think it would move to an elliptical orbit with major axis equal to the old radius and a minor axis equal to either 1/2 or sqrt(2)/2 times the old radius. I'm...
Modeling the time evolution of the sun and earth orbiting each other using ##F = \frac{GMm}{r^2}## is straightforward. However, it appears that modeling the time evolution of the same 2 body system using general relativity seems to be a hard/intractable problem?
There was in depth discussion by...
In MOND should two small masses placed at a great distance orbit faster than in Newtonian dynamics?
Would it be theoretically possible to do this experiment in the solar system, or would the presence of the sun and earth be a problem?
Einstein said, when describing someone falling off a building, that the Earth accelerating up to meet him/her. Without the Earth getting larger in all directions as the paradox goes, it curvature of space-time which is why you can have the acceleration up without the surface moving up as you...
I'm looking for a database of binary stars within 10 PC of Earth, including information such as eccentricity of orbits, their distance from one another, etc. I'm hoping to find a list with this information, or just a collection of pages with this information. I've tried Simbad but I can't find...
The wikipedia article on Lunar distance contains a confusing graph. The graph seems to suggest that the eccentricity of the lunar orbit is maximal in january and ~july, and minimal in april and ~october. I think the eccentricity should be constant. Is wikipedia right or wrong, or is there some...
Listen to the following arguments:
Earth's orbit isn't perfect ellipse because classically there is the gravitational field of moon and possibly of Mars and Venus which affect it
According to general relativity isn't perfect ellipse because there is the curvature of space time which doesn't...
I am attempting this homework exercise part b).
I've modified my code but I get error overflow message. My goal is to modify my code so it returns kinetic and potential energy of Earth's orbit.
I made a new f(z,t) and modified the rows 99 and 100 with dz[2]=-G*M*m/r, and dz[3]=0.5*m*y**2 but...
I am attempting this homework exercise but my plot does not show the whole trajectory. I don't know if it is something wrong with my equations or if it is a plotting matter.
Cheers!This is my code:import matplotlib.pyplot as plt
import numpy as np
import scipy.integrate as spi
G =...
Problem: a particle of mass m is in a circular orbit around a planet at a distance R from the center. The planet mass is M and it's radius is R_0.
What is the tangential impulse that will cause the particle to brush against the back of the planet? Describe the orbit.
The attempt at solution...
Epimetheus and Janus switch places periodically, because they follow a horseshoe orbit around Saturn, which is considered a "pseudo-orbit" around each other. I'm thinking that if you look at the conic sections - taking an elliptical orbit of two moons to greater and greater extremes until they...
We know that the centripetal force and the tangential velocity is responsible for the motion of Earth around sun. Newton’s second law says F=ma. If we all get together(whole population) in one place like parallel to the tangent to orbit and jump can we displace Earth from its orbit?
A station is orbiting a planet at a distance R1, a moon is orbiting the planet at distance R2 with the period T. The planet itself has a radius rp and a mass mp. We know that when an object adds its velocity at a point in the orbit, the height of the opposite orbit will increase. Determine the...
1) Considering the forces on one of the moons, I have: ##\frac{GMm}{(10R)^2}+\frac{Gm^2}{(20R)^2}=m\frac{v^2}{10R}\Leftrightarrow v=\sqrt{\frac{G}{10R}(M+\frac{m}{4})}.##
2) Considering the initial situation in which the satellite is at rest on the surface of the planet...
Does the Earth's orbit get very slightly larger over long periods of time due to the expansion of the universe? If so does it stay at the slightly larger distance or somehow does the energy to go back to the earlier orbit go somewhere else and if so where? If the orbit does not get slightly...
I was already puzzled by the concept of orbiting a Lagrange point and then I find out it's about the same size orbit as the Moon. I am thinking that if there was no Moon that the Earth and the Sun are far enough away to be treated as points and so that there would be an exact distance further...
I hope this is okay to ask here. I'm working on a sci-fi short story, and for the purposes of the story I want to have a small ship that maintains its position over a specific location on the Earth's surface.
Originally, I thought this would be easy. After all, that's what geostationary...
I actually have worked through the solution just fine by taking the derivative of \vec{L}:
\frac{d \vec{L}}{dt} = \dot{\vec{v}} \times \vec{M} - \alpha \left(\frac{\vec{v}}{r} - \frac{\left(\vec{v} \cdot \vec{r}\right)\vec{r}}{r^{3}}\right)
I permuted the double cross product:
\dot{\vec{v}}...
Hi :) This is a problem from David Tong's Classical Dynamics course, found here: http://www.damtp.cam.ac.uk/user/tong/dynamics.html. In particular it is problem 6ii in the first problem sheet.
Firstly we can see the lagrangian is ##L = \frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2) -...
Summary:: A question on a recent exam was, "At what eccentricity does an orbit experience a velocity at periapsis that is twice the velocity an apoapsis?" I don't know why the provided solution is correct.
On a recent exam, one of the questions was "At what eccentricity does an orbit...
An Analemma is the shape that arises from taking a picture of the sun everyday at the same time of the day. This shape also have depth to it since Earth's orbit is elliptical, showing what looks like an wobbly orbit. Isn't this the "real" orbit of Earth around the sun?
Summary:: What is the maximum safe mass of an asteroid in geostationary orbit before it causes problems?
Hello everyone,
If there was an asteroid in geostationary orbit around the Earth, over the Pacific Ocean, what would be the highest mass it could have before it would start having...
If a hypothetical spacecraft could keep station at the 2nd focus of Earth's orbit, what useful observations could be made?
Each minute sees a new solar triangle Earth Sun Craft (ESC). Swept area remains constant, as should length SC, and length CE + ES.
With the Sun as our point source...
I have read that an electron requires certain minimum energy of threshold frequency to move an orbit
However the energy needed decreases with increase in shell number
The transition energy is reduced with each orbit
For example
The energy to shift an electron from 1st to 2nd orbit is much...
Here's the problem setup, my student and I are stuck.
A disk is rotating at constant angular velocity ω, and we are watching a point on the rim, parameterized by the angular position θ, move. Because we are observing the motion from an inclination angle Ψ, we do not always observe the...
I recall some time ago seeing a GR equation describing the rate of orbital energy loss from the moving objects in orbit generating gravitational waves. I can no longer find this equation again. I am hoping someone can help me.
Hi guys,
just a short question of Hohmann transfer.
I got the derivation for the required ##\Delta v##, composed by the sum of two impulses, for establishing on a larger orbit... but how do we demonstrate it's actually the transfer which requires the smallest ##\Delta v##?
We can write the Newtonian metric in the form of
$$ds^2 = -(1 - 2M/r)dt^2 + (1+2M/r)[dr^2 + r^2d\Omega^2]$$
In order to obtain the orbit equation I have written the constant of motion,
$$e = (1 - 2M/r)(\frac{dt}{d\tau})$$
and
$$l = r^2sin^2(\theta)(\frac{d\phi}{d\tau})$$
I can divide the...
Looking at L&L's solution to problem four of section §106. Lagrangian for a system of particles:\begin{align*}
L = &\sum_a \frac{m_a' v_a^2}{2} \left( 1 + 3\sum_{b}' \frac{km_b}{c^2 r_{ab}} \right) + \sum_a \frac{m_a v_a^4}{8c^2} + \sum_a \sum_b' \frac{km_a m_b}{2r_{ab}} \\
&- \sum_a \sum_b'...
function Asteroid_Mining
clc
%Initial conditions
g0 = 9.81; %gravity (m/s^2)
p = 1.225; %atmospheric density at sea level (kg/m3)
Re = 6378; %radius of Earth (km)
Ra = 7.431e7; %distance of Bennu from Earth in (km) [August 2023]
G = 6.674e-11/1e9; % Gravitational constant (km3/kg.s2)
mu =...
Hi. can you help me to plot orbit around Earth with sphere in MATLAB?
I wrote the code that plot 3D orbit but without earth.
the code:
clear all
close all
clc
R=6371e+3;
r0=3e+5;
p0=[R+r0;0;0;0;10000;0];
[t,p]=ode45(@sattelite,[0,12000],p0);
xout=p(:,1);
yout=p(:,2);
zout=p(:,3);
vx=p(:,4)...
hi guys, I'm trying to write a program in MATLAB to solve and plot equation of motion of 2 body problem but it errors as i don't know what it says!
do you know how to help me? please!
The main equation is r(double dot)=-(mu)*r(hat)/r^3
The code that i wrote is:
clear all
close all
clc...
I've already found the potential and force that produce the given orbit. my results were:
##V=-\frac{al^2}{mr^3}##
##\vec{F}=-\frac{-3al^2}{mr^4}\hat{r}##
Now, I've been trying to find the period using the equation
##t=\sqrt{\frac{m}{2}}\int_{r_0}^{r}\frac{dr'}{\sqrt{E-V_{eff}}}##
Using...
I tried in the first place to use the effective potential of a parabolic orbit which is 0 to get the angular momentum L.
Evaluating the function U(r) at r = rP i get U(rP) = L^2/(2m(rP)^2) - GmM/rP = 0.
Here I get L = m√(2GMrP).
Now the relationship between angular momentum L and areal...
Recently, when reading an entry about Mercury's perihelion shift, someone mentioned a "hand-wavy" explanation as to why GR predicts the orbit so precisely. I was wondering if there was some elementary way to expound on what he was saying. Fundamentally, the comment said something to the effect...
Hi,
They gave me this formula T = 2piR / v, with T the revolution period of the satellite, R the distance between the center of masses and v the velocity.
They gave me the value of G and the eath's mass and asked to determine the value of R.
I don't even see fromwhere I should start...
Thank...