I looked at two back-of-the-envelope (BOE) design studies for lifting boosters, in the sense of hypersonic carrier aircraft for a rocket second stage to orbit. Both of these were delta-wing aircraft of identical overall size, with a ducted centerbody housing a ramjet engine. Booster rockets and all fuels and propellants were assumed to be housed within the 10% thickness wing. Inert weight fraction was 30%, reflecting an aircraft-like design for easy reusability, long life, and low maintenance.
No integral booster was assumed for the ramjet. The ramjet frontal area was fixed at 5.1% of the vehicle wing planform area, and it operates sequentially after the rocket boost is done. These two studies differed only in the trajectory to be flown by the vehicle: lots or little of the altitude gain to be accomplished during rocket boost to ramjet takeover (1600 feet/sec, or just about Mach 1.5).
Propulsive Components
Rocket performance was estimated from sea level potential for liquid oxygen (LOX) and liquid methane (LM) from data in an old Pratt & Whitney vest-pocket aeronautical handbook. Ramjet performance was taken from a circular-section engine design with a translating inlet spike for constant shock-on-lip operation from Mach 1.5 all the way to Mach 6. This design used RJ-5 synthetic “kerosene” fuel, although LM was assumed for this particular study. Properly-sized ramjet engine performance on LM should be quite similar, although some sized geometric details could differ somewhat. The original design data were posted previously, and published in the 20 Feb. 2010 post. These data were correlated with a relatively simple equation, presented in the 23 July 2010 post.
In both studies, the weight fractions of fuels and propellants were arbitrarily factored up by 10% to cover three items. These included (1) parallel-burn rocket operation for a second stage pullup-and release maneuver, (2) rocket power for landing go-around, and (3) ramjet power for cruise back to launch point. The target final condition was 6000 feet/sec (just about Mach 6) at 100,000 feet altitude on a US 1962 Standard Day.
Zoom Climb Study
The first of these studies examined pulling up sharply while still in rocket boost, so that ramjet takeover occurred at some variable middle altitude, depending upon the level of assumed average boost acceleration. This study predated the final form of the BOE analysis technique, so that adequacy of thrust over drag for the trajectory assumptions is less certain. In addition, drag was represented by a single average value, an assumption later found inappropriate for truly reliable thrust-drag modeling.
For purposes of BOE analysis, the flight was broken into two phases: accelerating rocket-powered climb at 15 degrees, followed by accelerating ramjet climb at a much-reduced angle to the target conditions.
The rocket boost was analyzed by BOE impulse methods that treat the vehicle as a constant mass particle over a short trajectory segment, using the average weight on that segment. Impulse terms are computed from the velocity and altitude endpoints of the trajectory interval. At average conditions, a drag impulse is estimated as average drag multiplied by time spent on that interval.
Ramjet takeover altitude was variable, set by a speed requirement of 1600 feet/sec (approximately Mach 1.5), the boost climb angle, and an assumed “sweep” of average boost acceleration levels. Boost propellant input has to be iteratively reconciled with output required boost propellant, at each acceleration level, in the spreadsheet.
The ramjet sustain was analyzed by BOE energy methods that also treat the vehicle as a constant mass particle at the average weight, over a short trajectory interval. Kinetic and potential energy terms are computed from the velocity and altitude endpoints of the trajectory interval. At average conditions over the interval, the drag work integral is estimated as average drag multiplied by interval flight path length.
Climb angle was reduced substantially before average thrust minus drag margin, over weight, was finally found to be adequate. This final average value was 2.5 degrees, which is really a declining-slope curve, not a straight-line path. Again, similar to the boost, sustain fuel input must be iteratively reconciled with required fuel output, for each and every design analyzed, in the spreadsheet. In this study, each ramjet design corresponded to a boost acceleration level. This was appropriate since propellant and fuel trade directly against each other with a variable takeover altitude.
The resulting trade study (vs boost acceleration) proved to be unreliable in terms of logically selecting a boost thrust level. Figured at 7.5 degrees average ramjet climb, the payload fraction trend was a curve-with-a-knee that was concave downward, but this angle exceeded thrust capabilities, when those were checked. At 5 degrees, the trend was flat, and the climb was still too steep for the ramjet. At 2.5 degrees, thrust seemed adequate, and the trend had returned to the curve-with-a-knee shape, but this time concave upward. There being no way to trust such a wildly-varying result, a value of 0.25 average boost gees (highest payload fraction on the curve) was arbitrarily selected as “representative”. The booster thrust to gross takeoff weight ratio is 0.75.
The corresponding weight statement for the selected design was arbitrarily factored up 10% on both rocket propellants (LOX and LM at 4:1) and ramjet fuel (LM alone). Given no transition ejecta (no integral booster), and 30% inert weight, this vehicle would seem capable of 5% “payload”, where that payload is the entire second stage and its ultimate payload. Not being done to the final recommended standard of a realistic drag and realistic thrust margin potentials, these data should be taken with at least a little “grain of salt”. However, such a design and trajectory combination do appear to be quite feasible.
Estimated Weight Statement
“Zoom Climb” Scenario
Item………………….fraction
Payload……………...0.0502
Inerts………………...0.3000
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RJ bo..……………….0.3502
RJ fuel……………….0.2141
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RJ ign………………..0.5643
Trans.ejecta………….0.0000
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Boost bo…….……….0.5643
Boost prop…………...0.4357
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Boost ign…………….1.0000
Second Stage Payload Effects
A very crude estimate for velocity increment 16,000 fps required out of a similar LOX-LM rocket second stage, yields a mass ratio requirement right about 5. Assuming this to be a one-shot throwaway rocket of conventional design, an inert fraction near 8-10% seems to be a reasonable assumption. That leaves 6-8% of the stage weight available for the final payload. Ratioing this to the initial aircraft takeoff weight (at stage weight 5% of takeoff weight) provides an overall payload fraction in the 0.3 to 0.4% range.
This low value is not unexpected for carrier-aircraft design approaches, but does illustrate how quickly the “big airplane” structural problem can be encountered: a 500,000 lb hypersonic aircraft with rocket and ramjet engines can send only 1500 to 2000 lb to orbit. Aircraft bigger than that tend to be (1) “water-balloons-resting-on-nails” as regards landing gear design, and also (2) very unsurvivable in a crash landing or any other abuse scenario.
Slow Climb Study
For the second study, a better estimate of drag characteristics was obtained from Hoerner’s old “Fluid Dynamic Drag” reference. These are in drag area format for the specific vehicle analyzed. By way of comparison, the average drag area figure used in the Zoom Climb Study was 20 square feet. The planform reference area in these studies was 1250 square feet. Converted to simple drag coefficient with that figure, they can used for any other size.
Combining these drag data with the ramjet thrust correlations made checking thrust margins far easier and more reliable. Such thrust margins can be used to accelerate pathwise at constant altitude, or to climb at some path angle at constant speed, or both. This study used a fixed ramjet takeover point of 1600 fps at 5000 feet altitude, again the US 1962 Standard Day. Thrust margin characteristics defined the climb angles and speeds before those trajectory segments were ever analyzed. That strategy then becomes the recommended procedure for future BOE analyses.
Once again, the average acceleration in boost was investigated as an independent tradeoff variable. Similar to the first study, a trend of boost propellant vs acceleration took the form of a curve with a knee in it. The design nearest that knee (0.75 gee) was selected as “representative”. This time however, with a fixed takeover point, there was no need to trade fuel for propellant as part of the overall design sizing. That one sized boost design became the basis of the subsequent trajectory definition and ramjet fuel sizing. The booster rocket thrust to takeoff weight ratio for this design is near 1.07.
The first ramjet sustain item investigated was the appropriateness of the 1600 feet/second takeover velocity. Thrust margin (as thrust minus drag, over weight) was calculated for a spread of speeds up to 2200 fps. Somewhat surprisingly, thrust margin decreases as speed increases, so the 1600 fps figure is the “best”. It is not possible to go much slower, because the inlet spike shock system detaches about Mach 1.4, causing a sharp drop in captured stream tube size, and thus a sharp drop in engine thrust. Drag at constant Mach number scales with air density, while thrust does not. Thus, at high altitudes, this thrust margin trend with speed changes dramatically, as shown below.
Because of the decreasing thrust margin with speed at low altitude, it was decided to climb on ramjet power at a constant 1600 fps speed, to high altitudes, pull over level, and then accelerate to high speeds. Accordingly, the thrust margin vs altitude at this speed is of great interest, since it sets the maximum achievable climb angle. The data show great potential (about 17 degrees) up to about 60,000 feet. Above that altitude, climb potential decreases rapidly due to the “thin air” effect. An average figure would be about 10 degrees from there to 100,000 feet, on a declining-slope curved path.
Once at 100,000 feet, thrust margin vs speed gives a measure of the final acceleration capability of the vehicle. For the data used in the study, thrust margin goes to zero at about 5800 fps, very close to the 6000 fps target. Unfortunately, the general level of the accelerations on this curve are quite low, due to the “thin air” effect (thrust and drag are less in thin air, weight is unaffected). Thus it will require a very long time (and range) to accelerate from 1600 fps to near 6000 fps at this altitude. The scale on the plot reads directly in gees of pathwise acceleration.
Using the BOE energy methods to investigate this flight path requires only takeover and “final target” speed and altitude data for the potential and kinetic energy terms. For each of the three flight segments, however, an average drag and path length must be computed and summed to estimate the drag work integral. For estimating purposes, a constant ramjet energy efficiency of 30% was assumed, per the 11 July 2010 post. The resulting weight statement, with fuel and propellant factored up 10%, shows two things: (1) payload fraction is higher at 12% (instead of 5% as in the zoom climb), and (2) there is insufficient range capability to return to launch site, because of the enormous range increment accelerating at 100,000 feet.
For the same second stage as described above, these data correspond to an overall payload fraction of 0.72 to 0.96%. For the same “maximum credible” 500,000 lb aircraft, these correspond to delivered payloads in the 3600 to 4800 lb range. For comparison, the one-man Mercury capsule of the early 1960’s was 4000 lb.
Estimated Weight Statement
“Slow Climb” Scenario
Item………………….fraction
Payload……………...0.1256
Inerts………………...0.3000
-----------------------------------
RJ bo..……………….0.4256
RJ fuel……………….0.3214
-----------------------------------
RJ ign………………..0.7470
Trans.ejecta………….0.0000
-----------------------------------
Boost bo…….……….0.7470
Boost prop…………...0.2530
-----------------------------------
Boost ign…………….1.0000
A final investigation traded away payload for the extra ramjet fuel to cruise back to launch site. Even with all fuel and no payload, such flight was impossible. To be flown as planned, this craft will have to land and refuel in order to return to its launch site. Clearly, the average acceleration at altitude must be far larger. There are two strategies that might accomplish this: (1) lower the release altitude below those levels afflicted with the “thin air” problem (perhaps closer to 60,000 feet), or (2) reduce payload in favor of the extra rocket propellant needed to accelerate rapidly in a parallel-burn rocket-and-ramjet scenario. Neither of these has yet been investigated; a “real” trajectory computer model would be more appropriate for such work.
Conclusions and Recommendations
Two-stage-to-orbit (TSTO) appears to be quite possible with a horizontal takeoff hypersonic aircraft as the fully-reusable “flyback” first stage, and a conventional throwaway rocket of mundane technology as the second stage.
As with most carrier-aircraft approaches, the overall payload fraction is quite low, which leads to low absolute payload sizes using aircraft sizes currently thought to be realistic and safe.
While low overall payload fraction tends to drive payload-to-orbit costs up, the reusability of the launch aircraft should overwhelm that effect and actually drive costs down.
Trajectory has a huge impact on results. Zooming up right off the runway leads to a first stage “payload” fraction near 5%, with easily-achieved fuel margin for return to launch site. Climbing slow in ramjet to a high-altitude acceleration leads to a first stage “payload” fraction nearer 12%, but also to an inability to return to launch site without refueling.
For a 500,000 lb aircraft at takeoff, and a LOX-LM second stage of mass ratio 5 and inert fraction 8-10%, the following payloads can be delivered to low Earth orbit (LEO):
Zoom climb 1500-2000 lb; slow climb 3600-4800 lb.
Integrating the second stage onto the aircraft without increasing the drag was not addressed at all.
Strategies for increasing the slow climb vehicle’s high altitude acceleration (to provide return-to-launch-site capability) were not investigated.
The full trajectory-shaping thrust and drag strategy used in the second (slow climb) study is recommended as the “standard” for all future BOE studies.
Other than basic feasibility and a (very) preliminary weight statement, BOE studies are not to be used for component and subsystem design trades. They are simply too crude to be reliable for that purpose.
A specific BOE analysis technique for parallel-burn rocket-ramjet has not yet been defined. It needs to be, for both the high-altitude launch aircraft acceleration problem, and for the ramjet-assisted vertical-launch ballistic stage rocket.
