Estimating Starship dry mass using IFT flight data
The calculations are done on an Excel spreadsheet.
Starship = the Booster (the first stage) and the Ship (the second stage)
The flight data are extracted from the information SpaceX provides on the video. The time step for analyzing the Booster flight data is 5 seconds and 10 seconds for the Ship. The time interval during the flight for the Booster extends from liftoff to staging and, for the Ship, from staging to the second-stage engine cutoff (SECO-1). I don't analyze the Booster return to launch site (RTLS) data. It's enough work just to analyze the launch to SECO-1 flight data.
That flight data includes the time after launch (Time At Launch (TAL) + x, seconds), altitude (km), speed (km/sec), flight path angle (FPA, radians), the propellant mass remaining at each time step during the engine burn (t, metric tons).
The flight path angle is used to calculate the gravity drag in meters per second using numerical integration. In IFT-12, time at staging is TAL + 135 seconds and SECO-1 time is TAL + 560 seconds.
The propellant mass flow rate (t/sec) is calculated from the propellant mass remaining divided by the time step. Engine thrust can be calculated from propellant mass flow rate and the engine specific impulse (sec), however, thrust is not explicitly needed in this analysis.
On the IFT-12 launch the Starship trajectory was vertical (FPA = 90 degrees) from liftoff to TAL + 20 sec and then the Booster flight computer started the gravity turn. The Booster flew on that trajectory until staging. Then the Ship flight computer took over and continued the flight on a gravity turn trajectory until SECO-1.
Since the Ship is the payload for the Booster, the Ship performance has to be calculated first, i.e. we need to estimate the Ship’s dry mass before the Booster analysis can begin. So, the calculation proceeds in a top-down fashion.
To calculate the Ship’s dry mass, we need an equation of motion (EoM) for the Starship. That’s the Rocket Equation, which is a result from Newtonian physics via the conservation of momentum equation. For this analysis the Rocket Equation, which is a transcendental equation (contains exponentials or logarithms), is written in its exponential form. The exponential is on the left side of the EoM and its argument contains the Starship dynamics (speeds and engine parameters). The right side of the EoM contains all of the masses that define the Starship (dry masses, payload mass, propellant masses, header tank mass, crew-related masses, etc.).
The two unknowns are the Booster dry mass and the Ship dry mass. You plug in all of the flight information on the left side and on the right side you plug in all of the mass information. Then you set up the equation
Diff = Left side - Right side.
Then the Excel “Goal Seek” operator uses iteration to drive the Diff to zero by changing the only independent variable, the dry mass. The Ship is analyzed first to calculate its dry mass. And then the Booster dry mass calculation is done in a similar fashion. Since the EoM is satisfied as long as the engines are providing thrust and the velocity is changing, you can select any time interval you like during the engine burn and run the analysis. For IFT-12, I chose the interval between TAL+25 sec and TAL+105 sec, which is the entire part of the Booster’s gravity turn trajectory. For the Ship I chose TAL+330 sec and TAL+450 seconds that’s roughly in the middle part of the Ship’s gravity turn.
Block 3 Ship IFT-12 flight data analysis IFT-12 AO98
Analysis starting time TAL + (sec) 330 AO99
Analysis ending time TAL + (sec) 450 AO100
Ship velocity at start of time interval (m/sec) 2628 AO101
Ship velocity at end of time interval (m/sec) 4383 AO102
Ship gravity drag during time interval (m/sec) 168 AO103
Boca Chica TX latitude (deg) 25.99 AO104
Earth rotation delta V at Boca Chica (m/sec) 418 AO105
AO106
Header tank mass (t) 35 AO107
Ship payload (t) 38.5 AO108
AO109
g0 (m/sec^2) 9.807 AO110
AO111
Number of Raptor 3 sealevel engines 3 AO112
Number of Raptor 3 vacuum engines 3 AO113
Raptor 3 sealevel engine Isp in vacuum (sec) 350 AO114
Raptor 3 vacuum engine in vacuum Isp (sec) 380 AO115
Average Raptor 3 Isp for Block 3 Ship (sec) 365 AO116
Ship propellant mass at end of interval (t) 661.3 AO117
AO118
Ship delta V in the time interval (m/sec) 1,755.0 AO119
Ship gravity loss during time interval (m/sec) 168.10 AO120
Isp (sec) 365 AO121
Ship propellant mass at start of time interval (t) 1,305.1 AO122
Ship propellant mass at end of time interval (t) 662.0 AO123
AO124
Block 3 Ship dry mass estimate (t) 168.7 AO125
Left 1.71128671 AO126
Right 1.71130161 AO127
AO128
Diff -1.4897E-05 AO129
LEFT=EXP((AO119+AO120)/(AO110*AO116)) RIGHT=(AO125+AO107+AO108+AO122)/(AO125+AO107+AO108+AO123) DIFF = LEFT - RIGHT
Since the information extracted from the flight data inevitably has noise contamination (errors), the dry mass estimate calculation is repeated numerous times by varying the propellant-remaining measurement at the end of the time interval by +/- 5% to determine a final estimate of the average dry mass and the standard deviation.
Block 3 Booster IFT-12 flight data analysis IFT-12
Analysis starting TAL + (sec) 25 AO27
Analysis ending TAL + (sec) 105 AO28
Booster velocity at start of time interval (m/sec) 405 AO29
Booster velocity at end of time interval (m/sec) 671 AO30
Booster delta V analysis TALs (m/sec) 266 AO31
Booster gravity loss TAL starting to ending (m/sec) 132 AO32
Booster propellant mass at analysis start TAL (t) 2,311 AO33
Booster propellant mass at analysis ending TAL (t) 1,851 AO34
AO35
IFT-12 Block 3 Ship mass at liftoff (t) 2,523 AO36
AO37
g0 (m/sec^2) 9.807 AO38
Isp (sec) 350 AO39
Booster dV at analysis TAL starting to ending (m/sec) 266 AO40
Booster gravity loss TAL starting to ending (m/sec) 132.2 AO41
Booster atmospheric drag loss (m/sec) 13 AO42
AO43
AO44
Booster propellant mass at analysis start TAL (t) 2,311 AO45
Booster propellant mass at analysis end TAL (t) 1,736 AO46
Ship liftoff mass (t) 2,523 AO47
Interstage ring mass (t) 11 AO48
AO49
AO50
IFT-12 Booster Block 3 estimated dry mass (t) 267 AO51
AO52
Left 1.12731 AO53
Right 1.12684 AO54
AO55
Diff 0.00047 AO56
LEFT: = EXP((AO40+AO41+AO42)/(AO38*AO39)) RIGHT: = (AO51+AO45+AO47+AO48)/(AO51+AO46+AO47+AO48) DIFF = LEFT - RIGHT