Ten Cool Things Seen in the First Year of LRO.

LRO stands for Lunar Reconnaissance Orbiter, an orbital satellite that is mapping the Lunar Surface.

The Moon is approximated as a two dimensional surface, a 2-Sphere, with an altitude recorded as a function of the lunar latitude (THETA) and longitude (PHI). THETA and PHI are periodic polar coordinates. In time, LRO will cover the entire lunar surface by following its trajectory, a one dimensional curve in space.

This raises an interesting problem in differential geometry. The distance along the curve is a scalar function of time, S(t), obtainable from the metric. At each instant the LRO is located at a particular point on the 2-Sphere, [THETA(S(t)), PHI(S(t))].

So here’s the tricky question: How can the two dimensional surface of the moon, a 2-Sphere, be completely covered by a one dimensional curve (the LRO orbit)?

Comments and discussion welcome. Answer to follow in this space.

Mathview's Channel on YouTube: http://www.youtube.com/user/Mathview

Hints: “differential” "Ergodic Curves"

Another Hint: Consider an ergodic orbit.

Take a spool of thread and tack the end to the north pole of the moon. Then start wrapping on great circles through both poles. Pick any point on the lunar surface, as you keep winding you can always come arbitrarily close to that chosen point, IF the winding law gives an ergodic covering of the 2D surface. So a one dimensional string of infinite length covers a 2D surface.

Each point on the surface has a "string length coordinate" given by the distance along the string. The "coordinate of the point" is the distance along the string between the origin at the north pole (say) and the point on the string that is "sufficiently close" to the point on the surface.

The reason this is not a "good" coordinate system is that it is non-local. An adjacent point in the neighborhood of a point may have a wildly different string length coordinate. So the string length coordinate system cannot be used for calculus on the 2D surface.

Anyhow, I think these strings are quite interesting. Poincare used a shrinking loop of string to determine the connectedness of manifolds. Also, it would be interesting to know the geometric flow PDEquation for a string embedded on a higher dimensional manifold. And is the geometric flow equivalent to a classical string theory? Specifically, is there a Lagrangian formulation leading to the geometric flow equations? And so on... Of course, experts know the answer to this one, I suppose.

But I digress...

MORE LRO DISCUSSION

Watch this video first:

A few observations. The surface is covered with fine dust. I guess it must be micro-meteoric and ejecta from macro-meteor impacts. Micrometeors don't reach us here on Earth, they burn up. It seems to me that excavations on the moon would tell us alot about the history of the solar system. Events such as epochs of cometary bombardment would leave a stratigraphic record unlike anything we can get on Earth. In short send drill core crew to the Moon. Also, looks to me like there was an ocean on the back side.

@Mathview From memory... That ocean looking thing on the far side is actually the remnants of a molten ocean after a collision or impact from a comet or meteor. Same as on the near side. And the core drill would have to be VERY long to get anything useful as most of the surface is dust... Very fine dust. And that is rather deep in places.

@gadget133

TY Useful stuff on the Maria. As to Dust, Dust is a good thing to study. It seems likely that lunar stratigraphy would be a great way of getting a ~10^9 year record of dust flux and composition in our solar system. Further, there will be a record of ejecta of earth origin, e.g. big volcanic and big impact. The lack of wind and water erosion on the moon suggests an undisturbed stratigraphy and geochronology of dust deposits. A real treasure trove.

@Mathview

Makes sense... I would suppose that you would get a chalky limestone (in texture only of course) stratification once you went down a meter or so... Yes, it would be a huge bonus to the understanding of how the solar system formed. The fact that there is no geological movement gives us an unprecedented ability to study impacts also. Makes you cry at the whole "never been back" thing.

@gadget133

Makes sense... I would suppose that you would get a chalky limestone (in texture only of course) stratification once you went down a meter or so... Yes, it would be a huge bonus to the understanding of how the solar system formed. The fact that there is no geological movement gives us an unprecedented ability to study impacts also. Makes you cry at the whole "never been back" thing.

@gadget133

Oh yes, now I'm Sad. But policy can change, President and Congress will change. So it's likely we will go back after all. LRO and LCROSS Amazing! Rocket from earth to the moon, creating an orbiting robotic laboratory, water impact experiments, precision maps of the the lunar surface of unprecedented quality, so smooth it is almost routine to go to the Moon. TY for the discussion.

DrDoe1

I wonder how the coldest place in the solar system is on the moon? You would think one of those ice moons or even something closer to the edge of the solar system would be colder....

Hello Mathview, nice to see you already have a blog. I see it's been idle for a while, so it looks like a good time to bring it back to life! I also started a blog, years ago, but it was extremely short lived :-(

ReplyDeletehttp://top-quark.blogspot.com/

As for the LRO orbit, I guess there is a practical answer and a theoretical one. Since each orbit covers a measurable strip of the lunar surface, after you complete an orbit, rotate its plane just enough to make the next one overlap a little bit with the previous. Eventually you'll cover all the surface; it's like covering an orange with duct tape.

If we make the strip narrower and narrower I think we could be as precise as we wanted. But I'm not sure we could map *any* point on the surface to a point on the orbit. I looks like this would be the case for "rational orbits", but probably not for "real orbits". So, I'm waiting for your geometrical answer here!