white dwarfs neutron stars black holes

White DwarfsNeutron StarsBlack Holes

INPE Lectures in Sao Jose dos Campos, October 2007Feryal 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.

Lives of Stars

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:

€

n =df

d3x d3 p∫ d3 p

For an ideal fermion/boson gas in equilibrium,

€

f (E) =1

exp[(E − μ) /kT] ±1

All averages, such as energy density are given by

€

ε = Edf

d3x d3 p∫ d3 p ;

€

E = (p2c 2 + m2c 4 )1/ 2

Fermion (half-integer spin particles)

Boson (integer-spin particles)

includes particle rest mass

Some limits of f(E):

High temperature, low density:

€

f (E) ≈ exp(μ − E

kT)

Fermions at zero temperature (complete degeneracy):

{f(E) ~ 1 (E EF)

0 (E > EF)

For fermions, chemical potential (energy cost of adding one particle) is the Fermi energy

€

f (E) =1

exp[(E − EF ) /kT] +1

where Fermi energy EF is defined such that

€

f (EF ) =1

2

For comparison, let’s look at bosons:

Unlike Fermions, as T--> 0, an unlimited number of bosons condense to the ground state.

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.

• We can write the available number of cells in terms momentum:

€

N( p)dp = 2 ×V

h34π p2dp

or in terms of energy by using E=p2/2m

€

N(E)dE =8π V

h3(2m3)1/ 2 E1/ 2dE

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

€

Pe =1

3Ne

0

∞

∫ ( p) f (p)v pdp

Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give

€

Pe =(3π 2)2 / 3

5

h2

me

ne5 / 3, v << c

€

Pe =(3π 2)1/ 3

4hcne

4 / 3, v ~ c

Notice there is no dependence on me.

€

Pe =1

3Ne

0

pF

∫ ( p)v pdp

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:

M ≥0.031 M

(ii) For He-burning at 2x108 K:

M ≥0.439 M

Smaller masses do not become MS

Just the mass if the He core; smaller mass starsbecome degenerate before triple- kicks in.

Note: P- relations of the type P=K are called polytropic equations of state.

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)

Now back to the fate of the evolving stars:

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.

Hydrodynamics Preliminaries:

€

∂∂t

+∂

∂x i

(ρ ui) = 0 Continuity equation

€

∂u j

∂t+ ui

∂u j

∂ x i

= a j −1

ρ

∂Ψij

∂ x iEuler (momentum) equation--a fancy F=ma for pliable materials)

What is ij ?

€

ij = Pδij −σ ij

diagonal terms: Pressure

off-diagonal terms: Viscosity-stress

So Euler equation becomes

€

∂u j

∂t+ ui

∂u j

∂ x i

= a j −1

ρ

∂P

∂ x j

+1

ρ

∂σ ij

∂ x i}}pressuregradient

viscosity

In steady state:

€

ui = 0

To study the structures of compact objects, we assume hydrostatic equilibrium (as is the case for stars in general)

€

∂∂t

= 0

Static:

So the momentum eqn reads:

€

−∂ψ∂x j

−1

ρ

∂P

∂ x j

= 0

€

1

ρ

∂P

∂r= −

GM(r)

r2

(with

€

a j =∂ψ

∂ x j

)

Which in spherical coordinates is

and

€

M(r) = 4π r2

0

r

∫ ρ(r)dr

3 equations for 3 unknowns P, M, and . We can specify boundary conditions and solve this setof equations. Typically the density at the center

and the density derivative at the center d/dr=0 are specified as the two boundary conditions.

€

(r = 0) = ρ c

Now we’re ready to solve our set of equations for the structure of a polytrope:

€

1

ρ

∂P

∂r= −

GM(r)

r2

€

dM(r)

dr= 4π r2ρ(r)

€

P = K ρ Γ

€

P = K ρ1+

1

n(alternatively )

It is common to express radius in terms of a unit length

€

a ≡(n +1)K ρ c

1

n−1

4π G

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

1/ 2

€

ξ =r

a

So that the solution for R and M become

€

R = ξ1

(n +1)K

4π G

⎡

⎣ ⎢

⎤

⎦ ⎥

1/ 2

ρ c

1−n

2n

where ξ1 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.

Whether M increases or decreases with R depends on the polytropic index n!

We can also write a relation for M and R:

€

M ∝ R3−n

1−n

Note that for n=3 ( = 4/3), M=constant! (independent of R)

€

M = f (ξ1) 4π(n +1)K

4π G

⎡

⎣ ⎢

⎤

⎦ ⎥

3 / 2

ρ c

3−n

2n

Similarly,

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

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

n p + e + 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

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

A supernova simulation from Burrows et al.

Formation of Neutron Stars



Dividing Lines

N.B. This is mostly to show uncertainties and possibilities than hard numbers!

From Fryer et al. 99

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.

Isolated Black Holes:

Not visible!!(except in gravity waves)

Isolated Neutron Stars:

Radio pulsars, millisecond radio pulsars,

magnetars (AXPs and SGRs), CCOs, nearby dim

isolated stars

Accreting Neutron Stars:

low-mass X-ray binaries, high-mass X-ray binaries,

bursters

Isolated White Dwarfs:

Cooling white dwarfs

Accreting White Dwarfs:

Cataclysmic variables, dwarf novae,…..

Accreting Black Holes:

X-ray binariesAGN, low-luminosity AGN

This is a non-trivial statement because accretion seems to alter the properties of the compact object permanently.

Cooling white dwarfs in globular cluster M4



The Crab Pulsar -- A “Prototypical” Rotation Powered Pulsar

Chandra Hubble

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

QuickTime™ and aVideo decompressor


P=Pdot diagram of pulsars

A Galactic Black Hole

QuickTime™ and aSorenson Video decompressorare needed to see this picture.

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

Structures of White Dwarfs

And Neutron Stars

INPE Advanced Course on Compact Objects

Lecture 2

Structure of White Dwarfs and the Chandrasekhar Limit:

Electron degeneracy pressure:

Remember that pressure of a degenerate gas is given by

€

Pe =1

3π 2h3v(p)p3dp

0

pF

∫

€

dPe

dr= −ρ (r)

GM(r)

r2

We solve this equation along with the continuity and force equations:

Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give

€

Pe =(3π 2)2 / 3

5

h2

me

ne5 / 3, v << c

€

Pe =(3π 2)1/ 3

4hcne

4 / 3, v ~ c

To obtain

€

R∝ ρ c

1−n

2n

€

M ∝ ρ c

3−n

2n

Structure of White Dwarfs and the Chandrasekhar Limit:

We can plug in some numbers for low-density white dwarfs

€

=5

3, n =

3

2

and the constants to obtain

€

R =1.122 ×104 ρ c

106 gcm−3

⎛

⎝ ⎜

⎞

⎠ ⎟

−1/ 6

km

€

M = 0.4964ρ c

106 gcm−3

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

M

€

R = 3.347 ×104 ρ c

106 gcm−3

⎛

⎝ ⎜

⎞

⎠ ⎟

−1/ 6

km

€

M =1.457 M

€

=4

3, n = 3

In the relativistic limit

Remember there was no dependence on me, c, or R in the extreme relativistic limit.

Chandrasekhar limit

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.

€

M ~ ρ c

3−n

2n

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

€

hn1/ 3

so that

€

EF ~hcN1/ 3

R

while the gravitational energy per baryon is

€

EG ~ −GMmB

R

Setting EF + EG = 0 gives

€

Nmax ≈ 2 ×1057

Mmax ≈ NmaxmB ≈1.5 M

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) becauseinverse -decay kicks in at lower densities.

4. Neutron stars can exceed “their Chandrasekhar limit” because there are othersources of pressure (not just pressure of degenerate neutrons)

White Dwarf Cooling

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)

€

L = −4π r2 c

3κρ

d

dr(aT 4 )

where opacity is provided mainly by free-free and bound-free transitions.

For values appropriate for a white dwarf, we find

€

L ≈ 2 ×106erg s−1 M

M

⎛

⎝ ⎜

⎞

⎠ ⎟T*

7 / 2

€

T* ≈106 −107K

How long does it take the White Dwarf to Cool?

€

U =3

2k T*

M

A mu

€

L =dU

dt

Combining this with our expression for L and solving for the cooling time gives

€

τ ∝ L

M

⎛

⎝ ⎜

⎞

⎠ ⎟−5 / 7

Or about ~109 yrs for typical white dwarf luminosities.

Two (most important) effects that we neglected:

1. When T falls below the melting temperature Tm, the liquid crystallizes and releases q ~ kTm per ion.

2. 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.

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.

magnitude

N

A typicalluminosityfunction

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

Neutron Stars

Density Regimes in Neutron Stars

1. Atmosphere ( 104 g /cm3):

2. 104 ≤ ≤ 107 g /cm3 :

3. 107 ≤ ≤ 1011 g /cm3 :

5. 5x1012 g /cm3:

Matter in gaseous form, filamentary if B 1010 G)

Matter as in white dwarfs. A lattice of nuclei embedded in a degenerate relativistic electron gas.

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.

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:

€

dM(r)

dr= 4π r2ρ(r)

€

dP(r)

dr= −

GM(r)

r2ρ(r)

€

M(r) = 4π r2ρ(r) dr0

r

∫(enclosed mass)

Need a third equation relating P(r) and (r ) (called the equation of state --EOS)

€

P = P(ρ)

Solve for the three unknowns M, P,

Equations in General Relativity:

€

dM(r)

dr= 4π r2ρ(r)

€

dP(r)

dr= −

G M(r) + 4π r3P(r)[ ]

r2 1−2GM(r)

rc 2

⎡ ⎣ ⎢

⎤ ⎦ ⎥

ρ(r) +P

c 2

⎛

⎝ ⎜

⎞

⎠ ⎟} Oppenheimer-

Volkoff Equations

Two important differences between Newtonian and GR equations:

1. 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

2. 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:

P ~ {5/3 (non-relativistic neutrons) 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 are lots of reasons why NS matter is non-ideal

(so that pressure is not provided only by degenerate neutrons)

(there would be no neutron stars in nature)

Some additional effects 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)

I. -stability

Neutron matter is formed by inverse -decay

p + e n + e

And is also unstable to -decay

escape

n p + e + e _

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:

a) 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.

b) 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.

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

n p + e + e _

yielding a relation between the chemical potentials of n, p, and e:

€

μn − μ p = μ e

And they can also decay through a different channel

n p + _

when the Fermi energy of neutrons exceeds the pion rest mass

€

EF ,n ≈ mπ c 2 ≈140 MeV

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:

It is very difficult for — to be present in the centers of neutron stars. How about other particles?

Nucleon reactions of the form

€

N + N → N + Λ + Κ

n + n → n + Σ+ + Κ−

are possible and lead to the creation of other particles with different decay properties.

For example, for the K mesons,

€

K 0 → 2γ

Κ− → μ− + ν μ → e− + ν e + ν μ + ν μ

Κ+ + μ− → 2γ + ν μ

which means that K0 and K+ will spontaneously decay, but

€

μΚ− =μ

e −

so K- can be present.

Hyperons in neutron-star matter

Hyperons and the masses of neutron stars

V. Quark Matter or Strange Matter

Exceeding a certain density, matter may preferentially be in the form of free (unconfined) quarks. In addition, because the strange quark mass is close to u and d quarks, the “soup” may contain u, d, and s.

Quark/hybrid stars: typically refer to a NS whose cores contain a mixed phase of confined and deconfined matter. These stars are bound by gravity. Strange stars: refer to stars that have only unconfined matter, in the form of u, d, and s quarks. These stars are not bound by gravity but are rather one giant nucleus.

Mass-Radius Relation for Neutron Stars

•We will discuss how accurate M-R measurements are needed to determine the correct EOS. However, even the detection of a massive (~2M) neutron star alone can rule out the possibility of boson condensates, the presence of hyperons, etc, all of which have softer EOS and lower maximum masses.

Stars withcondensates

Strange Stars

Normal Neutron Stars

Effects of Stellar Rotation on Neutron Star Structure

Spin frequency (in kHz)

Using Cook et al. 1994

Effects of Magnetic Field on Neutron Star Structure

Magnetic fields start affecting NS equation of state and structure when B ≥ 1017 G. by contributing to the pressure. For most neutron stars, the effect is negligible.

Masses and Radii of

Neutron Stars


Lecture 3

Mass-Radius Relation for Neutron Stars

Baryonic vs. Gravitational Mass

Important point about what we mean by NS mass:

We measure “gravitational” mass from astrophysical observations: the quantity that determines the curvature of its spacetime. This is different than “baryonic” mass: the sum of the masses of the constituents of the NS.

Remember the equation of structure for the NS:

€

dMgrav (r)

dr= 4π r2ρ (r)

€

Mgrav = 4π r2ρ (r) dr0

R

∫

Here, “r” is not the proper radius (the one a local observer would measure) but the Schwarzschild radius (which is smaller)

The baryonic mass can be calculated from

€

Mb = 4π 1−2GM(r)

rc 2

⎛

⎝ ⎜

⎞

⎠ ⎟r2ρ(r) dr

0

R

∫And is larger than Mgrav.

Why is Mgrav< Mb ?

Classically, the total energy in the volume of the NS is

€

Εtot = Mbc2 + Ε pot

Epot < 0

The mass seen by a test particle outside the neutron star is related to the total energy,

€

Mgrav ≈Ε tot

c 2≈ Mb −

|Ε pot |

c 2< Mb

This potential energy is released during the formation of the neutron star and is converted into heat. The heat escapes (mostly) in the form of neutrinos and (a small fraction) as photons.

• Dynamical mass measurements (very important but mass only)

• Neutron star cooling (provides --fairly uncertain-- limits)

• Quasi Periodic Oscillations

• Glitches (provides limits)

• Maximum spin measurements

Methods of Determining NS Mass and/or Radius

Dynamical Mass Measurements

€

( ˙ P b )GR = f (m1,m2,sin(i))The observed binary period derivative can be expressed in terms of the binary mass function. Need a short binary period, preferably a fast pulsar, a long baseline to get accurate timing parameters.

Also use Shapiro delay,

€

Δt = f (m2,sin(i))

Use the general relativistic decay of a binary orbit containing a NS

(For black holes, measurements are more approximate and rely on the binary mass function)

from Nice et al. 05

Limits on PSR J0751+1807

M = 2.1 M

Neutron Star CoolingWhy is cooling sensitive to the neutron star interior?

The interior of a proto-neutron star loses energy at a rapid rate by neutrino emission.

Within ~10 to 100 years, the thermal evolution time of the crust, heat transported by electron conduction into the interior, where it is radiated away by neutrinos, creates an isothermal core.

The star continuously emits photons, dominantly in X-rays, with an effective temperature Teff that tracks the interior temperature.

The energy loss from photons is swamped by neutrino emission from the interior until the star becomes about 3 × 105 years old.

The overall time that a neutron star will remain visible to terrestrial observers is not yet known, but there are two possibilities: the standard and enhanced cooling scenarios. The dominant neutrino cooling reactions are of a general type, known as Urca processes, in which thermally excited particles alternately undergo - and inverse- decays. Each reaction produces a neutrino or antineutrino, and thermal energy is thus continuously lost.

Neutron Star Cooling

The most efficient Urca process is the direct Urca process.

This process is only permitted if energy and momentum can be simultaneously conserved. This requires that the proton to neutron ratio exceeds 1/8, or the proton fraction x ≥ 1/9.

If the direct process is not possible, neutrino cooling must occur by the modified Urca process

n + (n, p) → p + (n, p) + e− + νe p + (n, p) → n + (n, p) + e + + νe

Which of these processes take place, and where in the interior, depend sensitively on the composition of the interior.

Neutron Star Cooling

Caveats: Very difficult to determine ages and distances Magnetic fields change cooling rates significantly

Quasi-periodic Oscillations

Accretion flows are very variable, with timescales ranging from 1ms to 100 days!

VAN DER KLIS ET AL. 1997

Power Spectra of Variability:

QUASIPERIODICOSCILLATIONS

BROAD-BAND VARIABILITY

HIGH FREQUENCIES

Quasi-periodic Oscillations

from Miller, Lamb, & Psaltis 1998

Limits from Maximum Neutron Star Spin

The mass-shedding limit for a rigid Newtonian sphere is the Keplerian rate:

€

PminN = 2π

R3

GM

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

= 0.545Mo

M

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2R

10km

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

ms

Fully relativistic calculations yield a similar result:

€

Pmin = 0.83Mo

M

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2R

10km

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

ms

for the maximum mass, minimum radius configuration.

Depending on the actual values of M and R in each equation of state, the obtainable maximum spinfrequency changes.







More promising methods (entirely in my opinion):

• Thermal Emission from Neutron Star Surface

• Eddington-limited Phenomena

• Spectral Features

Methods to Determine M and/or R

Radius for a thermally emitting object from continuum spectra:

R2 = F D2

T4

RX J1856

Mass from the Eddington limit:


LEdd= 4 G c M

(1+X)

At the Eddington Limit, radiation pressure provides support against gravity


Globular Cluster BursterKuulkers et al. 2003

M/R from spectral lines:

E = E0 ( )

Cottam et al. 2003

2M

R1


NS

n

GR

neutron stars lens their own surface radiation due to their strong gravity

In reality, Mass and Radius are always coupled because

Gravitational Lensing

€

ϑ =4GM

c 2b

deflection angle

b impact parameter

A perfect ring of radiation: R/M = 3.52

NS

max= 900+deflection angle

Gravitational Self-Lensing

Self-Lensing

The Schwarzschild metric:

€

ds2 = dt 2 1−2M

R

⎛

⎝ ⎜

⎞

⎠ ⎟ − dr2 1−

2M

R

⎛

⎝ ⎜

⎞

⎠ ⎟−1

− f (ϑ ,φ)

Photons with impact parameters b<bmax can reach the observer:

€

bmax = R(1− 2M

R)−1/ 2

General Relativistic Effects



Lensing of a hot spot on the neutron star surface

Pulse Amplitudes

* Normalized to DC

Two antipodal hot spots at a 45 degree angle from the rotation axis

€

bmax = R(1− 2M

R)−1/ 2

Apparent Radius of a Neutron Star

Because of lensing, the apparent radius of neutron stars changes

R2 = F D2

T4

2M

R(1 )

-1

2M( )

1/2

R1LEdd =

4 G c M

(1+X)

The correct expressions (lowest order)

GR Modifications

Effects of GR

Modifications to the Eddington limit

What if the NS is rotating rapidly?

v = 0.1 c

R

~ 600 Hz

Doppler Boosts

E =E0

R/c)

Time delays

t = R/c

Frame dragging

Oblateness

(Stergioulas, Morsink,Cook)

Other effects:

Equation of State

Özel & Psaltis 03

Effect of Rotation on Line Widths

E/Eo M/R FWHM R

May affect the inferred redshift and detectability BUT

A fourth method!

Combining the Methods

2. Surface emission gives a maximum NS mass!!

3. Eddington limit gives a minimum radius!!

1. The methods have different M-R dependences: they are complementary!

Özel 2006

gravity effects can be undone

Surfaces of

Neutron Stars

&

Observations of their Masses, Radii and Magnetic Fields


Lecture 4

Determining Mass and Radius

2. Surface emission gives a maximum NS mass!!

3. Eddington limit gives a minimum radius!!

1. The methods have different M-R dependences: they are complementary!

Özel 2006

gravity effects can be undone

A Unique Solution for Neutron Star M and R

M and R not affected by source inclination because they involve flux ratios

Effect of Systematic Errors

A 15% systematic error in the assumed distance will prohibit a unique solution

Applying the Methods to Sources:

For isolated sources: Can use surface emission from cooling to get area contours (and possibly a redshift)

For accreting sources: Can possibly apply all these methods, especially if there is Eddington limited phenomena

Pros and Cons of Surface Emission from Isolated vs. Accreting:

Isolated: Accreting:

Pros:

Cons: Heavy elements --atmospheres complicated

Accretion luminosity can be high

Strong magnetic fields --atmospheres complicated

Non-thermal emission often dominates

Heavy elements may not be present -- redshifted lines unlikely

Surface emission non-uniform

Eddington-limited phenomena

(Redshifted) spectral features more likely

Surface emission likely to be uniform

Bright

No heavy elements --atmospheres simple

No accretion luminosity

Good Isolated Candidates

• Nearby neutron stars with no (or very low) pulsations

• No observed non-thermal emission (as in a radio pulsar)

• (Unidentified) spectral absorption features have been observed in some

Sources have been dubbed “the magnificient seven” initially, even though they are more numerous now.

Good Accreting Candidates: LMXBs Showing Thermonuclear Bursts

Sample lightcurves, with different durations and shapes.

Spectra look pretty featureless and are traditionally fit with blackbodies of kT~few keV.

Thermonuclear Bursts and Eddington-limited Phenomena

QuickTime™ and aBMP decompressor


Burst proceeding by deflagration

Bursts propagate and engulf the neutron star at t << 1 s.

from Spitkovsky et al.


QuickTime™ and aVideo decompressor


from Zingale et al.

Emission from neutron stars during thermonuclear bursts are likely to be uniform and reproducible

Theoretical reasons to think that the emission is uniform and reproducible

Magnetic fields of bursters (in particular 0748-676) are dynamically unimportant

(for EXO 0748: Loeb 2003)

--> fuel spreads over the entire star



An Eddington-limited (i.e., a radius-expansion) Burst

A flat-topped flux, a temperature dip, a rise in the inferred radius


The Constant Peak Luminosity

The peak luminosity is constant to 2.8% for 70 bursts of 4U 1728-34

Galloway et al. 2003

Measuring the Eddington Limit: The Touchdown Flux

An “H-R” diagram for a burst

Temperature (keV)

Lum

inos

ity

(arb

itra

ry)

3 2

R

2R

4R

Constant inferred radius from

Fcool

Tc

4

Savov et al. 2001

Reliability of the Inferred “Radius”

Need a model for the surface emission!

N.B. This is both to determine NS mass and radius but also to understand a wide range of phenomena happening on neutron stars!

Emission from the Surfaces of Neutron Stars: Isolated NS

I. Composition of the Surface:

Ne= Np and dτ = Ne Tdz ==> τ = Ne T z (assuming electron density is independent of depth)

€

m =τ

σ T

mp 4π RNS2

For typical values, m=10-17 M for an unmagnetized neutron star.

1. How much material is necessary to cover the surface and dominate the emission properties?

Assume zero magnetic field, need material to optical depth τ=1.

€

m = ρV

= ρ 4π RNS2 h

= N pmp 4π RNS2 h

2. How long does it take the cover the NS surface with a 10-17 M hydrogen or helium skin by accreting from the ISM?

Using Bondi-Hoyle formalism:

€

˙ M =4π (GM)2 ρ ISM

v 3

If we take

€

v ≈107 cm /s

ρ ≈ mp /cm3 ≈1.7×10−24 g /cm3

M ≈1.5 × 2 ×1033 g

€

˙ M ISM ≈ 7 ×108 g /s ≈10−17 Mo / yr taccr= 1 yr.

Assuming magnetic fields do not prevent accretion, very quickly, NS surfaces can be covered by H/He.

3. Settling of Heavy Elements

Heavy elements settle by ion diffusion, as they are pulled down by gravity and electron current.

(Bildsten, Salpeter, & Wasserman)

€

tsettle ≈13sg

1014

⎛

⎝ ⎜

⎞

⎠ ⎟−1

kT

1keV

⎛

⎝ ⎜

⎞

⎠ ⎟

−3 / 2

How long does it take for them to settle below optical depth ~1 (where they no longer affect the spectrum?)

(T enters because it affects the speed of ions and the inter-particle distances)

II. Ionization State of the Atmosphere and Magnetic Fields:

1. The ionization state of a gas is given by the Saha equation:

€

nH

npne

=VZH

ZeZ p

Partition function Z defined for each species:

€

Ze =V

2π λ e3

, λ e = (2π h2

mekT)1/ 2

Z p =V

2π λ p3

e−χ / kT

When we consider H atoms at kT ≈ 1keV, <<kT so the atmosphere is completely ionized.For lower temperatures (kTeff ~ 50 eV), need to consider the presence of neutral atoms.

2. Magnetic Fields

At B ≥ 1010 G, magnetic force is the dominant force, >> thermal, Fermi, Coulomb energies.

Photon-Electron Interaction in Confining Fields

e-

-

B

parallel mode

perp mode

Magnetic Opacities

expect non-radial beaming and deviations from a blackbody spectrum

Ts

Tbs EE

≈

≈

)/(

Energy, angle & polarization dependence

2

1

1

1

s /

Ne T

Vacuum Polarization Resonance

-- at B ~ Bcr virtual e+ e- pairs affect photon transport

-- resonance appears at an energy-dependent density

Plasma-dominatedVacuum-dominated

-- proton cyclotron absorption features appear at ~keV, and are weak

Emission from the Surfaces of Neutron Stars: Accreting Case

I. Composition of the Surface:

A steady supply of heavy elements from accretion as well as thermonuclear bursts

Atmosphere models need to take the contribution of Fe, Si, etc.

II. Ionization State:

Temperatures reach ~few keV. Magnetic field strengths are very low (108--109 G)

Light elements are fully ionized. Bound species of heavy elements.

III. Emission Processes: Compton Scattering

Most important process is non-coherent scattering of photons off of hot electrons Bound-bound and bound-free opacities also important for heavy elements

Compton Scattering

“Compton” scattering is a scattering event between a photon and an electron where there is some energy exchange (unlike Thomson scattering which changes direction but not the energies)

By writing 4-momentum conservation for a photon scattering through angle , we find

€

E f

E i

=1− β i cosα i

1− β i cosα f +E i

γmc 2(1− cosθ)

Energy gainfrom the electron

Recoil term

Typical to expand this expression in orders of , and average over angles.

To first order, photons don’t gain or lose energy due to the motion of the electrons (angles average out to zero)

Compton Scattering

To second order, we find on average

€

ΔE

E i

=1

3β i

2 −E i

mc 2

Energy gainfrom K.E. of electron

Energy lossfrom recoil

If electrons are thermal,

€

i2 =

3kT

mc 2

€

ΔE

E i

=kT − E i

mc 2

If Ei < kT, photons gain energy

If Ei > kT, photons lose energy

Model Atmospheres:

Hydrostatic balance:

Gravity sustains pressure gradients

€

dP

dτ=

gρ

yG2 Neσ T

(τ = Neσ T dz)0

h

∫

yG is the correction to the proper distance in GR

€

yG = 1−2GM

Rc 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Equation of State:

Assume ideal gas P = 2NkT

yGμdIE

i

dτes

=χai I i −χa

i BE

2+χs

i I i − χsij(μ, ′ μ )∫

j=1,2∑ ′ I jd ′ μ

for i = 1, 2

H(τ)=σ Teff4 = I(τ,μ,E)μ dμ∫ dE

Radiative Equilibrium :

Techniques for solving the Transfer equation (with scattering):Feautrier Method, Variable Eddington factors, Accelerated Lambda Iteration…

Techniques for achieving Radiative Equilibrium:Lucy-Unsold Scheme, Complete Linearization…

Equation of Transfer:

Typical Temperature Profiles:

magnetic field

strengths

Typical Spectra (Isolated, Non-Magnetic):

From Zavlin et al. 1996

Typical Spectra (Isolated, Magnetic):

T=0.5 keV

B=4•1014 GB=6•1014 GB=8•1014 G

B=10•1014 G

B=12•1014 G

Typical Spectra (Accreting, Burster):

From Madej et al. 2004, Majczyna et al 2005

• Comptonization produces high-energy “tails” beyond a blackbody• Heavy elements produce absorption features

Color Correction Factors

From Madej et al. 2004, Majczyna et al 2005

EXO 0748-676

Redshifted lines with XMM:

Cottam, Paerels, & Mendez 2003

Four Eddington-limited bursts with EXOSAT and RXTE:

Gottwald et al. 1986, Wolff et al. 2005

z=0.35

€

Fcool /σ Tc4 =1.14 ± 0.10

€

FEdd = 2.25 ± 0.23×10−8 ergcm−2s−1

Slow rotation

= Hz

Villareal & Strohmayer 2004

Mass and Radius of EXO 0748-676

Özel 2006

M-R limits:

M = 2.10 ± 0.28 M

R = 13.8 ± 1.8 km

Neutron star equations of state need to allow heavy and large neutron stars

Future Prospects

- Monitoring and long exposure observations of bursters and isolated stars necessary

- Distance determination to sources (or their companions) would eliminate the need for the redshift

- Atmosphere models are getting more and more sophisticated

-In the meantime, we can try to understand emission mechanisms and magnetic field strengths of isolated NS, bursters, AXPs, SGRs, and even surface properties of some radio pulsars.

Spectral Analysis

Fits to seven epochs of XMM data on XTE J1810-197

Guver, Ozel, Gogus, Kouveliotou 07

Guver, Ozel, Gogus, Kouveliotou 07

Temperature Evolution and Magnetic Field of XTE J1810-197

Magnetic field remains nearly constant; is equal to spindown field! Temperature declines steadily and dramatically

No changes in magnetospheric parameters during these observations

Black Holes


Lecture 5

General relativity is a relativistic theory of gravity.

The invariant distance between two (nearby) events is given by

€

ds2 = −c 2dt 2 + dr2 + r2dϑ 2 + r2 sin2 ϑ dφ2

In special relativity, we specify events by 4 spacetime coordinates

€

(ct,r x )

€

ds2 = η μν dx μ dxν

or

General Relativity Basics

In GR, events are still specified by 4 coordinates, with the invariant distance given by

€

ds2 = gμν (xα ) dx μ dxν

€

gμνis the metric that specifies the properties of the (curved) spacetime in the presence of matter (energy).

Minkowski space

GR has two ingredients

The Equivalence Principle

€

˙ ̇ x μ +Γρσμ ˙ x ρ ˙ x σ = 0

Einstein’s Equation

€

Gμν = 8π Tμν

Newtonian Gravity Analog

Newton’s Second Law

€

˙ ̇ x μ −mG

mI

gμ = 0

Poisson’s Equation

€

∇2φ = −4πρ

Predictions of General Relativity

€

Gμν = 8π Tμν

Tμ contains all forms of energy

Solutions of this equation yield metrics that describe the properties of the spacetime.

Let’s first look at validations / tests of GR.

The Equivalence Principle Has Been Tested to a Very High Degree

Will 2001

What measures the strength of the gravitational field?

€

<<1

€

<<1

€

u /cVelocity,

€

ε ~GM

Rc 2

Potential

Weak Field Strong Field

€

≅1

€

≅1

Gravity

Matter

No scale in the theory! No field is either weak or strong!

The field equations have only been tested in the weak field limit.

Slide credit: D. Psaltis

€

1−E∞

E0

Redshift:

GENERAL RELATIVISTIC PHENOMENA

EclipseHulse-Taylor

Mercury

Moon

Neutron StarsGalactic Black Holes

AGN

LISAGP-B

LIGO

The Schwarzschild Metric

€

ds2 = − 1−2GM

rc 2

⎛

⎝ ⎜

⎞

⎠ ⎟dt 2 + 1−

2GM

rc 2

⎛

⎝ ⎜

⎞

⎠ ⎟−1

dr2 + r2dϑ 2 + r2 sin2 ϑ dφ2

• However, if the enclosed mass is so concentrated that it is within M<r/2, and you extend the vacuum solution to r=2M, this equation predicts an “event horizon”

customary to set G=c=1

€

ds2 = − 1−2M

r

⎛

⎝ ⎜

⎞

⎠ ⎟dt 2 + 1−

2M

r

⎛

⎝ ⎜

⎞

⎠ ⎟−1

dr2 + r2dϑ 2 + r2 sin2 ϑ dφ2

• This region of spacetime cannot communicate with the external universe.

• At r=0 (all the way inside a black hole), field equations predict a singularity of infinite density.

€

Gμν = 8π Tμν

Solution of the Einstein field equation

in spherical symmetry, in the absence of any matter (in the region of solution)is the Schwarzschild metric:

• The Schwarzschild metric describes the exterior spacetime of any spherical mass distribution (not just black holes)

Event Horizons

Proper time in this metric:

€

dτ = 1−2M

r

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

dt gravitational time dilation

At r=2M, proper time is infinite; gravitational redshift is infinite

==> it takes infinite amount of time for a signal emitted at the event horizon to reach a distant observer

Event Horizons

What happens to an observer getting close to the event horizon?

Collapse to a Black Hole

QuickTime™ and aAnimation decompressor


Spacetime during Collapse

QuickTime™ and aAnimation decompressor


Astrophysical Evidence for Black Holes & Event Horizons

Evidence for Event Horizons

Orbital period is a measure of the mass accretion rate M.

Menou et al. 98 Garcia et al. 01

Charles 1999

CAUSALITY LIMIT (?)

Signatures of a Black Hole

A Very Heavy Compact Object


Kinematics of gas around black hole


Water MASERs


Reverberation mapping

All kinematic measurements indicate ~billion solar mass black holes

The Case for Supermassive Black Holes



Using Kepler’s law, Ghez et al. and Gebhart et al. find 4x106 M inside this volume

from Stellar Dynamics

The mass of the black hole in the centerof the Milky Way

The Case for the Existence of Galactic Black Holes

•Variability timescale: Δt ~ 1-10 ms ==> emission zone size R ≤ c Δt ≤ 108 cm.

(coherent variable phenomena must occur over the size of the compact object/emission zone)

• Orbital parameters obtained from the Doppler curve give a mass function and a minimum primary mass

IMAGING OF SELF LENSING

C. REYNOLDS

EVEN FOR ACTIVE GALACTIC NUCLEI REQUIRES μarcsec X-RAY INTERFEROMETRY: THE BLACK HOLE IMAGER

Seeing Black Holes

What do Black Holes look like?

Broderick & Loeb 2006

Other Strange Phenomena

The existence of an Innermost Circular Stable Orbit

€

Veff = −G M

r+

1

2

L2

r2

For orbits in central potentials, we define

€

VeffGR = −

G M

r+

1

2

L2

r2(1−

2GM

r)

In GR,

Other Strange Phenomena

For any L, there are no stable orbits inside r=6M.

Correlations between black hole and galaxy properties

From Kormendy et al.

white dwarfs neutron stars black holes

Documents

degenerate stars

normal stars

structures of stars

degenerate fermions

type p

polytropic equation

number density

energy density