# Velocity in planetary orbits

by AakashPandita
Tags: orbits, planetary, velocity
 P: 137 Is the speed of different bodies in the same planetary orbit equal to each other? I think it should be the same because the acceleration is always the same in such a case. But it was not so when i compared the speed of moon per m/s with that of the international space station.
P: 476
 Quote by AakashPandita Is the speed of different bodies in the same planetary orbit equal to each other?
yes

 Quote by AakashPandita But it was not so when i compared the speed of moon per m/s with that of the international space station.
not the same orbit
 Mentor P: 11,889 The Moon is MUCH further away than the ISS. The ISS orbits between about 375 - 400 km from Earth's surface while the moon has an average orbital radius of about 384,000 km. So the Moon is about 1000 times further away than the ISS is.
 P: 137 Velocity in planetary orbits sorry..actually i intended to ask why the speed of the ISS as well as the moon is not the same...they are both accelerated and have the same "g".
 P: 2,179 The accelerations are not the same. The force of gravity drops off as 1/r^2, and since the moon is about 60 times farther from the Earth's center than the ISS, its acceleration is about 1/3600 as large.
Mentor
P: 15,170
 Quote by AakashPandita Is the speed of different bodies in the same planetary orbit equal to each other?
In general, NO.
 I think it should be the same because the acceleration is always the same in such a case.
You are ignoring that the orbiting body also attracts the body that it is orbiting. That the Earth is many orders of magnitude more massive than even the biggest artificial satellite orbiting the Earth means that the acceleration of the Earth toward these artificial satellites will be negligible; two artificial satellites in the same orbit will have the same period. That the Moon is about 1/81 the mass of the Earth means that the acceleration of the Earth toward the Moon is not negligible.

Kepler's third law is only approximately correct. A better formula for the period at which two objects, one of mass M and the other of mass m, orbit one another is

$$P=2\pi\sqrt{\frac{a^3}{G(M+m)}}$$
P: 476
Quote by D H
 Quote by AakashPandita Is the speed of different bodies in the same planetary orbit equal to each other?
In general, NO.
With different velocities it wouldn't be the same orbit.
Mentor
P: 15,170
 Quote by DrStupid With different velocities it wouldn't be the same orbit.
Yes, it would. Consider the Moon. Suppose you magically replace the Moon with an object several orders of magnitude smaller in mass than the Moon but keep the position and velocity the same as the Moon's. This object will be in a different orbit. To get the same orbit as the Moon's current orbit you will have to change the velocity.

This is an important consideration in the formation of a planetary system from an accretion disk. Given a planetesimal in a circular orbit of radius a about a nascent star amidst some particles orbiting at the same distance, the planetesimal will be moving slightly faster than the individual particles. The planetesimal will have an orbital velocity of $\sqrt{G(M+m)/a}$ where M is the mass of the nascent star and m is the mass of the planetesimal; the orbital velocity small particles co-orbiting with the planetesimal will only be $\sqrt{GM/a}$. The planetesimal will plow through and sweep up the surrounding particles. This can lead to the planetesimal migrating toward the star.
 Mentor P: 11,889 Why would the orbit change when the mass of the object changes? Is it because of the objects reduced attraction of the Earth towards it?
Emeritus
PF Gold
P: 2,361
 Quote by Drakkith Why would the orbit change when the mass of the object changes? Is it because of the objects reduced attraction of the Earth towards it?
Because both the object and the Earth actually orbit their common barycenter.

As mass of the object increases, the barycenter moves closer to the object and away from the center of the Earth. The radius of its orbit around the barycenter decreases, while the distance between object and Earth remains the same.. Since the centripetal force needed to maintain a circular path is equal to

$$\frac{mv^2}{r}$$

A decease in v is needed to maintain the same object-Earth distance.
 Mentor P: 11,889 Ah ok, that makes sense Janus. Thanks.
P: 137
 Quote by Janus Because both the object and the Earth actually orbit their common barycenter. As mass of the object increases, the barycenter moves closer to the object and away from the center of the Earth.
how?
wouldn't the distance from the barycenter remain the same when mass increases?
P: 476
Quote by D H
 Quote by DrStupid With different velocities it wouldn't be the same orbit.
Yes, it would.
Is the orbit really characterized by its shape of the only?
Mentor
P: 11,889
 Quote by AakashPandita how? wouldn't the distance from the barycenter remain the same when mass increases?
No, as the object would pull the Earth towards it more than it did before, hence the barycenter would move.
 P: 137

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