Question about orbits and Kepler's problem

[L0r3n20]

I've been looking at the Kepler's problem, and it looks like your orbit (elliptic, parabolic or hyperbolic) are given in terms of energy and angular momentum. I was wondering: if I have a central attractive potential (such as the Sun) and a projectile starting from an infinite distance at a given velocity and impact parameter, would it be possible to obtain the orbit equation for such an object? I mean: I would like to solve Kepler's problem for a projectile after a given velocity and impact parameter.

 

[Orodruin]

The quantities you quoted are directly relatable to the energy and angular momentum, so yes.

 

[L0r3n20]

The quantities you quoted are directly relatable to the energy and angular momentum, so yes.

The module of the angular momentum can be expressed as L = m v b (being b the impact parameter) but what about the energy? I mean if the projectile is at an infinite distance it has only kinetic (therefore positive) energy, so I cannot find any bounded state. What am I missing?

 

[Orodruin]

If the particle can reach infinity, it obviously is not in a bound state by definition.

 

[L0r3n20]

If the particle can reach infinity, it obviously is not in a bound state by definition.

But what if it starts from infinity moving towards the Sun?

 

[Orodruin]

If it is at infinity it is not in a bound orbit, precisely because of what you mentioned.

 

[Orodruin]

Look at it this way: The bound orbits are ellipses of finite major and semi-major axis. Therefore a particle at infinity cannot be on one of those ellipses.

 

[L0r3n20]

Look at it this way: The bound orbits are ellipses of finite major and semi-major axis. Therefore a particle at infinity cannot be on one of those ellipses.

Ok, i get it but I was wondering if a particle coming from infinite be captured by an attractive potential and bound into a closed orbit.

 

[Orodruin]

If it is moving solely under the influence of a central potential: No, that would violate conservation of energy.

Of course it can be captured if it interacts with another body and transfers enough energy to it (part of my research deals with the possible capture of dark matter particles by the Sun.), but this requires additional interactions beyond the motion in the central potential.

 

[L0r3n20]

If it is moving solely under the influence of a central potential: No, that would violate conservation of energy.

Of course it can be captured if it interacts with another body and transfers enough energy to it (part of my research deals with the possible capture of dark matter particles by the Sun.), but this requires additional interactions beyond the motion in the central potential.

Thanks, I really understood! :)