## A DayDream: On Magic Leap's Recruitment Pamphlets

May 17, 2016

In April 2016, I attended Games Developer Conference to take a look at recent developments in virtual reality technology. The famously mysterious company Magic Leap had a booth among the game studios, and was handing out recruitment pamphlets.

A funny picture, I thought, and laughed at the ridiculousness of a balloon on the lunar surface. Because there is no atmosphere, and therefore no buoyancy force, the balloon would simply fall to the ground. When discussing the photo with a friend of mine, she smartly pointed out that the moon does in fact have an atmosphere, albeit a very thin one. Sensing an interesting problem, I asked myself: Could a balloon actually float in the thin lunar atmosphere?

As with any physics problem, the first step is to choose an approach. Bouyancy seemed the logical strategy, so I started with the simple equation

That represents the equilibrium condition for floating, where the mass of the gas in the balloon and the mass of the balloon skin equal the mass of displaced atmosphere. In order to create an ideal situation, I chose to ignore the string in the picture, as it is just dead weight.

The next step was to calculate the masses of the components. For simplicity (and the best volume to surface-area ratio) I chose to approximate the balloon as a sphere, which gives us the following equations based in spherical surface area and volume formulas:

Note that density for foil is mass per square area, but the density of the gas and atmosphere is mass per cubic area.

Plugging back into the first equation,

We come up with the following:

Thankfully, a lot of these values cancel out (isn’t it great when physics works out like that?), granting us a much simpler equation:

It is worth noting that the balance point is linearly dependent on the radius of the balloon, which matches my intuition (A larger balloon can carry more weight than a smaller one).

Solving for the radius, we get:

Now it is time to start plugging in values. NASA provides the composition of the lunar atmosphere, captured by experiments left by the Apollo missions. If you would like to experiment with different values, I’ve made an excel document available here. Using this data and some basic calculation, we calculate the lunar atmosphere to have a particle density of

And a mass density of

We want to use the most favorable conditions possible for this thought experiment, so we use the lightest gas (molecular Hydrogen) and the same particle density of the atmosphere (no need to over-inflate the balloon) to get:

This is about one tenth the weight of the atmosphere. Note that we make these calculations assuming the gasses behave as ideal gasses, and that the temperature of gasses inside the balloon is the same as the atmosphere.

To further improve the favorability of this thought experiment, we want to use the thinnest, lightest substance known as the skin of the balloon, in this case graphene, which has a density of:

Plugging in these values grants us a balloon radius of:

This is a very large value, almost twice the distance of the earth to the moon, so clearly the balloon in the picture is impossible. At such scales, variations in the moon's gravity with distance take play, the mass of the hydrogen gas may begin to measurably pull in on itself, and the earth's gravity becomes important as well.

Furthermore, there are other engineering problems that need to be addressed: graphene may not have the structural integrity to contain that magnitude of gas (or even its own weight) and has never been manufactured in sizes of more than a few square centimeters. Transporting and assembling the delicate atom-thick structure in space would prove an immense challenge. Lastly, graphene is permeable to hydrogen gas, meaning that the entire balloon would leak, and likely rather quickly.

It seems that the most implausible and technically challenging achievement in this photograph is not the man sent to the moon, as was clearly the photographer's intention, but the balloon, which could only float if the moon was terraformed to have a much thicker atmosphere.