How much mass to orbit? How much mass to space?
All facts are from the Saturn V press kit of the Marshall Space Flight Center, Huntsville Alabama at http://history.msfc.nasa.gov/saturnV, more specifically from a copy of the "Saturn V News Reference", December 1968.
|
Stage
|
weight loaded
|
weight dry
|
time of burn
|
final height
|
final velocity
|
| Apollo |
98,000 lb
|
||||
| Instr. Unit |
4,500 lb
|
||||
|
3 to moon
|
???
|
5.2 min
|
24,500 miles/h
|
||
|
3 to orbit
|
257,900 lb
|
25,900 lb
|
2.75 min
|
115 miles
|
17,500 miles/h
|
| 3 aft interstage |
7,700 lb
|
||||
|
2
|
1,037,000 lb
|
95,000 lb
|
6 min
|
114.5 miles
|
15,300 miles/h
|
|
1
|
4,881,000 lb
|
303,000 lb
|
2.5 min
|
38 miles
|
6,000 miles/h
|
| Total: |
6,262,500 lb
|
The aft interstage of stage three is separated shortly after stage separation and was therfore separated from the weight of stage three.
Let's convert this to easier units:
| 1 lb = 0.454 kg |
| 1 mile = 1.61 km |
gives:
|
Stage
|
weight loaded
|
weight dry
|
time of burn
|
final height
|
final velocity
|
| Apollo |
44,492 kg
|
||||
| Instr. Unit |
2,043 kg
|
||||
|
3 to moon
|
???
|
312 s
|
11.00 km/s
|
||
|
3 to orbit
|
117,087 kg
|
11,759 kg
|
165 s
|
185 km
|
7.83 km/s
|
| 3 aft interstage |
3,496 kg
|
||||
|
2
|
470,798 kg
|
43,130 kg
|
360 s
|
184 km
|
6.84 km/s
|
|
1
|
2,215,974 kg
|
137,562 kg
|
150 s
|
61 km
|
2.68 km/s
|
| Total: |
2,843,175 kg
|
What is the weight of the third stage after the first burn?
Assuming all fuel is completely consumed after the second burn, it is:
120,582 kg - (120,582 kg - 15,254 kg) · (165 s / (165 s + 312 s)) = 84,147
kg
The third stage J-2 engine uses 449 lb/s oxygen and 81.7 lb/s hydrogen, which
is 241 kg/s:
120,582 kg - 165 s · 241 kg/s = 80,827 kg
It seems the above numbers are somewhat imprecise (e.g. they indicate that the third stage has fuel for only 437 s but also say it burns for 477 s).
Approximately 44,492 kg + 2,043 kg + (84,147 kg+80,827 kg)/2 = 129 t go into orbit.
How high could the Saturn-V go if only the first stage was used?
Most of the atmosphere is below 61 km. Air pressure is too high for an orbit, but already so low that we can ignore it for this estimate. Earths gravity is already slightly lower, but only a few percent. Ignoring this too will somewhat compensate errors from ignoring atmosphere.
After the first stage has burnt out, the space craft goes up at 2.68 km/s and is decellerated by earths gravity. This means it travels up for 2680 m/s / 10 m/s² = 268 s. It will reach a height of 61 + 1/2 · 10 m/s² · (268 s)² = 420 km. Then it would take another 268 s to fall down to a height of 61 km again, which gives a total time of 2 · 268 s = 536 s = 9 minutes in free fall.
The space craft still has its full weight minus the used fuel of stage 1, i.e. 2,843,175 kg - (2,215,974 kg - 137,562 kg) = 764 t. A crosscheck counting the other way gives 44,492 kg + 2,043 kg + 117,087 kg + 3,496 kg + 470,798 kg + 137,562 kg = 775 t.
Approximately 770 t go to space.
129 / 770 = 17% of the effort are for going to space. 6 times more mass to space than to orbit.
Slightly different numbers are given in the book "Projekt Apollo" from Werner Büdeler, Bertelsmann Sachbuchverlag, 1969, Library of Congress Catalog Card Number 71-96070.
Weight to orbit: 113 t
Total weight at lift off: 2837 t
Fuel of first stage: 2035 t
This gives a total weight to space of 2837 t - 2035 t = 802 t
With these numbers, only 113 / 802 = 14% of the effort are for going to space. 7 times more mass to space than to orbit.
Note: from numbers in the book I computed a maximal acceleration for the first stage of 4.3 g. This is more than what the current space shuttle uses (3 g), but not much more. Saturn V peak acceleration of 6 g does not happen during stage 1 burn.
Note that a height of 420 km and 9 minutes of free fall may be more than inverted aerobraking needs. But the major motivation for this example was to use data from a realistic, technically proven ascend through atmosphere. Any hypothetical earlier engine cut off of stage 1 would need exactly those difficult estimates about atmoshperic properties that I want to avoid by using this example.
Anyone who has a better example?