Is there a cooler object in the sky than Saturn? Maybe Jupiter, but I like Saturn a little better. If you know what you are looking for, you can see the rings of Saturn even with a pair of binoculars. Personally, I love getting out the telescope and showing Saturn to people. Their expressions when they see it show their amazement. Most people don't realize that you can ACTUALLY see the rings.
Enough about viewing Saturn. Here is the one thing that bothers me. You will often see in textbooks and other media that Saturn has a low density and would actually float in water. No. This is wrong. Well, it's kind of wrong.
The Density of Saturn
Let's assume that Saturn is a sphere. We can easily calculate the density now. Well, easily assuming that we look up values for the radius and the mass. According to Wikipedia, Saturn has a mass of 5.68 x 1026 kg and a radius of about 5.6 x 107 meters. Knowing the volume of a sphere, we get the following calculation for the density.
The density of water is 1000 kg/m3. What does this mean? Well, if I have a block of some material underwater then I can draw the following two forces on it:
On the surface of the Earth, the magnitude of the gravitational force can be written as:
Here I just wrote the mass of the object as the product of the density of the object (ρo) and the volume of the object (Vo). For the buoyancy force, I can calculate this as the weight of the water displaced. This would be written as:
Both the weight and the buoyancy force have the same Vog term. The only thing that is different is the density. So, if the density of water is greater than the density of the object, the buoyancy force when the object is fully submerged will be greater than the weight. In order to be at equilibrium, the object would be just partially submerged. We commonly call this "floating". And here you see that if the density of an object is less than the density of water, that object will float.
If you want a more detailed derivation of the buoyancy force - check out this post about the Magdeburg Water Bridge.
Would Saturn Float?
Saturn's density is less than water. Things with a density less than water float - things like ducks, tiny rocks and gravy. So it seems logical that Saturn would also float. Right? Wrong.
How much water would you need for Saturn to float? Let's assume for now that this is some ginormous planet with as much water as we need. Also, I will assume that in this region of water, the gravitational field is constant and pointing straight down since the planet is so large.
If the planet could float (see below), how deep would the water need to be? For a floating object, the buoyancy force is equal to the gravitational force. This means that only part of the planet would be underwater. But how much? If I call the volume of the planet underwater Vd (d is for displacement), then I can write:
This means that the volume of water displaced will be the volume of Saturn multiplied by the ratio of the densities. Using my density of Saturn, 77.2% of it would be underwater. How deep would this be? Here is a picture.
You can see I need to find the value for h which is the depth the planet would go underwater. It's clear it will be larger than the radius of the planet, but by how much? Instead of deriving the formula for the volume of a partial sphere - I will use this Wikipedia page for a spherical cap. This says that the volume of the cap (the top part) would be:
If I set this cap volume to 0.228 the volume of the full sphere, then I can solve for a. I will skip the details - you can do this for a homework problem if you like. It's not too difficult to solve, but I get a value for a of 0.6189*R. This means that h would be 1.38*R. With the radius of Saturn, you would need water that is 7.7 x 107 meters deep. Maybe you would like this depth in different units. How about a water depth of 6 Earth diameters?
Let me make a sketch of this. I am just going to draw a water planet that looks big enough to be mostly "flat" around our floating Saturn.
I left the inside of the planet empty - I don't know why. However, based on this sketch the water surface planet would have a radius 8 times larger than Saturn's radius. This makes the water planet on the same order as the size of the Sun - except water. Water is hydrogen and oxygen. You know what else has a lot of hydrogen? Yes, the Sun. I haven't done the calculations, but it seems like a planet the size of our water planet would have enough pressure in the core to start nuclear fusion.
Oh, so that's why I made it hollow. Still, the pressure at the bottom of this ocean would be way too high for the stuff at the bottom to still be liquid water. Really, I don't know what would happen to it. I just don't think you could make any body of water this deep no matter what you try.
Saturn Still Wouldn't Float
Ok, maybe you found some awesome way to make water really really deep but still water. Maybe you have devoted the resources of the entire solar system just to make a giant sea of water. Ok, I get it. Still Saturn wouldn't float.
If you take a ping pong ball and toss it in your tub, it will float. A ping pong ball is a rigid object. Saturn is not rigid. The large bulk of Saturn's outer volume is filled with molecular hydrogen. The interior is something much denser - maybe metallic hydrogen and/or a rocky core. The denser materials are in the center because a gravitational interaction. If you like, you could think of the collective gravitational force of all the bits of Saturn pulling such that the denser stuff is in the middle supporting the lower density materials.
But what would happen if you put this non-rigid object on the giant water planet? If the planet is very very large in mass, the net gravitational field will be towards the center of the water planet and not towards the center of Saturn. This means that all of that material - especially the rocky core will also be pulled to the center of planet water. Let me change my floating Saturn diagram to show the core.
What is going forces are going to be acting on the core? Well, there the gravitational force of planet water pulling on it. But what pushes up on it? The hydrogen in the atmosphere of Saturn pushes up - but not very much, it's just not dense enough. That means this core will "fall" towards the surface of the water planet. The hydrogen atmosphere will then move up and probably become part of the water planet's atmosphere. This would kind of be like trying to hold a raw egg without the shell. It just doesn't stay together.
In the end, you would have a giant rocky core at the bottom of the water planet's ocean. If you want to call the destruction of a planet "floating", well I guess that's ok. Or maybe we could keep the old definition of floating and leave Saturn where it is.
So, what should you say about Saturn's density? How about something like this:
Oh, I think I should talk about how humans find the mass and volume of Saturn. That will be another post though.
Don't forget that today (July 19, 2013) is Smile and Wave at Saturn Day. At around 21:30 UTC, the Cassini spacecraft will take a picture of Earth and Saturn at the same time. So, wave and comb your hair.
