### What’s the mass of that planet?

Sometimes folks use the word “weight” instead of mass, but I know what you mean.  Having spent my whole life on Earth, I’ve come to the point of equating the two in casual conversation.  So astronomers tend to talk about “mass” as one aspect of a celestial object; planet, moon or star.

So what’s the mass of that planet?  Good question.  There are many good resources to answer that and other astronomical questions.  Many times I refer to the most recent edition of “Observer’s Handbook” from The Royal Astronomical League of Canada.  The 2012 edition says that Saturn is the equivalent of 95.161 Earth masses (M⊕).  And many other planets, moons and stars are so described.

But how do we calculate their masses?  Where do those numbers come from?

My students recently finished an astronomy lab making those very calculations for themselves.  The bottom line will be Newton’s Version of Kepler’s Third Law. This formula gives us the ability to determine the masses of an object and an orbiting body:

## P2 = (4 π 2 / GM ) R3

We observe the moon and determine it’s orbital period, p.  Then we use that orbital period and Kepler’s Third Law  to learn R, the semi-major axis of the orbit of the moon.   G is the gravitational constant.  M will be the mass of the planet and moon, but the mass of a moon is many times negligible in comparison.

That information plugged into Newton’s Version of Kepler’s Third Law will deliver the mass of the moon-planet pair.  Since moons are usually much smaller than the planets, we might just as well ignore the moon’s mass and consider M to be the mass of the planet

As far as I know this is the most reliable way to calculate the mass of an object.  Venus has no moon, so how do we calculate the mass of that planet?

(Great Question.  Here’s a NASA sticker!) If we put a space probe in orbit around the planet we can use it as an artificial moon.   NASA’s Magellan mission spent more than five years studying our sister planet.  Since we can know the mass of the artificial moon and be even more precise in out calculation.

We can do that for any planet, moon or star.  Isn’t that wild!  Thanks to Kepler and Newton.