Stars

Something there is more immortal even than the stars,
(Many the burials, many the days and nights, passing away,)
Something that shall endure longer even than lustrous Jupiter
Longer than sun or any revolving satellite,
Or the radiant sisters the Pleiades.

-- Walt Whitman, On the Beach at Night

My plan is for this to be my final entry in the "Sirius B & White Dwarfs" series of blog entries.

In my last entry I discussed the nature and origin of electron degeneracy pressure. In this entry I will (briefly) discuss a program that I wrote to solve for the structure of a white dwarf star -- that is, to calculate the mass and density vs. radius profiles of such a star, starting from the core and working towards the surface.



Some of the most critical work in this area was done by Chandrasekhar and is addressed in his book (which I mentioned in my last entry) or in his 1935 paper The Highly Collapsed Configurations of a Stellar Mass, Monthly Notices of the Royal Astronomical Society, Vol. 95, p.207-225.

Deriving the equation of state (i.e., pressure / density relationship) for a degenerate electron gas is the first and probably most theoretically challenging step towards solving for the structure of a white dwarf. This equation of state is represented parametrically by Chandrasekhar in equation 3 of his paper. After deriving the equation of state, solving for the structure of the star is conceptually simple. One simply has to solve a pair of first-order ordinary differential equations simultaneously (Equation 6 in Chandrasekhar's paper). Those equations are the equation of hydrostatic equilibrium:



...and a second differential equation that is used to simply add up all the mass interior to a given radius:



In a "normal" model of stellar structure (i.e., a model of a star where fusion is still actively occuring), there are additional differential equations that describe the rate of energy generation and transport to the surface, but since a white dwarf is a dead star -- i.e., there is no fusion going on -- these terms do not apply. Furthermore, since the equation of state for a degenerate electron gas does not depend on temperature (since the electrons are already moving faster -- due to quantum-mechanical considerations -- than they should be going based on the range of temperatures encountered in white dwarfs), there is no differential equation to account for temperature either. Even if temperature were considered, the degenerate electron gas is such a good conductor of heat that the star is essentially isothermal anyway. All things considered, this is a very simple model.

After defining the problem with the equation of state (which I have not presented, but is available in the paper that I linked to above) and the two differential equations, Chandrasekhar performs some substitutions and combines all these equations into a single equation -- Equation 16 in his paper:



This equation is written with dimensionless variables, but eta is related to the radius and phi is related to the density. For exact definitions, please see Chandrasekhar's paper (it's free online -- just click the link above). As this is a second-order differential equation, it requires two boundary conditions for unique solution. These are given immediately after Equation 16, but basically amount to "density goes to zero at the surface" and "d-phi/d-eta = 0 at the core due to symmetry".

From my standpoint, I wished to solve Equation 16 above in order to have a complete density vs. radius and mass vs. radius model of a white dwarf star. I accomplished this numerically in Fortran 95, and put the results online here. By supplying the two parameters that describe the star and two additional parameters that control the integration, one can create profiles of white dwarfs of any given mass. If all goes well, a plot similar to the one above should appear after submitting the form. Please see the bottom of the front page for some additional technical details regarding the code. Also, the actual code can be downloaded from the bottom of that page. Compiling it will require a Fortran 95 compiler, and g95 should work (it's what I used).

As one tries various values for 1/y0^2 (a quantity that is related to the central density), it should become apparent that as this quantity approaches zero, the size of the star also approaches zero while the overall mass approaches a finite limit -- about 1.4 solar masses. This, of course, is the Chandrasekhar limit -- the maximum possible mass for a white dwarf.

I will not go into further details here, but suffice to say that this will conclude my series on the physical nature of white dwarfs. There is quite a lot of additional theory, of course. The results that my program can generate are 70+ years old, and the field has progressed a great deal in that time.

One point that does deserve some mention is what happens to white dwarfs that approach the Chandrasekhar limit. The equation of state used in this model does not account for the possibility of inverse beta decay. As the mass of the star (and hence the centeral density) becomes very large, electrons and protons combine to form neutrons (inverse beta decay). In a star close to the Chandrasekhar limit, this process can lead to a star with fewer electrons than might otherwise be expected, so the model becomes increasingly inaccurate as one approaches the 1/y0^2=0 limit. If this process progresses to completion, the entire star is converted to neutrons leading to a neutron star rather than a white dwarf. Such an entity requires a different equation of state and is not dealt with in this model.

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.

At this time of year, the star Sirius is high in the southern sky (to Northern Hemisphere observers) after dark. It's easy to spot -- it's the brightest star visible in the evening sky at an apparent magnitude of about -1.44. Unlike the Sun, Sirius is actually two stars that are fairly close to the Earth. The larger & brighter of these stars, Sirius A, is a class A main sequence star that is about twice as massive as the sun and about 1.7 times as luminous. The other star, however, is truly remarkable.

The mass of Sirius B is about the same as the Sun, or slightly less, but all that mass is squeezed into a star that is about the size of the Earth. This means that its mean density is on the order of 2,000,000 times that of water -- 1 liter would weigh in excess of 2000 metric tons. Even though Sirius B is much hotter than its larger companion, it is much less luminous (by a factor of about 10,000) due to its small size. The two stars orbit their mutual center of gravity with an orbital period of about 50 years, but they never get very far apart from one another as seen from Earth. Even with a decent amateur telescope, "splitting" the pair is difficult due to their proximity and their huge difference in brightness. With the Hubble Space Telescope, the situation is different. The image below was taken by Hubble. The small star in the lower-left of the image is Sirius B, while the much larger star in the center is Sirius A.



Without the clear view provided by the vacuum of space, the light from Sirius A tends to spread out & overwhelm its tiny companion, so many amateur astronomers have never seen Sirius B. For those who wish to try, there is an interesting article and map in the February 2008 issue of Sky & Telescope (page 30).

What is it that makes Sirius B so remarkable though? Why is it the same mass as the Sun, but only about 1/100 of the Sun's diameter? Sirius B is a white dwarf. That is, it's a star that has exhausted its fuel supply and is now slowly cooling off, just as a hot stove burner or lamp filament may continue to glow for a few seconds after the power is removed. Unlike its companion, Sirius B is not generating any new energy through fusion, but simply radiating away what energy it has left-over. Long before humans started observing Sirius B, it was a normal main sequence star like the Sun or Sirius A -- although probably larger & brighter than either one. After running out of fuel, its core began to shrink & its outer layers most likely drifted out into space (although there is no planetary nebula visible today). Without the outward pressure provided by nuclear fusion in its core, the gravity of Sirius B squeezed it into its present size -- and then stopped.

In a normal star, there are nuclear reactions occurring in the core that liberate energy and raise the temperature to millions of degrees. This energy then flows outward -- from hot to cold -- until it reaches the surface. As a consequence of its high temperature, there is a high pressure in the core. That pressure is simply the ideal gas pressure, P = kT/v (where k is Boltzmann's constant and v is the specific volume -- the inverse of density). When fusion ceases, however, the core would tend to cool since there is no new energy being generated. Under other circumstances, this would result in a pressure drop due to the P = kT/v law. In the core of a star, however, this cannot be allowed to occur. The pressure in the core must be high enough to counter-act the gravitational force exerted by the mass of the outer layers of the star in order to keep the star in balance ("hydrostatic equilibrium"). In order to maintain the high central pressure, the outer layers squeeze in until a new equilibrium is reached. This causes the v (the specific volume) term in the ideal gas law to decrease & the temperature term to increase. The core is still at a very high pressure, but it has decreased in size. In shrinking, it has turned some of its gravitational potential energy into heat and thereby maintained a high enough core pressure to remain in equilibrium.

As this process continues, the core grows smaller and hotter, and its constituent particles become closer and closer together. In reality, this process occurs before the entire star is dead -- while fusion is still going on in a layer outside the core. There is a limit to how dense the core can become however. As its size decreases and its density increases, the particles in the core (particularly the electrons, as all atoms are completely ionized at the temperatures involved) begin to interact with each other in ways that are not accounted for by the ideal gas pressure. These interactions provide an additional pressure -- the pressure that supports Sirius B against its own gravity -- called "electron degeneracy pressure". I will discuss the nature of this pressure in a future entry.

When this process reaches completion -- when the core has ceased fusion and become as small and hot as it can, and when the outer layers have swelled so much that they have floated off into space -- a white dwarf is all that remains. Sirius B, like all white dwarfs, is the dead core of a former star -- incredibly dense and propped up against gravity by an entirely different kind of pressure than during its former life.