# How do you find the mass of planets?

*Dear Urban Astronomer*

*At the risk of opening myself to ridiculously complicated maths, how do you actually calculate the mass of a planet, or a sun, or, for that matter, any astronomical body.*

Thanks,

Andrew

Thanks,

Andrew

So it all becomes a bit of a detective game, where you have to start with a known mass and work your way across the Solar System till eventually you find the mass of the body you’re interested in.

We normally start with the mass of the Earth. In this case, we can use simple experiments to measure how powerful gravity is at the Earth’s surface (it causes an acceleration of 9.82 meters per second squared). Since the Earth is so much vastly more massive than any single object we could put on a scale, that value doesn’t change in any measurable way as the mass of the object changes, which is why all objects fall at the same speed (unless there’s a lot of wind resistance, like with a feather, or a guy wearing a parachute!) Knowing the strength of the gravitational field, and the radius of the Earth, it’s a VERY simple sum to calculate the Earth’s mass (answer: a lot. A LOT lot. The number of kilogrammes has 24 zeroes in it).

Now that we know the Earth’s mass, and if we know how far away the Sun is, and if we know how long it takes to Orbit the Sun, we can use Kepler’s laws to calculate the mass of the Sun. And now that we know all about the Sun, and have timed the orbits of all the planets, we can use the same calculation to find THEIR masses. And now that we know their masses, we can figure out the masses of their satellites.

What about other objects, where there is no chain we can follow, like a galaxy far away? Well… that’s where the detective work starts including more guesses. For example, we might measure the light coming from the galaxy and from that get an idea of how many stars it must have. From studying stars in our own neighbourhood, we have an idea of what the average star weighs, and so we can work out what the galaxy probably weighs. But it’s more complicated that that – we know that galaxies have vast clouds of gas and dust which hide stars from us, and we have to make educated guesses about how many stars are hidden. We also have to consider how much Dark Matter the galaxy has. To get past this, and to check how good our guesses were, we then try other methods: For example, we can measure the speeds that stars in a galaxy are orbiting its centre, and then plug those speeds into simulations on supercomputers to see what sort of mass would give the speeds that we’re measuring. This can also be problematic, because while the maths to calculate 2 things orbiting each other is relatively simple, doing it for a few hundred BILLION stars is… well it’s practically impossible. We have to close our eyes and simplify it down to what our computers can cope with and hope that it doesn’t mess the results up too badly. For a third method, we might come up with clever theoretical models of how we think galaxies work, and plug in the numbers that we do know (brightness, speed of stars, distance) and see what sort of mass the theory predicts. And we then take ALL these numbers from all the different methods and we compare them: If they’re in the same region then we pat ourselves on the back because it means that we’ve at least got a number that’s close to the real value.

Another example is working out the mass of distant plutoids (Distant bodies that are almost-but-not-quite planets – like Pluto). Most of them have only been discovered in the past decade or so, and they move so slowly that we haven’t been able to track their orbits with any reliability. This means that we don’t know how long their years are yet, which means that we can’t use any of these methods to figure them out. Then, we have to apply a different method.

First, we’ve been watching Pluto for almost a hundred years, so we’ve got its orbit mapped out pretty accurately. And since we discovered its gigantic moon Charon in the 1970’s, we’ve got a pretty accurate idea of its mass. We’ve also managed to measure its size, thanks to the Hubble Space Telescope being able to see it so clearly (although it’s really only just clear enough to measure the size and hint at the fact that it’s not perfectly smooth and featureless). From that we measure its density. We also know how bright it is.

As for the Plutoids, the only reliable measure is their brightness – we know how much light reflects away. We also have VERY unreliable figures for their size – they’re so far away that even the best telescopes only show them as dots. So we assume that they’re made of the same stuff as Pluto, which would mean that they’re just as reflective. So we calculate backwards how big they would have to be to have the brightness we measure… but it’s hard because we don’t know if that guess about what they’re made of was right, and we don’t have their distance accurately measured yet. But, having worked out a possibly-accurate value for size and density, we can then calculate the mass that way. But the good news with all these new distant bodies is that they DO move, and over time we’ll be able to work out their orbits more accurately until we have pretty good values for their mass. And then we’ll be able to do that whole process backwards, and see how good our guesses were ðŸ™‚

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