Elon Musk Letter

Dear Elon Musk;
I am writing to you primarily to point out a natural synergism between two of your interests and mine.
First, the Falcon 9 or the BFR rocket could increase its payload to orbit by about a factor of 2 by being accelerated prior to firing the rocket. In particular, a tunnel boring machine can create a sort of mortar with a large enough diameter so that it could shoot a rocket, even one as large as the BFR, out at an initial velocity that is supersonic. This idea is developed in my patent application (Which I included in my original letter, sent through an intermediary on January 23rd).
I also want to point out that I was until recently Vice President for Research and Development at Alevo, Inc., which is now in bankruptcy. I believe you know about Alevo’s inorganic-electrolyte lithium-ion battery -- the implementation of manufacturing was all screwed up, but the basic technology is really groundbreaking. You might want to take an interest, as I know you have in the past.
I was hired by Alevo not because of the battery, but rather for my inventions relating to long-distance power transmission, and future DC grids: the elpipe and the Ballistic Breaker. These were to be second and third generation products for Alevo.  I believe supergrids, which are continental-scale DC grids, are an essential feature of a decarbonized economy; please take a look at my technology at www.Elpipe.com.
Also, I have ALS and I'm barely able to type. In fact I used voice-to-text to prepare these documents; so please forgive any stylistic errors.
Yours sincerely,
Roger Faulkner
Gas Powered Mortar Launch of a Heavy-Lift Rocket
Much of the fuel and oxidizer of any rocket that is launching a satellite is burned just getting up to about one thousand miles an hour.  The first part of the launch is particularly inefficient compared to a gun. You can think of the fact that rocket thrust depends upon the momentum change of the expelled gas, while the relative kinetic energy contained in the gas versus the rocket changes as the speed increases. When the velocity of the rocket equals the velocity of the gas expelled from the rocket engine, the thrust and therefore most of the chemical energy of the propellant is going to  either raise the rocket Against Gravity or to increase the kinetic energy of the rocket (in a vacuum), whereas when the rocket is sitting still nearly all the kinetic energy due to the gas velocity is carried away with the exhaust gas, in a sense.
I propose a Jules Verne style big cannon, or more descriptively a mortar to launch a rocket. The pressures involved are much lower than the pressures in a military cannon, or even an ordinary mortar. I envision pressures typically around 20 bar   behind the rocket.
  1. Consider as an example of a large rocket to be launched by the proposed large gas mortar the SpaceX rocket called the BFR. This rocket is 9 meters diameter for  most of its length, but has fins On it's upper stage that bring out the total diameter To about 14 m. I know  from a talk by Elon Musk,The October 2017 update to the planetary Society concerning plans for the 2022 Mars mission. from that talk, I learned that talk that the BFR upper stage is designed to carry 1200 tons of fuel plus oxidizer, with an estimated 85 tons dry weight. The cited Mass for the fully loaded bfr is 4400 tons, which means that the first stage would weigh 3115 tons. If the same ratio between wet weight and dry weight exists for the first stage rocket as for the upper stage, then the lower stage would be carrying about 2865 tons of fuel and oxidizer.  Musk said in his talk that the initial thrust is to be 5400 tons, which If we take gravitational acceleration on Earth to be 9.81 meters per second squared, means the thrust would be 53,000,000 newtons. This implies that the  total thrust of the rocket is 1.227  times normal Earth gravity; therefore initial vertical acceleration of the rocket is only about  2.23 meter per second squared because the accelerating thrust of 9810 kilonewtons represents only about (22.7/1.227)% of the total thrust, most of which is going to simply counter gravity at the beginning of the launch.  Consider how much more gravitational work would be done at the beginning of rocket firing if the overall rocket has a high vertical  velocity when the rocket is ignited.
An optimum vertical rate of ascent of the bfr rocket at the time of rocket ignition exists such that the rocket is neither accelerating or decelerating when it leaves the launch tube, right after rocket ignition. In other words the vertical velocity should be such that the aerodynamic resistance is equal to the thrust supplied by the rocket over and above that which is needed to balance the gravitational force due to the mass of the rocket. That is an aerodynamic calculation, to which I do not have the answer right now. However it is obvious  from the high net thrust of 9810 kilonewtons that this represents a great speed, probably supersonic, and as the rocket rises the decreasing air drag and the reduced mass of the rocket would allow the rocket to accelerate further. If the initial rocket velocity from the launch tube is supersonic, then the vibration that occurs as the rocket passes the speed of sound would not occur, which simplifies the structural design of the rocket.
By  accelerating the rocket in a mortar launch prior to igniting the rocket, then simply holding the speed constant just after the time the rocket ignites upon its vertical exit from the launch tube, the rate of doing gravitational work in lifting the rocket becomes much higher.  Remember that two forms of work are being done as the rocket is lofted into space: first there is an increase in gravitational potential energy as the rocket is lifted, and second there is the increasing kinetic energy of the rocket. At any time one can describe the total work being done by the rocket motor  as a sum of the acceleration work and the lifting work. As the velocity of the rocket increases,  and as the direction of motion of the rocket approaches horizontal, an increasing portion of the work is going towards the acceleration of the rocket, but in the first stage a large part of the work is the lifting of the rocket against gravity.
In the very first part of flight, more of the total work is going into lifting the rocket compared to accelerating the rocket, except in the case of high initial acceleration above two standard Earth gravities by the rocket. The gravitational work component scales with initial velocity, and so is quite low at the beginning of flight of the rocket for a launch from a standing start. If however the rocket is given a high initial vertical velocity at the time that the rocket begins firing, then the rate of working against gravity is very high. That means that the energetic efficiency of the rocket is greatly increased by means of accelerating the rocket prior to firing it.
In order to fit the SpaceX BFR rocket into a launch tube, the tube needs to be wide enough to accommodate the fins on the upper stage. That implies an approximate diameter for the launch tube of about 14 m. In addition, the rocket must sit on a sabot that forms a gas tight seal below the rocket and which is able to support the mass of the rocket while it is being accelerated. I estimate that this sabot would weigh 200 tons, which implies a total launch weight of 4600 tons.  Given this and the other assumptions outlined above, I calculate that 113 million Newtons would be needed to cause an upward acceleration of 4 G's for the BFR rocket in a mortar launch.  That implies that a passenger on board the rocket would experience an effective gravity of 5 times normal gravity on Earth at the beginning of the launch. This implies a pressure difference of 1.05 megapascals, about 160 pounds per square inch, between the pressure behind the rocket and in front of the rocket.
In one possible scenario, at the very beginning of the launch, non-combusting gas pressure alone would begin the acceleration. Thereafter combustion would begin within the gas phase behind the rocket. Acceleration would vary at different times after the rocket first begins to move. I believe an average acceleration corresponding to 3 G’s, 32.4 meters per second squared is achievable.
As an example, consider a barrel length in front of the rocket of 5 kilometers with an effective average acceleration of 32.4 meters per second squared.  We can solve for the time of flight in the barrel as the square root of the quantity 2 x barrel length / acceleration; the answer is that the rocket is in the barrel for 17.6 seconds. This corresponds to an exit velocity of 569 meters per second. This is way less than orbital velocity, but it is comfortably supersonic, and would greatly increase the initial efficiency of the rocket and substantially increase the cargo mass that could be delivered to orbit by the rocket.  By using higher accelerations and longer launch tunnels it is possible to get much higher speeds, potentially approaching orbital velocity, though that would require a very strong rocket capable of withstanding  both the accelerations and the aerodynamic forces of hypersonic travel through the lower atmosphere.
Because the air drag will be very high in the lower atmosphere, it is not desirable to accelerate substantially after such a proposed mortar launch until one is above the troposphere. Somewhere between 25,000 to 40,000 meters of elevation, the atmospheric pressure will be low enough for more efficient, higher expansion ratio rockets to be used, such as the Raptor Rocket that has been developed by SpaceX for the second stage of the Falcon 9 rocket and the BFR. I do not  think it would be practical to use the mortar launch alone to get the rocket high enough so that the more efficient high-expansion-ratio Raptor vacuum rockets could be used for the rest of the trip. Perhaps the high-expansion-ratio rockets may be used in the lower troposphere provided that the rocket is traveling fast enough to create a vacuum behind the rocket.  I do not know what speed would be required to make the vacuum type Raptor Rockets work in the troposphere; I intuit that a minimum speed well above the speed of sound in air would be necessary, possibly corresponding to the exit velocity of the gas coming out of the rocket.  I do not believe that such a hypersonic launch would be practical in any case for a rocket carrying humans.
I visualize a combination of a gas gun with no combustion and a gas gun in which combustion occurs. In particular, I envision a launch tube that is 5.5 km long, and has been dug by a tunnel boring machine.  The projectile is a BFR rocket and the diameter of the tunnel is 14 m, in which some of the annular clearance between the rocket and the tube launcher which will be occupied by sliding bushings that may also serve the function of a secondary backup sabot and lateral reinforcement of the rocket  against buckling. The primary sabot would support the rocket and also have the function of forming a gas tight seal between the rocket and the launch tube. The rocket projectile is located 5 km from the exit end of the launch tube, with half a kilometer of compressed combustible gas behind the rocket. It is highly desirable that the 5 km of the launch tube in front of the rocket be evacuated. Some means of vacuum sealing at the exit end of the launch tube would be provided such that the end seal is intact until the rocket punches through. This basic setup is used as the basis for a series of examples below for various launch scenarios.
Examples of the invention
Let's consider a very high velocity launch.  This could not apply to any current rocket, because present-day rockets are not designed to allow the very high accelerations that are considered here, for example an average acceleration of 1000 meters per second squared. Though this would be a very high acceleration for a rocket, it would be low for a gun. For maximum velocity, the mass of the rocket would have to be much less than the mass of the fully-loaded bfr rocket, for example 500 tons or so. the bottom 0.5 km behind the rocket prior to launch would be filled primarily with hydrogen gas,  preferably a mixture of hydrogen and oxygen. The pressure in the gas chamber would be raised up to a level which is able to accelerate the rocket at the maximum allowed acceleration strictly by gas pressure, prior to combustion of the gas mixture. I suggest this could be as high as 100 G's. If one builds a system capable of a hundred G's, one need not use that capability for every launch. In particular it might be practical to operate the gas launcher at 100 G’s for robotic cargo ships, while you also launch space tourists at maximum acceleration of 3 G's, for example.
In the case of a high-acceleration hypersonic launch, it is particularly desirable that the compressed gas chamber behind the rocket prior to launch should be filled with primarily hydrogen gas, perhaps containing oxygen at a sub-stoichiometric ratio, so that the post combustion gas is still predominantly hydrogen. Several variants on this theme are possible, for example one could fill the entire 0.5 km compressed-gas region behind the rocket prior to launch with hydrogen at the beginning, then introduce a stoichiometric mixture of methane and oxygen from the bottom; this guarantees that the gas right behind the rocket is pure hydrogen while still enabling a combustion driven pressurization from behind, after the initial part of the launch which would be accomplished strictly by the stored PV energy in the compressed gas. At the very beginning of launch some form of restraint would be released that would enable the high pressure hydrogen gas to begin pushing the rocket up the launch tube. As the hydrogen expands, the acceleration will decrease until the time where the combustible-gas mixture below the hydrogen zone is ignited.  At that time acceleration will increase again.
I envision another hypersonic launch scenario in which oxygen is blown into the launch tube from jets on the sides of the launch tube, only after the rocket is fully loaded and ready to go, or perhaps even after the rocket begins to move. In the very beginning all the gas behind the rocket could be hydrogen but as the hydrogen expands there could be oxygen injected into the hydrogen at points behind the moving rocket. The injection of the oxygen increases the pressure, leading to a greater acceleration of the rocket even before ignition of the hydrogen oxygen mixture, as long as the rocket velocity remains subsonic. It is also possible to inject oxygen in such a way as to create a pocket of an explosive hydrogen and oxygen mixture at the back end of the launch tube, with the portion of the launch tube just behind the rocket comprising mainly hydrogen or practically pure hydrogen. This hydrogen and oxygen mixture would be ignited after the rocket has been moving for a second or so.
The importance of the hydrogen content of the propellant gas is that the speed of sound scales with the inverse of the square root of molecular weight. As a result the speed of sound in hydrogen is around 3 times the square root of 2 times faster in hydrogen compared to oxygen. The speed of sound in the propellant gas limits the maximum muzzle velocity from a gas powered gun. I think it is therefore desirable for a maximum velocity launch to have the gas immediately behind the rocket be nearly pure hydrogen. I think it will be possible to nearly achieve that by using a long launch tube with a combustible gas mixture only in the end of the tube farthest away from the rocket. note that the combustion velocity in hydrogen oxygen mixtures can be above 2000 m per second, depending on temperature and pressure.
A desirable high velocity launch scenario involves a hydrogen oxygen mixture with more than stoichiometric amounts of hydrogen. One can visualize a hybrid launch system in which a high-hydrogen gas mixture with oxygen content close to the minimum concentration of oxygen to make the gas mixture combustible initially fills the part of the launch tube behind the rocket. In this scenario, the combustible gas mixture initially accelerates the rocket like an ordinary air gun, but then the hydrogen-oxygen mixture is ignited sometime after the rocket begins its forward motion in the launch tube, after the acceleration is reduced from its initial maximum to some selected level. The ignition would occur at the opposite end of the launch tube from the now rapidly moving rocket,  perhaps a few seconds after the rocket is released. After ignition, a combustion wave will travel up the column of combustible gas causing compression of the gas in front of the wave, leading to a second maximum of acceleration. Even after the combustion wave reaches the rocket, one could still use additional oxygen injection from the sides into the hot steam and hydrogen mixture created by of the initial burn of the hydrogen-rich hydrogen and oxygen mixture for an additional boost. This form of launch is intrinsically more risky than one in which a non-explosive mixture of gases is behind the rocket prior to launch, since a premature explosion could cause greater acceleration than would be planned for or tolerable.
A hybrid concept would have the rocket loaded and ready for firing based on a non-combustible pure hydrogen gas pressure behind the rocket but then perhaps only minutes before launch the requisite amount of oxygen would be blown in. It would take only seconds or at most a minute for the gas to mix and be ready for the burning of the gas mixture. If there is excess hydrogen in the mix then the oxygen doesn't have to be thoroughly distributed to get the entire benefit of the burn of the gas, though in this case the energy yield due to the gas burning would be less than if there is a stoichiometric mixture of hydrogen and oxygen. I like this scenario better than premixing the fuel and oxidizer as the lower chamber is pressurized.
A low-cost launch scenario would start with compressed air behind the rocket. After the rocket is fully loaded and ready to be launched, natural gas would be injected from the sides to create a combustible mixture behind the rocket. As with other methods described above involving hydrogen, the initial acceleration of the rocket would begin with simply the stored energy in the pressurized combustible gas mixture behind the rocket. Ignition of the combustible gas mixture would occur after the rocket has been released and is moving down the launch tube.  Ideally the combustion wave would reach the backside of the rocket just as the rocket exits the launch tube into the atmosphere. The velocity of the combustion wave in burning methane air mixtures is far too slow to catch up with the moving rocket; however one can manipulate the effective velocity of the combustion front by igniting the gas mixture from the sides of the launch tube.
A tunnel boring machine can certainly bore a 10 m diameter vertical tunnel, and it appears to be possible to stretch that to a 14 m diameter tunnel. If the rocks are selected correctly, boring the tunnel would be a small part of the total project cost. I believe a relatively thin metal inner tube could be cemented inside the rock tunnel with adequate smoothness to work as a launch tube.
I envision the rocket being surrounded by a sort of sabot sleeve comprised of a slippery polymer like ultra-high molecular weight polyethylene or PTFE at the outer perimeter where it contacts the launch tube. Delrin would also be very cost-effective; basically this is poly formaldehyde, but I think it would be unavoidable in that case that you would produce formaldehyde emissions at launch, so I think UHMW polyethylene and PTFE are the most logical candidates for the sliding surface between the sabot and the launch tube. It is also probable that some form of thermal or plasma spray process could be used to coat the steel launch tube on the inside to reduce sliding friction and/or corrosion. It is even possible that sputtering, PVD, CVD or some other coating technology that must be applied in a vacuum could be applied to the inner surface of the of the launch tube, to minimize sliding friction and abrasion against the polymer-surfaced sabot.
Using a sabot as a gas-tight seal has the unique potential advantage of enabling the rocket to be pulled from the front rather than pushed from the back. Since structural components are much stronger in tension than in compression due to buckling behavior, one could probably achieve greater acceleration of the rocket within the launch tube because it is pulled from near the front instead of pushed at the rear by the gas pressure. It is necessary in this case to be cautious about the possibility that internal parts of the rocket could contain pressurized gas when it exits the launch tube.
The shape of the Falcon F9 rocket with its Dragon  capsule from SpaceX would be particularly well suited to the concept of pulling the rocket from the front. In this case, the sabot would be placed just behind the  dragon capsule at the front end of the rocket, which has a larger diameter than the rest of the rocket. It is feasible to use the same launch tube to launch both the BFR and F9/Dragon rockets, though the sabot would be quite different for the two different rockets.
The optimum place to locate a launch tube would be at high elevation right on the equator. I suspect that a slight slanting of the launch tube towards the east would be desirable since using the Earth's rotation saves quite a bit of energy, and so the most economical launches are always towards the east. The Andes look interesting except for one thing: in most cases it's very important to have an ocean to the east of the launch site or at least an area with low occupancy by humans, since there is always a chance that a rocket launch will fail in such a way that the rocket will fall down. This is far less dangerous if it falls into the ocean compared to any landmass on Earth.  This is why a slight tilt eastwards would be desirable so that even if all control is lost, the rocket would still fall into the sea in an accident.
A recent book proposes Kenya as the location for a major launch site, but that might not be true if the rock there is not suitable for the deep-tunnel-launch-tube aspect of this invention, and in view of political instability in that region.  I have been imagining a launch tube that is about 5.5 km long. The maximum pressure within the launch tube depends upon the maximum acceleration planned for various rockets that would be launched. For human launches and for rocket ships with an average areal density similar to that planned for SpaceX's BFR rocket, this corresponds to a pressure differential from behind the rocket to in front of the rocket between 10 to 20 atmospheres. If the same launch tube is to also be used for unmanned cargo craft, accelerations as high as 100 G’s would be desirable; this could correspond to pressures as high as 200 atmospheres, which would likely be created through combustion.
In order to achieve supersonic launch, it is very desirable that the portion of the launch tube in front of the rocket is evacuated. I can visualize three possible methods to keep the launch tube in front of the rocket evacuated as the rocket is accelerated by gas pressure behind it;
 1.  A dome holding back atmospheric pressure could seal the end of the launch tube, and be designed so that it bonds to the front end of the rocket upon impact.  In this case the ceiling dome would effectively become part of the spaceship, fulfilling the function of a heat shield during the ascent of the rocket through the atmosphere. If it is properly designed, this shield could later be used in the construction of a space station for example.
2. The dome holding back atmospheric pressure could be blown out of the way using shaped charges of high explosive just before the front end of the rocket exits the launch tube.
3. The dome holding back atmospheric pressure could be designed in conjunction with a leading edge of the rocket so that the rocket pushes the dome aside. This necessarily entails that the dome break into pieces. The dome could consist of a fiber-reinforced elastomer film, similar to those used to seal the end of a subsea missile launch tube until the nose cone of the missile rips through the membrane. Alternatively, the dome could be made up of solid pieces which are designed to part as the rocket passes through; in this case the dome would likely have straight sections that would in cross-section form a regular geometric shape such as an octagon or a dodecahedron for example, and would match the shape of the rocket nose-cone exactly.
It is energetically desirable to minimize the speed of the rocket through the thick part of the atmosphere. This is so in part because aerodynamic resistance scales with at least the square of the velocity and linearly with the gas pressure. I think it is most energetically desirable for the thrust at the time when the rocket breaks through the ceiling dome and into the atmosphere to be approximately equal to the thrust required to prevent deceleration from atmospheric drag.
To take a specific example, if the rocket is traveling at 1000 meters per second at the time it bursts through the ceiling dome of the tunnel launcher, it will take about 40 seconds to reach an altitude where the atmospheric pressure is low enough to enable vacuum-type rockets to work. If the rocket is neither accelerating or decelerating at the time just after it first enters the atmosphere and ignites its rocket engines, then if the thrust is maintained constant the acceleration will increase with altitude for two reasons. First, the aerodynamic drag would decrease with altitude. Second, the mass of the rocket would be decreasing due to consumption of rocket fuel.
Let's consider for a moment the optimum location for such a tunnel based mortar launcher. Here are some of the considerations:
  • ·         The launcher should ideally be near the equator to maximize the effect of the earth’s rotation to aid achieving orbital velocity
  • ·         The launch should point east, to make use of the rotation of the Earth
  • ·         The launcher should be located so that the end of the barrel is well above sea level
  • ·         The launcher should be located in a country that is politically stable
  • ·         The region to the east of the launcher should be an ocean
  • ·         The launch site should be near a suitable port from which the rocket can be transported to the launch site.
I believe that these considerations argue for the mountains in Costa Rica, Nicaragua, Panama, or Columbia as candidate launch sites.
There are a wide variety of potential gas mixtures to power the tunnel based launcher. At the very low end of feasible velocity would be a purely compressed air based launch mechanism. The next level up I believe would comprise an air plus natural gas mixture, in which the initial acceleration would be simply due to the expansion of the combustible gas mixture; next, the mixture would be ignited at the bottom and a combustion wavefront would move up the tube towards the rocket. It would be particularly desirable if the wavefront reaches the backside of the rocket approximately when the rocket exits the launch tunnel. In this scenario, one can vary the energy yield from combustion by varying the stoichiometry of the combustible mixture, from the lowest explosive stoichiometry ranging up to exact stoichiometry for full oxidation of the methane.
The next level up would substitute oxygen methane gas mixtures for the air natural gas mixtures described in the previous paragraph.
One can also envision more exotic gas mixtures such as hydrogen and oxygen as previously mentioned above. This is particularly of interest for the case in which maximum acceleration is allowed to be much higher than that which would be tolerable to human beings.
This invention is aimed at increasing the payload to low-earth orbit for a conventional rocket, such as the SpaceX F9/Dragon or BFR rockets. An extremely important aspect of the invention is that a tunnel boring machine would be used to create the launch tunnel. This greatly reduces the cost compared to prior proposals in which the launch tube is made of metal and is in essence a large cannon barrel.
Note that the pressures that are required behind the rocket in the present invention are quite low compared to the pressures that would be used in prior art cannon launched spacecraft proposals, such as the launch cannons developed in the HARP program funded by the US Army in the early 1960s. Even though the envisioned launch speed of 300-1000 m per second is only a small fraction (about one-sixth at best) of orbital velocity, the payload delivered to orbit would be greatly increased because of the very low efficiency of rockets for the low velocity portion of the flight.
In order to achieve a launch velocity of 1000 meters per second, constrained by the requirement that the average vertical acceleration is only 3 G's, one would need a 15 km launch tube. If higher acceleration can be tolerated, the launch tube can be shorter. I envision an initial project in which the launch tube is 5.5 km long, with half a kilometer of combustible gas behind the rocket. I have described above several launch protocols in which the gas beneath the rocket is initially in a compressed state. There is a second possibility in which the gas mixture underneath the rocket is not compressed but rather at atmospheric pressure or even a slight vacuum. This particular option, as I will describe below, appears to be an optimum solution.
Consider the case of a launching mortar in which the gas below the rocket is comprised of primarily hydrogen gas add a pressure well below ordinary atmospheric pressure. The hydrogen gas contains an amount of oxygen that will react with  only part of the hydrogen, perhaps between 5 to 20% of all the hydrogen present.  Since the atmosphere is normally only about 20% oxygen, a reaction of one mole of oxygen with 10 moles of hydrogen would be very similar to a stoichiometric reaction of hydrogen with air. I adopt this for my example below.
Hydrogen and oxygen mixtures have higher combustion velocity than any other gas mixture I have examined. They also exhibit very high pressure ratios between the pre combustion gas and the pressure following combustion, in spite of the fact that there are fewer moles of water formed then the moles of reactive gas. For stoichiometric hydrogen oxygen mixtures, a pressure ratio of about 70 is observed. I don't have the data for the pressure ratio from burning of non stoichiometric hydrogen oxygen mixtures but I assume that The pressure ratio will be less and that dilution with extra hydrogen would be about equivalent to dilution with nitrogen as in hydrogen air mixtures.
Consider the example of a mixture of 10 moles of hydrogen with 1 mol of oxygen. the initial pressure is in the vacuum range; let us stay that the pressure is about half of an atmosphere. The launch would be initiated by sparks or a single spark at the bottom of the hydrogen oxygen mixture, and would move with a speed close to 1000 meters per second. Even before this combustion wave reaches the surface of the sabot, the hydrogen oxygen mixture next to the sabot will be compressed by the combustion wave below. Half a second after ignition of the combustion wave would reach the surface of the moving sabot. After the initial combustion is complete, additional oxygen could be injected from the sides of the launch tube for continuing to maintain near constant acceleration. This is a complex situation which I think can be manipulated so that the combustion wave propagation followed by oxygen injection is optimized to use of the generated PV energy while also controlling the acceleration of the projectile, the rocket.
One can also envision  a totally combustion driven launch, similar to that envisioned above with hydrogen oxygen mixtures, but fueled by methane air mixtures. The pressure ratio for combustion of methane air mixtures is much less than then for hydrogen oxygen mixtures, on the order of 5x increase in pressure. On the other hand, the much lower velocity of the flame front in methane air mixtures allows a desirable degree of control that would not be possible for hydrogen oxygen mixtures, since ignition from the sides of the launch tube can control the movement of the combustion front. Thus a region composed of compressed methane and air could exist behind the rocket initially, with a pressure just below what is required to lift the rocket. Igniting combustion at the far end of the column of methane and air would compress the gas below the rocket and begin the acceleration. The low combustion wave velocity in methane and air would make it easier to control the advance of the combustion front, and potentially to even out the acceleration during the 5 km motion of the rocket down the launch tube.
Even the lowest cost gas mixtures and a launch tube that is only 5.5 km long could still achieve a supersonic initial velocity for the rocket and so greatly increase the load delivered to low-Earth orbit for a rocket. this is believed to be highly practical for the scenario where hundreds of launches year are anticipated.
There would be a massive muzzle blast when the rocket exits the launch tube. Therefore, some form of muzzle silencer would be required.

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