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White dwarfs : The galactic fossil population. Jordi Isern Institut de Ciències de l’Espai. IAC , October 3 th 200 6. Sirius B. # White dwarfs were discovered by Alvan Clark in 1862 L ~ 3x10 -3 L o, T , ~ 29500 K  R ~ 7.4x10 -3 R o

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White dwarfs:

The galactic fossil population

Jordi Isern

Institut de Ciències de l’Espai

IAC, October3th 2006


Sirius B

# White dwarfs were discovered by Alvan Clark in 1862

L ~ 3x10-3 Lo, T, ~ 29500 K R ~ 7.4x10-3 Ro

# Sirius is a binary system and it is possible to obtain the

mass, M ~ 1.053 Mo  ~ 3x106 g/cm3


The Fermi temperature is:

Tf ~109 K

Electrons are degenerate!


White dwarf structure

Energy flux


Energy reservoir

tcool ~ 10 Gyr !


The luminosity function

Observations: Stars/pc3/magnitude

Galactic properties

Stellar properties

Good observational properties


Reliable white dwarf models

Galactic properties

can be obtained: , tGal


Surveys are more and more accurate and signifivcative

Nevertheless, the cutoff is still poorly defined

Sloan sample of WD


~ 300,000 disc WD

~ 90,000 halo WD

white dwarf cooling
White Dwarf Cooling

To solve this equation it is necessary a L(TC) relationship

that depends on the properties of the envelope

the cooling process i
The cooling process (I)
  • Neutrino cooling [log(L/Lo) > -1.5]
    • Is the must complicated phase because the initial conditions are unknown.
    • Neutrinos dominate & thermal structures converge
    • Very short epoch ( < 108 yr)
  • Fluid cooling [-1.5 > log(L/Lo) > -3]
    • Gravothermal energy
    • Coulomb plasma
    • The main uncertainty comes from the C/O abundances that depend on the 12C(,)16O reaction , Z, & the treatment of convection

---- Oxygen

___ Carbon

Salaris et al 1997

Domínguez, Höflich & Straniero, 2001

the cooling process ii
The cooling process (II)
  • Crystallization [-3 > log(L/Lo) > -4.5]
    • Latent heat ( kTs per particle)
    • Sedimentation upon crystallization that depends on the chemical profile and the phase diagrams
  • Debye cooling [-4.5 > log(L/Lo) ]
    • At low temperatures, the specific heat follows the Debye law
    • Compression of outer layers is the main source of energy & prevents the sudden disappearence of the white dwarf

Change of the chemical profile because of solidification

Delay in the cooling

process introduced by

sedimentation of oxygen

upon crystallization

(DA atmosphere)


After solidification


white dwarf envelopes
White dwarf envelopes
  • DA: Pure H layers.
    • 90,000 K > Te > 6,000 K, below this T Balmer lines are not seen
  • DO: spectrum dominated by He II
    • 100,000 K > Te > 45,000 K. They are the hottest
    • C,N,O,Si are present in the photosphere
    • The coolest are H-poor
  • DB: He dominated armospheres
    • 30,000 K > Te > 12,000 K
    • There is a gap betwee DO and DB!!!
  • DQ: He dominated atmospheres
    • 12,000 K > Te > 6,000 K
    • C abundances in the range of 10-7 - 10-2
  • DZ: only metallic features (Ca II H-K)
    • T to small to show the lines of the dominant elements
  • DC: So cool that the dominant component is not seen
    • No lines deeper than 5%


Accretion from ISM

(H,He, metals)

Light elements float




Particle diffusion



Radiative levitation

Heavy elements sink


Two families of white dwarf envelopes

  • The H layer:
  • Acts as a source of opacity
  • If its mass is larger than 2x10-4 Mo, H-burning
  • Evolution predicts 10-4 Mo




150,000 K

  • The He layer
  • Important source of energy at very low Te
  • Low opacity (n-Das cool much faster)
  • Controls the diffusion of H inwards (DA-nDA)
  • Controle the diffusion of C outwards (DB-DQ)
  • Evolution predicts 10-2 Mo
  • Is the origin of the DA, n-DA
  • character:
  • primordial ?
  • mixing?
  • both?

6,000 K


Luminosity versus time

(dotted lines without sedimentation)




Long period waves  102 - 103 s

  • Gravity is the restoring force

Non-radial g-modes

Brunt-Väisälä frequency

Period increases as the star

cools down

The characteristic drift is ~ 10-15 ss-1



  • * Discovered by McCraw & Robinson
  • * Main properties
    • ·M 0.53 - 0.59 Mo
    • ·Te  11620 K
  • * Pulsation periods
    • ·0bs = 215.2, 271, 304.4 s
    • & harmonics + linear combinations

Data from 1975-90 

Data from 1975-00 

fiducial case mode identification
Fiducial case & mode identification

Parameters range

Additional constraints


Te=11620 K

Salaris et al’97 profile

log MHe/MWD =-2



The most critical parameters are:



Mode identification:

M=0.50 Mo k=1,2,3 log MH/MWD=-6.6

M=0.55 Mo k=1,2,3 log MH/MWD=-7.0

k=2,3,4 log MH/MWD=-4.0

M=0.60 Mo No satisfactory fit

We adopt:

M*=0.55 Mo

log(MH/M*) = -4.04

fiducial model error budget
Fiducial model & error budget

PF=210.4 s

dP/dt =3.9x10-15ss-1

Error budget


Source P(s) dP/dt (ss-1)x1015


Mode identification 6 1.0

M* 6 1.0

Chemical profile 4 0.1

Teff 2 0.2


The luminosity function

If the evolutive models are reliable it is possible to invert the integral

and to obtain .

, the initial mass function, is the statistical weight that connects the

stellar and the galactic properties and it is assumed constant along the

life of the Galaxy (there is not any example in the contrary) .

The solution is not unique and depends on the shape of the trial function


# The bulk of stars in the galactic plane have been formed during the last 9.5 Gyr

# The observed luminosity function is compatible with long tails as long as 20 Gyr

# It is necessary to improve the dim part of the luminosity function with deep and

large surveys in the red and infrared !

how did the halo form
How did the halo form?
  • By the monolithic collapse of an initial mass of gas?
  • By accretion of tidally disrupted satellite galaxies?
  • A mixture of both?

Probably the halo WD population can provide some

insight about the problem

  • WDs may have three origins:
  • Primitive halo
  • Capture from the exterior
  • Expelled from the disc
  • Identification criteria:
  • Kinematics
  • Age

The age depends on the lifetime in the main sequence: tMS(Z)

and on the cooling time cool (Z).


The halo luminosity function

*Halo WD cannot be identified by Z

*Radial velocities cannot be measured

because lines are to shallow

* The identification is only based on the

tangential component of the velocity

*Mixing between the halo and thick disk

*Only the bright part is known. Still

incomplete and doubtful




It is clear that if it was possible to detect the halo peak

it would be possible to determine the age of the galaxy!


The bright branch of the luminosity function is homologous respect to

the SFR and the information about the temporal properties is lost

during the process of normalization

In order to determine the history

of the halo it is necessary to obtain

the dim branch


TH = 12 Gyr

Dt: 0.1, 1 & 3 Gyr

He WD cool much faster!




Discovery functions: Number of white dwarfs per pc3

and interval of magntude (mV < 20)

The chances to detect the dim part

is very small for pure He WD

Deep & wide surveys are needed

IR colors




Because of the blocking of the H2, some colors become bluer as

the WD cools down. This offers a good opportunity to detect them!

Can pure He WD exist?

What about the H accreted from

the ISM?


Pre-WD lifetime

Hurley et al

Salaris et al

Lifetime for 1M and different


Lifetime for Z=0.02 & 0.0005

(upper and lower curves respectively)

as a function of the mass

The lifetime decreases when the metallicity decreases !

influence of the metallicity
Influence of the metallicity
  • Pre-white dwarf lifetime
  • Initial-final mass relationship
  • C/O profiles:
    • Larger specific heat
    • Lower latent heat
    • Higher sedimentation energy
    • Energy release at lower temperatures
  • Sedimentation of impurities




Tcool (WD)

Straniero et al)

Ages compatible with the

error bars as a functionof

the metallicity

Possible ages as a function of the


Halo location


Salaris et al

13 Gyr

11 Gyr


T=13 Gyr





T=13 Gyr

dt = 13 Gyr

m > 0.90

Metallicity plays an important role at the moment to assign an age, but not in the LF
  • The present uncertainties in Mf-Mi do not allow to rule out any suspected halo WD on the basis of the age alone.
  • Some relationships predict an excellent agreement
  • If the IMF is universal, the halo is still producing bright, low mass WDs

A key problem:

is the IMF


Easy to check:

Salpeter’s like IMFs are still producing bright, young WD in the halo!

Biased IMFs will produce an

excess of dim WDs

the initial final mass function is required for
The initial – final mass function is required for:
  • Determination of supernova rates
    • Core collapse supernovae (Fe & O/Ne cores)
    • Thermonuclear supernovae (CO cores)
  • Chemical evolution of the Galaxy
  • Star formation and feedback processes in galaxies
  • Understanding the properties of the galactic population of white dwarfs:
    • Field white dwarfs (halo & disk)
    • Clusters (open & globular)

Despite its importance, this function is poorly known both from the observational and theoretical point of views.

First attempt: Weidemann (1977)



Initial Final Mass


(Weidemann 2000)


Is it a single valued function?

mWD (MSP, Z, , NHe/NH, B, binariety...)

# Magnetic white dwarfs are systematicaly more massive (Ferrario, 1998)

(systematic bias in the measurement of the mass?)

# Certainly, close binaries can evolve diferently


MSP = 6.5 Mo

# Because of lifting effects,

rotation can modify the final size

of the core! (Dominguez et al’93)


Initial - Final mass

Hurley et al

Salaris et al

Domínguez et al



IFMR: Wood

SFR = exp

IFMR: Wood

SFR = const

Observational data from the PG survey (LBH’05)


IFMR: Domínguez

SFR = const

IFMR: Domínguez

SFR = exp

Observational data from the PG survey (LBH’05)


Fraction of white dwarfs with mass larger than m0

(from the PG survey – LBH’05)

• m > 0.6 Mo

• m > 0.7 Mo


IFMR: Wood

SFR (unit volume) const

Single stars

IFMR: Wood

SFR (unit volume) const

Single stars


IFMR: Domínguez et al

SFR (unit volume) const

Single stars

IFMR: Domínguez et al

SFR (unit volume) exp

Single stars


Mass distribution

(single stars)

Normalization from

the LF

Wood, exp

Domínguez, exp

Dominguez, cnst


Mass distribution of DA

white dwarfs

(Palomar Green survey)

Liebert et al (2005)

Excess of WD in the

mass range 1.0 – 1.4

Defect of WD in the

mass range 0.65 – 0.85

(Common proper motion pair:

REJ0317-853 & LB9802;

The most massive is much

hotter than the less massive)

Helium white dwarfs

produced by binary


Birthrate calculationIsern et al, Thermonuclear Supernovae, Ed. Ruiz-Lapuente, Canal, Isern, Kluwer p. 127 (1997)
  • Models obtained with FRANEC. Solar metallicity
  • Dmínguez/Straniero IFMR
  • Common envelope treatment: Iben & Tutukov (1984)
  • Salpeter’s IMF for the primary,
  • F(q)  q; q = M2/M1
  • Distribution of initial separations: H(A0)  1/A0
  • During the merging ALL the mass of the secondary is transfered to the primary

0.6+0.8 case

Guerrero et al 2004

Position of the particles

Velocities of the particles


Mass distribution


Binaries, merging

allowed, all population

Binaries, merging

allowed, hot

Normalization to the tota

population independently


Fraction of WD versus luminosity

• m > 0.6 Mo

• m > 0.7 Mo

Binary effects (continuous line)

Single WD (dotted line)

Clusters & non-interacting binaries suggest an IFMR with a large dispersion. A large part of this dispersion may be due to models
  • The LF of massive WD can only be reproduced if the IFMR favors the formation of massive WD
  • Binary interaction favors the formation of massive WD
  • Merging seems to be the responsible of the bumps observed in the mass distribution
double degenerate binary systems
Double degenerate binary systems

The search of DD systems is difficult because of the

character intrinsically dim of white dwarfs.

  • Several strategies are possible:
  • Radial motions in spectroscopic binaries (Robinson & Schafter, 1987)
  • White dwarfs of very low mass (Bergeron et al 1991)
  • White dwarfs showing peculiar, composite spectra and colors
    • (Bergeron et al 1990)
  • Common proper motion binaries (Sion et al 1991)

Several systems (>15) are known and the search continues:

Napiwotzki et al, Marsh et al...

The interest relies on the fact that these systems can merge

in less than a Hubble time if roughly a < 3 R

coalescence of two white dwarfs
Coalescence of two white dwarfs

There are several double degenerate binary systems able to merge

in a time shorter than the Hubble time because they loose

angular momentum as a consequence of gravitational wave emission

  • Type Ia supernovae?
  • Accretion induced collapse?
  • Peculiar WD?

Which are the observational


The outcome depends on:

# The chemical composition of both white dwarfs

# The effective accretion rate from the disc to the white dwarf

# Total mass of the final system (M > MCh ?)


The merging process

The less massive star transfers mass to the most massive

( R  M-1/3 ) As R2 increases when M2 decreases,

transfer accelerates

Conservative transfer

Since dM1 > 0, da >0 and the separation increases

  • There is a critical value M2 = 0.3 - 0.4
    • If M2 > Mc dynamic merging
    • If M2 < Mc self regulated merging
method and input physics
Method and input physics
  • SPH (Smoothed Particle Hydrodynamics)
    • Artificial viscosity (Balsara 1995)
    • Binary tree (Barnes & Hut 1986)
    • Polynomic kernel (Monaghan & Lattanzio 1985)
    • Predictor corrector (Serna et al 1996)
    • Energy & angular momentum conserved at 0.01% in 104 steps
    • 20,000 particle per star
  • EOS: Degenerate electrons + Coulomb plasma + photons
  • Full set of nuclear reactions (α-chain from He to Zn) plus the C+C and O+O reactions

3D density evolution

Case: 0.6+0.8 Mo

Unit mass : 1 Mo

Unit length: 0.1 Ro

G = 1


3D temperature evolution

Case: 0.6+0.8 Mo


0.6+0.8 case

Position of the particles


0.6+0.8 case

Velocities of the particles


0.6+0.8 case

Equatorial temperature



0.6+0.8 case

Density profiles of the disk

1000 g/cm3

Z (10-1 Ro)



R (10-1 Ro)


0.6+0.8 case

Evolution of the tangential

velocity respect to the center

of masses

The critical point is the

intrinsic viscosity of the

SPH methods


Preliminary results

  • The total mass expelled is very small.
  • Temperature is never high enough to initiate a prompt
  • explosion.
  • The mass remains roughly confined around the equatorial
  • belt
  • The outcome will depend on the evolution of the disk
    • At which rate the disk transfer mass to the WD?
    • At which rate the WD can accept mass?

Gravitational waves

  • Accelerated masses produce GW
  • GW stretch & squeeze objects because of their quadrupolar nature
  • Two polarization states: “+” and “x”
  • If ldetector << λGW both polarizations act as:

+ polarization

x polarization





Chirping binaries




From LIGO pages


Maximum distances of detectability

S/N = 5

Detection probability

rmer ~ 0.03 - 0.008 yr-1

Vaccs ~ 1.2x1014 pc3

Vgal ~ 3x1011 pc3

rdiscovery ~ rmer

During merging the major part of the mass of the system remains bounded to it (at least for the adopted initial conditions.
  • There is not a prompt explosion.
  • A significant part of the gravitational signal is out from the expected confusion limit.
  • However, the signal is to weak to have any chance to be detected during the lifetime of LISA

Long period waves  102 - 103 s

  • Gravity is the restoring force

Non-radial g-modes

Brunt-Väisälä frequency

Period increases as the star

cools down

If the characteristic drift , ~ 10-15 ss-1, is known for a bunch of stars

anomalous cooling or heating could be detected



  • Axions
    • (Isern et al 1993, Althaus et al 2001)
  • Gravity constant
    • (Garcia-Berro et al 1996, Isern et al 2001)
  • Compactification scales of extra dimensions
    • (Biesiada & Malec 2001)
  • Neutrino magnetic momentum
    • (Blinnikov & Dunina-Barkovskaya 1994)
axion case
Axion case
  • Axions were proposed as a solution to the strong CP problem
    • KVSZ model -> Axions coulple to hadrons & photons
    • DFSZ model -> Axions also couple to charged electrons
  • Coupling is determined by the Peccei-Quinn scale f which is related to the mass of the axion
  • Experiments have failed to detect axions
  • Constraints from astrophysical arguments
    • Solar properties
    • Red giants (HB & AGB stars)
    • Gravitational supernovae
    • Cosmological considerations

10-2 - 10-5 eV

Asteroseismological properties

of white dwarfs ?


White dwarf cooling

Bremsstrahlung is dominant

L  T7

La  T4

Lph  T2.5



Mode identification:

M=0.50 Mo k=1,2,3 log MH/MWD=-6.6

M=0.55 Mo k=1,2,3 log MH/MWD=-7.0

k=2,3,4 log MH/MWD=-4.0

M=0.60 Mo No satisfactory fit

We adopt:

M*=0.55 Mo

log(MH/M*) = -4.04


Axions influence

Fiducial model + axions

Axions do not change the thermal


Rate of change of P

k=3 is trapped

Models are able to reproduce the evolution and seismological properties of DAVs
  • It is not necessary to invoke additional cooling to account for the behavior of G117-B15A
  • This allows to set a strong limit the mass of axions (DFSZ) macos2<4 meV
  • The interaction axion-electron cannot be ruled out
  • We need:
    • A set of DAVs with well defined fundamental parameters
    • Precise measurements of dP/dt
    • A good theory of the evolution of the outer layers