How Do We Know That Jupiter is 318 Times More Massive Than the Earth?

Question posed by COOL DIVYA.

This question is really asking 'how do we know the mass of Jupiter?' You're probably not going to like the answer to this one because what it essentially boils down to is one word:


But don't stop reading yet. Really, any real life problem can be boiled down to maths, and this is especially important in a lot of astronomy problems because actually going to places and measuring stuff is prohibitively difficult to say the least.

Without going into too much detail*, back in the late 17th century Isaac Newton published what was to become one of the most influential books in the field of physics, Philosophae Naturalis Principia Mathematica. In it was contained his Universal Law of Gravitation; a scientific law which describes the way two bodies interact by way of the force of gravity. Specifically, it involves an equation relating the masses of the two objects and the distance between them with the force that each exerts on the other (that force is what we call 'gravity').

As an example, if we could find the mass of Jupiter and one of its moons (say, Europa), and the average distance between them as they orbit, we can use Newton's law to calculate the force of gravity between them.

With a bit of mathematical ingenuity and knowledge about such things as 'force', it's possible to rejig Newton's equation to give us one that can help us find the mass of Jupiter, or any other body. The really snazzy bit is that in the re-jigging of the equations, the mass of the second object becomes unnecessary, and we can find the mass of Jupiter if we know just the following things:
  • The distance between it and whichever moon we're considering.
  • The period of the moon's orbit (i.e. the time it takes to do a lap).
That's it! Here's the equation we need:
Now, if that looks confusing just remember that, like any equation, the letters just tell us what numbers we need and their positions tell us what to do with those numbers. The letters in this equation refer to:
  • M: The mass of the body we're trying to find, measured in kilograms. In this case, that's Jupiter.
  • π: The Greek letter 'pi' stands in for a particular number that's just a bit bigger than 3.
  • r: The radius of the orbit of a moon around Jupiter, measured in metres. We can find this using trigonometry if we take a few photographs and have already worked out the distance to Jupiter.
  • T: The period (or Time) of the same moon's orbit around Jupiter, measured in seconds.
  • G: The Gravitational Constant. This is just a number that is there to adjust the calculation to make it work every time. Its value is about 0.000000000067**.
If we plug all this information into the equation we get an answer of 1.8987 x 1027 kg***. We can do the same thing to find the mass of the Earth (using values for the radius and period of the Moon's orbit instead), and we get 5.9742 x 1024 kg. Dividing one by the other we find that Jupiter is almost 318 times more massive than Earth.

* If you'd like too much detail please, please let me know and I'll be happy to oblige.
** Its units are "metres-cubed per kilogram per second-squared." Well, you did ask.
*** That's 1,898,700,000,000,000,000,000,000,000 kilograms.


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