AstrochemistryUniversity of Helsinki, December 2006
Lecture 3
T J Millar, School of Mathematics and PhysicsQueen’s University Belfast,Belfast BT7 1NN, Northern Ireland
Grain Surface Time-scales
Collision time: tc = [vH(πr2nd)]-1 ~ 109/n(cm-3) years
Thermal hopping time: th = ν0-1exp(Eb/kT)
Tunnelling time: tt = v0-1exp[(4πa/h)(2mEb)1/2]
Thermal desorption time: tev = ν0-1exp(ED/kT)
Here Eb ~ 0.3ED, so hopping time < desorption time
For H at 10K, ED = 300K, tt ~ 2 10-11, th ~ 7 10-9 s
Tunnelling time < hopping time only for lightest species (H, D)
For O, ED ~ 800K, th ~ 0.025 s.
For S, ED ~ 1100K, th ~ 250 s, tt ~ 2 weeks
Heavy atoms are immobile compared to H atoms
Grain Surface ChemistryZero-order approximation:
Since H atoms are much more mobile than heavy atoms, hydrogenation dominates if n(H) > Σn(X), X = O, C, N
Zero-order prediction:
Ices should be dominated by the hydrogenation of the most abundant species which can accrete from the gas-phase
Accretion time-scale:
tac(X) = (SXvXσnd)-1, where SX is the sticking coefficient ~ 1 at 10K
tac (yrs) ~ 109/n(cm-3) ~ 104 – 105 yrs in a dark cloud
Interstellar Ices
Mostly water ice
Substantial components:
- CO, CO2, CH3OH
Minor components:
- HCOOH, CH4, H2CO
Ices are layered
- CO in polar and non-polar
ices
Sensitive to f > 10-6
Solid H2O, CO ~ gaseous H2O, CO
Grain Surface Chemistry
• Deterministic (Rate Coefficient) Approach:
Basics: Define an effective rate coefficient based on mobility (velocity) and mean free path before interaction (cross-section). Let ns(j) be surface
abundance (per unit volume) of species i which has a gas phase abundance n(i). Then we can write the usual differential terms for formation and loss of grain species allowing for surface reaction, accretion from the gas phased and desorption from the grain.
Technique: Solve the set of coupled ODEs which describe grain surface and gas phase abundances (approximately doubles the no. of ODEs)
Problem: Rate equations depend on an average being a physically meaningful quantity – ok for gas but not for grains
4 grains + 2 H atoms – average = 0.5 H atoms per grain
BUT reaction cannot occur unless both H atoms are actually on the same grain
Grain Surface Chemistry
• Stochastic (Accretion Limit) Approach:
Basics: Reaction on the surface can only occur if a particle arrives while one is already on the surface – the rate of accretion limits chemistry
Technique: Monte-Carlo method – attach probabilities to arrival of individual particles and fire randomly at surface according to these probabilities
Caselli et al. 1998, ApJ, 495, 309
Agreement between rate and MC poor for low values of n(H) – as expected
Grain Surface Chemistry
• Stochastic (Accretion Limit) Approach:
Solution?: Improve method of calculating surface rate coefficients
Problem: Modifications cannot be a priori – you need a MC calculation – and these are ‘impossible’ for large numbers of species
Caselli et al. 1998, ApJ, 495, 309
Fully modified rate approach
Grain Surface Chemistry
• Stochastic (Accretion Limit) Approach:
Solution?: Master Equation
Reaction depends on the probabilities of a particular number of species being on the grains e.g. PH(0), PH(1), PH(2), … PH(N), PO(0), PO(1), …
Biham et al. 2001, ApJ, 553, 595
Green et al. 2001, A&A, 375, 1111
Technique: Integrate the rates of change of probabilities, eg dPH(i)/dt
Problem: Formally, one has to integrate an infinite number of equations
For a system of H only:
dP(i)/dt = kfr[P(i-1) - P(i)]
+ kev[(i+1)P(i+1) – iP(i)]
+0.5kHH[(i+2)(i+1)P(i+2) –i(i-1)P(i)]
for all I = 0 to infinity
For larger systems, eg O, OH, H2O, H, H2, the ODEs get very complex – even the steady state solution is difficult to solve
Protoplanetary Disks
Thin accretion disks from which protostar forms
Inflow from large radii (100 AU) onto central protostar
Temperature of outer disk is cold (10 K)
n(H2) ~ 1016 – 1021 m-3
Molecular gas is frozen on to dust grains in outer disk
Temperature of inner disk is ~ 100 K at 10 AU, ~1000 K at 1 AU
Ices evaporate in inner disk
Density and temperature profiles
Hotter surface layerThicker disk
Some processes – deuterium fractionation, freeze-out, thermal desorption – very sensitive to low T regime
Some processes – H2 reactions – very sensitive to high T regime
Disk ionization degree at 1 Myr
Surface (U
V, X-ra
ys)
Intermediate
(X-ra
ys)
Midplane (CR, RN)
Semenov, Wiebe, Henning
Chemical differentiation in z-direction
Surface layer (hot): PDR-like chemistry (X-rays and UV), H+, He+, C+, CN, C2H
Intermediate layer (warm): Rich molecular chemistry (X-rays), surface reactions,
desorption, CS, CO, NH3, H2CO, HCO+, HCNH+, NH4
+, H3CO+, S+, He+
Midplane (cold): Dark chemistry (CR and RN), ‘total’ freeze out, Metal ions, H3
+, HCO+, N2H+ , H2D+, D2H+, D3+
Molecular Ice Distributions
Molecular Distributions
Markwick, Ilgner, Millar, Henning, Astron. Astrophys., 385, 632 (2002)
Vertical Diffusion
Radial accretionNo vertical mixing
Radial accretionVertical diffusion
Ilgner, Henning, Markwick, Millar, Astron. Astrophys., 415, 613 (2004)
Modelling scheme
HCO+(1-0): n0=4105 cm-3 (3-2)/ (1-0): p10.3CS(5-4): only ‘‘clumpy’’ model works!Total mass: 1 Msun
Accretion rate: 4·10-8
Msun / yrLifetime: 25 Myr
Density structure of the envelope
Star-Forming Hot CoresDensity: 106 - 108 cm-3
Temperature: 100-300 K
Very small UV field
Small saturated molecules: NH3, H2O, H2S, CH4
Large saturated molecules: CH3OH, C2H5OH, CH3OCH3
Large deuterium fractionation
Few molecular ions - low ionisation ?
f(CH3OH) ~ 10-6
Modelling G34.3+0.15
Use 2-D continuum radiative transfer code to fit dust spectrum – gives Td(r) and n(r)
Use these to calculate Tgas(r)
Adopt initial molecular ice abundances (inner core)
and elemental abundances (outer envelope)
Follow chemistry at several depth points as mantles evaporate due to (time-dependent) heating by central source.
Parents and Daughters(Chemical Clocks)
Evaporated mantle molecules (parents) are protonated and become reactive
Form more complex species (daughters) on time-scale of 103-104 yr
Surface Trapping
Detailed spatial (and temporal) distributions depend on details of surface binding energies, the detailed process by which species evaporate, and the grain temperature
Can induce lots of small scale structure amenable to interferometers (particularly ALMA).