# Velocity in planetary orbits

1. Oct 13, 2011

### AakashPandita

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.

2. Oct 13, 2011

### DrStupid

yes

not the same orbit

3. Oct 13, 2011

### Drakkith

Staff Emeritus
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.

4. Oct 17, 2011

### AakashPandita

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".

5. Oct 17, 2011

### phyzguy

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.

6. Oct 17, 2011

### D H

Staff Emeritus
In general, NO.
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

$$P=2\pi\sqrt{\frac{a^3}{G(M+m)}}$$

7. Oct 17, 2011

### DrStupid

With different velocities it wouldn't be the same orbit.

8. Oct 17, 2011

### D H

Staff Emeritus
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.

9. Oct 17, 2011

### Drakkith

Staff Emeritus
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?

10. Oct 17, 2011

### Janus

Staff Emeritus
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.

11. Oct 17, 2011

### Drakkith

Staff Emeritus
Ah ok, that makes sense Janus. Thanks.

12. Oct 18, 2011

### AakashPandita

how?
wouldn't the distance from the barycenter remain the same when mass increases?

13. Oct 18, 2011

### DrStupid

Is the orbit really characterized by its shape of the only?

14. Oct 21, 2011

### Drakkith

Staff Emeritus
No, as the object would pull the Earth towards it more than it did before, hence the barycenter would move.

15. Oct 31, 2011