Influence of chemi-ionization and chemi-recombination processes on the population of hydrogen Rydberg states in atmospheres of late type dwarfs

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1 A&A 403, (2003) DOI: / : c ESO 2003 Astronomy & Astrophyscs Influence of chem-onzaton and chem-recombnaton processes on the populaton of hydrogen Rydberg states n atmospheres of late type dwarfs A. A. Mhajlov 1,2,D.Jevremovć 3,4, P. Hauschldt 5,6, M. S. Dmtrjevć 2,4, Lj. M. Ignjatovć 1,2,andF.Alard 7 1 Insttute of Physcs, PO Box 57, Belgrade, Serba and Montenegro 2 Insttute Isaac Newton of Chle, Yugoslava Branch, Belgrade, Serba and Montenegro 3 APS Dvson, Phys. Depart., Queen s Unversty Belfast, BT7 1NN, Northern Ireland, UK 4 Astronomcal Observatory, Volgna 7, Belgrade, Serba and Montenegro 5 Depart. of Phys. and Astronomy and Center for Smulat. Physcs, Unversty of Georga, Athens, GA 30602, USA 6 Hamburger Sternwarte, Unverstaet Hamburg, Gojenbergsweg 112, Hamburg, Germany 7 CRAL/ENS, 46 Allée d Itale, Lyon Cedex 07, France Receved 22 July 2002 / Accepted 4 February 2003 Abstract. We study the nfluence of a group of chem-onzaton and chem-recombnaton processes on the populatons of hgher states of hydrogen n the layers of a stellar atmosphere. The group of processes ncludes onzaton: H (n) + H(1s) = H + 2 +e,h (n)+h(1s) = H(1s)+H + +e, and nverse recombnaton: H + 2 +e = H (n)+h(1s), H(1s)+H + +e = H (n)+h(1s), where H (n) s the hydrogen atom n a state wth the prncpal quantum number n 1, and H + 2 s the hydrogen molecular on n a weakly bound rho-vbratonal state of the ground state. These processes have been treated wthn the framework of the sem-classcal approxmaton, developed n several prevous papers, and have been ncluded n the general stellar atmosphere code PHOENIX. We present results for an M dwarf atmosphere wth T eff = 3800 K and fnd that the ncluson of chem-onzaton and chem-recombnaton processes s sgnfcant n the low temperature parts of the atmosphere. Key words. atomc processes molecular processes stars: late type 1. Introducton For several years we have studed the chem-onzaton atomatom collsonal processes and nverse chem-recombnaton processes that may nfluence the populatons of the hydrogen atom Rydberg states n the weakly onzed layers of stellar atmospheres (Mhajlov et al. 1996a, 1997). In the case of hydrogen, these processes are: H (n) + H(1s) = H e, H (n) + H(1s) = H(1s) + H + + e, H e = H (n) + H(1s), H(1s) + H + + e = H (n) + H(1s), (1a) (1b) (2a) (2b) where H (n) s a hydrogen atom n a state wth the prncpal quantum number n 1, and H + 2 s a hydrogen molecular on n a weakly bound rho-vbratonal state of the ground electronc state. In the case of onzaton (1a,b), the lower lmt of the Rydberg state range that was consdered was set to n = 4 (Mhajlov et al. 1996a, 1997). The block of rho-vbratonal Send offprnt requests to: M. S. Dmtrjevć, e-mal: mdmtrjevc@aob.bg.ac.yu states wth a dssocaton energy less than 1 ev has an essental role n the case of formaton of the molecular on (processes (1a) and (2a)) and wthn the range of quantum numbers that was consdered. In the case of cool stellar atmosphere layers (3000 K T 5000 K), the block of such weakly bound rho-vbratonal states can be consdered to be n equlbrum wth the collsonal H(1s) +H + complexes, wthout the requrement that the lower rho-vbratonal states of molecular on are n equlbrum (Mhajlov & Ljepojevć 1982; Mhajlov et al. 1992). Ths fact makes the descrpton of recombnaton processes (2a) and (2b) easer, snce t enables us to avod the drect determnaton of the densty of molecular ons n weakly onzed rho-vbratonal states. In our prevous work (Mhajlov et al. 1997, 1998) we estmated the nfluence of the processes (1a,b; 2a,b) for the low temperature layers of the Solar atmosphere. We compared the onzaton and recombnaton fluxes wth and wthout ncluson of the above processes and found a sgnfcant dfference n the populaton of the H (n) atomc states wth 4 < n < 8. Here we dscuss the mportance of these processes n stellar atmospheres of lower temperature. Ths paper s organzed as follows: n Sect. 2 we summarze the theoretcal framework and n Sect. 3 we descrbe the changes n the stellar atmosphere Artcle publshed by EDP Scences and avalable at or

2 788 A. A. Mhajlov et al.: Influence of chem-onzaton/recombnaton processes code PHOENIX and parameters of the test model atmosphere. We present our results and dscusson n Sect. 4 and draw our conclusons n Sect Theoretcal remarks Let I (a,b) (n) andi r (a,b) (n) denote onzaton and recombnaton fluxes affectng populaton and depopulaton of excted hydrogen atom atomc levels due to reactons. By defnton, onzaton fluxes I (a,b) (n) determnng the rate of depopulaton of H (n) states are gven by I (a) (n) = K (a) (n) N(H) N(H (n)) (3a) I (b) (n) = K (b) (n) N(H) N(H (n)) (3b) where N(H) and N(H (n)) are H(1s) and H (n) atom denstes, and K a (n) andkb (n) are rate coeffcents of the reactons (1a) and (1b) respectvely. The recombnaton flux I r (a) (n), determnng the velocty of populaton of H (n) states due to the reacton (2a), s gven by defnton as: I r (a) = K dr N(H + 2 ) N(e), where N(e) and N(H + 2 ) are the electron densty and the densty of molecular on H + 2 n the abovementoned weakly bound rhovbratonal states, and K dr s the correspondng rate coeffcent of the dssocatve recombnaton reacton (2a). The expresson for the recombnaton flux I r (a) may be wrtten by: I r (a) (n) = K r (a) (n) N(H) N(H + ) N(e) (4a) where K r (a) = K dr [N(H) N(H + )/N(H + 2 )] 1. The advantage of expresson (4a) for the recombnaton flux I r (a) (n) s the fact that t (n), determnng the rate of populaton of H (n) states due to the reacton (2b), s gven, by defnton, by: ntroduces a new effectve rate coeffcent of the reacton (2a). If the onzaton rate coeffcent K a (n) s known, the K(a) r (n) may be determned drectly, under assumpton of the equlbrum between H + 2 weakly bound rho-vbratonal states and the H + H+ collsonal complex (Mhajlov & Ljepojevć 1982; Mhajlov et al. 1992). Ths makes the treatment of the problem easer and enables us to avod the determnaton of the K dr (n)ratecoeffcent. The recombnaton flux I (b) r I r (b) (n) = K r (b) (n) N(H) N(H + ) N(e), (4b) where K r (b) (n) stheratecoeffcent of the reacton (2b). The nfluence of the processes (1a,b) and (2a,b) on H (n) atomc states populatons may be estmated usng only the N(H), N(H (n)), N(H + )andn(e) populatons. Rate coeffcents for the processes (1, 2) have been determned wthn the sem-classcal theory descrbed n the prevous papers (Janev & Mhajlov 1980; Mhajlov & Ljepojevć 1982; Mhajlov et al. 1992). Ths theory takes nto account atomc collsons, where the H (n) + H system n the processes (1a,b) as well as the e + H + + H system n the processes (2b) go through the phase where they may be treated as a (H + + H) + esystem.here,(h + + H) denotes that the on-atom subsystem may be treated as a quas-molecular onatom complex under assumpton that ts electronc states may be descrbed as superpostons of the ground and the frst excted states of the H + 2 molecular on. Furthemore, ths means that durng the quas-molecular phase the average electron dstancefromthecenterofmassoftheh + + H subsystem s much larger than the nternuclear dstance between H + and H. Consequently, the dpole nteracton between the (H + + H) and an electron (Smrnov & Mhajlov 1971) s the orgn of the chem-onzaton/recombnaton processes (1a,b) and (2a,b). One can notce that wthn the frame of ths approach, the processes (2a) and (2b) may be descrbed practcally n the same way. Expressons for the rate coeffcents n the processes (1a,b) and (2a,b) whch are used here as well as the method of determnaton of the rate coeffcents are descrbed n detals n Mhajlov et al. (1996b). 3. Modelng We ncluded procesess (1a,b) and (2a,b) nto the general stellar atmosphere code PHOENIX descrbed n Hauschldt et al. (1999), Short et al. (1999), Baron & Hauschldt (1998). PHOENIX solves smultaneously the equatons of hydrostatc equlbrum, mult-level and mult-speces statstcal equlbrum and NLTE radatve transfer. The major advantage of PHOENIX s that t uses extensve atomc and molecular databases and can be used for modelng of very cool objects. The man change n the code s n the addton of a new term n the statstcal equlbrum equaton whch descrbes addtonal collsonal processes. Ths change allows us to treat chem-onzaton/recombnaton processes consstently wth all the other processes. Due to lmts of the sem-classcal approach, we nclude addtonal collsonal term only for levels larger than n = 4. We use for our calculaton the plane-parallel, statc opton of PHOENIX whch has been adopted for use n chromospherc type of problems. As a test atmosphere we use a schematc chromospherc structure wth an effectve temperature of 3800 K, logarthm gravty of 5.0, log m 0 = 2.5 beng the column mass at temperature mnmum, and log m TR 4.8, m TR standng for the column mass n the transton regon. The choosen parameters represent a typcal M0 dwarf of weak actvty. The procedure of buldng ths model s dentcal to that descrbed n Jevremovć et al. (2000) and ts structure s shown n Fg. 1. As we are nterested n behavor of hydrogen populatons, we solve the NLTE problem for the 30 lowest levels. The standard collsonal data for hydrogen are those from Vrens & Smeets (1980). 4. Results and dscusson In Fgs. 2 4 we present the behavor of populaton ratos ζ wthout and wth ncluson of chem-onzaton (1a,b) and chem-recombnaton (2a,b) processes for the levels 3 to 20. The ratos are close to unty for n = 3, and start to dffer for n = 4 and 5 (Fg. 2). For n = 5thelargestdfference s around 10%. Ths s to be expected as we nclude the procesess for n 4, whch s based on the valdty of the sem-classcal approxmaton. The populaton of lower levels (n = 2, 3) s nfluenced by other processes ncluded n the statstcal equlbrum equaton.

3 A. A. Mhajlov et al.: Influence of chem-onzaton/recombnaton processes 789 Fg. 1. Temperature and electron densty structure for the model of test atmosphere. Fg. 2. The behavour of the populaton rato ζ(n)for3 n 9 as a functon of the column mass. As expected, the addtonal collsonal term n statstcal equlbrum brngs the soluton closer to local thermodynamcal equlbrum (LTE). In terms of our populaton ratos, n the parts of atmosphere where populatons, determned n our calculatons usng PHOENIX, are larger than LTE populatons (departure coeffcents b > 1), our parameter ζ s larger than one. Ths means that n such a case, chem-onzaton processes depopulate excted states and domnate n comparson wth chem-recombnaton processes. The result s that populatons determned wth processes (1, 2) ncluded are smaller n comparson wth populatons calculated wthout them and consequently ζ s larger than one. In the reverse case (b < 1), parameter ζ s smaller than one, whch means that n ths case chem-recombnaton processes domnate over chem-onzaton ones and ncrease the populaton of excted levels. As a result, n both cases, processes (1, 2) act as mportant factors closng the dfference between the calculated and LTE populatons.

4 790 A. A. Mhajlov et al.: Influence of chem-onzaton/recombnaton processes Fg. 3. The behavour of the populaton rato ζ(n) for 10 n 15 as a functon of the column mass. Fg. 4. The behavour of the populaton rato ζ(n) for 16 n 20 as a functon of the column mass. As can be seen from Fg. 2, the behavour of ζ(n) for6< n < 9 shows a tendence to converge to a pattern, whch may dffer up to 40% from unty. For hgher levels (Fgs. 3 and 4) the convergence to a pattern s complete and t can be understood by assumng that hghly excted hydrogen level populatons have quas-boltzman dstrbutons dfferent from dstrbutons for partal LTE. It s not suprsng that the populaton ratos for the levels up to 30 and the proton densty show the same behavour as levels wth n > 15. One also has to remember that n the layers wth temperatures lower than 4500 K the man electron contrbutors are metals so, as expected, there s very lttle change n electron densty. 5. Concluson We have demonstrated the mportance of the ncluson of chem-onzaton (1a,b) and chem-recombnaton (2a,b) processes n modelng of atmospheres of late type stars.

5 A. A. Mhajlov et al.: Influence of chem-onzaton/recombnaton processes 791 In prevous work (Mhajlov et al. 1996a, 1997, 1998), t had been assumed that these processes were mportant up to n = 10. Here we have demonstrated that ncluson of processes (1a,b) and (2a,b) sgnfcantly changes the populaton of all Rydberg levels due to PLTE. However, n some atmospherc layers selfconsstent modelng shows dfferences of up to 40% between calculatons wth and wthout ncluson of these processes. We plan to contnue these nvestgatons for atmospheres wth dfferent effectve temperatures, ncludng stars wth hgher T eff, such as the Sun and Sun-lke stars. Namely, on the bass of our prevous results and those presented here, one could expect that for such stars, processes (1a,b) and (2a,b) should nfluence not only the populaton of hgher levels of hydrogen, but also the electron densty. Furthermore, we plan to nclude n PHOENIX the chem-onzaton/recombnaton processes smlar to (1a,b) and (2a,b) n the He(1s 2 )atomwhch wouldbeusednmodelngofhelum-rchstars. Also, there s a need for a developement of a theory of processes smlar to processes (1a,b) and (2a,b) for the lower levels of hydrogen, wth the prncpal quantum numbers n = 2 and 3. Ths extenson would enable more sophstcated modelng of stellar atmospheres as the populaton of lower levels may sgnfcantly alter the electron densty and consequently the atmospherc structure. Acknowledgements. Ths work s a part of the projects Radaton and transport propertes of the non-deal laboratory and onospherc plasma (Project number 1466) and Influence of collsonal processes on astrophyscal plasma lneshapes (Project number GA 1195) and was supported by the Mnstry of Scence, Technologes and Development of Serba. D.J. was supported at QUB through the PPARCgrant PPA/G/S/1998/ Authors acknowledge the care of revewer to mprove ths paper. References Baron, E., & Hauschldt, P. H. 1998, ApJ, 495, 370 Hauschldt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377 Janev, R. K., & Mhajlov, A. A. 1980, Phys. Rev., 21, 819 Jevremovć, D., Doyle, J. G., & Short, C. I. 2000, A&A, 358, 575 Mhajlov, A. A., & Ljepojevć, N. N. 1982, Proc. Symp. Phys. Ionz. Gases, Dubrovnk, Contrbuted Papers, p. 385 Mhajlov, A. A., Ljepojevć, N. N., & Dmtrjevć, M. S. 1992, J. Phys. B: At. Mol. Phys., 25, 5127 Mhajlov, A. A., Dmtrjevć, M. S., Ignjatovć, Lj. M., & Vasljevć, M. M. 1996a, Int. Conf. n the memory of I. S. Shklovsky, S. A. Kaplan, S. B. Pkel ner: Current problems n Astrophyscs, Moscow, Russa, Mhajlov, A. A., Dmtrjevć, M. S., & Djurć, Z. 1996b, Phys. Scr., 33, 159 Mhajlov, A. A., Ignjatovć, Lj. M., Vasljevć, Lj. M., & Dmtrjevć, M. S. 1997, A&A, 324, 1206 Mhajlov, A. A., Ignjatovć, Lj. M., & Dmtrjevć, M. S. 1998, Novyj ckl aktvnosty Solnca, Trudy, Sankt Peterburg, Pulkovo, p. 52 Short, C. I., Hauschldt, P. H., & Baron, E. 1999, ApJ, 525, 375 Smrnov, V. A., & Mhajlov, A. A. 1971, Opt. Spektrosk., 30, 984; Opt. Spectrosc., 30, 525 Vrens, L., & Smeets, A. H. M. 1980, Phys. Rev. A, 22, 940