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Every so often a claim comes along that is so simultaneously amusing and absurd that we have to take pause and appreciate it. Our current favorite is Apollo Moon Flights: NASA Hoax -- Or Alien Technology!, found (for the time being) here. It was written by Der Voron of UFOlogy fame. It was so incredibly wrong that we just had to figure out what its author was thinking. To our surprise, Mr. Voron responded to our inquiry. Well, first he tried to sell us an anti-gravity device. But when he realized we weren't interested, then he responded to our inquiry.
Fig. 1 -The Ariane 5 heavy-lift booster on its launch pad. (Arianespace)

Mr. Voron's bifold thesis is quite simple: (1) the stated figures for the Apollo lunar module describe a spacecraft that couldn't possibly have made it to orbit from the lunar surface, and (2) any vehicle that could have done that would have been too heavy to place on the moon using any rockets from the 1960s.

Mr. Voron's method for determining the minimum mass of a lunar module ascent stage is particularly amusing. Here's how it goes: Take any old joe rocket built to launch big commercial payloads into earth orbit (Fig. 1). It has a certain amount of power. It also has a certain mass when fully fueled. So if you need only one-sixth the power from a lunar module (because of one-sixth the gravity) then the lunar module must have one-sixth the mass of your reference booster. A lunar module with less than one-sixth the mass of some particular earth booster won't reach orbit.

Yes, that's his theory.

Mr. Voron hastens to point out that the Weekly Universe article is wrong, and that he should have divided the Ariane's mass by 36, the square of the gravity ratio. That gives a "minimum" spacecraft mass of 20 tons. As we'll see, this correction is a bit like rearranging the deck chairs on the Titanic. Der Voron's problem is with his completely unjustifiable method, not the numbers he plugs into it.

Gee, where to begin? First we can dispense with the notion that NASA would have to build a Cape Kennedy on the moon. Rockets are launched "in the field" all the time without elaborate facilities. It's convenient but not required to have one. Generally the smaller the rocket, the less ground support it needs. Compared to the Saturn V, the lunar module ascent stage is a pretty small rocket.


COULD THE LUNAR MODULE HAVE MADE IT TO ORBIT?

Fig. 2 -The force of the ascent engine's thrust lifting the spacecraft is about twice as great as the force of gravity holding it down. (NASA)
It's not that hard to figure out. First we have to toss out some figures[1]. Der Voron didn't know the lunar module split into two stages and that only the upper stage returned to lunar orbit. That's important to know if you plan to argue it wouldn't have worked. The ascent engine produces 15,570 N thrust. Its total mass is 4,547 kg including fuel but excluding crew. NASA budgeted 144 kg for crew, for a total takeoff mass of 4,691 kg. 2,358 kg of this total mass is fuel. Assuming it burnt all the fuel[2] its final dry mass would have been 2,333 kg.

Lunar gravity pulls down on a mass of 4,691 kg with a force of 7,662 N. The engine pushes up with a force of 15,570 N. It doesn't take a rocket scientist to predict which direction the lunar module will go (Fig. 2).

But after ten seconds it is hard. That's when the ascent stage stops going straight up and "pitchover" occurs. The thrust is no longer directly opposing gravity, so it becomes a vector problem. But luckily the standard reference materials provide a method of computing a spacecraft's total capacity to change its velocity, and that will give us some impression of the LM ascent stage's ability to go places. It's

Dv = g Isp ln l
where g is the standard SI mass constant (9.8 m/sec2)[7], Isp is the specific impulse of the engine, and l is the vehicle's mass ratio -- total mass over dry mass. The specific impulse of Aerozine 50 and nitrogen tetroxide in the ascent engine is 312 sec. according to Grumman's data sheet. The mass ratio is 4,691 kg (fully loaded) divided by 2,333 kg (no propellant remaining), or 2.011. So the total ability of the lunar module ascent stage to change velocity according to this formula is 2,136 m/sec.

But is this enough?

The target orbit is 14 km by 72 km above the lunar surface, or 1,752 km by 1,810 km from the moon's center. Now the actual ascent trajectory departs significantly from simple textbook examples and so it's hard to be scrupulously accurate about it while at the same time keeping it simple enough to understand here.

But first let's characterize the necessary orbit. We know the apolune and perilune of the orbit that NASA says the astronauts were aiming for. In order to get into that orbit the spacecraft would have to be at least 14 km above the regularized level of the lunar surface. And it would have to be moving downrange (i.e., along the orbit) at a speed sufficient to keep it up there.

Now an elliptical orbit doesn't have one constant orbital speed. The satellite will "dive" toward the low point of the orbit, picking up speed. And it will "climb" toward the high point, losing speed as it goes. But there are standard equations in orbital mechanics to give us the required orbital velocity of the high and low points.

vp = SQRT ( ( 2 GM Rp) / ( Rp( Ra + Rp ) ) )
where
  • vp is the velocity at perilune (the low point of our orbit, where we want the orbital insertion point to be);
  • GM is a convenience constant: the universal gravitational constant G multiplied by the mass of the primary around which we are orbiting, in this case the moon[3];
  • Rp is the radius of the desired orbit at perilune (the low point); and
  • Ra is the radius of the desired orbit at apolune (the high point).
We solve this equation for our desired orbit around the moon and arrive at a perilune velocity of 1,686 meters per second. So if we can put the lunar module 14 kilometers above the surface, moving downrange at 1,686 m/s, it will stay in the orbit planned above.

There is no one singular ascent trajectory to achieve that insertion point. But in general an ascent trajectory goes straight up for a little, then gradually begins to tilt and accelerate downrange. The lunar module did this in a series of linear steps: it went straight up for a few seconds, then tilted through a set of predefined angles each for a certain number of seconds. These steps were prearranged to put the spacecraft at the right spot above the moon, moving in the right direction at the right speed[4].

But this will pose too complicated a question for our example. So let's simplify the actual method down to a piecewise approach that we can solve simply in a few paragraphs. If you want to get to a point 14 km above the lunar surface, you can just operate the engine in order to give you enough vertical velocity to achieve that altitude. Then you can turn downrange and "floor it" in order to achieve your 1,686 meters per second along the orbital path parallel to the ground.

If you throw a baseball upward, it rises to a certain height and then falls back down. The height it reaches is determined solely by the velocity of the ball as it leaves you hand. With every second of the ball's ballistic flight, the earth's gravity will take away 9.8 meters per second of that ball's upward velocity until it's all gone, at which point the ball reaches the apex of its ballistic arc. Then gravity adds 9.8 meters per second of downward velocity to bring it back to your hand.

Fortunately physics provides us a way to compute just how much velocity our lunar module "baseball" would need in order to peak at an altitude of 14 kilometers. From basic Newtonian physics we recall

Dx = ½ a t2
when the initial velocity is zero. We know the displacement Dx is 14,000 meters, and the moon's gravitational acceleration is one-sixth of earth's so we solve for t to arrive at 131 seconds. That is, if we give the lunar module just enough upward velocity to reach 14 km, it will take 131 seconds to get there. We can then use
v = a t
to determine just how much upward velocity would be exactly eroded to zero by the steady pull of gravity over the interval of 131 seconds. It's 213 meters per second. If the ascent stage were given that much velocity upwards, it would peak at 14 kilometers high, 131 seconds after departing the surface. (But of course the ascent stage's upward velocity isn't imparted "instantly" by the engine -- so the ascent will require longer than 131 seconds. But as long as you provide that change in velocity of 213 meters per second along the way, you'll get there.)

So much for the needed vertical velocity. How about the horizontal velocity? Well, the ascent stage starts from a standstill and eventually needs to be going 1,686 m/s horizontally. If we add in the 213 m/s of upward velocity that it needs, we have 1,899 meters per second of total change in velocity needed to achieve lunar orbit in our highly simplified (and very inefficient) ascent trajectory.

Now above we computed the lunar module ascent stage's total ability to change velocity: 2,136 meters per second. But we only need 1,899 meters per second to achieve the rendezvous orbit that NASA planned for the ascent stage. So we've got some to spare.

And keep in mind that our ascent profile is the worst you can imagine. In reality you don't separate the horizontal and vertical accelerations into two separate maneuvers. You go straight up for a while to avoid local terrain obstacles (and on earth, to get above the densest layers of air). Then you start tilting downrange so that an increasing fraction of your thrust is used to accelerate you horizontally. As that horizontal velocity achieves a significant fraction of the necessary orbital velocity, your momentum will help you stay aloft vertically. So our elementary trajectory is a worst-case scenario.

Real rocket science (albeit with our ballpark figures) confirms that a vehicle with the Apollo lunar module's specifications could indeed make it to lunar orbit.


BOOSTER VERSUS MOON LANDER

So where does Der Voron get this wacky idea that a lunar spacecraft would have to weigh at least 20 tons in order to make it into lunar orbit? Frankly we don't know. Which is to say, we know how he arrived at the number (Ariane-5 has a mass of 750 tons and 1/36 of that is 20 tons). But we don't understand just exactly how that number is supposed to express anything important or useful.

The number of qualitative differences between the lunar module and an Ariane (or any other earth-launch booster) is staggering, and Mr. Voron doesn't seem to appreciate any of them. But they all have drastic effects on how the respective spacecraft are built, and therefore how much they weigh.

We could go for days just talking about air. Earth has it, and the moon doesn't. Boosters launched from earth have to push through a lot of air before getting to orbit. The air pushes back -- it's called "drag" -- and that means you need more force from the engines to compensate for it and arrive at that same velocity. More force means more fuel you have to add to get the same speed. Extra fuel adds mass to the booster. (Drive a long distance with a car-top carrier and you'll get an object lesson in how much gas it takes to compensate for drag.)

And the booster has to be made stronger to withstand that drag. Stronger structures have more mass and require more fuel to lift. To make the vehicle more aerodynamic it has to be made long and skinny. But structures to support long skinny shapes are less efficient[5] than structures to support a centralized shape of the same volume.

The designers of the lunar module didn't have to worry about air. They didn't have to make their spacecraft long and skinny to slip through the air easier. Since it could almost any shape, the designers were able to choose an optimum shape for which an efficient structure could be made. The lunar module's skin didn't have to be thick to withstand the blast of the slipstream, and the structure didn't have to be strengthened (and therefore made heavier) to prevent it from caving in from air resistance.

Now if you want to orbit earth you have to do it above the atmosphere. If you didn't, the aforementioned drag would slow you down and you'd plummet back to earth. There's a minimum altitude for an earth orbit (about 160 km) and so an earth-launch booster has to be able to get its payload to at least that altitude -- and moving fast enough to stay up there. Lunar orbits aren't concerned with air. You don't have to push through it on your way up so you don't have to budget fuel to compensate for the drag. And you can orbit at a much lower altitude. So you don't have to push your payload up so high.

Those big monster boosters are made to lift tons of essentially deadweight cargo to earth orbit. Anything that has to move tons has to be built to handle tons. Ten tons multiplied by 3 or 4 g's is a hefty load, and a hefty load requires a hefty structure that weighs a lot no matter what shape it is. The payload for the lunar module ascent stage was basically two astronauts and a box of rocks -- about 200 kg. That's itty-bitty compared to what an Ariane or a space shuttle is asked to lift, and the lunar module payload was inside the structure, not on top of it. Directly comparing an Ariane and a lunar module implies they're expected to lift the same loads in the same way.

It gets more insidious. Take this excerpt:

"'Eagle' weighed about 16 tons, or about 5,500 pounds if we mean lunar gravity. To launch a satellite of such a mass, at least an Ariane-5 class rocket is required."
Now wait a minute. The Eagle ascent stage wasn't just dead payload like what usually sits atop a booster. It was itself the booster. Half of Eagle's mass was the fuel it was about to burn up as it ascended. And let's not get started on Mr. Voron's habitual confusion between weight and mass.

Sooner or later we had to know just where Mr. Voron learned to analyze spacecraft. Nowhere, it turns out. Although he claims a measured I.Q. of 165, he freely admits having no training in aeronautical engineering. Or, apparently, in any kind of scientific or technical field. Even more suspicious, Mr. Voron has, until recently, stubbornly resisted informing his readers of his true qualifications.

If you buy an anti-gravity device from Der Voron, we suggest you keep the receipt.
It seems geniuses don't need to do much research either. After claiming in his article that there was no booster capable of delivering the lunar module to the moon, Mr. Voron asked for the payload capacity of the Saturn V rocket. Shouldn't he have asked that before drawing his conclusion? The answer can be found in minutes on the web. It can also be found in the Apollo press kit we sent him, which he refused even to look at.

Caveat emptor. If you buy an anti-gravity device from this man, we suggest you keep the receipt.

But, says, Mr. Voron

"... the American flag, and a plaque with inscriptions on it next to the flag, are reported, by many persons who visit observatories, to be clearly seen on the Moon surface."
So if the lunar module was a bust, how did those flags[6] get there? Obviously NASA has already developed anti-gravity propulsion, says Mr. Voron. Or maybe, he suggests, they just borrowed the anti-grav engines from the friendly Zeta Reticulans or Orionites. Mr. Voron is unclear whether these are the same space aliens he alleges shot down the space shuttle Columbia. (We swear we're not making this up.)

Like all conspiracy theorists, Mr. Voron can't help noting all those "impossible" shadows in photography, but we'll let that pass for now. There's no need to cajole him into admitting he doesn't know anything about photographic analysis either. We frankly recommend a career in comedy writing.


THE DARKER SIDE OF DER VORON

This would simply be another crackpot case if it weren't for the plight of Denise M. Clark. Very shortly after we contacted Mr. Voron by e-mail to ask him questions, the Clavius mailbox began to fill with solicitations from a "Denise M. Clark" asking us to publish her glowing review of Der Voron's UFO book Starcraft, and to publish other articles written by Mr. Voron. We responded in typical fashion that we weren't interested.

We didn't think any more of this until one of our associates stumbled onto Ms. Clark's web site. We contacted her via the posted e-mail address and told her politely that she was wasting her time submitting her writings to us. Little prepared we were for the torrent of frustration that followed.

Someone, it seems, has been impersonating Ms. Clark through e-mail, asking every editor and webmaster on the planet (and perhaps -- knowing Mr. Voron -- some from other planets) to publish her Starcraft review and Mr. Voron's own writings. And predictably, the review appears all over the web, including places that Ms. Clark does not necessarily want her writings to appear. She has had to track down and contact every editor to whom this imposter has provided her material and try to have it withdrawn. She gets hate mail and angry phone calls from people she doesn't even know because of what has been done in her name.

Denise wants her life back.

Naturally we asked Mr. Voron about this. He flatly denied having impersonated Ms. Clark, and suggested it must be some other person who is trying to discredit him. However, the "imposter" is acting in Mr. Voron's interest -- not discrediting him -- and if anyone is the victim here it's Ms. Clark. Further, Mr. Voron declined to explain why Clavius received Voron-related solicitations from "Denise M. Clark" mere hours after we informed Mr. Voron -- and only Mr. Voron -- of our interest in his work. Hm.

Clearly the one-armed man responsible for the "Denise M. Clark" impersonations remains at large, but based on our experience and those of other webmasters, we believe we don't have to look any farther than to Der Voron. We hope he has the good sense to try to correct the damage he has done to her and her reputation. Good luck, Denise.


NOTES

  1. Where did we get all these figures? No two lunar modules were the same. No two used the same amount of RCS fuel, for example, and that factors into the capability for liftoff. Those built for Apollos 15 through 17 had greater payload capacity. We could either do all the numbers for all six successful landing missions, or we could just average the numbers for the early lunar modules. That's what we've done. Publications such as Apollo By The Numbers and the Apollo Mission Reports give us the basic loadout numbers, and from them we've arithmetically concocted a "typical" Apollo lunar module.
  2. In practice spacecraft can't always run their fuel tanks dry. The physical limitations of the plumbing make some of the fuel inaccessible.
  3. Conspiracists often forget, or don't realize, that the moon's gravity is much less than earth's. This affects orbital parameters such as orbital period, altitude vs. velocity, etc.
  4. This is akin to finding buried treasure by going so many steps in a certain direction from a starting point, then so many steps in a different direction, and so forth. If you follow that procedure with sufficient precision, you will arrive at the treasure. But your inaccuracy will compound along the way and get worse with each step. So a better way -- and the way we use now -- is to steer the rocket according to an actual predetermined path. The rocket guidance systems today are smart enough to follow the path and steer left or right, up or down, as necessary to return to the ideal path. The ascent stage's trajectory was a masterpiece of simplification. The computer had to be told simply to follow a predetermined table of durations and corresponding angles of deflection from the local vertical. This could be worked on in advance by powerful computers on earth and then uploaded to the lunar module. The lunar module's digital autopilot already had the ability to hold the spacecraft at a certain attitude, even with the crew moving about and the fuel sloshing in the tanks. So with the ascent program generating a timed sequence of attitudes to hold, and the digital autopilot holding them, the lunar module could achieve a reasonably stable orbit with a minimum of onboard computer power.
  5. Efficiency in terms of a structure means its load-bearing capacity compared to the mass of the structure itself. An efficient structure is one which is light itself, but which can bear a large load. For cylindrical rockets there is a compression load from the payload mass, as well as sheer and torsion loads from steering and weather. These loads often induce oscillations in long skinny structures, which must be damped out. Other structures have their problems too, but not like this.
  6. As a matter of fact the American flag is not visible from earth, or even with the Hubble Space Telescope. The flag is just too small, as is the lunar lander descent stage. The plaque was actually affixed to the leg of the lunar module, some distance from the flag.
  7. A number of readers have written to remind us that the value 9.81 (in SI) is valid only for the earth and that we should be using one-sixth that value in the lunar environment. This is a misunderstanding of what is intended by the mass constant. The same equation holds unmodified on earth, on the moon, or in deep space. The mass constant relates the unit of mass to the unit of force for certain computations, standardized as the amount of force exerted upon that mass by earth's gravity. The force of earth's gravity on one slug of mass is 32 pounds-force. Similarly the force of earth's gravity on one kilogram of mass is 9.8 newtons. This constant is introduced so that values computed according to the English system are directly comparable to values in SI units, an vice versa. Without this, specific impulse values would be different in the different systems and not directly comparable.

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