1. White Dwarf Cooling and Cosmochronology Jeffrey M. Silverman Astro 252: Stellar Structure & Evolution 11/30/2006

2. What are White Dwarfs (WDs)? (Stuff you should already know) Evolutionary endpoint of stars with ZAMS mass of 0.07M  to ~8M  97% of Milky Way stars will end up as WD No longer producing energy through fusion reactions Held up by electron degeneracy pressure (i.e. can’t contract) and P independent of T  WDs evolve at constant R ~ R  T eff ~ 150,000 K – 4000 K  T c ~ 2×10 7 K – 5×10 6 K

3. What are White Dwarfs (WDs)? (More stuff you should already know) M ~ 0.59 ± 0.13 M  (mass distribution sharply peaked) Large , large g, coolest WDs have very low L Most WDs composed of:  A mixture of C and O cores  Thin H and/or He atmospheres Observed radiation from WDs comes from the thin atmosphere (photosphere) Most WDs have H atmospheres (DA WDs), ~25% do not (non-DA WDs)

4. The Basics of WD Cooling (Even more stuff you should already know) Degenerate e - are great conductors  WD cores are quickly thermalized WDs are made of:  a (nearly) isothermal core (containing ~0.99m WD )  a thin, opaque, insulating, nondegenerate outer envelope (or atmosphere) e - do not contribute to the energy reservoir (they’re already degenerate) WD cools  ions lose kinetic energy  core goes from gas to liquid to (eventually) solid Liquid to solid transition is a 1 st order phase transition  releases latent (~1 keV per ion)  decreases the rate of cooling WDs eventually become a cool, crystallized object known as a “black dwarf”

5. WD Cooling and Convection If present, convection in the envelope affects the emergent flux we observe It also directly affects the cooling rate:  If the base of the convection zone reaches into the degenerate core, they become thermally coupled  This drastically increases the energy transfer (and thus increases the rate of cooling)  This is called “convective coupling”

6. The Cooling Rate Cooling rate depends on:  Amount of thermal energy stored Must know the internal thermodynamics of the fluid/solid core  How rapidly it escapes through the thin atmosphere Solve in detail the energy transfer across the star; must know the thermodynamics and radiative and conductive opacities of the gas/fluid envelope; must also worry about convection

7. Cooling Rates  is the energy generation rate minus neutrino losses (= 0 for WD) 2 nd term is rate of gravitational energy release (= 0 for WD) 3 rd term is rate of thermal energy release C v = specific heat per gram at constant volume Assume T = T c = constant L(t 1 ) = L 1 L(t 2 ) = L 2 t cool = t 2 – t 1 

8. Cooling Rates: A 1 st Approximation A = atomic weight of core material  env = mean molecular weight of envelope material  e = mean molecular weight of e - Assumptions made:  Neglect e - heat capacity  Ions are an ideal gas  Envelope is in radiative equilibrium  Kramer’s Law for opacity  t cool  A -1  env -2/7  e 4/7 M 5/7 L -5/7

9. Cooling Rates: A 1 st Approximation Mestel first did this in 1952 We basically did this in class Was the first analysis of WD cooling As T eff decreases, all of the assumptions quickly become invalid!!  t cool  A -1  env -2/7  e 4/7 M 5/7 L -5/7

10. The History of WD Cooling Theory Difficult to model due to high  and low T eff Mestel (1952) – first derived t cool Schmidt (1959) – first recognized usefulness of WDs as cosmochronometers Van Horn (1971) – improved on Mestel’s model Lamb (& Van Horn) (1975) – relaxed Mestel’s approximations; full evolutionary code Winget et al. (1987) – computed L of local WDs (pure C cores); estimated an age of the Galactic disk Wood (1990 & 1995) – more realistic compositions (C & O) but has some flaws at low L Most models have trouble with T eff < 4000 K (this has been studied a lot recently)

11. Fontaine (2001) – PASP Invited Review Most accurate and detailed WD cooling model to date It includes:  Details of the crystallization process  Diffusion of different elements  Convective mixing  Nuclear process (i.e. residual fusion) Works down to T eff = 1500 K They admit that the weakest item is the “lack of reliable low-temperature and high-density radiative opacities”.

12. DA WD, 0.6 M , pure C core, T eff next to each curve Thin (but opaque) atmosphere/envelope Vertical dashed lines – H/He and He/C transition zones e - degenerate right of dots Ions are a fluid right of circles Even in the hottest model, C core is completely liquid (i.e. ideal gas law fails for core) Thick part of curves – solid core; by 2000 K >99% of the mass of the star has solidified Cores never perfectly isothermal (another failed assumption) Dotted part of curves is where convection is important Models with T eff ≤ 15,000 K have a convection zone that will affect emergent flux Fontaine (2001)

13. DA WD, 1.3 M , pure C core Kinks in curve seen in models at all masses The atypical large M is shown here because the kinks are well spaced Kinks for lower mass WDs overlap (i.e. kinks occur at similar times) 1 st kink due to the release of latent heat upon crystallization  It is gradual and continues until most of the WD has solidified 2 nd kink due to convective coupling  Before this the core is thermally insulated by the opaque envelope  Bump – convection breaks into thermal reservoir (i.e. degenerate core), now more energy must be radiated by the WD, cooling stalls  Flattening out – after initial stalling, convection speeds up cooling (as compared to radiative transport alone) since core is now less insulated Fontaine (2001)

14. WD Cooling & Mass At higher luminosities, higher mass WDs take longer to cool (since they have a larger energy reservoir) Once crystallized, the specific heat of the ions drops like a rock (i.e. the ions can’t store Hansen (2003) heat anymore) due to “simple” Debye theory of solids in quantum statistical mechanics This leads to a rapid final phase of cooling known as “Debye cooling” Since more massive WDs crystallize sooner (due to the higher  ), they actually cool faster than low mass WDs at low luminosities! DA WD

15. Pure C takes ~2.4 Gyr longer than pure O to cool to 10 -4.5 L  Pure C takes ~3.6 Gyr longer than pure O to cool to 10 -6 L  The mixed core model has intermediate cooling times, as expected Mestel’s approximation predicts this (t cool  A -1 ) Fontaine (2001) C vs. O Cooling Curves Model 2: uniformly mixed C/O core (X C = X O = 0.5) Figure shows only the lowest L tail of the cooling curves

16.  Treatment of core composition and separation of elements  Non-ideal effects of the equation of state of the atmosphere Hansen (2003) Cooling Curves vs. Cooling Curves Qualitative agreement is encouraging Quantitative differences at the 10% level Largest discrepancies occur at lowest L (i.e. lowest T eff ) Differences likely due to:

17. H vs. He Atmospheres & Metals Decreasing the fractional mass of the He layer increases cooling time:  X He = 10 -2  10 -3  t cool goes up by a factor of ~20 Bradley (2001):  5 out of 7 WDs consistent with the “standard” atmosphere used in Fontaine (2001): X He = 10 -2 X H = 10 -4 WDs with thicker H layers, or equivalently thinner He layers, cool slower because H is a better insulator than He:  Extreme case: DA WDs take ~2.5 Gyr longer to cool to 10 -5 L  than non-DA WDs Effects of varying Z have yet to be quantitatively studied with modern codes

18. Cooling Rate Problems: Spectral Evolution DA WDs can turn into non-DA WDs (and vice versa) by:  Convection and mixing of thin surface layers  Accretion  Diffusion of elements Cannot be certain a WD has cooled as a pure DA model or a pure non-DA model Many WDs “must” evolve as DA WDs since observations match DA models so well

19. Cooling Rate Problems: Accretion WDs can Bondi accrete ISM Might alter the composition of the outer layers of the WD (spectral evolution) Will increase the mass of the atmosphere and thus increase the mass of the convection zone  alter cooling rate Accretion can explain some observations of more- massive-than-predicted convection zones and strange metal abundances… …but it can’t explain all the weird observations

20. Cooling Rate Problems: Magnetic Fields?!?! Detected in a minority of WDs Strengths range from tens of kG to nearly a 10 9 G (= 1 GG ?) Fraction of WDs with strong magnetic fields is quite uncertain due to small samples Strong fields will alter cooling rates, although no one seems to want to say exactly how it will affect the WD cooling

21. Luminosity Functions n(L) = differential L function of a WD population in a stellar system of age t (i.e. the expected number of WDs in the system with luminosity L, per unit bolometric magnitude, per pc 3 Integrate over MS mass; M u is ~8 M  ; M ℓ is a function of L given by: t cool (L,M ℓ [m wd ]) + t ms (M ℓ ) = t t cool (L,M ℓ [m wd ]) = cooling time to L of a WD with MS mass M ℓ t ms (M ℓ ) = MS lifetime of progenitor with mass M ℓ M ℓ [m wd ] = initial-to-final mass relation (IFR) dt cool /dM bol = inverse of cooling rate  = time-dep. stellar formation rate (SFR)  = initial mass function (IMF)

22. Luminosity Functions for the Galactic Disk Fontaine (2001) used:   = SFR = constant   = Salpeter IMF = M -2.35  M[m wd ] = IFR = 8 * log(2.5 m wd )  t ms (M) = 10M -2.5 Gyr M and m wd are in solar units Here, only DA WDs with pure C cores are used to calculate luminosity functions for the Galaxy

23. WDs pile up with decreasing L At low L, less massive (and more numerous) WDs don’t exist since they haven’t had enough time to cool Break occurs at ~10 -4 L  All ages are identical on the “ascending branch” Ages can be distinguished only on the “descending branch” Descending branch is populated by massive, less numerous, and oldest WDs in a population (recall that at low L, higher mass WDs cool quicker) Bend near peak comes from combination of crystallization and convective coupling (since these both stall cooling, at least temporarily) Fontaine (2001)

24. Observations of Local WDs WDs are often difficult to observe due to low L Pre-2001 (i.e. for the purposes of the models in Fontaine (2001)) there was spectroscopy for only a few thousand WDs  Most within ~500 pc of the Sun (local disk stars) There was also some pre-2001 photometric observations of WDs in distant open and globular clusters. Pre-2001 complete local surveys:  Leggett, Ruiz, & Bergeron (1998) – proper motion survey; 43 objects  Knox, Hawkins, & Hambly (1999) – colorimetric survey; 58 objects

25. Both surveys indicate the theorized peak near ~10 -4 L  Bend near peak is also (sort of) seen  first direct proof of convective coupling in WDs The steep drop-off is also seen and is likely populated by a few of the most massive and oldest WDs that have cooled relatively quickly to the lowest L observed Fontaine (2001) This gives a local Galactic disk age of 10 – 11 Gyr (consistent with other estimates) Recall that the models shown are for a pure C core and thus the age is likely an upper limit to the actual value  A pure O core gives an age of ~8.5 Gyr

26. Observations of WDs in Distant Clusters Observed in both globular and open clusters Open clusters are easier to study since they’re closer and younger (i.e. brighter WDs) A handful of clusters have been aged, ranging from 1 – 11 Gyr Observations are two-band photometry, which give luminosity Data from Richter et al. (1998):  Canada-France-Hawaii Telescope  Old open cluster M67  88 WDs  Converted observed photometry to L

27. Definitely a peak, maybe near ~10 -4 L  Bend near peak could exist, but not enough data points (only four bins were used to improve statistics) The steep drop-off is again seen at low L This gives an age of a little under 6 Gyr Fontaine (2001) Again recall that the models shown are for a pure C core and thus the age is likely an upper limit to the actual value Measuring the cluster age independently (maybe using the MS turnoff method) would help constrain the cooling model

28. Observations of Ancient WDs in the Halo It has been suspected that there exists a very old population of WDs in the Galactic halo Could possibly contribute to galactic baryonic dark matter (MACHOs) Supported by observations of a few dozen WDs with large tangential velocities (possibly interlopers from the halo)  Most of these WDs are found to be ~9 Gyr old (significantly younger than the estimated disk age of ~11 Gyr)  However, there have been recent observations that put the ages of a few of these WDs at 12.7 ± 0.7 Gyr

29. Observations of Ancient WDs in the Halo The MACHO microlensing experiment – Alcock et al. (1997, 1999, 2000):  MACHOs make up 8% – 20% of the galactic dark matter in the form of 0.5 M  objects Very possibly ancient, faint, cool WDs Fontaine (2001) used the microlensing results to model an ancient halo population of WDs:  They derived an age of ~13 Gyr  Older than the disk age of ~11 Gyr  Consistent with recent direct measurements of lone halo WDs (which gave an age of 12.7 ± 0.7 Gyr) Richer (2001):  WDs not found in the HDFN  Consistent with WD MACHOs being <20% of galactic dark matter

30. The Most Recent Models & Searches Schröder (2004):  Based on the model of Fontaine (2001)  Added in spectral evolution Kawka (2006) – using the VLT, found ~1000 WD candidates within ~200 pc (Galactic disk WDs) from proper motion observations Eisenstein (2006):  Found 9316 spectroscopically confirmed WDs in the SDSS Release 4  About 6000 of them are new discoveries, roughly doubling the number of spectroscopically confirmed WDs  There exists a set of He core DA WDs with estimated masses <0.3M  (including two that may be the lowest- mass yet found!)

31. Conclusions WD cooling has:  Aged the local Galactic disk to 8.5 – 11 Gyr, with the largest uncertainty coming from the unknown proportions of C and O in the WD cores  Aged a few globular and open clusters from 1 – 11 Gyr  Supported the idea that there exists an ancient (~12.7 Gyr) population of dim halo WDs (MACHOs) that could account for up to 20% of the galactic dark matter Fontaine (2001): “…immense progress has been made in the last decade [on WD cooling models and theories], but not to the point where we could seriously claim that white dwarf cosmochronology has superseded other methods [of astronomical aging]. Not yet.”

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