# 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.

## 8 thoughts on “Figure of Merit”

1. Virgil H. Soule says:

The next thing you need to discuss is specific impulse. To get to orbit you need a certain total impulse. To make space travel practical, you need a propellant combination that yields that impulse using as little propellant as possible, ergo, high specific impulse. The SSMEs produce about 350 sec using LH2/LO2, which is about as good as you’re going to get using rockets. The SRBs only produce, what, 200 or so? (I forget exactly.) So, the Ares I design has taken a big step backwards as regards productivity. Are people at NASA looking at propulsion and launch schemes that will produce higher specific impulse? The goal should be single-stage-to-orbit for which you need 700 or so. For example, has anyone considered something like the hypersonic sled facility at Holloman AFB as a first stage just to get the vehicle off the ground and up to, say, Mach 4?

2. Michael C. says:

I enjoyed reading your recent blog about the difficulties of getting into Low Earth Orbit. I hope the Review of U.S. Human Space Flight Plans Committee will consider these points before offering commercial access to Low Earth Orbit as an option in their final report. I hope commercial ventures into Low Earth Orbit will continue, however, turning over America’s human access to space to commercial entities may be a mistake. Until the for-profit community can successfully demonstrate their ability to place humans into space, they should not be considered a serious alternative to NASA.

3. Ron says:

One of the biggest problems in my opinion has been the focus on radical changes in launch vehicles to achieve a true space faring civilization. When Vasco da Gamma set out to explore or the Virginia company set out to start settlements in the new world, their mode of transportation had not radically changed in a few hundred years. What changed to allow exploration was navigational breakthroughs (compass) as well as a change in philosophy of Europe. Steam ships were not required to settle the new world, and locomotives were not the prelude to western settlement. It was only after a base of operation (forts/ports) and infrastructure was developed and demand was high enough for transportation that Trains went went and propeller propulsion was introduced. So just because your mode of transportation looks like it came from your textbook rather than Star Trek should not matter, it is where it goes and what it builds.

(of course I love how the orbiter looks, but I would settle on a capsule to Mars or an asteroid many times over!)

4. Jeff says:

I’m not willing to try to figure all of the math, but it seems discouraging that getting to orbit is only a thousand times as hard and yet takes tens of thousands times the amount of \$\$\$\$!

5. Sean says:

There are a fair number of engines with comparable ISPs and in fairness, the SpaceX Falcon 1 launched Razaksat to 685 km altitude, with an orbital velocity around 17,500 mph, so that would match the ISS LEO line in the table above.

But using energy as a FOM and leaving out the payload mass disguises the next challenge a bit. And that is that Quantity has a Quality all it’s own. Razaksat (SpaceX’s payload on its last Falcon 1 flight) only weighs 180 kg.

That’s still only 1/600th the total energy of a shuttle to ISS altitude.

6. JC says:

The figure for the Mercury Redstone seems too high to me. Being a suborbital mission, I think you should not add kinetic and potential energies: in the apogee, speed was almost null (well, not exactly, but I think it is a good simplification), so you should consider only potential energy at the apogee. If you add the kinetic energy related to the maximum velocity along the trajectory, the results are unrealistic. A figure around 2500 kJ/kg seems more accurate to me. Or maybe I am losing something…

Anyway, this is just a little detail. Good article, very enlightening.

7. Calli Arcale says:

One of the biggest problems in my opinion has been the focus on radical changes in launch vehicles to achieve a true space faring civilization.

From Akin’s Laws:

39. The three keys to keeping a new manned space program affordable and on schedule:
1) No new launch vehicles.
2) No new launch vehicles.
3) Whatever you do, don’t decide to develop any new launch vehicles.

That said, there are limitations to what we have, and it isn’t entirely unreasonable to want something that has better performance. It is also true that the advances in launch vehicles over the past sixty years have *not* been radically new. They’ve been incremental, and really, the Atlas V isn’t fundamentally all that different from the V-2. It’s more sophisticated, sure. But it’s not fundamentally different. It’s just more powerful and more reliable.

So, we haven’t really changed as radically as you may think. After having seen a V-2 engine side-by-side with a J-2 engine from a Saturn V, I am convinced of that.

BTW, it’s actually not true that shipbuilding didn’t change significantly between the time of Vasco de Gama and the Virginia Company. On the contrary, as the Age of Sail really got underway, the world saw a tremendous explosion in the number of ship types in production, and radical advances in sailing technology. And I’m not just talking about navigation here. I’m talking about the ships themselves. They got bigger, faster, more stable, more able to cope with light winds, and better able to defend themselves from other ships.

8. jccooper says:

I once did some quick figuring and determined that if LEO orbit was the equivalent of going from New York to Los Angeles, a commercial airliner made it to about Scranton, PA (in terms of height and speed). A quick glance at a map shows most anyone there’s a significant leap in difficulty between the air and space.

And really, it’s even harder than that, since you can’t stop on the way there; think of making the trip with a giant bungee cord tied between you and the Empire State Building.