![Page 1: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/1.jpg)
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
![Page 2: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/2.jpg)
22
Molecular Formation in the ISMMolecular Formation in the ISM
![Page 3: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/3.jpg)
33
Horse-Head Nebula
Molecular Formation in the ISMMolecular Formation in the ISM
![Page 4: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/4.jpg)
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
![Page 5: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/5.jpg)
55
The SolutionThe Solution
![Page 6: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/6.jpg)
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
![Page 7: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/7.jpg)
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
![Page 8: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/8.jpg)
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
![Page 9: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/9.jpg)
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
![Page 10: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/10.jpg)
1010
RRHH vs. Grain Size vs. Grain Size2
FH = 10-10S (atoms s-1)
E0 = 22 E1=32 (meV)
Tsurface = 10 K
![Page 11: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/11.jpg)
1111
OH O2
H2
O
H
H2O OH
The parameters: Fi ; Wi ; Ai
(i=1,2,3)
1
3 2
Complex ReactionsComplex Reactions
![Page 12: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/12.jpg)
1212
OH O2
H2
O
H
H2O OH
1
3 2
The Master Disaster:
P(N1,N2,N3)
Exponential Growth
Complex ReactionsComplex Reactions
![Page 13: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/13.jpg)
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
![Page 14: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/14.jpg)
1414
We need more knowledge…
Imposing a cutoff on P(N)
The Daring Imposition:
P(N>2) = 0
Truncating the EquationsTruncating the Equations
![Page 15: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/15.jpg)
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
![Page 16: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/16.jpg)
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
![Page 17: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/17.jpg)
1717
RRHH vs. Grain Size vs. Grain Size2
![Page 18: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/18.jpg)
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
![Page 19: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/19.jpg)
1919
Production Rates vs. Grain SizeProduction Rates vs. Grain Size
![Page 20: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/20.jpg)
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
![Page 21: By Baruch Barzel and Prof. Ofer Biham Efficient Simulations of Gas-Grain Chemistry Using Moment Equations](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0e858/html5/thumbnails/21.jpg)
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.