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“FERMION STARS, BOSON
STARS AND POLYTROPES”
presented at
COSMOSUR III OBSERVATÓRIO ASTRONÓMICO DE CÓRDOBA – CÓRDOBA, ARGENTINA –
AUGUST 3 - 7, 2015
CLAUDIO M G DE SOUSA
UNIVERSIDADE CATÓLICA DE BRASÍLIA (UCB) – WWW.UCB.BR
UNIVERSIDADE FEDERAL DO OESTE DO PARÁ (UFOPA) - WWW.UFOPA.EDU.BR
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Compact Objects
As they barely emit visible light, most of the compact objects are considered as part of the dark matter in cosmology.
Dark matter is the parcel of the Universe mass that can only be detected by its gravitational effects.
Hence, the concept of dark matter is a reference dependent definition.
Despite that, the subject has increasingly been accepted as decisive for understanding the evolution of the Universe.
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Fermion Stars
Fermion stars is a general name
denoting particular ones like
neutron star and white dwarf stars.
Since Oppenheimer and Volkoff
(Oppenheimer & Volkoff 1939),
these compact objects have
received large attention.
Many of their properties are
already determined.
Astronomers used theoretical
information to detect them.
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Boson Stars
In contrast to fermion star there is the so-called boson star (Ruffini & Bonazzola 1969; Liddle & Madsen 1992; Mielke 1991).
Built up with self-gravitating bosons.
Bosons are of the free kind, instead of virtual bosons (those that take place when particles interact in Field Theory).
Despite this compact object has not been detected yet, there are several theoretical efforts to understand its properties.
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Polytropes
In thermodynamics, polytropes
can be considered as a
special path in the Carnot
graphic, usually expressed by a
relation concerning pressure
and energy density;
Within the graphical scenario,
this is referred as a polytropic
path.
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White Dwarf
Compact Objects with radius 5000 km approximatelly;
Mass density of order 106 g/cm³
Hydrostatic support given by Pauli pressure.
Neutron Star
Compact Objects with radius 10 km approximatelly;
Mass density of order 1014 g/cm³
Hydrostatic support given by Pauli pressure.
Estrelas de Bósons
Compact Objects with radius 10 km approximatelly;
Mass density of order 1022 g/cm³
Hydrostatic support given by Heisemberg pressure.
Present Subject
The aim of this seminar is to show that there is an
intrinsic relation between the statistical mechanics of the gas inside the star and the polytropic model for both fermions and bosons.
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References
Amsler C. et al. (Particle Data Group), 2008, Phys. Lett. B, 667, 1
Chandrasekhar S., 1939, An Introduction to the Study of Stellar Structure. Chicago Univ. Press,
Chicago, IL
Chavanis P. H., 2002, A&A, 381, 709
Chavanis P. H., 2008, A&A, 483, 673
de Sousa C. M. G., 2006, preprint (astro-ph/0612052)
de Sousa C. M. G., Tomazelli J. L., Silveira V., 1998, Phys. Rev. D, 58, 123003
Edwards T. W., Merilan P. M., 1981, ApJ, 244, 600
Ferrel R., Gleiser M., 1989, Phys. Rev. D, 40, 2524
Gleiser M., 1988, Phys. Rev. D, 38, 2376 (erratum: 1989, Phys. Rev. D, 39, 1257)
Honda M., Honda Y. S., 2003, MNRAS, 341, 164
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References Ingrosso G., Ruffini R., 1988, Nuovo Cimento B, 101, 369
Kolb E. W., Turner M. S., 1990, The Early Universe. Westview Press, Boulder, CO
Kusmartsev F. V., Mielke E. W., Schunck F. E., 1991, Phys. Rev. D, 43, 3895
Liddle A. R., Madsen M. S., 1992, Int. J. Mod. Phys. D, 1, 101
Merafina M., 1990, Nuovo Cimento B, 105, 985
Natarajan P., Linden Bell D., 1997, MNRAS, 286, 268
Oppenheimer J. R., Volkoff G. M., 1939, Phys. Rev., 55, 374
Pathria R. K., 1972, Statistical Mechanics. Pergamon, Oxford
Portilho O., 2009, Braz. J. Phys., 39, 1
Ruffini R., Bonazzola S., 1969, Phys. Rev., 187, 1767
Schunck F. E., Mielke E. W., 2003, Class. Quant. Grav., 20, R301
Silveira V., de Sousa C. M. G., 1995, Phys. Rev. D, 52, 5724
Wald R. M., 1984, General Relativity. Chicago Univ. Press, Chicago, IL
Weinberg S., 1972, Gravitation and Cosmology. Wiley, New York
Zeldovich Ya. B., Novikov I. D., 1971, Stars and Relativity. Dover Press, New York (Original text: 1971, Relativistic Astrophysics, Vol. 1, Chicago Univ. Press, Chicago, IL)
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Quantum Statistics for Bosons and
Fermions
In physics there are two types of
quantum statistics: Bosonic and
Fermionic ones.
On the right side, “p” is the pressure,
“ ” is the energy density,
They obey the statistics related to
temperature and average energy.
“m” is the particle mass
“s” is the spin of the considered particle
(fermion or boson).
“T” is the temperature, in Kelvins.
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The functions F( , ) are
given in statistical physics
studies, like in Pathria (1972).
The functions g(x, ) represents
Fermions (+) and/or Bosons (-)
statistics.
Quantum Statistics for Bosons and
Fermions
potentialchemical
constantBoltzmann Bk
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Ingrosso G., Ruffini R., 1988, Nuovo Cimento B, 101, 369
Relativistic Stars In order to consider star’s geometry we use the spherical metric
element:
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Chandrasekhar proposed that a compact object could be treated
as a perfect fluid described by an energy-momentum tensor:
Where u represents the four-velocity vectors.
Relativistic Stars Using Einstein’s Field equations one reaches:
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Those equations, together with the hydrostatic equilibrium :
Allows us to look for the solution for the pressure and for energy.
Relativistic Stars The solution for pressure and energy density opens way to determine
several star’s properties.
One of them Is the total mass:
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Then, using this and the following
approximations, we obtain:
Relativistic Stars 18
This equation could be a linear differential equation (LDE), except for the
fact that it is to be solved both on p(r) and on ρ(r).
However, if p and ρ have a relation (known as the ‘equation of state’),
then equation becomes an actual Differential Equation, with the solution
depending only upon p or ρ.
Polytropic Equation
The problem can be solved if we use the so-called polytropic
equation of state ( = adiabatic constant ):
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Kp
nKp
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Or, in the version where we use the polytropic index (n), taking
=1+(1/n):
Polytropic Equation
Some thermodinamic states give us known examples:
γ = 6/5 is in the range of super large gaseous stars;
Fermion stars usually present 4/3 ≤ γ ≤ 5/3, where γ∼=4/3
correspond to largest-mass white dwarfs and γ∼=5/3 to
small-mass white dwarfs;
Incompressible stars have very high adiabatic indices, γ
→ . As an LDE it is studied in Lane-Emden equation solutions.
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Polytropic Equation
Lane-Emden equation solutions:
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(from wikipedia)
Chandrasekar Isothermal Spheres Isothermal gas spheres are important in astrophysics because they serve
as a starting point for understanding composite stars.
For a standard star (Chandrasekhar 1939), we have from the theorems of
the equilibrium of the star:
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•Notice that both K and D depend on temperature.
•Isothermal spheres is correspond closely to a polytropic equation with γ = 1,
if we take D→0 (or Kρ >>D).
What we did?
Putting together theories by INGROSSO-RUFFINI and CHANDRASEKHAR
we found an interesting connection!
From Ingrosso-Ruffini, we can relate directly pressure and energy
density:
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Which is, apart from the terms, a polytropic equation.
New physics can be inferred if we take the polytropic-like relation: 24
And Taylor expand it around the central density 0
Gives us new variables:
25 Thus, comparing with Ingrosso-Ruffini, the expansion:
NUMERICAL RESULTS
The value of δ brings relevant
information, as we can associate it
with the polytropic index.
The function is new in this
context, and relates thermal
energy with rest mass energy of the
particles,
Hence, results for bosons and
fermions can present numerical
differences
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NUMERICAL RESULTS FERMIONS - neutrons
mass = 938.3 MeV c−2
spin = 1/2
BOSONS – Z0
mass = 91.19 GeV c-2
Spin = 1.
TEMPERATURE range: 0K to 3K
CHEMICAL POTENTIAL range:
From 10-26 to 10-25
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(From the Particle Data Group;
Amsler et al 2008)
FERMIONS 28
FERMIONS 29
BOSONS 30
BOSONS 31
Limits One can also study the results that are close to the case:
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In fact, this result is numerically achieved when one find the following
limit for small , which is to say:
This means we have to work out these functions:
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Statistical Mechanics vs. Polytropic Model
Hence, we found that the Polytropic Model and the
Statistical Mechanics (for Bosons and/or for Fermions) are
intrinsicly related!
This opens a way for more questions:
Whats the influence of that for Chaplygin gas cosmological models?
What about standart barionic stars? How that model fits it?
Boson stars are usually studied as a zero temperature
compact objects... Is it possible they exist at finite
temperature? Do they still “Dark Matter”?
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Muchas gracias a todos!
Claudio M G de Sousa
Universidade Catolica de Brasilia -
UCB
Universidade Federal do Oeste do
Pará – UFOPA
Emails:
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FIN
