Math for the Win

So I was watching the Mars Science Laboratory Lander (@MarsCuriosity) land and I thought about some of the discussions that had occurred during the NASA Social meeting to celebrate the 50th Anniversary of Cape Canaveral. We had people from the launch teams join us for some discussions. One of the comments that @NASA_Caley made resonated with me, “There are 7 minutes of terror for landing. For launch, we had 44 minutes of terror.” She then went on to explain all the things that were thought of in the process of planning the route for Curiosity.

I am sure that my post will gloss over some things, but I am still reeling at how many things had to be accounted for.

Let’s say you want to send a projectile (Curiosity) from a source object (Earth) to a destination object (Mars), what do you need to consider. Well first you need to think about the location of the objects relative to each other so that you can point the projectile in the proper direction. But what is the right direction?

If the two objects are stationary it is pretty easy, just aim from one toward the other. If the objects are rotating at different speeds, it gets more challenging. The Earth revolves around the Sun in basically 365.25 days, Mars since it is farther away makes the same revolution in approximately 688 days. As a result the relative position of the two objects changes constantly. The knee-jerk reaction is to say to send the projectile when the two objects are closest to each other. So let’s declare two basic variables, location of source, location of destination.

Oh wait, that is really a more complex variable than those simple statements of source location and destination location would indicate. These objects are not just spinning around the star at the center of our solar system, they are also spinning around their own axis. Their axis and orbital paths are also not perpendicular or parallel to each other. So these locations are not in one plane with just X & Y coordinates but actually in three axises.

Seems to make sense, the locations relative to each other will be the shortest distance and the shortest travel time. Yes, travel time, a new variable. The travel time has to be accounted for as the objects will move relative to each other as the projectile travels from the source to the destination. The easiest way to think about it, is leading a target, many of us do it while playing video games, or shooting rubber bands at people. So you have to account for the amount of time to get from Point A to Point B while both points are moving. You definitely need to know the speed of the projectile to know the travel time.

The speed of the projectile is not a simple value, it is also another variable of the equation that needs to be considered. The obvious answer is to make things travel as fast as possible. However there is always a trade off. To make things move quickly requires lots of thrust. So you need to have more fuel to allow for more thrust. Fuel has mass which means that you need more fuel to get that mass in motion. The designers need to determine the proper balance between speed, launch weight, and time to travel to destination. Don’t forget to include the mass of the projectile in there. Hopefully the mass has been determined but it might still be in flux. Is that one variable or four variables? I would say it is three interrelated variables, as the designers will work within a mass budget. What about the time it takes to get up to speed? I guess we can just use average speed… that 44 minutes of launch is not that significant compared to the approximately 253 days and 14 hours of cruise. It is just 0.012002182215% of the total time. That is tongue in check.

While leaving Earth there are some other things to consider. Such as not hitting the International Space Station or other satellites. Plus one also has to consider the biggest satellite out there along the way, the Moon. We have to steer clear of that, and its gradational pull. Wait we can use its gravitational pull to speed us up some more or to adjust the course.

So as you can see just a little math.

This math was done and allowed for landing within 1.5 miles of the desired landing site after a journey of approximately 350,000,000 miles. Yes, they can do some adjustments in flight, but those have to be calculated as well. To put this in perspective  to get the same resolution you would need to land on a single human hair (0.04mm) from 14km (8.7 miles) and not hit anything, seen or unseen, during the flight.

Did you remember to compensate for the satellites going around Mars?

Pretty impressive. It isn’t rocket science, it is math and physics.

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