Interstellar travel

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A Bussard ramjet, one of many possible methods that could serve as propulsion of a starship.

Interstellar travel is the term used for hypothetical crewed or uncrewed travel between stars or planetary systems. Interstellar travel will be much more difficult than interplanetary spaceflight; the distances between the planets in the Solar System are less than 30 astronomical units (AU)—whereas the distances between stars are typically hundreds of thousands of AU, and usually expressed in light-years. Because of the vastness of those distances, interstellar travel would require a high percentage of the speed of light; huge travel time, lasting from decades to millennia or longer; or a combination of both.

The speeds required for interstellar travel in a human lifetime far exceed what current methods of spacecraft propulsion can provide. Even with a hypothetically perfectly efficient propulsion system, the kinetic energy corresponding to those speeds is enormous by today's standards of energy production. Moreover, collisions by the spacecraft with cosmic dust and gas can produce very dangerous effects both to passengers and the spacecraft itself.

A number of strategies have been proposed to deal with these problems, ranging from giant arks that would carry entire societies and ecosystems, to microscopic space probes. Many different spacecraft propulsion systems have been proposed to give spacecraft the required speeds, including nuclear propulsion, beam-powered propulsion, and methods based on speculative physics.^[1]

For both crewed and uncrewed interstellar travel, considerable technological and economic challenges need to be met. Even the most optimistic views about interstellar travel see it as only being feasible decades from now—the more common view is that it is a century or more away. However, in spite of the challenges, if interstellar travel should ever be realized, then a wide range of scientific benefits can be expected.^[2]

Most interstellar travel concepts require a developed space logistics system capable of moving millions of tons to a construction / operating location, and most would require gigawatt-scale power for construction or power (such as Star Wisp or Light Sail type concepts). Such a system could grow organically if space-based solar power became a significant component of Earth's energy mix. Consumer demand for a multi-terawatt system would automatically create the necessary multi-million ton/year logistical system.^[3]

Challenges

Interstellar distances

Distances between the planets in the Solar System are often measured in astronomical units (AU), defined as the average distance between the Sun and Earth, some 1.5×10⁸ kilometers (93 million miles). Venus, the closest other planet to Earth is (at closest approach) 0.28 AU away. Neptune, the farthest planet from the Sun, is 29.8 AU away. As of January 2018, Voyager 1, the farthest man-made object from Earth, is 141.5 AU away.^[4]

The closest known star Proxima Centauri, however, is some 268,332 AU away, or over 9,000 times farther away than Neptune.

Object	A.U.	light time
Moon	0.0026	1.3 seconds
Sun	1	8 minutes
Venus (nearest planet)	0.28	2.41 minutes
Neptune (farthest planet)	29.8	4.1 hours
Voyager 1	141.5	19.61 hours
Proxima Centauri (nearest star and exoplanet)	268,332	4.24 years

Because of this, distances between stars are usually expressed in light-years, defined as the distance that a ray of light travels in a year. Light in a vacuum travels around 300,000 kilometres (186,000 mi) per second, so this is some 9.461×10¹² kilometers (5.879 trillion miles) or 1 light-year (63,241 AU) in a year. Proxima Centauri is 4.243 light-years away.

Another way of understanding the vastness of interstellar distances is by scaling: One of the closest stars to the Sun, Alpha Centauri A (a Sun-like star), can be pictured by scaling down the Earth–Sun distance to one meter (3.28 ft). On this scale, the distance to Alpha Centauri A would be 276 kilometers (171 miles).

The fastest outward-bound spacecraft yet sent, Voyager 1, has covered 1/600 of a light-year in 30 years and is currently moving at 1/18,000 the speed of light. At this rate, a journey to Proxima Centauri would take 80,000 years.^[5]

Required energy

A significant factor contributing to the difficulty is the energy that must be supplied to obtain a reasonable travel time. A lower bound for the required energy is the kinetic energy ${\displaystyle K={\tfrac {1}{2}}mv^{2}}$ where ${\displaystyle m}$ is the final mass. If deceleration on arrival is desired and cannot be achieved by any means other than the engines of the ship, then the lower bound for the required energy is doubled to ${\displaystyle mv^{2}}$.^[6]

The velocity for a manned round trip of a few decades to even the nearest star is several thousand times greater than those of present space vehicles. This means that due to the ${\displaystyle v^{2}}$ term in the kinetic energy formula, millions of times as much energy is required. Accelerating one ton to one-tenth of the speed of light requires at least 450 petajoules or 4.50×10¹⁷ joules or 125 terawatt-hours^[7] (world energy consumption 2008 was 143,851 terawatt-hours),^{[citation needed]} without factoring in efficiency of the propulsion mechanism. This energy has to be generated onboard from stored fuel, harvested from the interstellar medium, or projected over immense distances.

Interstellar medium

A knowledge of the properties of the interstellar gas and dust through which the vehicle must pass is essential for the design of any interstellar space mission.^[8] A major issue with traveling at extremely high speeds is that interstellar dust may cause considerable damage to the craft, due to the high relative speeds and large kinetic energies involved. Various shielding methods to mitigate this problem have been proposed.^[9] Larger objects (such as macroscopic dust grains) are far less common, but would be much more destructive. The risks of impacting such objects, and methods of mitigating these risks, have been discussed in the literature, but many unknowns remain^[10] and, owing to the inhomogeneous distribution of interstellar matter around the Sun, will depend on direction travelled.^[8] Although a high density interstellar medium may cause difficulties for many interstellar travel concepts, interstellar ramjets, and some proposed concepts for decelerating interstellar spacecraft, would actually benefit from a denser interstellar medium.^[8]

Hazards

The crew of an interstellar ship would face several significant hazards, including the psychological effects of long-term isolation, the effects of exposure to ionizing radiation, and the physiological effects of weightlessness to the muscles, joints, bones, immune system, and eyes. There also exists the risk of impact by micrometeoroids and other space debris. These risks represent challenges that have yet to be overcome.^[11]

Wait calculation

It has been argued that an interstellar mission that cannot be completed within 50 years should not be started at all. Instead, assuming that a civilization is still on an increasing curve of propulsion system velocity and not yet having reached the limit, the resources should be invested in designing a better propulsion system. This is because a slow spacecraft would probably be passed by another mission sent later with more advanced propulsion (the incessant obsolescence postulate).^[12] On the other hand, Andrew Kennedy has shown that if one calculates the journey time to a given destination as the rate of travel speed derived from growth (even exponential growth) increases, there is a clear minimum in the total time to that destination from now.^[13] Voyages undertaken before the minimum will be overtaken by those who leave at the minimum, whereas those who leave after the minimum will never overtake those who left at the minimum.

Prime targets for interstellar travel

There are 59 known stellar systems within 40 light years of the Sun, containing 81 visible stars. The following could be considered prime targets for interstellar missions:^[12]

System	Distance (ly)	Remarks
Alpha Centauri	4.3	Closest system. Three stars (G2, K1, M5). Component A is similar to the Sun (a G2 star). On August 24, 2016, the discovery of an Earth-size exoplanet (Proxima Centauri b) orbiting in the habitable zone of Proxima Centauri was announced.
Barnard's Star	6	Small, low-luminosity M5 red dwarf. Second closest to Solar System.
Sirius	8.7	Large, very bright A1 star with a white dwarf companion.
Epsilon Eridani	10.8	Single K2 star slightly smaller and colder than the Sun. It has two asteroid belts, might have a giant and one much smaller planet,[14] and may possess a Solar-System-type planetary system.
Tau Ceti	11.8	Single G8 star similar to the Sun. High probability of possessing a Solar-System-type planetary system: current evidence shows 5 planets with potentially two in the habitable zone.
Wolf 1061	~14	Wolf 1061 c is 4.3 times the size of Earth; it may have rocky terrain. It also sits within the ‘Goldilocks’ zone where it might be possible for liquid water to exist.[15]
Gliese 581 planetary system	20.3	Multiple planet system. The unconfirmed exoplanet Gliese 581g and the confirmed exoplanet Gliese 581d are in the star's habitable zone.
Gliese 667C	22	A system with at least six planets. A record-breaking three of these planets are super-Earths lying in the zone around the star where liquid water could exist, making them possible candidates for the presence of life.[16]
Vega	25	A very young system possibly in the process of planetary formation.[17]
TRAPPIST-1	39	A recently discovered system which boasts 7 Earth-like planets, some of which may have liquid water. The discovery is a major advancement in finding a habitable planet and in finding a planet that could support life.

Existing and near-term astronomical technology is capable of finding planetary systems around these objects, increasing their potential for exploration

Proposed methods

Slow, uncrewed probes

Slow interstellar missions based on current and near-future propulsion technologies are associated with trip times starting from about one hundred years to thousands of years. These missions consist of sending a robotic probe to a nearby star for exploration, similar to interplanetary probes such as used in the Voyager program.^[18] By taking along no crew, the cost and complexity of the mission is significantly reduced although technology lifetime is still a significant issue next to obtaining a reasonable speed of travel. Proposed concepts include Project Daedalus, Project Icarus, Project Dragonfly, Project Longshot.,^[19] and more recently Breakthrough Starshot.^[20]

Fast, uncrewed probes

Nanoprobes

Near-lightspeed nano spacecraft might be possible within the near future built on existing microchip technology with a newly developed nanoscale thruster. Researchers at the University of Michigan are developing thrusters that use nanoparticles as propellant. Their technology is called "nanoparticle field extraction thruster", or nanoFET. These devices act like small particle accelerators shooting conductive nanoparticles out into space.^[21]

Michio Kaku, a theoretical physicist, has suggested that clouds of "smart dust" be sent to the stars, which may become possible with advances in nanotechnology. Kaku also notes that a large number of nanoprobes would need to be sent due to the vulnerability of very small probes to be easily deflected by magnetic fields, micrometeorites and other dangers to ensure the chances that at least one nanoprobe will survive the journey and reach the destination.^[22]

Given the light weight of these probes, it would take much less energy to accelerate them. With onboard solar cells, they could continually accelerate using solar power. One can envision a day when a fleet of millions or even billions of these particles swarm to distant stars at nearly the speed of light and relay signals back to Earth through a vast interstellar communication network.

As a near-term solution, small, laser-propelled interstellar probes, based on current CubeSat technology were proposed in the context of Project Dragonfly.^[19]

Slow, manned missions

In crewed missions, the duration of a slow interstellar journey presents a major obstacle and existing concepts deal with this problem in different ways.^[23] They can be distinguished by the "state" in which humans are transported on-board of the spacecraft.

Generation ships

A generation ship (or world ship) is a type of interstellar ark in which the crew that arrives at the destination is descended from those who started the journey. Generation ships are not currently feasible because of the difficulty of constructing a ship of the enormous required scale and the great biological and sociological problems that life aboard such a ship raises.^{[24][25][26][27]}

Suspended animation

Scientists and writers have postulated various techniques for suspended animation. These include human hibernation and cryonic preservation. Although neither is currently practical, they offer the possibility of sleeper ships in which the passengers lie inert for the long duration of the voyage.^[28]

Frozen embryos

A robotic interstellar mission carrying some number of frozen early stage human embryos is another theoretical possibility. This method of space colonization requires, among other things, the development of an artificial uterus, the prior detection of a habitable terrestrial planet, and advances in the field of fully autonomous mobile robots and educational robots that would replace human parents.^[29]

Island hopping through interstellar space

Interstellar space is not completely empty; it contains trillions of icy bodies ranging from small asteroids (Oort cloud) to possible rogue planets. There may be ways to take advantage of these resources for a good part of an interstellar trip, slowly hopping from body to body or setting up waystations along the way.^[30]

Fast missions

If a spaceship could average 10 percent of light speed (and decelerate at the destination, for manned missions), this would be enough to reach Proxima Centauri in forty years. Several propulsion concepts have been proposed ^[31] that might be eventually developed to accomplish this (see also the section below on propulsion methods), but none of them are ready for near-term (few decades) developments at acceptable cost.

Time dilation

Assuming faster-than-light travel is impossible, one might conclude that a human can never make a round-trip farther from Earth than 20 light years if the traveler is active between the ages of 20 and 60. A traveler would never be able to reach more than the very few star systems that exist within the limit of 20 light years from Earth. This, however, fails to take into account relativistic time dilation.^[32] Clocks aboard an interstellar ship would run slower than Earth clocks, so if a ship's engines were capable of continuously generating around 1 g of acceleration (which is comfortable for humans), the ship could reach almost anywhere in the galaxy and return to Earth within 40 years ship-time (see diagram). Upon return, there would be a difference between the time elapsed on the astronaut's ship and the time elapsed on Earth.
For example, a spaceship could travel to a star 32 light-years away, initially accelerating at a constant 1.03g (i.e. 10.1 m/s²) for 1.32 years (ship time), then stopping its engines and coasting for the next 17.3 years (ship time) at a constant speed, then decelerating again for 1.32 ship-years, and coming to a stop at the destination. After a short visit, the astronaut could return to Earth the same way. After the full round-trip, the clocks on board the ship show that 40 years have passed, but according to those on Earth, the ship comes back 76 years after launch.

From the viewpoint of the astronaut, onboard clocks seem to be running normally. The star ahead seems to be approaching at a speed of 0.87 light years per ship-year. The universe would appear contracted along the direction of travel to half the size it had when the ship was at rest; the distance between that star and the Sun would seem to be 16 light years as measured by the astronaut.

At higher speeds, the time on board will run even slower, so the astronaut could travel to the center of the Milky Way (30,000 light years from Earth) and back in 40 years ship-time. But the speed according to Earth clocks will always be less than 1 light year per Earth year, so, when back home, the astronaut will find that more than 60 thousand years will have passed on Earth.

Constant acceleration

This plot shows a ship capable of 1-g (10 m/s² or about 1.0 ly/y²) "felt" or proper-acceleration^[33] can go far, except for the problem of accelerating on-board propellant.

Regardless of how it is achieved, a propulsion system that could produce acceleration continuously from departure to arrival would be the fastest method of travel. A constant acceleration journey is one where the propulsion system accelerates the ship at a constant rate for the first half of the journey, and then decelerates for the second half, so that it arrives at the destination stationary relative to where it began. If this were performed with an acceleration similar to that experienced at the Earth's surface, it would have the added advantage of producing artificial "gravity" for the crew. Supplying the energy required, however, would be prohibitively expensive with current technology.^[34]

From the perspective of a planetary observer, the ship will appear to accelerate steadily at first, but then more gradually as it approaches the speed of light (which it cannot exceed). It will undergo hyperbolic motion.^[35] The ship will be close to the speed of light after about a year of accelerating and remain at that speed until it brakes for the end of the journey.

From the perspective of an onboard observer, the crew will feel a gravitational field opposite the engine's acceleration, and the universe ahead will appear to fall in that field, undergoing hyperbolic motion. As part of this, distances between objects in the direction of the ship's motion will gradually contract until the ship begins to decelerate, at which time an onboard observer's experience of the gravitational field will be reversed.

When the ship reaches its destination, if it were to exchange a message with its origin planet, it would find that less time had elapsed on board than had elapsed for the planetary observer, due to time dilation and length contraction.

The result is an impressively fast journey for the crew.

Propulsion

Rocket concepts

All rocket concepts are limited by the rocket equation, which sets the characteristic velocity available as a function of exhaust velocity and mass ratio, the ratio of initial (M₀, including fuel) to final (M₁, fuel depleted) mass.

Very high specific power, the ratio of thrust to total vehicle mass, is required to reach interstellar targets within sub-century time-frames.^[36] Some heat transfer is inevitable and a tremendous heating load must be adequately handled.

Thus, for interstellar rocket concepts of all technologies, a key engineering problem (seldom explicitly discussed) is limiting the heat transfer from the exhaust stream back into the vehicle.^[37]

Ion engine

A type of electric propulsion, spacecraft such as Dawn use an ion engine. In an ion engine, electric power is used to create charged particles of the propellant, usually the gas xenon, and accelerate them to extremely high velocities. The exhaust velocity of conventional rockets is limited by the chemical energy stored in the fuel’s molecular bonds, which limits the thrust to about 5 km/s. This gives them high power^{[clarification needed]} (for lift-off from Earth, for example) but limits the top speed. By contrast, ion engines have low force, but the top speed in principle is limited only by the electrical power available on the spacecraft and on the gas ions being accelerated. The exhaust speed of the charged particles range from 15 km/s to 35 km/s.^[38]

Nuclear fission powered

Fission-electric

Nuclear-electric or plasma engines, operating for long periods at low thrust and powered by fission reactors, have the potential to reach speeds much greater than chemically powered vehicles or nuclear-thermal rockets. Such vehicles probably have the potential to power Solar System exploration with reasonable trip times within the current century. Because of their low-thrust propulsion, they would be limited to off-planet, deep-space operation. Electrically powered spacecraft propulsion powered by a portable power-source, say a nuclear reactor, producing only small accelerations, would take centuries to reach for example 15% of the velocity of light, thus unsuitable for interstellar flight during a single human lifetime.^[39]

Fission-fragment

Fission-fragment rockets use nuclear fission to create high-speed jets of fission fragments, which are ejected at speeds of up to 12,000 km/s (7,500 mi/s). With fission, the energy output is approximately 0.1% of the total mass-energy of the reactor fuel and limits the effective exhaust velocity to about 5% of the velocity of light. For maximum velocity, the reaction mass should optimally consist of fission products, the "ash" of the primary energy source, so no extra reaction mass need be bookkept in the mass ratio.

Nuclear pulse

Modern Pulsed Fission Propulsion Concept.

Based on work in the late 1950s to the early 1960s, it has been technically possible to build spaceships with nuclear pulse propulsion engines, i.e. driven by a series of nuclear explosions. This propulsion system contains the prospect of very high specific impulse (space travel's equivalent of fuel economy) and high specific power.^[40]

Project Orion team member Freeman Dyson proposed in 1968 an interstellar spacecraft using nuclear pulse propulsion that used pure deuterium fusion detonations with a very high fuel-burnup fraction. He computed an exhaust velocity of 15,000 km/s and a 100,000-tonne space vehicle able to achieve a 20,000 km/s delta-v allowing a flight-time to Alpha Centauri of 130 years.^[41] Later studies indicate that the top cruise velocity that can theoretically be achieved by a Teller-Ulam thermonuclear unit powered Orion starship, assuming no fuel is saved for slowing back down, is about 8% to 10% of the speed of light (0.08-0.1c).^[42] An atomic (fission) Orion can achieve perhaps 3%-5% of the speed of light. A nuclear pulse drive starship powered by fusion-antimatter catalyzed nuclear pulse propulsion units would be similarly in the 10% range and pure matter-antimatter annihilation rockets would be theoretically capable of obtaining a velocity between 50% to 80% of the speed of light. In each case saving fuel for slowing down halves the maximum speed. The concept of using a magnetic sail to decelerate the spacecraft as it approaches its destination has been discussed as an alternative to using propellant, this would allow the ship to travel near the maximum theoretical velocity.^[43] Alternative designs utilizing similar principles include Project Longshot, Project Daedalus, and Mini-Mag Orion. The principle of external nuclear pulse propulsion to maximize survivable power has remained common among serious concepts for interstellar flight without external power beaming and for very high-performance interplanetary flight.

In the 1970s the Nuclear Pulse Propulsion concept further was refined by Project Daedalus by use of externally triggered inertial confinement fusion, in this case producing fusion explosions via compressing fusion fuel pellets with high-powered electron beams. Since then, lasers, ion beams, neutral particle beams and hyper-kinetic projectiles have been suggested to produce nuclear pulses for propulsion purposes.^[44]

A current impediment to the development of any nuclear-explosion-powered spacecraft is the 1963 Partial Test Ban Treaty, which includes a prohibition on the detonation of any nuclear devices (even non-weapon based) in outer space. This treaty would, therefore, need to be renegotiated, although a project on the scale of an interstellar mission using currently foreseeable technology would probably require international cooperation on at least the scale of the International Space Station.

Nuclear fusion rockets

Daedalus interstellar vehicle.

Fusion rocket starships, powered by nuclear fusion reactions, should conceivably be able to reach speeds of the order of 10% of that of light, based on energy considerations alone. In theory, a large number of stages could push a vehicle arbitrarily close to the speed of light.^[45] These would "burn" such light element fuels as deuterium, tritium, ³He, ¹¹B, and ⁷Li. Because fusion yields about 0.3–0.9% of the mass of the nuclear fuel as released energy, it is energetically more favorable than fission, which releases  less than 0.1% of the fuel's mass-energy. The maximum exhaust velocities potentially energetically available are correspondingly higher than for fission, typically 4–10% of c. However, the most easily achievable fusion reactions release a large fraction of their energy as high-energy neutrons, which are a significant source of energy loss. Thus, although these concepts seem to offer the best (nearest-term) prospects for travel to the nearest stars within a (long) human lifetime, they still involve massive technological and engineering difficulties, which may turn out to be intractable for decades or centuries.

Early studies include Project Daedalus, performed by the British Interplanetary Society in 1973–1978, and Project Longshot, a student project sponsored by NASA and the US Naval Academy, completed in 1988. Another fairly detailed vehicle system, "Discovery II",^[46] designed and optimized for crewed Solar System exploration, based on the D³He reaction but using hydrogen as reaction mass, has been described by a team from NASA's Glenn Research Center. It achieves characteristic velocities of  greater than 300 km/s with an acceleration of ~1.7•10⁻³ g, with a ship initial mass of ~1700 metric tons, and payload fraction above 10%. Although these are still far short of the requirements for interstellar travel on human timescales, the study seems to represent a reasonable benchmark towards what may be approachable within several decades, which is not impossibly beyond the current state-of-the-art. Based on the concept's 2.2% burnup fraction it could achieve a pure fusion product exhaust velocity of ~3,000 km/s.

Antimatter rockets

An antimatter rocket would have a far higher energy density and specific impulse than any other proposed class of rocket.^[31] If energy resources and efficient production methods are found to make antimatter in the quantities required and store^[47][48] it safely, it would be theoretically possible to reach speeds of several tens of percent that of light.^[31] Whether antimatter propulsion could lead to the higher speeds (>90% that of light) at which relativistic time dilation would become more noticeable, thus making time pass at a slower rate for the travelers as perceived by an outside observer, is doubtful owing to the large quantity of antimatter that would be required.^[31]

Speculating that production and storage of antimatter should become feasible, two further issues need to be considered. First, in the annihilation of antimatter, much of the energy is lost as high-energy gamma radiation, and especially also as neutrinos, so that only about 40% of mc² would actually be available if the antimatter were simply allowed to annihilate into radiations thermally.^[31] Even so, the energy available for propulsion would be substantially higher than the ~1% of mc² yield of nuclear fusion, the next-best rival candidate.

Second, heat transfer from the exhaust to the vehicle seems likely to transfer enormous wasted energy into the ship (e.g. for 0.1g ship acceleration, approaching 0.3 trillion watts per ton of ship mass), considering the large fraction of the energy that goes into penetrating gamma rays. Even assuming shielding was provided to protect the payload (and passengers on a crewed vehicle), some of the energy would inevitably heat the vehicle, and may thereby prove a limiting factor if useful accelerations are to be achieved.

More recently, Friedwardt Winterberg proposed that a matter-antimatter GeV gamma ray laser photon rocket is possible by a relativistic proton-antiproton pinch discharge, where the recoil from the laser beam is transmitted by the Mössbauer effect to the spacecraft.^[49]

Rockets with an external energy source

Rockets deriving their power from external sources, such as a laser, could replace their internal energy source with an energy collector, potentially reducing the mass of the ship greatly and allowing much higher travel speeds. Geoffrey A. Landis has proposed for an interstellar probe, with energy supplied by an external laser from a base station powering an Ion thruster.^[50]

Non-rocket concepts

A problem with all traditional rocket propulsion methods is that the spacecraft would need to carry its fuel with it, thus making it very massive, in accordance with the rocket equation. Several concepts attempt to escape from this problem:^[31][51]

Interstellar ramjets

In 1960, Robert W. Bussard proposed the Bussard ramjet, a fusion rocket in which a huge scoop would collect the diffuse hydrogen in interstellar space, "burn" it on the fly using a proton–proton chain reaction, and expel it out of the back. Later calculations with more accurate estimates suggest that the thrust generated would be less than the drag caused by any conceivable scoop design.^{[citation needed]} Yet the idea is attractive because the fuel would be collected en route (commensurate with the concept of energy harvesting), so the craft could theoretically accelerate to near the speed of light. The limitation is due to the fact that the reaction can only accelerate the propellant to 0.12c. Thus the drag of catching interstellar dust and the thrust of accelerating that same dust to 0.12c would be the same when the speed is 0.12c, preventing further acceleration.

Beamed propulsion

This diagram illustrates Robert L. Forward's scheme for slowing down an interstellar light-sail at the star system destination.

A light sail or magnetic sail powered by a massive laser or particle accelerator in the home star system could potentially reach even greater speeds than rocket- or pulse propulsion methods, because it would not need to carry its own reaction mass and therefore would only need to accelerate the craft's payload. Robert L. Forward proposed a means for decelerating an interstellar light sail in the destination star system without requiring a laser array to be present in that system. In this scheme, a smaller secondary sail is deployed to the rear of the spacecraft, whereas the large primary sail is detached from the craft to keep moving forward on its own. Light is reflected from the large primary sail to the secondary sail, which is used to decelerate the secondary sail and the spacecraft payload.^[52] In 2002, Geoffrey A. Landis of NASA's Glen Research center also proposed a laser-powered, propulsion, sail ship that would host a diamond sail (of a few nanometers thick) powered with the use of solar energy.^[53] With this proposal, this interstellar ship would, theoretically, be able to reach 10 percent the speed of light.

A magnetic sail could also decelerate at its destination without depending on carried fuel or a driving beam in the destination system, by interacting with the plasma found in the solar wind of the destination star and the interstellar medium.^[54][55]

The following table lists some example concepts using beamed laser propulsion as proposed by the physicist Robert L. Forward:^[56]

Mission	Laser Power	Vehicle Mass	Acceleration	Sail Diameter	Maximum Velocity (% of the speed of light)
1. Flyby – Alpha Centauri, 40 years
outbound stage	65 GW	1 t	0.036 g	3.6 km	11% @ 0.17 ly
2. Rendezvous – Alpha Centauri, 41 years
outbound stage	7,200 GW	785 t	0.005 g	100 km	21% @ 4.29 ly[dubious – discuss]
deceleration stage	26,000 GW	71 t	0.2 g	30 km	21% @ 4.29 ly
3. Manned – Epsilon Eridani, 51 years (including 5 years exploring star system)
outbound stage	75,000,000 GW	78,500 t	0.3 g	1000 km	50% @ 0.4 ly
deceleration stage	21,500,000 GW	7,850 t	0.3 g	320 km	50% @ 10.4 ly
return stage	710,000 GW	785 t	0.3 g	100 km	50% @ 10.4 ly
deceleration stage	60,000 GW	785 t	0.3 g	100 km	50% @ 0.4 ly

Interstellar travel catalog to use photogravitational assists for a full stop

The following table is based on work by Heller, Hippke and Kervella.^[57]

Name	Travel time (yr)	Distance (ly)	Luminosity (L☉)
Sirius A	68.90	8.58	24.20
α Centauri A	101.25	4.36	1.52
α Centauri B	147.58	4.36	0.50
Procyon A	154.06	11.44	6.94
Vega	167.39	25.02	50.05
Altair	176.67	16.69	10.70
Fomalhaut A	221.33	25.13	16.67
Denebola	325.56	35.78	14.66
Castor A	341.35	50.98	49.85
Epsilon Eridiani	363.35	10.50	0.50

• Successive assists at α Cen A and B could allow travel times to 75 yr to both stars.
• Lightsail has a nominal mass-to-surface ratio (σ_nom) of 8.6×10⁻⁴ gram m⁻² for a nominal graphene-class sail.
• Area of the Lightsail, about 10⁵ m² = (316 m)²
• Velocity up to 37,300 km s⁻¹ (12.5% c)

Pre-accelerated fuel

Achieving start-stop interstellar trip times of less than a human lifetime require mass-ratios of between 1,000 and 1,000,000, even for the nearer stars. This could be achieved by multi-staged vehicles on a vast scale.^[45] Alternatively large linear accelerators could propel fuel to fission propelled space-vehicles, avoiding the limitations of the Rocket equation.^[58]

Theoretical concepts

Faster-than-light travel

Artist's depiction of a hypothetical Wormhole Induction Propelled Spacecraft, based loosely on the 1994 "warp drive" paper of Miguel Alcubierre. Credit: NASA CD-98-76634 by Les Bossinas.

Scientists and authors have postulated a number of ways by which it might be possible to surpass the speed of light, but even the most serious-minded of these are highly speculative.^[59]

It is also debatable whether faster-than-light travel is physically possible, in part because of causality concerns: travel faster than light may, under certain conditions, permit travel backwards in time within the context of special relativity.^[60] Proposed mechanisms for faster-than-light travel within the theory of general relativity require the existence of exotic matter^[59] and it is not known if this could be produced in sufficient quantity.

Alcubierre drive

In physics, the Alcubierre drive is based on an argument, within the framework of general relativity and without the introduction of wormholes, that it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed by a local expansion of spacetime behind the spaceship and an opposite contraction in front of it.^[61] Nevertheless, this concept would require the spaceship to incorporate a region of exotic matter, or hypothetical concept of negative mass.^[61]

Artificial black hole

A theoretical idea for enabling interstellar travel is by propelling a starship by creating an artificial black hole and using a parabolic reflector to reflect its Hawking radiation. Although beyond current technological capabilities, a black hole starship offers some advantages compared to other possible methods. Getting the black hole to act as a power source and engine also requires a way to convert the Hawking radiation into energy and thrust. One potential method involves placing the hole at the focal point of a parabolic reflector attached to the ship, creating forward thrust. A slightly easier, but less efficient method would involve simply absorbing all the gamma radiation heading towards the fore of the ship to push it onwards, and let the rest shoot out the back.^[62][63][64]

Wormholes

Wormholes are conjectural distortions in spacetime that theorists postulate could connect two arbitrary points in the universe, across an Einstein–Rosen Bridge. It is not known whether wormholes are possible in practice. Although there are solutions to the Einstein equation of general relativity that allow for wormholes, all of the currently known solutions involve some assumption, for example the existence of negative mass, which may be unphysical.^[65] However, Cramer et al. argue that such wormholes might have been created in the early universe, stabilized by cosmic string.^[66] The general theory of wormholes is discussed by Visser in the book Lorentzian Wormholes.^[67]

Hyperdrive

If the conjecture of Felber is correct, ie any mass moving at 57.7% C generates an anti-gravity beam then an Orion drive may be capable of acting as the initial boost, and as-yet-undiscovered technology based on theorized anti-matter repulsion effects under exotic conditions such as entanglement that directly manipulates space-time to open a window into hyperspace. This could feasibly exploit physics in extradimensional space to travel very quickly through the galaxy. The side effect here is that due to the need for a second vehicle to slow down it would be a one-way trip, although other concepts such as a solar sail could be used to slow down near the destination. ^[68] All the components could be feasibly re-used and also provide very effective SETI targets if other species are using this technology. It is also likely that hyperdrive jumps could be limited to discrete areas such as near stars due to gravitation wells.

Designs and studies

Enzmann starship

The Enzmann starship, as detailed by G. Harry Stine in the October 1973 issue of Analog, was a design for a future starship, based on the ideas of Robert Duncan-Enzmann. The spacecraft itself as proposed used a 12,000,000 ton ball of frozen deuterium to power 12–24 thermonuclear pulse propulsion units. Twice as long as the Empire State Building and assembled in-orbit, the spacecraft was part of a larger project preceded by interstellar probes and telescopic observation of target star systems.^[69]

Project Hyperion

Project Hyperion, one of the projects of Icarus Interstellar.^[70]

NASA research

NASA has been researching interstellar travel since its formation, translating important foreign language papers and conducting early studies on applying fusion propulsion, in the 1960s, and laser propulsion, in the 1970s, to interstellar travel.

The NASA Breakthrough Propulsion Physics Program (terminated in FY 2003 after a 6-year, $1.2-million study, because "No breakthroughs appear imminent.")^[71] identified some breakthroughs that are needed for interstellar travel to be possible.^[72]

Geoffrey A. Landis of NASA's Glenn Research Center states that a laser-powered interstellar sail ship could possibly be launched within 50 years, using new methods of space travel. "I think that ultimately we're going to do it, it's just a question of when and who," Landis said in an interview. Rockets are too slow to send humans on interstellar missions. Instead, he envisions interstellar craft with extensive sails, propelled by laser light to about one-tenth the speed of light. It would take such a ship about 43 years to reach Alpha Centauri if it passed through the system. Slowing down to stop at Alpha Centauri could increase the trip to 100 years,^[73] whereas a journey without slowing down raises the issue of making sufficiently accurate and useful observations and measurements during a fly-by.

100 Year Starship study

The 100 Year Starship (100YSS) is the name of the overall effort that will, over the next century, work toward achieving interstellar travel. The effort will also go by the moniker 100YSS. The 100 Year Starship study is the name of a one-year project to assess the attributes of and lay the groundwork for an organization that can carry forward the 100 Year Starship vision.

Harold ("Sonny") White^[74] from NASA's Johnson Space Center is a member of Icarus Interstellar,^[75] the nonprofit foundation whose mission is to realize interstellar flight before the year 2100. At the 2012 meeting of 100YSS, he reported using a laser to try to warp spacetime by 1 part in 10 million with the aim of helping to make interstellar travel possible.^[76]

Other designs

• Project Orion, manned interstellar ship (1958–1968).
• Project Daedalus, unmanned interstellar probe (1973–1978).
• Starwisp, unmanned interstellar probe (1985).^[77]
• Project Longshot, unmanned interstellar probe (1987–1988).
• Starseed/launcher, fleet of unmanned interstellar probes (1996)
• Project Valkyrie, manned interstellar ship (2009)
• Project Icarus, unmanned interstellar probe (2009–2014).
• Sun-diver, unmanned interstellar probe^[78]
• Breakthrough Starshot, fleet of unmanned interstellar probes, announced in April 12, 2016.^[79][80][81]

Non-profit organizations

A few organisations dedicated to interstellar propulsion research and advocacy for the case exist worldwide. These are still in their infancy, but are already backed up by a membership of a wide variety of scientists, students and professionals.

• 100 Year Starship ^[82]
• Icarus Interstellar ^[75]
• Tau Zero Foundation (USA) ^[83]
• Initiative for Interstellar Studies (UK) ^[84]
• Fourth Millennium Foundation (Belgium) ^[85]
• Space Development Cooperative (Canada) ^[86]

Feasibility

The energy requirements make interstellar travel very difficult. It has been reported that at the 2008 Joint Propulsion Conference, multiple experts opined that it was improbable that humans would ever explore beyond the Solar System.^[87] Brice N. Cassenti, an associate professor with the Department of Engineering and Science at Rensselaer Polytechnic Institute, stated that at least 100 times the total energy output of the entire world [in a given year] would be required to send a probe to the nearest star.^[87]

Astrophysicist Sten Odenwald stated that the basic problem is that through intensive studies of thousands of detected exoplanets, most of the closest destinations within 50 light years do not yield Earth-like planets in the star's habitable zones.^[88] Given the multi-trillion-dollar expense of some of the proposed technologies, travelers will have to spend up to 200 years traveling at 20% the speed of light to reach the best known destinations. Moreover, once the travelers arrive at their destination (by any means), they will not be able to travel down to the surface of the target world and set up a colony unless the atmosphere is non-lethal. The prospect of making such a journey, only to spend the rest of the colony's life inside a sealed habitat and venturing outside in a spacesuit, may eliminate many prospective targets from the list.

Moving at a speed close to the speed of light and encountering even a tiny stationary object like a grain of sand will have fatal consequences. For example, a gram of matter moving at 90% of the speed of light contains a kinetic energy corresponding to a small nuclear bomb (around 30kt TNT).

Interstellar missions not for human benefit

Explorative high-speed missions to Alpha Centauri, as planned for by the Breakthrough Starshot initiative, are projected to be realizable within the 21st century.^[89] It is alternatively possible to plan for unmanned slow-cruising missions taking millennia to arrive. These probes would not be for human benefit in the sense that one can not foresee whether there would be anybody around on earth interested in then back-transmitted science data. An example would be the Genesis mission,^[90] which aims to bring unicellular life, in the spirit of directed panspermia, to habitable but otherwise barren planets.^[91] Comparatively slow cruising Genesis probes, with a typical speed of ${\displaystyle c/300}$, corresponding to about ${\displaystyle 1000\,{\mbox{km/s}}}$, can be decelerated using a magnetic sail. Unmanned missions not for human benefit would hence be feasible ^[92]

Discovery of Earth-Like planets

In February 2017, NASA has announced the discovery of 7 Earth-like planets in the TRAPPIST-1 system orbiting an ultra-cool dwarf star 40 light-years away from our solar system.^[93] NASA's Spitzer Space Telescope has revealed the first known system of seven Earth-size planets around a single star. Three of these planets are firmly located in the habitable zone, the area around the parent star where a rocky planet is most likely to have liquid water. The discovery sets a new record for greatest number of habitable-zone planets found around a single star outside our solar system. All of these seven planets could have liquid water – the key to life as we know it – under the right atmospheric conditions, but the chances are highest with the three in the habitable zone.

Bekenstein bound

From Wikipedia, the free encyclopedia

In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.^[1] It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In computer science, this implies that there is a maximum information-processing rate (Bremermann's limit) for a physical system that has a finite size and energy, and that a Turing machine with finite physical dimensions and unbounded memory is not physically possible.

Upon exceeding the Bekenstein bound a storage medium would collapse into a black hole.^[2] This finds parallels with the concept of a kugelblitz, a concentration of light or radiation so intense that its energy forms an event horizon and becomes self-trapped: according to general relativity and the equivalence of mass and energy.

Equations

The universal form of the bound was originally found by Jacob Bekenstein as the inequality^[1][3][4]

${\displaystyle S\leq {\frac {2\pi kRE}{\hbar c}}}$

where S is the entropy, k is Boltzmann's constant, R is the radius of a sphere that can enclose the given system, E is the total mass–energy including any rest masses, ħ is the reduced Planck constant, and c is the speed of light. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain the gravitational constant G.
In informational terms, the bound is given by

${\displaystyle I\leq {\frac {2\pi RE}{\hbar c\ln 2}}}$

where I is the information expressed in number of bits contained in the quantum states in the sphere. The ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states.^[5] Using mass energy equivalence, the informational limit may be reformulated as

${\displaystyle I\leq {\frac {2\pi cRm}{\hbar \ln 2}}\approx 2.5769082\times 10^{43}mR}$

where ${\displaystyle m}$ is the mass of the system in kilograms, and the radius ${\displaystyle R}$ is expressed in meters.

Origins

Bekenstein derived the bound from heuristic arguments involving black holes. If a system exists that violates the bound, i.e., by having too much entropy, Bekenstein argued that it would be possible to violate the second law of thermodynamics by lowering it into a black hole. In 1995, Ted Jacobson demonstrated that the Einstein field equations (i.e., general relativity) can be derived by assuming that the Bekenstein bound and the laws of thermodynamics are true.^[6][7] However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.^{[3][4][8][9][10][11][12][13][14][15][16]}

Examples

Black holes

It happens that the Bekenstein-Hawking Boundary Entropy of three-dimensional black holes exactly saturates the bound

${\displaystyle S={\frac {kA}{4}}}$

where A is the two-dimensional area of the black hole's event horizon in units of the Planck area, ${\displaystyle \hbar G/c^{3}}$.
The bound is closely associated with black hole thermodynamics, the holographic principle and the covariant entropy bound of quantum gravity, and can be derived from a conjectured strong form of the latter.

Human brain

An average human brain has a mass of 1.5 kg and a volume of 1260 cm³. If the brain is approximated by a sphere, then the radius will be 6.7 cm.

The informational Bekenstein bound will be ${\displaystyle \approx 2.6\times 10^{42}}$ bits and represents the maximum information needed to perfectly recreate an average human brain down to the quantum level. This means that the number ${\displaystyle O=2^{I}}$ of states of the human brain must be less than ${\displaystyle \approx 10^{7.8\times 10^{41}}}$.

Tolman–Oppenheimer–Volkoff limit

From Wikipedia, the free encyclopedia

The Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Observations of GW170817, the first gravitational wave event due to merging neutron stars (which are thought to have collapsed into a black hole^[1] within a few seconds after merging^[2]), suggest that the limit is close to 2.17 solar masses.^[3][4][5][6] A neutron star in a binary pair has been measured to have a mass close to or slightly above this limit, 2.27+0.17
−0.15 M_☉.^[7] Earlier theoretical work placed the limit at approximately 1.5 to 3.0 solar masses,^[8] corresponding to an original stellar mass of 15 to 20 solar masses. In the case of a rigidly spinning neutron star, the mass limit is thought to increase by up to 18%.^[2]

History

The idea that there should be an absolute upper limit for the mass of a cold (as distinct from thermal pressure supported) self-gravitating body dates back to the work of Lev Landau. In 1932, he reasoned based on the Pauli exclusion principle. Pauli's principle shows that the fermionic particles in sufficiently compressed matter would be forced into energy states so high that their rest mass contribution would become negligible when compared with the relativistic kinetic contribution (RKC). RKC is determined just by the relevant quantum wavelength λ, which would be of the order of the mean interparticle separation. In terms of Planck units, with the reduced Planck constant ħ, the speed of light c, and the gravitational constant G all set equal to one, there will be a corresponding pressure given roughly by P = 1/λ⁴. That pressure must be balanced by the pressure needed to resist gravity. The pressure to resist gravity for a body of mass M will be given according to the virial theorem roughly by P³ = M²ρ⁴, where ρ is the density. This will be given by ρ = m/λ³, where m is the relevant mass per particle. It can be seen that the wavelength cancels out so that one obtains an approximate mass limit formula of the very simple form M = 1/m². From this, m can be taken to be given roughly by the proton mass. This even applies in the white dwarf case (that of the Chandrasekhar limit) for which the fermionic particles providing the pressure are electrons. This is because the mass density is provided by the nuclei in which the neutrons are at most about as numerous as the protons. Likewise the protons, for charge neutrality, must be exactly as numerous as the electrons outside.

In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7 solar masses, ^[9][10] which was less than the Chandrasekhar limit for white dwarfs. Taking account of the strong nuclear repulsion forces between neutrons, modern work leads to considerably higher estimates, in the range from approximately 1.5 to 3.0 solar masses.^[8] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known. The mass of the pulsar PSR J0348+0432, at 2.01±0.04 solar masses, puts an empirical lower bound on the TOV limit.

Applications

In a neutron star less massive than the limit, the weight of the star is balanced by short-range repulsive neutron-neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons, preventing collapse. If its mass is above the limit, the star will collapse to some denser form. It could form a black hole, or change composition and be supported in some other way (for example, by quark degeneracy pressure if it becomes a quark star). Because the properties of hypothetical, more exotic forms of degenerate matter are even more poorly known than those of neutron-degenerate matter, most astrophysicists assume, in the absence of evidence to the contrary, that a neutron star above the limit collapses directly into a black hole.

A black hole formed by the collapse of an individual star must have mass exceeding the Tolman–Oppenheimer–Volkoff limit. Theory predicts that because of mass loss during stellar evolution, a black hole formed from an isolated star of solar metallicity can have a mass of no more than approximately 10 solar masses.^{[11]:Fig. 16} Observationally, because of their large mass, relative faintness, and X-ray spectra, a number of massive objects in X-ray binaries are thought to be stellar black holes. These black hole candidates are estimated to have masses between 3 and 20 solar masses.^[12][13]

Chandrasekhar limit

The Chandrasekhar limit (/tʃʌndrəˈʃeɪkər/) is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M_☉ (2.765×10³⁰ kg).^[1][2][3]

White dwarfs resist gravitational collapse primarily through electron degeneracy pressure (compare main sequence stars, which resist collapse through thermal pressure). The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, a white dwarf with a mass greater than the limit is subject to further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole. Those with masses under the limit remain stable as white dwarfs.^[4] Collapse is not inevitable: most white dwarfs explode rather than undergo collapse.

The limit was named after Subrahmanyan Chandrasekhar, the Indian astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 20, in India by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E.C. Stoner for a uniform density star. Importantly, the existence of a limit, based the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. That the roles of Stoner and Anderson are often forgotten in the astronomy community has been noted.^{[5] [6]}

Physics

Radius–mass relations for a model white dwarf. The green curve uses the general pressure law for an ideal Fermi gas, while the blue curve is for a non-relativistic ideal Fermi gas. The black line marks the ultrarelativistic limit.

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons increases on compression, so pressure must be exerted on the electron gas to compress it, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture, relieving the pressure.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P = K₁ρ^5/3, where P is the pressure, ρ is the mass density, and K₁ is a constant. Solving the hydrostatic equation leads to a model white dwarf that is a polytrope of index 3/2 – and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.^[7]

As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, the equation of state takes the form P = K₂ρ^4/3. This yields a polytrope of index 3, which has a total mass, M_limit say, depending only on K₂.^[8]

For a fully relativistic treatment, the equation of state used interpolates between the equations P = K₁ρ^5/3 for small ρ and P = K₂ρ^4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at M_limit. This is the Chandrasekhar limit.^[9] The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. μ_e has been set equal to 2. Radius is measured in standard solar radii^[10] or kilometers, and mass in standard solar masses.

Calculated values for the limit vary depending on the nuclear composition of the mass.^[11] Chandrasekhar^{[12], eq. (36),[9], eq. (58),[13], eq. (43)} gives the following expression, based on the equation of state for an ideal Fermi gas:

${\displaystyle M_{\rm {limit}}={\frac {\omega _{3}^{0}{\sqrt {3\pi }}}{2}}\left({\frac {\hbar c}{G}}\right)^{\frac {3}{2}}{\frac {1}{(\mu _{\text{e}}m_{\text{H}})^{2}}}}$

where:

• ħ is the reduced Planck constant
• c is the speed of light
• G is the gravitational constant
• μ_e is the average molecular weight per electron, which depends upon the chemical composition of the star.
• m_H is the mass of the hydrogen atom.
• ω⁰
  ₃ ≈ 2.018236 is a constant connected with the solution to the Lane–Emden equation.

As √ħc/G is the Planck mass, the limit is of the order of

${\displaystyle {\frac {M_{\text{Pl}}^{3}}{m_{\text{H}}^{2}}}}$

A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature.^[11] Lieb and Yau^[14] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

History

In 1926, the British physicist Ralph H. Fowler observed that the relationship between the density, energy, and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei that obey Fermi–Dirac statistics.^[15] This Fermi gas model was then used by the British physicist Edmund Clifton Stoner in 1929 to calculate the relationship among the mass, radius, and density of white dwarfs, assuming they were homogeneous spheres.^[16] Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37×10³⁰ kg.^[17] In 1930, Stoner derived the internal energy–density equation of state for a Fermi gas, and was then able to treat the mass–radius relationship in a fully relativistic manner, giving a limiting mass of approximately 2.19×10³⁰ kg (for μ_e = 2.5).^[18] Stoner went on to derive the pressure–density equation of state, which he published in 1932.^[19] These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.^[20] Frenkel's work, however, was ignored by the astronomical and astrophysical community.^[21]

A series of papers published between 1931 and 1935 had its beginning on a trip from India to England in 1930, where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas.^[22] In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state,^[7] and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.^{[8][9][12][23]} Chandrasekhar reviews this work in his Nobel Prize lecture.^[13] This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,^[24] who, however, did not apply it to white dwarfs.

Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur Eddington. Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible. However, he was unwilling to accept that this could happen. After a talk by Chandrasekhar on the limit in 1935, he replied:

The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. … I think there should be a law of Nature to prevent a star from behaving in this absurd way!^[25]

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P = K₁ρ^5/3 universally applicable, even for large ρ.^[26] Although Niels Bohr, Fowler, Wolfgang Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.^{[27], pp. 110–111} Through the rest of his life, Eddington held to his position in his writings,^{[28][29][30][31][32]} including his work on his fundamental theory.^[33] The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar.^[27] In Miller's view:

Chandra's discovery might well have transformed and accelerated developments in both physics and astrophysics in the 1930s. Instead, Eddington's heavy-handed intervention lent weighty support to the conservative community astrophysicists, who steadfastly refused even to consider the idea that stars might collapse to nothing. As a result, Chandra's work was almost forgotten.^[27]:150

Applications

The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various stages of stellar evolution, the nuclei required for this process are exhausted, and the core collapses, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.^[34]

If a main-sequence star is not too massive (less than approximately 8 solar masses), it eventually sheds enough mass to form a white dwarf having mass below the Chandrasekhar limit, which consists of the former core of the star. For more-massive stars, electron degeneracy pressure does not keep the iron core from collapsing to very great density, leading to formation of a neutron star, black hole, or, speculatively, a quark star. (For very massive, low-metallicity stars, it is also possible that instabilities destroy the star completely.)^{[35][36][37][38]} During the collapse, neutrons are formed by the capture of electrons by protons in the process of electron capture, leading to the emission of neutrinos.^{[34], pp. 1046–1047.} The decrease in gravitational potential energy of the collapsing core releases a large amount of energy on the order of 10⁴⁶ joules (100 foes). Most of this energy is carried away by the emitted neutrinos.^[39] This process is believed responsible for supernovae of types Ib, Ic, and II.^[34]

Type Ia supernovae derive their energy from runaway fusion of the nuclei in the interior of a white dwarf. This fate may befall carbon–oxygen white dwarfs that accrete matter from a companion giant star, leading to a steadily increasing mass. As the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of compressional heating, its temperature also increases. This eventually ignites nuclear fusion reactions, leading to an immediate carbon detonation, which disrupts the star and causes the supernova.^{[40], §5.1.2}

A strong indication of the reliability of Chandrasekhar's formula is that the absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, M_V is approximately −19.3, with a standard deviation of no more than 0.3.^{[40], (1)} A 1-sigma interval therefore represents a factor of less than 2 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

Super-Chandrasekhar mass supernovae

In April 2003, the Supernova Legacy Survey observed a type Ia supernova, designated SNLS-03D3bb, in a galaxy approximately 4 billion light years away. According to a group of astronomers at the University of Toronto and elsewhere, the observations of this supernova are best explained by assuming that it arose from a white dwarf that grew to twice the mass of the Sun before exploding. They believe that the star, dubbed the "Champagne Supernova" by University of Oklahoma astronomer David R. Branch, may have been spinning so fast that a centrifugal tendency allowed it to exceed the limit. Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily. Nevertheless, they point out that this observation poses a challenge to the use of type Ia supernovae as standard candles.^[41][42][43]
Since the observation of the Champagne Supernova in 2003, more very bright type Ia supernovae have been observed that are thought to have originated from white dwarfs whose masses exceeded the Chandrasekhar limit. These include SN 2006gz, SN 2007if and SN 2009dc.^[44] The super-Chandrasekhar mass white dwarfs that gave rise to these supernovae are believed to have had masses up to 2.4–2.8 solar masses.^[44] One way to potentially explain the problem of the Champagne Supernova was considering it the result of an aspherical explosion of a white dwarf. However, spectropolarimetric observations of SN 2009dc showed it had a polarization smaller than 0.3, making the large asphericity theory unlikely.^[44]

Tolman–Oppenheimer–Volkoff limit

After a supernova explosion, a neutron star may be left behind. These objects are even more compact than white dwarfs and are also supported, in part, by degeneracy pressure. A neutron star, however, is so massive and compressed that electrons and protons have combined to form neutrons, and the star is thus supported by neutron degeneracy pressure (as well as short-range repulsive neutron-neutron interactions mediated by the strong force) instead of electron degeneracy pressure. The limiting value for neutron star mass, analogous to the Chandrasekhar limit, is known as the Tolman–Oppenheimer–Volkoff limit.