The solar systems and space probes

[vidmar]

OK the title is a bit practical, I've made the question somewhat more theoretical (technical, whatever), in particular:
One is given three bodies: two relatievly massive (for example the Sun and a massive planet like Jupiter) and one relatievly small (for example a satellite) - meaning that its gravitational effect on the other two may be neglected or is "turned off" (depending on how you wish to formulate the problem). The two "massive" bodies, denote them by M1 and M2, are in a closed orbit around their centre of mass (CM). Does there exist a set of initial conditions (fot time t0) involving (velocities of M1 and M2, position of satellite relative to the centre of mass) which will "thrust" the satellite ad infinitum (i.e. for every distance from the centre of mass of the system (CMS), there exists a time t from t0, for which the satellite achieves this distance), provided that it does not move relative to the CMS at the biginning (t0). Basically what I am asking is, can I place the satellite in the gravitational field of two objects and thrust it ad infinitum with no beginnig kinetic energy relative to the CMS taking advantege only of the gravitational pulls of the objects (with a few limitations of course:) Naturally I would like only Newton to "bear on this problem" (general relativity would be slightly to much.
Also I've come across an interesting fact that given 5 objects (gravitation, Newton) they can go to infinity in finite time! If anyone has anything specific on that, feel free to post.

 

[CarlB]

Yes.

let's suppose that the orbital system, like the sun and jupiter, has two stages of massiveness. There is a very massive sun in the center which we will assume is stationary. And there is a massive planet which orbits the sun in a circular manner. The speed of the planet in its orbit is constant and that speed is determined by its distance to the sun and the sun's mass as follows:

$$F/m = a = GM/r^2 = v^2/r$$

or

$$v = \sqrt{GM/r}$$

Now the slingshot effect can give a velocity to the test mass of 2v, (a result you can obtain by considering the problem as a scattering in the rest frame of the massive particle and then translating back to the rest frame of the sun) so its kinetic energy afterwards will be:

$$KE = 0.5mv^2 = 0.5 \times 4 GmM/r = 2 GmM/r$$.

This is twice the object's gravitational potential energy, so it is the case that the test object has achieved escape velocity.

Carl

 

[charng]

I can say that this is a very interesting and complex problem that requires a thorough understanding of Newton's laws of motion and gravitational forces. To answer your question, it is possible to place a satellite in the gravitational field of two massive objects and thrust it ad infinitum with no initial kinetic energy relative to the center of mass. This can be achieved by carefully selecting the initial conditions of the satellite's velocity and position relative to the center of mass.

However, there are limitations to this scenario. The satellite's trajectory will be affected by the gravitational forces of the two objects, and it will eventually reach a point where it will either collide with one of the objects or escape the system entirely. This is due to the complex nature of the three-body problem, where the interactions between the three bodies cannot be solved analytically.

Additionally, the concept of five objects going to infinity in finite time is known as the "three-body scatter problem" and it has been studied extensively by scientists. In this scenario, the five objects would have to be carefully arranged and have specific initial conditions for this phenomenon to occur. This is a rare and complex occurrence that requires further study and understanding.

In conclusion, while it is possible to place a satellite in the gravitational field of two massive objects and thrust it ad infinitum, there are limitations and complexities involved in this scenario. Further research and understanding of the three-body problem is necessary to fully comprehend and predict the behavior of such systems.