Conditions for 2 Elliptical Orbits w/ Same Peri/Aphelion

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In summary, to have 2 elliptical orbits with the same perihelion distance and aphelion distance, the ratio of their semimajor axes must be (1 + e1) / (1 - e2). This only guarantees that the perihelion of one orbit is the same as the aphelion of the other, but not necessarily in the same point in space. To ensure this, there are two possible cases that involve the inclinations, longitudes of the ascending node, and arguments of the perihelion of the orbits. However, there may be other cases that need to be considered as well. Additionally, if two planets with the mass of Earth and eccentricities of 0.5 were
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goleafsgo113
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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?
 
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goleafsgo113 said:
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
 
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1. What are the conditions for two elliptical orbits to have the same periapsis and aphelion?

The conditions for two elliptical orbits to have the same periapsis and aphelion are that both orbits must have the same eccentricity and semi-major axis. This means that the shapes and sizes of the ellipses must be identical.

2. Can two elliptical orbits with different eccentricities have the same periapsis and aphelion?

No, two orbits with different eccentricities cannot have the same periapsis and aphelion. The eccentricity determines the shape of the ellipse, meaning that if the eccentricities are different, the shapes of the ellipses will also be different.

3. What is the significance of having the same periapsis and aphelion for two orbits?

Holding the same periapsis and aphelion means that the two orbits have the same closest and farthest distances from the center of mass, respectively. This can have implications for gravitational interactions between the two objects in the orbits.

4. Is it common for two objects to have the same periapsis and aphelion in elliptical orbits?

No, it is not common for two objects to have the same periapsis and aphelion in elliptical orbits. This is because it requires very specific conditions and parameters for the orbits to align in this way.

5. How do the velocities of objects in two elliptical orbits with the same periapsis and aphelion compare?

The velocities of the objects in these orbits will be different, as the velocities depend on the distance from the center of mass. However, at the periapsis and aphelion points, the objects will have equal velocities due to the conservation of angular momentum.

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