# Why is the value of the Isp of the Rocketdyne J-2 Saturn V engine in my calculation so off?

While calculating the delta V the Saturn V has on its second stage, I wanted to also calculate the Isp of the J2 rockets engines. However, the value difference between my results and the values given on Wikipedia were larger than I expected. I wanted to know why there is such a large difference between my calculations and the numbers given on Wikipedia. Is it just that in real life other factors come into play which change the actual Isp or have I made a mistake with my calculation?

Here are the values given on this wikiepdia page:

Variables values
Thrust in a vacuum 1,033.1 kN (232,250 lbf)
Thrust at sea-level 486.2 kN (109,302 lbf)
Specific impulse in a vacuum 421 seconds (4.13 km/s)
Specific impulse at sea-level 200 seconds (2.0 km/s)

Unfortunately Wikipedia didn't share the mass flow rate of the engine so I got the values from this site

Mass flow rate values
Oxidizer 449 lb/s (203.663 kg/s)
Fuel 81.7 lb/s (37.0585 kg/s)
Total 240.7215 kg/s

I used this formula to calculate the Isp:

$$I_{sp} = \frac{ Force\ [in\ N]}{gravitational\ acceleration\ [in\ m/s²] * mass\ flow\ rate\ [in\ kg/s]}$$

Using the data from the charts I get 2 answers. One for the Isp of the J2 engine at Sealevel and the other for the Isp of the engine in a vacuum.

Isp at Sealevel:

$$I_{sp} = \frac{ 486200\ N}{9.81\ m/s² * 240.7215\ kg/s} = 205.88\ s$$

Isp in a vacuum:

$$I_{sp} = \frac{ 1033100\ N}{9.81\ m/s² * 240.7215\ kg/s} = 437,48\ s$$

The value at Sea level is quite close to the actual value (200s vs 205s), but the Isp value that I got for the engine in a vacuum is quite different than what it says on wikipedia (421s vs 437). I wanted to use the Isp value to calculate the delta V the second stage of the Saturn V rocket had, but with a gap of this size, my calculations of the delta V were quite off.

While calculating the delta V second stage I used these values as the mass

Stages of the Saturn V Mass
Total weight 2 822 000 kg
First stage gross mass 2 280 000 kg
Second stage gross mass 480 000 kg
Second stage propellant mass 443 000 kg

sources:

$$m_{before} = m_{total} - m_{first\ stage} = 2 822 000 kg - 2 280 000 = 542 000\ kg$$ $$m_{after} = m_{before} - m_{second\ stage\ propellant\ mass} = 542 000 kg - 443 000\ kg = 99 000\ kg$$

edit M after is the total mass of the Saturn V with everything on top included minus the first stage and the propellant mass of the second stage.

I used the Tsiolkovsky rocket equation to calculate the delta V of the second stage.

$$\Delta v = Isp * g * ln( \frac{m_{before}}{m_{after}})$$ The delta V using the Isp value I calculated:

$$\Delta v = 437,48\ s * 9.81\ m/s * ln( \frac{542 000\ kg}{99 000\ kg}) = 7296.48\ m/s$$ The delta V value when using the value on wikipedia: $$\Delta v = 421 * 9.81\ m/s * ln( \frac{542 000\ kg}{99 000\ kg}) = 7021.62\ m/s$$

Just from the difference between the Isp value given by wikipedia and the one from my calucalation, I get a difference of about 200 m/s of delta V which is a quite large margin of error.

My question: Why is there such a large gap between the Isp value of my calculation and the one given on Wikipedia. Is it just that in real life other factors come into play or is it because I made an error in my calculations?

• There was some variation between missions (see Apollo by the numbers, p284 and subsequent. Are you sure you're putting the right numbers together?
– Ludo
Dec 15, 2023 at 12:26
• @Ludo I added the sources where I got the numbers so if you think that the numbers are mixed up, you can check it, if you want. Also since the ISP is different in my calculations and the only factors which could change were the force and mass flow rate, any mixed up numbers would come from there. I think the number for the force is correct since it was given on the same page as the ISP, but maybe the mass flow rate is incorrect? Idk Dec 15, 2023 at 12:31
• The document I referenced gives the launch weights and propellant burn rates for each mission. You'll see there's quite some variation. I don't know if it's enough to explain the discrepancy, but you should make sure that all the numbers you're using belong to the same mission.
– Ludo
Dec 15, 2023 at 12:44
• If you want to calculate the delta V of the second stage of the Saturn V, you need not only the wet and dry mass of the second stage but also the wet mass of the third stage, the service module, the command module and the lunar module. All the mass that is accelerated by the thrust of the second stage.
– Uwe
Dec 15, 2023 at 19:13
• @Uwe I think there was a misunderstanding. The dry mass value given included those things. I will edit the question to make that clear. Dec 15, 2023 at 20:42