Calculating second stage performance and mass from Inmarsat mission

[LouScheffer]

The Inmarsat mission in May, since it was a burn to depletion, provides yet another way to estimate second stage characteristics.

Starting with the SpaceX video on YouTube, I recorded the velocity at each frame when the clock ticked over a second.  Subtracting each from the previous gives the acceleration in that second.  This gives the blue line below.  While noisy, it is just what you would expect - acceleration builds up until it reaches about 5G. then limits at that value to the end of the burn.

The next step was to fit a model to this data.   This has 3 free parameters - the initial mass, the rate of mass loss (consumption by the engine), and the maximum acceleration allowed.  The mass loss is turned into thrust by the engine, by a multiplier of ISP*g (ISP is 348).  Doing a least square fit yields (red line on the plot):
Starting mass:  28868 kg
Mass loss:  270.5 kg/second
Acceleration limit:  48.93 m/s
These are all very close to what you might expect, although it seems they may not be running the second stage at full maximum throttle (this mass rate x 348*g = 922 kN, while SpaceX says the second stage can do 934 kN)

So using these numbers, the second stage starts limiting acceleration when it masses 18858 kg.  Next, we can calculate the final mass.   We know F/M = A, and A is constant.  If we let M(t) be the mass as a function of time, then M'(t) is the rate of mass loss, and ISP*g*M'(t) is the force.  So
    ISP*g*(-M'(t))/M(t) = A, or
    M'(t) = -M(t)*A/(ISP*g)
Since A, ISP, and g are constants, we have a derivative that is proportional to the value, so the solution is an exponential.  Fitting the initial value, we get
M(t) = 18858*exp(-t*A/(ISP*g)), where t = seconds since start of constant acceleration.
The constant acceleration phase is 23 seconds long, so the final mass is 0.719 of the initial mass, or 13558 kg.
Subtracting the mass of the satellite (6070 kg) gives a second stage mass of 7488 kg (no fuel mass is subtracted since it's a burn to depletion).

This estimate is quite a bit higher than that derived from comparing performance of different mass missions (where the estimate was about 4500 kg).   I'm not at all sure where the difference comes from.  Both methods seem plausible in their own way...

[DAZ]

Is ISP content when the engine throttles down to stay under G load limit?

[S.Paulissen]

PLF jettison taken into account?

Also it's unclear to me how you derived your starting mass of 28868.

[LouScheffer]

PLF jettison taken into account?

Also it's unclear to me how you derived your starting mass of 28868.

PLF jettison is not included since this burn is from parking orbit to GTO, so PLF jettison happened before the start.

To find the starting mass, we find the the rate of mass loss from the increase in acceleration during the burn.  That times the ISP gives the thrust.  The thrust and the initial acceleration gives the initial mass.

[hans_ober]

IMO there are too many variables to get an accurate figure (a figure accurate enough to tell whether they've been changing S2 between blocks).
We don't know whether an ISP of 348 still stands.
They published a thrust of 934kN, but we don't know the throttle profile.
Webcast data might be fudged.

Nice work on the calculations.

An error in S2 mass affects payload capability directly. 2 tons of additional weight on S2 means payload drops by
2 tons.

[Proponent]

This is a nice piece of analysis.  Regarding the inferred Isp seeming a bit low, I wonder whether that might be explained by gravity and steering losses.  Estimating the steering losses might be pretty tough, given the date provided, but the change in altitude during the burn should give a pretty good figure for gravity losses.

[edkyle99]

This estimate is quite a bit higher than that derived from comparing performance of different mass missions (where the estimate was about 4500 kg).   I'm not at all sure where the difference comes from.  Both methods seem plausible in their own way...

There will be unburned residuals and unusable residuals, even in a burn to depletion.  There is also the unaccounted mass of the spacecraft adapter, which is going to be a few hundred kg.  The 4.5 tonne number is for the Block 5 variant performance, so this stage was likely heavier.  The numbers shown on the SpaceX webcast may not have been accurate.  Etc.

 - Ed Kyle

[Jim]

And what was the inclination at sc sep?

[edkyle99]

And what was the inclination at sc sep?

This one reportedly ended up at 381 x 69,839  km x 24.5 deg

 - Ed Kyle

[LouScheffer]

This estimate is quite a bit higher than that derived from comparing performance of different mass missions (where the estimate was about 4500 kg).   I'm not at all sure where the difference comes from.  Both methods seem plausible in their own way...

There will be unburned residuals and unusable residuals, even in a burn to depletion.  There is also the unaccounted mass of the spacecraft adapter, which is going to be a few hundred kg.  The 4.5 tonne number is for the Block 5 variant performance, so this stage was likely heavier.  The numbers shown on the SpaceX webcast may not have been accurate.  Etc.

Looking at both estimates, I think this one is more reliable.

The old estimate (4500 kg) relied on the difference between LEO (22.8t claimed) and GTO (8.3t claimed) capabilities.  This has more assumptions (the LEO and GTO orbits are not specified) and seems internally inconsistent (from the current performance, you would expect a rocket that can put 8.3t into GTO to put more than 22.8t into LEO).  Plus the 22.8 to LEO would require a new super-heavy PAF (the current "heavy" PAF is only good to 10.8t).  This muddies the calculation further.

For the new estimate, the physics is very clean.   The only assumption is the ISP of 348, which SpaceX has explicitly stated.  The SpaceX telemetry numbers seem plausible - they show 26419 km/hr before the GTO burn, and 36096 after.  That's a delta V of 2688 m/s, close to what you would expect for the SSTO obtained (though an exact calculation is hard since SpaceX does not specify their parking orbit, in particular the inclination).  Also the calculations are internally consistent with a simpler model that uses only the rocket equation.  The beginning and ending masses above (28868/13558 kg) give a mass ratio of 2.129.  They do not model the thrust ramp-up and tail-off.  So if we take from the webcast the velocity at 27:02 and 28:02 (beginning and end of full acceleration) we get 26733 km/hr and 36046 km/hr.  This gives a delta V of 2586 m/s, and an almost identical mass ratio of 2.134 at an ISP of 348.

So although uncertainties remain, as Ed stated above, I think the empty mass is closer to 7000kg than the previous estimate of 4500 kg.  However, this is not the final version - this version looks capable of putting 21.4t in LEO (LEO mass above - estimated stage mass), and about 6.7t to a minimal GTO (extrapolated from above).  Those are 6% and 20% less than the SpaceX website claims, so more changes are to be expected.

[cambrianera]

Something is not correct here.
You can't use Newton's second law for a variable mass system.
Also you lack information on the potential energy at beginning and end of constant acceleration burn.
Second stage and spacecraft are orbiting bodies, and raising (or lowering) the orbit, or part of it, requires (or releases) energy.

[acsawdey]

Something is not correct here.
You can't use Newton's second law for a variable mass system.
Also you lack information on the potential energy at beginning and end of constant acceleration burn.
Second stage and spacecraft are orbiting bodies, and raising (or lowering) the orbit, or part of it, requires (or releases) energy.

Looks ok to me. At any instant in time, F/M=A is going to be true. He's computed F(t) as a function of the derivative of M(t) using the known ISP, then solved the resulting differential equation.

I think this also makes the assumption that the amount of velocity (kinetic energy) converted to gravitational potential energy by the trajectory is small compared to the amount of velocity the burn added, is that what you're saying about potential energy?

[launchwatcher]

And what was the inclination at sc sep?

This one reportedly ended up at 381 x 69,839  km x 24.5 deg

since the cape is somewhere around 28.5 degrees north of the equator do we have any way to estimate how much performance went into reducing inclination?

[cambrianera]

https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law

I liked some neat analysis done by LouScheffer, but I think in this case assumptions on masses and DV are doubtful.


Edit: looking better, LouScheffer handling of the formula is correct (escape velocity is considered through ISP).
I think the problem is acceleration can't be considered constant, it is a vector.

[LouScheffer]

  The only assumption is the ISP of 348, which SpaceX has explicitly stated.

That could be it.  SpaceX has provided numbers, but they've turned out to be for Block 5.  What happens to the calculations when a lower ISP is assumed? 

Lower ISP causes more fuel to be burned to match the acceleration, so lower mass at the end, so lower estimated second stage mass.  But the effect is not big.  Lowering the ISP from 348 to 330 (almost surely too far) reduces the ending mass from 13558 kg to 13023 kg.  This in turn reduces the estimated second stage mass to just under 7000 kg.  So it reduces but does not solve the difference from the LEO-GTO method.

[Owlon]

This estimate is quite a bit higher than that derived from comparing performance of different mass missions (where the estimate was about 4500 kg).   I'm not at all sure where the difference comes from.  Both methods seem plausible in their own way...

There will be unburned residuals and unusable residuals, even in a burn to depletion.  There is also the unaccounted mass of the spacecraft adapter, which is going to be a few hundred kg.  The 4.5 tonne number is for the Block 5 variant performance, so this stage was likely heavier.  The numbers shown on the SpaceX webcast may not have been accurate.  Etc.

Looking at both estimates, I think this one is more reliable.

The old estimate (4500 kg) relied on the difference between LEO (22.8t claimed) and GTO (8.3t claimed) capabilities.  This has more assumptions (the LEO and GTO orbits are not specified) and seems internally inconsistent (from the current performance, you would expect a rocket that can put 8.3t into GTO to put more than 22.8t into LEO).  Plus the 22.8 to LEO would require a new super-heavy PAF (the current "heavy" PAF is only good to 10.8t).  This muddies the calculation further.

For the new estimate, the physics is very clean.   The only assumption is the ISP of 348, which SpaceX has explicitly stated.  The SpaceX telemetry numbers seem plausible - they show 26419 km/hr before the GTO burn, and 36096 after.  That's a delta V of 2688 m/s, close to what you would expect for the SSTO obtained (though an exact calculation is hard since SpaceX does not specify their parking orbit, in particular the inclination).  Also the calculations are internally consistent with a simpler model that uses only the rocket equation.  The beginning and ending masses above (28868/13558 kg) give a mass ratio of 2.129.  They do not model the thrust ramp-up and tail-off.  So if we take from the webcast the velocity at 27:02 and 28:02 (beginning and end of full acceleration) we get 26733 km/hr and 36046 km/hr.  This gives a delta V of 2586 m/s, and an almost identical mass ratio of 2.134 at an ISP of 348.

So although uncertainties remain, as Ed stated above, I think the empty mass is closer to 7000kg than the previous estimate of 4500 kg.  However, this is not the final version - this version looks capable of putting 21.4t in LEO (LEO mass above - estimated stage mass), and about 6.7t to a minimal GTO (extrapolated from above).  Those are 6% and 20% less than the SpaceX website claims, so more changes are to be expected.

I can say with certainty that 4500kg is much closer to the current second stage mass than 7000kg.

[IainMcClatchie]

Lou,

I really like your analysis.  There must be some assumption that you've made which is off somewhere.

Altitude went from 295 km to 315 km during the burn, a change of 178,500 J/kg.

I think that's equivalent, at this altitude, to a change in delta-V of 23 m/s.  That's not much.  I don't think this effect can be the problem.

If you initial fit was off a bit, and the max acceleration peaked at 5 G rather than 4.8 G, would that make much of a difference to your results?

[LouScheffer]

There must be some assumption that you've made which is off somewhere.

Aha!  I believe I have found the missing assumption.  It's that the second stage starts the burn at full throttle.

Suppose instead the second stage starts at 80% throttle, keeps this until acceleration hits 5Gs, then holds to 5G max.  Then if you reduce all the masses by a factor of 0.8, you will get EXACTLY the same acceleration profile, with the exact same mean squared error.    Then the final mass is 11100 kg, and the empty mass 5030 kg.

Worse, once you relax the full throttle assumption, the same solution applies to ANY second stage mass.  Adjusting the throttle correspondingly generates the exact same acceleration profile, which can be obtained with any second stage mass, from 0 kg (requiring about 50% throttle) to 7500 kg (needs full throttle).  So unfortunately this method cannot be used to estimate the second stage mass.

To me this seems the likely solution to this conundrum, with SpaceX using perhaps 80% throttle with a 5000kg empty mass.  I has assumed the GTO burn uses full throttle, since that maximizes ISP and makes the maneuver more instantaneous, maximizing the Oberth effect.  But these effects are small, and presumably running at 80% helps reliability.  So I bet that's what they do...

If you initial fit was off a bit, and the max acceleration peaked at 5 G rather than 4.8 G, would that make much of a difference to your results?

But g=9.801, so 5G = 49 m/s.  That's one of the reasons I thought the fit was acceptable - it's almost exactly 5G.

[IainMcClatchie]

Good.

But it should be possible to add another constraint, too.  You can look at the first second-stage burn and fit the acceleration you see there as well.  Now it might use a different throttle setting on this burn, but if it does, that'll change the acceleration.

Over the entirety of the two second stage burns, this leaves two variables: the relative throttle setting (just a single parameter), and the amount of initial propellant.  The stage final mass may be quite sensitive to the throttle setting, but then so is the corresponding initial propellant load -- and you have some better constraints on the initial propellant load, right?  You should be able to propagate those constraints to the throttle setting and then to the stage final mass.

[Proponent]

What's the duration of the burn?