presented at cosmosur iii - famaf unc · quantum statistics: bosonic and fermionic ones. on the...

“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

CMGSOUSA@GMAIL.COM

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:

cmgsousa@gmail.com

claudio@unb.br

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FIN