By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations.

byby

Baruch BarzelBaruch Barzelandand

Prof. Ofer BihamProf. Ofer Biham

Efficient Simulations Efficient Simulations of Gas-Grain Chemistry of Gas-Grain Chemistry

Using Moment Using Moment EquationsEquations

22

Molecular Formation in the ISMMolecular Formation in the ISM

33

Horse-Head Nebula

Molecular Formation in the ISMMolecular Formation in the ISM

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HH22 Production in the gas phase: Production in the gas phase:

H + H → HH + H → H22

Gas-Phase Reactions Cannot Account for the Observed Production Rates

Observed Production Rates in ISC:Observed Production Rates in ISC:

RRHH ~ 10-15 (mol cm ~ 10-15 (mol cm-3-3ss-1-1))2

The HThe H22 Puzzle Puzzle

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The SolutionThe Solution

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kBT

-E0

AH = (1/S) e

= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt

Incoming fluxDesorption

Recombination

WH = e kBT

-E1

The Production Rate of HThe Production Rate of H22 Molecules: Molecules:

RH = AH‹NH›2 (mol s-1)2

The Rate EquationThe Rate Equation

77

Mean-field approximation

= FH - WH‹NH› - 2AH‹NH›2d‹NH› dt

When the Rate Equation FailsWhen the Rate Equation Fails

•Neglects fluctuations•Ignores discretization

Not valid for small grains and low flux

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P(0)P(0)

P(1)P(1)

P(NP(NHH-1)-1)

P(NP(NHH))

P(NP(NHH+1)+1)

P(NP(NHH+2)+2)

P(NP(Nmaxmax))

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

Probabilistic ApproachProbabilistic Approach

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= 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

The Master EquationThe Master Equation

1010

RRHH vs. Grain Size vs. Grain Size2

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

E0 = 22 E1=32 (meV)

Tsurface = 10 K

1111

OH O2

H2

O

H

H2O OH

The parameters: Fi ; Wi ; Ai

(i=1,2,3)

1

3 2

Complex ReactionsComplex Reactions

1212

OH O2

H2

O

H

H2O OH

1

3 2

The Master Disaster:

P(N1,N2,N3)

Exponential Growth

Complex ReactionsComplex Reactions

1313

‹NHk› = NH

kPH(NH)NH=0

8

After applying the summation:

‹NH› = FH + (2AH - WH)‹NH› - 2AH‹NH2›

‹NH2› = FH + (2FH + WH - 4AH)‹NH›

+ (8AH - WH)‹NH2› - 4AH‹NH

3›

The Moment EquationsThe Moment Equations

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We need more knowledge…

Imposing a cutoff on P(N)

The Daring Imposition:

P(N>2) = 0

Truncating the EquationsTruncating 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›

And after imposing the cutoff…

Moment Equations for HMoment Equations for H22 Production Production

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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 HMoment Equations for H22 Production Production

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RRHH vs. Grain Size vs. Grain Size2

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‹N1›, ‹N3›‹N2›,

OH O2

H2

O

H

H2O OH‹N1N2›

‹N1N3›

‹N22›

‹N12›

3 vertices + 2 edges + 2 loops = 7 equations

A View to Complex NetworksA View to Complex Networks

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Production Rates vs. Grain SizeProduction Rates vs. Grain Size

2020

O

H3CO

CO

DCO OD

D3CO

HDCO

H2DCO

HD2CO

HCO OH

D2CO

H

H2CO

D

15 vertices

30 edges

+ 3 loops

48 equations

Multi-Specie NetworkMulti-Specie Network

2121

SummarySummaryThe advantages of the moment equations:The advantages of the moment equations:

Reliable even for low coverageReliable even for low coverageEfficient Efficient LinearLinearEasy to incorporate into rate equation modelsEasy to incorporate into rate equation modelsDirectly generate the required momentsDirectly generate the required moments

Further applications should be tested.Further applications should be tested.