1. Electron-impact rotational excitation of H 3 + : relevance for thermalization and dissociation Alexandre Faure* Laurent Wiesenfeld* & Jonathan Tennyson # *Laboratoire d’Astrophysique de Grenoble, France # University College London

2. Electron-molecule collisions Rotational excitation of molecules by electron-impact is very efficient: Rotational excitation of molecules by electron-impact is very efficient: –k(e) ~ 10 -6 cm 3 s -1 By comparison: By comparison: –k(H, H 2 ) ~ 10 -10 cm 3 s -1 Electrons are the dominant exciting partners as soon as: Electrons are the dominant exciting partners as soon as: – n(e)/n(H) > 10 -4

3. H 3 + in the diffuse ISM Unexpected high abundance of H 3 + in diffuse clouds Unexpected high abundance of H 3 + in diffuse clouds Three uncertain key parameters: k e, n(e) and  Three uncertain key parameters: k e, n(e) and  Observations suggest Observations suggest –n(e)/n(H 2 ) ~ 4 10 -4 –high CR ionization rate (  ~ 10 -15 s -1 ) Laboratory and space spectra of H 3 +, from McCall et al. Nat. 2003

4. H 3 + toward the galactic center Large column densities in the (3, 3) metastable state Large column densities in the (3, 3) metastable state Very low column densities in the (2, 2) state Very low column densities in the (2, 2) state Provide evidence of Provide evidence of –high T (~ 250 K) –low n (~ 100 cm -3 ) –high  (> 10 -15 s -1 ) H 3 + and CO spectra toward GCS3-2, from Oka et al. ApJ 2005

5. Rotation and DR measurements H 3 + internal excitation known to influence DR rate measurements H 3 + internal excitation known to influence DR rate measurements Influence of electron- impact excitation? Influence of electron- impact excitation? Rotational cooling and heating by electrons observed at TSR (talk by A. Wolf) Rotational cooling and heating by electrons observed at TSR (talk by A. Wolf) DR rate coefficients, from Lammich et al. 2005 CRYRING (McCall et al. 2003) TSR, short storage time TSR, long storage time

6. Electron-impact (de-)excitation Experiments extremely difficult Vibrational excitation: negligible at relevant temperatures (first threshold at 0.3 eV) Rotational excitation: standard theory is the long- range Coulomb-Born approximation (Chu & Dalgarno 1974, Chu 1975) However, short-range forces are crucial! (Rabadán et al. 1998, Faure & Tennyson 2001)

7. The R-matrix method Internal region : exchange, correlation (adapt quantum chemistry codes) External region : Multipolar potential (adapt electron-atom codes) electron R-matrix sphere internal region external region

8. Electron-H 3 + calculations H 3 + wavefunction taken from R-matrix calculations of Faure & Tennyson (2002) H 3 + wavefunction taken from R-matrix calculations of Faure & Tennyson (2002) Ground-state quadrupole: 0.914 ea 0 2 (close to 0.9188 ea 0 2 calculated by Meyer et al. 1986) Ground-state quadrupole: 0.914 ea 0 2 (close to 0.9188 ea 0 2 calculated by Meyer et al. 1986) Scattering model includes four target states, via CI expansion. Scattering model includes four target states, via CI expansion. Continuum functions represented by Gaussian- type basis functions with l  4 (Faure et al. 2002). Continuum functions represented by Gaussian- type basis functions with l  4 (Faure et al. 2002). Resonances in good agreement with Orel’s results Resonances in good agreement with Orel’s results

9. Rotational excitation calculations H 3 + is taken at its equilibrium geometry The adiabatic nuclei rotation (ANR) method (sudden approximation) is employed Cross sections are expressed as a partial wave expansion with high partial waves deduced from long range approximations Excitation cross sections are corrected (forced to zero) near threshold (Morrison & Sun 1995)

10. Rotational cross sections and selection rules Cross sections computed from 10 meV to 10 eV Cross sections computed from 10 meV to 10 eV Entirely dominated by short range interactions Entirely dominated by short range interactions Selection rules: Selection rules: –  J=(0), 1, 2, (3, …) –Ortho  para forbidden –  K=0, (3)  J=1, 2 comparable in magnitude  J=1, 2 comparable in magnitude Faure & Tennyson JPB 2002

11. Rate coefficients Rates obtained from 100 to 10,000K Rates obtained from 100 to 10,000K No dipole and large rotational thresholds: No dipole and large rotational thresholds: –Excitation rates generally peak above 1,000K, at about 10 -7 cm 3 s -1 –Deexcitation rates increase slightly below 1,000K Faure & Tennyson MNRAS 2003

12. Comparison with DR rate coefficients Latest measurements with rotationally cold H 3 + : Latest measurements with rotationally cold H 3 + : –k(23K)=2.6 10 -7 cm 3 s -1 –k(300K)=6.8 10 -8 cm 3 s -1 Two possible regimes: Two possible regimes: –Rotational cooling important below 100K –Rotational heating important above 100K McCall et al. PRA 2004

13. Thermalization of H 3 + in space Centrifugal distorsion causes « forbidden » transitions:  J=0, 1  K=3 Centrifugal distorsion causes « forbidden » transitions:  J=0, 1  K=3 Spontaneous emission times comparable to collision intervals Spontaneous emission times comparable to collision intervals Nonthermal rotational distribution expected (Oka & Epp 2004) Nonthermal rotational distribution expected (Oka & Epp 2004) Forbidden rotational transitions, from Pan & Oka ApJ 1986

14. Reactive collisions with H 2 In contrast to standard neutral collisions, collisions between H 3 + and H 2 are reactive: In contrast to standard neutral collisions, collisions between H 3 + and H 2 are reactive: H 3 + + H 2  (H 5 + )*  H 3 + + H 2 Random selection rules: ortho/para conversion is allowed Random selection rules: ortho/para conversion is allowed Langevin potential: rates expected to lie between between 10 -10 cm 3 s -1 and 10 -9 cm 3 s -1 Langevin potential: rates expected to lie between between 10 -10 cm 3 s -1 and 10 -9 cm 3 s -1 Rigorous quantum (or even classical) calculations greatly needed! Rigorous quantum (or even classical) calculations greatly needed!

15. Thermalization by H 2 (Oka & Epp 2004) Collision rates based on Langevin rate and detailed balance Collision rates based on Langevin rate and detailed balance Steady state approximation: Steady state approximation: –Lifetime ~ 10 9 s –Collision time ~ 10 7 s Results consistent with observations for Results consistent with observations for –T ~ 250K –n(H 2 ) ~ 100cm -3 Population ratios and T ex as a function of n(H 2 ) and T, from Oka & Epp 2004

16. Thermalization by e-impact? The electron effect is estimated by Oka & Epp to be 2 orders of magnitude less than that of H 2 : The electron effect is estimated by Oka & Epp to be 2 orders of magnitude less than that of H 2 : –k(e)/k(H 2 ) ~ 10 2 –n(e)/n(H 2 ) ~ 10 -4 However, it is not unreasonable to assume: However, it is not unreasonable to assume: –k(e)/k(H 2 ) ~ 10 3, i.e. k(H 2 ) ~ 10 -10 cm 3 s -1 –n(e)/n(H 2 ) ~ 10 -3, i.e. high ionization rate In such conditions, might electrons compete with neutrals? In such conditions, might electrons compete with neutrals?

17. Steady-state approximation Solve the rate equation: Solve the rate equation: Ortho/para conversion forbidden: Ortho/para conversion forbidden: –Initial n(1, 0)/n(1, 1) is crucial The steady state solution is NOT compatible with observations! The steady state solution is NOT compatible with observations! Population ratios as a function of T and n(e) Obs ~ 0.7! Obs ~ 0.5! n(e)

18. Time dependent approach? However, steady state approximation is NOT valid: However, steady state approximation is NOT valid: –t(lifetime)~3 10 8 s –t(steady-state)>10 9 s Proper modelling needs inclusion of rates for: Proper modelling needs inclusion of rates for: –formation (H 2 + +H 2 ) –destruction (H 3 + + e) Level populations as a function of time for T=300K, n(e)=5 10 -2 cm -3 t(lifetime)

19. Conclusions Electron-impact rotational (de)excitation rates of H 3 + are comparable in magnitude to the DR rate at 300K, i.e. about 10 -7 cm 3 s -1 Electron-impact rotational (de)excitation rates of H 3 + are comparable in magnitude to the DR rate at 300K, i.e. about 10 -7 cm 3 s -1 Ortho-para conversion is collisionally forbidden Ortho-para conversion is collisionally forbidden We now provide rotational rates for all allowed transitions up to (5, 4) from 100 to 10,000K We now provide rotational rates for all allowed transitions up to (5, 4) from 100 to 10,000K Future works: Future works: –Modelling of H 3 + thermalization by electrons in space –Modelling of H 3 + cooling and heating by electrons in storage rings –Isotopologs of H 3 +