Hop's Blog

Mf is a Mofo

The Tyranny of the Rocket Equation is an excellent article by astronaut Don Pettit written while he was aboard the I.S.S..

The foundation of the article is this version of Tsiolkovsky’s rocket equation:

Mf = 1 - e^{-delta v/v_exhaust}

Mf is the fraction of the spaceship’s mass that is propellant.

V_exhaust for chemical propellant ranges from 3 to 4.5 km/s. Delta V to get into orbit is around 9 km/s.

Plugging these in we can see a spaceship must be more than 80 percent propellant. Pettit notes The 3-stage Saturn V rocket was 85 percent propellant and the Soyuz rocket 91 percent.

Pettit explains this sort of mass fraction is very challenging and drives up engineering expense:

If a vehicle is 10 percent propellant, it is typically made from billets of steel. Changes to its structure are possible without detailed engineering anyalysis: you simply weld on another hunk of steel to reinforce the frame according to what your intuition might indicate. I can easily overload my three quarter ton pick-up truck by a factor of two... 

Once vehicles become airborne, engineering structures become more serious. Lightweight structures made of aluminum, magnesium, titanium, and composites of epoxy-graphite are the norm. To alter a structure requires significant analysis: one does not simply weld on another chunk to your airframe or drill a hole though some convenient section if you want to live. ... Overloading an airplane by a factor of of two results in disaster. Even though these vehicles are 30 to 40 percent propellant (and thus, 60 to 70 percent structure and payload), there is eough ‘wiggle rooom’ to comfortably operate aircraft, which is how we have a robust, safe, and cost effective aviation industry.  

Rockets at 85 percent propellant and 15 percent structure and payload are on the extreme edge of our ability to fabricate, not to mention pay for. They require constant work to keep flying. The seemingly smallest modifications require monumental analysis and testing of prototypes in vacuum chanbers, shaker tables and test launches... 

The common soda can... is 94 percent soda and 6 percent can by mass. Compare that to the external tank (ET) for the space shuttle at 96 percent propellant and thus, 4 percent structure. The ET is big enough to hold a barn dance inside and contains cryogenic fluids at 20 degrees above absolute zero pressurized to 60 pounds per square inch, four times atmospheric pressure...”

ET photo courtesy of NASA

One very expensive aspect of Mf Pettit didn’t talk about: Delta V budgets for reaching low earth orbit and beyond strongly encourage multi-stage expendable rockets. Here is a video explaining how multi-stage rockets are a way to deal with large delta V budgets. Expendable is another word for disposable. After transporting it’s payload, much or all of the engineering marvel is typically thrown away. Imagine how expensive a transcontinental plane ticket would be if we threw away a 747 each trip.

Pettit suggests a way to break the tyranny of the rocket equation:

A rudimentary and basic skill to master is to learn how to use raw materials of space to create new capabilites there. Our nearest planetary neighbor, the moon, is close and useful – it contains the material and energy resources we need to build a permanent space transportation system. Extracting and producing useful materials useful products from the raw materials of the Moon (particularly water, useful for life suport, rocket propellant and many other space applications) would relieve us from the need to drag everything we need in space from the bottom of Earth’s deep gravity well. This eventuality would significantly alter the consequences of the rocket equation in our favor.

I’d like elaborate on this. Here's a delta V map of our neighborhood, cislunar space:

A propellant tanker from the Moon to EML1 would have a round trip delta V budget of 5 km/s. Mf is 67 percent (assuming hydrogen/oxygen propellant).

A propellant tanker from EML1 to LEO would have a round trip delta V budget of 4.5 km/s. Mf is 63 percent.

A tanker from EML1 to GEO would have a round trip delta V budget of 2.6 km/s and a 43 percent Mf.

With propellant available at these locations, vehicles for moving about cislunar space would have mass fractions ranging from 25 percent to 58 percent

No herculean feats of engineering needed to meet these mass fractions. And they allow single stage reusable vehicles.

Lunar water isn’t the only way to cut Mf. Planetary Resources wants to move a water rich asteroid into high lunar orbit or EML1. Whether propellant comes from the Moon or a Near Earth Asteroid, it would revolutionize space transportation in our own neighborhood as well as deep space destinations such as Mars or asteroids.

Pettit's article appeared in the Fall 2012 issue of Ad Astra published by the National Space Society.