# Thread: education, math and celestial mechanics

1. ## education, math and celestial mechanics

Hopefully this post will be topical. Pardon the question dump.

Today I was wondering about the possible effects of Mars being at L3 or L5. Would Mars become tidally locked with Earth? Vice versa? Would Mars's mass be unstable at either of those Lagrange points? Why would it be stable or unstable? Would tides increase? Would Mars become geologically active once again having Earth, the Moon and the Sun massaging its surface and recreate its magnetic field? Would Mars be visible in the daytime from Earth? Would the Moon overcome its tidal lock with Earth? How would Earth's rotation be affected by the presence of Mars at L5 or L3?

Anyway the one question I'd like answered and most likely it'd be more advice is how do I go about learning about this kind of stuff and figuring it out for myself? I'm just a layperson who completed some college and wasn't even a science major.

Thanks for looking tho.

2. I'm guessing you mean L4 and L5, the Trojan points? L3 is away over on the opposite side of the Earth's orbit from Earth.
There are criteria for the stability of a couple of objects of similar mass in mutual Trojan positions; I'd need to dig them out. What can happen is that the two objects end up in "horseshoe" orbits instead, swapping energy back and forth, like the Saturnian moons Janus and Epimetheus.
Mars would be as far from Earth as Earth is from the Sun, so it would actually be farther away than it can approach at present: 1AU in Trojan position versus 0.5AU at opposition. So no interesting tidal or rotational effects on either body (or the Moon), or we'd have noticed them every time we passed Mars.
Likewise, Mars couldn't get as bright as it does at present: we're seeing it full illuminated from a distance of 0.5AU at opposition, and we'd be seeing it in 60-degree phase at 1AU when it was in the Trojan position. So around a fifth as bright as at opposition.

Grant Hutchison

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The L3 point is at 1 AU from the Sun, opposite Earth's position. So an object placed there would be behind the solar disk. This point is not stable. An object placed there will stay put for perhaps 10s to 100s of years, but ultimately will be perturbed away from the point. Very slowly, orbit by orbit it will approach Earth. When it gets close, Earth's gravity will seemingly repel it in a rotating frame of reference. Actually, gravity never repels. If its orbit is slightly faster than Earth's, Earth pulls it into higher, slower orbit, where it becomes slower than Earth and seemingly reverses direction. This object would be in either an L4 or L5 tadpole orbit, depending on which side of the L3 point it rolled off of, or it will be in a horseshoe orbit, where it returns to and crosses the L3 point, only to approach Earth from the other side and repeat the whole process over again. Over its lifetime, it may jump between the 2 tadpole orbits or the horseshoe orbit many times. But the configuration is stable. It will never collide with Earth.

If the object were at the L5 (or L4) point, it would remain stationary with respect to Earth. It would always appear a fixed 60 degrees from the Sun. It's doubtful that it would ever be visible in the daytime sky, as it would remain a fixed distance of 1 AU from Earth, and never present a full face to Earth. By contrast, Mars in its current position gets much closer to Earth during opposition than an L4 or L5 Mars would get. Because of this, tides, geologically active due to Earth, Moon overcome its tidal lock... doubtful to any of these.

My guesses are based on an intuitive feel I have for orbital configurations from using Gravity Simulator, a program I wrote. Here's an animation of Venus being placed directly on Earth's L3 point. It sits there for a little while before rolling off the L3 point into a horseshoe orbit.

This simulation is discussed here: http://www.orbitsimulator.com/cgi-bi...num=1189814191

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Originally Posted by grant hutchison
...There are criteria for the stability of a couple of objects of similar mass in mutual Trojan positions; I'd need to dig them out...
I remember we discussed this in the Celestia forum once. The combined mass of the planet + trojan needs to be about 4% or less than the mass of the Sun. The actual answer is surprising as there's an unstable zone from 2.86% to 3.1% . Here's a recent BAUT thread on this issue:
http://www.bautforum.com/astronomy/5...tml#post987277

5. Augh yes I meant L4 and L5. I did not mean to use L3 because it's opposite the Earth and unstable. Sorry

6. Originally Posted by tony873004
Here's a recent BAUT thread on this issue:
http://www.bautforum.com/astronomy/5...tml#post987277
Ah, thanks. You give a link there to the Laughlin & Chambers paper I was thinking of.

Grant Hutchion

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Originally Posted by phaishazamkhan
Hopefully this post will be topical. Pardon the question dump.

Today I was wondering about the possible effects of Mars being at L3 or L5. Would Mars become tidally locked with Earth?
No. Too distance if Earth and Mars were in the same orbit around the sun at the L4 and L5 Lagrange points.

Would Mars's mass be unstable at either of those Lagrange points?
Mars' mass has nothing to do with stability at Lagrange points. It certainly wouldn't fluctuate or become "unstable."

Would the Moon overcome its tidal lock with Earth? How would Earth's rotation be affected by the presence of Mars at L5 or L3?
Again, L4 and L5.

The Earth-Moon system is so small compared to the distance between L4 and L5 Lagrange points in either an Earth or Mars orbit, our system would remain the same.

8. Originally Posted by mugaliens
Mars' mass has nothing to do with stability at Lagrange points. It certainly wouldn't fluctuate or become "unstable."
Yes, Mars wouldn't destabilize Earth's Trojan points, but it's probably worth pointing out that in general the mass of the object placed at the Trojan point has to be reckoned into the stability calculation. There's more detail in the Laughlin and Chambers paper.

(As a concrete example, we might consider what would happen if Earth's moon were twice as massive as it is in reality. It would still have stable regions at its Trojan points for low-mass objects, but anything much more massive than the real moon would not find a stable Trojan equilibrium.)

Grant Hutchison

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## Unbalanced

There is a recent article, I think in UT, discussing the instability of a Mars sized mass body at the leading or following Trojan points eventually resulting in the impact that formed the mostly silicate Moon and left the Earth with most of the iron.

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