Strike a Pose, Gravity

What's the deal with gravity?

We don't know what causes it, but we do know it's a force that occurs when two objects -- such as the Earth and the moon -- attract to one another. We also know that if each object is treated as a perfect sphere, the math describing gravity is fairly simple.

But reality bites, and the Earth is not a perfect sphere. It has deep oceans and towering mountain peaks, and is thicker around the equator. Each little rock, pebble, grain of sand and molecule of water pulls on the moon, planets, the sun and everything else in the universe. Needless to say, accounting for these disturbances is a difficult problem.

Since before the first satellite launch, humanity has studied gravity anomalies, and those investigations have brought us to the Gravity Recovery and Climatology Experiment, aka "GRACE," which uses special satellites to measure this mysterious force.

See right for a 3-D map of these anomalies that the GRACE mission has teased out. The red indicates anomalies stronger than Earth's average pull of gravity (about 32.1740486 ft/s²), while the blue shows weaker-than-average anomalies.

Looks cool, but what do we do with this kind of information?

From modeling satellite orbits to monitoring the melting of glaciers and the polar ice caps, the benefits are numerous. To elaborate on the glacier example: If one is melting, the water drains away. That change in water means a change in mass -- and that translates to a small change in gravity around the glacier. So we can use gravity to study climate change.

Here's where I fit in: Scientists and engineers have used one type of gravity model for the past 100 years -- it's called spherical harmonics and is a bit dense to describe here (just look at the equation). My research looks at gravity from a totally different perspective, and to understand it, let's talk photography.

A photograph helps us preserve a memory, e.g. you as a kid. Years later, the picture reminds us of what we looked like when we were younger. That retrospective approach is similar to the gravity modeling I'm studying: Let's take our knowledge of the Earth's gravity, and store that information like a photograph so we can use it later. And instead of taking a picture of what the Earth visually looks like, we take a snapshot of what the Earth gravitationally looks like.

But one photograph doesn't show us the whole picture. If we take a picture of one side of the Earth, what does the other side look like? How do we know what gravity is like there? Even two pictures don't lend us a clear picture of the edges. We actually need six photographs to give us the Earth as a cube (see right).

So, why this new way versus the "old" way?

We've all sat there, waiting for a computer game or some piece of software to run... the more complex it is, the longer it takes for the computer to crunch the numbers. The "cubed sphere" model of gravity actually makes the software scientists use run faster -- so you get the speed of a rough test with the accuracy of an incredibly complex one. For people trying to model how their shiny new satellites will react to the Earth's anomalies while in orbit, that's a valuable improvement.

Modeling gravity is especially important for missions to the moon, since the spacecraft will be carrying people, and the moon's gravity is much more irregular than the Earth's. Better mapping of the moon's gravity not only improves simulations for planning trips there, but also helps astronauts figure out precisely where their spaceship is without asking for help from Earth.

It's also useful because, quite simply, someone can easily zoom in on a region they're interested in. Like looking at pictures of ourselves from when we were younger, we can use this cube-like model to easily see changes over time in things such as melting glaciers, mountains and more.

Brandon Jones is PhD student in aerospace engineering at the University of Colorado, Boulder.

Photos, top to bottom: Brandon Jones; NASA. Wikimedia Commons; Brandon Jones

Comments

[De Bunker]

Ok, cool, but don't you get huge edge effects at the corners? Are you just doing a fourier transform in cartesian coordinates? Spherical harmonics works because the earth approximately a sphere (e.g. l=0 is the biggest term). But to approximate a sphere using a cartesian basis requires an infinite number of Fourier modes...

[Dave Mosher]

From Brandon Jones in response to the comment by "De Bunker":

Good questions.

Actually, you don't get any edge effects. In order to avoid a bunch of mathematical details, let's keep the photography analogy. If you zoom in on a single pixel of a photograph, you don't care about the other pixels. You only care about the color, or in this case the gravity, represented by that pixel. If you are between pixels, you can guess (interpolate) based on the gravity in the pixels around you.

Now, what if you are exactly on the edge? Don't you have to use information from two different pictures? Actually, we store the information so that pictures overlap. That way, you only have to use information stored in a single picture. Problem solved!

Commonly the spherical harmonics model is represented in spherical coordinates (latitude, longitude, and radius). I've never seen it represented in Cartesian for the evaluation of gravity (but that doesn't mean people don't do it). The result is then converted to Cartesian coordinates, but the evaluation itself is spherical. For current cubed sphere gravity models, the gravity is still represented in the spherical coordinates. This makes switching between the spherical harmonics and the cubed sphere easier, among other things. After all, we are taking a "picture" of a spherical object. When you look at a picture of the Earth, you see a 2D representation of a 3-D, spherical object. The cubed sphere model is similar. Why pretend it is anything else? :)

Here is a question to ponder, how do we model the gravity for an object that is not a sphere, such as many comets and asteroids? Spherical harmonics starts to break down because the model is not spherical. The cubed sphere doesn't care what the shape of the object is, you only have to be able to take a "picture" of it.

[Tom Manning]

Personally I think this kid is just plain brilliant but I have a couple of questions.

What are we doing currently to provide such excellent maps of the lunar gravity field? Also what are we doing for Mars? Are these maps really critical to the success of manned missions to Mars and to the Moon? Are they more important because we are sending humans and if so why? Also will they help us to map out the mineral content of Mars?

[Tom Manning]

I realize you answered the question about the Lunar surface but I mean really is it that important? We landed on the Moon previously without this and I wonder if it really makes it that much safer or more reliable. And also while on the subject is the Martian gravity field all that irregular?

[Tom Manning]

Are U.S. tax dollars paying for this?

[Brandon Jones]

Oh Tom, did you get bored during your lunch break :) (Tom is an old friend of mine, to those that may think I’m belittling a stranger)

With NASA returning to the Moon, the interest in the Moon’s gravity field is growing considerably. The Japanese have SELENE (Kaguya), NASA is about to send LRO, and there are plans for a GRACE like mission to the Moon (GRAIL). As for the need for more accurate models, there are major differences between what NASA did in the Apollo days and what is planned with the Constellation program. While two Apollo astronauts explored the lunar surface, the command module was in orbit for about three days with a crewmember aboard. Gravity anomalies won’t cause as much of a problem after three days as they do after 6 months (which is one of the possible mission durations considered by NASA). What if the Orion vehicle loses contact with the ground? How will it know its position and velocity? How will it make corrections? Higher fidelity gravity model aboard the vehicle help mitigate the problem.

Are these better models absolutely critical to manned mission to Mars or the Moon? No. The success of the Apollo missions proves that. However, they make things considerably easier, safer, and cheaper. Not to mention a faster gravity model makes ground simulations more accurate for little or no change in computation time. This helps in mission planning.

As for your question on whether better gravity models are more important because we are sending humans, the answer is no. Better gravity models are more necessary for scientific missions, and have been a topic of research for over 100 years. Ask oceanographers relying on the Jason-1 or OSTM satellites how better navigation solutions (which rely on accurate gravity models) have improved tracking of the Earth’s oceans. This of course helps with predicting and monitoring hurricanes (for example), which is something I’m sure you care about if you live along the Gulf of Mexico…

As for monitoring the mineral content of Mars, I do not know. That is more of a question for a scientist. However, changes in the gravity field do allow you to track changes in mass distribution, i.e. the seasonal melting and growth of the polar ice caps of Mars. See research by Maria Zuber of MIT, and possibly others.

As for the source of the funding for this research, I’m not at liberty to divulge that information :)

[Steve Provence]

So you are saying that the source of the funding for this is shrouded in secrecy?

[Steve Provence]

So you are saying that the source of the funding for this is shrouded in secrecy?

[Brandon Jones]

Hey Steve. Of course not! It's just me giving Tom a hard time. This project is currently funded by the NASA Graduate Student Researchers Program (GSRP) through JSC.

[Tom Manning]

Well I have to say I think the "higher fidelity gravity model aboard the vehicle" might be a bit ambitious. Building a guidance system with that sort of complexity sounds like a great research project but the increased cost and complexity of building and maintaining that beast would be greater than its corresponding benefit. (just a guess, what would I know?). However I do see the point about using it in training and in simulation to help the humans on board the landing vehicle cope with the anticipated gravity environment. Because we succeeded with Apollo does not mean we cannot fail with Orion.

As far as unmanned vehicles go I think its still best left as a preprocessing enterprise until we have the ability to make radiation hardened computers that can do this sort of thing along with all of their other responsibilities. Right now to automate landing I believe you need a higher frequency feedback response than adding this software would allow. Of course in the future who knows. I know you are on the bleeding edge more than I am so maybe you know of some deep space systems that can do this. Or maybe you have a minimal version of it that can be used for such purposes. That would be something that would peak my interest.

And no I did not put the comment about the taxpayer dollars. That was posted by my favorite bureaucrat.