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1 Recombiatio Lies Itroductio to Radio Spectral Lies Á Ü Spectral lies are arrow ( ) emissio or absorptio features i the spectra of gaseous sources. Examples of radio spectral lies iclude the Õ = cm hyperfie lie of iterstellar HI, recombiatio lies of ioized hydroge ad heavier atoms, ad rotatioal lies of polar molecules such as carbo mooxide (CO). Spectral lies are itrisically quatum pheomea. Classical particles ad waves are idealized cocepts like ifiitesimal poits or perfectly straight lies i geometry; they do't exist i the real world. Some thigs are early waves (e.g., radio waves) ad others are early particles (e.g., electros), but all share characteristics of both particles ad waves. Ulike ideal waves, real radio waves do ot have a cotiuum of possible eergies. Istead, electromagetic radiatio is quatized ito photos whose eergy is proportioal to frequecy: E = h. Ulike ideal particles, real particles of mometum p have wave fuctios whose De Broglie wavelegth is Õ = h=p. A electro's orbit about the ucleus of a atom must allow stadig waves, so its circumferece must be a iteger umber of wavelegths. The Plack's costat h Ù 6:6607  0 À7 erg s i these two equatios is a quatum of actio; its dimesios are (massâ legth  time ), the same as (eergyâ time) or (agular mometum) or (legthâ mometum). Spectral lies have defiite frequecies resultig from trasitios betwee discrete eergy states i physical systems, ad these discrete states arise from quatizatio of agular mometum. Aother quatum effect importat to spectral lies, particularly at radio wavelegths where h Ü kt, is stimulated emissio. Fortuately, the fudametal characteristics of radio spectral lies from iterstellar atoms ad molecules ca be derived from fairly simple applicatios of quatum mechaics ad thermodyamics. Spectral lies are powerful diagostics of physical ad chemical coditios i astroomical objects. Doppler shifts of lie frequecies measure radial velocities. These velocities yield the redshifts ad Hubble distaces of extragalactic sources, as well as the rotatio curves ad radial mass distributios for resolved galaxies. Collapse speeds, turbulet velocities, ad thermal motios cotribute to lie broadeig i Galactic sources. Temperatures, desities, ad chemical compositios of HII regios, dust-obscured dese molecular clouds, ad diffuse iterstellar gas are costraied by spectral-lie data. Some characteristics of radio spectral lies iclude: () The "atural" lie widths are much smaller tha Doppler-broadeed lie widths, so very small chages i radial velocity ca be measured. () Stimulated emissio is importat because h Ü kt. This causes lie opacities to vary as T ad favors the formatio of atural masers. (3) The ability to peetrate dust i our Galaxy ad i other galaxies allows the detectio of lie emissio emergig from dusty molecular clouds, protostars, ad molecular disks orbitig AGNs. (4) I practice, frequecy (iverse time) ca be measured with much higher precisio tha of 0 /06/008 :4 A

2 wavelegth (legth), so very sesitive searches for small chages i the fudametal physical costats over cosmic timescales ca be made. ost of the iterstellar medium (IS) i our Galaxy is i rough pressure equilibrium because mass motios with speeds up to the speed of soud act to reduce pressure gradiets quickly. Temperatures equilibrate more slowly, so there are wide rages of the (temperatureâ desity) product cosistet with a give pressure. Cosequetly, there are four importat phases of the IS havig comparable pressures: () cold (0's of K) dese molecular clouds () cool ( Ø 0 K) eutral HI gas (3) warm ( Ø 0 4 K) ioized HII gas (4) hot ( Ø 0 6 K) low-desity ioized gas (i bubbles formed by expadig superova remats, for example). All but the hot phase are sources of radio spectral lies. Recombiatio Lie Frequecies The semiclassical Bohr atom cotais a ucleus of protos ad eutros aroud which oe or more electros move i circular orbits. The uclear mass is always much greater tha the sum of the electro masses m e, so the ucleus is early at rest i the ceter-of-mass frame. The wave fuctios of the electros have De Broglie wavelegths p h h Õ = = ; p m v where is the electro's mometum ad is its speed. Oly those orbits whose circumfereces equal a iteger umber of wavelegths correspod to stadig waves ad are permitted. Thus the Bohr radius a of the th permitted electro orbit satisfies the quatizatio rule The requiremet that Ùa = Õ = h : m v a = v h Ùm v e implies that the orbital agular mometum be a iteger multiple of. a v The relatio betwee ad is provided by balacig the Coulomb ad cetrifugal forces. For a hydroge atom, e e = Öh m v e L = m e va Ö h Ñ h=(ù) e a m e v = a so of 0 /06/008 :4 A

3 v = e m e a a = Ö h m e e (7A) Example: What is the Bohr radius of a hydroge atom whose electro is i the th electroic eergy level? a = Öh [6:63=(Ù)  0 À7 erg s] = Ù 0:53  0 À8 cm  m e 9:  0 À8 g  ( 4:8  0 0 statcoul) e = The Bohr radius of a hydroge atom i its groud electroic state ( ) is oly cm, but sice a highly excited ( ) radio-emittig atom i the a Ù 0:53  0 À8 a / Ù 00 a00 Ù 0 À4 = Ö IS ca be remarkably large: cm m, which is bigger tha most viruses! The radius of the th Bohr orbit is proportioal to, so radio-emittig hydroge atoms with are much bigger tha ordiary hydroge atoms i the groud state. Ø 00 = The electro i a Bohr atom ca fall from the level to, where ad are ay ; ; 3; ::: atural umbers ( ) by emittig a photo whose eergy equals the eergy differece ÁE betwee the iitial ad fial levels. Such spectral lies are called recombiatio lies because formerly free electros recombiig with ios quickly cascade to the groud state by emittig photos. Astroomers label each recombiatio lie by the ame of the elemet, the fial level umber, ad successive letters i the Greek alphabet to deote the level chage Á : Ë for Á =, Ì for Á =, Í for Á = 3, etc. For example, the recombiatio lie produced by the trasitio betwee the = 9 ad = 9 levels of a hydroge atom is called the H9 lie. Ë The total electroic eergy the th circular orbit: E ( + Á) Á is the sum of the kietic ad potetial eergies of the electro i 3 of 0 /06/008 :4 A

4 The electroic eergy chage goig from level to level is equal to the eergy h of the emitted photo: E = T + V = À T = V = = À e e mee = À a = À m e e4 Ö h Ö ÁE ( + Á) 4 m e Ô Õ e ÁE = Öh À = h ( + Á) h so the photo frequecy is Ù mee 4 Ó Ô Õ = c h3c À ( + Á) The factor i large paretheses is called the Rydberg costat to the limit of ifiite uclear mass. R Ñ R Ù 3:8984 Â 0 5 R Ù mee 4 Ó Ù Á 9: Â 0 À8 g Á ( 4:8 Â 0 0 esu) 4 Ù h3c (6:63 Â 0 À7 erg s) 3 Á 3 Â 0 0 cm s R = : : : : Â 0 5 cm R c, where the subscript refers The dimesios of are legth, ad the product is the Rydberg frequecy Hz. Allowig for the fiite uclear mass ad repeatig the aalysis above i the atomic ceterof-mass frame yields the same frequecy formula with R replaced by R : Ô Õ Ó = R c À m e where R ( + Á) Ñ R + (7A) mp Ù 836:m e (H) Ù 836:m The hydroge ucleus is a sigle proto of mass, so e. Ë + Á = 0 = 09 Ô Õ Ó = R c À m e where R ( + Á) Ñ R + Example: What is the frequecy of the photo produced by the H09 trasitio from to? R c = 3 :8984 Â 0 5 Hz + Ó = 3:8805 Â 0 5 Hz 836: 4 of 0 /06/008 :4 A

5 = 3:8805 Â 0 5 Hz : À Ó Ù 5 Â 9 Hz 0 4 The mass of a eutro is about equal to the mass of a proto so the He ucleus cosistig of two protos ad two eutros has mass, the isotope of carbo with six 4 ( He) Ù 4(H) ( C) Ù (H) protos ad six eutros has, ad so o. Electros recombiig oto sigly ioized atoms with ay umber N p of protos ad N p À electros orbit i the potetial produced by a et charge of oe proto, so the recombiatio-lie spectra o f heavier atoms are very similar to that of hydroge, but the lies of heavier atoms are at slightly higher frequecies (Eq. 7A) ad may be detected idividually. For example, the primordial abudace 3 of the rare helium isotope He is importat because it reflects the desity of baryos i the 3 early uiverse. The abudace of He i galactic HII regios has bee measured via radio recombiatio-lie emissio ad idicates that baryos accout for oly a few percet of the critical desity eeded to close the uiverse. Observed recombiatio-lie spectra from the 9 ad 9 trasitios of hydroge, helium, ad carbo observed i a HII regio (Quireza et al. 006, ApJS, 65, 338). The stroger radio recombiatio lies are produced by trasitios with sometimes use the approximatio Ë Ë Á Ü, so we ca 5 of 0 /06/008 :4 A

6 Ô Õ À ( + Á) + Á À + Á + ( Á) À Á Á = [ Ù + Á + ( Á) ] 4 = 3 ( ) Ù ( + Á ) Thus a simpler (but ot extremely accurate) approximatio for the radio frequecy is Ù (R c)á 3 ad the frequecy separatio Á = () À ( + ) Á 3 Ù betwee adjacet lies is about (7A3) Adjacet high- (low ) radio recombiatio lies have small fractioal frequecy separatios, so two or more lies ca ofte be observed simultaeously. Á = The frequecy. Ë hydroge recombiatio lies, show here as vertical bars, are closely spaced i The H09 lie was first detected by P. ezger i 965, despite (icorrect) theoretical predictios that pressure broadeig would smear out the lies i frequecy ad make them udetectable. It is true that atomic collisios i the iterstellar medium sigificatly disturb the eergy levels of large atoms, but this disturbace is about the same for adjacet eergy levels, so the differetial disturbace that alters the lie frequecy is actually much smaller. His coclusio: Do't abado a observatio just because you have bee told that it wo't work. Recombiatio Lie Stregths 6 of 0 /06/008 :4 A

7 Next we cosider the spotaeous emissio rate how quickly does a isolated atom with µ decay to a lower eergy level? A rigorous aswer requires quatum mechaics. However, we ca get a fairly good aswer by otig that radio photos are emitted by atoms with µ ad use the correspodece priciple, Bohr's hypothesis that systems with large quatum umbers behave almost classically. The time-averaged radiated power for µ trasitios is give by the classical Larmor's equatio for a dipole with dipole momet ea. hp i e = (! a ) hcos (!t)i 3c 3 The photo emissio rate (s ) is this average power emitted by oe atom divided by the eergy of the emitted photo. This rate is called the spotaeous emissio rate. The spotaeous emissio rate for trasitios from level to level ( À ) is deoted by A ;. where e 4 a hp i = (Ù ) 3c 3 hp i A ; = ; h Ù R 3 cá i the limit. Also i that limit,. Recall that µ Á A +; Ù A ; a Ù h 4Ùmee so hp i e 6Ù 4 3 Ó a A +; Ù Ù h 3c 3 h 6Ù 4 e A +; Ù 3 c h 6Ù 4 e A +; Ù 3 c h 3 3 R 3 Ó 3 c h 4Ùmee Ó 4Ù mee 3 h Ó h 3 4Ùmee 5 A +; Ù 6 64Ù m e 3 c h e (7A4) 7 of 0 /06/008 :4 A

8 Evaluatig the costats gives A +; Ù Ô 64Ù 6 Á 9: Â 0 À8 g Á ( 4:8 Â 0 0 statcoul) 0 Õ 3 Á ( 3 Â 0 0 cm s ) 3 Á ( 6:63 Â 0 À7 erg s) 6 5 Ó A +; Ù 5:3 Â 0 9 s 5 (7A5) Example: The GHz H09 trasitio rate is s. Ë A 0;09 Ù 0 :3 The associated atural or itrisic lie width follows from the ucertaity priciple: ÁEÁt Ù Öh. Substitutig há for ÁE ad A for Át for each eergy level ivolved i the +; trasitio ad summig these two ucertaities yields Á Ø A +; =Ù Ø 0: Hz This is egligibly small at the large that produce radio-frequecy photos. Collisios of the emittig atoms cause collisioal broadeig, where Á is a small fractio of the collisio rate whe Á Ü. Except for very large, collisioal broadeig is usually small ad the actual lie profile (ormalized itesity as a fuctio of frequecy) is primarily determied by Doppler shifts reflectig radial velocities v r. These motios may be microscopic (thermal) or macroscopic (idicatig large-scale turbulece, flows, or rotatio). I the orelativistic limit v r Ü c, the Doppler equatio relatig the observed frequecy to the lie rest frequecy 0 is Ù 0 À c v r Ó so the radial velocity ca be estimated from The thermal compoet of the lie profile from a source i LTE is determied by the axwellia speed distributio (Eq. 4B) of atoms with mass ad temperature T. The speed i ay oe = coordiate of a isotropic distributio is of the total speed i three dimesios so R is the ormalized ( f (v r)dv r = ) radial velocity distributio. The ormalized lie profile ( ) for thermal emissio is c( 0 À ) v : r Ù 3 0 Ó = v f(v r ) = exp Ó À r ÙkT kt j ( )d j = f(v r)dv r 8 of 0 /06/008 :4 A

9 Ó = ( ) = expô À ÙkT c ( À 0 ) Õ dv r Ì d Ì kt 0 Ó = ( ) = c 0 ( À ) expô Õ 0 À ÙkT kt c 0 (7A6) R ( )d = ( ) 0 Á ( ) The parameters of the ormalized ( ) lie profile are the ceter frequecy, the FWH lie width, ad the profile value at the ceter frequecy 0. This is a Gaussia lie profile. Its full width betwee half-maximum poits (FWH) solutio of Ô exp À kt (Á =) Õ = c 0 Á is the c Á kt 4 0 = l 8 l k Ó = Ó = T Á = 0 (7A7) c Ë 0 = 5:0089 T Ù 0 4 Example: What is the FWH of the H09 lie ( GHz) i a quiescet (o macroscopic motios) HII regio with temperature K? 9 of 0 /06/008 :4 A

10 Ô 8 l Á :38 Â 0 6 Õ = erg K 0 4 Ó = K Á Ù Á 5:0089 Â 0 9 Hz (3 Â 0 0 cm s ) 836 Á 9: Â 0 À8 g Note that Á µ A 0;09 Ù 0:3 Hz. Á Ù 3:6 Â 0 5 Hz R = Normalizatio (requirig ( )d ) implies that the value of at the lie ceter ( = 0 ) is Ó = ( 0 ) = c 0 ÙkT ( 0 ) = c Á 8 l kt c Ó = ÙkT ( ) 0 = l Ó= Ù Á (7A8) Note that, for a give itegrated (over frequecy) lie stregth, the lie stregth per uit frequecy at ay oe frequecy (e.g., at ) is iversely proportioal to the lie width. 0 Á 0 of 0 /06/008 :4 A