# Understanding the multi-stage mass tables in Wiesel's Spaceflight Dynamics

In Wiesel's Spaceflight Dynamics, 3rd Edition (2010) Chapter 7.5 (pg 220), there is a table of mass values for Von Braun's 1951 design for a three-stage space station supply rocket. In the book, the author defines several quantities of interest with respect to stage masses:

$m_{0k}$ is the initial mass of the kth stage, which includes structural and propellant mass of the current stage and everything else after it.

$m_{fk}$ is the final mass after burnout of all the propellant of the kth stage, which includes the structural mass of the current stage and everything else after it (i.e. it omits the propellant of the kth stage).

$m_{sk}$ is the structural mass of the current stage only (not including structural masses of all the other stages after it).

$m_{pk}$ is the propellant mass of the current stage only.

The author provides values for each of the stages of the rocket as follows:

           Stage 1    Stage 2  Stage 3   Payload
m_{0k}  6,349,000   897,930   129,700    35,400
m_{fk}  1,587,250   199,540    60,225
m_{sk}    698,390    69,840    21,950
m_{pk}  4,761,750   698,400    69,480


However, when analyzing this number, it seems like they are off quite significantly. If we analyze Stage 1, for example, the difference between the total initial mass $m_{01}$ and the first stage propellant mass $m_{p1}$ is exactly equal to the final 1st stage mass $m_{f1}$ as expected; that is, $m_{01}-m_{p1} = m_{f1}$. However, if we take the difference between the final stage 1 mass $m_{f1}$ and the initial mass of stage 2 $m_{02}$, we should expect this to equal the structural mass of stage 1 $m_{s1}$. Instead, when we subtract, we get an absolute error of 9070 kg. This seems to be a rather large error in the calculation and I'm wondering if I'm missing something fundamental or if this is just a simple typo or something else that I'm not factoring into the mass calculations.

Edit

In the event that this was just a transcription error of some sort on the part of the author, I looked further to another similar table (same book, pg 222) for the space shuttle, where the following (nominal) values are provided:

           Stage 1    Stage 2  Stage 3   Payload
m_{0k}  2,015,600   670,500   108,300    29,500
(1,169,200) (846,400)
m_{fk}    934,300   143,300   104,700
m_{sk}    163,800    35,000    68,000
m_{pk}  1,181,300   527,200     3,600
(1,005,400) (703,100)


where the values in parentheses refer to components of the shuttle and are not equivalent parallel stages. In this table, the values seem more consistent. The values satisfy the equation $m_{fk}-m_{0k+1}=m_{sk}$ for interstages 1-2 and 2-3, but not for 3-payload. Is there a reason why the stage 3 structural mass is not supposed to be equal to the payload mass? On a side note, what exactly is meant by "components of the shuttle" as opposed to the equivalent parallel stages?

• My first thought was that it represented an interstage -- Saturn V launch weight is ~6,500,000 lbs, and the 1-2 interstage is ~11,000 lbs -- but if I understand that table correctly the error is in the wrong direction. Since there's no similar discrepancy between stages 2-3, I suspect this is just an error. (I note that 9070 is not divisible by 9, so the error is not a simple 2-digit transposition.) Sep 27 '17 at 2:43
• Are the units specified as kg in your text, or lbs? Sep 27 '17 at 2:48
• @RussellBorogove: yes, they are all in kilograms.
– Paul
Sep 27 '17 at 3:00
• According to this article: wired.com/2014/09/… , the widely publicized version of WvB's 1952 design was specified in imperial units for the comfort of an American audience, so it's possible that an error was introduced in converting to metric for Wiesel. Sep 27 '17 at 3:09
• @RussellBorogove: If it was correctly tabulated in the appropriate units, shouldn't we expect $m_{f1}-m_{02}=m_{s1}$ ?
– Paul
Sep 27 '17 at 3:15