1. Astrochemistry University of Helsinki, December 2006 Lecture 3 T J Millar, School of Mathematics and Physics Queen’s University Belfast,Belfast BT7 1NN, Northern Ireland

2. Grain Surface Time-scales Collision time: t c = [v H (πr 2 n d )] -1 ~ 10 9 /n(cm -3 ) years Thermal hopping time:t h = ν 0 -1 exp(E b /kT) Tunnelling time:t t = v 0 -1 exp[(4πa/h)(2mE b ) 1/2 ] Thermal desorption time: t ev = ν 0 -1 exp(E D /kT) Here E b ~ 0.3E D, so hopping time < desorption time For H at 10K, E D = 300K, t t ~ 2 10 -11, t h ~ 7 10 -9 s Tunnelling time < hopping time only for lightest species (H, D) For O, E D ~ 800K, t h ~ 0.025 s. For S, E D ~ 1100K, t h ~ 250 s, t t ~ 2 weeks Heavy atoms are immobile compared to H atoms

3. Grain Surface Chemistry Zero-order approximation: Since H atoms are much more mobile than heavy atoms, hydrogenation dominates if n(H) > Σ n(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: t ac (X) = (S X v X σn d ) -1, where S X is the sticking coefficient ~ 1 at 10K t ac (yrs) ~ 10 9 /n(cm -3 ) ~ 10 4 – 10 5 yrs in a dark cloud

4. Interstellar Ices Mostly water ice Substantial components: - CO, CO 2, CH 3 OH Minor components: - HCOOH, CH 4, H 2 CO Ices are layered - CO in polar and non-polar ices Sensitive to f > 10 -6 Solid H 2 O, CO ~ gaseous H 2 O, CO

5. 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 n s (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

6. 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

7. 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

8. 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 = k fr [P(i-1) - P(i)] + k ev [(i+1)P(i+1) – iP(i)] +0.5k HH [(i+2)(i+1)P(i+2) – i(i-1)P(i)] for all I = 0 to infinity For larger systems, eg O, OH, H 2 O, H, H 2, the ODEs get very complex – even the steady state solution is difficult to solve

9. 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(H 2 ) ~ 10 16 – 10 21 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

10. Density and temperature profiles Hotter surface layer Thicker disk Some processes – deuterium fractionation, freeze-out, thermal desorption – very sensitive to low T regime Some processes – H 2 reactions – very sensitive to high T regime

11. Disk ionization degree at 1 Myr Surface (UV, X-rays) Intermediate (X-rays) Midplane (CR, RN) Semenov, Wiebe, Henning

12. Chemical differentiation in z- direction  Surface layer (hot): PDR-like chemistry (X-rays and UV), H +, He +, C +, CN, C 2 H  Intermediate layer (warm): Rich molecular chemistry (X-rays), surface reactions, desorption, CS, CO, NH 3, H 2 CO, HCO +, HCNH +, NH 4 +, H 3 CO +, S +, He +  Midplane (cold): Dark chemistry (CR and RN), ‘total’ freeze out, Metal ions, H 3 +, HCO +, N 2 H +, H 2 D +, D 2 H +, D 3 +

13. Molecular Ice Distributions

14. Molecular Distributions Markwick, Ilgner, Millar, Henning, Astron. Astrophys., 385, 632 (2002)

15. Vertical Diffusion Radial accretion No vertical mixing Radial accretion Vertical diffusion Ilgner, Henning, Markwick, Millar, Astron. Astrophys., 415, 613 (2004)

16. Modelling scheme

17.  HCO + (1-0): n 0 =4  10 5 cm -3  (3-2)/ (1-0): p  1  0.3  CS(5-4): only ‘‘clumpy’’ model works!  Total mass:  1 M sun  Accretion rate:  4·10 -8 M sun / yr  Lifetime:  25 Myr Density structure of the envelope

18. Star-Forming Hot Cores Density: 10 6 - 10 8 cm -3 Temperature: 100-300 K Very small UV field Small saturated molecules: NH 3, H 2 O, H 2 S, CH 4 Large saturated molecules: CH 3 OH, C 2 H 5 OH, CH 3 OCH 3 Large deuterium fractionation Few molecular ions - low ionisation ? f(CH 3 OH) ~ 10 -6

19. Modelling G34.3+0.15 Use 2-D continuum radiative transfer code to fit dust spectrum – gives T d (r) and n(r) Use these to calculate T gas (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.

20. Parents and Daughters (Chemical Clocks) Evaporated mantle molecules (parents) are protonated and become reactive Form more complex species (daughters) on time-scale of 10 3 - 10 4 yr

21. 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).