In article ,
says...

So glad this group is still alive, so I had someone to ask.

The Saturn V second stage was mostly troublesome since it was performance critical

No. That is not why the S-II was most troublesome. It _was_ both of those things but one did not lead to the other.

Performance critical because, as the only stage still in development, any time the weight needed to go up in the LES, CSM, LM or SLA Marshall insisted that it be made up in the S-II. "You haf made the CSM too heavy, make up for it vith the part of za Saturn you are responsible for building"

Troublesome because North American decided to invent explosive forming for the gore segments, because they had insulation bonding problems, because the common bulkhead assembly required a precision which manufacturing at the time was unable to provide--so they had to invent that too.




Note that the Saturn IB first stage was heavier and more complex than it
needed to be due to its clustered tank design.


First stages just aren't good upper stages. They're built for high strength/high thrust at sea level to get off the ground

Technically this is not a constraint. The only details about a stage that are relevant to it doing its job is what delta v can it deliver to what payload.



engines were
not at all suited for an upper stage, being LOX/kerosene engines *not*
optimized for vacuum use


This is also not relevant technically to getting the job done. While it is true that the H-1 engine bells were not optimized for vacuum expansion, that doesn't mean they wouldn't work. It just means you'd have to burn more propellants for the same level of thrust.

I was hoping I wouldn't have to but okay, I'll do the math for us.

Saturn IB
Fueled Mass: 448,648 kgs
Empty Mass: 41,594 kgs

S-II
Fueled Mass: 480,000 kgs
Empty Mass: 36,000 kgs

So it passes the first test, a fully fueled IB weighs less that a fully fueled S-II, therefore
Yes, an S-IC can get an S-IB off the ground.

Now, I need to do the rocket equation for both cases.
First we need the Specific Impulses which since this is possibly the most studied engineering in history are actual and not theoretical.

S-IB H-1's burning RP-1/LOX

Isp(vacuum): 296 secs
Isp(sea level): 260 secs

S-II J-2's burning LH2/LOX
Isp(vacuum): 420 secs
Isp(sea level): 200 secs

and the S-IC data
Fueled Mass: 2,300,000 kgs
Empty Mass: 131,000 kgs
Isp(sea level): 263 secs

and I need to find the everything-everything-else-as-payload weight to be able to properly calculate the "second stage" final mass.

?v = v(exhaust) * ln (mass initial / mass final)
v(exhaust) = Isp * g0

To handle the non-vacuum optimized complaints I'll use the vacuum Isp.

?v of S-IB = 296s * 9.8 m/s^2 * ln ( 448,648 kg / 41,594 kg )
= 2900 m/s * ln ( 10.78 )
= 2900 m/s * 2.378 = 6,896 m/s

?v of S-II = 420s * 9.8 m/s^2 * ln ( 480,000 kg / 36,000 kg)
= 4116 m/s * ln ( 13.33_ )
= 4116 m/s * 2.590 = 10,661 m/s

so individually as single stages the S-II is a more efficient stage, but how is it as part of the Saturn V configuration?
the payload mass for the S-II can be calculated from the Saturn V total mass by subtracting the total mass of the S-IC and S-II. The payload to orbit mass isn't suitable in this instance because the S-IVB partially to get to that orbit, so the propellants it consumed count as payload to the S-II.

Total Mass Saturn V stack: 2,800,000 kgs
Second Stage Payload Mass = Saturn V Mass - S-IC Mass - S-II Mass
= 2,800,000 - 2,300,000 - 480,000 = 20,000 kg

lets see how adding that to our final masses changes the calculations
S-IB:
= 2900 m/s * ln ( 468648 kg / 61594 kg ) = 5,885 m/s
S-II:
= 4116 m/s * ln ( 500000 kg / 56000 kg ) = 9,011 m/s

so this is the actual delta v imparted to the Saturn V payload by the S-II stage. It looks like the S-IB is about 3,100 m/s short. Except for one thing. We forgot about the S-IC and the fact that the S-IB initial mass if lower than the S-II initial mass

So lets calculate the delta v of the S-IC as flown with the S-II stage:
S-1C with S-II second stage:
= 263 s * 9.8 m/s^2 * ln ( 2,800,000 kg / 500000 kg )
= 2577 m/s * ln ( 5.6 )
= 2577 m/s * 1.7227 = 4,440 m/s

Now lets replace the S-II with the lighter S-1B:
= 263 s * 9.8 m/s^2 * ln ( 2,768,648 kg / 468648 kg )
= 2577 m/s * ln ( 5.90773 )
= 2577 m/s * 1.7762 = 4,577 m/s

Okay, so wow. No way. A 10% lighter second stage results in a 3% increase in first stage velocity. We lose more than 3,100 m/s from the second stage but only pick up 137 m/s with the first. Even a better Oberth Effect isn't going to provide the missing 3,000 m/s.

So we lost 1200 m/s by switching from Hydrogen to RP-1 and lost the other 1,800 to a poorer mass fraction. I wonder what the mass fraction of a stage using H-1 engines with RP-1 would have to be?
required delta v = 9,011 m/s
9,011 m/s = 2,900 m/s * ln ( mass fraction )
9,011 m/s / 2,900 m/s = ln ( mass fraction )
3.107 = ln ( mass fraction )
e^3.107 = mass fraction
22.36 = mass fraction ~= impossible
using the above initial stage mass with the payload, the empty stage weight of an S-IB would have to be about 1000 kgs. I'll just attach 8 H-1s to the back of my Jeep and we can go to the Moon.

The math doesn't lie and points damning fingers at the culprits (lower
ISP LOX/kerosene engines and poorer mass fraction). In other words, we
were right about the Saturn IB's first stage being inadequate as a
replacement second stage for the Saturn V.

Lower ISP doesn't hurt a first stage nearly as much as a second stage.
A first stage engine needs high thrust more than high ISP. Once you're
out of the atmosphere, higher ISP is better.

That's not to say that you can't build an upper stage with a lower ISP
engine (e.g. SpaceX), but that you'll just have to make everything else
(first stage) bigger to compensate. In the case of SpaceX, they are
trying to optimize cost, not performance. Having an upper stage engine
that has much in common with your first stage engine is good for many
reasons, many of which tend to reduce costs.

Jeff
--
"the perennial claim that hypersonic airbreathing propulsion would
magically make space launch cheaper is nonsense -- LOX is much cheaper
than advanced airbreathing engines, and so are the tanks to put it in
and the extra thrust to carry it." - Henry Spencer