Astronomy:Habitability of red dwarf systems

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Short description: Possible factors for life around red dwarf stars
An artist's impression of a young red dwarf surrounded by three planets.

The theorized habitability of red dwarf systems is determined by a large number of factors. Modern evidence indicates that planets in red dwarf systems are unlikely to be habitable, due to their low stellar flux, high probability of tidal locking and thus likely lack of magnetospheres and atmospheres, small circumstellar habitable zones and the high stellar variation experienced by planets of red dwarf stars, impeding their planetary habitability. However, the ubiquity and longevity of red dwarfs could provide ample opportunity to realize any small possibility of habitability.

A major impediment to life developing in these systems is the intense tidal heating caused by the proximity of planets to their host red dwarfs.[1][2] Other tidal effects reduce the probability of life around red dwarfs, such as the extreme temperature differences created by one side of habitable-zone planets permanently facing the star, and the other perpetually turned away and lack of planetary axial tilts. Still, a planetary atmosphere may redistribute the heat, making temperatures more uniform.[3][2] Non-tidal factors further reduce the prospects for life in red-dwarf systems, such as extreme stellar variation, spectral energy distributions shifted to the infrared relative to the Sun, (though a planetary magnetic field could protect from these flares) and small circumstellar habitable zones due to low light output.[2]

There are, however, a few factors that could increase the likelihood of life on red dwarf planets. Intense cloud formation on the star-facing side of a tidally locked planet may reduce overall thermal flux and drastically reduce equilibrium temperature differences between the two sides of the planet.[4] In addition, the sheer number of red dwarfs statistically increases the probability that there might exist habitable planets orbiting some of them. Red dwarfs account for about 85% of stars in the Milky Way[5][6] and the vast majority of stars in spiral and elliptical galaxies. There are expected to be tens of billions of super-Earth planets in the habitable zones of red dwarf stars in the Milky Way.[7]

M-type stars are also considered possible hosts of habitable exoplanets, even those with flares such as Proxima b. Determining the habitability of red dwarf stars could help determine how common life in the universe might be, as red dwarfs make up between 70 and 90% of all the stars in the galaxy. However, it is important to bear in mind that flare stars could greatly reduce the habitability of exoplanets by eroding their atmosphere.[8]

Background

Red dwarfs[9] are the smallest, coolest, and most common type of star. Estimates of their abundance range from 70% of stars in spiral galaxies to more than 90% of all stars in elliptical galaxies,[10][11] an often quoted median figure being 72–76% of the stars in the Milky Way (known since the 1990s from radio telescopic observation to be a barred spiral).[12] Red dwarfs are usually defined as being of spectral type M, although some definitions are wider (including also some or all K-type stars). Given their low energy output, red dwarfs are almost never naked-eye visible from Earth: the closest red dwarf to the Sun, Proxima Centauri, is nowhere near visual magnitude. The brightest red dwarf in Earth's night sky, Lacaille 8760 (+6.7) is visible to the naked eye only under ideal viewing conditions.

Photosynthesis

Size is not the only factor in making red dwarfs potentially unsuitable for life, however. On a red dwarf planet, photosynthesis on the night side would be impossible, since it would never see the sun. On the day side, because the sun does not rise or set, areas in the shadows of mountains would remain so forever. Photosynthesis as we understand it would be complicated by the fact that a red dwarf produces most of its radiation in the infrared, and on the Earth the process depends on visible light. There are potential positives to this scenario. Numerous terrestrial ecosystems rely on chemosynthesis rather than photosynthesis, for instance, which would be possible in a red dwarf system. A static primary star position removes the need for plants to steer leaves toward the sun, deal with changing shade/sun patterns, or change from photosynthesis to stored energy during night. Because of the lack of a day-night cycle, including the weak light of morning and evening, far more energy would be available at a given radiation level.

Longevity and ubiquity

Red dwarfs have one advantage over other stars as abodes for life: far greater longevity. It took 4.5 billion years before humanity appeared on Earth, and life as we know it will see suitable conditions for 1[13] to 2.3[14] billion years more. Red dwarfs, by contrast, could live for trillions of years because their nuclear reactions are far slower than those of larger stars, meaning that life would have longer to evolve and survive.

While the likelihood of finding a planet in the habitable zone around any specific red dwarf is slight, the total amount of habitable zone around all red dwarfs combined is equal to the total amount around Sun-like stars given their ubiquity.[15] Furthermore, this total amount of habitable zone will last longer, because red dwarf stars live for hundreds of billions of years or even longer on the main sequence.[16] However, combined with the above disadvantages, it is more likely that red dwarf stars would remain habitable longer to microbes, while the shorter-lived yellow dwarf stars, like the Sun, would remain habitable longer to animals.

Luminosity and spectral composition

Relative star sizes and photospheric temperatures. Any planet around a red dwarf, such as the one shown here (Gliese 229A), would have to huddle close to achieve Earth-like temperatures, probably inducing tidal lock. See Aurelia. Credit: MPIA/V. Joergens.

For years, astronomers have been pessimistic about red dwarfs as potential abodes for life. The low masses of red dwarves (from roughly 0.08 to 0.60 solar masses (M☉)) cause their nuclear fusion reactions to proceed exceedingly slowly, giving them low luminosities ranging from 10% of Sol's to just 0.0125%.[17] Consequently, any planet orbiting a red dwarf would need a low semi-major axis in order to maintain an Earth-like surface temperature, from 0.268 astronomical units (AU) for a relatively luminous red dwarf like Lacaille 8760 to 0.032 AU for a smaller star like Proxima Centauri.[18] Such a world would have a year lasting just 3 to 150 Earth days.[19][20]

At those distances, the star's gravity would cause tidal locking. One side of the planet would eternally face the star, while the other would always face away from it.[citation needed] The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet.[citation needed] It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.[citation needed]

Photosynthesis would be more difficult, as much of the low luminosity falls under the lower energy infrared and red part of the electromagnetic spectrum, and would therefore require additional photons to achieve excitation potentials.[21] Potential plants would likely adapt to a much wider spectrum (and as such appear black in visible light).[21]

In addition, because water strongly absorbs red and infrared light, less energy would be available for aquatic life on red dwarf planets.[22] However, a similar effect of preferential absorption by water ice would increase its temperature relative to an equivalent amount of radiation from a Sun-like star, thereby extending the habitable zone of red dwarfs outward.[23]

Another fact that would inhibit habitability is the evolution of the red dwarf stars; as such stars have an extended pre-main sequence phase, their eventual habitable zones would be for around 1 billion years a zone where water was not liquid but in its gaseous state. Thus, terrestrial planets in the actual habitable zones, if provided with abundant surface water in their formation, would have been subject to a runaway greenhouse effect for several hundred million years. During such an early runaway greenhouse phase, photolysis of water vapor would allow hydrogen escape to space and the loss of several Earth oceans of water, leaving a thick abiotic oxygen atmosphere.[24]

This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side.[25] This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[26]

Tidal effects

At the close orbital distances, which planets around red dwarf stars would have to maintain for liquid water to exist at their surfaces, tidal locking to the host star is likely. Tidal locking makes the planet rotate on its axis once every revolution around the star. As a result, one side of the planet would eternally face the star and another side would perpetually face away, creating great extremes of temperature.

For many years, it was believed that life on such planets would be limited to a ring-like region known as the terminator, where the star would always appear on or close to the horizon. It was also believed that efficient heat transfer between the sides of the planet necessitates atmospheric circulation of an atmosphere so thick as to disallow photosynthesis. Due to differential heating, it was argued, a tidally locked planet would experience fierce winds with permanent torrential rain at the point directly facing the local star,[27] the sub-solar point. In the opinion of one author this makes complex life improbable.[28] Plant life would have to adapt to the constant gale, for example by anchoring securely into the soil and sprouting long flexible leaves that do not snap. Animals would rely on infrared vision, as signaling by calls or scents would be difficult over the din of the planet-wide gale. Underwater life would, however, be protected from fierce winds and flares, and vast blooms of black photosynthetic plankton and algae could support the sea life.[29]

In contrast to the previously bleak picture for life, 1997 studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibar, or 10% of Earth's atmosphere, for the star's heat to be effectively carried to the night side, a figure well within the bounds of photosynthesis.[30] Research two years later by Martin Heath of Greenwich Community College has shown that seawater, too, could effectively circulate without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Additionally, a 2010 study concluded that Earth-like water worlds tidally locked to their stars would still have temperatures above 240 K (−33 °C) on the night side.[31] Climate models constructed in 2013 indicate that cloud formation on tidally locked planets would minimize the temperature difference between the day and the night side, greatly improving habitability prospects for red dwarf planets.[4] Further research, including a consideration of the amount of photosynthetically active radiation, has suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[32]

The existence of a permanent day side and night side is not the only potential setback for life around red dwarfs. Tidal heating experienced by planets in the habitable zone of red dwarfs less than 30% of the mass of the Sun may cause them to be "baked out" and become "tidal Venuses."[1] The eccentricity of over 150 planets found orbiting M dwarfs was measured, and it was found that two-thirds of these exoplanets are exposed to extreme tidal forces, rendering them uninhabitable due to the intense heat generated by tidal heating.[33]

Combined with the other impediments to red dwarf habitability,[3] this may make the probability of many red dwarfs hosting life as we know it very low compared to other star types.[2] There may not even be enough water for habitable planets around many red dwarfs;[34] what little water found on these planets, in particular Earth-sized ones, may be located on the cold night side of the planet. In contrast to the predictions of earlier studies on tidal Venuses, though, this "trapped water" may help to stave off runaway greenhouse effects and improve the habitability of red dwarf systems.[35]

Moons of gas giants within a habitable zone could overcome this problem since they would become tidally locked to their primary and not their star, and thus would experience a day-night cycle. The same principle would apply to double planets, which would likely be tidally locked to each other.

An artist's impression of GJ 667 Cc, a potentially habitable planet orbiting a red dwarf constituent in a trinary star system.

Note however that how quickly tidal locking occurs can depend upon a planet's oceans and even atmosphere, and may mean that tidal locking fails to happen even after many Gyrs. Additionally, tidal locking is not the only possible end state of tidal dampening. Mercury, for example, has had sufficient time to tidally lock, but is in a 3:2 spin orbit resonance.[36]

Variability

Red dwarfs are far more variable and violent than their more stable, larger cousins. Often they are covered in star-spots that can dim their emitted light by up to 40% for months at a time. On Earth life has adapted in many ways to the similarly reduced temperatures of the winter. Life may survive by hibernating and/or by diving into deep water where temperatures could be more constant. Oceans would potentially freeze over during extreme cold periods. If so, once the dim period ends, the planet's albedo would be higher than it was prior to the dimming. This means more light from the red dwarf would be reflected, which would impede temperatures from recovering, or possibly further reduce planetary temperatures.[citation needed]

At other times, red dwarfs emit gigantic flares that can double their brightness in a matter of minutes.[37] Indeed, as more and more red dwarfs have been scrutinized for variability, more of them have been classified as flare stars to some degree or other. Such variation in brightness could be very damaging for life. Flares might also produce torrents of charged particles that could strip off sizable portions of the planet's atmosphere.[38] Scientists who subscribe to the Rare Earth hypothesis doubt that red dwarfs could support life amid strong flaring. Tidal-locking would probably result in a relatively low planetary magnetic moment. Active red dwarfs that emit coronal mass ejections (CMEs) would bow back the magnetosphere until it contacted the planetary atmosphere. As a result, the atmosphere would undergo strong erosion, possibly leaving the planet uninhabitable.[39][40][41] It was found that red dwarfs have a much lower CME rate as expected from their rotation or flare activity, and large CMEs occur rarely. This suggests that atmospheric erosion is caused mainly by radiation rather than CMEs.[42]

Otherwise, it is suggested that if the planet had a magnetic field, it would deflect the particles from the atmosphere (even the slow rotation of a tidally locked M-dwarf planet—it spins once for every time it orbits its star—would be enough to generate a magnetic field as long as part of the planet's interior remained molten).[43] This magnetic field should be much stronger compared to Earth's to give protection against flares of the observed magnitude (10–1000G compared to the terrestrial 0.5G), which is unlikely to be generated.[44] But actual mathematical models conclude that,[45][46][47] even under the highest attainable dynamo-generated magnetic field strengths, exoplanets with masses like that of Earth lose a significant fraction of their atmospheres by the erosion of the exobase's atmosphere by CME bursts and XUV emissions (even those Earth-like planets closer than 0.8 AU, affecting also G and K stars, are prone to losing their atmospheres). Atmospheric erosion even could trigger the depletion of water oceans.[48] Planets shrouded by a thick haze of hydrocarbons like the one on primordial Earth or Saturn's moon Titan might still survive the flares as floating droplets of hydrocarbon are particularly efficient at absorbing ultraviolet radiation.[49]

Actual measurements reject the presence of relevant atmospheres in two exoplanets orbiting a red dwarf: TRAPPIST-1 b and TRAPPIST-1 c are bare rocks or have as much thinner atmospheres.[50]

Another way that life could initially protect itself from radiation, would be remaining underwater until the star had passed through its early flare stage, assuming the planet could retain enough of an atmosphere to sustain liquid oceans. The scientists who wrote the television program "Aurelia" believed that life could survive on land despite a red dwarf flaring. Once life reached onto land, the low amount of UV produced by a quiet red dwarf means that life could thrive without an ozone layer, and thus never need to produce oxygen.[21]

It is worth noting that the violent flaring period of a red dwarf's life cycle is estimated to only last roughly the first 1.2 billion years of its existence. If a planet forms far away from a red dwarf so as to avoid tidal locking, and then migrates into the star's habitable zone after this turbulent initial period, it is possible for life to have a chance to develop.[51]

It has been found that the largest flares happen at high latitudes near the stellar poles so if exoplanets orbits are aligned with the stellar rotation then they are less affected by the flares than previously thought.[52]

Flare activity

For a planet around a red dwarf star to support life, it would require a rapidly rotating magnetic field to protect it from the flares. A tidally locked planet rotates only very slowly, and so cannot produce a geodynamo at its core. The violent flaring period of a red dwarf's life cycle is estimated to only last roughly the first 1.2 billion years of its existence. If a planet forms far away from a red dwarf so as to avoid tidal locking, and then migrates into the star's habitable zone after this turbulent initial period, it is possible that life may have a chance to develop.[53] However, observations of the 7 to 12-billion year old Barnard's Star showcase that even old red dwarfs can have significant flare activity. Barnard's Star was long assumed to have little activity, but in 1998 astronomers observed an intense stellar flare, showing that it is a flare star.[54]

Abundance

A major advantage that red dwarfs have over other stars as abodes for life is their longevity. It took 4.5 billion years before humans appeared on Earth, and suitable conditions for life will last 1.5 billion more years.[55] Red dwarfs exist for trillions of years, because their nuclear reactions are far slower than those of larger stars, meaning that life would have far longer to potentially evolve and survive. And given the ubiquity of red dwarves, the total habitable zone around all red dwarfs combined is likely equal to the total amount around Sun-like stars, even if individual habitable zones are rarer or narrower.[56] The first super-Earth with a mass of a 3 to 4 times that of Earth's found in the potentially habitable zone of its star is Gliese 581g, and its star, Gliese 581, is indeed a red dwarf. Although tidally locked, it is thought possible that at its terminator liquid water may exist.[57] The planet is thought to have existed for approximately 7 billion years and has a large enough mass to support an atmosphere.

Another possibility could come in the far future, when according to computer simulations a red dwarf becomes a blue dwarf as it is exhausting its hydrogen supply. As this kind of star is more luminous than the previous red dwarf, planets orbiting it that were frozen during the former stage could be thawed during the several billions of years this evolutionary stage lasts (5 billion years, for example, for a 0.16 M☉ star), giving life an opportunity to appear and evolve.[58]

Water retention

Planets can retain significant amounts of water in the habitable zone of ultra-cool dwarfs, with a sweet spot in the 0.08 – 0.11 M⊙ range, despite FUV-photolysis of water and the XUV-driven escape of hydrogen.[59]

Water worlds orbiting M-dwarfs could have their oceans depleted over the Gyr timescale due to the more intense particle and radiation environments that exoplanets experience in close-in habitable zones. If the atmosphere were to be depleted over the timescale less than Gyr, this could prove to be problematic for the origin of life (abiogenesis) on the planet.[48]

Methane habitable zone

If methane-based life is possible (similar to the hypothetical life on Titan), there would be a second habitable zone further out from the star corresponding to the region where methane is liquid. Titan's atmosphere is transparent to red and infrared light, so more of the light from red dwarfs would be expected to reach the surface of a Titan-like planet.[60]

Frequency of Earth-sized worlds around ultra-cool dwarfs

TRAPPIST-1 planetary system (artist's impression)

A study of archival Spitzer data gives the first idea and estimate of how frequent Earth-sized worlds are around ultra-cool dwarf stars: 30–45%.[61] A computer simulation finds that planets that form around stars with similar mass to TRAPPIST-1 (c. 0.084 M⊙) most likely have sizes similar to the Earth's.[62]

In fiction

The following examples of fictional "aliens" existing within Red Dwarf star systems exist:

  • Ark: In Stephen Baxter's Ark, after planet Earth is completely submerged by the oceans a small group of humans embark on an interstellar journey eventually making it to a planet named Earth III. The planet is cold, tidally locked and the plant life is black (in order to better absorb the light from the red dwarf).
  • Draco Tavern: In Larry Niven's Draco Tavern stories, the highly advanced Chirpsithra aliens evolved on a tide-locked oxygen world around a red dwarf. However, no detail is given beyond that it was about 1 terrestrial mass, a little colder, and used red dwarf sunlight.
  • Nemesis: Isaac Asimov avoids the tidal effect issues of the red dwarf Nemesis by making the habitable "planet" a satellite of a gas giant which is tidally locked to the star.
  • Star Maker: In Olaf Stapledon's 1937 science fiction novel Star Maker, one of the many alien civilizations in the Milky Way he describes is located in the terminator zone of a tidally locked planet of a red dwarf system. This planet is inhabited by intelligent plants that look like carrots with arms, legs, and a head, which "sleep" part of the time by inserting themselves in soil on plots of land and absorbing sunlight through photosynthesis, and which are awake part of the time, emerging from their plots of soil as locomoting beings who participate in all the complex activities of a modern industrial civilization. Stapledon also describes how life evolved on this planet.[63]
  • Superman: Superman's home, Krypton, was in orbit around a red star called Rao which in some stories is described as being a red dwarf, although it is more often referred to as a red giant.
  • Ready Jet Go!: In the children's show Ready Jet Go!, Carrot, Celery and Jet are a family of aliens known as Bortronians who come from Bortron 7, a planet of the fictional red dwarf Bortron. They discovered Earth and the Sun when they picked up a "primitive" radio signal (Episode: "How We Found Your Sun"). They also gave a description of the planets in the Bortronian solar system in a song in the movie Ready Jet Go!: Back to Bortron 7.
  • Aurelia This planet, seen in the speculative documentary Extraterrestrial (also known as Alien Worlds), details what scientist theorize alien life could be like on a planet orbiting a red dwarf star.

See also

Template:Wikiversity inline

References

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  36. ↑ Kasting, James F.; Whitmire, Daniel P.; Reynolds, Ray T. (1993). "Habitable Zones around Main Sequence Stars". Icarus 101 (1): 108–128. doi:10.1006/icar.1993.1010. PMID 11536936. Bibcode: 1993Icar..101..108K. http://www.as.utexas.edu/astronomy/education/spring02/scalo/kasting.pdf. 
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  38. ↑ Guinan, Edward F.; Engle, S. G.: "Future Interstellar Travel Destinations: Assessing the Suitability of Nearby Red Dwarf Stars as Hosts to Habitable Life-bearing Planets"; American Astronomical Society, AAS Meeting #221, #333.02 Publication Date:01/2013 Bibcode: 2013AAS...22133302G
  39. ↑ Khodachenko, Maxim L. (2007). "Coronal Mass Ejection (CME) Activity of Low Mass M Stars as An Important Factor for The Habitability of Terrestrial Exoplanets. I. CME Impact on Expected Magnetospheres of Earth-Like Exoplanets in Close-In Habitable Zones". Astrobiology 7 (1): 167–184. doi:10.1089/ast.2006.0127. PMID 17407406. Bibcode: 2007AsBio...7..167K. 
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  41. ↑ Garcia-Sage, K. (2017). "On the Magnetic Protection of the Atmosphere of Proxima Centauri b". The Astrophysical Journal Letters 844 (1): L13. doi:10.3847/2041-8213/aa7eca. Bibcode: 2017ApJ...844L..13G. 
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  43. ↑ Alpert, Mark (November 1, 2005). "Red Star Rising: Small, cool stars may be hot spots for life". Scientific American 293 (5): 28. doi:10.1038/scientificamerican1105-28. PMID 16318021. Bibcode: 2005SciAm.293e..28A. http://www.sciam.com/article.cfm?id=red-star-rising. Retrieved 2013-01-19. 
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  46. ↑ See, V.; Jardine, M.; Vidotto, A. A.; Petit, P.; Marsden, S. C.; Jeffers, S. V.; do Nascimento, J. D. (30 October 2014). "The effects of stellar winds on the magnetospheres and potential habitability of exoplanets". Astronomy & Astrophysics 570: A99. doi:10.1051/0004-6361/201424323. Bibcode: 2014A&A...570A..99S. 
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  54. ↑ Croswell, Ken (November 2005). "A Flare for Barnard's Star". Astronomy Magazine. Kalmbach Publishing Co. http://www.astronomy.com/news/2005/11/a-flare-for-barnards-star. 
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  57. ↑ Vogt, Steven S.; Butler, R. Paul; Rivera, E. J.; Haghighipour, N.; Henry, Gregory W.; Williamson, Michael H. (2010). "The Lick-Carnegie Exoplanet Survey: A 3.1-M🜨 Planet in the Habitable Zone of the Nearby M3V Star Gliese 581". The Astrophysical Journal 723 (1): 954–965. doi:10.1088/0004-637x/723/1/954. Bibcode: 2010ApJ...723..954V. 
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  59. ↑ Bolmont, E.; Selsis, F.; Owen, J. E.; Ribas, I.; Raymond, S. N.; Leconte, J.; Gillon, M. (21 January 2017). "Water loss from terrestrial planets orbiting ultracool dwarfs: implications for the planets of TRAPPIST-1". Monthly Notices of the Royal Astronomical Society 464 (3): 3728–3741. doi:10.1093/mnras/stw2578. Bibcode: 2017MNRAS.464.3728B. 
  60. ↑ Cooper, Keith (10 November 2011). "The Methane Habitable Zone". Astrobiology Magazine. https://www.astrobio.net/alien-life/the-methane-habitable-zone/. 
  61. ↑ He, Matthias Y.; Triaud, Amaury H. M. J.; Gillon, Michaël (2017). "First limits on the occurrence rate of short-period planets orbiting brown dwarfs". Monthly Notices of the Royal Astronomical Society 464 (3): 2687–2697. doi:10.1093/mnras/stw2391. Bibcode: 2017MNRAS.464.2687H. 
  62. ↑ Alibert, Yann; Benz, Willy (26 January 2017). "Formation and composition of planets around very low mass stars". Astronomy & Astrophysics 598: L5. doi:10.1051/0004-6361/201629671. Bibcode: 2017A&A...598L...5A. 
  63. ↑ Stapledon, Olaf Star Maker 1937 Chapter 7 "More Worlds" Part 3 "Plant Men and Others"

Further reading

  • Stevenson, David S. (2013). Under a crimson sun : prospects for life in a red dwarf system. New York, NY: Imprint: Springer. ISBN 978-1461481324. 

External links



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