Friday, January 25, 2013

Degenerate Matter: White Dwarfs

This post deals with white dwarfs and their composition. For an introduction to Degenerate Matter, see here.

Stellar remnants, which no longer produce the huge amounts of energy necessary to counteract the compressive force of gravity, succumb to it. A star becomes a stellar remnant when nuclear fusion, and therefore energy output, ceases. At what point this occurs depends on the size of the star.

For the smallest stars, fusion ends when all of the hydrogen is consumed and converted into helium.

The composition of a low-mass star.

The core has become rich in helium after the completion of hydrogen fusion. When the outer layers are shed, a so-called white dwarf of pure helium remains. It is estimated that, for stars with less than .5 solar masses, this type of white dwarf is the eventual fate. Such stars, during their lifetime, never achieve the temperature necessary (100 million Kelvin) to form elements heavier than helium. However, such small stars also burn their hydrogen fuel very slowly, so such a star would have a lifetime greater than 13.7 billion years, the current age of the Universe. As such, none of this type of dwarf should be found in the Universe. Despite this, some have been observed, and it is proposed that such objects form when another body sucks the outer hydrogen layers from a low-mass star, making it too small to continue fusion and abruptly ending its lifetime.

However, a slightly heavier star, such as our Sun, is able to fuse helium near the end of its life, forming heavier elements such as carbon and oxygen. The white dwarf that will be the end stage of our Sun's lifetime will be composed mainly of these elements.

The size of a white dwarf also depends on its mass, but not in the expected fashion. More massive white dwarfs actually have smaller radii. To be more precise, the cube of the radius varies inversely with the mass, or, for some proportionality constant k, R3 = k/M.

A mass-radius diagram for a white dwarf (click to enlarge).

A Fermi gas is a gas composed of particles that exert degeneracy pressure. The name distinguishes it from an ideal gas; models assuming gases to be ideal ignore particle size and atomic forces. In the case of a white dwarf, the electrons exert the degeneracy pressure, and they are the particles of the Fermi gas. The curve labeled "Non-relativistic Fermi gas" models the inverse cube law that relates the mass and radius. A typical white dwarf (from the mass-radius chart), therefore has a density hundreds of thousands of times greater than that of Earth.

At first, it is nearly coincident with the actual variation, that modeled by relativity (the green curve). The gradual, and then rapid, disassociation of these curves stems from the velocity of the electrons. By Heisenburg's uncertainty principle, the velocity of the electrons that cause the degeneracy pressure increases as the radius of the white dwarf decreases, due to the increased certainty of position. However, as the speed of the electrons approaches c, the speed of light, the theory of relativity predicts that their mass will actually increase. This, from a relativistic frame of reference, augments the mass of the white dwarf and causes it to contract further before stabilizing due to increased degeneracy pressure.

However, this can only continue up to a certain point before the relativistic model would spiral out of control: the added degeneracy pressure caused by the contraction of the white dwarf would not be enough to combat the increased velocity, and therefore mass, of the electrons. The contraction would then be self-reinforcing, and the radius would shrink to zero, as illustrated on the graph.

Therefore, there is an upper limit on white dwarf mass, known as the Chandrasekhar Mass Limit (marked CM on the diagram), equal to about 1.44 solar masses. Some white dwarfs have been observed as being more massive, but this may be due to another factor: spin. If a white dwarf is rotating, the centrifugal force is added to the degeneracy pressure to combat the force of gravity. Theoretically, a rapid spin could allow a white dwarf to even exceed 2 solar masses.

Such an event is somewhat rare though, and in the case of a parent star over 8 solar masses, the remnant is usually too massive to become a white dwarf. In this event, however, the stellar remnant does not simply contract to nothing; it becomes another type of degenerate matter, which will be discussed in the next post.

Sources: White Dwarf at Wikipedia,,

Thursday, January 17, 2013

Degenerate Matter: Introduction

Degenerate matter, simply speaking, is matter at very high densities. Environments where such matter exists are relatively rare throughout the Universe, and they most commonly occur in the cores of stars, or stellar remnants.

Formally, ordinary matter assumes a "degenerate" state when atoms are packed in such close proximity that the very structure of atoms, and eventually, their constituent particles, breaks down. This happens in stages, and at each stage, there is a different type of degenerate matter.

Two major quantum mechanical principles come into play in determining the nature of degenerate matter. The first is the Heisenberg Uncertainty Principle. It states that there are certain pairs of properties of a particle has that can only be simultaneously known to a certain accuracy. In the context of degenerate matter, the pair of position and momentum is the most important. Therefore, as the position of a particle is known with greater and greater accuracy, the momentum is more and more uncertain, and vice versa. The principle is represented by the equation below.

In this equation, ∆χ is the error in the position, and ∆ρ is the error in momentum. The right side of the equation is simply a constant, specifically the reduced Planck Constant divided by 2. The product of the two errors is always greater than this value. It is clear that, if ∆χ is reduced, ∆ρ will increase, and vice versa.

The second principle of quantum mechanics necessary for an analysis of degenerate matter is the Pauli Exclusion Principle. This principle establishes that no two identical particles can be in the same quantum state simultaneously, i.e. have the same quantum numbers. Since these numbers can take only specific values, one particle literally "excludes" another from having the same properties. The most well-known example of this is the principle's application to elections, which defines the structure of the atom.

The above is a visualization of the electron structure of the element Xenon. For each electron shell, i.e. each concentric ring in the above image, the Pauli Exclusion Principle precludes "stuffing in" any more electrons beyond full capacity. The first three shells here have a capacity of 2, 8, and 18 electrons respectively, all of which are filled; no more electrons could fit within them, and the rest therefore had to next occupy the fourth and fifth shells. The key idea is that electron states are discrete; only a finite number of electrons can fill a certain level. There is also a minimum energy level, below which no electron can exist.

Armed with these ideas, one can now understand how degeneracy comes about. Consider a gas under normal conditions stored in a container (below).

The individual particles (molecules or atoms) have varying momenta, with some moving faster, and some slower, but the average speed of all the particles in the container can easily be found. It is commonly known as the temperature of the gas. Also, the gas produces a pressure pushing on the container, created when particles bounce off all sides. This pressure is practically uniform, due to the minuteness of the particles involved, and is called the thermal pressure. This pressure is what keeps a balloon inflated, for example.

Clearly, this thermal pressure varies with temperature. When the temperature is increased, the particles involved are more energetic, move faster, and therefore exert a higher pressure on the sides of the container. Similarly, if one shrinks the container, more particles will bounce off the sides of the container per unit time, and consequently, the pressure on the container will increase. Therefore, pressure on the container is proportional to the temperature of the gas and inversely proportional to the space which it occupies. These phenomena are part of the kinetic theory of gases, and are summarized in the equation below:

p = kt/s

where p is pressure on the container, t is temperature, s is size of the container, and k is the associated proportionality constant, which varies from gas to gas.

By the above theory, when the temperature of a gas is lowered, the size would have be lowered correspondingly to keep the pressure constant. If the temperature were lowered to absolute zero, however, the particles of the gas would no longer be in motion, and no pressure would be exerted on the sides of the container no matter how small it was. The gas could then be compressed into an arbitrarily small volume with no resistance.

However, this does not occur, as the kinetic theory of gases in an approximation, and does not take quantum phenomena into account. When the electrons associated with particles of gas are put into a very small space, their positions are easily measured, and almost precisely pinpointed. Returning to the Heisenburg Uncertainty Principle, this implies that to compensate for the very low uncertainty in position, there must be a high uncertainty in momentum, meaning that the electrons must have very high velocities despite a temperature of near absolute zero.

Note that this principle does not dictate that a particular electron is suddenly imparted with a higher momentum, but merely states that, on average, the uncertainty must increase, and there therefore are some electrons traveling faster. At the same time, many electrons are being pushed into the same space as the container is compressed, and, by the Pauli Exclusion Principle, cannot occupy the same quantum state. They are then pushed into higher energy states, generally producing an outward force.

The combination of these two effects produces an intense pressure against further compression not predicted by the kinetic theory. This is known as electron degeneracy pressure. When more of the counter-force pushing outward on the container is supplied by electron degeneracy pressure than by thermal pressure, then the matter involved is known as electron-degenerate.

To find a naturally occurring example of electron-degenerate matter, one must look no further than the night sky, see the next post.


Wednesday, January 9, 2013

Behavior of the Gamma Function

In the previous post, the idea of the Gamma function, or the extension of the factorial beyond the positive integers, was introduced. In addition, an expression for it, in the form of an infinite product, was discussed. But how does the function actually behave outside of the integers? First, we shall consider some basic facts:

We have already confirmed that the factorial function grows faster than the polynomial and exponential functions. In addition, it grows slower than the function xx, a fact which can be confirmed without difficulty in a similar manner.

Also of importance is the total determination of the function on the real numbers given only its value on the interval (0,1) (the real numbers from 0 to 1 not including 0 or 1). This is due to one of the two fundamental properties of the Gamma function: Γ(x + 1) = xΓ(x). Since any real number can be expressed as the sum of an integer and a number between zero and one, iterated applications of the above property will yield the desired value. For example,

Γ(3.5) = (2.5)Γ(2.5) = (2.5)(1.5)Γ(1.5) = (2.5)(1.5)(0.5)Γ(0.5), thus reducing the problem of calculating the value of the function over the real numbers to that of calculating it only between zero and one.

Of the values on (0,1), all can be approximated numerically to any degree of accuracy with the infinite product, but only one, Γ(1/2), is known in closed form. Its exact value is

Γ(1/2) = (π)1/2, a fact that was discovered when a previously known infinite product that equaled π was identified with the expression when 1/2 is substituted in for the definition of Γ(x). Thus, any real number with fractional part 1/2 can be expressed in closed form. The graph of the Gamma function on the real line is as shown below.

It is clear that the Gamma function is undefined at each non-positive integer, as is consistent with the definition. In addition, there is no number x such that Γ(x) = 0. The graph alternates between positive and negative on the intervals between the negative integers because going from one interval to the next is equivalent to multiplying (or dividing) by a negative number in accordance with the above rule.

However, the Gamma function is not even limited to the real line. It is also defined for all complex values using the same infinite product as before. The product is defined at all complex numbers off the real line, i.e. there are no additional discontinuities beyond the non-positive integers. The behavior of the Gamma function over the entire complex plane is illustrated below.

To graph such a function on the complex plane would normally require four axes, two for the input z, the real part and the imaginary part, and two for the output Γ(z). Since there are only two dimensions available when graphing on a plane, hue and lightness are used in addition to illustrate the behavior of the function. The two physical dimensions of the graph illustrate the position of the input on the complex plane in accordance with the numbered axes (horizontal: real axis, vertical: imaginary axis). The output is colored according to the scheme illustrated below:

The modulus of the complex number is indicated by its lightness; numbers closer to zero are indicated by darker colors, while complex numbers far from zero are whiter. Furthermore, the argument, or angle made with the positive horizontal axis, is indicated by hue. For example, a complex number with an argument near zero appears red, while one with an argument near π (180°) is blue. Returning to the complex "graph" of the Gamma Function, it is easy to see how the graph on the real line is a cross section of the full graph: along the horizontal real axis in the middle, the output is red (positive real number, argument 0) for positive input, and alternating between red and light blue (negative real number, argument π) for negative integers. Also, as one travels further negative, the outputs become very small (darker), except for the spike at each negative integer where the function goes to infinity (white dots). The points at which the Gamma function is undefined are called poles.

Over the rest of the complex plane, it can be seen that the function's value at any complex number is the conjugate (reflection over the real axis) of the value at its conjugate. In other words, the output at two points that are reflections over the real axis have the "opposite" colors indicated by the scheme above. In equation form,

All of these properties are illustrated on a copy of the complex graph above:
Due to its nature as an extension of the factorials, the Gamma function allows an analogous broadening of the field of probability theory. It also finds applications related to binomials and the evaluation of coefficients in a product. Finally, it is related to other functions involved in number theory, and emerges in computations in a variety of mathematical fields.

Sources: Gamma Function, Wikipedia, Mathematical Thought from Ancient to Modern Times (Vol. 2) by Morris Kline

Tuesday, January 1, 2013

The Gamma Function

The concept of the "factorial" function in mathematics is well-known. For any positive integer n, n factorial, denoted n!, is the product of n with all smaller positive integers. In equation form,

n! = (n)*(n-1)*(n-2)*...*3*2*1

The first few values are 1! = 1, 2! = 2*1 = 2, 3! = 3*2*1 = 6, 4! = 4*3*2*1 = 24,... In addition, one can easily see that dividing the factorial of a number by the number itself yields the previous factorial, that is, n!/n = (n-1)!. As an example, 4!/4 = 24/4 = 6 = 3!. Extrapolating backward, 1!/1 = 1 would be the value of zero factorial. Thus we consider 0! = 1.

The problem with which we are concerned is how to extend the idea of factorials to values other than the positive integers, a procedure called interpolation. In other words, we seek a function, f(x) that generates the factorials for the positive integers but also is defined on non-integers.

First to be considered are the polynomial and exponential functions, i.e. those of the form xn and ax, respectively, for natural numbers n and general positive numbers a, where x is the variable quantity. Can either of these types, or additions/subtractions thereof, yield a function that matches the factorials? The answer is no, for the following reason: the factorial function grows faster than any polynomial or exponential function. This means that if one increases x high enough, x! (assuming x is an integer) will always exceed a function of the above types. To see this, we shall examine the growth of polynomial and exponential functions in turn.

For any function xn, xn exceeds x! at x = n, as nn is a product of n n's, while n! is a product of n numbers less than or equal to n. However, consider the comparison of values at x = n2:

(n2)n = n2n, while
(n2)! = n2*(n2-1)*...*n*...*2*1.

A close examination of the expansion of the factorial allows one to see that there are n2 - n terms exceeding n (those that occur prior to n), while the polynomial value is a product with 2n terms, all equaling n. So as long as
n2 - n > 2n, the factorial will be greater. This is true for all n > 3. Also, for any xn, the ratio of the value of this function at x = a + 1 to that at a for positive integral a approaches 1 as a increases, as this ratio is

Since (a + 1)/a clearly approaches 1 as a increases, so does this quantity taken to a constant positive power. In contrast, the ratio (a + 1)!/a! always equals a + 1, and this continues to increase as a increases. Thus, once a factorial exceeds a polynomial, its exceptionally large rate of change insures that it stays above the polynomial. To treat the cases where n = 2,3 and n2 - n ≤ 2n, it is easy to see that any polynomial with an x4 term exceeds one with either an x2 or an x3 as its highest term, and since the factorial exceeds x4, it is also greater than the other two.

For exponential functions, it suffices to treat those ax where a is an integer. If the factorial exceeds these, it also exceeds any exponential function for real a. Going as before, at x = a, ax is greater than a! for the same reasons as for the polynomials. However, each increase of x by 1 beyond this is equivalent to multiplying the exponential function by a, but the factorial by a number greater than a:
aa + 1 = a*aa, while (a + 1)! = (a + 1)*a! and so on. The factorial clearly increases faster, and thus "catches up with" and exceeds the exponential function. To further illustrate this, consider the graph below (click to enlarge):

The polynomial function x10 (green), the exponential function 10x (red), and the factorial function x! (blue) compared on a logarithmic graph. The factorial function is interpolated here, but for the time being, we still consider only integer values. Initially, when x is between 2 and 10, the polynomial is the greatest. At 10, the polynomial and exponential functions are obviously equal. Finally, x = 25 is the first point at which the factorial exceeds both functions, the value of 25! being over ten trillion trillion!

The factorial, however, can be shown to increase less rapidly than functions such as xx. In fact, no combination of elementary functions can represent the factorial. To find an expression for the factorial, one must consider more precisely the conditions that must be met. There are an infinite number of curves through any set of points, and therefore the condition that the function coincide with the factorial for positive integers is insufficient to define it uniquely. Therefore, the following two conditions are given for the function f:

f(0) = 1, and
f(x + 1) = (x + 1)f(x).

The second condition generalizes the standard rule for factorials, (n + 1)! =
(n + 1)n!, to any x.

The function that satisfies these conditions is the Gamma function, denoted Γ(x). However, the Gamma function is actually slightly different from the factorial function, as it is translated one unit. What this means is that, for positive integers n,
Γ(n) = (n - 1)!. Therefore, the Gamma function actually satisfies

Γ(1) = 1, and
Γ(x + 1) = xΓ(x)

for all x. There are several ways to actually formulate the Gamma function, one being an infinite product:

In evaluating this product, the expression after the Π must be calculated for every positive integer k. Then, all of these values must be multiplied. As the upper bound on k increases to infinity, the product converges to the actual value of n!. Note that the above product is defined for every n, except for negative integers. This is because, if n is a negative integer, in the term k = -n, the denominator of k/(k + n) becomes zero. Since once term is undefined, the whole product is. In addition, due to the 1/n term outside of the infinite product, Γ(0) is undefined. Thus the Gamma function is defined for all numbers except the nonpositive integers.

Next, we must confirm that the Gamma function satisfies the required conditions, in order to see that it truly is an extension of the factorial function. It is clear that Γ(1) = 1, because, in this case, ((k + 1)/k)n = (k + 1)/k, and k/(k + n) = k/(k + 1), which is precisely the reciprocal of the first term. When these are multiplied, they yield 1, no matter the value of k. Thus the infinite product takes the form (1/1)(1*1*1*...) = 1, as desired. Next, it must be proven that Γ(n + 1) = nΓ(n). When n + 1 is substituted for n in the above product, it yields the following expression:
After some manipulation, this becomes
, (2)
as an infinite product of two terms multiplied together is the product of the infinite products of each term. The first of the two expressions in parentheses is simply the expression for Γ(n), and the value of the second can be inferred from its expansion:
The fraction (n + 1)/(n + 2) in the first term (k = 1) of the expansion is canceled by its reciprocal, which appears later as the value of (k + 1)/k when k = n + 1. Whenever the first and second fractions of the infinite product can assume reciprocal values for a positive integer value of k, they cancel, for any (k + 1)/k can be cancelled by some expression of the form (k + n)/(k + n + 1) as long as k > n. Therefore, with all of these canceled, all that remains is a product of the fractions (k + 1)/k, as k ranges from 1 to n, or, in numerical form, (2/1)*(3/2)*...*((n + 1)/n). The value of this is (n + 1)/1 = n + 1, since every other number appears once in the numerator of a term, and once in the denominator. Finally, the n + 1 that results cancels with the fraction before the second infinite product in (2), yielding simply n. Thus, finally,

Γ(n + 1) = nΓ(n),

and the infinite product formula agrees with the (shifted) factorial function for all positive integers n. The actual behavior of the Gamma function and its values are discussed in another post.

Sources: Gamma Function, Wikipedia, Mathematical Thought from Ancient to Modern Times, vol. 2 by Morris Kline