Conditions for 2 Elliptical Orbits w/ Same Peri/Aphelion

[goleafsgo113]

hey all, I'm just wondering, what conditions are necessary so that we could have 2 elliptical orbits such that the perihelion distance of one is the same as the aphelion of the other?

 

[Jenab]

hey all, I'm just wondering, what conditions are necessary so that we could have 2 elliptical orbits such that the perihelion distance of one is the same as the aphelion of the other?

Let orbit 1 be the smaller orbit. Let orbit 2 be the larger orbit.

r1 is the aphelion distance of orbit 1.
r1 = a1 ( 1 + e1 )

r2 is the perihelion distance of orbit 2.
r2 = a2 ( 1 - e2 )

Insist:
r2 = r1

Therefore:
a2 (1 - e2) = a1 (1 + e1)

Solved for ratio of semimajor axes.
a2/a1 = (1 + e1) / (1 - e2)

This only ensures that the perihelion distance of orbit 2 is equal to the aphelion distance of orbit 1. It does not require that the aphelion of orbit 1 and the perihelion of orbit 2 occur in exactly the same point of space.

To make the aphelion of orbit 2 and the perihelion of orbit 1 occur at the same point in space requires, further, that either of two conditions prevail regarding their inclinations, longitudes of the ascending node, and arguments of the perihelion, e.g.,

Case 1.
1. The inclinations of orbit 1 and of orbit 2 be equal to each other.
2. Their longitudes of the ascending node be equal to each other.
3. Their arguments of the perihelion differ by pi radians.

Case 2.
1. The inclinations of orbit 1 and of orbit 2 sum to pi radians.
2. Their longitudes of the ascending node differ by pi radians.
3. Their arguments of the perihelion differ by pi radians.

Somebody should check me that these are the only two cases, because I seem to be fuzzy minded at the moment.

Okay, I was wrong. Those two cases above are the cases for which the two orbits are insisted to be coplanar. You can twirl either (or both) orbits about their major axes (in either case) and keep their respective apsides where they are.

If both planets, each with the mass of Earth, had eccentricities of 0.5 and met in a head-on collision at their mutual apside at a heliocentric distance of 1 AU, the impact energy would be equal to how many days' worth of the sun's total energy output? Answer: 321 days at 3.826E+26 Watts.

Jerry Abbott