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I began to wonder if the Falcon first stage with modifications would be capable of a single stage to orbit.

https://en.wikipedia.org/wiki/Falcon_9_Full_Thrust

Mass (without propellant)[39]    22,200 kg (48,900 lb)
Mass (with propellant)    433,100 kg (954,800 lb)
Liquid oxygen tank capacity 287,400 kg (633,600 lb)
Kerosene tank capacity 123,500 kg (272,300 lb)
Payload fairing      1,700 kg (3,700 lb)

Thrust (stage total)[4]    7,607 kN (1,710,000 lbf) (sea level)
Specific impulse    Sea level: 282 seconds[

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EXACTLY those technologies were in ASALM-PTV,  a ramjet cruise missile prototype,  flight tested successfully in 1980.  This thing was boosted to ramjet takeover (NOT scramjet!!!) at Mach 2.5,  then flew in ramjet power to cruise at Mach 4 and 80,000 feet.  It would then dive onto its target at an average Mach 5. In one flight test,  we had a throttle runaway failure,  and it reached an unintended Mach 6 at only about 20,000 feet!  It would have melted and broken up,  had this been more than a several-seconds-long transient event. 
…

Dr Clark,

In that case, I'm going to go way out on a limb and say this isn't likely to prove feasible, even if it's theoretically possible.  If there are no actual examples where something similar to what's required has been done in the past, then there's probably a good reason.  I would opine that the reason it hasn't been done before, is that the wings would get ripped off of a hypersonic airframe with a wing loading that low.

…
That being said, I don't think there is a practical solution for the Skylon airframe (as it is usually depicted) to survive re-entry, not with those engines on its wingtips. I think better thought went into the engine than did the vehicle design. The Wikipedia article about Skylon has the BS about reduced ballistic coefficient reducing entry heating. Use my entry spreadsheet and run the trades yourselves. Yes, lower ballistic coefficient raises deceleration altitude. But only by a little bit here at Earth. So the peak heating is only lowered a little bit. The effect is dramatic at Mars. Not here…

At $20M per copy, SpaceX can afford to "blow up" 7 Starships and Super Heavy boosters before they reach the cost of a single RS-25E engine for the Space Launch System.  Nominally, SpaceX would not blow up any vehicles, but if that is what they must do to learn and push the hardware to its limits, then I think NASA and the US Air Force are getting their money's worth out of the test program.  Sometimes you have to "break things" to learn what not to do.  The flight testing phase of development is the correct time to intentionally try to break everything.

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You may want to use a bit higher.  Fine.  Do so.  How about 440 sec?  Corresponds to Vex =  4.315 km/s.  We'll use that.

The mass ratio required to reach destination is exp(dV/Vex) = 8.630.  The corresponding propellant mass fraction Wp/Wig = 1 - 1/MR = 0.884,  or some 88.4% of the liftoff mass is propellant.  THERE IS NO WAY AROUND THAT.  My other numbers are right in that same ballpark.  Go look for yourself.  Go do the calculation for yourself.  I just gave you the equations.

That leaves 0.116 (or 11.6%) of the liftoff mass to be both inert mass plus payload mass.  THERE IS NO WAY AROUND THAT,  EITHER!  And my other numbers from the other studies are also right in that same ballpark. 

Just remember,  the sum of payload plus inert is your dry-tanks burnout mass.  Add the propellant to it for your filled-tanks ignition mass.  MASS MUST BE CONSERVED!  Cannot play games here!

Now if you have 11.6% allowance,  and you believe you can build an all-expendable stage with 4.5% inert,  which has been done,  then that leaves 7.1% max payload fraction.  …
GW

…

What I am looking for in this topic is a solid mathematical presentation that will stand up to the most intense scrutiny, that will show that SSTO from the surface of the Earth is possible.

(th)

A LO2/Kerosene SSTO Rocket Design Study

In February of 1997, Mitchell Burnside-Clapp posted a vehicle dsign study to the sci.space.policy Usenet Newsgroup. It was based on a NASA Access To Space study and looked at the same vehicle design but using higher density LO2/Kerosene propellants and engines rather than the LOX/Liuid Hydrogen propellant and engines of NASA's design. The results were quite interesting.
A LO2/Kerosene SSTO Rocket Design (long)

Mitchell Burnside Clapp
Pioneer Rocketplane

(view with a fixed pitch font such as courier or monaco)

Abstract

The NASA Access to Space LO2/hydrogen single stage to orbit rocket was examined, and the configuration reaccomplished
with LO2/kerosene as the propellants. Four major changes were made in assumptions. First, the aerodynamic
configuration was changed from a wing with winglets to a swept wing with vertical tail. The delta-V for ascent
was as a result recalculated, yielding a lower value due to different values for drag and gravity losses. The engines
were changed to LO2/kerosene burning NK-33 engines, which have a much lower Isp than SSME-type engines used in the
access to space study, but also have a much higher thrust-to-weight ratio. The orbital maneuvering system on
the Access to Space Vehicle was replaced with a pump-fed system based on the D-58 engine used for that purpose now
on Proton stage 4 and Buran. Finally, the wing of the vehicle was allowed to be wet with fuel, which is a
reasonable practice with kerosene but more controversial with oxygen or hydrogen. Additionally, in order to reduce
the technology development needed, the unit weights of the tankage were allowed to increase by 17 percent.

After the design was closed and all the weights recalculated, the empty weight of the LO2/kerosene vehicle
was 35.6% lighter than its hydrogen fuelled counterpart.

Introduction

NASA completed a study in 1993 called Access to Space, the purpose of which was to consider what sort of vehicle
should be operated to meet civil space needs in the future.  The study had three teams to evaluate three different broad
categories of options. The Option 3 team eventually settled on a configuration called the SSTO/R. This vehicle was a
LO2/hydrogen vertical takeoff horizontal landing rocket.  The mission of the Access to Space vehicle was to place a
25,000 pound payload in a 220 n.mi. orbit inclined at 51.6 degrees. The vehicle had a gross liftoff weight of about
2.35 million pounds. The thrust at liftoff was 2.95 million pounds, for a takeoff thrust to weight ratio of 1.2. The
empty weight of the vehicle was 222,582 pounds, and the propellant mass fraction (defined here as [GLOW-empty]/GLOW)
was 90.5%.

Main power for this vehicle was provided by seven SSME derivative engines, with the nozzle expansion ratio reduced
to 50. This resulted in an Isp reduction from 454 to 447.3 seconds. Each engine weighed 6,790 lbs, for an engine sea
level thrust to weight ratio of 62.

Aerodynamically the vehicle was fairly squat, with a fineness ratio (length:diameter) of 5. The overall length
of the vehicle was 173 feet and its diameter was 34.6 feet.  It had a single main wing (dry of all propellants) of about
4,200 square feet total area, augmented by winglets for directional control at reentry. The landing wing loading
was about 60 lb/ft2. The oxygen tank was in the nose section. The payload was mounted transversely between the
oxygen and hydrogen tanks, and was 15 feet in diameter and 30 feet long.

This design exercise was among the most thorough ever conducted of a single stage to orbit LO2/LH2 VTHL rocket.
It was probably the single greatest factor in convincing the space agency that single stage to orbit flight was
feasible and practical, to borrow from the title of Ivan Bekey's paper of the same name.

A LO2/kerosene alternative

A number of people have been asserting for some time that higher propellant mass fractions available from dense
propellants may make single stage to orbit possible with those propellants also. The historical examples of the
extraordinary mass fractions of the Titan II first stage, the Atlas, and the Saturn first stage are all persuasive.
Further, denser propellants lead to higher engine thrust to weight ratios, for perfectly understandable hydraulic
reasons.

It has not usually been observed that higher density also leads to significant reductions in required delta-v.
There are two major reasons that this is so. First, the reduction in volume leads to a smaller frontal area and
lower drag losses. The second, and more significant, reason is that the gravity losses are also reduced. This is because
the mass of the vehicle declines more rapidly from its initial value. The gravity losses are proportional to the
mass of the vehicle at any given time, and hence the vehicle reaches its limit acceleration speed faster.

NASA itself has implicitly recognized this effect. When the Access to Space Option 3 team examined tripropellant
vehicles, the delta-v to orbit derived from their work was 29,127 ft/sec, for precisely the reasons described in the
previous paragraph. This compares to a delta-v of 30,146 ft/s for the hydrogen-only baseline, as reported in a
briefing by David Anderson of NASA MSFC dated 6 October 1993. To be clear, these delta-v numbers include the back
pressure losses, so that no "trajectory averaged Isp" number is used. They did not, however, report any results
for kerosene-only configurations.

To come to a more thorough understanding of the issues involved in SSTO design, I have used the same methodology
as the Access to Space team to develop compatible numbers for a LO2/kerosene SSTO. There are four major changes in
basic assumption between the two approaches, which I will identify and justify here:

1: The ascent delta-v for the LO2/kerosene vehicle is 29,100 ft/sec, rather than 29,970 ft/sec. The reason for
this is argued above, but I ran POST to verify this value, just to be sure. The target orbit is the same: 220 n.mi.
circular at 51.6 degrees inclination. The detailed weights I have for the NASA vehicle are based on a delta-v of
29,970 ft/sec rather than the 30,146 ft/sec reported in Anderson's work, but I prefer to use the values more
favourable to the hydrogen case to be conservative. The optimum value of thrust to weight ratio turns out to be
slightly less than the hydrogen vehicle: 1.15 instead of 1.20.

2: The aerodynamic configuration is that of Boeing's RASV.  Without arguing whether this is optimal, the fineness ratio
of 8.27 and large wing lead to a much more airplane-like layout, better glide and crossrange performance, and
reduced risk. The single vertical tail is simpler and safer than winglets as well. Extensive analysis has justified the
reentry characterisitics of this aircraft. The wing is assumed to be wet with the kerosene fuel, as is common on
most aircraft. The fuel is also present in the wing carry-through box. The payload is carried over the wing
box, and the oxidizer tank is over the wing. This avoids the need for an intertank, which in the NASA Access to
Space design is nearly 6,600 pounds.

3. The main propulsion system is the NK-33. The engine has a sea level thrust of 339,416 lbs, a weight of 2,725 lbs
with gimbal, and a vacuum Isp of 331 seconds. Furthermore, it requires a kerosene inlet pressure of only 2 psi
absolute, which dramatically reduces the pressure required in the wing tank. It also operates with a LO2 pressure at
the inlet of only 32 psi. The comparable values for the SSME are about 50 psi for both propellants. This will have
a substantial effect on the pressurization system weight.

4. The OMS weight is based on the D-58 engine. This engine is used for the Buran OMS system and the Proton stage 4. As
heavy as it is the Isp is an impressive 354 seconds. NASA's vehicle used a pressure fed OMS, which is a sensible design
choice if you're stuck with hydrogen and you wish to minimize the number of fluids aboard the vehicle. But
because both oxygen and kerosene are space-storable, there is no reason to burden the design with a heavy pressure fed
system.

Using the same methodology for calculating masses, and accepting the subsystems masses as given in the Access to
Space vehicle, a redesign with oxygen and kerosene was accomplished. The results appear in Table 1.

Table 1: Access to Space vehicle and LO2/kerosene
alternative

Name                           O2/H2      LO2/RP
Wing                          11,465      11,893 lb
Tail                           1,577       1,636 lb
Body                          64,748      33,741 lb
            Fuel tank         30,668           - lb
          Oxygen tank         13,273      17,271 lb
      Basic Structure         14,610      10,274 lb
  Secondary Structure          6,197       6,197 lb
Thermal Protection            31,098      21,238 lb
Undercarriage, aux. sys        7,548       5,097 lb
Propulsion, Main              63,634      36,426 lb
Propulsion, RCS                3,627       1,234 lb
Propulsion, OMS                2,280         823 lb
Prime Power                    2,339       2,339 lb
Power conversion & dist.       5,830       5,830 lb
Control Surface Actuation      1,549       1,549 lb
Avionics                       1,314       1,314 lb
Environmental Control          2,457       2,457 lb
Margin                        23,116      16,105 lb
Empty Weight                 222,582     141,682 lb

Payload                       25,000      25,000 lb

Residual Fluids                2,264       1,911 lb
     OMS and RCS               1,614       1,261 lb
      Subsystems                 650         650 lb
Reserves                       7,215       8,895 lb
     Ascent                    5,699       7,587 lb
        OMS                      679         541 lb
        RCS                      837         767 lb
Inflight losses               13,254      17,445 lb
         Ascent Residuals     10,984      15,175 lb
      Fuel Cell Reactants      1,612       1,612 lb
  Evaporator water supply        658         658 lb
Propellant, main           2,054,612   3,034,972 lb
     Fuel                    293,604     843,048 lb
     Oxygen                1,761,008   2,191,924 lb
Propellant, RCS                2,814       2,556 lb
     Orbital                   2,051       1,756 lb
     Entry                       763         800 lb
Propellant, OMS               19,357      15,452 lb
GLOW                       2,347,098   3,246,156 lb
Inserted Weight              292,486     211,185 lb
Pre-OMS weight               271,482     186,152 lb
Pre-entry Weight             252,125     170,700 lb
Landed Weight                251,362     169,900 lb
Empty weight                 222,582     141,682 lb

Sea Level Thrust           2,816,518   3,733,080 lb
Percent margin                 11.6%       12.8%
Assumed Isp(vac)               447.3       331.0 s
Ascent Delta-V                29,970      29,100 ft/s
OMS delta-V                    1,065         987 ft/s
RCS delta-V                      108         107 ft/s
Deorbit Delta-V                   44          53 ft/s
Reserves                       0.28%       0.25% lb/lb
Residuals                      0.53%       0.50% lb/lb
Wing Parameter                 4.56%       7.00% lb/lb
TPS parameter                 12.37%      12.50% lb/lb
Undercarriage parameter        3.00%       3.00% lb/lb
Wing Reference Area            4,189       5,528 ft2
Density of fuel                  4.4        50.5 lb/ft3
Density of oxygen               71.2        71.2 lb/ft3
Volume of fuel                66,276      16,694 ft3
Volume of oxygen              24,733      30,785 ft3
Fuel tank parameter             0.42           - lb/ft3
Oxygen tank parameter           0.48        0.56 lb/ft3

Some discussion of the results and justification is in order.

The wing is about 40 percent heavier as a percentage of landed weight than for the hydrogen fueled baseline. When
considered as a tank, it is about 60 percent heavier for the volume of fuel it encloses. Its weight per exposed area
is about the same and the wing loading is half at landing.  No benefit is taken explicitly for the lack of a
requirement for kerosene tank cryogenic insulation.

The tail is assumed to have the same proportion of wing weight for both cases. This is conservative for the
kerosene wehicle because its single vertical tail is structurally more efficient.

The body of the kerosene vehicle has three components. The oxidizer tank has an increased unit weight of about 17
percent. This is done in order to avoid the need for aluminum-lithium, which was assumed in the Access to Space
vehicle. The basic structure group is unchanged, except that the intertank is deleted and the thrust structure is
increased in proportion to the change in thrust level.  The secondary structure group is mostly payload support
related, and was not changed.

The thermal protection group is in both cases about 12.5% of the entry weight. This works out to 1.107 lbs/ft2 of
wetted area for the kerosene vehicle, which is common to many SSTO designs.

The undercarriage group is 3% of landed weight for both vehicles. There is no benefit taken for reductions in gear
loads for the kerosene vehicle due to lower landing speed and lower glide angle at landing.

The main propulsion group includes engines, base mounted heat shield, and pressurization/feed weights. The engines
are far lighter for their thrust than SSME derivatives. The pressurization weights are reduced in proportion to the
pressurized volume for the kerosene vehicle. No benefit is taken for reduced tank pressure.

Here is as good a place as any to point out the erroneous assertion that increased hydrostatic pressure is going to
lead to increased tankage weights. There is no requirement for a particular ullage pressure except for the need to
keep the propellants liquid. It is the pressure at the base of the fluid column rather than the top of the column that
is of engineering interest. The column of fluid exerts a hydrostatic load on the base of the tank, but this load
does not typically exceed the much more adverse requirement for engine inlet pressurization. For the kerosene vehicle,
the hydrostatic load at the base of the oxygen tank is 49 psi, which is compatible with the pressures normally seen
in oxygen tanks for rocket use. The load declines after launch because the weight goes down faster than the
acceleration goes up.

The bottom line here is that dense propellants may require you to alter a tank's pressurization schedule, but not to
overdesign the entire tank. Structures are sized by loads and tankage for rockets is sized principally by volume, and if
the vehicle is small, by minimum gauge considerations.  This is not completely true for wet wings, however, as
discussed previously. In this particular example, there is no need for high pressure in the wing tank either, because
of the low inlet pressure required by the NK-33.

The OMS group is the only other major change, as discussed above. The reliable D-58 engine has been performing space
starts for decades and will serve well here. The acceleration available from the OMS is about 0.12 g, which
is standard.

All the other weights are pushed straight across for the most part. A brief inspection suggests that this is very
conservative. Control surface actuation requirements are certainly less, electrical power requirements less, much
better fuel cells available than the phosporic acid type assumed here, and reduced need for environmental control.
Nonetheless, rather than dispute any of these values it is easier simply to accept them.

The margin is applied to all weight items at 15% execpt for the engine group at 7.5%. The justification for this is that
the main and OMS engine weights are known to high accuracy.

The vehicle has an overall length of 1955 inches, and a diameter of 236.4 inches. The wing has a leading edge sweep
of 55.5 degrees and a trailing edge sweep of -4.5 degrees.  Its reference area is 5,632 square feet, of which 3,992
square feet is exposed. The wing encloses 16,694 ft3 of fuel, with a further 5% ullage. The carry-through is also
wet with fuel. The wing span is 1293 inches, and the taper ratio is 0.13.

The payload bay has a maximum width and height of 15 feet.  It sits on top of the wing carry through box. The thrust
structure from the engines passes through and around the payload bay to the forward LO2 tank. The payload bay is 30
feet in length. It has a pair of doors, the aft edge of which is just forward of the vertical tail leading edge.

The engine section encloses 11 NK-33 engines, with a 4 - 3 - 4 layout. The engines are each 12.5 feet long, and
additional structure and subsystems take up another 6.5 feet.

The oxygen tank comprises the forward fuselage, which encloses 30,785 ft3 of oxygen, with a further 5% ullage.
The length of the tank is about 100 feet. The ventral surface of the tank is moderately flattened as it moves
aft, to fair smoothly with the wing lower surface. This flattening reduces its length by about 5% with respect to a
strictly cylindrical layout. The aft edge of the oxygen tank is about even with the forward payload bay bulkhead. A
compartment of about 13.9 feet provides room for some subsystems and a potential cockpit in future versions.

Conclusion

The methods of the NASA Access to Space study were used to design a single stage to orbit vehicle using existing
LO2/kerosene engines. An inspection of the final results shows that the vehicle weighs about 36.5% less than its
hydrogen counterpart, with reductions in required technology level and off the shelf engines. The center of
mass of the vehicle is about 61% of body length rather than 68% for the Access to Space vehicle, which should improve
control during reentry. The landing safety is considerably improved by lower landing speed and better glide ratio.
Structural margins are greater overall. The vehicle designed here appears to be superior in every respect:
smaller, lighter, lower required technology, improved safety, and almost certainly lower development and
operations cost.