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).