Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: seth on 17/06/2008 02:52:20

OK, so my physics teacher posed this question, which is completely theoretical, but I'd like to figure out:
There are two space ships orbiting the earth, Spaceship A and Spaceship B. Spaceship B's orbit is larger than A's, and thus goes it slower. An ignorant scientist attaches a giant spring to the ships in order to make them orbit at the same speed. The idea being that the force of the spring wanting to coil will pull Spaceship B and make it orbit faster (while maintaining is orbital distance). Why wont this work?
The question has to do with moons and how they orbit. Any help would be appreciated, especially laws etc. that explain why this is impossible. Thanks
[diagram=354_0]

I can't go into the maths of it as I'm just a dumb rodent, but it's all down to mass, speed, distance, gravity & angular momentum.
The easy answer is that the orbital speed of an object is determined by its mass and the radius of its orbit (which determines the gravitational attraction between the orbiting object and what it is orbiting). For any given mass & velocity, there is only 1 stable orbit. You can also say that for any given mass and radius of orbit, there is only 1 velocity that will result in a stable orbit. An increase in velocity will cause the object to fly off out of its orbit due to angular momentum (gravity will no longer hold the orbit stable), a decrease will cause the orbit to decay (its speed will no longer be sufficient to balance the gravitational attraction).
By connecting 2 objects that have different orbits you will cause the inner of the 2 to decelerate and the outer one to accelerate. Consequently, their orbit will become unstable.
Connecting them with a spring may introduce extra complications (check out Hook's Law).
I'll leave it to our esteemed resident physics whizzos to dot the Is & cross the Ts.

An interesting thought experiment.
Even without the spring, you need to remember that they are already affecting each other  all masses do because of the gravitational force between them. They are actually orbiting around each other in a very 'loose' elliptical orbit whilst their common Centre of Mass is orbiting around the Earth, in the same way that the Earth and Moon orbit each other but still go round the Sun. It may take hundreds and hundreds of Earth orbits to happen but the Earth orbits of the ships will gradually change and the inner one will become the outer one and back again. Orbiting satellites are constantly affecting each other in this way and fuel is used up in keeping them in the right places ('on station'). It's an effect which limits how many you can have up there and their permitted spacing.
When the spring is attached, the two masses will be pulled together and the actual stiffness of the spring and their masses will determine how fast they come together. They would come together with a bang at their common centre of mass BUT, because they have different speeds (angular momentum of their orbits) they will rotate around the CM. Their paths around each other will oscillate in speed and separation. The characteristic of their mutual orbits, because the attraction is governed by Hooke's law and not the inverse square law, will not be a nice, stable looking ellipse but a whole range of possible shapes (petal shaped orbits), depending upon the initial conditions. If the spring were 'damped' in some way, they would end up in circular orbits around their CM  i.e. no oscillation because of the conservation of angular momentum.
The CM will carry on in orbit around the Earth at the same rate as it did before (before they were joined). It will have to because there is nothing to alter that.

show off [:(!]

It seems a little paradoxical but for objects in orbit to speed up or slow down they have to do the opposite.
If the outer spaceship B speeds up it will move out, but moving out puts it into a slower orbit. If spaceship B wants to speed up, it must first slow down, which will drop it into a smaller orbit, which is faster.
When something like the Soyuz or Shuttle has to rendevouz with the ISS it'll catch up, or wait for the ISS by changing it's orbit, not by trying to directly accelerate to catch it, or slow itself down to wait for it.
Tethered Satellites are a current area of experimentation, and apart from breakages of the tether, have been considered largely successful.

If the outer spaceship B speeds up it will move out, but moving out puts it into a slower orbit. If spaceship B wants to speed up, it must first slow down, which will drop it into a smaller orbit, which is faster.
But there is a difference between velocity and angular velocity.
To move out, you have to gain Potential Energy. All your engine can give you is Kinetic energy. More confusion.

Isn't this similar to the situation on Mars where the spring is replaced by gravity?
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fen%2Fthumb%2Ff%2Ff8%2FMoonofmars.gif%2F615pxMoonofmars.gif&hash=ccb572d1201945e9a324ac3de5f86ba9)

A body in orbit is travelling along a straight line in curved space sophie. I don't think potential energy due to gravity comes into it  that is, it doesn't have any potential energy until it's stopped. And the energy needed to stop it then becomes... [;D]

Sorry turnipsock, but I can't help thinking of a fried egg on a plate, with a cherry and two peas on the side. Oh yeah, and a 'bad' big pea that's been put to the side of the plate, not to be eaten.

Egg, cherry & peas? Interesting dinner [:D]

A body in orbit is travelling along a straight line in curved space sophie. I don't think potential energy due to gravity comes into it  that is, it doesn't have any potential energy until it's stopped. And the energy needed to stop it then becomes...
Nope; you have both GPE and KE all the time.
In orbit, your GPE depends only on your position wrt the Earth's centre. Your KE depends upon your speed. In a circular orbit, the two remain constant. In an elliptical orbit, there is a continual transfer between GPE (max at apogee and min at perigee) and KE (min at apogee and max at perigee).
If you choose to model it using curved space then you have to make you analysis entirely using GR. You're welcome to try but you would have to include a treatment of time and mass changes which would lead you to changes in Kinetic Energy, in any case.
The above is the classical, Newtonian, explanation which is near enough for most Astronomical observations. You should get that right in your mind before going for the hard stuff.

Isn't this similar to the situation on Mars where the spring is replaced by gravity?
No, because the spring doesn't follow an inverse square law so the details of the motion will be very, very different.
The ISL is there because of the geometry of Space and the resulting orbits are very simple. If you have anything other than an ISL, the orbits can go wild.

The Newtonian concept of gravitation was an immense achievement and serves as a good approximation, as does the concept that atoms are indivisible, which I believe is still taught to young children.
GR isn't hard stuff  not even thinking of gravitational 'force' as time gradients  GR is one of the most simple and elegant things ever conceived.

GR isn't hard stuff  not even thinking of gravitational 'force' as time gradients  GR is one of the most simple and elegant things ever conceived.
It may not be hard to 'get' GR qualitatively but I bet it's harder to derive the elliptical orbit that way than it is using Newton.
Perhaps you'd like to give us a demo? I wouldn't know where to start.

The Newtonian concept of gravitation was an immense achievement and serves as a good approximation, as does the concept that atoms are indivisible, which I believe is still taught to young children.
GR isn't hard stuff  not even thinking of gravitational 'force' as time gradients  GR is one of the most simple and elegant things ever conceived.
You lucky! I still don't even know what are time, space, velocity, mass, energy, in GR.

sophiecentaur: I'm not sure I see the problem with an ellipical orbit. A body in a circular orbit remains at the same point on the gradient, and so follows a constant path. A body in an elliptical orbit is moving up and down the gradient, as shown by it's changing path.
Newton gives a very good approximation but it's incomplete, and it's because of that incompleteness that it appears to be simpler. However, it can never give you an exactly correct answer. GR is more complete (but still incomplete) and that greater degree of completeness necessarily requires more things to be taken into consideration. There's nothing magical or mystical about that.
What would be the point of me giving you a demo? What would that achieve? And I'm sure you do know where to start  perhaps you don't know how to off the top of your head, but you must certainly know how to find out.
lightarrow: Time, space (spacetime) and energy appear to be fundimentals whichever model you use.

The point I was making about deriving the equation of an orbit is that, using Newton and an inverse square law, you get a perfect ellipse by straightforward analytical processes. That's just a bit of maths, of course but it relates pretty closely to what we see up there.
The GR model is not so clear because it doesn't use simple cartesian geometry and introduces brain ache.
If someone asks what happens when . . . . then the best explanation is to talk in familiar terms. Newton works so use Newton.
Simple laws like Newton and Hooke's give a very good comparison between the two situations in the original posts.
I am not a complete dolt where these things are concerned but I would choose the simple explanation any day rather than one which involves another layer of difficulty (if you get the right answer both ways).
A lot of the questions on these forums are meant in a straightforward way because they are often posed by non (or beginner) Scientists. I am sure they would rather have an answer which converges onto a bit of understanding than an answer which merely introduces another level of complexity. Most people, including me, would rather have the short answer first!
As far as the GR answer is concerned, it is still only half way there. You might just as well supply an answer in terms of String Theory and really confuse us all.
In cases like this, I think it would be better to open another thread with the harder / more complete, discussion and I would read avidly, with a possible contribution where I feel competent.

What would be the point of me giving you a demo? What would that achieve?
For a start it would make me shut my gob!!

You speak very eloquently and I enjoy reading what you say  why would I want you to shut your gob?

lightarrow: Time, space (spacetime) and energy appear to be fundimentals whichever model you use.
To compute a moving body's energy, you have to know, at least, its mass and its velocity. How you define velocity in a curved space?

You can't do a simple conservation of energyanalysis of things in GR. You need a full field theory treatment, using a stressenergy tensor (http://en.wikipedia.org/wiki/Stressenergy_tensor#In_general_relativity). GR is certainly the more elegant theory, but I can't imagine a good reason for wanting to use it to answer this question. You need to employ it when you need extreme precision, or when you're working with supermassive objects or huge scales that make Newtonian mechanics inaccurate.
Lightarrow: I think your KE is contained in that stressenergy tensor, which in turn depends on the interaction of the mass with the curved spacetime. (Since you fold time into the equations, you know its velocity from its path in spacetime).
sophiecentaur: I took a graduate seminar on GR a few years back, and we did derive the elliptical orbits from GR. The way we did it was to show that in the smallmass, largedistance limit, GR reduces to Newton's equations, and you can solve it from there. [;)] GR ends up not having simple, closedform solutions in most cases.

sophiecentaur: I took a graduate seminar on GR a few years back, and we did derive the elliptical orbits from GR. The way we did it was to show that in the smallmass, largedistance limit, GR reduces to Newton's equations, and you can solve it from there. [;)] GR ends up not having simple, closedform solutions in most cases.
That's interesting  so, basically, you get the answer via Newton rather than going directly there? It's reassuring, somehow.
SO, Newton is certainly the answer for someone who's question starts off "My Physics Teacher . . ."

The way we did it was to show that in the smallmass, largedistance limit, GR reduces to Newton's equations, and you can solve it from there.
Even for light's deflection near the mass?

The way we did it was to show that in the smallmass, largedistance limit, GR reduces to Newton's equations, and you can solve it from there.
Even for light's deflection near the mass?
We worked in particular with massive objects orbiting other massive objects. I don't know off the top of my head if you get zero deflection for light using the same limits, but I'd have to guess that you don't.

Hi guys,
You have all responded better than I could,
But this is my 1 cent peace.
If they were joined by great springs they would act no differently from objects held together by springs on earth but on an immense scale.
A lot of really frightening bouncing, jumping contortion, pogo stick madness. [;)]
The universe has a better idea gravity, but it is a nice thought experiment.
Alan

A lot of really frightening bouncing, jumping contortion, pogo stick madness. wink
With a bit of maths we could do better than that.