In my previous post about a floating Saturn, I hinted that I could write about the methods we can use to find the density of Saturn. Oh, and once again, the density of Saturn is lower than the density of water on Earth - but it wouldn't float.
Just as a reminder, we define density as:
This means that we really need to determine two things. First, we need the mass of Saturn. Second, we need the volume. We can get the volume if we know the radius of Saturn.
Volume
Technically, Saturn isn't perfectly spherical. The distance from the center to the equator is greater than the distance from the center to the pole. This is because Saturn is spinning and it isn't a rigid object. Think of spinning pizza dough - same thing except it's Saturn. You can actually measure both the polar and equatorial radius using the same idea - but I am just going to pretend Saturn is a sphere.
If it is a sphere, then the volume would be:
But how do you get the radius (or diameter). The first step is to look at the angular size. If you know the angular size of an object and the distance to that object, you can find the size. Here is a picture I have used several times that shows this relationship.
So, if the object is far enough away or small enough then the height (or length) will approximately be the arc length of a circle with a radius the same as the distance. The size of the object will just be the angular size multiplied by the distance the object.
But how do you even measure angular size? Well, if you have a picture you need to know the angular field of view for your camera - I did this experimentally with an iPhone. In days before cameras, you could just use a telescope. It isn't too difficult measure the angular size with a lens. You just need to determine the angular field of view for the lens and then put some markings on there so you can estimate the fraction of the field for the object's angular size.
This is great, but it depends on something rather important. How far away is Saturn? This is where Johannes Kepler comes into the story. By using available data, Kepler came up with three models for the motion of objects in the solar system.
- The path of an object in the solar system is an ellipse with the Sun at a foci.
- As an object moves closer to the Sun, it goes faster. Kepler went even further and said that for a given time interval, the object would sweep out the same area no matter where it was in its orbit.
- The orbital period is related to the orbital distance (semi-major axis). In fact, the square of the period is proportional to (but not equal) to the cube of the semi-major axis.
Kepler's Laws of planetary motion aren't new physics. If you like, you could get the same set of laws using the momentum principle and gravitational force that proportional to one over the distance squared. However, the laws work and it is the last law that is useful here. If I know the orbital period of Saturn and the Earth, then I can write:
The T is the common physics symbol for the period and the time units don't really matter. The proportionality constant, k cancels when I divide one equation by the other. In the end, I have an expression for semi major axis for Saturn. If Saturn were in a circular orbit, this would be the radius and the distance to the Sun. Ah ha! But I don't actually have the distance from the Earth to Saturn. I can get the distance to Saturn in terms of the distance from the Sun to the Earth. Just to make things easier, we call this Earth-Sun distance 1 Astronomical Unit (AU). That's great and all, but if I use that unit (AU) for the size of Saturn, I would get the density in some weird units - kg/AU3. In order to compare the density of Saturn to water, we need the distance in something useful - like meters or maybe metres.
How do you find the value of 1 AU in meters? There are several ways. One way to find this distance is the Greek way. Yes, Greek astronomers did this sometime around 500 BC. Here is a short version of how they did it:
- Use shadows at different locations on the Earth to determine the radius of the Earth.
- Assume the moon moves in a circle around the Earth. Determine the difference between the calculated position (based on the center of the Earth) and the actual position (measured from the surface) to determine the distance (and size) of the moon.
- Measure the angle between the Sun and the moon when the phase of the moon is a quarter. This makes a right triangle. With the distance from the Earth to the moon already known, you can get the distance (and size) of the moon.
Here is an older post that shows more of the details in these measurements. Perhaps you can already see the problem with this method. If your measurements are off for the size of the Earth, then everything else is off. The Greek's determination of the distance to the Sun wasn't very accurate.
A better way to get the Earth-Sun distance is to use a transit of Venus. During this event, Venus passes between the Earth and the Sun. If you measure the start and end time from different locations on the Earth, you can get a value for the Earth-Sun distance. Here is an example with modern data.
I like the above ways to find the distance to Saturn because theoretically, you could do it yourself. Of course there are even better (more accurate) ways to find this, but the point is that you could indeed find the distance to Saturn and thus the size. With the radius, you could find the volume.
Mass
We can't just use Kepler's Laws to find the mass. No, we need to use some more fundamental physics. In short, we can find the mass of Saturn by looking at one of the moons of Saturn. If we know the orbital distance and the orbital period of one of the moons, we can find the mass. Notice that this is different than what we did above to find the volume. In that case, we used the orbital period of Saturn as it moved around the Sun to find the distance. Here we need both the distance and the period of the moon.
Let's start with some basic physics. Here is a diagram of the largest moon of Saturn, Titan, as it orbits.
The gravitational force depends on both the mass of Saturn and Titan as well as the distance between them. The magnitude can be written as:
Where G is just the universal gravitational constant. The momentum principle says that this gravitational force changes the momentum. Since this force is perpendicular to the momentum (p), then the force just changes the direction of the momentum and not the magnitude. It turns out that I can write the momentum principle in terms of the gravitational force and the angular speed of Titan as it orbits.
I know that I skipped some steps but the point is that there is a relationship between the mass of Saturn, the orbital size and the orbital speed. If I put in the period instead of the angular velocity (period = 2π/ω) I can solve for the mass of Saturn.
Now you just need three things: G, the size of the orbit, and the period of the orbit for Titan. The period is pretty easy. You just need to observe the planet through a telescope for some time and count the days until Titan makes a complete trip around the planet Saturn (about 16 days). The orbital size isn't too hard to get either. Essentially you do the same thing for this as the size of Saturn - use the distance and angular size.
The gravitational constant can be found with the Cavendish experiment. Basically, some small masses on a rotating rod are attracted to larger stationary masses. By looking at the twist in the rod you can determine the gravitational force and thus G.
And that's it. Once you have the mass and the volume, you can calculate the density. See, it's simple.
