Thursday, August 29, 2013

Mass Effect and Kepler's Third Law

For those of you who may be unfamiliar with the games, the Mass Effect series consists of 3 science-fiction action-RPGs (role-playing games) set in the not-quite-so-distant future (~1250 years into the future), at which point humanity has long since become a spacefaring race and quickly learned that we were not alone in the universe, much less the galaxy. Needless to say, because this is a science fiction game, there is a fair amount of really hilariously bad "science" in the game invoked to make things like FTL travel and the like work (just read the Mass Effect Wiki entry on Element Zero for an example). But that's not what I'm here to talk about. Rather, I'm going to talk about some of the science that they do right... sort of.

As part of Mass Effect 2 in particular, you fly your spaceship around to explore planets in other star systems (this happens in the other two games as well, though in a different sense). While I could probably do a whole post on the featured planets of the Mass Effect universe, I'll refrain for now, mostly because that would take a lot of work on my part. Some of the planets (a definite minority) are story-critical inhabited planets that you have to visit in order to progress the plot. There are still other planets on which you can find side-quests (for you non-gamers, a side-quest is optional and not really important to the story overall). The vast majority of planets, kind of unsurprisingly, are totally uninhabitable but can be mined for various resources (easily the most mind-numbingly dull part of the game, hence why I just edit my save files to give me more resources than I'll ever need) that are used for upgrading weapons/armor/your ship and so forth.

Most interestingly for me, all planets come with some sort of description. This can be as dull as "[planet name] is a standard Jovian gas giant with [some feature]..." or can feature an interesting description of the planet's history, interactions with other bodies in the system, and, once in a while, a description of the central star of the system. The best of these (as uncommon as they may be) even list the spectral type of the star, or give some indicator of what it may be. Beneath the verbal descriptions, you often get some numerical descriptions of the planet that can include its orbital period, semi-major axis (orbital distance from the star), surface gravity (measure of the gravitational force on the planet's surface; requires planet to be solid), surface temperature, the radius of the planet, day length, and so on.

What makes this really cool is that, through the full form of Kepler's Third Law, which you can derive from Newton's Law of Universal Gravitation (getting Kepler's laws right was one of the early successes of Newtonian gravity), we can actually calculate the mass of the star itself! Like I continuously told my students this past summer, the best way to measure an object's mass is to put something in orbit around it.

This requires the approximation that the mass of the star is much greater than the mass of the planet if the mass of the planet is unknown (it generally is not given in the Mass Effect entries) which is usually a pretty good assumption. As an example, Jupiter is about 1000 times less massive than the Sun. For an astronomer, a massless Jupiter is a pretty good approximation (and boy do we love our approximations!), so for this calculation, we'll pretend that planets are actually massless points.

So, we start with the Newtonian form of Kepler's Third Law.
where G is Newton's Gravitational Constant, T is the orbital period of the planet, R is the distance between the planet and the star it's orbiting, and M is the mass of the star. You can derive this yourself if you really want to, or you can take my word for it. G, π, and 4 are all constants, and T and R are given for the system in question, so the only unknown is the star's mass. When we cross-multiply and solve for M, we get
So we can measure the masses of the stars in the Mass Effect universe! As you could probably guess, I was very excited to figure this out because I am a huge geek.

Naturally, as soon as I realized this, I had to put it to the test. So let's take the example of the planet Gei Hinnom in the Sheol system. (Yeah, I have no clue how they came up with the names of most of these, though apparently "Sheol" translates as "grave" in Hebrew.) This is a direct screenshot from Mass Effect 2, except it has been modified so the full text description is shown. The description begins in the left panel and ends in the right panel (with a little overlap between the two).

So this planet is 0.83 Astronomical Units away from the star Sheol and orbits once every 0.8 years. We are also told that Sheol is a red dwarf star, which likely means it is an M-dwarf, though it could also be a late K-star. (Again, see the Wikipedia entry for stellar classification.) If we use the above equation (and Wolfram Alpha for our calculations), we find that Sheol has a mass of 0.89 times that of the Sun.

Wait, what?

A star with a mass of 0.89 M☉  is not a red dwarf. A reasonable upper limit on the mass of a red dwarf is probably around 50-60% the mass of the Sun. I think the writer for this blurb, whoever they may be, just didn't do their research properly. There are other systems in Mass Effect 2 that actually get their numbers right, but I don't remember what they were called or where to find them. Oh well.

One of the great things about the Mass Effect series is Bioware's attention to little details about the characters, the individual worlds, the continuity (turns out Conrad Verner has a Ph.D., as you learn if "Shepard bothered to interact with Conrad twice before and Conrad didn't die horrifically both times"). But naturally, they can't get everything right all the time, and this seems to be one of the cases where they didn't. Of course, this is something that almost no one would ever notice, and certainly no one reasonable. So in terms of doing stuff wrong, it may as well be something you can get away with, unlike the ending to Mass Effect 3.

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