Update 7-4-17: Events have since made this assessment of mine obsolete. Spacex is becoming ever more successful at recovering first stage boosters, and has even begun re-flying them. It would seem that end-on attitude control during entry is enough to prevent wind-pressure breakup. And, there seems to be some sort of limiting effect on hypersonic heating, produced by the retropropulsive rocket plume during what Spacex calls its "entry burn". I am unsure exactly how that works.
Be that as it may, Spacex is enjoying considerable success recovering used first stage boosters. There are persistent claims they want to attempt recovering second stages as well. The difference in difficulty is considerable: first stage entry speeds should be crudely similar to first stage speed at staging, which is around 10,000 feet/sec (~3 km/sec). A second stage would hit the atmosphere at essentially orbital speed: 25,000 feet/sec (~8 km/sec). That's over 6 times higher forces and energies to deal with.
Spacex is doing a wonderful job upending the conventional wisdom of space launch. Blue Origin is less publicized, but is also recovering and re-flying boosters from its suborbital vehicle. The future would seem to hold some amazing things yet to come.
A group of folks I correspond with (at the forums on NewMars.com) has been discussing reusable launch rocket possibilities. One of the names they use is “big dumb booster”, or BDB. My own opinion is that reusability is incompatible with the low inert mass fractions used in the stages of typical launch rockets today: too light is simply too fragile. I do know from their website that Spacex is interested in reusing the first stage of their Falcon-9 booster, but that their results so far are unsuccessful. So, my analysis results here should be of interest, both to my correspondees, and to Spacex.
Spacex’s Falcon-9 is a two-stage rocket with kerosene-oxygen engines in both stages. It features an interstage ring and a payload shroud (on the satellite version) that I assume both get jettisoned at staging. The same engines are used in both stages, except that the one in the second stage has a longer bell than the nine in the first stage, and the first stage engines see atmospheric backpressure.
Baseline Falcon-9 Performance Estimate
I looked up most of the basic engine and vehicle data from Spacex’s website, for Falcon-9 as a baseline case, and reverse-engineered the rest. Here it is, summarized, in Figure 1:
Figure 1 – Baseline Falcon-9 Data
These performance data were computed with the simple rocket equation, and some experiential “jigger factors” that knock down ideal velocity increments to more realistic values. The other choice for analysis is a real trajectory computer code, either two-dimensional or three-dimensional, which is a complicated thing to set up and to use. I used the simple analysis approach to set up actual computer trajectory analyses, for the Scout launch vehicle at LTV Aerospace, about 4 decades ago.
Here, I used a “jigger factor” of 1.10 to knock down the first stage ideal velocity increment, because that stage sees air drag, and flies mostly vertically, so that gravity drag is significant. For the second stage, I used 1.05, reflecting flight in vacuum, mostly but not entirely horizontal. The final summed velocity increment I estimate for Falcon-9 is about 26,900 feet/second, or 8.19 km/second, which is remarkably close to the orbital velocity at low altitudes (about 7.9 km/second). It’s close enough that any simplified design trades made under these assumptions are realistic enough to be useful.
I looked at two potential solutions to the trade-off between extra structural weight for reusability, and reduced payload fraction that increases the price per unit payload delivered to orbit. One was to retain the basic two-stage design, and increase the size of the first stage to compensate for added inert fraction, at constant mass ratio. The other approach was to replace the two-stage design with an equivalent three stage design, keep the top two stages as throwaways, and increase the first stage size to compensate for increased first stage inert weight fractions. Both were done at constant delivered payload weight.
Two Stage Analysis with Heavier Structural Inert Fractions in the 1st Stage
The payload is exactly the same as baseline. I assumed the payload shroud weight to be proportional to the maximum payload weight it contains at 15.18%. There are no changes to the second stage weight statement or performance values. The interstate ring weight I assumed proportional at 0.815% to the weight it carries, in this case the second stage ignition weight. It is the first stage weight statement that varies, but at constant mass ratio, so the propellant weight fraction is the same as baseline in all cases. The equation relating mass ratio MR and propellant weight fraction fprop is:
fprop = (MR – 1)/MR
Now, 1 – fprop is the total of the inert mass fraction and the stage payload mass fraction, where the first stage payload comprises the ready-to-ignite second stage, the interstage ring, and the payload shroud. I looked at the baseline, twice, and three times the first stage inert weight fraction, scaling up the first stage ignition weight to match. The resulting weight statements are given in Figure 2. Bear in mind that the delivered stage performance data are identical to baseline, since the mass fractions are identical to baseline.
Three Stage Analysis with Heavier Structural Inert Fractions in the 1st Stage (Only)
I had to allocate velocity increments among the three stages in some logical fashion. I chose to make the second and third stage mass ratios 5 like the Falcon-9 second stage, and my first stage mass ratio 4, like the Falcon-9 first stage. I used “jigger factors” of 1.10 and 1.05 on my first and third stages, similar to the Falcon-9 first and second stages. I used an intermediate factor of 1.07 for my second stage. My first stage Isp was 289.5 sec, like the Falcon-9 first stage. My second and third stages used Isp = 304 sec, like the Falcon-9 second stage. The corresponding exhaust velocities are 9314.4 and 9780.9 ft/sec.
I computed the sum of the estimated actual velocity increments to be factor 1.5406 too high, so I knocked down each stage’s velocity increment by this factor, and recomputed mass ratios as 2.45935 for my first stage, and 2.84251 in my second and third stages. I ran the design study to the same payload as Falcon-9, with the same shroud weight, and two interstage rings at 0.815% of the stage weights above each ring. I assumed that interstage ring 1-2 and the payload shroud drop off with stage 1, and that interstage ring 2-3 drops off with stage 2.
The payload is exactly the same as baseline at 23,050 lb. I assumed the payload shroud weight to be proportional to the maximum payload weight it contains at 15.18%, for 3500 lb. There are no changes to the second or third stage weight statements or performance values as I changed first stages. The interstate ring 2-3 weight I assumed proportional at 0.815% to the weight it carries (in this case the second stage ignition weight) for 606 lb. Interstage ring 1-2 is 0.815% of stage 2 ignition weight, for 1999 lb. It is the first stage weight statement that varies, but at constant mass ratio, so the propellant weight fraction is the same as baseline in all cases, and so is the performance.
For the “baseline” three-stage inert fractions, I assumed 5% for my first stage, very similar to the multi-engine first stage of Falcon-9. I used the same 4.2% for my third stage as for the single-engine second stage of Falcon-9. My second stage has an intermediate inert fraction of 4.6%, chosen to reflect only a few engines in the second stage. The weight statements for the trade study are given in Figure 3. Bear in mind that all three versions of the three-stage vehicle have exactly the same estimated velocity performance, also shown in the figure.
Figure 2 – Weight Statements for the Two-Stage Reusability Trade Study
Figure 3 – Weight Statements for the Three-Stage Reusability Trade Study
Note that in both Figure 2 and Figure 3, I have included the overall payload weight fraction, computed as payload weight delivered to orbit Wpay, divided by the stage 1 ignition weight, which is the launch weight WL. (In the context of this analysis, the term “weight” really refers to mass.) In both trade studies, payload fraction decreases as stage 1 inert weight increases, exactly as expected. I was surprised and pleased to see that the baseline throwaway 3-stage option had a slightly higher payload fraction than the corresponding baseline throwaway 2-stage option. This and the slopes of the trends did seriously impact the final conclusions.
The final trajectories are compared in Figure 4. Both the 2-stage and 3-stage vehicles follow similar paths to the same orbital insertion conditions, at the same altitude (in the vicinity of 200-300 miles, or 300-500 km, up). Only potential re-use of the first stage was considered, for either configuration. A first stage fallback is indicated for each. Reentry velocity is simply assumed the same as the first stage burnout velocity. They would be comparable, in any event. Noting that reentry gets really challenging much above 10,000 feet/second (near Mach 10), I see little point to trying to make the second stage of the 3-stage vehicle reusable. It simply comes back too fast to be readily survivable.
Figure 4 – Comparison of Trajectories for 2-Stage and 3-Stage Vehicles
Figure 5 – Comparison of 2-Stage and 3-Stage Results
Payload Fraction Results Comparison
The payload fraction vs first stage inert fraction data are plotted in Figure 5 for both the 2-stage and 3-stage vehicles. The trends are reasonably linear-looking over the ranges computed, but at different slopes. As expected, the 3-stage vehicle design is less sensitive to first stage inert fraction than the 2-stage design (3-stage having the shallower slope). I did not really expect to see the baseline 3-stage vehicle to have a slightly-higher payload fraction at the baseline throwaway inert value, but it did.
Between the higher baseline throwaway payload fraction, and the shallower slope with first stage inerts, it appears that at 10% inerts in the first stage, the 3 stage vehicle has a payload fraction near 2.8%, while the 2-stage vehicle is down near 2.2% at the same 10% inerts. 10% inerts in the first stage is of enormous interest, because that is close to the inert fraction of the Space Shuttle solid booster motors, which actually were reusable most (but not all) of the time. That’s about the level where your tankage becomes strong enough to be pressure vessel-capable, as well as survivable for ocean impact on parachutes. Tankage that is pressure vessel-capable might as well be used as a pressure-feed system, eliminating the weight, cost, and reliability risks of turbopump machinery.
It is clear the 3-stage option is more tolerant of higher inert weights in the first stage. Combine this with a lower first stage fall-back speed, and reusability seems more certain at 10% inerts, and with a higher payload fraction (nearly 3% 3-stage vs only a bit over 2% 2-stage).
Accordingly, 3 stages is a better option than 2 stages, if the first stage is to be reused. The drop from 2-stage non-reusable payload fraction is actually quite small (3.1% to about 2.8%). This is because the all-throwaway 3-stage vehicle actually has a better baseline throwaway payload fraction than the 2-stage (3.3% at 5% inerts, vs 3.1% at 4% inerts).
This does raise the question of whether 4 stages might allow first stage reusability at even better payload fraction, or else allow the same payload fraction with both first and second-stage reusability. I leave that for others to investigate.
The main lesson here is that you really do have to do something different in order to get a different result. Reusability will require a greater inert weight fraction to cover recovery gear, and to confer the strength to survive better. Practical reusability simply cannot happen in the 4-8% inert range.
This study points toward 10% inerts in the first stage, at the very least. The more first stage inerts you have to “cover”, the more stages you need to use, to be tolerant of lowered mass ratio in each stage.
But at least we know the job really can be done, and here is one well-proven way to do it (more stages).