Wow. In xkcd 681 comic, there is an impressive illustration of the common term “gravity well”. Here is a small part of that large image:

I can’t resist. I must talk about this awesome illustration. My goal for this post is to help someone understand that comic (although the comic itself does a pretty good job).

### Energy

Energy is the key here. Here, I will talk about two types of energy – kinetic energy and field energy. In this case, kinetic energy is basically just the energy associated with something moving. Field energy is the energy stored in the gravitational field. You could also think of the field energy as the gravitational potential energy stored in the configuration of a system. I know that I didn’t talk about particle energy (you know, the E = mc^{2} stuff because it doesn’t matter here)

In a closed system, energy is conserved. This means that I can write:

Just to say – a closed system is one that has no work done on it. Maybe the best way to explain a closed system is with an example. If I drop a ball and let it fall to the Earth, the ball by itself would be an open system. The ball plus the Earth would be a closed system. I really don’t want to go too much into work-energy principles, just enough to get to where I want to go (explaining xkcd).

So, back to the energy equation above. For this situation, I can write the kinetic energy (K) and the gravitational potential (U_{g}) as:

I guess I should say that G is the gravitational constant (the big G, not the little g). M_{E} is the mass of the Earth (change this if you are on a different planet) and the little m is the mass of the object you are looking at. Why is the gravitational potential negative? How about I just say that it is for now. How about a plot of U_{g}/m for an object somewhere around the Earth? (starting from r = Radius of Earth)

I plotted the distance in units of “radius of the Earth”. Also, I included a “zoomed-in” portion of the graph. This zoomed in part is a plot of the same thing except from r = radius of the Earth to 10,000 meters higher. You will notice in this part, it looks pretty darn linear. In fact, I could even fit a linear function to that part of the data. Here is that function (where r is now in units of meters and measured from the center of the Earth)

See anything familiar? I know you see “g” in there. Yes, that is the same g you know. This is where you get that function in the textbooks:

The y-intercept is left off because only the change in potential matters. Ok, now for an example. Suppose I throw a ball up from the ground. If I consider the time after the ball left my hand AND I consider the system to be the ball and the Earth, then there is no work done on the system and the energy is constant. I can write:

Notice that both K and U_{g} have an m term in it. So, the mass doesn’t really matter. Now let me represent this as a sketch of a graph.

The green line represents the total energy. This means that for any possible height, the difference between E and U is the kinetic energy. Notice that there is a maximum height for this given energy. If the ball were to exist in this energy plot to the right of that line, the kinetic energy would have to be negative. This is a problem in that it would imply an imaginary velocity. Also notice that this plot does not show you the trajectory of the thrown object. It just shows you what the speed will be for a given position.

Now back to the real potential energy plot. Here is the same thing as the above diagram for a ball that is thrown faster (ignoring the work done by air resistance). For this plot, I am going to pretend I throw a ball straight up with a speed of 10 km/s (yes, that is fast). Note that for this plot, the vertical axis is energy/mass.

In this case, the ball (or whatever it is) will get about 5 Earth-radii away from the surface before it starts to fall back down. But there is one big difference with this real potential function and the linear one from above. The linear function keeps increasing. If that was the potential, you could never get an infinite distance from the planet. However, with the real potential you can get an infinite distance away. If the total energy is

Since U_{g} goes to zero as r goes to infinity, then an object CAN escape. If the total energy is zero, then I can solve for the velocity needed to escape:

You can think of this velocity need tot escape as “the escape velocity”. Really, you should think about the “escape energy” which is the energy needed to get away from the planet and never return. The escape velocity assumes it is a free falling object. The problem is that it could be a combination of several things like the rotational motion of the object on the rotating planet or extra rockets or whatever.

How about a plot of Earth’s gravity well?

I added the Earth in there just to make it pretty.

### The xkcd version

My well looks different than Randall’s (the xkcd author). He writes that the planets are not space out to scale so I guess he just artistically drew the wells (to look like wells). Also, he writes:

“Each well is scaled such that the rising out of a physical well of that depth – in constant Earth surface gravity – would take the same energy as escaping from that planet’s gravity in reality”

Let me check and see if this works. First, I will need to take some measurements. Sure, you could use photoshop or gimp or something to measure, but I will use Tracker Video Analysis. It’s free and does images also. Now, which planet should I look at? How about Uranus, because it is fun to say.

Step one – use the radius of the Earth to scale the picture.

Now to measure the “height” of the Uranus gravity well. Using the same technique, I get that the well is about 3.8 Earth radii. So, what is the gravitational potential for the surface of Uranus? According to google, the mass of Uranus is 8.68 x 10^{25} kg and its radius is 2.55 x 10^{7} m. This gives a gravitational potential per mass of:

Now, how high would a “well” on Earth have to be to have the same change in potential per kg? (yes, this assumes the slope of the potential stays constant). Remember from before, on the surface of the Earth:

The real change in potential for Uranus is also positive since the final potential is zero. So, setting the U_{g}/m to the value for Uranus and solving for h:

Wow. It worked. So, you can see where Randall gets the general expression for the height of the well in his drawing. He sets the real potential to mass equal to the gh Earth potential and gets:

I love this drawing (or comic – not sure what to call it other than AWESOME).

The rest of this image could be left alone and be part of Dan Meyer’s What Can You Do With This series. But I can’t contain myself. Here are some suggested homework problems.

- How big of a piece of paper would you need to include the Sun at this scale?
- What if you wanted to space the planets out in correct horizontal scale also – how big of a paper would you need?
- Do Randall’s sample escape speed calculations work?
- What if you wanted to redo the whole image and include the rotational effects of the planets AND the orbital effects. What would it look like?

### Update

Well, maybe this isn’t an update, but I thought I would share the python code I used to plot the potential well. Perhaps someone will find my sloppy code useful.

If you don’t have pylab module installed, the easiest thing to do is to get the Enthought Python Distro