Learn All That is Learnable

[Title from Star Trek: The Motion Picture, V'GER's mission.]

Earlier this week, the Voyager 2 spacecraft broke an 8-month silence to check in with NASA researchers. This is one of the most distant space probes from Earth, currently about 11.6 billion miles away. It was originally launched in 1977, with the primary goal of observing Neptune and Uranus. To get to those planets in 1989, it was given enough velocity to leave the solar system entirely, which it did 2 years ago. The antennas that communicate with Voyager, called the NASA Deep Space Network, were under maintenance for the past 8 months, preventing contact.

After seeing the news article about the renewed contact, I started reading more about the probes, and how they achieved their Sun-escaping speed. They key lies in the idea of a gravity assist, where by flying by a planet in the right way, an object can get a boost of speed:

Wikipedia
   Voyager 2 ·   Earth ·   Jupiter ·   Saturn ·   Uranus ·   Neptune ·   Sun

The principle at work in a gravity assist is actually something usually taught in intro Physics: an elastic collision. This is when two (or more) objects interact in a way that conserves both energy and momentum. An everyday example is a rubber ball that bounces to its original height when dropped. The energy and momentum of an object are given by

where m is the mass and v is the velocity. Conserving these quantities means that before and after the collision, the sums of each one for all the objects involved stays the same. To simplify things, we can consider the case in one dimension: Initially the probe and planet are moving toward each other, the probe slingshots around the planet, and leaves in the opposite direction it came. Then we can use the elastic collision equations to get

where capital variables are for the planet, lowercase for the probe, and i and f for initial and final. Since planets are generally a lot bigger than space probes (though maybe not in the case of the source of this post's title), we can Taylor expand around m/M ≪ 1 to get

Jupiter's mean orbital velocity is about 13.1 km/s, or about 30,000 mph, and we could get double that! Unfortunately, the angles typically don't work out that way, as you can see in the animation above.

I feel that any discussion of the Voyager Spacecraft has to include the hope that went into creating them, and to that end, I want to end this post with a picture of one of the Golden Disks. Each of the spacecraft was loaded with a message to the stars about Earth and its people. We looked for a way to represent ourselves to the cosmos, and found what was best in us.