What I find interesting about Pokrovsky’s theory is how easy it is to debunk with so little information. All one needs to know is the thrust-to-weight ratio of the rocket at liftoff, how long the engines burned, and the specific impulse of the engines. Two of the three – thrust-to-weight ratio and burn time – can be easily verified from video of the launch, and therefore impossible to lie about. From there it’s just a matter of applying a little math.
I know from published thrust and mass data that the initial thrust-to-weight ratio of the Saturn V was about 1.19, but suppose I don’t know that. How do I figure this out from video? Answer: by measuring the initial acceleration of the rocket.
To lift the rocket the thrust must first cancel out the weight of the rocket, and then what thrust remains accelerates the rocket upward. Suppose analysis of video shows the rocket lifting off the pad with an acceleration of 1.9 m/s
2. This is the acceleration after the thrust has already cancelled out the 9.8 m/s
2 acceleration of gravity; therefore the total acceleration is 11.7 m/s
2. We convert this to g’s by dividing by standard gravity, 11.7 / 9.8 = 1.19 g. The thrust-to-weight ratio of the rocket is, therefore, 1.19.
Next we need to know the burn time of the engines. The total burn time of the outboard engines was about 162 seconds, which should be easily verified by examining archive footage of the launches. Center engine cutoff occurred at about 135.5 seconds. It is easier if we just say all five engines burned for 157 seconds, which gives us the equivalent amount of impulse and propellant use as the actual burn times.
For the last item, specific impulse, or Isp, we just have to trust the documentation. The F-1 engines of the Saturn V had a sea level Isp of 265 seconds, which is entirely normal for engine of its type and propellant. There is no reason to disbelieve this figure, but we’ll come back to that later.
The propellant mass flow rate is simply the thrust divided by the Isp,
1.19 / 265 = 0.00449
I haven’t put any units on the above because it really doesn’t matter. What this tells us is that the rocket consumes 0.00449 times its own mass every second. If the Isp of 265 s is correct, the vehicle must consume this much of its mass to produce the observed acceleration. This is from firmly established physics and mathematics and can be no other way.
So, if the rocket burns its engines for 157 s, the amount of total mass consumed as a ratio of the rocket’s initial mass is,
0.00449 x 157 = 0.705
From
Tsiolkovsky’s rocket equation we can calculate the total change in velocity, or Δv,
Δv = V
e x LN[ m
o / m
f ]
Where V
e = exhaust gas velocity, m
o = initial mass, and m
f = final mass. Effective exhaust gas velocity can be determined by the product of the Isp and standard gravity. Therefore we have,
Δv = 265 x 9.8 x LN[ 1 / (1 – 0.705) ] = 3,170 m/s
There are a couple things to point out here. First, 265 s is the Isp of the F-1 engines at sea level. As the rocket climbs through the atmosphere, the Isp increases as the outside air pressure drops. In a vacuum, the Isp is 304 seconds. My own simulations have shown that the effective Isp over the totality of the S-IC burn is about 295 s; therefore, the Δv is actually more than calculated above. Second, the launch vehicle experiences loses from gravity and drag as it must counteract the downward pull of gravity and push its way through the atmosphere. Taking all this into account, my simulations have shown that the actual Earth-fixed velocity at S-IC burnout is about 2,400 m/s.
I haven’t read all of Pokrovsky’s paper, but it appears he has applied two methods to determine the Saturn V’s velocity at S-IC burnout. In one case he determines the velocity falls between 1,200–1,600 m/s, and in the other case between 1,100–1,450 m/s.
My numbers don’t lie. It is mathematically impossible to have the observed liftoff acceleration, burn the engines for the observed duration at the required propellant mass flow rate, and then be traveling as slowly as the velocities given by Pokrovsky.
Pokrovsky’s claim requires that the rocket accelerate far more slowly than claimed by NASA. So, under what conditions can Pokrovsky be correct in his observation? Well, the liftoff acceleration is observed and verifiable and cannot be in dispute; therefore, it is the acceleration at the middle and the end of the burn that must be less. Typically the acceleration increases as the rocket burns off propellant. For the Saturn V to be accelerating more slowly than claimed, it must be retaining more of its initial mass by burning propellant at a much lower rate.
The thrust-to-weight ratio at liftoff is fixed based on the observed acceleration and cannot be changed. Therefore, since we cannot reduce the thrust, and since Pokrovsky’s claim requires a lower propellant mass flow rate, we must produce the same thrust using less propellant. Consequently, Pokrovsky is essentially claiming that the F-1 engines were more efficient with a higher Isp than claimed in official documents. Let’s calculate the Isp required to match Pokrovsky’s observation.
I stated previously that my simulations yield an S-IC burnout velocity of about 2,400 m/s. If we take the median of Pokrovsky’s numbers we have 1,325 m/s. My calculated Δv of 3,170 m/s would indicate that we have about 770 m/s in gravity and drag losses. A vehicle with a lower acceleration, as Pokrovsky requires, will have a greater gravity loss. Let’s assume the losses are 1,000 m/s. This means the launch vehicle Δv in Pokrovsky scenario is about 2,325 m/s.
(I could provide a better estimate of Pokrovsky’s gravity and drag losses by simulating the launch, but that requires more time than I’m willing to devote to the problem at the moment.)
Pokrovsky’s propellant mass flow rate is,
q = 1.19 / Isp
And from Tsiolkovsky’s equation we have,
Isp x 9.8 x LN[ 1 / ( 1 – 157 x q ) ] = 2,325
We now have two unknowns, Isp and q, and two equations, so we can simultaneously solve equations to arrive at our answer. I’m not going to show all the math, but the answer is,
Isp = 476 s (sea level)
Of course this number is ridiculously high; even the best liquid hydrogen engine can’t attain this efficiency.
It’s quite ironic Pokrovsky’s claim that the Saturn V was seriously underpowered necessitates that its engines be impossibly efficient. Pokrovsky’s argument leads to other problems as well. For instance, if the F-1 engines were really so efficient, why did the rocket carry so much propellant that it weighed so much in the first place. With more efficient engines, the propellant load could have been reduced, resulting in a lighter vehicle that would have attained the higher burnout velocity that we’d expect. There’s a bizarre circularity to this that makes Pokrovsky’s claim entirely illogical. The head spinning illogic forces one to conclude Pokrovsky is wrong.
