Correspondence principle

From Wikipedia, the free encyclopedia

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

In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agree with classical calculations.

The principle was formulated by Niels Bohr in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom.

The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal limit under which the new theory reduces to the older, thanks to the existence of a deformation parameter.

Classical quantities appear in quantum mechanics in the form of expected values of observables, and as such the Ehrenfest theorem (which predicts the time evolution of the expected values) lends support to the correspondence principle.

Quantum mechanics

The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large. Arnold Sommerfeld referred to the principle as "Bohrs Zauberstab" (Bohr's magic wand) in 1921.

The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered.

The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws.

The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are broad: states of a physical system form a complex vector space and physical observables are identified with Hermitian operators that act on this Hilbert space. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.

Other scientific theories

The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit".

For example,

• Einstein's special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below);
• General relativity reduces to Newtonian gravity in the limit of weak gravitational fields;
• Laplace's theory of celestial mechanics reduces to Kepler's when interplanetary interactions are ignored;
• Statistical mechanics reproduces thermodynamics when the number of particles is large;
• In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding genes.
• In mathematical economics, as formalized in Foundations of Economic Analysis (1947) by Paul Samuelson, the correspondence principle and other postulates imply testable predictions about how the equilibrium changes when parameters are changed in an economic system.

In order for there to be a correspondence, the earlier theory has to have a domain of validity—it must work under some conditions. Not all theories have a domain of validity. For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically dominant for 18 centuries, does not have any domain of validity (on the other hand, it can sensibly be said that the falling of objects through the air ("natural motion") constitutes a domain of validity for a part of Aristotle's mechanics).

Examples

Bohr model

Main article: Bohr model

If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit does not decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.

In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of 1/T, so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level,

$${\displaystyle \mathrm{\Delta }{E}_n=\frac{h}{T(E_n)}.}$$

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state $E_n$, or $E_{n+1}$, or some average—in hindsight, this model is only the leading semiclassical approximation.

Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r^3/2. The energy scales as 1/r, so the level spacing formula amounts to

$${\displaystyle \mathrm{\Delta }E\propto \frac{1}{r^{3/2}}\propto E^{3/2}.}$$

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut.

The angular momentum L of the circular orbit scales as √r. The energy in terms of the angular momentum is then

$${\displaystyle E\propto \frac{1}{r}\propto \frac{1}{L^2}.}$$

Assuming, with Bohr, that quantized values of L are equally spaced, the spacing between neighboring energies is

$${\displaystyle \mathrm{\Delta }E\propto \frac{1}{(L+\hslash {)}^2}-\frac{1}{L^2}\approx -\frac{2\hslash }{L^3}\propto -E^{3/2}.}$$

This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be ħ, so the angular momentum should be an integer multiple of ħ,

$${\displaystyle L=\frac{nh}{2\pi }=n\hslash .}$$

This is how Bohr arrived at his model. Since only the level spacing is determined heuristically by the correspondence principle, one could always add a small fixed offset to the quantum number— L could just as well have been (n+.338) ħ.

Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he was able to get so much from what is only the leading order approximation. A less heuristic treatment accounts for needed offsets in the ground state L², cf. Wigner–Weyl transform.

One-dimensional potential

Bohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define a quantity J(E) which is a function only of the energy, and has the property that

$\frac{dJ}{dE}=T$

This is the analogue of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey J = nh for n integer, since

${\displaystyle \mathrm{\Delta }E=E_{n+1}-E_n=\frac{dE}{dJ}(J_{n+1}-J_n)=\frac{1}{T}\mathrm{\Delta }J}$

This quantity J is canonically conjugate to a variable θ which, by the Hamilton equations of motion changes with time as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable θ increases steadily from 0 to 1 over one period.

The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J,θ coordinates is that of a half-cylinder, capped off at J = 0, which is the motionless orbit at the lowest value of the energy. These coordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space.

The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as in J-θ. But in the J-θ coordinates, this area is the area of a cylinder of unit circumference between 0 and J, or just J. So J is equal to the area enclosed by the orbit in x-p coordinates too,

${\displaystyle J={\int }_0^{T}p\frac{dx}{dt}dt.}$

The quantization rule is that the action variable J is an integer multiple of h.

Multiperiodic motion: Bohr–Sommerfeld quantization

Bohr's correspondence principle provided a way to find the semiclassical quantization rule for a one degree of freedom system. It was an argument for the old quantum condition mostly independent from the one developed by Wien and Einstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action.

Bohr was reluctant to generalize the rule to systems with many degrees of freedom. This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system,

$J_k=h{n}_k.$

Each action variable is a separate integer, a separate quantum number.

This condition reproduces the circular orbit condition for two dimensional motion: let r,θ be polar coordinates for a central potential. Then θ is already an angle variable, and the canonical momentum conjugate is L, the angular momentum. So the quantum condition for L reproduces Bohr's rule:

${\int }_0^{2\pi }Ld\theta =2\pi L=nh.$

This allowed Sommerfeld to generalize Bohr's theory of circular orbits to elliptical orbits, showing that the energy levels are the same. He also found some general properties of quantum angular momentum which seemed paradoxical at the time. One of these results was that the z-component of the angular momentum, the classical inclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotational invariance. This was called space quantization for a while, but this term fell out of favor with the new quantum mechanics since no quantization of space is involved.

In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It is possible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and this resolves the intuitive paradoxes of the Sommerfeld model.

The quantum harmonic oscillator

Main article: Quantum harmonic oscillator

Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior.

Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values,

$E=(n+1/2)\hslash \omega ,n=0,1,2,3,\dots ,$

where ω is the angular frequency of the oscillator.

However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of macroscopic systems falls within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is

$E=\frac{m{\omega }^2{A}^2}{2}.$

Thus, the quantum number has the value

$n=\frac{E}{\hslash \cdot \omega }-\frac{1}{2}=\frac{m\omega A^2}{2\hslash }-\frac{1}{2}$

If we apply typical "human-scale" values m = 1kg, ω = 1 rad/s, and A = 1 m, then n ≈ 4.74×10³³. This is a very large number, so the system is indeed in the correspondence limit.

It is simple to see why we perceive a continuum of energy in this limit. With ω = 1 rad/s, the difference between each energy level is ħω ≈ 1.05 × 10⁻³⁴J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent classical limit.

Relativistic kinetic energy

Here we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classical expression, for speeds that are much slower than the speed of light, v ≪ c.

Albert Einstein's mass-energy equation

$E=\frac{m_0}{\sqrt{1-v^2/c^2}}{c}^2,$

where the velocity, v is the velocity of the body relative to the observer, ${\displaystyle m_0}$ is the rest mass (the observed mass of the body at zero velocity relative to the observer), and c is the speed of light.

When the velocity v vanishes, the energy expressed above is not zero, and represents the rest energy,

$E_0=m_0{c}^2.$

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, by definition, the kinetic energy,

${\displaystyle T=E-E_0=\frac{m_0{c}^2}{\sqrt{1-v^2/c^2}}-m_0{c}^2.}$

Using the approximation

$${\displaystyle (1+x{)}^n\approx 1+nx}$$

for ${\displaystyle |x|\ll 1}$, we get, when speeds are much slower than that of light, or v ≪ c,

$${\displaystyle \begin{array}{rl}T& =m_0{c}^2\left(\frac{1}{\sqrt{1-v^2/c^2}}-1\right)\\ & =m_0{c}^2\left({\left(1-v^2/c^2\right)}^{-\frac{1}{2}}-1\right)\\ & \approx m_0{c}^2\left((1-(-\begin{array}{c}\frac{1}{2}\end{array})v^2/c^2)-1\right)\\ & =m_0{c}^2\left(\begin{array}{c}\frac{1}{2}\end{array}{v}^2/c^2\right)\\ & =\begin{array}{c}\frac{1}{2}\end{array}{m}_0{v}^2,\end{array}}$$

which is the Newtonian expression for kinetic energy.

Single-stage-to-orbit

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Single-stage-to-orbit

The VentureStar was a proposed SSTO spaceplane.

A single-stage-to-orbit (SSTO) vehicle reaches orbit from the surface of a body using only propellants and fluids and without expending tanks, engines, or other major hardware. The term usually, but not exclusively, refers to reusable vehicles. To date, no Earth-launched SSTO launch vehicles have ever been flown; orbital launches from Earth have been performed by either fully or partially expendable multi-stage rockets.

The main projected advantage of the SSTO concept is elimination of the hardware replacement inherent in expendable launch systems. However, the non-recurring costs associated with design, development, research and engineering (DDR&E) of reusable SSTO systems are much higher than expendable systems due to the substantial technical challenges of SSTO, assuming that those technical issues can in fact be solved. SSTO vehicles may also require a significantly higher degree of regular maintenance.

It is considered to be marginally possible to launch a single-stage-to-orbit chemically fueled spacecraft from Earth. The principal complicating factors for SSTO from Earth are: high orbital velocity of over 7,400 metres per second (27,000 km/h; 17,000 mph); the need to overcome Earth's gravity, especially in the early stages of flight; and flight within Earth's atmosphere, which limits speed in the early stages of flight due to drag, g, and influences engine performance.

Advances in rocketry in the 21st century have resulted in a substantial fall in the cost to launch a kilogram of payload to either low Earth orbit or the International Space Station, reducing the main projected advantage of the SSTO concept.

Notable single stage to orbit concepts include Skylon, which used the hybrid-cycle SABRE engine that can use oxygen from the atmosphere when it is at low altitude, and then using onboard liquid oxygen after switching to the closed cycle rocket engine at high altitude, the McDonnell Douglas DC-X, the Lockheed Martin X-33 and VentureStar which was intended to replace the Space Shuttle, and the Roton SSTO, which is a helicopter that can get to orbit. However, despite showing some promise, none of them has come close to achieving orbit yet due to problems with finding a sufficiently efficient propulsion system and discontinued development.

Single-stage-to-orbit is much easier to achieve on extraterrestrial bodies that have weaker gravitational fields and lower atmospheric pressure than Earth, such as the Moon and Mars, and has been achieved from the Moon by the Apollo program's Lunar Module, by several robotic spacecraft of the Soviet Luna program, and by China's Chang'e 5.

History

Early concepts

ROMBUS concept art

Before the second half of the twentieth century, very little research was conducted into space travel. During the 1960s, some of the first concept designs for this kind of craft began to emerge.

One of the earliest SSTO concepts was the expendable One stage Orbital Space Truck (OOST) proposed by Philip Bono, an engineer for Douglas Aircraft Company. A reusable version named ROOST was also proposed.

Another early SSTO concept was a reusable launch vehicle named NEXUS which was proposed by Krafft Arnold Ehricke in the early 1960s. It was one of the largest spacecraft ever conceptualized with a diameter of over 50 metres and the capability to lift up to 2000 short tons into Earth orbit, intended for missions to further out locations in the solar system such as Mars.

The North American Air Augmented VTOVL from 1963 was a similarly large craft which would have used ramjets to decrease the liftoff mass of the vehicle by removing the need for large amounts of liquid oxygen while traveling through the atmosphere.

From 1965, Robert Salkeld investigated various single stage to orbit winged spaceplane concepts. He proposed a vehicle which would burn hydrocarbon fuel while in the atmosphere and then switch to hydrogen fuel for increasing efficiency once in space.

Further examples of Bono's early concepts (prior to the 1990s) which were never constructed include:

• ROMBUS (Reusable Orbital Module, Booster, and Utility Shuttle), another design from Philip Bono. This was not technically single stage since it dropped some of its initial hydrogen tanks, but it came very close.
• Ithacus, an adapted ROMBUS concept which was designed to carry soldiers and military equipment to other continents via a sub-orbital trajectory.
• Pegasus, another adapted ROMBUS concept designed to carry passengers and payloads long distances in short amounts of time via space.
• Douglas SASSTO, a 1967 launch vehicle concept.
• Hyperion, yet another Philip Bono concept which used a sled to build up speed before liftoff to save on the amount of fuel which had to be lifted into the air.

Star-raker: In 1979 Rockwell International unveiled a concept for a 100-ton payload heavy-lift multicycle airbreather ramjet/cryogenic rocket engine, horizontal takeoff/horizontal landing single-stage-to-orbit spaceplane named Star-Raker, designed to launch heavy Space-based solar power satellites into a 300 nautical mile Earth orbit. Star-raker would have had 3 x LOX/LH2 rocket engines (based on the SSME) + 10 x turboramjets.

Around 1985 the NASP project was intended to launch a scramjet vehicle into orbit, but funding was stopped and the project cancelled. At around the same time, the HOTOL tried to use precooled jet engine technology, but failed to show significant advantages over rocket technology.

DC-X technology

Main article: McDonnell Douglas DC-X

The maiden flight of the DC-X

The DC-X, short for Delta Clipper Experimental, was an uncrewed one-third scale vertical takeoff and landing demonstrator for a proposed SSTO. It is one of only a few prototype SSTO vehicles ever built. Several other prototypes were intended, including the DC-X2 (a half-scale prototype) and the DC-Y, a full-scale vehicle which would be capable of single stage insertion into orbit. Neither of these were built, but the project was taken over by NASA in 1995, and they built the DC-XA, an upgraded one-third scale prototype. This vehicle was lost when it landed with only three of its four landing pads deployed, which caused it to tip over on its side and explode. The project has not been continued since.

Roton

Main article: Rotary Rocket

From 1999 to 2001 Rotary Rocket attempted to build a SSTO vehicle called the Roton. It received a large amount of media attention and a working sub-scale prototype was completed, but the design was largely impractical.

Approaches

There have been various approaches to SSTO, including pure rockets that are launched and land vertically, air-breathing scramjet-powered vehicles that are launched and land horizontally, nuclear-powered vehicles, and even jet-engine-powered vehicles that can fly into orbit and return landing like an airliner, completely intact.

For rocket-powered SSTO, the main challenge is achieving a high enough mass-ratio to carry sufficient propellant to achieve orbit, plus a meaningful payload weight. One possibility is to give the rocket an initial speed with a space gun, as planned in the Quicklaunch project.

For air-breathing SSTO, the main challenge is system complexity and associated research and development costs, material science, and construction techniques necessary for surviving sustained high-speed flight within the atmosphere, and achieving a high enough mass-ratio to carry sufficient propellant to achieve orbit, plus a meaningful payload weight. Air-breathing designs typically fly at supersonic or hypersonic speeds, and usually include a rocket engine for the final burn for orbit.

Whether rocket-powered or air-breathing, a reusable vehicle must be rugged enough to survive multiple round trips into space without adding excessive weight or maintenance. In addition a reusable vehicle must be able to reenter without damage, and land safely.

While single-stage rockets were once thought to be beyond reach, advances in materials technology and construction techniques have shown them to be possible. For example, calculations show that the Titan II first stage, launched on its own, would have a 25-to-1 ratio of fuel to vehicle hardware. It has a sufficiently efficient engine to achieve orbit, but without carrying much payload.

Dense versus hydrogen fuels

Hydrogen fuel might seem the obvious fuel for SSTO vehicles. When burned with oxygen, hydrogen gives the highest specific impulse of any commonly used fuel: around 450 seconds, compared with up to 350 seconds for kerosene.

Hydrogen has the following advantages:

• Hydrogen has nearly 30% higher specific impulse (about 450 seconds vs. 350 seconds) than most dense fuels.
• Hydrogen is an excellent coolant.
• The gross mass of hydrogen stages is lower than dense-fuelled stages for the same payload.
• Hydrogen is environmentally friendly.

However, hydrogen also has these disadvantages:

• Very low density (about 1⁄7 of the density of kerosene) – requiring a very large tank
• Deeply cryogenic – must be stored at very low temperatures and thus needs heavy insulation
• Escapes very easily from the smallest gap
• Wide combustible range – easily ignited and burns with a dangerously invisible flame
• Tends to condense oxygen which can cause flammability problems
• Has a large coefficient of expansion for even small heat leaks.

These issues can be dealt with, but at extra cost.

While kerosene tanks can be 1% of the weight of their contents, hydrogen tanks often must weigh 10% of their contents. This is because of both the low density and the additional insulation required to minimize boiloff (a problem which does not occur with kerosene and many other fuels). The low density of hydrogen further affects the design of the rest of the vehicle: pumps and pipework need to be much larger in order to pump the fuel to the engine. The result is the thrust/weight ratio of hydrogen-fueled engines is 30–50% lower than comparable engines using denser fuels.

This inefficiency indirectly affects gravity losses as well; the vehicle has to hold itself up on rocket power until it reaches orbit. The lower excess thrust of the hydrogen engines due to the lower thrust/weight ratio means that the vehicle must ascend more steeply, and so less thrust acts horizontally. Less horizontal thrust results in taking longer to reach orbit, and gravity losses are increased by at least 300 metres per second (1,100 km/h; 670 mph). While not appearing large, the mass ratio to delta-v curve is very steep to reach orbit in a single stage, and this makes a 10% difference to the mass ratio on top of the tankage and pump savings.

The overall effect is that there is surprisingly little difference in overall performance between SSTOs that use hydrogen and those that use denser fuels, except that hydrogen vehicles may be rather more expensive to develop and buy. Careful studies have shown that some dense fuels (for example liquid propane) exceed the performance of hydrogen fuel when used in an SSTO launch vehicle by 10% for the same dry weight.

In the 1960s Philip Bono investigated single-stage, VTVL tripropellant rockets, and showed that it could improve payload size by around 30%.

Operational experience with the DC-X experimental rocket has caused a number of SSTO advocates to reconsider hydrogen as a satisfactory fuel. The late Max Hunter, while employing hydrogen fuel in the DC-X, often said that he thought the first successful orbital SSTO would more likely be fueled by propane.

One engine for all altitudes

Some SSTO concepts use the same engine for all altitudes, which is a problem for traditional engines with a bell-shaped nozzle. Depending on the atmospheric pressure, different bell shapes are required. Engines designed to operate in a vacuum have large bells, allowing the exhaust gasses to expand to near vacuum pressures, thereby raising efficiency. Due to an effect known as Flow separation, using a vacuum bell in atmosphere would have disastrous consequences for the engine. Engines designed to fire in atmosphere therefore have to shorten the nozzle, only expanding the gasses to atmospheric pressure. The efficiency losses due to the smaller bell are usually mitigated via staging, as upper stage engines such as the Rocketdyne J-2 do not have to fire until atmospheric pressure is negligible, and can therefore use the larger bell.

One possible solution would be to use an aerospike engine, which can be effective in a wide range of ambient pressures. In fact, a linear aerospike engine was to be used in the X-33 design.

Other solutions involve using multiple engines and other altitude adapting designs such as double-mu bells or extensible bell sections.

Still, at very high altitudes, the extremely large engine bells tend to expand the exhaust gases down to near vacuum pressures. As a result, these engine bells are counterproductive due to their excess weight. Some SSTO concepts use very high pressure engines which permit high ratios to be used from ground level. This gives good performance, negating the need for more complex solutions.

Airbreathing SSTO

Skylon spaceplane

Some designs for SSTO attempt to use airbreathing jet engines that collect oxidizer and reaction mass from the atmosphere to reduce the take-off weight of the vehicle.

Some of the issues with this approach are:

• No known air breathing engine is capable of operating at orbital speed within the atmosphere (for example hydrogen fueled scramjets seem to have a top speed of about Mach 17). This means that rockets must be used for the final orbital insertion.
• Rocket thrust needs the orbital mass to be as small as possible to minimize propellant weight.
• The thrust-to-weight ratio of rockets that rely on on-board oxygen increases dramatically as fuel is expended, because the oxidizer fuel tank has about 1% of the mass as the oxidizer it carries, whereas air-breathing engines traditionally have a poor thrust/weight ratio which is relatively fixed during the air-breathing ascent.
• Very high speeds in the atmosphere necessitate very heavy thermal protection systems, which makes reaching orbit even harder.
• While at lower speeds, air-breathing engines are very efficient, but the efficiency (Isp) and thrust levels of air-breathing jet engines drop considerably at high speed (above Mach 5–10 depending on the engine) and begin to approach that of rocket engines or worse.
• Lift to drag ratios of vehicles at hypersonic speeds are poor, however the effective lift to drag ratios of rocket vehicles at high g is not dissimilar.

Thus with for example scramjet designs (e.g. X-43) the mass budgets do not seem to close for orbital launch.

Similar issues occur with single-stage vehicles attempting to carry conventional jet engines to orbit—the weight of the jet engines is not compensated sufficiently by the reduction in propellant.

On the other hand, LACE-like precooled airbreathing designs such as the Skylon spaceplane (and ATREX) which transition to rocket thrust at rather lower speeds (Mach 5.5) do seem to give, on paper at least, an improved orbital mass fraction over pure rockets (even multistage rockets) sufficiently to hold out the possibility of full reusability with better payload fraction.

It is important to note that mass fraction is an important concept in the engineering of a rocket. However, mass fraction may have little to do with the costs of a rocket, as the costs of fuel are very small when compared to the costs of the engineering program as a whole. As a result, a cheap rocket with a poor mass fraction may be able to deliver more payload to orbit with a given amount of money than a more complicated, more efficient rocket.

Launch assists

Many vehicles are only narrowly suborbital, so practically anything that gives a relatively small delta-v increase can be helpful, and outside assistance for a vehicle is therefore desirable.

Proposed launch assists include:

• sled launch (rail, maglev including Bantam, MagLifter, and StarTram, etc.)
• air launch or aircraft tow
• in-flight fueling
• Lofstrom launch loop/space fountains

And on-orbit resources such as:

• Space tether
• tugs

Nuclear propulsion

Main article: Nuclear propulsion

Due to weight issues such as shielding, many nuclear propulsion systems are unable to lift their own weight, and hence are unsuitable for launching to orbit. However, some designs such as the Orion project and some nuclear thermal designs do have a thrust to weight ratio in excess of 1, enabling them to lift off. Clearly, one of the main issues with nuclear propulsion would be safety, both during a launch for the passengers, but also in case of a failure during launch. As of December 2021, no current program is attempting nuclear propulsion from Earth's surface.

Beam-powered propulsion

Main article: Beam-powered propulsion

Because they can be more energetic than the potential energy that chemical fuel allows for, some laser or microwave powered rocket concepts have the potential to launch vehicles into orbit, single stage. In practice, this area is not possible with current technology.

Design challenges inherent in SSTO

The design space constraints of SSTO vehicles were described by rocket design engineer Robert Truax:

Using similar technologies (i.e., the same propellants and structural fraction), a two-stage-to-orbit vehicle will always have a better payload-to-weight ratio than a single stage designed for the same mission, in most cases, a very much better [payload-to-weight ratio]. Only when the structural factor approaches zero [very little vehicle structure weight] does the payload/weight ratio of a single-stage rocket approach that of a two-stage. A slight miscalculation and the single-stage rocket winds up with no payload. To get any at all, technology needs to be stretched to the limit. Squeezing out the last drop of specific impulse, and shaving off the last pound, costs money and/or reduces reliability.

The Tsiolkovsky rocket equation expresses the maximum change in velocity any single rocket stage can achieve:

$\mathrm{\Delta }v=I_{\text{sp}}\cdot g_0\ln (MR)$

where:

$\mathrm{\Delta }v$ (delta-v) is the maximum change of velocity of the vehicle,

$I_{\text{sp}}$ is the propellant specific impulse,

$g_0$ is the standard gravity,

$MR$ is the vehicle mass ratio,

$\ln$ refers to the natural logarithm function.

The mass ratio of a vehicle is defined as a ratio the initial vehicle mass when fully loaded with propellants $\left(m_i\right)$ to the final vehicle mass $\left(m_f\right)$ after the burn:

$MR=\frac{m_i}{m_f}=\frac{m_p+m_s+m_{\text{pl}}}{m_s+m_{\text{pl}}}$

where:

$m_i$ is the initial vehicle mass or the gross liftoff weight $\left(GLOW\right)$,

$m_f$ is the final vehicle mass after the burn,

$m_s$ is the structural mass of vehicle,

$m_p$ is the propellant mass,

$m_{\text{pl}}$ is the payload mass.

The propellant mass fraction ($\zeta$) of a vehicle can be expressed solely as a function of the mass ratio:

$\zeta =\frac{m_p}{m_i}=\frac{m_i-m_f}{m_i}=1-\frac{m_f}{m_i}=1-\frac{1}{MR}=\frac{MR-1}{MR}$

The structural coefficient ($\lambda$) is a critical parameter in SSTO vehicle design. Structural efficiency of a vehicle is maximized as the structural coefficient approaches zero. The structural coefficient is defined as:

Comparison of growth factor sensitivity for Single-Stage-to-Orbit (SSTO) and restricted stage Two-Stage-to-Orbit (TSTO) vehicles. Based on a LEO mission of Delta v = 9.1 km/s and payload mass = 4500 kg for range of propellant Isp.

$\lambda =\frac{m_s}{m_p+m_s}=\frac{m_s}{m_i-m_{\text{pl}}}=\frac{\frac{m_s}{m_i}}{1-\frac{m_{\text{pl}}}{m_i}}$

The overall structural mass fraction $\left(\frac{m_s}{m_i}\right)$ can be expressed in terms of the structural coefficient:

$\frac{m_s}{m_i}=\lambda \left(1-\frac{m_{\text{pl}}}{m_i}\right)$

An additional expression for the overall structural mass fraction can be found by noting that the payload mass fraction $\left(\frac{m_{\text{pl}}}{m_i}\right)$, propellant mass fraction and structural mass fraction sum to one:

$1=\frac{m_{\text{pl}}}{m_i}+\frac{m_p}{m_i}+\frac{m_s}{m_i}=\frac{m_{\text{pl}}}{m_i}+\zeta +\frac{m_s}{m_i}$

$\frac{m_s}{m_i}=1-\zeta -\frac{m_{\text{pl}}}{m_i}$

Equating the expressions for structural mass fraction and solving for the initial vehicle mass yields:

$m_i=GLOW=\frac{m_{\text{pl}}}{1-\left(\frac{\zeta }{1-\lambda }\right)}$

This expression shows how the size of a SSTO vehicle is dependent on its structural efficiency. Given a mission profile $\left(\mathrm{\Delta }v,m_{\text{pl}}\right)$ and propellant type $\left(I_{\text{sp}}\right)$, the size of a vehicle increases with an increasing structural coefficient. This growth factor sensitivity is shown parametrically for both SSTO and two-stage-to-orbit (TSTO) vehicles for a standard LEO mission. The curves vertically asymptote at the maximum structural coefficient limit where mission criteria can no longer be met:

${\lambda }_{\text{max}}=1-\zeta =\frac{1}{MR}$

In comparison to a non-optimized TSTO vehicle using restricted staging, a SSTO rocket launching an identical payload mass and using the same propellants will always require a substantially smaller structural coefficient to achieve the same delta-v. Given that current materials technology places a lower limit of approximately 0.1 on the smallest structural coefficients attainable, reusable SSTO vehicles are typically an impractical choice even when using the highest performance propellants available.

Examples

It is easier to achieve SSTO from a body with lower gravitational pull than Earth, such as the Moon or Mars. The Apollo Lunar Module ascended from the lunar surface to lunar orbit in a single stage.

A detailed study into SSTO vehicles was prepared by Chrysler Corporation's Space Division in 1970–1971 under NASA contract NAS8-26341. Their proposal (Shuttle SERV) was an enormous vehicle with more than 50,000 kilograms (110,000 lb) of payload, utilizing jet engines for (vertical) landing. While the technical problems seemed to be solvable, the USAF required a winged design that led to the Shuttle as we know it today.

The uncrewed DC-X technology demonstrator, originally developed by McDonnell Douglas for the Strategic Defense Initiative (SDI) program office, was an attempt to build a vehicle that could lead to an SSTO vehicle. The one-third-size test craft was operated and maintained by a small team of three people based out of a trailer, and the craft was once relaunched less than 24 hours after landing. Although the test program was not without mishap (including a minor explosion), the DC-X demonstrated that the maintenance aspects of the concept were sound. That project was cancelled when it landed with three of four legs deployed, tipped over, and exploded on the fourth flight after transferring management from the Strategic Defense Initiative Organization to NASA.

The Aquarius Launch Vehicle was designed to bring bulk materials to orbit as cheaply as possible.

Current development

Current and previous SSTO projects include the Japanese Kankoh-maru project, ARCA Haas 2C, Radian One and the Indian Avatar spaceplane.

Skylon

Main article: Reaction Engines Skylon

The British Government partnered with the ESA in 2010 to promote a single-stage to orbit spaceplane concept called Skylon. This design was pioneered by Reaction Engines Limited (REL), a company founded by Alan Bond after HOTOL was canceled. The Skylon spaceplane has been positively received by the British government, and the British Interplanetary Society. Following a successful propulsion system test that was audited by ESA's propulsion division in mid-2012, REL announced that it would begin a three-and-a-half-year project to develop and build a test jig of the Sabre engine to prove the engines performance across its air-breathing and rocket modes. In November 2012, it was announced that a key test of the engine precooler had been successfully completed, and that ESA had verified the precooler's design. The project's development is now allowed to advance to its next phase, which involves the construction and testing of a full-scale prototype engine.

Starship

Main article: SpaceX Starship

Elon Musk, CEO of SpaceX, has claimed that the upper stage of the prototype "Starship" rocket, currently in development in Starbase (Texas), has the capability to reach orbit as an SSTO. However he concedes that if this was done, there would be no appreciable mass left for a heat shield, landing legs, or fuel to land, much less any usable payload.

Alternative approaches to inexpensive spaceflight

Many studies have shown that regardless of selected technology, the most effective cost reduction technique is economies of scale. Merely launching a large total number reduces the manufacturing costs per vehicle, similar to how the mass production of automobiles brought about great increases in affordability.

Using this concept, some aerospace analysts believe the way to lower launch costs is the exact opposite of SSTO. Whereas reusable SSTOs would reduce per launch costs by making a reusable high-tech vehicle that launches frequently with low maintenance, the "mass production" approach views the technical advances as a source of the cost problem in the first place. By simply building and launching large quantities of rockets, and hence launching a large volume of payload, costs can be brought down. This approach was attempted in the late 1970s, early 1980s in West Germany with the Democratic Republic of the Congo-based OTRAG rocket.

This is somewhat similar to the approach some previous systems have taken, using simple engine systems with "low-tech" fuels, as the Russian and Chinese space programs still do.

An alternative to scale is to make the discarded stages practically reusable: this was the original design goal of the Space Shuttle phase B studies, and is currently pursued by the SpaceX reusable launch system development program with their Falcon 9, Falcon Heavy, and Starship, and Blue Origin using New Glenn.