Chapter 4: Damping . (4-1)

Transcription

1 6 Chapter 4: Dampng 4.: The Absrbng Atm and ts Envrnment In chapter, we saw hw the absrptn ceffcent fr a nn-nteractng atmc speces at rest s gven by the cmbnatn f the strength f the transtn, the ppulatn f vald absrbers, a crrectn fr stmulated emssn and a lne prfle functn descrbng the frequency dependence f the transtn. The lne prfle functn fr an slated atm at rest s gven by equatn (-6): λ φ ( λ) = π λ λ λ +. (4-) Perturbng atms wll nt affect the basc physcal prcesses that gve rse t ths prfle, but they can alter the physcal parameters that cntrl t, namely λ (alterng the wdth f the lne) and λ (shftng the wavelength f the lne). Thus the lne prfle functn fr a sngle atm can be altered by nteractns wth ts envrnment. As we d nt have a sngle absrber, but an ensemble f absrbers wth each absrber n dfferent cndtns, each absrbng atm can have a dfferent lne prfle functn. The effect seen wll be a cnvlutn f the lne prfle functn as affected by the envrnment and the prbablty dstrbutn functn f the envrnment: φ ( λ) = φ slated λ W envrnment = φ slated all pssble cndtns ( wavelength shfted t λ by envrnment) ( envrnment) W d envrnment (4-)

2 6 Slar Lne Asymmetres 4.: The Lrentzan Lne Wdth The wdth f the Lrentz prfle f the lne wll be altered by any prcess whch changes the lfetmes f ether the upper r lwer energy states f the transtn. The natural wdth f the lne s gven by equatn (-6) as λ Γ λ = πc (4-) where the dampng cnstant Γ s the sum f the rates at whch atms n the upper and lwer levels change state (thus beng equal t the sum f the recprcals f the lfetmes fr each level). The nly way n whch a nn-nteractng atm can change state, r therwse affect the lfetme f the transtn, s by spntaneus emssn. Fr such an slated atm, Γ s the sum f the spntaneus emssn rates fr each level. Althugh Γ s dependent slely n the level lfetmes fr an slated atm, Γ s a prperty f the transtn between the tw levels, rather than a prperty f the levels. Thus, Γ can be cnsdered t depend n the lfetme f the transtn. Ths s an mprtant dstnctn when nteractns wth ther partcles (.e. cllsns) are cnsdered. Only a small fractn f cllsns wll deppulate the upper r lwer level (nn-adabatc cllsns), mst cllsns wll be adabatc. An adabatc cllsn wll, n general, alter the transtn energy (and thus the wavelength) fr the duratn f the cllsn. If ths shft n energy s great enugh, t can be cnsdered as an nterruptn n the exstence f the transtn, and wll thus affect the lfetme f the transtn, and therefre, Γ. Ths wll be cnsdered n mre detal n sectn 4.5. The transtn lfetme f a nn-slated atm can be affected n a number f ways: spntaneus emssn, nteractn wth the radatn feld (absrptn r stmulated emssn) and cllsns. The dampng cnstant fr the transtn wll be gven by the sum f the rates fr these prcesses: Γ = ΓR + ΓA + ΓC (4-4)

3 Chapter 4: Dampng 6 where Γ R s the ttal spntaneus emssn rate, Γ A s the cmbned absrptn and stmulated emssn rate and Γ C s the rate f ccurrence f sgnfcant cllsns. 4..: Spntaneus Emssn All dwnward transtns frm bth the upper and lwer energy states need t be cnsdered. The sum f all f the pssble spntaneus transtn rates frm a level s smply gven by the natural lfetme f the level: Γ R = t (4-5) N where t N s the lfetme. The ttal spntaneus emssn rate takng bth levels nt accunt wll be gven by a cmbnatn f the tw level lfetmes nvlved: Γ Γ Γ R Upper level = Lwer level R + R. (4-6) In the slar phtsphere, spntaneus emssn wll nt strngly affect the lne wdth, as the effect f cllsns wll be much greater. 4..: Absrptn and Stmulated Emssn When there s a radatn feld present, atmc level lfetmes wll be affected by stmulated emssn and absrptn. The rates f these prcesses wll be related t the ntensty f the radatn feld at the transtn wavelength. The stmulated absrptn rate wll be gven by Γ j = B I λd ω (4-7) j all drectns The radatve cntrbutns t the dampng cnstant Γ, namely Γ R (wth R fr radatn) and Γ A are small n the slar phtsphere. The dampng wdth f the lne wll be determned by cllsns. Ths wll nt be the case fr all cndtns, such as n nebulae, where cllsn rates are much lwer and spntaneus emssn rates becme crrespndngly mre mprtant. If the spntaneus emssn and radatve rates are nt small cmpared t cllsnal rates, LTE des nt ccur. The exstence f LTE thus guarantees small radatve rates.

4 64 Slar Lne Asymmetres where dω s the sld angle nt whch the specfc ntensty I λ passes and the ntegral s carred ut ver all drectns. The stmulated emssn rate s Γ j = B I λd ω. (4-8) j all drectns T estmate the cntrbutn f stmulated emssn and absrptn t the level lfetme, we can assume that I λ s strpc and apprxmately equal t B λ. The absrptn and emssn rates are then gven by = 4πB B Γ j j 8πhc = 5 λ λ e B j hc λkt (4-9) fr absrptn, and Γ j 8πhc = 5 λ e hc B j kt λ (4-) fr emssn. In LTE, the relatnshp between the spntaneus emssn rates and the rates abve can be btaned frm equatn (-), whch gves hc λkt Aj = e B jbλ. (4-) Cnsderng the rat f the spntaneus emssn rate fr ths transtn t the stmulated emssn rate, we btan Aj Γ = 4π j hc λkt ( e ). (4-) Fr vsble wavelengths and phtspherc temperatures, the spntaneus emssn rate s many tmes greater than the stmulated emssn rate. Transtns (spntaneus and stmulated) between the upper level and ther levels, and between the lwer level and ther levels wll als affect the level lfetmes. A smlar relatnshp between spntaneus and stmulated emssn and absrptn rates wll exst fr these transtns. Thus, n the slar phtsphere, the cntrbutn f stmulated emssn and absrptn t the wdth f the lne s much less than the cntrbutn due t spntaneus emssn (the natural lfetme). As the natural lne wdth (due slely t spntaneus emssn) s small cmpared t the lne wdth n the phtsphere (due manly t cllsns), the effects f stmulated emssn and absrptn n the lne wdth can be safely neglected.

5 Chapter 4: Dampng : Cllsns Interactns between the absrbng atm and the surrundng partcles dmnate the wdth f the Lrentz prfle f the lne. Of the mechansms cntrbutng t the wdth f the lne, as well as beng the mst mprtant, t s als the least well descrbed theretcally, as atms generally are suffcently cmplex s as t defy smple quantum mechancs. In general, each type f perturber must be taken nt accunt, alng wth any nteractns between the perturbers. Hwever, sutable apprxmatns can be made s as t make the prblem mre apprachable. 4..4: Transtn Lfetme and Asymmetry It s usually assumed that the cllsn rates wll be ndependent f the wavelength acrss the small wavelength range f the transtn. Ths gves a wavelength ndependent (as far as the partcular lne s cncerned) value f λ. Thus, althugh the wdth f the lne wll change f the lfetme changes, the Lrentzan prfle wll reman cmpletely symmetrc. Althugh determnatn f cllsn rates s n general mprtant t the thery f spectral lne frmatn, as they wll nt cause any asymmetry f the lne, ther accuracy s nt s mprtant here as any dscrepancy between thery and bservatn can be cmpensated fr by adjustng the value f Γ used s as t reprduce the bservatns. Fr ther purpses, accurate knwledge f cllsn rates s mprtant, such as when accurate abundances are beng determned, as the abundance determned frm a spectral lne depends n the dampng rates, partcularly fr strng lnes.

6 66 Slar Lne Asymmetres 4.: The Transtn Wavelength The wavelength f the transtn, λ, wll be affected by any external electrc feld; ths change n wavelength s called the Stark effect. In hydrgen and helum, spectral lnes are bserved t splt nt a number f cmpnents, wth the splttng prprtnal t the feld strength; ths s called the frst rder, r lnear Stark effect. Wth ther elements, the splttng s neglgble and what s bserved s a shft f the wavelength f the lne, usually twards lnger wavelengths. Ths shft s prprtnal t the square f the magntude f the electrc feld and s called the quadratc Stark effect. 4..: The Stark Effect 4 The energy E f an atm n state n an external electrc feld E can be wrtten as E = E + A E + B E + C E + (4-) where E s the unperturbed energy f the atm and A, B etc. are cnstants dependent n the state f the atm. When the frst perturbatn term s dmnant, a frst rder Stark effect wll be seen, and f the secnd s dmnant, a quadratc Stark effect wll be bserved. The frst rder Stark effect predmnates when levels f ppste party are nearly degenerate. These levels are then shfted n ppste drectns by the feld. When such levels are further apart (~ cm - ), they d nt nteract, and respnd dfferently t the external electrc feld, and n frst rder Stark effect s seen. Ths s the usual case fr atms ther than hydrgen and helum whch shw a quadratc Stark effect. Named after Stark wh frst bserved t n 9. 4 The Stark effect wll nly be brefly dscussed here. Many wrks n the quantum thery f atms nly brefly dscuss the Stark effect, r gnre t altgether. A mre cmplete, but stll smple, treatment can be fund n Whte, H.E. Intrductn t Atmc Spectra, McGraw-Hll (94).

7 Chapter 4: Dampng 67 As the energy shfts f the upper and lwer levels f a transtn wll dffer, the transtn wavelength wll be altered. 4..: The Lne Prfle and the Stark Effect The shft n wavelength f a sngle atm can be fund frm the lcal electrc feld, and the shft f the entre lne can be fund n terms f an average magntude f the electrc feld. The shape f the lne prfle wll als be affected, as each ndvdual absrbng atm wll have ts lne prfle shfted by a varus due t dfferng lcal electrc felds. The resultant lne prfle f the ensemble f absrbers wll be a cmbnatn f these ndvdual shfted prfles. 4..: Asymmetry and Wavelength Shft As there wll be electrc felds present n the phtsphere due t bth charged partcles (electrns and ns) and dples (neutral atms, especally hydrgen), there wll be changes n the wavelengths f lnes. The electrc felds are due t mcrscpc fluctuatns n the dstrbutn f partcles, and the dstrbutn f the electrc feld wll be very asymmetrc, and any effect wll cntrbute t the asymmetry n spectral lnes. The effects wuld be expected t be qute small n verall effect, but shuld be taken nt accunt as a pssble surce f asymmetry. 4.4: Dampng Thery Frmulatng a cmplete thery f dampng s a frmdable prblem, and s nt even usually attempted. Numerus apprxmatns are usually ntrduced t make the prblem mre tractable. Sme f the apprxmatns tradtnally made are sund, but thers excessvely reduce the accuracy f the resultng thery, r lmt ts applcablty t specal cases nly.

8 68 Slar Lne Asymmetres Dampng thery s usually apprached frm ne f tw vewpnts: the mpact apprxmatn wheren ndvdual encunters wth perturbers are cnsdered, and the quas-statc apprxmatn where all perturbers are cnsdered smultaneusly but the mtn f the perturbers s neglected. Each f these appraches has ts wn strengths and drawbacks. 4.4.: The Impact Apprxmatn In the mpact apprxmatn, nly the effects f clse encunters wth perturbng partcles are cnsdered. Impact bradenng thery thus cnssts f determnng the rate f sgnfcant encunters, and the cumulatve effect f such sgnfcant encunters. Each encunter s assumed t be wth a sngle perturbng partcle, and the tme taken fr an encunter s assumed t be small cmpared t the tmes between encunters (whch s necessary fr encunters t be wth sngle perturbers). A number f cndtns must be satsfed fr the mpact apprxmatn t be vald. The perturber densty must be lw enugh s that the average perturber dstance s large enugh fr the effect f dstant perturbers t be small, the effect f a clse encunter wth a perturber must be greater than the effects f dstant perturbers, and the clse encunter must take place suffcently rapdly. The frst cndtn (perturber densty) s mst readly satsfed by the less abundant perturbers (any ther than neutral hydrgen), the secnd by perturbers wth predmnantly shrt-range effect (such as neutral atms), and the thrd (hgh speed) by partcles f lw mass (partcularly electrns). We can thus expect bradenng by electrns t be well descrbed by the mpact apprxmatn, whle that due t pstve ns t be less s. Partcle denstes n the phtsphere are lw, and temperatures are hgh, s we can expect bradenng by neutral atms, partcularly lght atms, t be treatable n the mpact regme. The smplest mpact bradenng theres assume the effects f all encunters are dentcal, reducng the prblem t fndng the cllsn rate. Mre sphstcated theres accunt fr dfferences between ndvdual encunters. The prblem f the

9 Chapter 4: Dampng 69 nteractn between the perturber and the absrber s qute cmplex; a successful mdern thery (the Brueckner-O Mara thery) 5 s dscussed n sectn 4.5, alng wth smpler theres. Mst theres assume that the perturbng partcle mves n a straght lne and ts mtn s unaffected by the perturbed absrber (the straght classcal path apprxmatn). The effects f remvng ths apprxmatn are qute small as the perturber mtn must be clse t a straght path when the mpact apprxmatn regme s vald. 4.4.: The Quas-Statc Apprxmatn The mpact apprxmatn gnres the effects f dstant perturbers. These perturbers d, hwever, have an effect, whch can be mprtant f the effects f the clsest dstant perturbers are suffcently large. Ths s mst lkely fr abundant perturbers, such as neutral hydrgen, r lng range perturbers such as ns and electrns. Quas-statc dampng thery, r statstcal bradenng thery, s an attempt t accunt fr smultaneus effects f multple perturbers. As such, apart frm determnng the effect f a sngle perturber, the pstns f all f the perturbers must be taken nt accunt. Althugh the prbablty f dfferent perturber cnfguratns can be readly determned, t wuld be a much mre dffcult prblem t deal wth the evlutn f the perturber pstns ver tme, s t s assumed that the mtn f the perturbers s small durng the tme n whch the perturbatn s mprtant. As many perturbers are cnsdered at nce, and emphass s n the effect f dstant perturbers, the nteractn between a sngle perturber and the absrbng atm s usually cnsdered n smple terms. Fr clse encunters between perturber and absrber, we can expect a smple pcture f the nteractn t be nvald, but the quasstatc apprxmatn tself ceases t be vald n such cases. 5 See, fr example, Anstee, S.D. and O Mara, B.J. An Investgatn f Brueckner s Thery f Lne Bradenng wth Applcatn t the Sdum D Lnes Mnthly Ntces f the Ryal Astrnmcal Scety 5, pg (99).

10 7 Slar Lne Asymmetres A prblem wth standard statstcal bradenng thery s that t s ncrrect. Mst wrk n quas-statc bradenng has been fr the bradenng f spectral lnes (partcularly frst rder Stark bradenng f hydrgen) 6 by hghly nsed dense plasmas, and whle t s generally crrect fr such cases, t s nt crrect fr quadratc Stark bradenng f heaver elements n the phtsphere. A mre apprprate thery s develped n sectn : Cmbnng the Impact and Quas-Statc Apprxmatns The mpact apprxmatn fals t take multple smultaneus nteractns nt accunt; such nteractns wll dmnate whle the nearest perturbng partcle s a lng way frm the absrber. The quas-statc apprxmatn cmpletely gnres the tme dstrbutn f events and fals fr clse encunters wth perturbers. A smple frstrder cmbnatn f the tw appraches can be made by usng mpact thery t deal wth clse encunters and usng statstcal bradenng thery t accunt fr the effects f dstant perturbers between clse encunters. As the quas-statc thery fals t predct any effect n the level lfetme, mpact thery must be used t determne Γ and the Lrentzan lne wdth λ. If quas-statc thery s used t fnd the lne shft, the resultant lne prfle (as fund usng equatn 4- ) becmes the dstrbutn f lne shfts W(λ ) as gven by the quas-statc thery cnvluted wth the Lrentzan prfle gven by the mpact thery: = (,, ) φ λ φ L λ λ λ W λ dλ = λ π λ + ( λ λ ) W ( λ ) dλ. (4-4) In sectn 4-6, t wll be shwn that the lne shft dstrbutn W(λ ) s asymmetrc, and thus shuld be taken nt accunt n any study f asymmetry, but s dffcult t 6 See Grem, H.R. Spectral Lne Bradenng by Plasmas Academc Press, New Yrk (974).

11 Chapter 4: Dampng 7 calculate. The pssble cntrbutn t the ttal asymmetry f the lne due t dampng s then examned n sectn 4-7. In the phtsphere, we can expect W(λ ) t be a narrw functn, centred n a wavelength very clse t the unperturbed λ. Thus, the damped lne prfle wll be clse t Lrentzan, and fr mst purpses, t wll be suffcent t assume that t s exactly Lrentzan. The nn-lrentzan cntrbutn under ther cndtns can be much larger, but, n such cases, bth the mpact and quas-statc apprxmatns wll tend t fal (due t smultaneus multple clse encunters wth fast mvng perturbers) s ths smple cmbnatn f the tw theres wll als nt be vald : Dampng by Varus Types f Perturbers The nteractns due t dfferent types f perturbers are usually treated separately. The tw man types f cllsnal dampng, namely dampng due t nteractns wth charged partcles (ns and electrns) and dampng due t neutral atms (especally hydrgen). Dampng caused by charged partcles s called Stark bradenng, whle dampng by neutral hydrgen s termed van der Waals bradenng. Despte ths nmenclature, bth frms f dampng have ther rgns n the Stark effect. 7 Any frm f dampng where the perturber affects the absrber va electrc felds can be cnsdered t be effected by the Stark effect. As the fundamental mechansm gvng rse t the dampng n these tw cases (whch nvlve Culmb felds and dple felds respectvely) s the same, t shuld be pssble t treat them smultaneusly wth a unfed thery. Ths s explred later n ths chapter. Wth the quadratc Stark effect, the shft due t an external electrc feld f magntude E can be gven n terms f a cnstant fr the level, s fr a transtn frm a level t a level j, the energy shfts are gven by E E j = C E = C E j. (4-5) 7 Whte, fr example, makes ths pnt n Whte, H.E. Intrductn t Atmc Spectra McGraw-Hll (94) (see pages 4 and 46).

12 7 Slar Lne Asymmetres The cnstant s nrmally gven n unts f cm - per kvcm -. The transtn energy s then changed by the dfference between the shfts f each level, s E = E E j j ( C j ) = C E = C E j (4-6) and as the shft s small cmpared t the ttal energy, the wavelength shft s gven by λ = C s E (4-7) where the Stark cnstant fr the wavelength shft s related t the cnstant fr the energy shft by C s Cj = λ hc. (4-8) The shft f the upper level wll generally be greater than that f the lwer level. 8 Fr electrc felds due t charged partcles, the verall effect wll vary as the nverse furth pwer f the separatn λ = C 4 4 r (4-9) and fr neutral atms, wth predmnantly dple electrc felds, the wavelength shft wll be λ = C 6 6 r. (4-) These are the usual equatns fr Stark bradenng by charged partcles and fr van der Waals bradenng. Whle the Stark effect s well understd fr unfrm electrc felds, the felds due t very clse encunters wll be far frm unfrm. Further dffcultes arse as at shrt ranges, the perturber and absrber wll strngly affect each ther, and the electrc feld prduced by the perturber wll be affected by the absrbng atm. Shrt range encunters (fr whch mpact bradenng thery s used) wll therefre be mre cmplcated, and great care must be taken n descrbng the nteractn n terms f a sngle cnstant. Fr quas-statc bradenng, where lng range nteractns are dealt wth, a smple treatment f the perturbng partcle shuld suffce. In the phtsphere, 8 See, fr example, Babcck, H.D. The Effect f Pressure n the Spectrum f the Irn Arc The Astrphyscal Jurnal 67, pg 4-6 (98).

13 Chapter 4: Dampng 7 as the majr neutral atmc speces s hydrgen, the smple dple descrptn (equatn (4-) ) wll be adequate. Ths wll nt necessarly be the case fr ther perturbers, such as helum, where nn-dple cntrbutns t the nstantaneus electrc feld are mprtant : Bradenng f Hydrgen Lnes As hydrgen energy levels wll be splt nt a number f cmpnents by the lnear Stark effect rather than shfted by the quadratc Stark effect, the prcedure fr the cmputatn f bradenng f hydrgen lnes s smewhat dfferent t that fr spectral lnes f ther elements. The resultant lne prfle can be fund as a cmbnatn f the separate Stark cmpnents nt whch the lne s splt, each wth ts wn wavelength shft dstrbutn. Each f the Stark cmpnents can be fund n a manner smlar t ther lnes, wth the nstantaneus shft beng prprtnal t the perturbng electrc feld rather than ts square. 9 As n hydrgen lne prfles are examned n ths wrk, the bradenng f hydrgen lnes s nt f great nterest here. 4.5: Impact Bradenng Thery The usual mpact apprxmatn bradenng thery cnsders the atm t be a classcal scllatr at a partcular frequency. A perturbng partcle wll cause a shft n ths frequency f p ω = C p r (4-) where r s the dstance t the perturber and C and p are cnstants dependng n the type f nteractn nvlved (see table 4-). If the perturbng partcles travel n straght lnes (the straght classcal path apprxmatn), wth a clsest apprach t the 9 Fr a full treatment f the bradenng f hydrgen lnes, see Grem, H.R. Spectral Lne Bradenng by Plasmas Academc Press, New Yrk (974).

14 74 Slar Lne Asymmetres scllatr f ρ, the passage f such a perturber wll cause a phase shft η(t) n the scllatn, dependent n the ttal tme taken by the encunter and the strength f the nstantaneus frequency shft. perturber cnstant velcty v nstantaneus dstance r clsest apprach ρ absrber Fgure 4 - : The Straght Classcal Path The ttal phase shft nduced by a perturber at tme t s t η ( t) = ω( t ) dt t p ( ρ ) = C + v t dt p (4-) whch can be readly evaluated t gve Cpψ p η( t) = p vρ (4-) where ψ p s a cnstant dependent n p (see table 4-). Table 4 - : p and the Type f Interactn. p Absrber Perturber ψ p Hydrgen Charged partcle π Neutral atm Same atm 4 Atm ther than hydrgen Charged partcle π/ 6 Atm ther than hydrgen Neutral atm π/8

15 Chapter 4: Dampng 75 One apprach frm here s t assume that nly encunters causng a phase shft greater than sme phase shft η wll cntrbute t the bradenng f the lne. In the Wesskpf apprxmatn, t s assumed that η = and that the phase change s nstant, breakng the scllatn nt dscrete segments (see fgure 4-)..5 Ampltude Tme (cycles) Fgure 4 - : Effect f Instantaneus Impacts n Osclla tn The effect f ths n the frequency f the scllatr s the same as the effect f the fnte lfetme f the level; the lfetme f the transtn s reduced by the cllsns causng sgnfcant changes n phase. The rate f sgnfcant cllsns s Γ c = πnρ v (4-4) where ρ s the clsest apprach needed t prduce the mnmum effectve phase shft. Cpψ p ρ = η v p, (4-5) and the mean relatve speed f the perturber s gven by 8kT v = + π Mabsrber M perturber. (4-6) Fr the mpact apprxmatn t be vald, the cllsns must be rapd and nt verlap n tme wth each ther. Such separatn f the cllsns n tme requres ρ t be much less than the mean nterpartcle dstance. Ths smple thery gves results that are nly f the rght rder f magntude, even f the dampng cnstant C p s knwn, and as the cutff phase shft fr a cllsn t be effectve s arbtrary, and s assumed t be sharp, wth n cntrbutn at all frm

16 76 Slar Lne Asymmetres the (numerus) cllsns wth lesser phase shfts, the results can scarcely be expected t be mre accurate. 4.5.: Lndhlm-Fley Thery The frequency f an scllatr at any tme s gven by the unperturbed frequency ω and the frequency shft ω gven by equatn (4-), s ω( t) = ω + ω( t). (4-7) The nstantaneus phase f the scllatr s gven by η( t) = η( t = ) + t dη( t ) dt dt = η + ω + ω( t ) dt = η + ω t + η ( t) t (4-8) where η s cntrbutn t the phase due t cllsns, the cntrbutn fr a sngle cllsn beng gven by equatn (4-). The lne prfle s gven by the Furer transfrm f the scllatn, whch s gven by T η ( t ) ωt φ ( ω) = lm e e dt T πt T. (4-9) As the cllsns can be cnsdered t ccur separately n tme, we can cnsder the sum f Furer transfrms f ndvdual cllsns, s equatn (4-9) becmes T N ηn t ωt φ ( ω) = lm lm e e dt T πt N = T (4-) T ηn ( t ) ωt = lm e e dt. T πt T n As the encunters causng phase shfts (.e. the tmes when ω > ) are randmly dstrbuted n tme and the nstantaneus phase s uncrrelated wth the phase change caused by a partcular perturber, the average scllatn n equatn (4-) can be replace by an scllatn ncludng an average phase factr (ths s the Lndhlm-Fley apprxmatn). Then, we have

17 Chapter 4: Dampng 77 φ ω = lm T πt T ωt η ωt e e e dt T. (4-) It wuld prve dffcult t calculate the tme average f the phase, s t s replaced by a frequency average (va the ergdc hypthess). In ths case, the frequency average s actually an average ver mpact parameters, whch, as the rate f ccurrence f mpacts at an mpact parameter ρ s πρdρnv, s η η ρ e = πnv e ρdρ. (4-) ρ Ths can then be substtuted nt equatn (4-). The effects f the dampng can be readly seen by cmparng the resultant expressn t equatn (-5) whch was used t fnd the lne prfle functn fr an slated statnary atm. The magnary cmpnent f the cmplex term gves the frequency f a Lrentzan prfle, and the real part gves the Lrentzan wdth. Ths gves a dampng cnstant f η ρ Γ c = 8πNv sn ρ d ρ (4-) and a lne shft f ω = πnv sn η ρ ρ dρ. (4-4) These gve, fr the case f atms (ther than hydrgen) nteractng wth charged partcles (each type f whch needs t be cnsdered separately as they wll have dfferng mean speeds), Γ c = 6. C 4 v N (4-5) and ω = C 4 v N. (4-6) Fr the p = 6 case, (where hydrgen wll be the dmnant perturber due t ts abundance) we btan 5 5 Γ c = 8. 8 C 6 v N (4-7) and 5 5 ω =. 94 C 6 v N. (4-8) The prblems wth ths apprach are that t stll des nt cnsder verlappng cllsns, whch wll ccur especally fr encunters at large dstances causng small

18 78 Slar Lne Asymmetres perturbatns, and stll assumes that all partcles (f a partcular type) can be adequately descrbed as mvng at a unfrm speed. The frequency at whch strng cllsns ccur, whch determnes Γ, shuld be gven reasnably accurately, but as the tmes between strng cllsns, when a large number f dstant perturbers have a cumulatve effect, are nt cnsdered, the lne shft may nt be gven accurately by ths thery. The bradenng, beng purely Lrentzan (wth a wavelength shft), s symmetrc. 4.5.: Dampng Cnstants fr Cllsns wth Neutral Hydrgen Determnng dampng cnstants accurately s a dffcult task. It s made smewhat easer n the case f the phtsphere by the fact that almst all neutral atmc perturbers (namely hydrgen and helum) wll be n the grund state due t the hgh energy f the frst excted state abve the grund state. Ths smplfes the quantum mechancal prblem f the absrber-perturber nteractn cnsderably. The dffculty les n adequately descrbng the absrbng atm. As a crude apprxmatn, we can assume that the absrber n state can be descrbed by an effectve prncpal quantum number n *, gven by n * = Z χ H χ χ I (4-9) where Z s the effectve nuclear charge, χ H s the nsatn energy fr hydrgen, χ I s the nsatn energy fr the absrbng atm, and χ s the energy f the state. Ths hydrgenc apprxmatn can then be used t determne the energy shft f the state due t the perturbatn. The energy shft s e a R E = α 6 r (4-4) where α s the plarsablty f hydrgen (= cm ) and The hydrgenc apprxmatn s cmmnly used because t s partcularly smple. Fr a full dervatn, see, fr example, pg 97 n Mhalas, D. Stellar Atmspheres Freeman (97). Unfrtunately, the hydrgenc apprxmatn s nt always accurate.

19 Chapter 4: Dampng 79 R ( n *, l) * n * = ( n l( l ) ) Z. (4-4) Ths gves the dampng cnstant ( upper lwer ) C6 = 4. 5 R R. (4-4) Whle dampng cnstants derved n ths manner can be used as apprxmate values, they are generally nsuffcently accurate when the dampng must be wellknwn. Fr mre accurate dampng cnstants, mre sphstcated thery, such as that f Brueckner, as extended by O Mara must be used. 4.5.: Brueckner-O Mara Thery The Brueckner-O Mara thery has s far prved t be an accurate and relable methd fr determnng dampng cnstants, partcularly fr cllsns wth neutral atmc hydrgen n the grund state. The thery assumes that the nteractn between the absrber and the hydrgen atm s suffcently weak s that perturbatn thery can be used. Raylegh- Schrödnger perturbatn thery s used and exchange nteractns are neglected. The Unsöld apprxmatn s used n secnd-rder perturbatn thery t replace the energy denmnatr wth a sutable average energy (whch allws majr smplfcatn). Lastly, the classcal path apprxmatn s assumed t be vald. The Hamltnan fr the absrber-perturber system can be wrtten as H = H + V (4-4) where, as s usual, V s the nteractn between the atms. See Anstee, S.D. and O Mara, B.J. An Investgatn f Brueckner s Thery f Lne Bradenng wth Applcatn t the Sdum D Lnes Mnthly Ntces f the Ryal Astrnmcal Scety 5, pg (99), r the earler papers by O Mara (see Bblgraphy). Unsöld, A. Quantenthere des Wasserstffmlekülns und der Brn-Landéschen Abstßungskräfte Zetschrft für Physk 4, pg (97).

20 8 Slar Lne Asymmetres electrn (perturber) r electrn (absrber) r p θ r p θ perturber (hydrgen atm prtn) R absrber Fgure 4 - : Gemetry f Perturber - Absrber System The nteractn term V s gven by V = e V R r r ( p ) (4-44) where the absrbng atm s mdelled as a sngle ptcally actve electrn utsde a clsed nc cre apprxmated by a Thmas-Ferm-Drac n (TFD n). The ptental f the TFD nc cre can be expressed n terms f a sheldng functn f(p ) as V ( p ) = + f ( p ) p. (4-45) Far frm the nc cre, the sheldng functn s zer, and appraches (Z )/p as the nucleus f ttal charge Z s apprached. Fr cnvenence, atmc unts can be used wheren e =, dstances are n Bhr rad and energes are n Hartree. Raylegh-Schrödnger perturbatn thery wll break dwn fr small separatns R, especally f R s clse t r less than the dstance t the ptental mnmum. At lnger ranges, secnd-rder perturbatn thery wll be suffcently accurate. The frst-rder nteractn energy s V E = ψ ψ (4-46) and the secnd-rder nteractn energy s E V V = ψ ψ ψ ψ, (4-47) k k k E Ek whch, usng Unsöld s apprxmatn, reduces t

21 Chapter 4: Dampng 8 E ( ) ( ) = V E E p ψ ψ. (4-48) The atmc states here are fr the cmbned system, but as vrtually all hydrgen n the phtsphere wll be neutral and n the grund state, they wll n practce nly depend n the state f the absrber. The frst rder energy s ften assumed t be zer, as t s an expnentally decreasng functn f R. At ths pnt, gven sutable wavefunctns fr the atmc states, the ptentals fr the upper and lwer states can be fund. T calculate the Lrentzan lne prfle, we can assume that the lne wdth Γ and the shft ω can be btaned frm a cmplex crss-sectn σ(v), where Γ + ω = N vw( v) ( v) σ dv, (4-49) v s the relatve velcty, and W(v) s the prbablty dstrbutn f relatve velctes. In the mpact apprxmatn, the crss-sectn σ(v) s gven n terms f the dampng parameter Π(ρ,v) fr a sngle cllsn as = σ v π Π ρ, v ρdρ. (4-5) Fr perturbatns by grund state hydrgen, the dampng parameter becmes Π ρ,v l tr lwer = + lupper + tr ( Supper ) ( S ) lwer. (4-5) The S matrces can then be calculated usng the nteractn energy f the system. Once the dampng parameter s knwn, the cmplex cllsn crss-sectn can be fund usng equatn (4-5), and thus the lne shft and wdth can be fund. The lne wdth can readly be fund frm equatn (4-49) f the cmplex crss-sectn σ(v) s replaced by ts real cmpnent. Then Γ = N vw( v) σ( v) dv (4-5) where σ(v) s the real cllsn crss-sectn. As σ(v) s a slwly varyng functn f the relatve velcty v, ths expressn fr the lne wdth Γ can be well apprxmated by Γ = Nvσ ( v ) (4-5)

22 8 Slar Lne Asymmetres n terms f the average relatve velcty (as gven by equatn (4-6) ). Fr cnvenence, the crss-sectns calculated usng ths thery can be expressed n terms f the crss-sectn at a reference velcty and an nterplatn cnstant α such that σ - ( v) = σ( v = kms ) v kms - α. (4-54) Gven these σ and α values, calculatn f the lne wdth s straghtfrward. The relatnshp between ths crss-sectn and the usual van der Waals cnstant C 6 can be fund frm equatn (4-7), gvng v 5 C ( v ) 6 = σ 86 (4-55) where the mean velcty s n c.g.s. unts. Ths can be used t cmpare results. Dampng cnstants gven by the Brueckner-O Mara thery appear t be f reasnable accuracy. The lack f accurate expermental results hnders cmparsn, but, partcularly n vew f the naccuracy f ther cmmn methds f calculatng dampng cnstants, the theretcal dampng cnstants btaned n ths way may well be the best avalable : Statstcal Bradenng Thery Impact bradenng thery des nt adequately accunt fr perturbatns caused by multple partcles smultaneusly. Statstcal bradenng thery (the quasstatc apprxmatn) s an attempt t deal wth ths shrtcmng. The usual startng pnt s equatn (4-), n wavelength unts gvng Sftware fr the calculatn f these cnstants was suppled by J.E. Rss (Physcs Department, The Unversty f Queensland). 4 Anstee and O Mara cmpare varus theretcal and expermental results fr the wdths f the sdum D lnes n Anstee, S.D. and O Mara, B.J. An Investgatn f Brueckner s Thery f Lne Bradenng wth Applcatn t the Sdum D Lnes Mnthly Ntces f the Ryal Astrnmcal Scety 5, pg (99). See als Mlfrd, P.N. Lne Intensty Rats and the Slar Abundance f Irn PhD Thess, The Unversty f Queensland (987). Recent wrk by O Mara cnfrms the general accuracy f the thery.

23 Chapter 4: Dampng 8 p λ = C p r. (4-56) Each perturber s then assumed t prduce a /r p feld, and the prbablty dstrbutn f the cmbned feld s then fund. 5 Ths standard treatment s ncrrect; whle t wll prduce a reasnable result, as t wll crrectly gve the effect f the nearest perturber, t wll nt crrectly predct cmbned effects. (Cnsder the dfference between cmbnng tw nverse square felds and fndng the square f the magntude and cmbnng tw /r 4 felds. Nte that a + b a a + b b n mst cases.) There s n reasn nt t attempt a crrect calculatn f the dstrbutn f the wavelength shft. 4.6.: The Stark Effect and Statstcal Bradenng Thery Whether r nt the perturbng partcle s an n, electrn r a neutral atm, unless t s very clse t the absrber, the nteractn s caused by the effect f the electrc feld f the perturber at the absrber. Thus, the nteractns are all due t the Stark effect, even thse nrmally labelled as van der Waals bradenng (p = 6). If we knw the prbablty dstrbutn fr the magntude f the electrc feld, W(E)dE, the prbablty dstrbutn fr the wavelength shft W( λ)d λ can be fund. Fr a quadratc Stark effect 6, the wavelength shft s related t the feld by λ = C s E (4-57) where C s s the cnstant f prprtnalty fr the quadratc Stark effect fr the transtn n questn. Then, snce d λ = CsEdE, (4-58) the prbablty dstrbutn s gven by W( E) W( λ) d λ = C E d λ. (4-59) s 5 See, fr example, pg 65 n Mhalas, D. Stellar Atmspheres Freeman (97). Here, the prcedure s adapted frm a calculatn by Chandrasekhar f the prbablty dstrbutn f the gravtatnal feld caused by an nfnte number f dentcal randmly dstrbuted stars, n Chandrasekhar, S. Stchastc Methds n Astrphyscs, Revews f Mdern Physcs 5, pg (94). The calculatn f an electrc feld dstrbutn s smlar. 6 Here we wll gnre the fact that the external electrc feld at ur absrber s nt spatally unfrm, and thus assume that the shft can be gven smply n terms f the electrc feld magntude.

24 84 Slar Lne Asymmetres The dffculty arses when we try t calculate W(E)dE. If we assume that we have an nfnte number f perturbers unfrmly dstrbuted, we btan a Hltsmark dstrbutn 7. Dstrbutns f ths frm have been calculated, but generally nly fr ether charged partcles (ns and electrns) r dples (neutral atms) ndvdually, rather than bth smultaneusly. Als, neutral atms are usually treated as prducng a pure /r dple feld, rather than cnsderng the effects f varatn n the dple mment and drectn f the dple, resultng n an ncrrect dstrbutn (but crrect mean behavur). The apprxmatn f a perturbng neutral atm as a dple wll nly be adequate at large separatns, but at clser separatns, the quas-statc apprxmatn wll fal anyway, s t shuld prve adequate fr the purpse. 4.6.: Hltsmark Thery The dstrbutn f the partcles wll nt be randm (fr example, the pstn f charged partcles wll be affected by the electrc feld), s we must als determne hw gd an apprxmatn the Hltsmark dstrbutn wll actually be. Feld dstrbutns takng the nn-randm dstrbutn f partcles have been calculated 8, but, as under the cndtns n the sun, the Hltsmark dstrbutn, whch wll be easer t calculate, may be a qute gd apprxmatn. A plasma can be charactersed by ts Debye length whch s the effectve dstance ver whch the electrc feld f a partcle nteracts wth ther partcles befre beng cancelled ut by the felds f sheldng partcles. Snce ns can be shelded by bth electrns and ther ns, the Debye length can be fund by cnsderng a ttal charged partcle densty f N e (assumng that all f the ns have a sngle pstve charge) whch gves a Debye length f kt ( λ Debye ) = (4-6) ns 8πN e e where e s the charge f an electrn. Ths wll als be the Debye length fr dples. Only electrns wll act t sheld the feld due t ther electrns (due t ther hgher speeds), s the Debye length fr electrns s gven by 7 Hltsmark, J. Über de Verbreterung vn Spektrallnen Annalen der Physk 58, pg 577 (99) 8 See Mzer, B. and Baranger, M. "Electrc Feld Dstrbutns n an Inzed Gas. II", n Physcal Revew 8, pg 66 (96) where they calculate the feld dstrbutn fr a cmpletely nsed gas under varyng cndtns.

25 Chapter 4: Dampng 85 ( λ Debye ) electrns = kt. (4-6) πn e e The ttal number f ns r dples f number densty N wthn a Debye sphere (a sphere f radus equal t the Debye length), beng the number f partcles whch cntrbute t the feld at a pnt, s gven by ( n Debye ) N kt = π N e and the number f electrns wthn a Debye sphere s ( n Debye ) electrns = 6 πn e e kt e (4-6) (4-6) The number f partcles n the Debye sphere at varus depths n the phtsphere can then be fund. (See table 4 -.) Table 4-: The number f partcles wthn a Debye sphere fr varus speces f partcles (usng Hlweger-Müller atmsphere). Optcal Depth τ Hydrgen Atms Charged Partcles Electrns Ins Snce there are a large number f hydrgen atms wthn a Debye sphere, the Hltsmark dstrbutn wll be a very gd apprxmatn fr the dple feld dstrbutn, and the number f charged partcles s als hgh enugh s that the Hltsmark dstrbutn wll be a reasnable apprxmatn.

26 86 Slar Lne Asymmetres 4.6.: Hltsmark Thery Revsted If we cnsder a regn f the phtsphere t be unfrm and effectvely nfnte, wth a charged partcle number densty f N c and a (hydrgen atm) dple densty f N d, we can determne the prbablty dstrbutn f the feld due t bth charged partcles and dples. 9 See table 4- fr a summary f ntatn used n ths calculatn. Table 4-: Summary f ntatn used n calculatn f Hltsmark dstrbutn. Symbl Meanng f Symbl W(η)dη Prbablty dstrbutn functn fr η E = E (r ) Electrc feld due t th charged partcle r W(E )de E j = E j (r j,l j ) r j l j E W(E)dE E W(E)dE Pstn vectr fr th charged partcle Feld dstrbutn fr a charged partcle Electrc feld due t jth dple Pstn vectr fr jth dple Dple separatn vectr fr jth dple Ttal electrc feld Dstrbutn fr ttal electrc feld Magntude f ttal electrc feld Dstrbutn fr magntude f ttal feld Frst we wll cnsder the feld due t n c dentcal charged partcles and n d dples, each wth an dentcal dstrbutn fr ther dple separatn vectrs. (The dple separatn vectr gves the drectn f the dple mment and the dstance between the tw charges cmprsng the dple.) If we knw the feld dstrbutns fr the charged partcles and the dples, we can then wrte the ttal feld dstrbutn as the cnvlutn f all f the ndvdual partcle dstrbutns: ( E) = ( E= ) ( E= n ) ( E j= ) ( E j= n ) W W.. W W.. W. (4-64) c As there are a great many partcles, t s nt pssble t calculate ths drectly, but we can rewrte ths cnvlutn as a prduct n a Furer dman as d 9 Ths calculatn fllws Chandrasekhar (n Chandrasekhar, S. Stchastc Methds n Physcs and Astrnmy Revews f Mdern Physcs 5, pg (94 ) ), but deals smultaneusly wth bth knds f partcles (charged partcles and dples).

27 Chapter 4: Dampng 87 [ ] r. W A ρ A ρ e E dρ [ ] j n n c d ( E) = 8π all space (4-65) where A (ρ) and A j(ρ) are the Furer transfrms f W(E ) and W(E j ). The Furer transfrm fr charged partcles s gven by ρ E ( r ) ( ρ). A = W E e de, (4-66) all felds r, n terms f W(r ) and r nstead f W(E ) and E : ρ E ( r ) ( ρ) all space. A = W r e dr. (4-67) If the partcles are unfrmly dstrbuted ver a vlume V, then W A.E r ρ ( ρ) = e V all space dr. (4-68) Smlarly, fr dples, the Furer transfrm s ρ. E j ( rj, l j ) ρ A = W E e de j all felds j j (4-69) and we can wrte ths n terms f W(r j ), r j, W(l j ) and l j : ρ. E j ( rj, l j ) Aj ρ = W( rj ) W( l j) e dl jdrj. (4-7) all all space dple vectrs Agan, W( r j ) =, and W A j V l j = πa V πa e e l e a l a ρ. E r, l j j j ρ = all all space dple vectrs (where a s the Bhr radus), s dl dr. (4-7) Usng these expressns fr the Furer transfrms, W(E) s gven by j j r = and V Chandrasekhar (n Chandrasekhar, S. Stchastc Methds n Physcs and Astrnmy Revews f Mdern Physcs 5, pg (94 ) ) arrves at ths pnt va a dfferent rute, by startng wth the sum f the prbabltes f all cnfguratns gvng a partcular value fr the ttal feld, and then transfrmng ths sum nt an ntegral ver all space by usng the dscntnuus ntegral f Drchlet, and thus btanng equatn (5-5) fr the speces f perturber n questn. Ths result, the prbablty dstrbutn fr the lcatn f an electrn n the grund state f a hydrgen atm may be mre famlar n the frm W l dl l 4 a l e a = dl where dl s cnverted t plar crdnates and, as there s n angle dependence, ntegrated ver all angles. Ths s used fr W(l) as the nly atm we are cnsderng as a strng dple s hydrgen, and almst all hydrgen atms wll be n the grund state due t the large separatn between the grund state energy and the energy f the frst excted state.

28 88 Slar Lne Asymmetres where W( E) = A( ρ) e 8π ρ. E dρ N V = V V l a e πa c ρ. E ( r ) A e e ρ. E r, l j j j ρ dr dl jdrj (4-7) N dv (4-7) where N c and N d are the number denstes f charged partcles and dples, s that the ttal number f charged partcles s gven by n c = N cv, and the ttal number f dples s gven by n d = N dv. Snce we are cnsderng a very large vlume wth an effectvely nfnte number f partcles, we can cnsder the lmt when V appraches nfnty. Keepng n mnd that n x x lm + = e (4-74) n n we can rewrte equatn (4-7) as ρ N c e A( ρ) = +. E r dr N cv N cv l a Nd e e a π + NdV r l dl jdrj ρ. E j j, j N dv (4-75) whch, usng equatn (4-74), gves A e N c D c ρ N d D d ρ ρ = (4-76) where D c and D d are gven by and D c D d ρ.e ( r ) ( ρ) = - e all space πa dr (4-77) a ρ ρ =. Ej rj, l j e e all all space dple vectrs l dl jdr j. (4-78) If these expressns can be calculated, then W(E) can be fund. If equatn (4-77) s rewrtten n plar crdnates, wth ρ drected alng the z-axs, we then have π π ρe csφ D ( ρ ) = ( e ) r sn φ dr dφ dθ. (4-79) c θ = φ = r= The magntude f the electrc feld s gven by

29 Chapter 4: Dampng 89 Cc E = r (4-8) where C c s the feld strength cnstant. (Equal t the electrnc charge e n ths case.) Then π c D r ρ = ( e ) r sn φ dr dφ dθ. (4-8) c π θ = φ = r= ρc csφ Integratng wth respect t θ gves r D ρ = π ( e ) r sn φ dr dφ. (4-8) c π φ = r= ρcc csφ If we substtute y = csφ, dy = snφ dφ, we btan r D ρ = π ( e ) r dr dy (4-8) c y=- r= ρcc y whch can be (snce sn ρc y c s dd w.r.t. y) wrtten as r ρcc y Dc ( ρ ) = π ( cs ) r dr dy (4-84) r y= r= whch can then be ntegrated t gve sn ρcc r Dc ( ρ ) = 4π ρcc r Cc Substtutng z = ρ and r = ( γ ρc ) c c r dr. (4-85) ( C ) c r dr = ρ 5 dz, we btan z ( ρc ) sn z c Dc ( ρ ) = 4π 5 dz z z z sn z = π ( ρcc ) 7 dz z whch can be evaluated t gve 4 π Dc ( ρ ) = π ( ρcc ) 5 where π γ = 8 c 5 (4-86) (4-87).6. (4-88) Smlarly, we can fnd a smple expressn fr D d. If we rewrte equatn (4-78) n plar crdnates θ, φ, and r fr the pstn vectr r j, and ξ, ψ, and l fr the dple vectr l j, we have

30 9 Slar Lne Asymmetres D d πa a ρe j csφ ( ρ ) = e ( e ) π π π π θ = φ = r= ξ = ψ = l= l (4-89) l snψ dl dψ dξ r sn φ dr dφ dθ. The magntude f the dple feld s gven by Cdl E j = + cs ψ (4-9) r where C dl s the dple mment. (Here, C d s the electrnc charge e and l s separatn between the prtn and the electrn.) If we ntegrate wth respect t θ and ξ and use equatn (4-9) fr E j we btan π π 4π Dd ( ρ ) = e a φ = r= ψ = l= e l ρc d + a cs ψ csφ r l snψ dl dψ r sn φ dr dφ. (4-9) Substtutng x = csψ, dx = snψ dψ, y = csφ and dy = snφ dφ, ths gves l ρc ly d 4π x a + Dd r ρ = e e l r dl dx dr dy. a y= r= x= l= (4-9) As befre, the sne term n the e... can be drpped, and the expressn ntegrated n y: l ( C l x r ) a d Dd ( ρ ) = 8 + e - π sn ρ l r dl dx dr. a r x l Cd l + x r = = = ρ (4-9) ρcd l + x Cdl x If we nw substtute z = and r dr = ρ + ths becmes r z l 8πρC d a z z sn Dd ( ρ ) = l e + x dl dx dz a z where π ρc d a = l e + x dl dx a π a ρcd = + x dx 4 π a ρc ln d = + 4 = γ a ρc d d z= x= l= x= l= π γ = + + d 4 ln l ( + ) (4-94) (4-95) The values that have been btaned fr D c and D d can be substtuted nt equatn (4-76) t gve c ( c c ) d ( d d ) ( ρ ) = γ ρ γ ρ A e N C N a C (4-96)

31 Chapter 4: Dampng 9 We can nw fnd W(E): N c ( γ ccc ρ ) N d ( γ dacd ρ ) ρ. W E = e e E dρ, (4-97) 8π r, n plar crdnates, π π N ( C ) N ( a C ) E W e c γ E = c c ρ d γ d d ρ ρ csφ e ρ sn φ dρ dφ dθ. π 8 θ = φ = ρ = As befre, we can nw ntegrate ver θ, gvng π N ( C ) N ( a C ) E W e c γ c c ρ E = d γ d d ρ ρ csφ e ρ sn φ dρ dφ. π 4 φ = ρ= (4-98) (4-99) Substtutng z = csφ and dz = snφ dφ, and drppng the sne term frm the cmplex expnental, N c ( γ ccc ρ ) N d ( γ dacd ρ ) W E = e cs ( ρez) ρ dρ dz.(4-) π 4 z= ρ= Ths can nw be ntegrated t gve W E e Nc γ ccc ρ N d γ dacd ρ E = ρ sn ρ π E d ρ, (4-) nt whch we can nw substtute x = ρe, gvng W( E ) = π E γ cccx γ cacd x N N c d E E e x sn x dx. (4-) We nw defne a nrmal charged partcle feld as F = C N 9 γ 5. N (4-) c c c c c and a nrmal dple feld as F = γ a C N N, (4-4) d d d d d and W(E) n terms f these nrmal felds s then W( E ) = π E Fc x Fd x + E E e x sn x dx. (4-5) We d nt actually want W(E), the electrc feld dstrbutn functn, but rather W(E), the dstrbutn functn fr the magntude f the electrc feld. S, as W(E) = 4π E W(E), W E r, f preferred, wth y = x/e, Fc x Fd x + E e E E = x sn π x dx (4-6) E F y c + Fd y W( E) = e y sn ( Ey ) dy. (4-7) π If we defne a nrmal feld rat α as

32 9 Slar Lne Asymmetres α = F d Fc (4-8) and measure the feld n unts f the nrmal charged partcle feld strength (gvng the dstrbutn n as clse as pssble t the standard ntatn ), β = E F c. (4-9) Then W(β) = F cw(e), gvng us r x αx + β β W( β ) = e x sn x dx (4-) πβ y + αy β W( β ) = e y sn ( βy) dy. (4-) π These ntegrals (equatns (4-8), (4-9), (4-) and (4-)) can then be numercally ntegrated t determne the feld dstrbutn fr a set f gven cndtns frm whch the nrmal feld strengths are determned. (See table 4-4 fr nrmal feld strengths n the phtsphere.) Table 4-4: Nrmal Felds n the Phtsphere Optcal Depth τ F c F d (esu) (esu) α If we nly cnsdered charged partcles r dples ndvdually, ths wuld actually gve the standard frm, but here we have tw separate nrmal felds and n smple methd t measure E n terms f bth f them.

33 Chapter 4: Dampng 9 Frm these results, we can see that the felds caused by charged partcles and dples are bth mprtant; bth must be taken nt accunt fr an accurate treatment f bradenng. The smple classcal treatment f the hydrgen atm dple used here wll be an adequate apprxmatn when estmatng the mprtance f quas-statc dampng. If accurate numercal results are needed, the apprprate ntegrals can be replaced by sums ver dscrete states. 4.7: Spectral Lnes and the Hltsmark Dstrbutn The dstrbutn fr the lne shft wll be gven by equatn (4-59), gvng W d C e Fc y Fd y y y λ + λ λ = C dy sn d λ (4-) π s s usng equatn (4-7), r a smlar expressn usng any f the ther three frms f the feld dstrbutn. A sample Hltsmark dstrbutn s shwn n fgure Fgure 4-4: The Hltsmark Dstrbutn 4.7.: Cmbnng the Impact and Quas-statc Apprxmatns Fr phtspherc dampng, quas-statc dampng s usually gnred, as the cntrbutn t the wdth wll be neglgble. The mpact apprxmatn alne s used fr the dampng, and gven sutable nteractn cnstants, gves an accurate result fr

34 94 Slar Lne Asymmetres the wdth f the Lrentzan prfle. The quas-statc dampng, hwever, s strngly asymmetrc even f t cntrbutes neglgbly t the verall lne wdth. A sutable methd t fnd the cmbned effect f mpact and quas-statc bradenng s t use the mpact apprxmatn t determne the Lrentzan wdth Γ and the quas-statc thery t calculate the lne shft dstrbutn, as gven by equatn (4-). The result wll then be a cnvlutn between the tw prfles. A quck examnatn f such cmbnatns (see fgure 4-5) shws us that the quas-statc dampng can cntrbute sgnfcantly t the asymmetry f the lne whle effectng nly mnr changes n the lne wdth. Lrentz Prfle + Hltsmark Vgt Prfle + Hltsmark Fgure 4-5: Cmbnatn f Hltsmark prfle and Lrentz and Vgt Prfles The asymmetry caused by the dampng s als the same general type f asymmetry that s seen s phtspherc absrptn lnes. Thus, t s mprtant t be able t accurately estmate the cntrbutn t the lne prfle due t quas-statc dampng. 4.7.: The Quas-Statc Cntrbutn The quas-statc and mpact bradenng are related by the same nteractns beng respnsble fr bth. Thus, n the case f the phtsphere, we can determne the relatve mprtance f the quas-statc cntrbutn. In rder t nvestgate the mprtance f quas-statc dampng, we must be able t cmpare the Stark ceffcent C s used n the quas-statc calculatns wth the

35 Chapter 4: Dampng 95 dampng nteractn cnstants C 4 and C 6 used n the mpact apprxmatn calculatns. The mean value f the magntude f the dple feld gven by equatn (4-9) when averaged ver all rentatns and all dple separatns wll be apprxmately ea E =. (4-) r Ths gves a wavelength shft f λ = Cse a 6. (4-4) r The crrespndng shft n angular frequency s π ω = 4 Cse a c 6 (4-5) λ r and the dampng cnstant C 6 s thus 4 e a c C 6 = π Cs. (4-6) λ Smlarly, C 4 e c = π Cs λ (4-7) and C = a C. (4-8) 6 4 In rder t cmpare the effects f the Lrentzan mpact bradenng and the Hltsmarkan quas-statc bradenng, we can assume a (fcttus) Fe I spectral lne at 5Å wth C 6 = 5. - (f the same rder f magntude as C 6 fr the sdum D lnes). Then, C 4 = and C s = (n standard c.g.s. unts). The mpact dampng can then be calculated fr varus heghts wthn the phtsphere (see table 4-5). Table 4-5: Impact Dampng Cnstants Optcal Depth Hydrgen Ins and Ttal Mnmum Sgnfcant τ Γ 6 Electrns Dampng Wavelength Shft Γ 4 Γ λ (Å) The mprtance f cllsns wth neutral hydrgen can seen n table 4-5, as they clearly prvde the greatest cntrbutn t the Lrentzan lne wdth.

36 96 Slar Lne Asymmetres The Hltsmark dstrbutns at these heghts can be calculated numercally. The electrc felds needed t gve wavelength shfts equal t these Lrentzan lne wdths are shwn n table 4-6. Table 4-6: Electrc Felds Requred fr Shfts Lne wdth (Å) Feld fr shft (e.s.u.) The Hltsmark dstrbutns at these phtspherc heghts and feld strengths are extremely small. The felds requred t prduce these shfts are much greater than the felds whch ccur wth any real degree f prbablty. The mst lkely felds are even smaller. (See table 4-4 fr expected felds (.e. nrmal felds).) As the quas-statc dampng has such a small effect, t can be cmpletely neglected n the slar phtsphere. As ths estmate f the mprtance f quas-statc dampng neglected Dppler bradenng, the effect cmpared t the ttal wdth f the Vgt prfle wll be even smaller. Thus, dampng shuld ntrduce n asymmetry, and, as the dampng wll be symmetrc, unlke the Dppler bradenng, strngly damped lnes shuld shw less asymmetry than weakly damped lnes. Ths s bserved n the slar spectrum (see chapter ). 4.8: Dampng Cnstants The methds descrbed abve (such as the Brueckner-O Mara thery) can be used t determne dampng cnstants (.e. lne wdths) fr many cases. At ther tmes t may nt be pssble t accurately determne a dampng cnstant fr a partcular lne, such as when the electrnc cnfguratn f the element s such that t prves partcularly dffcult t apprprately descrbe the atm at the ranges at whch mprtant cntrbutns t the dampng are made. Sme slar lnes are stll undentfed, whch makes any such calculatn mpssble. Fr sme cases where the dampng cannt be theretcally calculated, expermental dampng cnstants have been measured. Expermental dampng cnstants fr cllsns wth atmc hydrgen, hwever, are few n number, and generally nt very accurate.