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## White Dwarfs Neutron Stars Black Holes

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**White Dwarfs**Neutron Stars Black Holes INPE Lectures in Sao Jose dos Campos, October 2007 Feryal Ozel University of Arizona**Some Warnings**• I will assume no background in fluid dynamics, general relativity, statistical mechanics, or radiative processes. (If you’ve seen them, some of this will be easy for you). • Because I’m charged with covering a wide range of topics, I made some choices based on personal preferences. (Really, neutron stars ARE very interesting). • Still, I am leaving out a lot. You can find more background material in e.g., “Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects” by Shapiro & Teukolsky. For current research on individual subjects, I’ll try to give references as we go, and you’re welcome to ask for more after the lectures. • I’m going to focus on their structure, their interiors, and their appearance as it relates to determining the properties of their interiors. • Please ask questions, interrupt, ask for more explanation, etc.**End Stages of Stellar Evolution**Main Sequence stars: H burning in the core, synthesizing light elements Heavier elements form in the later stages, after H in the core is exhausted and core contracts, central T rises to ignite “triple-” reaction 3 He4 --> C12 Which stars can ignite He? If they cannot, what happens during the contraction phase? The stellar mass determines if there is sufficient contraction (and thus heating) to ignite further nuclear reactions or if matter becomes degenerate (at very high densities) before nuclear reactions set in. Let’s first look at equation of state of degenerate Fermions.**Kinetic Theory Preliminaries**Let’s start with the distribution function and define number density: All averages, such as energy density are given by includes particle rest mass ; For an ideal fermion/boson gas in equilibrium, Fermion (half-integer spin particles) Boson (integer-spin particles)**Some limits of f(E):**High temperature, low density:**where Fermi energy EF is defined such that**1(E EF) { f(E) ~ 0(E > EF) For fermions, chemical potential (energy cost of adding one particle) is the Fermi energy Fermions at zero temperature (complete degeneracy):**For comparison, let’s look at bosons:**Statistical distributions of photons detected at different times following the startup of the laser oscillation. At short times the source is chaotic and the distribution is of Bose-Einstein type. At longer times the source is a laser and the distribution becomes Poissonian. Unlike Fermions, as T--> 0, an unlimited number of bosons condense to the ground state.**We can write the available number of cells in terms**momentum: or in terms of energy by using E=p2/2m Thus, at a given E and for fixed V, the phase space available to the system of particles decreases with the particle mass, and electrons can fill the phase space much more easily than protons. • The pressure associated with the degenerate electron gas is given by Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give Notice there is no dependence on me.**Now back to the fate of the evolving stars:**During the contraction of a star, nuclear reactions must start when P ≥ Pe. (otherwise, pressure due to degenerate electrons stops the contraction before necessary T for the reactions is reached) Because stellar temperatures scale with mass, this condition equates to a minimum mass for the onset of reactions: (i) For H-burning at 106 K: Smaller masses do not become MS M ≥0.031 M (ii) For He-burning at 2x108 K: Just the mass if the He core; smaller mass stars become degenerate before triple- kicks in. M ≥0.439 M Note: P- relations of the type P=K are called polytropic equations of state. We saw that this is exact for degenerate matter (e.g., inside a white dwarf ) and a good approximation for some normal stars.**Polytropes**Polytropes are simple stellar models applicable to degenerate stars as well as normal stars. To obtain the structures of stars, we need to solve hydrodynamic equations along with an “equation of state” relating pressure P to density . When we use a P- relation of the type P=K (i.e., a polytropic equation of state), we can obtain a simple polytropic stellar model.**Continuity equation**Euler (momentum) equation--a fancy F=ma for pliable materials) What is ij ? off-diagonal terms: Viscosity-stress diagonal terms: Pressure So Euler equation becomes } } pressure gradient viscosity Hydrodynamics Preliminaries:**To study the structures of compact objects, we assume**hydrostatic equilibrium (as is the case for stars in general) In steady state: Static: ) (with So the momentum eqn reads: Which in spherical coordinates is and**Now we’re ready to solve our set of equations for the**structure of a polytrope: ) (alternatively 3 equations for 3 unknowns P, M, and . We can specify boundary conditions and solve this set of equations. Typically the density at the center and the density derivative at the center d/dr=0 are specified as the two boundary conditions. It is common to express radius in terms of a unit length**So that the solution for R and M become**where 1 is the radial point at which density becomes zero (the “surface” of the star) and depends only on the polytropic index n. Note that for 0<n<1, R increases with increasing central density, while for N>1, R decreases. Similarly, We can also write a relation for M and R: Whether M increases or decreases with R depends on the polytropic index n! Note that for n=3 ( = 4/3), M=constant! (independent of R)**Possible Outcomes of Stellar Deaths**The possible end states of stellar evolution are (i) White dwarfs (ii) Neutron stars (iii) Black holes Which compact object will form depends on whether electron degeneracy is achieved at high or low Temperature (which in turn depends on the stellar mass). M ≤ 1.4 M : Electron degeneracy is reached at a relatively low T. Consequently, advanced nuclear burning is not reached. Support against gravity is provided by Pe. 1.4 M ≤ M ≤ 4 M : At the red giant phase, H burns in a shell, and He in another shell. Prad supports against gravity. Mass loss at this stage. Subsequent evolution to a white dwarf. M > 8 M : C12 ignites prior to the development of a degenerate core. Advanced burning stages can be reached. The core eventually collapses to form a compact object.**How many of each form??**A LARGE NUMBER OF UNCERTAINTIES: • late stages of evolution (especially some mass regimes) • mass loss during evolution • differential rotation of the star as the core collapses • explosion energies for supernovae • fallback during supernovae • whether dynamo and/or flux freezing play a role in generating magnetic fields • theoretical uncertainties in maximum NS mass**_**n p + e + e Inverse -decay One other reaction we should briefly talk about in the evolution of stars into compact objects is the inverse -decay p + e n + e In “ordinary” environments, -decay also proceeds efficiently and enables an equilibrium between electrons, protons, and neutrons. But at high densities, when electron Fermi energy is high and the electron produced by -decay does not have sufficient energy, the inverse decay proceeds to primarily create more neutrons.**Formation of Neutron Stars**A supernova simulation from Burrows et al.**Dividing Lines**From Fryer et al. 99 N.B. This is mostly to show uncertainties and possibilities than hard numbers!**Isolated White Dwarfs:**Accreting White Dwarfs: Cataclysmic variables, dwarf novae, ….. Cooling white dwarfs Isolated Neutron Stars: Accreting Neutron Stars: low-mass X-ray binaries, high-mass X-ray binaries, bursters Radio pulsars, millisecond radio pulsars, magnetars (AXPs and SGRs), CCOs, nearby dim isolated stars Accreting Black Holes: Isolated Black Holes: Not visible!! (except in gravity waves) X-ray binaries AGN, low-luminosity AGN How do Compact Objects Appear? The most important factor that determines the observable properties of a compact object is whether or not it is in a (interacting) binary. This is a non-trivial statement because accretion seems to alter the properties of the compact object permanently.**The Crab Pulsar -- A “Prototypical” Rotation Powered**Pulsar Hubble Chandra**An Accretion Powered Pulsar**• magnetic field of the primary (the neutron star) channels the flow to the polar caps: • an X-ray pulsar • the angular momentum transported to the neutron star causes it to spin up**Our Supermassive Black Hole**• this is the longest Chandra exposure image of our Galactic Center Sagittarius A* • supermassive black holes feed off of nearby stars and ISM gas • timescale of flaring events suggests they are occurring near the event horizon**INPE Advanced Course on Compact Objects**Lecture 2 Structures of White Dwarfs And Neutron Stars**Structure of White Dwarfs and the Chandrasekhar Limit:**Electron degeneracy pressure: Remember that pressure of a degenerate gas is given by Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give We solve this equation along with the continuity and force equations: To obtain****In the relativistic limit Chandrasekhar limit Structure of White Dwarfs and the Chandrasekhar Limit: We can plug in some numbers for low-density white dwarfs and the constants to obtain Remember there was no dependence on me, c, or R in the extreme relativistic limit.**M**Chandrasekhar Limit for White Dwarfs: A Quick Treatment The existence of a maximum mass for degenerate stars is very fundamental. Let’s understand it in two ways: I. So as matter under extreme density gets more and more relativistic, mass can no longer increase by increasing the central density but asymptotes to a constant. II. Another way to look at it is the Fermi energy at the quantum limit where the volume per fermion is 1/n = R3/N (Pauli principle), momentum per fermion is so that while the gravitational energy per baryon is Setting EF + EG = 0 gives**Important Note on the Chandrasekhar Limit:**White dwarfs and neutron stars have maximum masses for different reasons!! 1. MCh for degenerate neutron gas is ~0.7 M ! 2. Neutron stars have a maximum mass because of general relativity (as we will see) 3. White dwarfs do not reach the Chandrasekhar mass (the absolute maximum) because inverse -decay kicks in at lower densities. 4. Neutron stars can exceed “their Chandrasekhar limit” because there are other sources of pressure (not just pressure of degenerate neutrons)**Interiors of white dwarfs are roughly isothermal because of**high thermal conductivity of degenerate matter. No heat generation ==> outer layers are in radiative equilibrium, photons carrying the thermal flux There is also local thermodynamic equilibrium (electrons and photons are thermalized) Finally, hydrostatic equilibrium holds for the star Solve photon diffusion equation (along with hydrostatic equilibrium + EOS) where opacity is provided mainly by free-free and bound-free transitions. For values appropriate for a white dwarf, we find White Dwarf Cooling**How long does it take the White Dwarf to Cool?**Combining this with our expression for L and solving for the cooling time gives Or about ~109 yrs for typical white dwarf luminosities. Two (most important) effects that we neglected: When T falls below the melting temperature Tm, the liquid crystallizes and releases q ~ kTm per ion. Crystallization also changes the heat capacity, adding additional 1/2 kT per mode from the lattice potential energy. The overall effect is to increase the thermal lifetime of the white dwarf.**N**magnitude Observations of White Dwarf Cooling • Very detailed studies of white dwarfs in globular clusters are carried out • Detailed cooling models are applied to, e.g., HST data • One such study of NGC 6397 (Hansen et al. 2007) finds a cluster age of Tc=11.47 ± 0.47 Gyrs. A typical luminosity function**Observations of White Dwarf Cooling**• Sloan Digital Sky Survey discovers “ultracool” WDs • At some arbitrarily low T, we start calling them “black dwarfs” • Spectral fits (and in some cases binary companions) allow us to determine WD masses as well from Kepler et al. 07**Density Regimes in Neutron Stars**1. Atmosphere ( 104 g /cm3): Matter in gaseous form, filamentary if B 1010 G) 2. 104 ≤ ≤ 107 g /cm3 : Matter as in white dwarfs. A lattice of nuclei embedded in a degenerate relativistic electron gas. 3. 107 ≤ ≤ 1011 g /cm3 : Inverse -decay transforms protons into n in nuclei. As nuclei get n-rich, the most stable configuration is no longer A=56 but shifts to higher values. 4. 1011 ≤ ≤ 5x1012 g /cm3 : Nuclei become so heavy (A~122) and so neutron-rich (n/Z=83/39) that they “drip” neutrons, forming a free neutron gas. 5. 5x1012 g /cm3: Mixture of degenerate n gas, ultrarelativstic electrons and heavy nuclei. Pn ~ Pe at this density. 6. 5x1012 g /cm3: Nuclei disappear, p, e, and n exist in -equilibrium. These density regimes are found in the “crust” of the neutron star, which is ~few hundred km thick and makes up a few percent of the star’s mass.**7. 1013 ≤ ≤ 5x1015 g /cm3 :**Free neutrons dominate. 8. 1015 g /cm3: ???**Neutron Star Structure and Equation of State**Structure of a (non-rotating) star in Newtonian gravity: (enclosed mass) Need a third equation relating P(r) and (r ) (called the equation of state --EOS) Solve for the three unknowns M, P, **Equations in General Relativity:**} Oppenheimer- Volkoff Equations Two important differences between Newtonian and GR equations: Because of the term [1-2GM(r)/c2] in the denominator, any part of the star with r < 2GM/c2 will collapse into a black hole Gravity ≠mass density Gravity = mass density + pressure (because pressure always involves some form of energy) Unlike Newtonian gravity, you cannot increase pressure indefinitely to support an arbitrarily large mass Neutron stars have a maximum allowed mass**Equation of State of Neutron Star Matter**We saw for degenerate, ideal, cold Fermi gas: { 5/3(non-relativistic neutrons) P ~ 4/3(relativistic neutrons) Solving Oppenheimer-Volkoff equations with this EOS, we get: R~M-1/3 As M increases, R decreases --- Maximum Neutron Star mass obtained in this way is 0.7 M (there would be no neutron stars in nature) --- There are lots of reasons why NS matter is non-ideal (so that pressure is not provided only by degenerate neutrons) Some additionaleffects we need to take into account : (some of them reduce pressure and thus soften the equation of state, others increase pressure and harden the equation of state)**_**n p + e + e I. -stability Neutron matter is formed by inverse -decay escape p + e n + e And is also unstable to -decay escape In every neutron star, -equilibrium implies the presence of ~10% fraction of protons, and therefore electrons to ensure charge neutrality. The presence of protons softens the EOS and reduces the maximum mass**II. The Strong Force**The force between neutrons and protons (as well as within themselves) has a strong repulsive core**II. The Strong Force**At very high densities, this interaction provides an additional source of pressure. The shape of The potential when many particles are present is very difficult to calculate from first principles, and two approaches have been followed: The potential energy for the interaction between 2-, 3-, 4-, .. particles is parametrized and and the parameter values are obtained by fitting nucleon-nucleon scattering data. A mean-field Lagrangian is written for the interaction between many nucleons and its parameters are obtained empirically from comparison to the binding energies of normal nucleons.**_**n p + e + e _ n p + III. Isospin Symmetry The Pauli exclusion principle makes it energetically favorable for a system of nucleons to have approximately equal number of protons and neutrons. In neutron stars, there is a significant difference between the neutron and proton fraction and this costs energy. This interaction energy is usually added to the theory using empirical formulae that reproduce the (A,Z) relation of stable nuclei. IV. Presence of Bosons, Hyperons, Condensates As we saw, neutrons can decay via the -decay yielding a relation between the chemical potentials of n, p, and e: And they can also decay through a different channel when the Fermi energy of neutrons exceeds the pion rest mass**The presence of pions changes the thermodynamic properties**of the neutron star interior significantly. WHY? Because pions are bosons and thus follow Bose-Einstein statistics ==> can condense to the ground state. This releases some of the pressure that would result from adding additional baryons and softens the equation of state. The overall effect of a condensate is to produce a “kink” in the M-R relation: