When a rocket leaves its launch pad, it fights gravity, but also the atmosphere. If you view a televised or webcasted rocket launch from Cape Canaveral or Vandenberg, you generally get to listen to the launch control team. During ascent, an engineer may call out “*max Q*“, the point of maximum dynamic atmospheric pressure. As the rocket’s speed increases, the dynamic pressure increases as well. But as it gains altitude, air density goes down, reducing pressure. Max Q is where the two balance out, and the impact of reduced air density starts to outweigh the impact of speed. Typically, it occurs 50 to 70 seconds after launch.

What if an airplane or balloon could carry the rocket above the bulk of the atmosphere? Wouldn’t this reduce the impact of dynamic pressure, lower the value of max Q, and make it easier for the rocket to reach space?

In fact, this is true. This is the basis of some air launches, including rockets which are lofted by aircraft or by balloon.

## Air pressure vs altitude

Air pressure is measured as force per unit area, e.g., pounds per square inch (psi), or newtons per square meter (Pascals, abbreviated “Pa”). At the Earth’s sea level, the pressure is 14.6959 psi or 101325 Pa, or simply 1 atmosphere (atm).

The atmosphere is in effect a thin shell that surrounds the Earth. While a precise thickness cannot be assigned, the rate at which it falls off with altitude is instructive.

Altitude (km above SL) |
Pressure (% atm) |
---|---|

40 km | 0.27% |

30 km | 1.1% |

20 km | 5.4% |

14 km | 14% |

12 km | 19% |

10 km | 26% |

8 km | 35% |

6 km | 47% |

4 km | 61% |

2 km | 78% |

0 km | 100% |

Half the atmosphere is below 5.5 km above sea level. The diameter of the Earth is 12,742 km, and the radius, 6371 km. That 5.5 km is less than 0.1% of the radius. The highest flying birds have been seen at 29,000 feet (8.8 km). Going the other direction, the ocean’s Mariana Trench is 11 km deep. So from the highest birds to the depths of the ocean, the Earth’s biosphere is limited to 0.31% of the Earth’s radius.

On October 14, 2012, when Felix Baumgartner made his record-breaking dive from 38 km (125,000 feet), the air pressure was 0.36% of sea level. (Recorded altitude was 127,852 ft => 38.969 km, pressure = 0.31%) This rarified atmosphere supported a helium-filled balloon, a gondola and Baumgartner — a mass of about 1350 kg (2976 km). With virtually no atmospheric drag to slow him down, when he finally hit denser air, he was travelling 1.25 times the speed of sound.

Getting a rocket above most of the atmosphere by balloon seems worthwhile. At 31 km, you are above 99% (pressure is 1% of sea level). Even at 16 km, you are above 90% (pressure is 10% of sea level).

If the payload has no passenger, you might even use hydrogen gas instead of helium. Hydrogen has half the density of helium, and should thus provide even more buoyancy, reaching even higher altitudes.

## Where is space? Getting to orbit?

While there isn’t a clean break between the atmosphere and space, a generally accepted definition is the Kármán line, at 100 km above sea level. There is still atmospheric drag at the altitude, but the air density is about 0.005% of sea level.

To stay in orbit in space, a satellite needs horizontal speed. For low Earth orbit, this is 7.8 km/sec. At this speed, although the satellite is falling, the Earth is also curving away at the same rate so that the satellite never falls to the surface.

Thus, for a rocket launched from an aircraft or balloon, while it still needs to climb out of the sensible atmosphere, above the Kármán line, it needs to put most of its thrust to imparting horizontal speed to its payload.

## Going suborbital

If the payload simply needs to go straight up and come straight down, the delta V required is considerably less.

The delta V can be modeled as an instantaneous impulse at the given altitude. That is, the payload immediately starts to coast at the start of the trajectory. Below is the delta V required to reach the edge of space at 100 km from a given altitude above the ground. (These are ballistic computations, with no atmospheric drag.)

Altitude (km above SL) |
Remaining (km) |
delta V (km/sec) |
---|---|---|

40 km | 60 km | 1.08 km/sec |

30 km | 70 km | 1.17 km/sec |

20 km | 80 km | 1.24 km/sec |

10 km | 90 km | 1.33 km/sec |

0 km | 100 km | 1.40 km/sec |

For sounding rockets, there is a very short burst of thrust lasting a few seconds. During this time, the rocket’s propulsion needs to compensate for 1 G (9.8 m/sec^{2}) pull of gravity. The acceleration is high, and generally not suitable for living organisms. But the distance traveled during the thrust period is very short. Once the rocket’s propulsion has burned out, the coast calculations apply.

For human space flight, the G level is considerably lower (e.g., 3.5 Gs). Thus, the thrust period is longer, and the rocket is fighting some fraction of gravity all the way to orbit.

## Hydrogen from a balloon

Can the hydrogen from a balloon then be used as propellant in an upper stage of the rocket? Frankly, I don’t know how to do it. (I don’t think it is possible without some very exotic physics.)

Rather than lifting it by an external balloon with tethers, could you put the balloon inside the rocket and lift it that way? No, but it presents some intriguing observations for what keeps a rocket grounded before launch and what gives a balloon lift.

If you want to use hydrogen gas from a balloon for a rocket, you need to take gas out of the balloon, and feed it to the propulsion system. Once you remove gas from the balloon, it starts lose buoyancy.

If it was stable at a certain altitude, it will start to drop.

If you try to carry the balloon with the rocket as it goes up, it needs to be arranged so that the engine exhaust does not impinge on the balloon for a couple of reasons.

- It creates drag which counteracts the thrust the engine is generating.
- Hot engine exhaust will tend to rip open the balloon. Hydrogen inside the balloon will then ignite when combined with oxygen in the atmosphere, resulting in a very spectacular explosion.

So whatever clever design is pursued, the exhaust has to stay away from the balloon.

How much gas is in the balloon? Is it enough to get you to orbit? Probably not. It would depend on the mass of the payload and rocket. If the payload is to reach orbit, the system of propellant gasses and rocket has to impart 7.8 km/sec of “delta V” (additional speed) to the payload. The larger the mass, the less the attainable delta V. This approach would only work for very small payloads.

## A lighter-than-air rocket

To get around the problems of having to pump the hydrogen from the balloon into the rocket, why not put the balloon inside the rocket itself? This won’t work. The reason has to do with buoyancy, but it is perhaps still worth examining what the impact of moving the hydrogen from the balloon to the rocket would be.

So here is a thought experiment. It involves a rocket with a structural (dry) mass of 10 kg, and a volume of 22.4 cubic meters. In this example, the rocket engine weighs nothing and takes no volume. (Why 22.4 cubic meters? See “Tricks of the trade” at the bottom.)

We’re at sea level and the air is rather chilly. Measurements are made at standard temperature and pressure (STP). Here, we will consider what happens when the rocket is filled with different kinds of gasses.

**Air**. Sitting on the ground, the rocket is full of air. The air weighs 28.8 kg. The rocket structure enclosing the 22.4 cubic meters is 10.0 kg. The total weight is 38.8 kg, but since it is also displacing 28.8 kg of air, the scale says 10.0 kg.

**Nitrogen**. The 22.4 cubic meters of nitrogen weighs 28.0 kg. On the scale, the 10.0 kg rocket now weighs 9.2 kg.

**Helium**. The 22.4 cubic meters of helium weighs 4.0 kg. The 10.0 kg rocket and 4.0 kg of helium now weigh 14.0 kg. But since it is replacing 28.8 kg of air, it now floats! If you could weigh it, the rocket at sea level now weighs -14.8 kg. The density of the floating rocket is 14.0 kg / 22.4 cu m = 0.625 kg/cu m. Looking at standard atmosphere tables, the rocket would float to about 6.5 km above sea level. *(In this case, the rocket is lighter than air.)*

**Hydrogen**. 22.4 cubic meters of hydrogen weighs 2.0 kg. Combined this the 10 kg of structure, the rocket weighs 12 kg and has a density of 0.536 kg/cu m. This time it floats to 7.8 km above sea level. *(Again, lighter than air.)*

We could pump more hydrogen into it, but it would make the rocket heavier. At double the amount of hydrogen, the gas now weighs 4.0 kg, and the overall rocket 14.0 kg (but floating off the scale).

At 10 times the amount of hydrogen, it now weighs 20 kg, the combined rocket and hydrogen weigh 30 kg, and it registers on the scale at 1.2 kg and no longer floats. To get this much hydrogen into the 22.4 cubic meter rocket, we’ve had to start chilling down the gas.

**Liquid hydrogen**. If it were full of 22.4 cubic meters of liquid hydrogen, which weighs 71.0 kg, the hydrogen + rocket – displaced air would show on the scale as 52.2 kg. In practice, a rocket with liquid hydrogen also carries liquid oxygen, which is far denser, and would result in a much greater rocket mass.

## Tricks of the trade

The choice of 22.4 cubic meters for the volume of the rocket was no accident. At standard temperature and pressure (STP), 1 mole of gas occupies 22.4 liters. What is a mole? A mole of hydrogen atoms has a mass of 1 gram. How many atoms is that? It is 6.022 x 10^{23} atoms.

A hydrogen gas molecule has 2 hydrogen atoms; a mole of hydrogen gas weighs 2 grams, and occupies 22.4 liters. A cubic meter is 1000 liters. A kilogram is 1000 grams. And 1000 moles of gas occupies 22,400 liters or 22.4 cubic meters.

So the atomic and molecular masses used for a mole translate seemlessly to this scheme of 1000 moles.

The delta V for suborbital flight was computed using a little piece of Python code. Below is the interactive session that shows how it was done:

>>> from math import sqrt >>> def speed(height): ... grav = 9.8 ... t = sqrt(2.0*height/grav) ... v = grav * t ... print "t (sec):", t, "v (m/sec)", v >>> speed(60000.0) t (sec): 110.656667034 v (m/sec) 1084.43533694 >>>

**Revision History**

2013 Dec 31 10:45pm – Original, during New Year’s Eve.