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16. White Dwarfs, Neutron Stars and Black Holes
I Introduction
In this section we will look at the physical mechanisms responsible for the formation compact stellar objects. Compact objects such as white dwarf stars, neutron stars, and ultimately black holes, represent the final state of a star's evolution. Stars are born in gaseous nebulae in which clouds of hydrogen coalesce becoming highly compressed and heated through the gravitational interaction. At a temperature of about 10^{7} K, a nuclear reaction begins converting hydrogen into the next heavier element, helium, and releasing a large quantity of electromagnetic energy (light). The helium accumulates at the center of the star and eventually becomes compressed and heated enough (10^{8} K) to initiate nuclear fusion of helium into heavier elements.
So far, the star is held in "near-equilibrium" by the countervailing forces of gravity, which compresses the star, and pressure from the vast electromagnetic energy produced during nuclear fusion, which tends to make it expand. However, as the star burns hotter and ignites heavier elements which accumulate in the core, electromagnetic pressure becomes less and less effective against gravitational collapse. In most stars, this becomes a serious problem when the core has reached the carbon rich phase but the temperature is still insufficient to fuse carbon into iron. Even if a star has reached sufficient temperature to create iron, no other nuclear fusion reactions producing heavier elements are exothermic and the star has exhausted its nuclear fuel. Without electromagnetic energy to hold the core up, one would think that the core would become unstable and begin to collapse---but another mechanism intervenes.
II The Electron Gas
But there is another "force" that holds the core up; now we will turn to a study of this force and how the balance between this force and gravity lead to the various stellar compact objects: white dwarfs, neutron stars and black holes.
The stabilizing force that keeps the stellar core from collapsing operates at terrestrial scales as well. All solid matter resists compression and we will trace the origin of this behavior in a material that turns out to most resemble a stellar compact object: ordinary metal. Although metal is "hard" by human standards, it is to some degree elastic---capable of stretching and compression. Metals all have a similar atomic structure. Positively charged metal ion cores form a regular crystalline lattice and negatively charged valence electrons form a kind of gas that uniformly permeates the lattice.
Suprisingly, the bulk properties of the metal such as heat capacity, compressibility, and thermal conductivity are almost exclusively properties of the electron gas and not the underlying framework of the metal ion cores. We will begin by studying the properties of an electron gas alone and then see if it is possible to justify such a simple model for a metal (or a star).
To proceed, two very important principles from Quantum Mechanics need to be introduced:
Pauli Exclusion Principle: Electrons cannot be in the same quantum state. For our purposes, this will effectively mean that electrons cannot be at the same point in space.
Heisenberg Uncertainty Principle: A quantum particle has no precise position, x, or momentum, p. However, the uncertainties in the outcome of experiment aimed at simultaneously determining both quantities is constrained in the following way. Upon repeated measurements, the "spread" in momentum, p, of a particle absolutely confined to a region in space of sizex, is constrained by
xp | 2 |
where 6.610^{-34} Joule-sec is a fundamental constant of nature (the Planck constant).
Here is how these two laws act together to give one of the familiar properties of metals. The Pauli Exclusion Principle tends to make electrons stay as far apart as possible. Each of N electrons confined in a box of volume R^{3} will typically have R^{3}/N space of its own. Therefore, the average interparticle spacing is a_{0} = R/N^{1/3}. (The situation is actually a bit more complicated than this.) Since the electrons are spatially confined within a region of linear size a_{0}, the uncertainty in momentum is p /a_{0}. The precise meaning of p^{2} is the variance of a large set of measurements of momentum. Denoting average by angle brackets,
Therefore, the average value of p^{2} must be greater than or equal top^{2}.
Based on these results, let us calculate how the energy of an electron gas depends upon the size of the box containing it. The kinetic energy of a particle of mass m and speed v is
= | 1 2 |
mv^{2} | = | 2m |
Now, taking the minimum value of momentum, p^{2} p^{2} ^{2}/a_{0}^{2}, we arrive at the energy, = ^{2}/m_{e}a_{0}^{2,}, for a single electron of mass m_{e}. The total kinetic energy of N electrons is then E_{e} = N. Finally, putting in the dependence of a_{0} on N and the system size, R, we get for E_{e},
E_{e} | m_{e}R^{2} |
N^{5/3}. |
As the system size R is reduced, the energy increases. Even though the electrons do not interact with one another, there is an effective repulsive force resisting compression. The origin of this force is the uncertainty principle! (neglecting e-e interactions and neglecting temperature.)
Let us test out this model by calculating the compressibility of metal. Consider a metal block that undergoes a small change in volume, V, due to an applied pressure P.
The bulk modulus, B, is defined as the constant of proportionality between the applied pressure and the fractional volume change.
P | = | B | V |
. |
The outward pressure (towards positive R) exerted by the electron gas is defined in the usual way in terms of a derivative of the total energy of the system:
P | = | A |
= | - | A |
R |
. |
he bulk modulus is then defined as
B | = | V | V |
= | - | 5 9 |
m_{e} |
V |
5/3 |
10^{-10} - 10^{-11} N/m^{2}. |
(We've taken the volume per electron to be 1 nm^{3}.) The values of B for Steel and Aluminum are B_{steel} 610^{-10} N/m^{2} and B_{Al} 210^{-10} N/m^{2}. It is hard to imagine that this excellent agreement in magnitude is wholly fortuitous (it is not). Having seen that the Heisenberg uncertainty principle is the underlying physics behind the rigidity of metal, we will now see that it is also physical mechanism that keeps stars from collapsing under their own weight.
III Compact Objects
A star can only be in a condition of static equilibrium if there is some force to counteract the compressive force of gravity. In large stars this countervailing force is the radiation pressure from thermally excited atoms emitting light. But in a white dwarf star, the force counteracting gravity has its origin in the uncertainty principle, as it did in a metal. The elements making up the star (mostly iron) exist in a completely ionized state because of the high temperatures. One can think of the star as a gas of positive charge atomic nuclei and negative charge electrons. Each metal nucleus is a few thousand times heavier than the set of electrons that were attached to it, so the nuclei (and not the electrons) are responsible for the sizable gravitational force holding the star together. The electrons are strongly electrostatically bound to core of the star and therefore coexist in the same volume as the nuclear core---gravity pulling the nuclei together and the uncertainty principle effectively pushing the electrons apart.
We will proceed in the same way as in the calculation of the bulk modulus by finding an expression for the total energy and taking its derivative with respect to R to find the effective force.
The gravitational potential energy of sphere of mass M and radius R is approximately
E_{g} | - | R |
swhere G 7 10^{-11} Nm^{2}/kg^{2} is the gravitational constant. (The exact result has a coeffient of order unity in front; we are doing only "order-of-magnitude" calculations and ignoring such factors.) The negative sign means that the force of gravity is attractive---energy decreases with decreasing R. We would like to express E_{g} in terms of N, like E_{e}---this will make the resulting expressions easier to adapt to neutron stars later on. The mass M of the star is the collective mass of the nucleons, to an excellent approximation. As you may know from chemisty, the number of nucleons (protons and neutrons) is roughly double the number of electrons, for light elements. If is the average number of nucleons per electron, for the heavier elements making up the star, The mass of the star is expressed as M=m_{n}N. Putting the expressions for the electron kinetic energy and the gravitational potential energy together, we get the total energy E:
E | = | E_{e} + E_{g} | m_{e}R^{2} |
- | R |
The graph of the function E(R)
reveals that there is a radius at which the energy is minimum---that is to say, a radius R_{0} where the force F = -E/R is zero and the star is in mechanical equilibrium. A rough calculation of R_{0} gives:
R_{0} | = | Gm_{e}^{2}m_{n}^{2} |
10^{7} m = 10,000 km. |
where we have used N 10^{57}, a reasonable value for a star such as our sun. R_{0} corresponds to a star that is a little bigger than earth---a reasonable estimate for a white dwarf star! The mass density may also be calculated assuming the radius R_{0}: 10^{9} kg/m^{3} = 10^{5} density of steel. On the average, the electrons are much closer to the nuclei in the white dwarf than they are in ordinary matter.
Under some circumstances, the star can collapse to an object even more compact than a white dwarf---a neutron star. The Special Theory of Relatvity plays an important role in this further collapse. If we calculate the kinetic energy of the most energetic electrons in the white dwarf, we get:
m_{e}a_{0}^{2} |
= | m_{e} |
R_{0} |
2/3 |
100^{-14} Joules. |
This energy is actually quite close to the rest mass energy of the electron itself, m_{e}c^{2} = 10^{-13} Joules. Recall that the expression for the kinetic energy, = p^{2}/2m, is only a nonrelativistic approximation. Rest mass energy is a scalar formed from the product
p^{µ}p_{µ} | = | c^{2} |
- | p^{2} = (mc)^{2}. |
The exact expression for the energy of a relativistic particle is then:
= | [(pc)^{2}+(mc^{2})^{2}]^{1/2} | = | mc^{2} | + | 2m |
+ terms of order | mc |
4 |
. |
When p mc (or, equivalently, when p^{2}/m mc^{2} as above) the higher order terms cannot be neglected.
Since the full expression for is unwieldy for our simple approximation schemes, we will look at the extreme relativistic limit, p >> mc. In this case, pc. This limit is effectively the limit for extremely massive stars, where the huge compressive force of gravity will force the electrons to have compensatingly high kinetic energies and enter the extreme relativistic regime.
The different form for the energy of the electrons (now linear rather than quadratic in p) will have dramatic consequences for the stability equation for the radius R_{0} derived earlier. The calculation proceeds as before; according to the uncertainty principle the estimate for the momentum of an electron within the star is
p | = | a_{0} |
= | R |
Therefore, the total electron energy is given by
E_{e} N Npc | R |
The same expression as before for E_{g} results in the following expression for the total energy:
E | + | E_{e} | + | E_{g} | R |
- | R |
The energy E(R) has a completely different behavior than in the nonrelativistic case. If we look at the force F = -E/R it is just equal to E/R. If the total energy is positive, the force always induces expansion; if the total energy is negative, the force always induces compression. Thus, if the total energy E is negative, the star will continue to collapse (with an ever increasing inward force) unless some other force intervenes. These behaviors are suggested in the figure below.
The expression for total energy tells us that the critical value of N (denoted by N_{C}) for which the energy crosses over to negative value is
N_{C} | = | ^{3} |
Gm_{n}^{2} |
3/2 |
. |
This is conventionally written in terms of a critical mass for a star, M_{C}, that separates the two behaviors: expansion or collapse. The critical mass is
M_{C} = N_{c}m_{n} | = | ^{2}m_{n}^{2} |
G |
3/2 |
. |
If M > M_{C}, the star will continue to collapse and its electrons will be pushed closer and closer to the nuclei. At some point, a nuclear reaction begins to occur in which electrons and protons combine to form neutrons (and neutrinos which are nearly massless and noninteracting). A sufficiently dense star is unstable against such an interaction and all electrons and protons are converted to neutrons leaving behind a chargeless and nonluminous star: a neutron star.
You may be wondering: what holds the neutron star up? Neutrons are chargeless and the nuclear force between neutrons (and protons) is only attractive, so what keeps the neutron star from further collapse? Just as with electrons, neutrons obey the Pauli Exclusion Principle. Consequently, they avoid one another when they are confined and have a sizable kinetic energy due to the uncertainty principle. If the neutrons are nonrelativistic, the previous calculation for the radius of the white dwarf star will work just the same, with the replacement m_{e} m_{n}. This change reduces the radius R_{0} of the neutron star by a factor of 2000 (the ratio of m_{n} to m_{e}) and R_{0 } 10 km. One of these would comfortably fit on Long Island but would produce somewhat disruptive effects.
Finally, if the neutron star is massive enough to make its neutrons relativistic, continued collapse is possible if the total energy is negative, as before in the white dwarf case. The expression for the critical mass M_{C} is easily adapted to neutrons by setting = 1. Since 2 for a white dwarf, we would expect that a star about four times more massive than a white dwarf is susceptible to unlimited collapse. No known laws of physics are capable of interrupting the collapse of a neutron star. In a sense, the laws of physics leave the door open for the formation of stellar black holes.
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