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Degenerate Objects

The Structure and Evolution of Brown Dwarfs

In structure, brown dwarfs more resembles Jupiter and Saturn than the stars.  They are cool, which gives many of them atmospheres similar to the giant gaseous planets, and they all have about the same radius as Jupiter, despite being many times more massive.  This second characteristic is a consequence of degeneracy pressure, which supports the brown dwarfs, as well as the giant gaseous planets, against self-gravity.

The brown dwarfs reverse the stellar trend of low-mass main-sequence stars being physically smaller than their more-massive cousins.  The change in behavior reflects the difference between degeneracy pressure and kinetic pressure.  At the core of a main-sequence star, the kinetic motion of electrons and atomic nuclei supply a pressure that is proportional to the core temperature of the star.   The core temperature is set by the thermonuclear fusion of hydrogen, and it varies from about 5 million °K for stars of 0.1 solar masses to over 30 million °K for stars over 10 solar masses.  This temperature sets the gravitational potential of the star, so that the ratio of stellar mass to radius is proportional to the core temperature.  Because the temperature rises slowly with stellar mass, the stellar radius increases with mass.  At the low-mass end of the main sequence, between 0.072 solar masses, which is the lower limit on a main-sequence star's mass, and 1 solar mass, the radius of a star is roughly proportional to its mass.

The electron degeneracy pressure within the brown dwarfs destroys this relationship between mass and radius.[1]  When the electrons are fully degenerate, which means that they are cold, the pressure they exert is proportional to the density to the 5/3 power, and it is independent of temperature.  When the degeneracy pressure of cold electrons is balanced against a brown dwarf's self-gravity, the brown dwarf's radius is proportional to the inverse of the mass to the 1/3 power.  For this reason, a fully-degenerate low-mass brown dwarfs has a larger radii than a fully-degenerate high-mass brown dwarf.  Comparing a fully-degenerate brown dwarf at the low-mass end (0.012 solar masses) of the brown dwarf mass range to its counterpart at the high-mass end (0.072 solar masses) of the range, one finds that the high-mass brown dwarf has a 40% smaller radius than the low-mass brown dwarf.  This means that the most dense main-sequence stars and brown dwarfs are those hovering around a mass of 0.072 solar masses.

As a practical matter, a brown dwarf is somewhat larger than its fully-degenerate radius, because its core is generally hot enough for the kinetic motion of the electrons and ions to provide some pressure (the electrons are not fully degenerate in this case).  This added pressure is more important in a high-mass brown dwarf than in a low-mass brown dwarf, because the more-massive brown dwarf takes longer to cool.  Because of the added pressure, the radius of a brown dwarf at the high-mass limit is only about 25% smaller than that of a a brown dwarf at the low-mass limit.  For this reason, brown dwarfs have roughly the same radius.  This conclusion extends below the lower mass limit to the giant gaseous planets, so the brown dwarfs are about the same size as Jupiter.

Degeneracy pressure changes the evolution of brown dwarfs relative to the evolution of protostars and main-sequence stars.  The high-mass protostars evolve to the main sequence faster than the low-mass protostars.  A 1 solar mass protostar reaches the main-sequence after about 40 million years, a 0.1 solar mass protostar reaches it after 700 million years, and a 0.072 protostar reaches it after 3 billion years.  The length of time spent on the main-sequence also increases as mass decreases.  With the brown dwarfs, this trend of longer evolutionary timescales with smaller masses reverses, because a brown dwarf stops shrinking at a larger radius than that required by the thermonuclear fusion of hydrogen.  This means that less gravitational potential energy is liberated in creating a brown dwarf than would be liberated if the brown dwarf could shrink until hydrogen fusion commences, so a brown dwarf has less energy to radiate away than it would if it could shrink to the hydrogen fusion radius.  Because the high-mass brown dwarf is close to its hydrogen fusion radius, but the low-mass brown dwarf is much larger than its hydrogen fusion radius, the low-mass brown dwarf stabilizes and cools much more rapidly than the high-mass brown dwarf.

Early in its life, a brown dwarf shrinks like a protostar, with its core becoming hotter and more dense as it becomes smaller; the core temperature increases inversely with its radius in this phase.  Eventually its density becomes high enough for degeneracy pressure to dominate other sources of pressure.  At this point in its life, a brown dwarf is the hottest it will ever be, with an internal temperature that ranges from several million °K for the most-massive brown dwarfs, to a temperature of half a million °K for the least massive.  The maximum temperature within a massive brown dwarf is sufficient to burn both deuterium and lithium in thermonuclear fusion.  Within a brown dwarf at the low-mass end of the scale, only deuterium can burn.  The most-massive brown dwarfs reaches their maximum core temperatures in about 300 million years after their births, which matches the time for the least-massive protostars to reach the main sequence.  The lest-massive brown dwarfs, in contrast, reach their maximum core temperatures after only 10 million years.

 The core temperature of a brown dwarf slowly drop away from its maximum value as the deuterium is consumed.  This take of 5 billion years or more for the most massive brown dwarfs, but it takes only about 100 million years for the least-massive brown dwarfs.  A brown dwarf cools more rapidly once its deuterium is exhausted.

 The effective temperature of the photosphere behaves much differently than the core temperature.  The effective temperature is the black-body temperature that would produced the observed flux.  The spectrum of a brown dwarf is not a black body, so the effective temperature is not a direct measurement of the gas temperature at the photosphere; instead, it is a measure of the luminosity, with that the luminosity proportional to Te4.  One finds that the luminosity of a brown dwarf decreases throughout its life.  For the heaviest brown dwarfs, this decrease is steady throughout its life, but for the lightest brown dwarfs, the decrease slows nearly to a halt during the burning of deuterium.  Once the deuterium in these light brown dwarfs is consumed, the effective temperature once again decreases steadily.

The heaviest brown dwarfs have an effective temperature of around 3,000°K at the beginning of their lives (in contrast to the Sun's effective temperature of 5,778°K), but after 100 million years, this temperatures begin to fall, so that by 1 billion years, the effective temperature is about 2,000°K, and after 10 billion years it is about 1,500°K.  The least-massive brown dwarfs begin their lives with an effective temperature of about 2,500°K.  This temperature falls by a couple of hundred degrees after about 10 million years, after which it is stabilized by the burning of deuterium.  After about 50 million years, the effective temperature begins to fall again, so that 100 million years after its birth, the lowest-mass brown dwarfs have an effective temperature below 1,500°K, and after 1 billion years, a temperature below 1,000°K.

This evolution suggests that the astronomical surveys are biased towards finding the high-mass brown dwarfs.  The time required to reach 1,500°K implies that more than half of the most-massive brown dwarfs created within our Galaxy are hotter than 1,500°K, but only about half of a percent of the least-massive brown dwarfs created within the Galaxy are still above 1,500°K.  Most of the lowest-mass brown dwarfs are cold and dim, with effective temperatures in the hundreds of °K; they may not be observable with such a low luminosity.

[1]Chabrier, Gilles, and Baraffe, Isabelle.  “Theory of Low-Mass Stars and Substellar Objects.”   In Annual Reviews of Astronomy and Astrophysics, edited by G. Burbidge, A. Sandage, and F. Shu, vol. 38.   Palo Alto, California: Annual Reviews, 2000: 337–377.

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