Download Document

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.
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
3
d
p
3
3
d xd p
All averages, such as energy density are given by


df
E 3 3 d3 p
d xd p
;
E  ( p 2c 2  m 2c 4 )1/ 2
includes particle rest mass
For an ideal fermion/boson gas in equilibrium,

1
f (E) 
exp[( E  ) /kT]  1
Fermion (half-integer spin particles)
Boson (integer-spin particles)
Some limits of f(E):
High temperature, low density:
f (E)  exp(

 E
kT
)
For fermions, chemical potential (energy cost of adding one particle) is the Fermi energy
f (E) 
1
exp[( E  E F ) /kT] 1
where Fermi energy EF is defined such that
f (E F ) 
1
2


Fermions at zero temperature (complete degeneracy):
f(E) ~
{
1
(E  EF)
0
(E > EF)
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:
N( p)dp  2 
V
4 p2dp
3
h
or in terms of energy by using E=p2/2m

N(E)dE 
8 V
(2m 3 )1/ 2 E1/ 2dE
3
h
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
1
Pe 
3

 N ( p) f ( p)v pdp
e
0
1
Pe 
3
pF
 N ( p)v pdp
e
0
Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic
case (v~c) to give

(3 2 ) 2 / 3 2 5
Pe 
n e / 3 , v  c
5
me
(3 2 )1/ 3
Pe 
cn e4 / 3 , v ~ c
4
Notice
 there is no dependence on me.
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.
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:
M ≥0.031 M
Smaller masses do not become MS
(ii) For He-burning at 2x108 K:
M ≥0.439 M
Just the mass if the He core; smaller mass stars
become degenerate before triple- kicks in.
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:
 

(  ui )  0
t x i

Continuity equation
u j
u j
1  ij
 ui
 aj 
t
 xi
  xi
Euler (momentum) equation--a fancy F=ma for pliable materials)
What is ij ?


ij  Pij   ij
off-diagonal terms:
Viscosity-stress
diagonal terms:
Pressure
So Euler equation becomes
u j
u j
1  P 1  ij
 ui
 aj 

t
 xi
  x j   xi

}
}
pressure
gradient
viscosity
To study the structures of compact objects, we assume hydrostatic equilibrium
(as is the case for stars in general)
In steady state:

0
t
ui  0
Static:

So the momentum eqn reads:


(with a j 
 1  P

0
x j  x j

Which in spherical coordinates is

1 P
GM(r)

 r
r2
and

M(r) 
r
 4 r (r)dr
2
0

)
x j
Now we’re ready to solve our set of equations for the structure of a polytrope:
1 P
GM(r)

 r
r2
dM(r)
 4 r 2 (r)
dr

P  K 
(alternatively
1
P  K
1
n
)
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


 (r  0)  c
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
1 

1
n
(n

1)K



c
a


4 G




1/ 2


r
a
So that the solution for R and M become
1n
(n  1)K 
R  1 
 c 2n
 4 G 
1/ 2

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,
3n
(n  1)K 
M  f (1 ) 4 
 c 2n
4

G


3/2
We can also write a relation for M and R:

M R
3n
1n
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
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
_
n  p + e + e
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 a
YUV420 codec decompressor
are needed to see this picture.
A supernova simulation from Burrows et al.
Formation of Neutron Stars
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Dividing Lines
From Fryer et al. 99
N.B. This is mostly to show uncertainties and possibilities than hard numbers!
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 White Dwarfs:
Accreting White Dwarfs:
Cooling white dwarfs
Cataclysmic variables, dwarf novae,
…..
Isolated Neutron Stars:
Accreting Neutron Stars:
Radio pulsars, millisecond radio pulsars,
magnetars (AXPs and SGRs), CCOs,
nearby dim isolated stars
low-mass X-ray binaries,
high-mass X-ray binaries,
bursters
Isolated Black Holes:
Accreting Black Holes:
Not visible!!
(except in gravity waves)
X-ray binaries
AGN, 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
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Chandra
Hubble
An Accretion Powered Pulsar
QuickTime™ and a
Video decompressor
are needed to see this picture.
- 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
P=Pdot diagram of pulsars
A Galactic Black Hole
QuickTime™ and a
Sorenson Video decompressor
are 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
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
Pe 
pF
1
3 2
 v( p) p dp
3
3
0
Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic
case (v~c) to give
(3 2 ) 2 / 3 2 5 / 3
Pe 
n e , v  c
5
me

(3 2 )1/ 3
Pe 
cn e4 / 3 , v ~ c
4

We solve this equation along with the continuity and force equations:

dPe
GM(r)
 (r)
dr
r2
To obtain
R  c

1n
2n
M  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

1/ 6
 

R  1.122 10  6 c 3 
10 gcm 
4
km
 

M  0.4964 6 c 3  M

10 gcm 
1/ 2

In the relativistic limit

4
 , n3
3
1/ 6
 c

R  3.347 10  6
3 
10 gcm 
4

M  1.457 M


km
Chandrasekhar limit
Remember there was no dependence on me, c, or R in the extreme relativistic 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 ~

3n
2n
c
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 n1/ 3
so that
1/ 3
EF ~
cN
R
while the gravitational energy per baryon is

EG ~ 
GMmB
R
Setting EF + EG = 0 gives


N max  2 10 57
M max  N max mB  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) 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)
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)
c d
L   4 r
(aT 4 )
3 dr
2
where opacity  is provided mainly by free-free and bound-free transitions.
For values appropriate for a white dwarf, we find
M 
L  2 106 erg s1 T*7 / 2
M
T*  10 6 10 7 K
How long does it take the White Dwarf to Cool?
3
M
U  k T*
2
A mu
L

dU
dt
Combining this with our expression for L and solving for the cooling time gives

 L 5 / 7
   
M 
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.
A typical
luminosity
function
N
magnitude
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):
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:
dM(r)
 4 r 2 (r)
dr
M(r) 
r
2
4

r
(r) dr

0
dP(r)
GM(r)

(r)
2
dr
r


Need a third equation relating P(r) and (r ) (called the equation of state --EOS)
P  P( )

Solve for the three unknowns M, P, 

(enclosed mass)
Equations in General Relativity:
dM(r)
 4 r 2 (r)
dr
3
G
M(r)

4

r
P(r)

dP(r)
P 

(r)  2 
2GM(r)  
dr
c 
2 
r 1



rc 2 
}
OppenheimerVolkoff 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 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 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
escape
And is also unstable to -decay
_
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:
n   p  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  m 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  
n  n  n    
are possible and lead to the creation of other particles with different decay properties.

For example, for the K mesons,
K 0  2
      e  e     
     2    
which means that K0 and K+ will spontaneously decay, but

so K- can be present.
  e


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
Normal Neutron Stars
Stars with
condensates
Strange 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.
Effects of Stellar Rotation on Neutron Star Structure
Using Cook et al. 1994
Spin frequency
(in kHz)
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.
INPE Advanced Course on Compact Objects
Lecture 3
Masses and Radii
of
Neutron Stars
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:
dM grav (r)
 4 r 2 (r)
dr
M grav  4 
R
 r (r) dr
2
0
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
 2GM(r)  2
M b  4  1
r (r) dr
2


rc
0
R
And is larger than Mgrav.

Why is Mgrav< Mb ?
Classically, the total energy in the volume of the NS is
 tot  M b c   pot
2
Epot < 0
The mass seen by a test particle outside the neutron star is related to the total energy,

M grav 
| pot |
 tot

M

 Mb
b
c2
c2
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.
Methods of Determining NS Mass and/or Radius
• Dynamical mass measurements (very important but mass only)
• Neutron star cooling (provides --fairly uncertain-- limits)
• Quasi Periodic Oscillations
• Glitches (provides limits)
• Maximum spin measurements
Dynamical Mass Measurements
Use the general relativistic decay of a binary orbit containing a NS
(PÝb )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,

t  f (m2,sin( i))
(For black holes, measurements are more approximate and rely on the binary mass function)

Limits on PSR J0751+1807
from Nice et al. 05
M = 2.1 M
Methods of Determining NS Mass and/or Radius
• Dynamical mass measurements (very important but mass only)
• Neutron star cooling (provides --fairly uncertain-- limits)
• Quasi Periodic Oscillations
• Glitches (provides limits)
• Maximum spin measurements
Neutron Star Cooling
Why 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
Methods of Determining NS Mass and/or Radius
• Dynamical mass measurements (very important but mass only)
• Neutron star cooling (provides --fairly uncertain-- limits)
• Quasi Periodic Oscillations
• Glitches (provides limits)
• Maximum spin measurements
Quasi-periodic Oscillations
Accretion flows are very variable, with timescales ranging from 1ms to 100 days!
QUASI
PERIODIC
OSCILLATIONS
VAN DER KLIS ET AL. 1997
Power Spectra of Variability:
HIGH FREQUENCIES
BROAD-BAND VARIABILITY
Quasi-periodic Oscillations
from Miller, Lamb, & Psaltis 1998
Methods of Determining NS Mass and/or Radius
• Dynamical mass measurements (very important but mass only)
• Neutron star cooling (provides --fairly uncertain-- limits)
• Quasi Periodic Oscillations
• Glitches (provides limits)
• Maximum spin measurements
Limits from Maximum Neutron Star Spin
The mass-shedding limit for a rigid Newtonian sphere is the Keplerian rate:
N
Pmin
 R 3 1/ 2
M 1/ 2  R 3 / 2
 2 
 ms
  0.545  



M
10km
GM 
Fully relativistic calculations yield a similar result:

Pmin
M 1/ 2  R 3 / 2
 0.83  
 ms
 M  10km 
for the maximum mass, minimum radius configuration.

Depending on the actual values of M and R in each equation of state, the obtainable maximum spin
frequency changes.
Methods of Determining NS Mass and/or Radius
• Dynamical mass measurements (very important but mass only)
• Neutron star cooling (provides --fairly uncertain-- limits)
• Quasi Periodic Oscillations
• Glitches (provides limits)
• Maximum spin measurements
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:
2
F
D
R2 =
 T4
RX J1856
Methods to Determine M and/or R
Mass from the Eddington limit:
4GcM
LEdd
=
 (1+X)
At the Eddington Limit, radiation pressure provides support against gravity
Methods to Determine M and/or R
Globular Cluster Burster
Kuulkers et al. 2003
Methods to Determine M and/or R
M/R from spectral lines:
E = E0 (1
2M
R
)
Cottam et al. 2003
In reality, Mass and Radius are always coupled because
neutron stars lens their own surface radiation due to their strong gravity
NS
GR
n
Gravitational Lensing
4GM
 2
cb
  deflection angle
b  impact parameter

Gravitational Self-Lensing
NS
max= 900+deflection angle
A perfect ring of radiation:
 R/M = 3.52
Self-Lensing
The Schwarzschild metric:
1




2M
2M
2
ds2  dt 2 1
  dr 1
  f ( ,  )


R 
R 

Photons with impact parameters b<bmax can reach the observer:
bmax
M 1/ 2
 R(1 2 )
R
General Relativistic Effects
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Lensing of a hot spot on the neutron star surface
* Normalized to DC
Pulse Amplitudes
Two antipodal hot spots at a 45 degree angle from the rotation axis

Apparent Radius of a Neutron Star
bmax
M 1/ 2
 R(1 2 )
R
Because of lensing, the apparent
radius of neutron stars changes
GR Modifications
The correct expressions (lowest order)
R2 =
LEdd
=
D2
F
 T4
(1
4GcM
 (1+X)
(1
2M
R
-1
)
2M
R
)
1/2
Effects of GR
Modifications to the Eddington limit
What if the NS is rotating rapidly?
E =E0  (1R/c)
Doppler Boosts
R
t = / ~ R/c
 /2 ~ 600 Hz
v = 0.1 c
Time delays
Other effects:
Frame dragging
Oblateness
Equation of State
(Stergioulas, Morsink,Cook)
Effect of Rotation on Line Widths
Özel & Psaltis 03
May affect the inferred redshift and detectability BUT
E/Eo  M/R
FWHM  R
A fourth method!
Combining the Methods
1. The methods have
different M-R
dependences:
they are
complementary!
2. Surface emission
gives a maximum
NS mass!!
3. Eddington limit
gives a minimum
radius!!
gravity effects
can be undone
Özel 2006
INPE Advanced Course on Compact Objects
Lecture 4
Surfaces
of
Neutron Stars
&
Observations of their Masses, Radii and Magnetic Fields
Determining Mass and Radius
1. The methods have
different M-R
dependences:
they are
complementary!
2. Surface emission
gives a maximum
NS mass!!
3. Eddington limit
gives a minimum
radius!!
gravity effects
can be undone
Özel 2006
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:
Pros:
No heavy elements
--atmospheres simple
No accretion luminosity
Accreting:
Eddington-limited phenomena
(Redshifted) spectral features
more likely
Surface emission likely to be uniform
Bright
Cons:
Strong magnetic fields
--atmospheres complicated
Non-thermal emission often dominates
Heavy elements may not be present
-- redshifted lines unlikely
Surface emission non-uniform
Heavy elements
--atmospheres complicated
Accretion luminosity can be high
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 a
BMP decompressor
are needed to see this picture.
from Spitkovsky et al.
Burst proceeding by deflagration
Bursts propagate and engulf the neutron star at t << 1 s.
Thermonuclear Bursts and Eddington-limited Phenomena
QuickTime™ and a
Video decompressor
are needed to see this picture.
from Zingale et al.
Thermonuclear Bursts and Eddington-limited Phenomena
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
Emission from neutron stars during thermonuclear bursts are likely
to be uniform and reproducible
Thermonuclear Bursts and Eddington-limited Phenomena
An Eddington-limited (i.e., a radius-expansion) Burst
A flat-topped flux, a temperature dip, a rise in the inferred radius
Thermonuclear Bursts and Eddington-limited Phenomena
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
Luminosity (arbitrary)
An “H-R” diagram for a burst
4R
2R
R
3
2
Temperature (keV)
Reliability of the Inferred “Radius”
Fcool
Constant inferred radius from
4
 Tc
Savov et al. 2001
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:
1. How much material is necessary to cover the surface and dominate the emission properties?
Assume zero magnetic field, need material to optical depth =1.
m 
V
2

 4  RNS
h
2
 N p m p 4  RNS
h
Ne= Np and d = Ne Tdz ==>  = Ne T z (assuming electron density is independent of depth)

2
m
m p 4 RNS
T

For typical values, m=10-17 M for an unmagnetized neutron star.

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:
2
4

(GM)
 ISM
Ý
M
v3

If we take
v  10 7 cm /s
  m p /cm 3  1.71024 g /cm 3
M  1.5  2 10 33 g
ÝISM  7 10 8 g /s  1017 M / yr
M
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
(Bildsten, Salpeter, & Wasserman)
Heavy elements settle by ion diffusion, as they are pulled down by gravity and electron current.
How long does it take for them to settle below optical depth ~1 (where they no longer affect the spectrum?)
3 / 2
 g   kT 
tsettle  13s  14  

10  1keV 
1
(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
VZ H

n p ne Ze Z p
Partition function Z defined for each species:
2 2 1/ 2
Ze 
, e  (
)
2 3e
me kT
V
  / kT
Zp 
e
2 3p
V

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
B
e--
parallel mode
perp mode
Magnetic Opacities
2
s / NeT
1
  ( E / Eb )  T
1
s
1
2
  T
2
s
1
Energy, angle & polarization dependence
expect non-radial beaming and deviations from a blackbody spectrum
Vacuum Polarization Resonance
Vacuum-dominated
Plasma-dominated
-- at B ~ Bcr virtual e+ e- pairs affect photon transport
-- resonance appears at an energy-dependent density
-- 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
Ef
1 i cos i

E i 1  cos  E i (1 cos )
i
f
mc 2

Energy gain
from 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
E 1 2 E i
 i 
Ei 3
mc 2
Energy gain
from K.E. of electron

If electrons are thermal,
i2 

3kT
mc 2
E kT  E i

Ei
mc 2
If Ei < kT, photons gain energy
 If Ei > kT, photons lose energy
Energy loss
from recoil
Model Atmospheres:
Hydrostatic balance:
Gravity sustains pressure gradients
dP
g
 2
d yG N e T
( 
h
N
e
T
dz)
0
yG is the correction to the proper distance in GR

 2GM 1/ 2
yG  1

 Rc 2 
Equation of State:
 Assume ideal gas P = 2NkT
Equation of Transfer:
dIEi
i i
i BE
yG 
 a I  a
  si I i 
d es
2
ij
j

(

,


)
I

d
 s
j 1,2
for i = 1, 2
Radiative Equilibrium :
H( )   Teff 4   I( , , E)  d dE
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…
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 G
B=6•1014 G
B=8•1014 G
B=10•1014 G
B=12•1014 G
From Madej et al. 2004, Majczyna et al 2005
Typical Spectra (Accreting, Burster):
• 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:
z=0.35
Cottam, Paerels, & Mendez 2003
Four Eddington-limited bursts with EXOSAT and RXTE:
Fcool / Tc4  1.14  0.10
FEdd  2.25  0.23 108 erg cm2 s1

Gottwald et al. 1986, Wolff et al. 2005
Slow rotation
0= 44.7Hz
Villareal & Strohmayer 2004
Özel 2006
Mass and Radius of EXO 0748-676
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
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
Guver, Ozel, Gogus, Kouveliotou 07
INPE Advanced Course on Compact Objects
Lecture 5
Black Holes
General Relativity Basics
General relativity is a relativistic theory of gravity.
In special relativity, we specify events by 4 spacetime coordinates
(ct, x)
The invariant distance between two (nearby) events is given by

ds2  c 2 dt 2  dr 2  r 2 d 2  r 2 sin 2  d 2
or
ds2   dx  dx 

Minkowski space
In GR, events are still specified by 4 coordinates, with the invariant distance given by

ds2  g (x  ) dx  dx 
g is the metric that specifies the properties of the (curved) spacetime in the presence of
matter (energy).




GR has two ingredients
Newtonian Gravity Analog
Newton’s Second Law
 The Equivalence Principle

Ý
xÝ

 
mG 
Ý
xÝ
g 0
mI

 
xÝ xÝ  0
 Einstein’s Equation
G  8 T
Poisson’s Equation


 2  4 
Predictions of General Relativity
G  8 T
Solutions of this equation yield metrics that describe the properties of the spacetime.
T contains all forms of energy

Let’s first look at validations / tests of GR.
Will 2001
The Equivalence Principle Has Been Tested
to a Very High Degree
The field equations have only been tested in the weak field limit.
What measures the strength of the gravitational field?
Matter Gravity
No scale in the theory! No field is either weak or strong!

Weak Field
Strong Field
Potential
GM
~
Rc 2
1
1
Velocity, u /c
1
1


Slide credit: D. Psaltis
GENERAL RELATIVISTIC PHENOMENA
LIGO
Neutron Stars
Galactic Black Holes
LISA
GP-B
Eclipse
Hulse-Taylor
AGN
Moon
Mercury
Redshift: 1
E
E0
The Schwarzschild Metric
Solution of the Einstein field equation
G  8 T
in spherical symmetry, in the absence of any matter (in the region of solution)
is the Schwarzschild metric:

 2GM  2  2GM 1 2
ds   1
dt  1
dr  r 2 d 2  r 2 sin 2  d 2
2 
2 


rc 
rc 
2
customary to set G=c=1

 2M  2  2M 1 2
2
2
2
2
2
ds   1
dt  1
 dr  r d  r sin  d


r 
r 
2
• The Schwarzschild metric describes the exterior spacetime of any spherical mass distribution
(not
 just black holes)
• 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”
• 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.
Event Horizons
Proper time in this metric:
 2M 1/ 2
d  1
 dt

r 
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 a
Animation decompressor
are needed to see this picture.
Spacetime during Collapse
QuickTime™ and a
Animation decompressor
are needed to see this picture.
Astrophysical Evidence for Black Holes & Event Horizons
Evidence for Event Horizons
Menou et al. 98
Garcia et al. 01
.
Orbital period is a measure of the mass accretion rate M
Signatures of a Black Hole
Charles 1999
 A Very Heavy Compact Object
CAUSALITY LIMIT (?)
Signatures of a Black Hole
 Kinematics of gas around black hole
Signatures of a Black Hole
 Water MASERs
Signatures of a Black Hole
 Reverberation mapping
All kinematic measurements indicate ~billion solar mass black holes
The Case for Supermassive Black Holes
from Stellar Dynamics
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Using Kepler’s law, Ghez et al. and Gebhart et al. find 4x106 M inside this volume
The mass of the black hole in the center
of the Milky Way
The Case for the Existence of Galactic Black Holes
•Variability timescale: t ~ 1-10 ms ==> emission zone size R ≤ c t ≤ 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
Seeing Black Holes
IMAGING OF SELF LENSING
C. REYNOLDS
EVEN FOR ACTIVE GALACTIC NUCLEI REQUIRES
arcsec X-RAY INTERFEROMETRY:
THE BLACK HOLE IMAGER
Broderick & Loeb 2006
What do Black Holes look like?
Other Strange Phenomena
The existence of an Innermost Circular Stable Orbit
For orbits in central potentials, we define
Veff
G M 1 L2


r
2 r2

In GR,
GR
eff
V

G M 1 L2
2GM


(1
)
r
2 r2
r
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.