Expert: Courtney Seligman - 11/8/2010

Question
QUESTION: Dear expert

Let's suppose a motionless planet called A and its satellite B.
B revolves around A at the velocity v0.

Now, if A revolves around a center C,
would the angular velocity of B relative to A be higher than v0?

In other words, do the orbital speed of a planet affect the orbital speed of its satellite?




         Sincerely






ANSWER: The orbital speed of the planet does not affect the orbital speed of a satellite. This is a consequence of what is called Galilean Relativity -- normally, motions look the same whether you are moving or not. As an example, if you throw a ball upward it goes up (and then down) the same as if you were at rest, even though you're moving to the east at several hundred miles an hour as a result of the Earth's rotation. In the same way, the motion of the Moon around the Earth looks the same as it would if the Earth were stationary, even though we're actually moving around the Sun at over sixty thousand miles an hour.

So in the case you cite, the orbital speed would still be v0.

(P.S. Sorry for the delay in answering. I was out all day, and just picked up your note.)

---------- FOLLOW-UP ----------

QUESTION: Thank you for your answer.


Of course, if the Earth moved in a straight line, that wouldn't be difficult because the Geocentric frame of reference would be inertial.
but since it revolves in a circle, it put a centrifuge acceleration on the Moon
so, Earth's frame of reference would be accelerating and thus non-inertial.

So if Moon receives a centrifuge acceleration from Earth's orbital movement, can't its velocity be affected? Do you ocnfirm your answer?



         Sincerely


Answer
You are correct that the non-linear motion of the Earth would produce a (small) non-inertial acceleration of the Moon. But that ignores the fact that the reason for the Earth's non-linear motion (the gravity exerted on the Earth by the Sun) is also acting on the Moon, in almost exactly the same way. As a result, the original answer is still correct, to a high degree of accuracy.

In other words, as the Earth "falls" around the Sun, the Moon is doing the same thing. It is "falling" a little slower if it is on the outside of the orbit near full moon, and a little faster if it is on the inside of the orbit near new moon; but on the average, over the course of the month it is doing almost exactly the same thing as before (the effects on one side of the orbit "canceling out" those on the other side). So to a very good approximation, there is no difference between the motion of the Moon around the Earth when the Earth is in orbital motion around the Sun, and the motion it would have had if the Earth were moving in a straight line.

Now, if you want to look at very small effects, there are differences between the motion of the Moon as it exists, and as it would have been if the Earth did not move around the Sun. But as it happens, even taking those into account, the average motion (corresponding to the original v0) is the same as before. There are "perturbations" which cause the Moon to move faster or slower (or the same speed but in different directions) at different times, but they don't affect the overall orbital size or speed due to v0old (or if there was a difference, v0new).

That doesn't mean there are no effects at all. Although essentially unchanged, the orbit gradually rotates in space, in two ways. First, the plane of the orbit (which is slightly tilted relative to that of our orbit around the Sun) gradually rotates to the west (which is the opposite direction from the orbital motion of the Moon, which is to the east), so that over a period of 18 years, the orbit turns once around in space. (Note: This effect would not exist if the Moon's orbit were not tilted relative to ours.)

The other effect is that the place where the Moon is closest to us (perigee) also rotates once around, within the plane of the orbit, once every 18 years. (This effect would still exist, even if the Moon's orbit were not tilted; but would not exist if the orbit were a circle.)

Because of these effects, when the Moon has gone once around the Earth, it is not back in exactly the same place in space relative to us, but about one lunar diameter "off". This also leads to complications in defining the length of the lunar orbit, as we can measure it (1) relative to the stars, (2) relative to the Sun (which defines the cycle of phases), (3) relative to the place where the Moon's orbit crosses ours, and (4) relative to the place where the Moon is at perigee. So if you were to pick one of these as a reference, you would get a slightly different value for v0 than if you picked another one as a reference. But the maximum deviation in speed (for one reference frame relative to another) is less than the ratio of an hour to a month (a little over one tenth of one percent), and would be positive in some reference frames, and negative in others; meaning the average value of the difference between v0 and v0new for the various reference frames would still be very close to zero.

However, I should note that all the effects described in the previous three paragraphs would still exist even if the Earth were stationary in space, providing that we allowed the difference in gravity of the Sun on the Earth and Moon to still affect the lunar orbit. So they aren't due to our motion around the Sun, but to the fact that the Earth and Moon aren't in the same place. And although there is a small difference between v0 in one reference frame and v0x in some other reference frame, the difference is (as already stated) very small, and fluctuates back and forth around a value which is even closer to zero. So to a very high degree of accuracy, my original (albeit deliberately simplified) answer is still correct.

I apologize for having given you what turned out to be an oversimplified answer the first time, but most people who ask such questions are just looking for a basic understanding of things, and would just be confused by the more detailed discussion above. It's nice to be asked a question by someone who is aware of the possible complications, and can appreciate a more thorough answer. I just hope that I haven't swung too far the other way, and made things more complicated than you wanted. If I did, please feel free to ask for clarification of any point.