## Figure of Merit

Figure of Merit is a term that may be unfamiliar.  Engineers use this term to describe a number – based on a formula – which is useful in comparing different items.  An everyday “figure of merit” is MPG (miles per gallon) for automobile fuel efficiency.  If you have bought a household appliance recently you may have noted an energy efficiency “figure of merit” on the label.  That allows you to decide to pay more for a more efficient appliance, or conversely to decide that the increased efficiency is not worth the cost and go for cheaper model.  A figure of merit is always a simplification and your real world results may vary.  For example, on my two year old vehicle, I have yet to achieve the MPG average that the sticker said it would get.  Maybe I just have a heavy foot, or something.  But that rating allowed me to compare vehicles in a significant way before I made the decision to buy.  A figure of merit may not in itself be the deciding factor.  But having a figure of merit is good when making a comparison between options.

There are many folks who wish that the world is different than it is.  Science fiction movies in my childhood concentrated how the rocket worked in getting people to space rather than what they did when they got there.  Nowadays, Han Solo jumps in the Millennium Falcon and instantaneously is in space making the calculations for hyperdrive.  Kirk and Spock, if not using the transporter, ride a shuttlecraft effortlessly to the space dock where the new starship is ready for flight.  Because Hollywood can do it with blue screens or computer animation, the popular imagination believes such things can be done in real life.  Or should be able to do it.  Or maybe just wish that we could do them.

So we see some folks that talk a good talk about getting into earth orbit.  Unfortunately the state of the art of technology doesn’t quite match the state of the art of portrayed in some powerpoints.

So I propose a figure of merit exercise to illustrate the difficulty of getting to earth orbit.  My figure of merit based on the energy state.  (Hold on, this takes just a little bit of physics and mathematics – nothing that a high school graduate shouldn’t be expected to know).

So a High school physics refresher: total energy is the sum of kinetic and potential energy.

E=PE+KE.

Potential energy depends on how high up you are: height (or altitude) times gravity times mass:

PE=h x g x m.

For example, a commercial airliner cruises at roughly 35,000 ft.  Let’s call it 6 nautical miles high, just to use an antique measurement system (I’m an old guy).  A spacecraft in low earth orbit probably needs to be at about 120 miles altitude to have significant orbital lifetime before atmospheric drag causes decay.  In simple math:

PE orbit/PE airplane = 120 miles x g x mass/6 miles x g x mass

So to stay in low earth orbit you need to be about 20 times higher than a commercial airliner.  That means, you need 20 times the potential energy to get from an airliner altitude to an orbital spacecraft altitude.  Wow.  No wonder space travel is hard.

But wait, that’s not all.  What about the other part of the equation, kinetic energy.  Kinetic energy increases as the square of velocity:

KE = ½ x m x v x v.

A typical commercial airliner cruises at about 500 mph.  To be in earth orbit requires a speed of 17,500 mph.

KE orbit/KE airplane = ½ x m x 17,500 mph x 17,500 mph / ½ x m x 500 x 500 = 1250 !

So it takes more than a thousand times as much kinetic energy to be in earth orbit as it does to be at airliner cruise speed!

It might be interesting to compare some other vehicles with orbital energy.  For example, the SR-71 is the fastest military aircraft ever.  It could go Mach 3 at an altitude of 80,000 ft. That is quite a bit more energy than a piddling commercial airliner.  And the X-15 got to Mach 6.7 and an altitude of over 350,000 feet – well, not simultaneously, but let’s do that calculation just to make it easy.  Here is a short table of some interesting vehicles:

Commercial airliner energy state at cruise:                            159 kjoule/kg

SR-71 at max speed & max altitude:                                      748 kjoule/kg

Space Ship 1 at max speed & max altitude:                         1,658 kjoule/kg

X-15 record altitude & record speed:                                   3,237 kjoule/kg

Mercury-Redstone at max speed & max altitude:                  5,605 kjoule/kg

International Space Station (low earth orbit):                    194,775 kjoule/kg

If you ever wonder why flying in space is not as simple or as easy as going to your local airport and getting on a scheduled commercial airliner, think physics.  Going to orbit is not twice as hard or ten times as hard as an airliner; it is over a thousand times as hard.

Wishful thinking won’t make it easier.