arXiv:nucl-th/0309041v2 26 Nov 2003
Neutron Stars for Undergraduates
Richard R. Silbar and Sanjay Reddy
Theoretical Division,
Los Alamos National Laboratory,
Los Alamos, NM 87545
Abstract
Calculating the structur e of white dwarf and neu tr on stars would be a s uitable topic for an
undergraduate thesis or an advanced special topics or independent study course. The subject is
rich in many different areas of physics accessible to a junior or senior physics major, ranging from
thermodynamics to quantum statistics to nuclear physics to special and general relativity. The
computations for solving the coupled structure differential equations (both Newtonian and general
relativistic) can be done using a symbolic computational package, such as Mathematica. In doing
so, the student will develop computational skills and learn how to deal with dimensions. Along the
way he or she will also have learned some of the physics of equations of state and of degenerate
stars.
PACS numbers: 01.40.-d, 26.60.+c , 97.10 .Cv, 97.20.Rp, 97.60.J d
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I. INTRODUCTION
In 1967 Jocelyn Bell, a graduate student, along with her thesis advisor, Anthony Hewish,
discovered the first pulsar, something from outer space that emits very r egular pulses of
radio energy. After recognizing that these pulse trains were so unvarying that they would not
suppo r t an origin from LGM’s (Little Green Men), it soon became generally accepted that
the pulsar was due to radio emission from a rapidly rotating neutron star [1] endowed with a
very strong magnetic field. By now more than a thousand pulsars have been catalogued [2].
Pulsars are by themselves quite interesting [3], but perhaps more so is the structure of the
underlying neutron star. This paper discusses a student project dealing with that structure.
While still at MIT befo r e coming to Los Alamos, one of us (Reddy) had the pleasure of
acting as mentor for a bright British high school student, Aiden J. Parker. Ms. Parker was
spending the summer of 2002 at MIT as a participant in a special research program (RSI).
With minimal guidance she was able to write a Fortran program for solving the Tolman-
Oppenheimer-Volkov (TOV) equations [4] to calculate masses and radii of neutron stars
(!).
In discussing this impressive performance after Reddy’s arrival at LANL, the question
came up of whether it would have been possible (and easier) for her to have done the com-
putation using Mathematica (or some other symbolic and numerical manipulation package).
This was taken up as a challenge by the other of us, who also figured it would be a good
opportunity to learn how these kinds of stellar structure calculations are actually done. (Sil-
bar’s only previous experience in t his field of physics consisted of having r ead, with some
care, the chapter on stellar equilibrium and collapse in Weinberg’s treatise on gravitation
and cosmology [5].)
In the process of meeting the challenge, it became clear to us that this subject would
be an excellent topic for a junior or senior physics major ’s project or thesis. After all, if a
British high school student could do it. . . There is much more physics in the problem than
just simply integrating a pair of coupled non-linear differential equations. In addition to
the physics (and even some astronomy), the student must think a bout the sizes of things he
or she is calculating, that is, believing and understanding t he answers one gets. Another
side benefit is that the student learns about the stability of numerical solutions and how
to deal with singularities. In the process he or she also learns the inner mechanics of the
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calculational package (e.g., Mathematica) being used.
The student should begin with a derivation of the (Newtonian) coupled equations, and,
presumably, be spoon-fed the general relativistic (GR) corrections. Before trying to solve
these equations, one needs to work out the relation between the energy density and pressure
of the matter that constitutes the stellar interior, i.e., an equation of state (EoS). The first
EoS’s to try can be derived from the non-interacting Fermi gas model, which brings in
quantum statistics (the Pauli exclusion principal) and special relativity. It is necessary to
keep careful track of dimensions, and converting to dimensionless quantities is helpful in
working these EoS’s out.
As a warm-up problem the student can, at this point, integrate the Newtonian equations
and learn about white dwarf stars. Putting in the GR corrections, one can then proceed in
the same way to work out the structure of pure neutron stars (i.e., reproducing the results
of O ppenheimer and Volkov [4]). It is interesting at this point to compare and see how
important the GR corrections are, i.e., how different a neutron star is from that which
would be given by classical Newtonian mechanics.
Realistic neutron stars, of course, also contain some protons and electrons. As a first
approximation one can treat this multi-component system within the non-interacting Fermi
gas model. In the process one learns a bout chemical potentials. To improve upon this
treatment we must include nuclear interactions in addition to the degeneracy pressure from
the Pauli exclusion principle that is used in the Fermi gas model. The nucleon-nucleon
interaction is not something we would expect an undergraduate to tackle, but there is
a simple model (which we learned about from Prakash [6]) for the nuclear matter EoS.
It has parameters which are fit to quantities such as the binding energy per nucleon in
symmetric nuclear matter, the so-called nuclear symmetry energy (it is really an asymmetry)
and the (not so well known) nuclear compressibility. Working this out is also an excellent
exercise, which even touches on the speed o f sound (in nuclear matter). With these nuclear
interactions in addition t o the Fermi gas energy in the EoS, one finds (pure) neutron star
masses and radii which are quite different from those using the Fermi gas EoS.
The above three paragraphs provide the outline of what follows in this paper. In the
presentation we will also indicate possible “gotcha’s” that the student might encounter and
possible side-trips that might be taken. Of course, the project we outline here can (and
probably should) be augmented by the faculty mentor [7] with suggestions for byways that
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might lead to publishable results, if that is desired.
We should point out that there is a similar discussion of this subject matter in this journal
by Balian and Blaizot [8]. They, however, used this material (and other, related materials)
to form the basis for a full-year course they taught at the Ecole Polytechnique in France.
Our emphasis is, in contrast, more toward nudging the student into a research frame of mind
involving numerical calculation. We also note that much of the material we discuss here is
covered in the t extbook by Shapiro and Teukolsky [9]. However, a s the reader will notice,
the emphasis here is on students learning through computation. One of our intentions is
to establish here a framework for the student to interact with his or her own computer
program, and in the process learn about the physical scales involved in the structure of
compact degenerate stars.
II. THE TOLMAN-OPPENHEIMER-VOLKOV EQUATION
A. Newtonian Formulation
A nice first exercise for the student is to derive the following structure equations from
classical Newtonian mechanics,
dp
dr
= −
Gρ(r)M(r)
r
2
= −
Gǫ(r)M(r)
c
2
r
2
(1)
dM
dr
= 4πr
2
ρ(r) =
4πr
2
ǫ(r)
c
2
(2)
M(r) = 4π
Z
r
0
r
′ 2
dr
′
ρ(r
′
) = 4 π
Z
r
0
r
′ 2
dr
′
ǫ(r
′
)/c
2
. (3)
Here G = 6.6 73×10
−8
dyne-cm
2
/g
2
is Newton’s gravitational constant, ρ(r) is the mass den-
sity at the distance r ( in gm/cm
3
), and ǫ is the corresponding energy density (in ergs/cm
3
)
[10]. The quantity M(r) is the total mass inside the sphere of radius r. A sufficient hint for
the derivation is shown in Fig. 1. (Challenge question: the above equations actually hold
for any value of r, not just the large-r situation depicted in the figure. Can the student also
do the derivation in spherical coo rdinates where the box becomes a cut-off wedge?)
Note that, in the second halves of these equations, we have departed slightly from New-
tonian physics, defining the energy density ǫ(r) in terms of the mass density ρ(r) according
to the (almost) famous Einstein equation from special relativity,
ǫ(r) = ρ(r)c
2
. (4)
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dr
A
p(r+dr) = F(r+dr)/A
p(r) = F(r)/A
FIG. 1: Diagram for derivation of Eq. (1)
This allows Eq. (1) to be used when one takes into account contributions of the interaction
energy between the particles making up the star.
In what follows, we may inadvertently set c = 1 so that ρ and ǫ become indistinguishable.
We’ll try not to do that here so students following the equations in this presentation can
keeping checking dimensions as they proceed. However, they might as well get used to this
often-used physicists’ trick of setting c = 1 (as well as ¯h = 1).
To solve this set of equations for p(r) and M(r) one can integrate outwards from the
origin (r = 0) to the point r = R where the pressure goes to zero. This point defines R as
the radius of the star. One needs an initial value of the pressure at r = 0, call it p
0
, to do
this, and R and the total mass of the star, M(R) ≡ M, will depend on the value of p
0
.
Of course, to be able to perform the integration, o ne also needs to know the energy
density ǫ(r) in terms of the pressure p(r). This relationship is the equation of state (EoS)
for the matter making up the star. Thus, a lot of the student’s effort in this project will
necessarily be directed to developing an appropriate EoS for the problem at hand.
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