How far from perseverance did the descent land?

Dare powerful things. It was the message hidden in the parachute of the Mars Perseverance rover. It’s not as powerful, but I’ll dare something myself: I’ll try to figure out how far the downhill stage would land from the rover.

OK, let me get back very quickly. Just in case you don’t know how it works, here’s the basic landing sequence: The spacecraft entered the Martian atmosphere and then deployed a parachute. After that, a rocket-powered descent stage slowed the rover as it approached the surface. At the very end of the descent stage, a cable lowered the rover to the ground. Then, the descent stage used its remaining fuel to move away from the landing site.

It is this fly-away step that I want to analyze. If I can get the acceleration out of the way, then maybe I can model its trajectory to see where it would land. Yes, NASA knows exactly where it landed –they even have a picture of his crash site. But it’s fun to see if I can do it just from video from a single rover.

OK, let’s get started. The plan is to use the angular size of the descent stage to get the distance to the rover in each frame of the video. But what is angular size and what does it have to do with position? Here’s a quick experience for you. Take your thumb and hold it at arm’s length from your face and close one eye. Yes, really do that. Now find something in the room that your thumb is covering. What happens when you bring your thumb closer to your eye? It looks bigger and covers even more stuff in the background. The actual size of your thumb has not changed, just its angular size.

Suppose there is another object – maybe it is a stick of length L in your field of view. Imagine that you could draw a line from your eye to each end of the stick. It would look like this.

The stick is kind of like part of a circle with a radius r centered over your eye. This means that the length of the stick is approximately equal to the length of the arc which has an angle θ. Assuming the angle is measured in radians, then the following would be true.