chemical equilibriuminthe atmosphere ofcoolwhitedwarfs · chemical equilibriuminthe atmosphere...

CHEMICAL EQUILIBRIUM IN THEATMOSPHERE OF COOL WHITE DWARFS

Simon Blouin,PhD candidate at Université de Montréal

Current Challenges in the Physics of White Dwarf StarsSanta Fe – June 13, 2017

Part IMotivation

1. Cool white dwarfs are interesting

The coolest WDs are the oldest and thus the most constrainingfor cosmochronology.

[Kowalski 2006]

2

2. Cool white dwarfs have a dense photosphere

Cool He-rich WD atmospheres are non-ideal gases.

102 103 104 105 106

T (K)

10 8

10 6

10 4

10 2

100

102

104(g

/cm

3 )

Jupiter

Sun

WD (Teff =5000K)

WD (Teff =5000K)

He

H

rs = rHevdW

3

2. Cool white dwarfs have a dense photosphere

Towards observer

ρ ≈ 1 g/cm3

ρ ≈ 0.001 g/cm3

"Warm" He-rich WD

He− ff

Hee

Shallow photosphere

"Cool" He-rich WD

Deep photosphere

4

3. Everything relies on chemical equilibrium

Structure: p(τ), T(τ)

EOS: ρ(τ), ni(τ), eint(τ)

Opacities: κν(τ), χν(τ)

Radiative transfer + convection:Iν(τ), Sν(τ), Fν(τ), Fconv

Synthetic spectrum

5

Outline

1. Motivation2. Problem statement

◦ Chemical picture3. Classical approaches

◦ Occupation probability formalism4. Current state of the art

◦ Ionization potentials◦ Classical fluid theory◦ Ab initio techniques

5. Current challenges in the chemical equilibrium of coolwhite dwarfs

6

Part IIProblem statement

What needs to be fixed

# Saha equation predicts a neutral gas at any temperature ifthe density is very large,

n j+1

n jne �

(2πme kT

h2

)3/2 2Z j+1

Z je−χ j/kT

# To solve theses issues→ take interactions into account# How?

8

Physical vs chemical pictures

Physical picture

# Basic particles: nuclei andelectrons

# Formally exact# Hard to include in

atmosphere models(population of thousandsof levels required tocompute opacities)

Chemical picture

# Basic particles: includebound species

# Not exact, physicscontained in pair potentials

# The notion of boundspecies can be problematic

# More physical insight# Easier to relate to opacities

9


Physical picture


# Formally exact

# Hard to include inatmosphere models(population of thousandsof levels required tocompute opacities)

Chemical picture





9


[Holst+2008]

9


Physical picture


# Formally exact

# Hard to include inatmosphere models(population of thousandsof levels required tocompute opacities)

Chemical picture




# More physical insight

# Easier to relate to opacities

9


Physical picture


# Formally exact# Hard to include in

atmosphere models(population of thousandsof levels required tocompute opacities)

Chemical picture





9

Free energy minimization

Free energy minimization recipe

1. Given a model for F(T,V, {N}), minimize F givenstoichiometric constraints, T and V .

dF � 0⇒ ∂F∂N j− ∂F∂N j+1

− ∂F∂Ne

� 0

2. The solution yields populations {N} and allthermodynamic properties (p, E, cv).

⇒ The problem boils down to finding a good model forF(T,V, {N}).

10

How does F look like?

F � Fide +

∑j

∑k

Fidj,k +

∑j

∑k

Fintj,k + Fexc

# Internal structure: destruction of highest energy levels dueto interactions

# Excess free energy:◦ Electrostatic interaction Fee + Fie + Fii

◦ Neutral interaction (van der Waals)

11

Part IIIOccupation probability formalism

Occupation probability

Key insightUnder given physical conditions, an electron as a finiteprobability of being bound and a corresponding probabilityof being ionized.

N jk

N j� w jk

g jk

Z je−ε jk/kT , Z j �

∑k

w jk g jk e−ε jk/kT

1. w jk allow Z j to converge2. w jk are continuous→ thermodynamic properties are

continuous3. w jk are combined to account for various interactions

(e.g., w � wneutral × wcharged)13

w jk for neutral interactions

# Atoms are considered as hard spheres with a radius rk foran excitation level k

# Using the second virial coefficient in the van der Waalsequation of state,

wk � exp

(−4π

3

∑k′

n′k(rk + rk′)3)

InterpretationWhen a state occupies a volume of the order of the meanvolume allowed per particle, it is gradually destroyed.

14

Limitations of the occupation probability formalism

1. The excluded volume effect is only a caricature of the realinteraction potential between two neutral particles.

2. No theoretical prescription for rk (rn � f n2a0Z ?)

0.5 1.0 1.5 2.0 2.5 3.0 3.5Separation (Å)

10 4

10 3

10 2

10 1

100

101

102

103

V (e

V)He-He pair potential

Experiment (Young+1981)Van der Waals radius (Bondi 1964)Hydrogenic approximation

[Bergeron+1991]

15

Limitations of the occupation probability formalism

1. The excluded volume effect is only a caricature of the realinteraction potential between two neutral particles.

2. No theoretical prescription for rk (rn � f n2a0Z ?)

[Bergeron+1991] 15

Part IVCurrent state of the art

Ionization potentials

In the context of ionization equilibrium, we are interested in thevariation of F when ionization occurs.

Ionization Relaxation

⇒Must take configurational entropy change into account

17

Accounting for everything

There are 3 contributions to ∆F when ionization occurs,

1. Entropy change (exclusion volume effect)2. Ion interaction energy before/after3. Free electron interaction energy

18

1. Entropy change

# Given an interaction potential u(r), the Ornstein-Zernikeequation yields the radial distribution function g(r),

g(r) � 4πr2ndr

# ∆F associated with exclusion volume can be computedusing u(r) and g(r) [Kiselyov+1990].

0 1 2 3 4 5

r (A)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

g(r

)

0 1 2 3 4 5

r (A)

−0.10−0.05

0.000.050.10

u(r

)(e

V)

19

1. Ion interaction energy

I0 Entropygain

0

1

2

3

4

5

6

7En

ergy

(eV)

6.11

-0.59

-3.88

+3.27 4.91

T = 6000 K = 0.5 g/cm3

Effective ionization potential of Ca in a dense He medium

20


# When ionization occurs,1. The interaction potential changes2. The distribution of neighboring atoms changes

Computing the interaction energy

1. Extract atomic configurations from MD simulations2. Get the interaction energy for each configuration3. Compute the mean and the statistical error

� Eint

21


I0 Entropygain

Ion excessenergy loss

0

1

2

3

4

5

6

7En

ergy

(eV)

6.11

-0.59

-3.66

+3.27 5.13

T = 6000 K = 0.5 g/cm3


22

3. Free electron interaction energy

# Add one electron to a neutral simulation box and computethe ground-state energy difference using DFT[Kowalski+2007]

# Good agreement with low-temperature measurements[Broomall+1976]

23

3. Free electron interaction energy

I0 Entropygain


Electronexcess energy

0

1

2

3

4

5

6

7En

ergy

(eV)

6.11

-0.59

-3.66

+3.27 5.13

T = 6000 K = 0.5 g/cm3


24

Putting everything together

I0 Entropygain


Electronexcess energy

Ieff0

1

2

3

4

5

6

7En

ergy

(eV)

6.11

-0.59

-3.66

+3.27 5.13

T = 6000 K = 0.5 g/cm3


25

Putting everything together

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6(g/cm3)

2.5

2.0

1.5

1.0

0.5

0.0I(e

V)T = 6000K

Ionization potential depression of Ca in a dense He medium

[Blouin+ in prep]

26

Part VCurrent Challenges in the ChemicalEquilibrium of Cool White Dwarf

Atmospheres

How do we know if we are right?

Two ways to validate our theoretical predictions,

1. Fit cool helium-rich white dwarfs2. Comparison to experimental data

28

Comparisons to spectra are always indirect

0.5 1.0 1.5 2.0 2.5( m)

0.0

0.5

1.0

f(e

rgcm

2s

1Hz

1 ) 1e 25

Teff = 4450 K log g = 7.96

WD0552-041

3900 4000 4100 4200 4300 4400(Å)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Norm

alize

dflu

xlog Ca/He = 10.71

3900 4000 4100 4200 4300 4400(Å)

0.2

0.4

0.6

0.8

1.0

Norm

alize

d flu

x

log Ca/He = -10.7

Ca ICa II

ObservationsIdeal Ca ionization equilibrium

[Blouin+ in prep]

29

Comparisons to spectra are always indirect

0.5 1.0 1.5 2.0 2.5( m)

0.0

0.5

1.0

f(e

rgcm

2s

1Hz

1 ) 1e 25

Teff = 4450 K log g = 7.96

WD0552-041

3900 4000 4100 4200 4300 4400(Å)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Norm

alize

dflu

xlog Ca/He = 10.71

3900 4000 4100 4200 4300 4400(Å)

0.2

0.4

0.6

0.8

1.0

Norm

alize

d flu

x

log Ca/He = -10.7log Ca/He = -10.8

Ca ICa II

ObservationsIdeal Ca ionization equilibriumNon-ideal Ca ionization equilibrium

[Blouin+ in prep]

It is often hard to tell what is wrong (e.g., chemical equilibriumor line profiles?)

30

(Lack of) experimental data

# We are missing experimental data to validate ab initiopredictions.

# When experimental data are available, the comparison isoften inconclusive. For instance, concerning the He EOS ofKowalski+2007,◦ Excellent agreement with shock wave experiment data◦ Significant disagreement for conductivity data, indicating a

discrepancy for the density required to pressure ionize He

31

Summary

# Cool white dwarf atmospheres are dense mediums inwhich the chemical equilibrium is non-ideal.

# Although simple and easy to implement, the occupationprobability formalism has its limits.

# Ab initio simulation techniques can be used to addressthese shortcomings.

# Significant progress is being made, but some stars keepchallenging our models and experimental data is cruellylacking.

32

References

References – 1/4

# Bergeron, P., Wesemael, F., & Fontaine, G. (1991). Synthetic spectra andatmospheric properties of cool DA white dwarfs. The Astrophysical Journal, 367,253-269.

# Bergeron, P., Leggett, S. K., & Ruiz, M. T. (2001). Photometric and spectroscopicanalysis of cool white dwarfs with trigonometric parallax measurements. TheAstrophysical Journal Supplement Series, 133(2), 413.

# Bondi, A. (1964). Van der Waals volumes and radii. The Journal of PhysicalChemistry, 68(3), 441-451.

# Broomall, J. R., Johnson, W. D., & Onn, D. G. (1976). Density dependence of theelectron surface barrier for fluid He 3 and He 4. Physical Review B, 14(7), 2819.

# Chabrier, G., & Potekhin, A. Y. (1998). Equation of state of fully ionizedelectron-ion plasmas. Physical Review E, 58(4), 4941.

# Cox, A. N. (Ed.). (2015). Allen’s Astrophysical Quantities. Springer.

# Crowley, B. J. B. (2014). Continuum lowering–A new perspective. High EnergyDensity Physics, 13, 84-102.

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References – 2/4

# Dufour, P., Bergeron, P., Liebert, J., Harris, H. C., Knapp, G. R., Anderson, S. F., ...& Edwards, M. C. (2007). On the spectral evolution of cool, helium-atmospherewhite dwarfs: Detailed spectroscopic and photometric analysis of DZ stars. TheAstrophysical Journal, 663(2), 1291.

# Fontaine, G., Graboske, H. C., & Van Horn, H. M. (1977). Equations of state forstellar partial ionization zones. The Astrophysical Journal Supplement Series, 35, 293.

# Fontaine, G., Brassard, P., & Bergeron, P. (2001). The Potential of White DwarfCosmochronology. Publications of the Astronomical Society of the Pacific, 113(782),409.

# Fortney, J. J., Baraffe, I., & Militzer, B. (2010). Giant planet interior structure andthermal evolution. Exoplanets, 397-418.

# Hansen, J. P., & McDonald, I. R. (1990). Theory of Simple Liquids. Elsevier.

# Holst, B., Redmer, R., & Desjarlais, M. P. (2008). Thermophysical properties ofwarm dense hydrogen using quantum molecular dynamics simulations. PhysicalReview B, 77(18), 184201.

35

References – 3/4

# Hong, Q. J., & van de Walle, A. (2012). Direct first-principles chemical potentialcalculations of liquids. The Journal of Chemical Physics, 137(9), 094114.

# Hummer, D. G., & Mihalas, D. (1988). The equation of state for stellar envelopes.I-an occupation probability formalism for the truncation of internal partitionfunctions. The Astrophysical Journal, 331, 794-814.

# Kiselyov, O. E., & Martynov, G. A. (1990). Free energy in the statistical theory offluids. The Journal of Chemical Physics, 93(3), 1942-1947.

# Kowalski, P. M. (2006). On the Dissociation Equilibrium of H2 in Very Cool,Helium-Rich White Dwarf Atmospheres. The Astrophysical Journal, 641(1), 488.

# Kowalski, P. M. (2006). The Physics of the Atmospheres of Cool White Dwarfs(Doctoral dissertation).

# Kowalski, P. M. (2010, November). Understanding the Oldest White Dwarfs:Atmospheres of Cool WDs as Extreme Physics Laboratories. In K. Werner, & T.Rauch (Eds.), AIP Conference Proceedings (Vol. 1273, No. 1, pp. 424-427). AIP.

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References – 4/4

# Kowalski, P. M., Mazevet, S., Saumon, D., & Challacombe, M. (2007). Equation ofstate and optical properties of warm dense helium. Physical Review B, 76(7),075112.

# Landau, L. D., & Lifshitz, E. M. (1980). Statistical physics, vol. 5. Course ofTheoretical Physics, 30.

# Mihalas, D., Dappen, W., & Hummer, D. G. (1988). The equation of state forstellar envelopes. II-Algorithm and selected results. The Astrophysical Journal,331, 815-825.

# Widom, B. (1963). Some topics in the theory of fluids. The Journal of ChemicalPhysics, 39(11), 2808-2812.

# Winisdoerffer, C., & Chabrier, G. (2005). Free-energy model for fluid helium athigh density. Physical Review E, 71(2), 026402.

# Young, D. A., McMahan, A. K., & Ross, M. (1981). Equation of state and meltingcurve of helium to very high pressure. Physical Review B, 24(9), 5119.

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