Sirius B & White Dwarfs -- Part 2
January 06, 2008
In this entry I’ll be referring to the textbooks Principles of Stellar Evolution and Nucleosynthesis (1st Ed.) by Donald Clayton. It’s a good textbook on the physical nature of stars, and it’s available on Amazon for a good price (though I can’t guarantee that the version they sell will have identical section & equation numbering as my 1970’s hardcover version). Another classic text is Chandrasekhar’s An Introduction to the Study of Stellar Structure. This is not quite as modern or as thorough in discussing the computation of main-sequence stellar models, but the material on degenerate matter & white dwarfs is very well done (by the man who won a Nobel Prize for this work).
As I described previously, a white dwarf is supported by a force known as “degeneracy pressure”. Where does this pressure come from, and what is it? Before touching on those questions, it might be helpful to understand where “normal” pressure comes from. That is, when one fills a balloon, why does it push back? Normal, or ideal gas pressure, comes from the collisions of many particles with each other or with the walls of a container. The effect is similar to being hit with a tennis ball – when the ball bounces off, you feel a momentary force exerted on wherever it hit. That is, when one places more and more air molecules in a balloon, there are more collisions with the walls and the pressure increases. In the case of normal temperatures & densities, gas particles move with a distribution of speeds that follows the Maxwell-Boltzmann distribution. This is discussed with more mathematical rigor in section 2-1 of Clayton’s book. If the particles in question are photons, rather than molecules, atoms, or electrons, then the pressure is called radiation pressure.
Degeneracy pressure is fundamentally different, and much less intuitive. It is a consequence of the laws of quantum mechanics as they apply to electrons. One should note that in a white dwarf, degeneracy pressure is due entirely to the electrons. Because of the high temperatures & pressures, all of the atoms in the star are assumed to be totally ionized, so the electrons & nuclei move separately. The nuclei provide mass to the star and do participate in ideal-gas type collisions, but they will not become degenerate. That would require much higher densities – i.e., neutron stars. Therefore, the matter in a white dwarf will be referred to as a degenerate electron gas, as the electrons are the only degenerate component.
Degeneracy pressure is difficult to explain in a way that is intuitive, but I find that the easiest way to approach it is from the Heisenberg Uncertainty Principle:
In other words, the product of the uncertainty in position (?x) and the uncertainty in momentum (?p) must be greater than some numerical constant involving Planck’s constant. When a gas is very, very dense, the positions of the particles become “fixed” – that is, we know that if there are 10 particles in 1 unit of volume, that each particle is confined to 1/10 of a volume unit. Therefore, as the density increases, the uncertainty in position (?x) decreases. Eventually, if this trend continues, we will bump up against Heisenberg’s limit. In order to compensate, we will require that the uncertainty in momentum become higher & higher. This means that at very high densities, some particles may be traveling slowly, but some particles will also be traveling very fast – regardless of thermal temperature. The distribution of speeds will no longer follow the Maxwell-Boltzmann distribution, with many particles forced to travel faster than they “should” be going, even if the gas may be any temperature – even near absolute zero. In an extreme case, where some of the particles are traveling extremely fast, they can actually approach the speed of light. Such a "relativistic" degenerate gas requires additional mathematical considerations, as special relativity imposes an upper bound (c) on the velocity, even though the maximum momentum may be arbitrarily high, as required by the density.
In fact, this same condition will exist when the temperature of a gas is decreased – very low temperatures will have the same effect as very high densities. What really matters when considering degeneracy is the combination of the temperature & density, rather then either number taken in isolation. Even a very dense gas may “break” its degeneracy at a high-enough temperature (such that the normal thermal velocities of the particles become higher than the lower-limit imposed by the Heisenberg Uncertainty Principle). Additionally, the onset of degeneracy is not sudden as density is gradually increased. As long as the temperature is not zero, there will always be some contribution from ideal gas pressure. Therefore, degeneracy can be spoken of as partial vs. "total" (with the understanding that "total" is an approximation), or as non-relativistic vs. relativistic (with relativistic degeneracy occurring at very high densities). In fact, at conditions of moderate density & very high temperature, partial AND relativistic degeneracy could occur simultaneously. An interesting diagram can be found in Clayton's book, Figure 2-11.
To summarize, under "normal" conditions, particle velocities must obey a Maxwell-Boltzmann distribution, but Heisenberg's Uncertainty Principle says that there will be an uncertainty in the speed, and that uncertainty is related to the density (since density relates to uncertainty in particle position). If the required uncertainty in speed is much greater than the speed predicted by the Maxwell-Boltzmann distribution, then the matter may be called "degenerate", and some particles will be found to be traveling much faster than expected. These speedy particles will, in turn, exert a higher pressure when they collide with other particles or with a container wall.
It is not my intention to write a textbook here, so I will end the discussion without going into mathematical derivations. For those who are interested, I recommend chapter 2 of Clayton’s book, or Chapter 8 of Chandrasekhar’s book for a full, technical treatment.
As I described previously, a white dwarf is supported by a force known as “degeneracy pressure”. Where does this pressure come from, and what is it? Before touching on those questions, it might be helpful to understand where “normal” pressure comes from. That is, when one fills a balloon, why does it push back? Normal, or ideal gas pressure, comes from the collisions of many particles with each other or with the walls of a container. The effect is similar to being hit with a tennis ball – when the ball bounces off, you feel a momentary force exerted on wherever it hit. That is, when one places more and more air molecules in a balloon, there are more collisions with the walls and the pressure increases. In the case of normal temperatures & densities, gas particles move with a distribution of speeds that follows the Maxwell-Boltzmann distribution. This is discussed with more mathematical rigor in section 2-1 of Clayton’s book. If the particles in question are photons, rather than molecules, atoms, or electrons, then the pressure is called radiation pressure.
Degeneracy pressure is fundamentally different, and much less intuitive. It is a consequence of the laws of quantum mechanics as they apply to electrons. One should note that in a white dwarf, degeneracy pressure is due entirely to the electrons. Because of the high temperatures & pressures, all of the atoms in the star are assumed to be totally ionized, so the electrons & nuclei move separately. The nuclei provide mass to the star and do participate in ideal-gas type collisions, but they will not become degenerate. That would require much higher densities – i.e., neutron stars. Therefore, the matter in a white dwarf will be referred to as a degenerate electron gas, as the electrons are the only degenerate component.
Degeneracy pressure is difficult to explain in a way that is intuitive, but I find that the easiest way to approach it is from the Heisenberg Uncertainty Principle:
In other words, the product of the uncertainty in position (?x) and the uncertainty in momentum (?p) must be greater than some numerical constant involving Planck’s constant. When a gas is very, very dense, the positions of the particles become “fixed” – that is, we know that if there are 10 particles in 1 unit of volume, that each particle is confined to 1/10 of a volume unit. Therefore, as the density increases, the uncertainty in position (?x) decreases. Eventually, if this trend continues, we will bump up against Heisenberg’s limit. In order to compensate, we will require that the uncertainty in momentum become higher & higher. This means that at very high densities, some particles may be traveling slowly, but some particles will also be traveling very fast – regardless of thermal temperature. The distribution of speeds will no longer follow the Maxwell-Boltzmann distribution, with many particles forced to travel faster than they “should” be going, even if the gas may be any temperature – even near absolute zero. In an extreme case, where some of the particles are traveling extremely fast, they can actually approach the speed of light. Such a "relativistic" degenerate gas requires additional mathematical considerations, as special relativity imposes an upper bound (c) on the velocity, even though the maximum momentum may be arbitrarily high, as required by the density.
In fact, this same condition will exist when the temperature of a gas is decreased – very low temperatures will have the same effect as very high densities. What really matters when considering degeneracy is the combination of the temperature & density, rather then either number taken in isolation. Even a very dense gas may “break” its degeneracy at a high-enough temperature (such that the normal thermal velocities of the particles become higher than the lower-limit imposed by the Heisenberg Uncertainty Principle). Additionally, the onset of degeneracy is not sudden as density is gradually increased. As long as the temperature is not zero, there will always be some contribution from ideal gas pressure. Therefore, degeneracy can be spoken of as partial vs. "total" (with the understanding that "total" is an approximation), or as non-relativistic vs. relativistic (with relativistic degeneracy occurring at very high densities). In fact, at conditions of moderate density & very high temperature, partial AND relativistic degeneracy could occur simultaneously. An interesting diagram can be found in Clayton's book, Figure 2-11.
To summarize, under "normal" conditions, particle velocities must obey a Maxwell-Boltzmann distribution, but Heisenberg's Uncertainty Principle says that there will be an uncertainty in the speed, and that uncertainty is related to the density (since density relates to uncertainty in particle position). If the required uncertainty in speed is much greater than the speed predicted by the Maxwell-Boltzmann distribution, then the matter may be called "degenerate", and some particles will be found to be traveling much faster than expected. These speedy particles will, in turn, exert a higher pressure when they collide with other particles or with a container wall.
It is not my intention to write a textbook here, so I will end the discussion without going into mathematical derivations. For those who are interested, I recommend chapter 2 of Clayton’s book, or Chapter 8 of Chandrasekhar’s book for a full, technical treatment.
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