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By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment...
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Transcript of By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment...
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
44
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
55
The SolutionThe Solution
66
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
88
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
99
= 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
1414
We need more knowledge…
Imposing a cutoff on P(N)
The Daring Imposition:
P(N>2) = 0
Truncating the EquationsTruncating the Equations
1515
‹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
1616
‹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
1717
RRHH vs. Grain Size vs. Grain Size2
1818
‹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
1919
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.