A question about orbits and speed

Factual Questions
1

My understanding of how objects orbiting each other goes like this: the **faster **the smaller object (A) is moving relative to the larger object (B), the **higher ** A’s orbit will be around B. That is to say, A will orbit further from B if it’s going faster, and closer if it’s going slowly. If A eventually goes sufficiently fast, it will achieve escape velocity and no longer be orbiting B; if A slows down too much it will eventually crash into B.

Is this correct? Does the Moon move faster relative to the Earth than the International Space Station? Does Pluto move faster relative to the sun than Mercury?

2

No, it’s exactly the opposite. For two bodies of unequal mass in a circular orbit, the orbital speed is inversely proportional to (roughly) the square root of the orbital radius.

Any two masses in space will attract each other, head on; the closer they are to each other, the faster they need to move to get out of each other’s way.

3

And for the specific numerical case of the moon versus the ISS:

moon’s average distance from center of Earth: 385,000 km
same for the ISS: 6800 km, give or take

moon’s average orbital speed: 1023 m/s
same for the ISS: 7706 m/s

Note that (385000 km / 6800 km) = 57, and also [ (7706 m/s) / (1023 m/s) ][sup]2[/sup] = 57.

4

There is an element of truth to what you say. If one object is in a circular orbit and another object at the same position is traveling in the same direction and a higher velocity, it will continue into an orbit that is everywhere above the orbit of the original object. It will be in an elliptical orbit. At it’s maximum distance from the parent object, it will be going slower than the original object. However, as mentioned above, if two objects are in circular orbits, the one further away will be traveling at a slower speed.

5

I’m pretty sure it is in the book Space (James Mitchener, 1982) where the story is told of an early crop of pilots training to be astronauts - they are taken to a circular track and given jeeps that have been rigged up to behave in an orbital manner. As they accelerate, they turn tighter (the mechanics were a bit more complex than that, but I read the book 30 years ago). The aim was for one jeep to catch up to another.

The point was to make a bunch of hotshot flyboys realise that they could not eyeball or seat-of-the-pants anything in orbit, because it was counter-intuitive to the way they were used to flying. They had to rely on external information and timing.

Si

6

GeromeK, you can calculate the moon’s orbital speed fairly quick and dirty as follows: It’s a quarter of a million miles away, making the circumference of its orbit about pi/2 million miles, or about 1.5 million miles. It takes about a month to go round the earth so divide 1.5 million by 30, that’s 50,000 miles per day or about 2000 miles per hour.

Mind you, you could Google “moon’s orbital speed” in less time than it took me to type that out…

7

When you go faster, it makes you go higher. When you go higher, you slow down. When you slow down, it makes you go lower. When you go lower, you speed up.

For a concrete example, consider a Hohmann transfer orbit, a simple and efficient way to get from one orbit to another. You start off in a low, fast circular orbit (say, the orbit of Earth around the Sun) and want to get into a higher, slower circular orbit (say, Mars around the Sun).

First, from your low orbit, you fire your engines to increase your speed (we’ll approximate this process as instantaneous). This changes your orbit from the original circular orbit, to an elliptical orbit. The closest point of the ellipse to the Sun is tangent to your original orbit, and the furthest point is tangent to the orbit you’re trying to get to.

Then, you go halfway around this ellipse, to the furthest point. During this time, you’re not firing your engines at all: You’re just in a freefall orbit, with Mr. Newton in the driver’s seat. Because you’re going uphill all this time, you’re slowing down.

When you reach the furthest point of the elliptical orbit, you fire your engines again, to speed up again. This changes your orbit again: The high point of your orbit stays the same, but you’re moving the low point up to match it, so you’re again in a circular orbit, further out from your original orbit. And even though, both times you fired your rocket, it was to speed you up, that’s more than made up for by the slowing down during the long coasting phase in between, so your final orbit ends up being slower than your original one.

8

I’d previously assumed the opposite since I thought it would explain why orbits are stable.

If orbits are faster the closer they are, that implies that if I were to give the ISS a nudge, say so it’s moving 1 ft per second away from the earth, then its tangential velocity will be too fast for its current orbit and it will just move into higher and higher orbits indefinitely. Is this right?

ETA: I didn’t see Chronos’ post. Still parsing to see if it answers my question…

9

It does, Mijin.

Thanks Chronos; I found that very illustrative. Is this the ‘side of the envelope’ explanation for the “Delta-V” we read about as being the first thing to learn about orbital physics?

A lower orbit has to be faster because gravity is stonger, so you need more centrifugal force to counterbalance the centripital force of gravity. (Of course, centrifugal force is just an apparent force caused by the centripetal acceleration, but it’s an easy way to think about the balance of forces.)

10

Hohman transfer:
In the lower orbit, as you add energy, you go faster - therefore your orbit becomes more elliptical and takes you to a higher point, where you slow down - that speed has translated to potential energy. You’ve slowed down so much that if you don’t add even more energy, you’ll fall back down to the lower orbit, in an ellipse between the high and lwo points.

Add energy (in the right direction vector), and now you are a higher, circular orbit.

Remember grade school physics. Potential enegry - like picking up a rock and putting it on a higher shelf, is stored energy - in this casse, stored by putting it up high in a gravitation field. remove the support and that can turn into kintetic energy (rock falls off shelf).

If you’ve seen those coin-donation things, where you roll a coin down a slot, and it rolls into a flared or cone-shaped bowl… as the coin rolls around the top, it’s slow. As it loses (potential) energy and gets deepr down the slope, it goes faster and faster. An orbital environment is often described as a “gravity well” working in the same manner, except outside the atmosphere there is virtually no friction.

11

http://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html

12

More fun than a spirograph! :smiley:

Wow, I just managed to make the moon collide with Earth. Good thing I’m not in control of the real one.

13

I crashed it into the Sun!

14

Me too. I like how the sun cartoonishly fries the planet.

It’s fun to try and make a really complicated 4 body system, and have it start out really erratic, and try to get them to fall into a stable system with no bodies being ejected into eternity, or getting eaten by the sun.

15

I managed to get one where body 4 starts out close to the sun, gets flung into a much larger orbit, recaptured again, and crashes into the sun at time=~340

body1: 200, 0, 0, 0, -1
body2: 10, 142, 0, 0, 140
body3: 0.01, 166, 0, 0, 74
body4: 0.001, -84, 0, 1, -163
(maximum accuracy)

16

You monster!

Okay, I’m plugging it in too…

17

I accidentally left both bodies 3 & 4 at a mass of .001 and left b4’s x value at 0. It wasn’t long before body 4 scathed by body 2, getting a crazy boost and trajectory, then proceeded to collide right into body 2.

You should’ve seen the look on the inhabitant’s faces!

ETA: Okay, this time, using your exact settings, b4 just careened right into b2. Nice work of solar system billiards.

18

b4 collided with b2? That’s not what I saw with those settings; I wonder if there’s a rounding error or something that causes it to not be exactly repeatable. (I had the speed/accuracy slider set to full accuracy.)

19

Me too. Double checked the settings, and maxed out the accuracy. Must be a floating point thing depending on our CPUs? I’m on a 12-core MacPro, Intel Xeon.

Did you uncheck System Centered? Having it checked removes the systems center of mass. I kept it checked, unchecking it now. Not sure if it makes a diff.

20

For those who like seeing physical analogies, rather than computer simulations:

Play with one of those funnel shaped coin collection devices that charities often put out. They have a ramp that inserts the coin into a large “orbit” and then it decays until the coin falls into the funnel.

It is fairly obvious that a coin in an infinite orbit (a flat table top, say) would only need enough speed not to fall over. As the slope becomes steeper (stronger gravity) more speed is needed.

Higher orbits have more energy, but that energy is the potential energy due to their position in the gravity well. Lower orbits have more kinetic energy due to higher speed, but this is more than offset by the reduction in potential energy due to being farther down in the gravity well…so the total energy is reduced.

So it IS rather perverse that when an orbiting body loses energy (due to drag, say) it’s orbit decays into a lower one, which is faster. Yet if you add energy (by firing a rocket engine, say) then the body first accelerates, but eventually ends up in a higher, slower orbit.

Another way of looking at this is escape velocity. It is fairly intuitive, I think, that if you stay far, far away from the black hole, you don’t need much speed to avoid falling in. If you want to get closer though, you better be moving quick, so you move on past without getting sucked in. Of course orbital and escape velocities are different things, but they are closely related (possibly just off by a constant ratio ??)