Efficient Simulations Efficient Simulations of Gas-Grain Chemistry of Gas-Grain Chemistry Using Moment EquationsUsing Moment Equations

M.Sc. Thesisby

Baruch Barzelpreformed under the supervision of

Prof. Ofer Biham

2

Complexity in the Universe

3

Complexity in the Universe

Horse-Head Nebula

The Interstellar Clouds (ISC)

The Interstellar Clouds

•Molecular and atomic H

•Density: ~10 -1000 (atoms cm-3)

•Gas Temperature: 50 -150 K

6

The Role of H2

H2

Complexity

Complex Molecules

Star Formation

7

The H2 PuzzleH2 Production in the gas phase:

H + H → H2

Gas-Phase Reactions Cannot Account for the Observed Production Rates

Observed Production Rates in ISC:

RH ~ 10-15 (mol cm-3s-1)2

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The Solution

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The Interstellar Dust Grains•Composition:

Carbons, Silicates, Olivine, H2O, SiC

•Temperature: ~5-20 K

•Size Range:

10-6-10-3 (cm) → 100-108 sites

•Activation Energies: (meV) MaterialE0 (diffus)E1 (disorp)

Carbon44.056.7

Olivine24.732.1

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kBT

-E0

AH = (1/S) e

= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt

The Rate Equation

Incoming fluxDesorption

Recombination

WH = e kBT

-E1

The Production Rate of H2 Molecules:

RH = AH‹NH›2 (mol s-1)2

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Mean-field approximation

= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt

When the Rate Equation Fails

•Neglects fluctuations•Ignores discretization

Not valid for small grains and low flux

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Probabilistic ApproachP(0)

P(1)

P(NH-1)

P(NH)

P(NH+1)

P(NH+2)

P(Nmax)

Flux term:

FH[PH(NH-1) - PH(NH)]

Desorption term:

WH[(NH+1)PH(NH+1) - NHPH(NH)]

Reaction term:

AH[(NH+2)(NH+1)PH(NH+2) - NH(NH-1)PH(NH)]

FH

WHAH

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The Master Equation

= FH[PH(NH-1) - PH(NH)]

+ WH[(NH+1)PH(NH+1) - NHP(NH)]

+ AH[(NH+2)(NH+1)PH(NH+2) - NH(NH-1)PH(NH)]

dPH(NH)

dt

‹NH›= NHPH(NH)NH= 0

S

RH = AH (‹NH2› - ‹NH›)2

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RH vs. Grain Size2

FH = 10-10S (atoms s-1)

E0 = 22 E1=32 (meV)

Tsurface = 10 K

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Complex Reactions

OH O2

H2

O

H

H2O OH

The parameters: Fi ; Wi ; Ai

(i=1,2,3)

1

3 2

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The Rate Equations

= F1 - W1‹N1› - 2A1‹N1›2 - (A1+A2)‹N1›‹N2›

- (A1+A3)‹N1›‹N3›

d‹N1›

dt

= F2 – W2‹N2› - 2A2‹N2›2 - (A1+A2)‹N1›‹N2›d‹N2›

dt

= F3 - W3‹N3› - (A1+A3)‹N1›‹N3›+(A1+A2)‹N1›‹N2›d‹N3›

dt

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The Master Equation

P(N1,N2,N3) = Fi[P(…,Ni-1,…)-P(N1,N2,N3)]

+ Wi[(Ni+1)P(..,Ni+1,..)-NiP(N1,N2,N3)]

+ Ai[(Ni+2)(Ni+1)P(..,Ni+2,..)-Ni(Ni-1)P(N1,N2,N3)]

+ (A1+A2)[(N1+1)(N2+1)P(N1+1,N2+1,N3-1)-N1N2P(N1,N2,N3)

+ (A1+A3)[(N1+1)(N3+1)P(N1+1,N2,N3+1)-N1N3P(N1,N2,N3)

3

i=1

3

i=1

2

i=1

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P(N1,N2,N3) = Fi[P(…,Ni-1,…)-P(N1,N2,N3)]

+ Wi[(Ni+1)P(..,Ni+1,..)-NiP(N1,N2,N3)]

+ Ai[(Ni+2)(Ni+1)P(..,Ni+2,..)-Ni(Ni-1)P(N1,N2,N3)]

+ (A1+A2)[(N1+1)(N2+1)P(N1+1,N2+1,N3-1)-N1N2P(N1,N2,N3)

+ (A1+A3)[(N1+1)(N3+1)P(N1+1,N2,N3+1)-N1N3P(N1,N2,N3)

3

i=1

3

i=1

2

i=1

Rij = (Ai + Aj) ‹NiNj›

Rii = Ai (‹Ni2› - ‹Ni›)

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The Rate vs. The MasterRate equations:

•Mean field approximation

•High efficiency

•Not reliable for surface reactions (at low coverage)

Master equation:•Microscopic probability distribution

•Accurate model of grain surface reactions

•Low efficiency (exponential growth)

•Hard work

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The Moment Equations

‹NHk› = NH

kPH(NH)NH=0

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After applying the summation:

‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›

‹NH2› = FH + (2FH + WH - 4AH)‹NH›

+ (8AH - WH)‹NH2› - 4AH‹NH

3›

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Truncating the Equations 1. Set the cutoff

2. Express the (k+1)th moment by the first k moments

‹NH1› = PH(1) + 2PH(2) + +kPH(k)

‹NH2› = PH(1) + 22PH(2) + +k2PH(k)

‹NHk› = PH(1) + 2kPH(2) + +kkPH(k)

PH(NH > k) = 0

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Truncating the Equations 1. Set the cutoff

2. Express the (k+1)th moment by the first k moments

3. Plug into the first k moment equations

‹NH1› = PH(1) + 2PH(2) + + kPH(k)

‹NH2› = PH(1) + 22PH(2) + +k2PH(k)

‹NHk› = PH(1) + 2kPH(2) + +kkPH(k)

PH(NH > k) = 0

‹NHk+1› = Ci‹NH

i›i=0

k

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Moment Equations for H2 Production

‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›

‹NH2› = FH + (2FH + WH - 4AH)‹NH›

+ (8AH - WH)‹NH2› - 4AH‹NH

3›

1. Set the cutoff → k=2

‹NH3› = 3‹NH

2› - 2‹NH›2. Reduce excessive moments →

3. Plug into the equations…

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‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›

‹NH2› = FH + (2FH + WH - 4AH)‹NH›

+ (8AH - WH)‹NH2› - 4AH‹NH

3›

‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›

‹NH2› = FH + (2FH + WH + 4AH)‹NH›

- (4AH + 2WH)‹NH2›

Moment Equations for H2 Production

‹NH3› = 3‹NH

2› - 2‹NH›

1. Set the cutoff → k=2

2. Reduce excessive moments →

3. Plug into the equations…

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RH vs. Grain Size2

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Moments for Complex Networks

OH O2

H2

O

H

H2O OH

The probability: P(N1,N2,N3)

The moments: ‹N1aN2

bN3c›

The cutoff: Ni < ki

The challenge: Reduction of the excessive moments

‹N1aN2

bN3c› = Clnm‹N1

lN2nN3

m› lmn=0

k-1

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Reduction of Excessive MomentsThe probability: P(N1,N2)

V(a,b) M(N1,N2,a,b) P(N1,N2)

v = M p

‹N1aN2

b› = Cnm‹N1nN2

m› mn=0

k-1

‹N1aN2

b› = N1aN2

b P(N1,N2)N1N2=0

k-1

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‹N1›, ‹N3›‹N2›,

Setting the Cutoffs

OH O2

H2

O

H

H2O OH‹N1N2›

‹N1N3›

‹N22›

‹N12›

3 vertices + 2 edges + 2 loops = 7 equations

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Production Rates vs. Grain Size

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Multi-Specie NetworkH2CO H3CO

OH

HCO H

O CO

CO2 + H

O2

H2

HCO

H2CO

OH

CO2

H3CO CH3CO

H2O

7 vertices

8 edges

2 loops

17 equations

+

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Production Rates vs. Grain Size

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Summary• The advantages of the moment equations:

Reliable even for low coverage Efficient Linear Easy to incorporate into rate equation models Directly generate the required moments

• Further applications should be tested.

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Revealing the TrickThe moment equations validity -

For small grains For large grains

Cutoff justified PH(NH) is Poisson

•Second order: ( << 1) The equations are valid

•First order: ( ≈ 1) Production rate is accurate but population size may deviate

Moment equations valid under all circumstances