Launch to Low Earth Orbit: 1 or 2 Stages?

Although I have examined this question before,  I wanted to look at it again,  because there is still enthusiasm for the single-stage notion using chemical propulsion.  The problem with that is achieving a very high specific impulse (Isp) across a broad range of altitudes with the stage engines.  These must have adequate thrust at sea level,  but also average a high Isp all across the ascent.  Those requirements are in conflict because of fundamental physics.

The two-stage notion does not face that quandary as directly.  While the first stage engines show the reduced Isp typical of a sea level design,  which does not improve much at all going to high altitudes,  that penalty is compensated by the first stage shouldering only a minority fraction of the total delta-vee (dV) requirement to low Earth orbit (LEO).  The second stage can be a “vacuum” design,  featuring much higher Isp,  with much-relaxed thrust requirements.

Fixed-geometry rocket engines provide the shortest list of possible failure modes,  compared to variable -geometry designs that compensate by deployable expansion bell extensions.  Fixed-geometry rockets also show much higher performance out in actual vacuum than any of the free-expansion designs,  because of the very high streamline divergence the free-expansion designs inherently suffer when out in actual vacuum.  (They work “best” in the lower stratosphere.)

Accordingly,  what I looked at here were entirely fixed-geometry rocket engines.  For the two-stage notion,  the first stage engines were sea level designs,  and the second stage engines were “vacuum” designs,  although,  strictly speaking,  there is no such thing as a “vacuum” design,  there are only practical design constraints on how big the expansion can actually be (it has to fit behind the stage).

The sea level designs size the expansion ratio to be perfectly-expanded at sea level for no pressure term penalty,  and its dimensions also size there,  to meet a sea level max thrust requirement. This is because the vehicle is the heaviest at ignition,  and yet adequate net acceleration upward against gravity (around half a gee net) must be obtained!  In addition,  the sea level engines were presumed to use kerosene-oxygen propellants,  in order to minimize first stage tankage volume and frontal area,  so that drag losses are minimized.  Not to do so makes the needed mass ratio even larger.

The vacuum designs for second stages (and for the single-stage design) were presumed designed for expansion just short of backpressure-induced flow separation at sea level, at a suitable part-throttle condition:  some 85% of max chamber pressure.  In that way,  the actual flight engines can be tested open-air nozzle at sea level,  at 85%-and-above chamber pressure,  drastically reducing development test costs!  Similarly, in flight,  thrust can be reduced to 85% chamber pressure levels from sea level on up,  without risking flow separation in the expansion bell.  

For the two-stage design,  a vacuum thrust requirement can be used to set dimensions.  This second stage was presumed to use oxygen-hydrogen propellants,  since the smaller stage volume is compatible with the same or smaller frontal area,  despite the low density of liquid hydrogen.  For the single-stage design,  a sea level thrust requirement must be imposed.   The single stage design needs a higher-energy propellant combination,  but also suffers greatly from the enormous tankage volumes and frontal area of a hydrogen design.  So,  a compromise was used:  methane-oxygen.  

The launch trajectory was presumed to be a thrusting gravity turn affected by atmospheric drag,  to LEO at low inclination eastward,  as shown in Figure 1.  For the all-expendable designs presumed here,  staging would be somewhere near 50 km altitude and about 2 km/s achieved speed.  Circular orbit speed near 300 km altitude is about 7.7 km/s achieved.  Assuming 5% each for gravity and drag losses,  the mass ratio-effective dV is about 8.5 km/s.  The loss to be overcome is thus about 0.8 km/s,  all assigned to the first stage of a two-stage vehicle as a decent approximation,  and all borne by the single-stage vehicle.

Figure 1 – Launch Requirements

I did not actually size engines for the kerosene-oxygen and hydrogen-oxygen engines of the two-stage design,  because I have done this before,  and my results match general industry experiences.  These represent only modestly state-of-the art designs:  330 s Isp for the kerosene-oxygen,  and 450 s Isp for the hydrogen-oxygen.  These would be for chamber pressures in the 2000-3000 psia range,  and maybe 2% bleed.

I presumed a very state-of-the-art methane-oxygen engine of full flow cycle so that bleed was zero,  with a very high max chamber pressure of 4000 psia and a rather-demanding pressure turndown ratio (P-TDR) of 3.  I also presumed I would size its expansion from 85% chamber pressure down to 3.3 psia,  with the separation-inducing backpressure set at 14.70 psia.  For initial rough-sizing purposes,  I simply presumed it would average 370 s Isp across its full ascent.

For vehicle rough-out sizing,  I presumed a 5% inert fraction (f_inert) for all stages,  as loaded with payload.  The payload was presumed to be a dead-head 100 metric tons,  streamline-shaped,  and mounted out in the open,  atop the launch vehicle.   The ratio of dV to effective exhaust velocity (Vex) determines the stage mass ratio (MR).  The propellant mass fraction (f_prop) of the loaded stage is then 1 – 1/MR.  And the payload fraction (f_pay) is thus 1 – f_prop – f_inert.  For the two-stage launch vehicle,  the first stage “payload” is the fully loaded and fueled second stage mass.

I used a very simply laid-out spreadsheet to calculate these numbers for the two designs,  using the presumed Isp values and the relationship Vex = gc*Isp/1000,  to get Vex in km/s to match the dV values.   Those initial results are shown in Figure 2.  Note that the one-stage design has about half the overall payload fraction and twice the launch mass of the two-stage design!  I used thrust-to-weight (T/W) ratios of 1.5 at liftoff for good ascent kinematics,  and a T/W just over 1 for the exo-atmospheric,  nearly-horizontal portion of flight near the end of the ascent.  These sized some stage thrust requirements for me.   I used only half-a-gee for the second stage of the two-stage vehicle.

Figure 2 – First Vehicle Rough-Out

               Revisiting the Rough-Out

I really had no questions regarding the feasibility of the presumed engine Isp levels for the two-stage design.  There was concern about the Isp = 370 s presumption for the one-stage design.  Accordingly,  I actually ran some engine sizing and performance estimates,  using a convenient spreadsheet tool.  The ascent-averaged Isp fell closer to 360 s than the initially-presumed 370 s.  This is illustrated in Figure 3 below.  That includes a sketch and notations,  plus some copied sizing and point performance data from the spreadsheet.  The predicted performance vs altitude plots from that same spreadsheet are given in Figure 4 below. 

To find the ascent-averaged Isp from the calculation block in the spreadsheet,  which is performance vs altitude,  I simply summed the 100% Isp values over the ascent,  and divided that by the number of entries in the table.   This is not the “right” average value,  because the vehicle does not spend equal amounts of time at each altitude,  but it is somewhere in the ballpark.  The Isp out in vacuum is pretty near the initial presumption of 370 s Isp,  but the low altitude values are much lower,  and the vehicle does spend a lot of time there,  since it is still moving slowly at low altitude.

Therefore,  I reran the vehicle size-out and thrust requirements for that one-stage vehicle,  with the nominally-lower presumed average Isp = 360 s.  That revised vehicle rough-out is depicted in Figure 5 below,  which is just Figure 2 edited in some places.  The edits are in red text.  The effect of the small Isp change is more dramatic on the one-stage vehicle than it would be for either stage of the 2-stage vehicle.  This is precisely because it is only one stage,  and the payload is a fixed number.

Figure 3 – LOX-LCH4 Engine Sizing (Re-scalable With Thrust Rating)

Figure 4 – Predicted LOX-LCH4 Performance Fell Short

Figure 5 – Revised Vehicle Rough-Out Reflects Revised Isp For the Single-Stage Engines

This second version of the vehicle rough-out is more reliable,  after revising the one-stage average Isp value.  The two-stage vehicle is probably “pretty close” as it is,  especially since the second stage Isp is likely a slight underestimate,  which would offset any over-estimate of the first stage Isp.

For the two-stage vehicle,  we are probably looking at 8 or 9 engines of some 220 metric tons-force thrust each,  in the first stage.  The second stage needs very little pathwise acceleration capability,  and most of that at ignition where it is heaviest,  so the same 220 metric tons-force of thrust would work,  although for redundancy,  I would recommend two engines of 110 metric tons-force thrust each.   That way,  it still flies adequately even if one engine quits.

For the one-stage vehicle,  the same engines burn all the way through the ascent,  only shutting down those that are not needed as weight decreases.  This is a compromise between too many engines and too much thrust late in the ascent.   What the figure shows is that 15-16 engines of around 250 metric-tons-force each,  will lift off well,  with only one of those still burning very late in the ascent.

Overall,  the message is clear:  to do this one-stage cuts the achievable payload fraction in half or less,  while increasing the liftoff mass by a factor a bit over 2,  all for placing the same payload in eastward,  low-inclination LEO.  Lower payload fraction and higher ignition mass increase cost!

The two-stage vehicle does better,  because its two stages address the wildly-different requirements of ascent out of the atmosphere and exo-atmospheric acceleration to orbit speed,  with two entirely-different engine designs and propellant combinations!  The one-stage design lacks that advantage,  and must push its engines to the very outer limits of the state-of-the-art.

Extending to Reusable Vehicles

To do this reusably just makes the vehicles somewhat larger.  For the two stage vehicle,  the first stage gets larger in order to have the extra propellant required to recover it and land it.  Up to this date,  there have been no demonstrations of any recovery of second stages at all.  This is the partial recovery path taken by SpaceX with its Falcon-9 and Falcon-Heavy vehicles.

The inert fraction of any recoverable second stage would be much larger than the 5% presumed here,  because it must be not just a stage,  but also a survivable orbital re-entry vehicle.  It might as well carry the payload internally,  which likely increases its inert fraction even more.  That path is the one chosen by SpaceX with its Starship/Superheavy orbital transport design.

As for making the one-stage vehicle reusable,  with only 4% payload fraction,  it could only have an inert fraction of 9%,  even if it carried no payload at all!  To make the stage also an entry vehicle,  and to carry the payload internally,  would seem to push well past the bounds of any reasonable assumptions at all,  with chemical Isp.  This is the path attempted without any success by the X-33 “Venture Star” project,  and it used hydrogen-oxygen,  the best chemical combination available!

               Summary Remarks

Because launch price is sensitive to payload fraction and ignition mass,  I cannot recommend the single-stage-to-orbit approach with any conceivable chemical propulsion,  even in expendable vehicles.  The numbers are just not there,  regardless of what kind of “trick” engines one proposes,  because such always have performance shortfalls somewhere across the ascent.  Two-stage to orbit,  using two different propellant combinations in the two stages,  is likely the best,  but SpaceX has already shown rather good results with the same propellant combinations in both stages.

To add reusability,  the best approach is still two-stage,  with either (1) an expendable second stage and payload riding atop it,  or (2) a second stage that is also its own entry vehicle,  with payload riding inside.  The first is still more mature than the second,  at the time of this writing.

Switching to all-hydrogen instead of the denser methane is not the solution to the single-stage problem,  because the far-larger tankage volume and frontal area will increase the drag loss,  raising the dV penalty,  and thus make mass ratio-effective dV requirement still higher.  Such acts to offset the effects of the higher Isp of the hydrogen,  which still has to be ascent-averaged.

If you really want to do single-stage to orbit,  the most fruitful thing to do would be developing into maturity a nuclear thermal engine of significantly-higher Isp and substantially-higher engine thrust/weight than the NERVA design that was ready to flight test,  when it was cancelled in 1974.  Such an option is very likely some sort of gas core design.  One needs at least about Isp = 1000 s or so,  to make fully-reusable stages that are their own entry vehicle,  and can contain the payloads internally.  The vehicle inert fractions will fall in the 20-30% range,  unless high engine weight drives it even higher.  The vehicle launches vertically,  and could land horizontally.  If clean,  dV ~ 8.5 km/s.

At 1000 s:  Vex = 9.80667 km/s,  MR = 2.3792,  f_prop = 0.5797,  guess f_inert = .25,  f_pay = 0.1703.  For W_pay = 100 metric tons,  W_ign = 587 m.ton,  W_inert = 147 m.ton,  and W_prop = 340 m.ton.  At liftoff T/W = 1.5,  the required liftoff thrust is 881 m.ton-force.  Burnout is about 247 m.ton,  for about 3.57 gees at liftoff thrust. The thing is likely winged,  or a lifting body shape,  to land on a runway or dry lake bed.

               Follow-Up on the Nuclear Single-Stage Notion

I created another spreadsheet worksheet to evaluate the possibility of a nuclear thermal one-stage design.  I made the inert fraction iterative,  with an R-value to estimate LH2 tankage inerts,  and an engine thrust/weight ratio to estimate engine inerts based on liftoff thrust required. 

This crude analysis includes nothing for on-orbit maneuvering,  or deorbit,  which would probably be storable propellants!  I made the inerts analysis iterative so that the overall inert fraction input would give a realistic airframe inert fraction,  that does not include the engine or the tankage. 

This one is a lifting body,  with an engine not all that far improved over NERVA,  and it would land dead-stick like the shuttle,  probably on a dry lakebed,  or a very long runway indeed.  It would likely touch down at around 200 mph. 

This one had the highest payload fraction I have seen yet,  and would likely be fairly cheap to operate,  as long as it proves tolerable to return the idled nuclear engine back to Earth.  (That is a really big “if”!)  See the spreadsheet image in Figure 6,  and a sketch of the vehicle concept in Figure 7 below.  

I only had to increase my assumed inert fraction a little bit to achieve an airframe-only inert fraction that I considered to be believable.  Even so,  the payload fraction is about twice the payload fraction of the two-stage expendable chemical vehicle,  and almost 4 times the payload fraction of the one-stage expendable chemical vehicle.  And the nuclear one-stage vehicle is entirely reusable,  but if and only if you can accept returning its engine to Earth!

Figure 6 – Spreadsheet Image for the Single-Stage Nuclear Vehicle

               Final Remarks

For the nearer term,  using only well-developed,  ready-to-apply technologies,  the highest payload fraction option is the two-stage vehicle,  which can readily adapt the designs of its two stages to the different circumstances of ascent out of the atmosphere,  and acceleration exo-atmospheric and nearly horizontal to orbital speed.  Making its first stage reusable would not cost that much payload fraction.

Trying to do this,  even if expendable,  as a one-stage vehicle with chemical propulsion,  is unlikely to provide a payload fraction high enough to actually pay off.  It will likely underperform the two-stage expendable in terms of payload fraction,  no matter what propellants might be used.  And it will be heavier at liftoff under any conceivable circumstances,  for the same payload.  Thus it will cost more.

Longer term,  a fully reusable one-stage vehicle of even higher payload fraction than the two-stage expendable chemical vehicle,  might be feasible with some form of nuclear thermal propulsion that performs only slightly better than NERVA.  Key to its viability will be the acceptability of returning and landing with that engine aboard.

Figure 7 – The Nuclear One-Stage Vehicle Concept

Update 3-6-2024:  

I went ahead and looked more closely at the engines for the two-stage vehicle.  These would be LOX-LH2 in the second stage with vacuum bell designs,  and LOX-RP1 in the first stage,  with something suitable as a sea level bell design.  Neither would push the state of the art the way the LOX-LCH4 engines must do,  in the one-stage vehicle.  I used very modest modern-technology characteristics for the engines of both stages:  2500 psia max Pc,  with only a P-TDR = 2.5,  and a dumped bleed fraction BF = 0.02.  They use otherwise the same 18-8^o bell profile and C_D = 0.995.

As Figure 8 shows,  the traditional sea level design with perfect expansion to sea level pressure from max Pc,  shows an ascent-averaged Isp shortfall relative to what I wanted for the first stage.  But when I used the “compromise design” approach (see Figure 9) to size those engines,  trading away unseparated sea level operation at min-throttle setting,  for more expansion ratio and higher vacuum and ascent-averaged Isp values,  that ascent-averaged Isp exceeded the assumptions used for roughing out the first stage. 

Elsewhere,  I had looked at vacuum designs for LOX-LH2 engines,  sized to arbitrary expansion ratios of A/A* = 100,  150,  and 200,  with those same modest modern-technology characteristics.  The min expansion version (Figure 10) gets you the smallest physical length and exit diameter,  and its vacuum Isp substantially exceeded what I assumed for the rough-sizing of the second stage.  So,  I revisited the vehicle rough-out with a somewhat-higher second stage Isp (Figure 11, blue edits).  That increased the payload fraction,  and reduced the launch weight,  both acting to lower costs. 

Figure 8 – Traditional Sea Level Sizing Falls Short of Desired Ascent-Averaged Isp = 330 s

Figure 9 – Sea Level “Compromise” Design Exceeds Desired Ascent-Averaged Isp = 330 s

Figure 10 – Vacuum Design at A/A* = 100 Substantially Exceeds Desired Vacuum Isp = 450 s

Figure 11 – Revised Vehicle Rough-Outs Show 2-Stage To Be Even Slightly Better

Update 3-7-2024: 

The question came up of whether I demanded enough dV of the vehicles?  I had added 10% to the 7.7 km/s orbital velocity for 8.5 km/s.  Here is what the size-out produces with 20% added,  for 9.2 km/s.  The “best” engines and nozzles that I found earlier were retained just as they were revised.  Only the velocity requirement was increased.  I put all the increased burden on the first stage of the two-stage vehicle,  precisely because it has the lower Isp,  as a worst case.  See Figure 12. 

 

Figure 12 – Revised Launch Requirements for Higher dV Values for Rough-Out

The resulting vehicle size-outs show larger vehicles and lower payload fractions,  to be sure exactly as expected!  However,  the SSTO is now worse by about a factor of 3,  not just 2,  than the two stage vehicle.  Both were considered to be expendables for this,  as before.  See Figure 13.  The engine count is getting to be something to worry about,  as well.  That cluster has to fit behind a slender tankage set.  If you make the tanks fatter and shorter to “cover” the cluster,  you are no longer “long and slender”,  and that increases your drag loss.   

One thing readers should consider is the requirement for adequate kinematics right off the launch pad.  You need half a gee or more,  of effective net acceleration beyond gravity,  to be efficient,  and not spend most of your propellant just climbing the first few thousand feet.  For an Earth launch,  that’s an ignition thrust/weight of 1.5 or higher. 

In the real world,  you can use more and/or bigger engines to achieve this,  or you can add some solids (always of much higher frontal thrust density than a cluster of liquids).  I chose to just use more and bigger engines.  Why complicate the study?

This is a very strong effect,  almost to the point of being overwhelming!  It is precisely why vehicles with low launch thrust/weight also have historically had low payload fractions.  The poor acceleration kinematics drastically raise the gravity loss,  making the dV requirement effectively much larger,  and THAT lowers payload fraction rapidly.  To be “efficient”,  you really have to scoot off the pad!  And THAT is exactly what I enforced in this study!

Figure 13 – Revised Rough-Out Results for Higher dV Requirement

----------   

I should probably have followed my own recommendations and used the surface circular orbit speed of 7.9 km/s,  and not the speed at orbit altitude 7.7 km/s,  as the "ideal dV" to be factored up for gravity and drag.  But it really doesn't matter very much when doing a comparison analysis.  The factor would be 1.1 if one assumes 5% each for gravity and drag losses.  It is 1.20 if instead one assumes 10% each for gravity and drag.  7.7*1.1 = 8.5,  some 0.8 loss to cover,  while 7.7*1.2 = 9.2,  some 1.5 loss to cover.  If instead you use 7.9 km/s,  the numbers are only slightly different:  7.9*1.1 = 8.7,  for 0.8 loss,  and 7.9*1.2 = 9.5,  for some 1.6 loss.  

More important is arriving on orbit with something left to support doing rendezvous,  plus some sort of controlled de-orbit burn.  The former is likely on the order of 0.3-0.5 km/s,  and the latter is about 0.1 km/s,  for about an extra 0.5 km/s.  You add those unfactored to the total effective launch dV.  That would be around 8.5+0.5 = 9.0,  or 9.2+0.5 = 9.7.   I did NOT include anything like that in the dV requirements,  because I was only looking for relative trends.  

And those relative trends say the single-stage-to-orbit (SSTO) does factor 2-to-3 worse in terms of payload fraction and launch weight than the two-stage design.  That's for both designs being clean and slender for low drag loss,  and neither pushing the state-of-the-art on structure technologies (the fixed 5% inert in a loaded stage).  The 2-stage does not push the state-of-the-art on its engine technologies,  but the SSTO has to.  SpaceX has already had its troubles with that,  in its own LOX-LCH4 engines.

If you go to LOX-LH2 to improve past the Isp of LOX-LCH4 for the SSTO,  you will end up having to push the state-of-the-art on your structure technologies as well as your engine technologies.    And it may no longer qualify as "clean-and-slender,  so the drag factor may increase,  too. 

If you want something easily and less-expensive to develop,  then don't push the state-of-the-art.  If you do,  you will have higher development costs to amortize.  Everything is acting in the wrong direction on costs,  with a chemical SSTO.