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Hi all!

I've got a fun orbits question I've been toying with for a few days now, and I've hit something of a wall. I was hoping I might get some insight from you fine folks that might push me in the general direction of the solution.

The question is this: given two arbitrary orbits defined by static conic sections (that is, pure two body motion), find the closest approach point or points between the conics.

Assume orbit 1 is defined by: <a1, e1, i1, O1, w1>

Assume orbit 2 is defined by: <a2, e2, i2, O2, w2>

These elements are arbitrary. For the purposes of our discussion here, let's assume that we're dealing with elliptical orbits, though it should be straight-forward to expand any solutions to all conic sections: 0.0 <= e1,e2 < 1.0

The anomaly is not pertinent to this discussion as I am looking for the close approach point of the

My thought on the solution to date has been this: I believe that the closest approach point between two conics should be at their relative ascending/descending nodes. Therefore, to get inertial unit vectors to those points, I can do the following:

Define the orbital angular momentum unit vector of orbit 1: h1

Define the orbital angular momentum unit vector of orbit 2: h2

The ascending/descending nodes are then located along the unit vectors: n = h1 ^ h2/norm(h1 ^ h2) and n` = -(h1 ^ h2/norm(h1 ^ h2)

From there it's trivial to determine the true anomaly at each orbit. Something like this, for example:

theta1 = asin(n_z/sin(i1))

true1 = theta1 - w1

Where n_z is the "z" component of the unit vector "n".

So now the issue: this doesn't seem to work. I can define orbits which I know approach each other closely at some point, but the closest approach point doesn't seem to be as I've defined it. Is the issue in my assumption (closest approach at nodes)? Is there something wrong in my mathematics?

Input appreciated! Thanks everyone.

I've got a fun orbits question I've been toying with for a few days now, and I've hit something of a wall. I was hoping I might get some insight from you fine folks that might push me in the general direction of the solution.

The question is this: given two arbitrary orbits defined by static conic sections (that is, pure two body motion), find the closest approach point or points between the conics.

Assume orbit 1 is defined by: <a1, e1, i1, O1, w1>

Assume orbit 2 is defined by: <a2, e2, i2, O2, w2>

These elements are arbitrary. For the purposes of our discussion here, let's assume that we're dealing with elliptical orbits, though it should be straight-forward to expand any solutions to all conic sections: 0.0 <= e1,e2 < 1.0

The anomaly is not pertinent to this discussion as I am looking for the close approach point of the

*orbits*(or, the geometry of the orbits) and*not*the close approach point of two bodies on those orbits.My thought on the solution to date has been this: I believe that the closest approach point between two conics should be at their relative ascending/descending nodes. Therefore, to get inertial unit vectors to those points, I can do the following:

Define the orbital angular momentum unit vector of orbit 1: h1

Define the orbital angular momentum unit vector of orbit 2: h2

The ascending/descending nodes are then located along the unit vectors: n = h1 ^ h2/norm(h1 ^ h2) and n` = -(h1 ^ h2/norm(h1 ^ h2)

From there it's trivial to determine the true anomaly at each orbit. Something like this, for example:

theta1 = asin(n_z/sin(i1))

true1 = theta1 - w1

Where n_z is the "z" component of the unit vector "n".

So now the issue: this doesn't seem to work. I can define orbits which I know approach each other closely at some point, but the closest approach point doesn't seem to be as I've defined it. Is the issue in my assumption (closest approach at nodes)? Is there something wrong in my mathematics?

Input appreciated! Thanks everyone.

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