As the New Horizons spacecraft is approaching Pluto, it's getting some great shots. The animation below shows Pluto and Charon as they orbit each other.
When a planet (or dwarf planet) has a satellite, you can use the motion of that object to find the mass of the planet. It's not too difficult. In fact, let's use the animation above to estimate the mass of Pluto. Ok, there is just one thing we need that's not in the video. We need to know the distance from Charon to Pluto. Actually, if I knew the angular field of view of the camera, I could get the orbital distance from the angular separation between the two objects. However, for this example I will just look up this value and use a Pluto-Charon distance of 1.957 x 107 m.
Also, a couple of other notes about this Pluto-Charon video.
- The sequence of images shows Pluto and Charon as the spacecraft approaches the two objects. This means that the distance scale changes in each frame.
- This video is relative to the center of mass of the Charon-Pluto orbit. You can see that both Pluto and Charon are orbiting a common center.
- We are not viewing the orbit from a completely perpendicular angle. This means that Charon doesn't appear to move in a circle (even though it essentially is).
There are really only two physics principles we need for this problem—the gravitational force and the acceleration of an object moving in a circle. Let's start with gravity.
The model for the gravitational interaction says that there is an attractive for pulling on objects with mass. The magnitude of this force is proportional to the product of the masses of the two objects interacting and inversely proportional to the square of the distance between the objects. I can write the magnitude of this force as:
The G is just a constant, so don't worry about that. But there is one more thing that is important. The gravitational force is always pulling the two objects together and is in the direction of the other object.
The other important idea is to consider the acceleration of an object moving in a circle—yes, both Pluto and Charon are moving in a circle about a common center. Here is a diagram (but not to scale).
Any object moving in a circle (even at a constant speed) will be accelerating. The direction of this acceleration will be pointed towards the center of the circle and it will have a magnitude that depends on both the angular velocity and the radius of the circle. This acceleration can be written as:
Here the ω is the angular velocity of the object in radians per second. But I guess there is one more physics thing to look at: the relationship between force and acceleration. Simply, the greater the net force on an object the greater the acceleration.
Now to put it all together. The force is the gravitational force and the acceleration is due to circular motion. There is one catch. If I write this out for the motion of Charon then the gravitational force will use the distance from Charon to Pluto but the acceleration will use the distance from Charon to the center of rotational motion. Just to be clear, I will call the distance between the objects r and the radius of Charon's circular orbit will be rc. Also, since the gravitational force (the only force) is in the same direction as the acceleration, I can write this as a scalar equation.
The mass of Charon cancels and so I can solve for the mass of Pluto.
So all I need are the two distances and the angular speed of Charon's orbit.
I already said that I was going to look up the distance from Pluto to Charon. However, I need the distance from Charon to the center of the orbital circle. By skipping through the frames of the animation, I can find the one that shows Charon and Pluto the furthest apart—it's at this point that I can see the full size of the orbit (no perspective). Using this image and the distance from Pluto to Charon, I get a circular orbit radius of 1.69 x 107 m.
Now I just need the angular velocity of the orbit. If I pick a frame with Charon almost above Pluto and then when it is almost below it, that would be half an orbit. Looking at the time stamps on each frame, this would give a time of about 3.2 days (276,480 seconds). That would put the angular velocity as:
Using a gravitational constant of G = 6.67 x 10-11 N*m2/kg2 I can put in all my values to calculate the mass. Crunching the numbers, I get a Pluto mass of 1.24 x 1022 kg. BOOM. This is pretty close to the value listed on Wikipedia with a value of 1.3 x 1022. I consider this to be close enough to count as a win.
What about the mass of Charon? You can use this same idea and the orbital distance for Pluto's orbit and get a mass estimate. It's a homework assignment for you.
Just for fun, here is a numerical model showing the motion of Pluto and Charon.
The above code is on trinket.io. This means that you can both look at the code AND change the code. Try changing the masses of the planets and see what happens. Notice that if they initial momentum of the two planets don't add up to the zero vector, the center of mass doesn't stay stationary. It's fun stuff.