How big of a balloon do you need to get to 120,000 feet high?

I am still thinking about the Red Bull Stratos Jump. Sorry, but there is just tons of great physics here. Next question - how big of a balloon would you need to get up to 120,000 feet?

I am not going into the buoyancy details of Archimedes Principle - I think that was covered fairly thoroughly with the MythBusters floating lead balloon. However, in short, here is a force diagram for a floating balloon.

For a floating balloon, the buoyancy force must equal the weight of the whole thing. It turns out that the buoyant force is equal to the weight of the gas (or fluid) the object displaces. I can write that as:

Here this depends on the density of the air the object is floating in, the volume of the object and the gravitational field (g). For the gravitational force (the weight), it is important to remember that this is for the balloon, the stuff in the balloon and the payload.

Red Bull Balloon

What about the balloon that will be used for the Stratos Jump? According to the Red Bull Stratos site, here are some details.

• Made from polyethylene 0.002 cm thick.
• Uses helium (not hydrogen)
• At the highest altitude, the balloon will be roughly 80 meters in diameter
• The capsule (payload) is made of fiberglass. They don't list the mass.

So, what makes this high altitude balloon different than a normal balloon? First, the density of the air decreases as you get higher. This means that your ability to create buoyancy force goes down (you need a bigger balloon). Now, I will estimate how high this Stratos balloon will go. Let me start with the assumption that the balloon is a sphere with a diameter d and the mass of everything is m. Also, I will assume the buoyancy from the actual payload is small enough to ignore. This means that the following must be true.

This says that the object will rise until its density is equal to the density of the air. At least I don't have to worry about the gravitational field changing with height (since that canceled). What is next? Well, I know the density as a function of altitude (I calculated this before). I also know the volume. I can estimate the mass of the capsule and the balloon material. The thing I really don't know is the mass of the helium. Maybe this is small and I can ignore it - but probably not. The one thing I know about the helium is that it is at the same temperature and pressure as the atmospheric air. If I treat both gases as ideal gases, then:

Here, n is the number density, or how many particles per cubic meter. If both gases act as ideal gases and are at the same temperature and pressure, then they must have the same number density. I can write this as:

I really just need the ratio of helium mass (per particle) to the air mass. Air is a little tricky since it is not one type of molecule. Let me assume the air is 20% O₂ and 80% N₂. This would give an average particle mass of the air as 9.57 x 10^-26 kg. The particle mass for helium is simple since it is just He, this is a mass of 6.65 x 10^-27 kg. Let me write the total mass as:

Here the m_s stands for "mass of stuff" where stuff is the payload, the jumper, the balloon material etc. Now I get:

Now I want to solve for the density of air. I get:

Just calculate the density of air and I can look up the altitude that gives that density. Now for the values (some of this stuff I am just making up).

• Mass of jumper = 80 kg
• mass of capsule = 150 kg
• mass of balloon = 360 kg (using a density of polyethylene of 930 kg/m^3)
• volume of object = 2.68 x 10⁵ m^3

If I put in these values, I get a density of 0.0024 kg/m^3. Running my density calculation again, I get that this corresponds to an altitude of 34 km (112,000 feet). What about my estimates? With that density, the mass of the helium would be 44 kg - not too large compared to the mass of the material of the balloon. This tells me that I really need to know the mass of the capsule. Still, not too far off from the 120,000 feet Red Bull projects.