Table Of ContentThe Cooling of Neutron Stars - Neutrino
Emissions from Neutron Stars with a Special
Focus on the Direct Urca Process
Ronny Kjelsberg
2012
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ISBN: 978-1-300-00864-4
”...and it is feared that the French public, always
impatient to come to a conclusion, eager to know
the connections between general principles and the
immediate questions that have aroused their passions,
may be disheartened because they will be unable to move
on at once.
That is a disadvantage I am powerless to over-
come, unless it be by forewarning and forearming those
readers who zealously seek the truth. There is no royal
road to science, and only those who do not dread the
fatiguing climb of its steep paths have a chance of
gaining its luminous summits.”
Karl Marx in the preface to the french edition of
Capital (Progress Publishers, Moscow, 1954).
Forword
This book is mainly focused around neutrino emissions from neutron star
cores,withanoverweightonthedirectUrca1 process. Itbeginswithageneral
introduction to neutron stars, the physics around compact objects and the
historical development of neutron star physics. Furthermore i contains an
overview of general relativity theory and the so-called TOV-equation derived
by Tolman, Oppenheimer and Volko(cid:88), to give an insight into the physics
behind the development of an equation of state. Then we brie(cid:223)y look at
the composition of the various equations of state, before we examine the
consequences an inclusion of hyperons have on the equation of state, and the
interactions which then must be included.
Moreover, given an overview of the various cooling processes, in which we
will ourselves in the direct Urca process, which constitutes the main work of
this task. We make calculations on the neutrino emissivity from the direct
Urca process and makes plots of matter composition, chemical potential and
nøytrinoemissivitet from the process with di(cid:88)erent interactions included. Fi-
nally, the results are discussed, and the direct Urca process is compared with
other cooling processes.
This book is based on the work I did as a part of my cand.scient.-thesis
at the Department of Physics at the Norwegian University of Science and
Technology under the guidance of Prof.. Morten Hjorth-Jensen, University
of Oslo. I would therefore especially like to thank Prof. Morten Hjorth-
Jensenforgoodcooperation,andagoodacademicguidanceduringthatwork.
Without his assistence the thesis, and this book, would not have seen the
light of day. Furthermore, I wish also to thank my original supervisor, Prof..
ErlendØstgaardatNTNUwhichunfortunatelywasnotabletocompletethis
workwithme,andIwishtothankmyinternalsupervisoratNTNU,Professor
1The acronym URCA comes from the physicist George Gamow, and was the name of
a casino in Rio de Janeiro. The casino was said to be e(cid:88)ective in removing money from
the pockets of its visitors.According to Gamows russian dialect, urca can also mean a
pickpocket.
i
Sigmund Waldenstrøm for help with navigating within the bureaucracy at
NTNU.
In addition, I must also give a general thanks to my fellow students for
both the more and the less useful academic and extra-curricular discussions.
In particular, I thank Sverre G. Johnsen and Anette Wrålsen for reading
the original thesis and giving useful comments. Moreover, I thank various
old and not quite as old men in the (cid:222)fth (cid:223)oor of the the science building
"Realfagbygget" at NTNU, who sometimes have been helpful in giving me
little nudge in the right direction. I would also like to thank Rolf G. Lunder
for useful programming tips. Finally, I will give a general thanks to all
thosewhohavegivenmetheinspirationandknowledgeduringmyeducation,
and I have got to thank the website Google translate, without which this
translation into english would have been much more time-consuming.
ii
Contents
Forword i
1 Introduction to neutron star physics 1
1.1 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Neutron production . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The history of neutron star physics . . . . . . . . . . . . . . . 4
1.4 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.1 The observational basis of neutron stars . . . . . . . . 6
1.4.2 Identi(cid:222)cationof therelationshipbetweenneutronstars
and supernovae . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 Why pulsars are neutron stars . . . . . . . . . . . . . . 8
2 Equation of state and observables 9
2.1 The theory of general relativity and the TOV-equation . . . . 9
2.1.1 General relativity . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 The TOV equation . . . . . . . . . . . . . . . . . . . . 19
2.2 The equation of state and neutron star observables . . . . . . 21
2.3 The Equation Of State . . . . . . . . . . . . . . . . . . . . . . 23
3 The composition of equations of state 25
3.1 Ideal Fermi gas of neutrons. . . . . . . . . . . . . . . . . . . . 25
3.2 Quark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Super(cid:223)uid baryon matter . . . . . . . . . . . . . . . . . . . . . 29
3.4 Kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Pion condensation. . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 32
iii
4 Hyperon matter 35
4.1 Description of compact matter with hyperons . . . . . . . . . 35
4.2 Parameterization of the equation of state for nuclear matter . 39
4.3 Hyperon matter . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Cooling processes 51
5.1 Cooling processes in neutron stars . . . . . . . . . . . . . . . . 51
5.2 Direct Urca . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.1 Direct Urca processes . . . . . . . . . . . . . . . . . . . 53
5.2.2 Direct Urca with nucleons . . . . . . . . . . . . . . . . 54
5.2.3 Direct Urca with muons . . . . . . . . . . . . . . . . . 56
5.2.4 Direct Urca with hyperons . . . . . . . . . . . . . . . . 56
6 Neutrino emissivity 59
6.1 Neutrino emissivity from the Direct Urca process . . . . . . . 59
6.2 A more exact calculation of the neutrino emissivity . . . . . . 64
7 Results 73
8 Conclusion 85
A Program for calculating the emissivity 87
B Calculations in Maple 101
C Tables 103
iv
Chapter 1
Introduction to neutron star
physics
1.1 Neutron stars
Compact objects are the end points of the development of a star. When the
fusion of hydrogen into helium can no longer maintain the radiation pressure
ofthestar, andtheequilibriumthatexistsbetweenthisandthegravitational
force is disturbed, the star is pulled together by gravity until reaching a
temperature that allows the egnition of another nuclear process, which thus
creates a new equilibrium. Eventually the star has however completely run
out of nuclear fuel, and it is pulled together into a compact star; a white
dwarf, a neutron star or a black hole. If the core mass is smaller than the
so-called Chandrasekhar mass (about 1.4 M , where M is the solar mass),
(cid:4) (cid:4)
degenerate electrons create a pressure that balances gravity, and we get a
white dwarf. If the core mass is larger than the Chandrasekhar mass, the
electrons become so relativistic that hydrostatic equilibrium is impossible,
and the mass collapse continues until it is halted by nuclear forces. At this
point the electrons have been captured through the process
e +p n+(cid:11) , (1.1)
(cid:2) e
(cid:11)
so-called inverse (cid:3)-decay, and the star ends up with the same density as the
nucleus [Ph99]. This is a neutron star.
Under normal circumstances, the most stable form of nuclear matter is
the one that is close to 56Fe. Less massive cores have a higher proportion of
nucleons at the surface, and more massive ones get a large repulsion between
protons. This changes when the temperature becomes so high that the elec-
trons become relativistic. When the electrons get an energy greater than
1
