TSTO Launch Fundamentals

I have illustrated the fundamentals of making a TSTO launch design-sizing calculation as two figures.  This is for an eastward launch at low inclination,  to a typical circular low Earth orbit (LEO).  The first figure shows the trajectory and how the delta-vees (dV’s) are estimated.  It also shows a sketched free-body diagram,  and the associated Newton’s 3^rd Law solution for vehicle acceleration,  at 3 typical points coming up the trajectory.  Typical loss factor estimates are also shown. 

Update 6-21-2023: made slight corrections to this first figure to make the acceleration formulas accurate,  with only the sine component of weight resisting thrust.  That sine function is of the path angle above horizontal.

The nature of the non-lifting (but thrusted) gravity turn ascent puts the vertical rise portion,  most of the steeply-inclined trajectory,  and all of the trajectory experiencing aerodynamic drag,  onto the first stage burn.   The free-body diagrams are there to emphasize this.  If the first stage thrust is not “huge” in relation to the other forces,  the acceleration capability will be small or negative (meaning the design cannot work).  The losses to be covered during this first stage burn are quite significant.

The trajectory during the second stage burn is exo-atmospheric,  and very nearly horizontal,  so that (1) there are little in the way of losses to cover,  and (2) the thrust can be much smaller and still get good acceleration in a reasonable time,  because the weight statement is different,  with much smaller numbers.  If the burn reaches the value estimated for “surface circular LEO speed”,  before reaching orbit altitude (as often happens),  then the second stage coasts onward to orbit altitude,  and makes a small circularization burn there.  That circularization burn is quite modest,  on the order of 100 m/s = 0.1 km/s.

The gravity and drag losses are percentages based on surface circular orbit speed as a measure of the total mechanical energy needed to attain orbit speed (kinetic energy) at orbit altitude (potential energy).  These losses get applied almost entirely to the first stage burn.  As a “ballpark” configuration sizing tool,  they all get applied to the first stage burn.  At most,  there is a distinct minority of the gravity loss,  that might apply to the second stage (maybe 10% of the estimated loss).

The dV required of stage 1 is thus the staging velocity Vstg plus the sum of the gravity and drag losses.  The dV required of stage 2 is the burn value dV2 plus the circularization burn dV,  where the dV2 burn value is the surface circular orbit speed less the staging speed. 

Typically,  in first stages we are looking at denser propellants to reduce aerodynamic drag losses and inert tankage weights at large contained volume,  with reduced specific impulse (Isp) capability.  We are also looking at engine bells sized for sea level expansion,  which also reduces Isp capability a little further.  That,  plus shouldering all the losses,  means the staging velocity falls well short of halfway to orbit speed,  in any practical design.

Typically,  in second stages we can use lower density propellants without incurring extra drag loss and inert tankage weights,  because the total propellant mass quantities are far smaller than those in the first stage (which has to lift the entire loaded second stage as its “payload”).  Plus,  the engines can be “vacuum-optimized” (although in reality there is no such thing,  there are only engine bells that fit within the allowable room),  which makes specific impulse capability a bit better than the sea level designs.

This allows the second stage to shoulder the great majority of the total dV required to reach orbit,  as the second figure illustrates.  

Once you have determined which propellants each stage will use,  and what numbers of appropriate engines,  thrust levels per engine,  and sea level or vacuum bells,  you have some idea of the average Isp capability (and thus the effective exhaust velocity Vex = gc*Isp) of each of the two stages.  From there,  the remaining significant variable to set,  is the staging velocity Vstg.  You run iterative calculations at various Vstg values to maximize your overall payload fraction at the “best” staging velocity Vstg.   As noted in the figure,  there is NO SINGLE rocket equation calculation to give you that answer!  It is an inherently iterative search,  and the result is unique to every design!

Adding recovery and re-use of stages complicates this analysis further,  and by a huge amount.  You must have at staging sufficient unburned propellant still aboard the first stage,  to support whatever kind of re-entry procedure and landing scheme that you have in mind.  That will reduce the “optimal” Vstg value,  and cause the second stage to “shoulder” more of the overall dV,  driving its required mass ratio MR higher (thus reducing stage payload fraction).  Every design is unique!  Those details are beyond scope here.

There is no way to recover and re-use second stages,  except to turn them into fully re-entry-capable spacecraft,  capable of landing by some means!  This also involves some amount of unburned propellant still aboard upon reaching orbit,  to support the return flight operations.  That reduces ignition payload fraction further,  causing such designs to be attractive only in larger sizes,  where there is an economy-of-scale effect.  This is also beyond scope here.

If you choose a liquid core with solid strap-on boosters as your first stage,  the odds are very low that the solids will burn out at the same time as the liquid first stage core.  In that case,  you break the first stage burn into two burns.  The first uses a thrust-averaged Isp for the solids and liquid burning simultaneously.  The second uses a liquid-only Isp,  and deletes the inert weights of the solids from its weight statement.  Again,  that complication is beyond scope here.

The main point of this discussion is that the notion of a single rocket equation calculation,  representing accurately all of the things that must figure into a two-stage ascent,  is utter nonsense!  The rocket equations are separate for the two stages,  with different Isp’s and different weight statements,  and these must be iteratively re-done until overall payload fraction is optimized at the “best” staging speed. 

There is NO general rule-of-thumb here,  not even for simple expendable two-stage-to-orbit (TSTO) designs!

Update 6-24-2023 with Addendum 1:

This is what is important to TSTO launch,  and when it is important.  The rocket equation always applies,  BUT,  the dV it predicts is irrelevant if your thrust is insufficient to accelerate along the path (everything is “eaten up” by gravity loss,  in that case).  Newton’s 3^rd Law lets you determine whether your pathwise acceleration is adequate or not.

Staging typically occurs just exo-atmospheric,  at a rather low path angle,  and at a velocity only about 20-35% of the total velocity the launch vehicle must achieve.  It usually is about 2 km/s,  which is about 25% of orbital velocity at 8 km/s.  It is set to maximize the payload fraction out of a design concept.

The first stage shoulders virtually all the losses,  and has the highest thrust requirement,  since it must push the heaviest weight up vertically against drag.  If the first stage dV requirement is modest,  one can afford lower Isp levels in order to get the acceleration required,  with simple raw thrust,  per Newton.

The second stage has no drag loss and virtually no gravity loss.  The path is nearly horizontal,  so the path-wise weight component is very small.  And with the first stage separated,  the mass that must be accelerated is very much smaller.  The second stage has a very low thrust requirement,  but must supply the larger dV.  So the highest-possible Isp is critical to second stage designs.

Update 6-26-2023 with Addendum 2:

Here is a way to generate realistic stage inerts from scratch,  in the absence of any actual data:

There’s basically 2 cases to consider for book-keeping payload items:  the payload within an ascent shroud,  and payload carried by a recoverable item (a capsule or an entry-capable spacecraft): 

Weight statements are made up of inerts and payload,  which sum to stage burnout;  plus propellant,  which sums with burnout to ignition.  Jettisoned payload shrouds are generally counted as part of the upper stage inert.  A recoverable item is not.  The entire loaded upper stage is the “payload” for the lower stage. 

If the upper stage still burns after the shroud is jettisoned,  that has to be a separately-computed rocket equation calculation,  using a separate (and different) weight statement that zeroes the shroud from the inert,  and only has the remaining propellant aboard the stage.  That portion of the burn before shroud jettison uses the weight statement with the shroud included in the inert,  but expels only that propellant which is used up to the jettison point. 

This complication is not an issue if the payload is contained inside a recoverable item.  The recoverable item plus the contained payload is the “payload” carried by the upper stage.

Here’s a summary of things that make up weight statements:

What is needed to estimate performance with the rocket equation is the effective exhaust velocity Vex of the propulsion.  This is not the actual expansion velocity coming out of the nozzle,  although it is similar in magnitude.  It is estimated from the specific impulse Isp as Vex = gc * Isp,  where gc is the standard acceleration of gravity.  There are “delta-vee” values (dV) associated with the launch trajectory.  Suitable values of dV and Vex,  and a mass ratio MR from the weight statement,  are related by the rocket equation:  dV = Vex LN(MR),  where MR = ignition mass/burnout mass.  The real “trick” is getting the right dV values to use,  from the launch trajectory,  including the effects of real-world losses:

The rocket engine produces thrust with a propellant flow rate drawn from the tanks.  It does that by expanding the hot gases through a convergent-divergent nozzle,  that has a suitable bell expansion ratio. 

Where the atmospheric backpressure is high,  expansion potential is more limited,  and the thrust is less.  The same pump system and combustion chamber rig can “drive” a bell with larger expansion and thrust,  with the same flow rate,  out in vacuum. 

The usual design for a first stage features “perfect expansion” to sea level-equivalent expanded pressure,  exactly matching the ambient atmospheric pressure.  That same nozzle generates somewhat more thrust as one flies out into space,  because the backpressure term that reduces delivered thrust goes to zero. 

It is possible to design the nozzle (with the same chamber) at some intermediate altitude for perfect expansion.  That will raise the thrust out in vacuum a little higher than the sea level version,  but the thrust at sea level will be lower than the sea level design.  That is usually unacceptable for liftoff.

There really is no such thing as a truly “vacuum-optimized” nozzle design,  which would imply an infinite expansion to zero expanded pressure.  The size of the exit area and length of the bell would be infinite.  The max dimensions of the bell are actually set by what will fit in the allowable space at the rear of the stage.  That is what is termed a “vacuum-optimized” design,  but that is really a misnomer.  It is merely the max expansion that meets the limited space constraint into which the engine must fit.

These “vacuum-optimized” designs have significantly-higher vacuum thrust because of the greater expansion.   They would have a very low sea level thrust due to a very large backpressure term (driven by the large exit area),  except that such designs will usually suffer flow separation (and thus not produce any significant thrust).  You cannot operate with a separated bell:  the engine will self-destruct.  Clearly the compressible flow “ballistics” of the nozzle expansion are the main factor driving performance:

The specific impulse Isp = F_th/w_tot,  where w_tot is the total propellant flow rate drawn from the tanks.  Isp must be based on w_tot to be consistent with the mass ratio calculations used in the rocket equation.  Unless there is no pump drive gas dumped overboard,  w_tot and the nozzle flow rate w_noz are not the same.  If the dumped pump drive gas w_bleed is nonzero,  as it is in some engine “cycles”,  then the cycle can be represented easily enough by the bleed fraction BF = w_bleed/w_tot.  This can be used to determine the total flow from the nozzle flow coming from the nozzle ballistics:  w_tot = w_noz/(1-BF).

There are some cycles that have no pump drive gas dumped overboard.  All the flow,  including the pump drive gas,  ends up going through the nozzle.  For these cycles,  just use BF = 0. 

The calculation starts with an estimate of the nozzle kinetic energy efficiency η_KE from the average bell half-angle.  This models the average of the cosine-component factors for the exiting streamlines that diverge off the centerline axis.   There is no measurable friction loss in a proper nozzle design.

For a given chamber pressure,  one chooses the appropriate expanded pressure value Pe,  and forms the pressure ratio PR,  and from that the temperature ratio TR,  as shown just above.  Expanded Mach number Me can be computed from TR,  as shown.  Using TR and Me,  find the expansion area ratio Ae/At,  as shown.  This procedure works “once-through” for sea level (or any nonzero Pa) nozzle designs.

If instead a desired Ae/At is the given requirement,  then just iterate this very same computation procedure over several values of Pe to find the one that gives you the desired Ae/At.  That iterative procedure works for all vacuum designs,  where it is the size of Ae that is your design constraint.

These computed Me,  Ae/At,  Pe,  and Pc values get used to compute the vacuum thrust coefficient C_Fvac.  That plus the Pa,  Pc,  and Ae/At values of the backpressure term,  combine as thrust coefficient C_F at Pa.

There is some thrust requirement F_th associated with some backpressure value Pa.  Find the C_F for that Pa,  and plug it,  the Pc,  and the F_th value into the thrust equation F_th = C_F Pc At,  which sizes the nozzle throat area At,  and then with the Ae/At value,  also sizes the nozzle exit area Ae.  However,  if you are using the gas generating item from another calculation for this one,  you will want to keep the At and sized flow rate values of that other case.  In that case,  iteratively adjust the required F_th value at its Pa value,  until the desired At and flow rates are obtained.

So far the only thing dependent upon the propellant combination is the gas specific heat ratio γ,  and that dependence is very weak (nearly all rocket gases have γ ~ 1.20).  Once the nozzle throat area At has been sized,  the propellant combination gives you a chamber c* velocity as an empirical function of Pc,  and then the nozzle flow rate equation w_noz = Pc C_D At gc/c* lets you determine w_noz.

Your engine cycle gets modeled with the BF value.  Using it,  we have w_tot = w_noz/(1-BF).  Then,  the specific impulse that is consistent with the rocket equation is just Isp = F_th/w_tot.  

Or looked at another way,  Isp = [C_F c* (1-BF)] / [gc C_D]. 

The only thing about Isp that is truly sensitive to the propellant combination (and only the propellant combination) is the c* value.  The rest is all coming from the compressible flow ballistics of nozzles,  and Isp is very sensitive to that.

Doing the nozzle ballistics from a c* value is the only way to get a reliable Isp for your propellant combination,  as used with a specific nozzle design.  

Tabulated values versus propellant combinations reflect only some Pc level (yours may be different),  with a perfect nozzle efficiency η_KE = 1.000 (about 1 or 2% too high,  usually),  and either sea level Pe expansion,  or expansion to a specified Ae/At value in vacuum (where your Ae/At in vacuum may be different).  They are therefore inherently some significant percentage off,  as a result.  Further,  a lot of reported Isp data are based on w_noz,  not w_tot,  which introduces a overestimating error of the same size as the cycle BF!  While calculating corrections is possible,  just doing the scratch ballistics is just as easy,  and far more reliable.

Now,  if you know how to do this for yourself,  then you can tell if someone else’s performance prediction software is feeding you “garbage” instead of “good stuff”.  But,  if you don’t know how to do this,  you will never know “garbage” from “good stuff”.   Remember:  with computers,  it’s GIGO (“garbage in,  garbage out”).  Computers process bad numbers just as readily as good numbers.  It all looks the same.

Some typical values:

Conical bell,  15 deg half-angle..........η_KE = 0.98296

Curved bell 18 & 8 deg half angles....η_KE = 0.98719

Well-made throat with smooth profile..C_D = 0.99 to 0.995

“typical” rocket gases..........................γ ~ 1.20

Modeling c* variation with pressure....

.....delivered c* = K Pc^m where m ~ 0.01

.....use a K & m curve fit from real test data

.....includes c* efficiency = c*/c*^o,  generally 0.95+

 

Update 6-27-2023 with Addendum 3:

It is important to have a reliable estimate for the actually-delivered c* from a rocket engine chamber (true of the solids as well as the liquids,  and also the hybrids). 

The c* value has the units of velocity,  and can be calculated from chamber temperature as chamber gas properties.  These are reported as theoretical values from thermochemical methods or programs (of which the NASE ODE code is considered to be the “gold standard”). 

               c* = [ (gc R Tc/γ)(gfactor ^exponent]^0.5             where gfactor = (γ + 1)/2 and exponent = (γ + 1)/(γ – 1)

However,  no rocket ever delivers the theoretical value of c*,  all have some experimental c* efficiency η_c* = c*/c*^o,  where c*^o is the theoretical thermochemical value.  Typically,  these efficiencies are in the vicinity of 0.95+,  and they vary with pressure as a power function,  as does c*^o itself,  using a function of the form

c* = K Pc ^m          where the exponent m is usually on the order of 0.01

As a result that function can be used to correlate actual delivered c* from real engine test data.  Once obtained,  this c* value at the chamber pressure Pc,  goes into the nozzle flow rate equation:

               w_noz = Pc C_D At gc / c*

In the case of a liquid engine test (or tests),  you literally measure the propellant flow rates fed to the engine,  and you measure the bleed flow rate with a metering orifice and pressure transducers,  if there is bleed.  You also have a precise before-and-after measurement of the throat diameter,  so At is known.  The gc item is just the standard Earth gravity acceleration,  so the only other item to quantify is C_D.  You need a separate flow calibration result to quantify that as precisely as possible,  although well-designed nozzles have C_D’s that fall in the 0.990 to 0.995 range,  usually.  (A simple drilled orifice operated choked is closer to C_D = 0.80.)

You simply plug in the flow rates,  Pc values,  and your At,  your C_D,  and the gc factor,  and solve for the corresponding c*.  You do that for a lot of tests at a lot of different Pc values.  Then you correlate c* versus the various levels of Pc,  and then least-squares curve fit that data set with the power function given above.  If you do all this correctly,  Pearson’s r² parameter will always exceed 0.99.

Solids are different,  in that there is no way to directly measure the instantaneous mass flow rate.  Under the assumptions that c* and At are never far from their average values for the test,  and that the test design is such that Pc is crudely constant,  the time integral of the nozzle mass flow equation yields these results,  which can be solved for the average c* and average Pc:

               ∫w_noz dt = η_exp Wp = [C_D At gc/(avg c*)][∫Pc dt]  and (avg Pc) = [∫Pc dt] / t_B

where η_exp = expelled weight divided by the loaded propellant weight Wp (which implies a defined weighing configuration),  and C_D and At are their best-estimated average values. 

A test motor configuration without significant “sliver” is required for this easily.  Most of those are simple cylindrical segment internal burners.  Sizes from about 6 inch diameter on up usually get c* data that correlate very well indeed with full scale motors.  Smaller ones do not correlate very well.  2-inch burn rate motors are essentially useless for correlating c*,  although they are quite good for correlating burn rates. 

Again,  you correlate avg c* vs avg Pc,  and least-squares curve-fit the data set with the power function.  Also again,  Pearson’s r² parameter will exceed 0.99 if you do this carefully enough.