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The Physics of Chandrasekhar and Super-Chandrasekhar White Dwarfs

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The Physics of Chandrasekhar and Super-Chandrasekhar White Dwarfs The physics of Chandrasekhar and Super-Chandrasekhar White Dwarfs Author: Edgar Molina Lumbreras Advisor: Mario Centelles Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Abstract: We consider white dwarfs to be composed by a degenerate electron gas at zero temperature and study the underlying physics. The electron pressure can only balance the gravitational force up to a maximum mass of the white dwarf, the so-called Chandrasekhar mass. By writing our own code, we have found the value of this limiting mass to be 1.417 solar masses, in accordance with most observations. However, recent discoveries of overluminous type Ia supernovae explosions suggest white dwarf masses above the Chandrasekhar limit. Motivated by recent literature, we show in this work that the presence of an intense magnetic field in the star leads to a higher value of the maximum mass, which could explain these new observations. I. INTRODUCTION mass income results in an increase in the star temperature and, if it is high enough, a point is reached where White dwarfs (WDs) are the remnants of ordinary stars temperature allows for carbon fusion in the core. The weakly which are unable to continue with the nuclear fusion temperature dependent electron pressure is unable to cool the processes in their cores. After all the hydrogen is burnt up to star by expanding it just as happens with ordinary stars. This helium in the core of a star, the latter shrinks due to the leads to a runaway fusion reaction that ends up blowing up gravitational force not being balanced by the nuclear the whole star in a type Ia supernova explosion. Due to the reactions in the core. This shrinking leads to an increase in existence of a maximum WD mass, type Ia supernovae have temperature until it is high enough for helium nuclear fusion a very well defined luminosity peak and can be used as to occur, returning the star to a state of hydrostatic standard candles [9]. These explosions were of prime equilibrium where the inward gravitational force is balanced importance in the discovery of the accelerated expansion of by the outward force due to the thermal pressure gradient. the universe, awarded with the Nobel Prize in 2011. Similar processes can take place for heavier elements as long However, recent observations [1, 10] of peculiar type Ia as the star mass is above a certain value. For masses lower supernovae with exceptionally high luminosities suggest the than about 8M0, where M0 stands for solar mass, the existence of Super-Chandrasekhar WDs, i.e. WDs with a gravitational force is too weak to reach the minimum density higher mass than the Chandrasekhar limit. One proposed and temperature needed for carbon burning, as, before this explanation to these observations is the presence of a strong happens, the degeneracy pressure of electrons becomes magnetic field in the star, which would increase the important enough to counteract the core shrinking. At this maximum possible mass [7, 9]. point, the outer layers of the star are blown away due to In this work, the physics concerning WDs is studied. strong stellar winds, and what remains is a hot compact core Section II provides the structural equations of a star in made of carbon and oxygen, or what we know as a WD. hydrostatic equilibrium. The equation of state for both non- Due to the high densities that can be reached in WDs, the and highly- magnetized WDs are discussed in section III. pressure in this kind of objects can be assumed to come Section IV focuses on the obtained numerical results and we entirely from the degenerate electrons. Two fermions can not conclude with a summary and outlook in section V. have the same quantum state, and thus in a WD electrons have to move to higher energy states by increasing their II. STRUCTURE EQUATIONS momenta, giving origin to what is called a degeneracy pressure. Due to its specific dependence with density, II.1. Newtonian gravity electron degeneracy pressure can only balance the There are two main forces acting on a star. One of them gravitational force up to a maximum WD mass, known as the is the inward directed gravitation and the other one results Chandrasekhar mass [3]. from a pressure gradient. In hydrostatic equilibrium these A WD in a close binary system can accrete matter from two forces are equal in magnitude, giving the first structure its companion star, increasing its mass in the process. This equation of the star [4]. For Newtonian gravity we have: 1 Barcelona, June 2015 The physics of Chandrasekhar and Super-Chandrasekhar White Dwarfs where mN is the nucleon mass and A and Z are the mass and dP G ε(r )m(r) =− , (1) atomic numbers, respectively. The first term in this equation dr c2 r 2 corresponds to the contribution to the energy density of the where P is the pressure, G the Newton's gravitational rest mass of the nucleons assuming electrical neutrality in the constant, c the speed of light and r the distance from the star. The second term is the energy density of the electrons centre of the star. m(r) is the mass contained within a radius r themselves. We do not include here the kinetic energy of the and ε(r)=ρ(r)c2 is the energy density, with ρ(r) being the nucleons as it is very low compared to their rest mass. As a mass density. The second structure equation comes from consequence, nucleons do not give a significant contribution mass conservation: to the pressure of the system and we can consider it to come entirely from the fast moving electrons. On the other hand, as dm 4 π r2 ε(r) = . (2) we will see in the next section, electrons do not give an dr c2 important contribution to the mass density of the star unless extremely high central densities are attained, where their II.2. General relativity corrections Fermi momentum becomes very large. When the star is very compact, general relativity effects With the electron energy must be taken into account. In this case, eq. (1) is replaced by 2 2 2 4 the Tolman-Oppenheimer-Volkoff (TOV) equation [2]: E ( p)= √ p c +me c , (6) −1 dP G ε(r)m(r) P(r) 4 πr 3 P(r) 2G m(r) = − 1+ 1+ 1− we can write εelec as follows [4]: dr c2r 2 [ ε(r) ][ m(r)c2 ][ c2 r ] . p F (3) 8 π 2 εelec( pF )= 3 ∫ E ( p) p dp This equation adds three correction factors to Newtonian (2 πħ) 0 , (7) ε gravity. As the three factors are greater than 1, they = 0 [(2 x3+x)√1+x2 − ln(x+√1+x2)] 8 strengthen the effect of gravity on the star. The importance of these corrections will be evaluated numerically in section IV. where we have defined ε0 and x as: m4 c5 III. EQUATIONS OF STATE ε = e , (8) 0 π2 ħ3 p III.1. Non-magnetized WDs x= F . (9) m c In order to be able to solve the previous equations we e need to find a relation between the pressure and the energy The pressure of an electron gas with an isotropic density, i.e. an equation of state (EoS) of the matter. Matter distribution of momenta is given by [4]: in WDs can be considered to be composed by atomic nuclei p F 2 2 and an ideal Fermi gas of degenerate electrons. The energy 1 8π p c 2 P( pF )= 3 ∫ p dp distribution of the electrons is given by the Fermi-Dirac 3 (2 π ħ) 0 E ( p) . (10) ε statistic, where we can set the temperature to zero [4], even = 0 [(2 x3−3x)√1+x2 − 3ln (x+√1+x2)] 24 when WDs core temperatures can be as high as 107 K [5], due to the huge densities present in the star. Eqs. (7) and (10) relate the energy density and pressure with Ignoring the electrostatic interactions, the number the Fermi momentum, and thus, they provide the equation of density of electrons has the following expression [4]: state. pF 3 2 p n = ∫ d 3 p= F , (4) III.2. Magnetized WDs e (2 π ħ)3 3π2 ħ3 0 If a magnetic field is considered, the electron motion in where pF is the Fermi momentum. the plane perpendicular to it becomes determined by the field We can now write the expression for the total energy and quantized into Landau orbitals (see [6] or [7] for more density: details). For sufficiently high magnetic fields, the cyclotron energy, ħω = ħ(eB/m c), is comparable to the electron rest A c e ε= n m c2+ε (k ) , (5) mass, m c2, and the electrons become relativistic. This e N Z elec F e 2 Barcelona, June 2015 The physics of Chandrasekhar and Super-Chandrasekhar White Dwarfs defines a critical magnetic field: IV. NUMERICAL RESULTS m2 c3 B = e = 4.414 × 1013G . (11) We have written numerical codes to compute several c ħe physical relations for WDs using the structure equations (1)- The Fermi energy of the electrons now gets an extra (3) and the EoS for non-magnetized and magnetized WDs. contribution due to the magnetic field, such that for a Landau level ν it has the following expression [6]: IV.1. Non-magnetized WDs In this work, a fourth-order Runge-Kutta method has 2 2 2 2 4 E F = pF (ν) c + me c (1+2 ν B D) , (12) been implemented to integrate numerically the structure equations.
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