The Formation of H2 and HD

1

• The Formation of H2 and HD
• with the Master Equation Approach
• Ofer Biham
• The Hebrew University
• Azi Lipshtat
  Gian Vidali
• Hagai Perets
  Valerio Pirronello
• Baruch Barzel
  Joe Roser
• Itay Furman
  Giulio Manicó
• Nadav Katz
• 
  Israel Science
  Foundation
• 
  The Adler
  Foundation for Space Research
• 
  

2
Outline

• Introduction
• Laboratory Experiments
• ModelingFormation of H2 and HD
• Rate equations
• Master equation
• Multi-plane method
• Astrophysical Implications

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Interstellar Clouds
Diffuse clouds
Dense clouds
Hydrogen Atoms
Hydrogen Molecules
Bare Dust Grains
Ice Coated Grains

• nH 10 - 100 cm-3
• ngrains 10-12 n
• Tgas 100 K
• Tgrains 10 20 K

104 cm-3 ngrains 10-12 n Tgas 10 20
K Tgrains 10 K
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Interstellar Dust Grains

• Ejected from Red Giants and novae
• Approximately 1 of the clouds mass
• Consist of amorphous silicates and carbon
  particles
• Broad size distribution in the range between 1nm
  100nm

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Molecular Hydrogen Formation
Gas phase reaction - inefficient
More efficient channel - on dust grain surfaces
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The Role of Molecular Hydrogen

• The most abundant molecule in the universe
• Enables chemical reaction networks in the gas
  phase chemical complexity
• Molecules provide cooling during gravitational
  collapse enables star formation

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Rate of Recombination
R 0.5 nH vH s g ngrains
vH thermal velocity of H atoms in the gas s
cross section of the dust grains g
recombination efficiency
Gould and Salpeter (1963)
Hollenbach, Werner and Salpeter (1971)
g g(grain composition, surface morphology,
temperature, size, flux)
Laboratory experiments are needed
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Laboratory Experiments
Requirements samples (silicates,carbon,ice)
low temperatures vacuum low flux
long times efficient detection of
molecules
Sample temperature
Time
detector
Pirronello et al., ApJ 475, L69 (1997) Katz et
al., ApJ 522, 305 (1999)
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Temperature Programmed Desorption (TPD)
Experiment
Olivine sample
Irradiation times 0.55 (minutes) 0.2 0.07
Second order kinetics Thermal hopping no
tunneling
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Rate Equation
atom concentration
hopping rate a? exp(-E0/kT)
desorption coefficient W? exp(-E1/kT)
flux
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TPD Experiment on Carbon
E044meV E157meV
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TPD Experiment on Low Density Ice (LDI)

E044.5 meV E152 meV
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H2 Formation Efficiency Under Interstellar
Conditions
Astrophysical Implications
Steady state conditions Gas temperature 100
K Gas density 10 atoms/cm3
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Grain Parameters
Size d 1 100 nm Adsorption site
density s 1 site/nm2 No. of sites on grain
Spd2s Flux density f 10-9 10-6 ML/sec Flux
FfS Number of H atoms on grain n?S Scanning
rate by H atoms A a/S
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Rate Equation ? Master Equation
For small grains under low flux the typical
population size of H atoms on a grain may go
down to n
1 Thus the rate equation (mean field
approximation) fails One needs to take into
account The discreteness of the H
atoms The fluctuations in the
populations of H atoms on grains ? Master
Equation for the probability distribution
P(n), n1,2,3,
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Master Equation

Flux
Desorption
Recombination
Direct Integration Biham, Furman,
Pirronello and Vidali, ApJ 553, 595 (2001)
Green, Toniazzo, Pilling, Ruffle, Bell and
Hartquist, AA 375, 1111 (2001) Monte Carlo
Charnley, ApJ 562, L99 (2001)
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H2 Formation Rate

• Green et al. AA 375 (2001)
• Biham Lipshtat, PRE 66 (2002)

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H2 Production Rate vs. Grain Size
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Grain Size Distribution
A broad power-law distribution
ngrain(d) d -3.5 for 5nm lt d lt 250 nm
Mathis et al. (1977) Therefore small grains
should be taken into account. Competing effects
smaller grains larger surface/volume ratio
however lower
recombination efficiency.
H2 Production rate RH2 R nH n With rate
constant R 10-16 cm3 s-1

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Moment Equations

Lipshtat Biham, AA 400, 585 (2003)
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H2 in Photon Dominated Regions

• H2 molecules were detected in photon dominated
  regions (PDRs) where the grain temperature is
  around 40K
• This is in apparent contradiction with the
  experimental results
• Several explanations were proposed. One of them
  is based on the existence of chemisorption sites.
  It does not seem to apply in the relevant
  temperature range.

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Porous Grains
Porous grains may provide efficient
recombination at higher grain temperatures. In
the pores atoms desorb and re-adsorb many times.
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Rate Equations for Reaction Networks
HH?H2 OO?O2 HO?OH
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Master Equation for two Species
flux
desorption
HH?H2 OO?O2 HO?OH
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Deuteration Processes
Grain surface mechanism for enhanced formation
of HD and D2 molecules. The mechanism is based
On the longer residence time of D atoms on
grain Compared to H atoms May possibly apply to
more complex molecules such as NHD2 and ND3??
Lipshtat, Biham Herbst,MNRAS 348 (2004)
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Complex Reaction Networks
Number of equations increases Exponentially
with the number of reactive species
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Reaction Networks
Stantcheva, Shematovich and Herbst, AA 391, 1069
(2002) Lipshtat and Biham, PRL 93, 170601 (2004)
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The Multi-Plane Method
Solid lines master eq. Symbols
multi-plane Dashed lines rate eq.
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Summary

• Mobility of H atoms on grains is dominated by
  thermal diffusion not by tunneling. Thus the
  recombination efficiency strongly depends on
  grain temperature, composition and surface
  morphology
• Experiments on silicates, carbon and ice show
  narrow temperature windows of high efficiency
  between 6-18 K
• Recombination efficiency drops for very small
  grains the grain size distribution should be
  taken into account
• H2 molecules observed in photon dominated regions
  (PDRs) where grain temperature is around 40K
  How do they form? Chemisorption?Porosity?Temp.
  fluctuations?
• The master equation suitable for gas-grain
  models of interstellar chemistry