Hydrogen Masers. I: Theory and Prospects

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HYDROGEN MASERS. I: THEORY AND PROSPECTS

1;2 3 1

Vladimir S. Strelnitski , Victor O. Ponomarev , and Howard A. Smith

Lab oratory for Astrophysics, National Air and Space Museum, Smithsonian Institution,

Washington, DC 20560

New Mexico Institute of Mining and Technology, So corro, NM 87801

P.N. Leb edev Physical Institute, 117924 Moscow, Leninsky prosp ect 53, Russia

Submitted to the Astrophysical Journal, Part 1 1

ABSTRACT

The discovery of the rst naturally o ccurring high gain hydrogen recombination line (HRL) maser,

in the millimeter and submillimeter sp ectrum of the emission line star MWC349, requires an expansion

of current paradigms ab out HRLs. In this pap er we re-examine in general the physics of non-LTE

p opulations in recombining hydrogen and sp ecify the conditions necessary for high-gain masing and

lasing in HRLs. Todosowe use the extensive new results on hydrogen level p opulations pro duced by

Storey and Hummer (1995), and our calculations for the net (that is, line plus continuum) absorption

co ecient for the hydrogen, and we present results for the - and -lines whose principal quantum

numb ers n are b etween 5 and 100, for gas whose electron numb er density3log N (cm ) 11, at two

electron temp eratures, T =5;000 and 10,000 K. We show that the unsaturated maser gain in an HRL

is a sharp function of N , and thus to achieve high-gain masing each line requires a suciently extended

region over which the density is rather closely sp eci ed.

Saturation of masing recombination lines is a critical consideration. We derive a simple equation for

estimating the degree of saturation from the observed ux density and the interferometric and/or mo del

information ab out the ampli cation path length, avoiding the vague issue of the solid angle of masing. We

also present a qualitativeway to approach the e ects of saturation on adjacent emission lines, although

the detailed mo deling is highly case-sp eci c .

We draw attention to another non-LTE phenomenon activeinhydrogen: the overcooling of p opula-

5 3

> <

20, in gas where N 10 cm . Observationally the HRL overco oling tions. This o ccurs for HRLs with n

might manifest itself as an anomalously weak emission recombination line, or as a \dasar," that is, an

anomalously strong absorption line. In the simplest case of a homogeneous HI I region the absorption

can b e observed on the prop er free-free continuum of the region, if some conditions for the line or/and

continuum optical depths are satis ed.

We brie y discuss the prosp ects of detecting hydrogen masers, lasers and dasars in several classes of

galactic and extragalactic ob jects, including compact HI I regions, Be or Wolf-Rayet stars, and AGNs.

Subject headings: masers | hydrogen: recombination lines | stars: WR stars | stars: Be stars |

galaxies: AGN |stars:particular:MWC 349. 2

1. INTRODUCTION

Masing sp ectral lines are an ecient observational prob e in astrophysics: b ecause of their strength

and narrowness they often provide the b est opp ortunity for detailed studies of kinematics and structure of

the emitting source. Until recently only molecular astrophysical masers have b een known. Yet calculations

of the atomic hydrogen level p opulations in HI I regions whichhave b een done since the 1930's in order to

interpret optical hydrogen recombination lines (HRLs), showed that global population inversions probably

exist across the Rydb erg levels, thus raising in principle the p ossibility of lasing and masing in the HRLs.

Numerical results directly implying inversion are found in pap ers like Cillie (1936), Baker and Menzel

(1938), and, later, in Seaton's series of pap ers starting in 1959.

In spite of that, and also in spite of the fact that Menzel (1937) p ointed out (even b efore the discovery

of lab oratory masers and lasers) that under non-LTE conditions the integral of energy absorb ed bya

line may in principle b ecome negative, the p ossibility of lasing in the HRLs attracted no attention for

manyyears. It was p erhaps due partly to the deeply ro oted presumptions that the correction factor

for stimulated emission, 1 exp(h =k T ) , is negligible for optical lines b ecause of their high frequency,

and that HI I regions are always optically thin except p erhaps in the Lyman HRLs, so that no serious

ampli cation e ects are p ossible anyway. Only relatively recently did these assumptions change. After

mo deling of optical HRLs in extragalactic ob jects suggested HI I regions of unusually high densities (up to

10 3

N 10 cm ) and/or large extensions, Krolik and McKee (1978) considered the theoretical p ossibility

of lasing in HRLs. Smith et al. (1979), analyzing their observations of an unusually intense infrared

HRL in a dense HI I region around the young stellar ob ject BN in Orion, prop osed that if the inversions

p ersisted to the high densities exp ected in the source they could lead to laser ampli cation of this line.

In the mid-1960's radio recombination lines were discovered. For them the correction factor for

stimulated emission is much more imp ortant than for the optical and IR lines. Goldb erg (1966) showed

that even in HI I regions having optical thickness j j < 1 in b oth a radio recombination line and the

adjacent free-free continuum, the inversion of p opulations can increase the line-to-continuum intensity

ratio byasmuch as a factor of a few.

The rst high gain HRL maser was discovered in the millimeter recombination lines in the emission

line star MWC349 (Martin-Pintado et al. 1989a). At rst the maser emission was ascrib ed to the ionized

out ow from the star (Martin-Pintado et al. 1989a,b), but later it was partly (Ponomarev et al. 1989),

and then fully, attributed to the circumstel lar disk (Gordon 1992; Planesas et al. 1992; Thum et al. 1992).

Subsequent observational and interpretational pap ers enlarged on the list of observed masing lines in this

source, and elab orated on the mo del in terms of a partly ionized masing Keplerian disk [Martin-Pintado

et al. 1989b; Thum et al. 1994a,b; 1995; Gordon 1994; Ponomarev et al. 1994 (\Pap er 1"); Strelnitski et al.

1995a; 1995b (\Pap er 3")].

With the discovery of real, high gain, HRL masers in MWC349, many basic theoretical questions

arise ab out the physics of the pumping mechanism that maintains the inversion, the dep endence of maser

gain on the physical parameters (density, temp erature, gas velo city) of the recombining gas, and ab out

other p ossible non-LTE phenomena in HRLs. Directly related to these is the more practical question of

discovering these non-LTE phenomena in other lines and in other sources. Though brie y touched up on

in some pap ers (e.g.: Walmsley 1990, Ponomarev et al. 1991), these questions deserve to b e studied on a

more systematic basis.

In this pap er we readdress the theory of non-LTE phenomena in HRLs. Using the nomenclature

for the non-LTE states of a quantum transition originally prop osed by one of us (Strelnitski, 1983),

along with the recent and extensive calculations of hydrogen level p opulations by Storey and Hummer

(1995), we analyze this phenomena (Section 2). We demonstrate that the p opulation calculations predict

not only inversion and its weaker complement, \overheating", b oth of which increase the intensities of

emission lines, but also their opp osite | \overco oling" of p opulations | which tends to pro duce an

anomalously strong absorption in the line (the \dasar" e ect). We brie y discuss from a physical and

thermo dynamical p oint of view how these non-LTE states are created in recombining hydrogen.

We address in Section 3 the conditions necessary for pro ducing high maser gain in an HRL, and the

conditions (and observational criteria) of saturation . The \net" absorption co ecient which accounts for

b oth the negative absorption (ampli cation) in the line and the p ositive free-free absorption at the line

frequencies, proved to b e a useful parameter (Ponomarev 1994). Our calculations of the net absorption

co ecient allow us to derive the optimum electron density N for strong unsaturated masing in - and

e 3

12 3

-lines with 5 n 100 for gas with densities up to N =10 cm , and at two representativevalues

of electron temp erature, T =5;000 and 10,000 K. The results are illustrated graphically and tabulated.

Rough analytical approximations are provided for the lower n transitions. We derive a formula for

estimating the degree of saturation in an observed masing HRL, if the line is formed at its optimum

electron density.

Section 4 is devoted to the phenomenon of overco oling and \dasing" in HRLs. We use the solution of

the line-plus-continuum radiative transfer equation for a homogenous slab to show that direct evidence of

overco oling of a line would b e its app earance in absorption on the free-free continuum of the HI I region.

We derive the minimal optical depths in the line and/or the continuum necessary for this to happ en.

This is not, however, the only way to detect overco oling: the increment of HRLs seen in emission can

demonstrate it as well.

Finally, in Section 5, we discuss the prosp ects of detecting HRL masers, lasers and/or dasers in

galactic and extragalactic sources other than MWC349. MWC349 itself | the only high-gain HRL

maser detected so far | is addressed separately in a forthcoming pap er (Strelnitski et al. 1995b,\Pap er

3").

2. NON-EQUILIBRIUM STATES OF RYDBERG TRANSITIONS

IN ATOMIC HYDROGEN

2.1. Radiative Transfer

Our analysis of non-equilibrium e ects in HRLs is based on the standard solution of the radiative

transfer equation for the simplest case of a steady-state, homogeneous, plane-parallel medium of nite

optical thickness :

I = S [1 exp ( )] : (2:1:1)

Here I is the outgoing radiation intensity at the frequency and S is the source function. No external

background source is assumed. In the case of the HRLs, where b oth line and free-free continuum are

formed together, it is convinient to represent the sorce function as S = B (T ) (Goldb erg 1966),

where

0 0

S k + k b

= ; (2:1:2)

B (T ) k + k b

e 1 12

0 0

B (T ) is the Planck function for the electron temp erature T ; k and k are the continuum and the

e e

line absorption co ecients at LTE, b oth corrected for stimulated emission (the prime indicates LTE

values); b is the Menzel's co ecient of departure of the level n p opulation from its LTE value. in

n 12

Eq. (2.1.2) is formally the ratio of the actual (non-LTE) and the LTE correction factors for stimulated

emission (Bro cklehurst and Seaton 1972):

1 (b =b ) exp (h =k T )

2 1 0 e

; (2:1:3) =

1 exp (h =k T )

0 e

where is the rest frequency of the 1-2 transition and constants have their usual meanings. is a

0 12

measure of the deviation of the p opulation ratio of two levels from its LTE value.

The net optical depth at the line center for a homogeneous medium is (we omit the subindex ):

0 0 0

+ k ) L k L: (2:1:4) b L (k L + k = + = k

l net 1 12 net c l

c l c

where k k b is the actual (non-equilibrium) line absorption co ecient and k is the \net"

l 1 12 net

(continuum plus line) absorption co ecient.

Since

I ( )= B (T )[1 exp ( )] ; (2:1:5)

c e c

the relative line intensity at the line center is given by:

I ( ) I ( ) 1 exp( ) 1exp ( )

0 c 0 net c l

r ( ) = 1= 1 : (2:1:6)

I ( ) 1 exp( ) 1 exp( )

c 0 c c 4

All 's and here and following are at the central line frequency , with the subindex omitted.

2.2. Non-Equilibrium States of a Quantum Transition

There is another commonly used parameter, b esides , to describ e deviations of a quantum tran-

sition from equilibrium: the excitation temp erature T , which is de ned by the Boltzmann-like formula

for the p opulation ratio of the two levels:

N =g n

2 2 2

= exp (h =k T ) ; (2:2:1)

0 x

N =g n

1 1 1

h =k

: (2:2:2) T =

ln (n =n )

1 2

In this subsection and in Fig. 1 we denote by n the p opulation density p er one degeneracy sublevel:

n N =g , with g b eing the degeneracy of the level i; this should not lead to a confusion with the

i i i i

principal quantum number for which the script n (without subindex) is used throughout the pap er.

Subindex 1 denotes the lower and 2 the upp er level of the transition.

Since

n b n

2 2

; =

n b n

1 1

where primed symb ols denote again the LTE values, wehave the following relation b etween and T :

12 x

1 exp (h =k T )

0 x

: (2:2:3) =

1 exp(h =k T )

0 e

For low frequency transitions (h =k T ), and when also jT j T ,

0 e x e

: (2:2:4)

The b ehavior of T and , as functions of n =n , is illustrated in Fig. 1 for the case of T =

x 12 1 2 e

10; 000 K and h =k = 100 K. When the p opulation ratio n =n decreases from 1 (all the atoms in

0 1 2

the lower level) to 0 (all the atoms in the higher level), T rst increases from 0 to 1 , then undergo es

a jump to 1 at n = n , and then increases to 0. The parameter decreases monotonicly from

1 2 12

max

=1=(1 exp(h =k T )) to 1, passing through 0 at n = n .

0 e 1 2

The classi cation of the excitation states of a transition is most natural in terms of temp eratures

(Strelnitski 1983), see Fig. 1. T = T is obviously the state of thermalization, and = 1 in this

x e 12

state. When T

e x

corresp onds to 0 < < 1. The case T < 0 (with also b eing negative here) is the case of population

12 x 12

inversion. Finally, when T >T >0 (corresp onding to > 1), we call this condition overcooling

e x 12

of the transition 1-2. The choice of this nomenclature will b ecome clearer as we analyze the observable

e ects of the corresp onding non-LTE states b elow (see also Strelnitski 1983).

2.3. Non-Equilibrium Populations in Hydrogen

Storey and Hummer (1995) have recently completed a detailed set of calculations of the hydrogen

level p opulations for levels n 50, and presented the results as machine-readable les. For Menzel's

Case B (the Lyman lines are in nitely optically thick, all other lines are thin), calculations were made for

2 log N 14 and for 2:7 log T 5 . Figs. 2{5, plotted with the use of the Storey and Hummer's

e e

data, show that recombining hydrogen may acquire al l the typ es of non-LTE p opulation distributions

just describ ed | overco oling, overheating and inversion.

Fig. 2 presents the b co ecients versus principal quantum number n, and as function of electron

density, N , for T =10 K. The ma jor qualitative features of the b b ehavior are:

e e n 5

(1) b tends to one when n tends to 1. This thermalization of the highest n levels is due to the fact

that their p opulations are dominated by three-b o dy collisional recombinations and collisional ionizations,

b oth pro cesses involving the same Maxwellian gas with kinetic temp erature T .Thus, a reservoir of

quasi-thermalized levels is created at the highest n's.

(2) b decreases toward lower n, due to the ever increasing rate of sp ontaneous decay which is roughly

prop ortional to n . There is a di use separation b etween those levels which are quasi-thermalized and

those which are underp opulated and show a steep p ositive gradientdb /dn. This b oundary progressively

shifts toward lower n's, as N (and the e ects of collisions) increases.

(3) At the lowest n's the gradientdb /dn tends to some high p ositive limiting value. Here p opulations

are totally controlled by radiative pro cesses, and the intrinsic trends of radiative probabilities across the

levels determine this \radiative limit" in the slop e of b (n) (compare this part of the b (n) curves with

n n

our b calculation for the case N = 0, shown in Fig. 2 with the broken line). The passage to the radiative

n e

regime, at low n, o ccurs b ecause, at a given density, the rate of transitions induced by collisions with

4!5

electrons decreases with decreasing n approximately as n , whereas the rate of sp ontaneous radiative

decay increases as n .Thus the relative role of radiative pro cesses, at a given density, increases with

decreasing n very steeply, steep er than n .

5 3

(4) Atlow enough densities (N 10 cm ), db /dn changes sign b efore passing to the radiative

e n

limit. As seen from Eq. (2.1.3), this means that exceeds +1, and thus the corresp onding transitions

are overcooled.To our knowledge, this imp ortant fact has never b een p ointed out b efore, although it

has b een implicitly present in the results of earlier (n; l ) p opulation calculations. It is clearly seen, for

example, in Fig. 2 of Hummer and Storey (1992). The change of sign of db /dn is due to the onset

of partial \unblurring" of the levels' angular momentum l -structure p opultations at these values of n,

a distribution which is blurred by proton collisions at higher n.Weinterpret it as follows. Unblurring

b egins when the radiative decay rate of an (n; l ) sublevel b ecomes comparable with the collision rate with

protons. Since sp ontaneous decay rates increase with decreasing l , the low l sublevels of a given n level

are unblurred the rst. Atany given density, therefore, there is a transitional group of levels in which

more and more l -sublevels (b eginning with the smaller l values) are unblurred as one lo oks to smaller n.

Blurring of the l -sublevels enables high l atoms on a level n to decay faster, via lower l sublevels, than

the l = 1 selection rule normally allows. The more a level is unblurred the higher is the l which lo oses

this ability to decay more rapidly. As a result, the lower the principal quantum number n the more its

b factors re ect higher p opulations in its higher l sublevels. This e ect is seen in Fig. 3, plotted with

3 3

the numerical data of Storey and Hummer (1995) for T =10;000 K and N =10 cm . The e ective

e e

b factor for a level n is derived from the statisticall y weighted, calculated b factors:

n nl

(2l +1)

b = b : (2:3:1)

n nl

Since high l sublevels have higher statistical weights (2l + 1), the increase of their b factors is more

signi cant for the weighted value of b (2.3.1) than the decrease of the b factors of low l sublevels.

n nl

As a result, unblurring increases the e ective b value of the whole n-level. And b ecause the degree of

unblurring increases with decreasing n,anegative gradientdb /dn may result for the partly unblurred

group of levels, as is seen in Fig. 2. We note that those previous calculations which neglected the angular

momentum state p opulation distributions (e.g. Walmsley 1990) show no evidence for \overco oling."

These four features of the b b ehavior determine the shap e of the vs. n curves for -lines,

n n;n+1

shown in Figs. 4a,b for several values of N , and the shap e of the vs. N curves shown in Fig. 5

e n;n+1 e

for several -transitions. Comparing Figs. 4a,b with Fig. 2 is instructive.

(1) At the high values of n the p ositive slop e of b vs. n results in an inversion of p opulations |

< 0.

n;n+1

(2) In a transitional set of n values b levels o and then changes slop e due to the unblurring of l

states at progressively smaller n values; here the inversion rst diminishes, and then disapp ears. If the

< 5 3

densityislow enough (N 10 cm ), the unblurring of l states rst reduces the inversion to overheating

(0 < < 1 ), then leads to thermalization ( = 1 ), and nally | to overco oling ( > 1).

n;n+1 n;n+1 n;n+1

(3) The nal transition of b to the \radiative limit," at still lower n , stops the trend towards

overco oling and brings down to the slightly overheated values typical for the \radiative limit" at

n;n+1

low n. 6

Atlow densities, therefore, the (n) vs. n function has two extrema | a minimum due to

n;n+1

5 3

inversion at high values of n , and a maximum due to overco oling at lower n's. At densities N 10 cm

there is only the inversion minimum b ecause ion collisions at these high densities blur the l -structure

down to very low n's, and thereby remove the e ective cause of overco oling.

Fig. 5 illustrates the b ehavior of the factor as a function of N for some selected -transitions.

n;n+1 e

Qualitatively, the (N ) functions are similar to the (n) functions: high n lines (higher than n 20)

havetwo extrema | a minimum due to inversion at high densities and a maximum due to overco oling at

low densities. Lower n lines show only inversion, and this at increasingly higher densities (and diminishing

amplitude) for lines of decreasing n.

2.4. Thermodynamic Considerations

The creation of stable conditions for inversion, overheating, or overco oling requires cycles of p op-

ulation transfer which, from the thermo dynamic p oint of view, op erate as \heat pumps" (Strelnitski

1983). The system of p opulation transfer cycles in hydrogen is rather complicated but we can infer some

general principles. The two ultimate high temp erature and low temp erature reservoirs that interact with

the system of hydrogen levels and provide free energy to create and maintain non-equilibrium p opula-

tions are, resp ectively, the \hot" ionizing UV radiation eld and the \cold" radiation eld at the lower

frequency recombination lines. Note that this di erence in the high-frequency and the low-frequency ra-

diation temp eratures arises not b ecause the central star is hot: a diluted Planck sp ectrum with any color

temp erature p ossesses this gradient of radiative temp eratures (Strelnitski 1983, Sob olev et al. 1985). In

such a radiation eld the net cycle of quantum transitions is always direct (distributive) | high energy

photons are split up to lower energy photons | this constitutes the content of the Rosseland theorem

(Rosseland 1926; see also Strelnitski 1983, and Sob olev et al. 1985). This cycle of uorescence makes

up the grand radiative-radiative (RR) pumping cycle in the recombining hydrogen in HI I regions, and

tends to pro duce a global p opulation inversion due to the steep increase of the sink (sp ontaneous decay)

rate with the decreasing n.However, b esides the ultimate source and sink radiative reservoirs, there are

other, intermediate, reservoirs involved with the pro cess: the recombination transition levels themselves,

with their excitation temp eratures, and the Maxwellized gases of electrons and protons with their kinetic

temp eratures T . The heat exchange b etween these reservoirs provides secondary cycles of p opulation

transfer. Each of the two collisional reservoirs, free electrons and protons, have di erent degrees of e ec-

tiveness in their interaction with the di erenthydrogen levels and sublevels, and b ecause of this di erence

they mo dify the pattern of the primary (RR) cycles and can even change, lo cally, the sign of the cycles,

pro ducing overco oling.

We note that only one net link | the source link or the sink link | can b e collisional in these

secondary cycles; the other link must b e radiative. In other words: pumping cycles in hydrogen can b e

of the radiative-radiative (RR), collisional-radi ative (CR), or radiative-collis ional (RC) typ e, but they

can not b e of the collisional-coll is ional (CC) typ e b ecause only one temp erature, T , is asso ciated with

collisions in HI I regions.

3. HYDROGEN MASERS

3.1. Maser Gain

The overheating and inversion of p opulations o ccur when is smaller than its LTE value =1.

It is seen from Eqs. (2.1.2), (2.1.4) and (2.1.6) that is then higher than its LTE value =1, is

net

lower than its LTE value, and in either case it increases the relativeintensity of the line. Goldb erg (1966)

demonstrated that in this situation the relativeintensity of the line r can increase up to a factor of a few

even when j j and are < 1.

l c

However, to pro duce a maser ampli cation 1, p opulation inversion < 0 alone is not sucient.

The net optical depth must also b e negative and its absolute value must b e considerably higher than

net

unity. When < 0, b oth and the numerator in Eq. (2.1.6) are negative, and Eq. (2.1.6) b ecomes:

net

j j

r exp j j : (3:1:1)

net

1 exp( )

c 7

Thus the relativeintensity of a recombination line may b ecome very high if the exp onent j j (the

net

\gain" of the maser) is 1.

The maser gain is determined by the length of the ampli cation path, L, and the value of k =

net

0 0

k + k b [see Eq. (2.1.4)]. A negativevalue of k can only derive from a negative , b ecause all

1 12 net 12

the other quantities determining it are essentially p ositive. Since k , in particular, is p ositive, free-free

absorption always reduces the maser gain, with the amount of reduction dep ending on the frequency.

8 3

10 cm ), for a given transition can attain If the density of an HI I region is not very high (say,

high negativevalues at the b ottom of its inversion minimum (cf. Fig. 5), thereby increasing signi cantly

0 0

the absolute value of k = k b as compared with its LTE value k .Yet, the density at which maser

l 1 12

l l

gain in a transition attains its maximum do es not coincide with the density of this maximum inversion.

The reason is that b oth and k dep end, and dep end di erently, on electron density N , in particular

12 e

with the latter having a quadratic dep endence on N . Multiplication of (N )by the steep (/ N )

e 12 e

function k (N ) shifts the maximum of the pro duct toward higher N . It also makes the range of densities

e e

with high jk j values much narrower than the range of densities with high j j values. Furthermore, since

l 12

k and k dep end on N di erently, accounting for free-free absorption shifts (downward) somewhat the

l e

value of the density where k is minimum with resp ect to that for k alone.

net l

In order to derive the optimum conditions for high-gain HRL masing wehave p erformed extensive

calculations of the k (N ) dep endence for - and -lines with 5 n 100. Since the minimum value

net e

of k for any line o ccurs at a densitymuch higher than the densityofunblurring the l -structure, our

net

program calculating b co ecients neglects the l -structure as unimp ortant.

Several illustrative cases of the calculated k (N ) dep endence, for T = 10,000 K, are shown in

net e e

Figs. 6a{f ( -lines) and 6g ( -lines). In Fig. 7 the k (N ) dep endence is given for a large interval of

net e

higher n. A logarithmic scale on the ordinate has b een used to show all these lines on one graph. Only

the lo cus of the maximal jk j values for individual lines and the envelope of the jk j (N ) plots are

net net e

3 11 3

shown for T =5;000 K. The k (n) dep endence for di erent densities, from N =10 to 10 cm ,

e net e

and for the two temp eratures is presented in Fig. 8.

Comparison of the k (N ) for the 36 line (Fig. 6b) with the (N ) for this line (Fig.5; the

net e n;n+1 e

intervals of high jk j and jk j are indicated by the bars) shows that whereas is negativeover a

l net n;n+1

5 3

large densityinterval, with a maximum of inversion that p eaks at N 10 cm , the densityinterval of

7 3

high jk j values for this line is much narrower, and p eaks at much higher density|N 10 cm .

net e

max

Table 1 gives N , the electron numb er density at which jk j attains its maximum, and the

net

corresp onding value of k , for twenty -lines and twenty -lines at T = 10,000 and 5,000 K. The

net e

max

dep endence N (n) is shown graphically in Fig. 9. For the lines with n 40 (these are of particular

interest b ecause many of them have already b een observed as strong masers in MWC349), we tthe

max

following analytical approximation for N (n):

max 15 5:66 4

N 8:0 10 n (T =10 K) ; (3:1:2)

max 15 5:75 3

N 7:7 10 n (T =510 K) : (3:1:3)

max

The k (n) dep endence is approximated in the same interval of n by:

net

max 3 8:0 4

k 2:310 n (T =10 K) ; (3:1:4)

net

max 2 8:5 3

k 3:410 n (T =510 K) : (3:1:5)

net

The di erence b etween these approximations and more exact mo del calcuations is under 50% for

5 n 40. The approximation (3.1.2) is drawn in Fig. 9 as a broken line.

max

We p oint out that N for a given transition do esn't dep end strongly on the assumed temp erature.

It is seen in Table 1 and in Eqs. (3.1.2) and (3.1.3) which are, in fact, almost identical. For rough

max

estimates of N in the whole temp erature interval 5,000 { 10,000 K their \average,"

15 5:7 3 4 max

8 10 n (5 10 T 10 K) ; (3:1:6) N

can therefore b e used.

max

In contrast, k (N )mayvary signi cantly with temp erature, esp ecially at the higher n values.

net

The temp erature dep endence of k is more generally seen in Figs. 7 and 8.

net

The narrowness of the k (N ) p eaks for individual lines demonstrated in Figure 6 implies that to

net e

reach high gain masing in any small group of recombination lines around some n an extended region at 8

the density close to the \optimum" one for this n is required. However, if a value of N is xed, rather

max

than a value of n, the corresp onding maximum value of the gain, jk j (N ), is not that of the line

net

attaining its maximum gain at this density: there are several lines, of smaller n, whose gain is higher;

likewise, there are several lines, of higher n, whose gain is smaller. This can b e seen in several graphs of

Fig. 6, but esp ecially clearly in Figs. 7 and 8: the lo cus of jk j values for individual -lines in Fig. 7 (the

net

two broken lines, for two temp erature values) passes below the lo cus of maximal jk j values for given

net

densities (the envelop e of the individual plots). Note that b oth these lo ci are straight lines in the log jk j

net

{ log N graph, displaying that corresp onding dep endencie s are of the p ower-lawcharacter. Fig. 8 also

shows that the lower the density, the atter the maximum of the jk j (n) function. Thus, in contrast to

net

the case of a given line where a narrow densityinterval is required to attain the maximum p ossible gain,

7 3 <

10 cm ) many lines are allowed to in the case of a given density (if the densityislow enough, say,

mase with comparable gains.

L jk j is the length corresp onding to j j = 1. Figures 6{8, Table 1 and Eqs. (3.1.4){(3.1.5)

1 net net

thus allow us to estimate how extended the medium of a given electron density should b e along the

line of sight to provide a maser gain > 1 in a corresp onding HRL. We note, however, that all these

give only an upper limit to a jk j or a jk j b ecause the calculations were done assuming the lo cal line

l net

width is only due to the thermal broadening, ignoring p ossible microturbulent broadening. If the latter

is present and describ ed by an equivalent temp erature T , the value of jk j decreases by a factor of

tur b l

1=2

[(T + T )=T ] . The relative decrease of jk j due to any microturbulent brao dening (and the

e tur b e net

corresp onding increase of L )iseven larger than that for jk j, b ecause of the contribution from k as seen

1 l

in Eq. (2.1.4). The value of L may additionally b e increased by the presence of large scale motions in

the source (expansion, contraction) which diminish the column densities of atoms at every radial velo city

(frequency), and thereby increase the value of L by a factor of approximately V=v, where V is the

disp ersion of radial velo cities due to the large scale motions and v is the lo cal radial velo city disp ersion

due to the thermal motions and microturbulence.

All the data for Fig. 6, Table 1, and Eqs. (3.1.2) { (3.1.5) were obtained assuming the media are

optical ly thin in all recombination lines (except for the Lyman lines which, for the Menzel's Case B, are

taken to b e in nitely optically thick; we do not consider these lines here). Yet, since we deal here with

high-gain masers, that is with gas which is supp osed to b e optical ly thick in masing lines, and since, in

hydrogen, the masing and pumping lines are intermixed and have comparable values of absorption co e-

cients, the assumption of optical thinness in all but masing lines is internally contradictory. Qualitatively,

the e ect of radiation trapping is similar in the \ordinary" (non-inverted) and in the inverted transition

cases: the more the radiation is trapp ed (the larger the mo dulus of the line optical depth), the more the

transitions induced by the trapp ed photons comp ete with sp ontaneous and collision-induc ed transitions

in controlling the p opulations. The results of p opulation calculations for a multilevel system with and

without account for radiative trapping may, of course, di er drastically. It is imp ossible in practice to

account in a general way for radiation trapping, not so much b ecause it is a mathematically complicated

nonlinear problem, but rather b ecause the pattern of optical depths is p eculiar to each concrete source.

Fortunately, in the imp ortant particular case of masing in a Keplerian disk (the case of MWC349

and similar ob jects) the problem of the pumping radiation trapping can b e nessed by supp osing that

the medium is optically thick in only one direction, that of the chord of maximum velo city coherence (see

Fig. 1 in our Pap er 1). In the directions p erp endicular to the disk, and also in those directions within

the disk which are far from that chord, the optical depth is smaller, and it can actually b e smaller than

unity,even if the chord optical depth is considerably larger than unity. Since these directions o ccupy, all

together, much larger solid angle than the directions close to the chord, sp ontaneously radiated photons

can escap e from the medium almost freely, providing an ecient pumping for high-gain, optically thick

masing into relatively small solid angle along the chord. There is one further fact providing con dence

in this general approach: the successful mo deling of the masing observed in the case of MWC349 (cf.

Pap er 3).

If the mo dulus of the optical depth along the chord of maximum ampli cation is large enough, the

rapid increase of maser intensity creates a sp ecial kind of radiation trapping problem | maser saturation.

3.2. Saturation 9

Saturation o ccurs when the growing maser radiation b egins to in uence the p opulations of the

transitions signi cantly. No general solution exists for the problem of maser saturation for the same reason

why no general solution is p ossible for radiation trapping in a system with p ositive optical depth. Yet,

as it will b e seen b elow and in Pap er 3, saturation may b e o ccuring in these systems, and understanding

its role will play a crucial part in interpreting the observed prop erties of masing HRLs, and in predicting

the observability of other lines. In this section we consider two basic questions concerning saturation in

HRLs: (1) how saturation in a masing line a ects the level p opulations (thus the gain) of the masing

transition itself; and (2) how, in a multi-line HRL maser, saturation in one line a ects the p opulations

and the maser gains in adjacent lines. We also derive a criterion for susp ecting when saturation has

o ccured in a masing HRL.

The rst question has b een intensively studied in the context of molecular masers. The condition

of saturation is that the maser radiation density in uences \noticeably" the p opulation di erence N

between the upp er (2) and lower (1) maser levels. Ideally one needs an analytical expression for N as

a function of all the basic pro cesses in uencing the p opulations. In practice this goal is unattainable when

the number of interacting levels is > 3 4. Several simplifying approaches have therefore b een prop osed

to get around this problem (see the discussion in Strelnitski, 1993). In all these approaches the two

relaxation pro cesses, radiative (maser) and collisional transitions b etween the maser levels themselves,

are accounted for almost exactly.Itismuch more dicult, however, to provide for a complex system a

simple yet adequate approximation for all the pumping pro cesses.

In one approach, pumping is represented by four phenomenological co ecients for the upp er and

3 1

lower maser levels: the \income" co ecients and (cm s ) and the \outlay" co ecients and

2 1 2

(s ). Omitting subindex \12" where it cannot lead to misunderstanding, and taking for simplicity

the degeneracies of the maser levels to b e equal, the solution of the statistical equilibrium equations in

this \; "-approach can b e represented as follows (e.g.: Strelnitski, 1993):

N N

0 0

N = = ; (3:2:1)

BJ J

1+ 1+

+C J

21 s

where

2 1

1+ = 1+ =

2 1 2 1

N = (3:2:2)

+ C

1 1

is the unsaturated p opulation di erence; ( + ) is the harmonic mean value of the decay

2 1

rate of the two levels; B is the Einstein co ecient for induced radiative transitions b etween the maser

levels (B B = B since we assumed the degeneracies of the two levels to b e equal); J is the actual

12 21

radiation intensityaveraged over the pro le of the absorption co ecient and over directions;

J ( + C )=B (3:2:3)

s 21

is the so called \saturation intensity;" and C is the rate of collisional relaxation b etween the maser

levels. To obtain this form of eqs. (3.2.1) and (3.2.2) we ignored the slight di erences b etween C and

C . Strictly sp eaking, then, these equations are only valid for transitions with h kT which, at

21 e

T 10 K, comprise all hydrogen -lines down to 5 .

Eq. (3.2.1) shows that maser radiation b egins to in uence the maser level p opulations when the rate

of its interaction with the maser levels, BJ , b ecomes comparable to the higher of the two rates: the rate

of collisional relaxation b etween the levels, C , and the rate of the decay of the p opulations from the

maser levels in the pumping pro cesses, . (It is worth noting that harmonic mean is always smaller than

the smallest of the twoaveraged numb ers, and equal to one half of each when they are equal.) The net

e ect of the maser radiation is the transfer of p opulation from the upp er to the lower state. It is seen in

Eq. (3.2.1) that in the limit J J the p opulation di erence N ! 0, and thus (N =N ) ! 1. Note,

s 2 1

however, that in this limit, N / J [see Eq. (3.2.1) with J J ]. This dep endence of N on J

reduces the maser ampli cation from an exp onential [Eq. (3.1.1)] to a linear dep endence. The e ect is

easily understo o d by noting that the radiative transfer equation for a suciently strong maser (where

the sp ontaneous term is negligible when compared to the induced term) reduces to:

dI h

= B NI:

dz 4 10

When N is indep endent of the intensity I (unsaturated regime), integration gives an exp onential growth

of I with z ; when N / I (saturation), I dep ends on z linearly (see also general reviews of maser

theory, e.g., Strelnitski, 1974; Reid and Moran, 1981; Elitzur, 1992).

The ma jor simplifying assumption of the \; "-approach is that and do not dep end on the

2 1

maser level p opulations. This leads eventually to the expression (3.2.1) which states that (N=N )

do es not dep end on 's but only on 's and C | a radical simpli cation. The fewer the numb er of the

levels involved with the pumping, the worse the approximation, b ecause in such a few-level system the

non-maser level p opulations | the source of \income" for the maser levels | dep end sensitively on the

p opulations of the maser levels.

In the case of HRLs pump ed by recombinations, cascading and collisions, every level is fed by many

other levels. If there is only one masing transition in the system, the change of its p opulations caused

by saturation should not pro duce any serious distortion of the reservoir of pumping. Moreover, even if

masing and saturation o ccur in several adjacent lines in the same region of space, this will hardly change

the reservoir of pumping for individual transitions b ecause saturation only re-distributes p opulations

between two adjacent levels, without bringing p opulations in or out of the reservoir as a whole. Thus

the \; "-approximation should give an adequate qualitative description of the N (J ) dep endence for

individual masing recombination lines, in spite of the fact that in hydrogen the reservoir of p opulations

for pumping may heavily overlap with masing transitions.

The hydrogen levels we consider here decay primarily through sp ontaneous and collision-induce d

transitions to other levels; thus,

A + C ; (3:2:4)

t t

1 1 1 1

1 1

where A (A + A ) and C (C + C ) are the harmonic mean values of the total Einstein

t t

t1 t2 t1 t2

A co ecient and of the total collision rate for the two maser levels resp ectively. Using Eq. (3.2.4) and

the relation b etween the Einstein A and B co ecients, we can rewrite Eq. (3.2.3) as

2h A + C + C

t t 21

J ; (3:2:5)

c A

where A is the Einstein co ecient for the maser transition, is the transition frequency and the

21 0

constants have their usual meanings.

To estimate the degree of saturation in an observed masing line, wehave to compare an estimate

of J in the source with J given by Eq. (3.2.5). A key and complex issue is that of the radiation's solid

angle, but we are able to sidestep the matter using the following argument (cf., Strelnitski, 1984). On

the one hand, unless the geometry of the source is very sp ecial, the solid angle of the maser b eam should

b e close to

; (3:2:6)

where is the observed emitting surface area, and l is the ampli cation path length. On the other hand,

the solid angle under whichwe see the source is

! ; (3:2:7)

where D is the distance to the source.

Thus, cancels out in the equation connecting J with the observed ux density S :

S D

J = I = = S ; (3:2:8)

4 ! 4 4l

where I is the sp eci c radiation intensityaveraged over the pro le of the absorption co ecient. From

Eqs. (3.2.5) and (3.2.8) we obtain the degree of maser saturation:

2 2

A J c D S

: (3:2:9)

J 8h A + C + C l

s t t 21

Eq. (3.2.9) gives an estimate of the degree of saturation in terms of the transition constants

;A ;A , the mo del parameters C ;C (determined by the mo del density of the emitting region),

0 21 t t 21 11

the observed ux parameter S , and the \semi-observed" parameter l which, though not directly observ-

able, can in principle b e reliably estimated from direct interferometry of the source structure in the plane

of the sky and the geometrical mo del of the source. We emphasize that transition to saturation is a

gradual pro cess. It is seen from Eq. (3.2.1) that to pass from an unsaturated regime (J=J 1) to a

heavily saturated one (J=J 1), the intensity needs to increase by ab out two orders of magnitude.

Thus the saturation criterion J>J is no more than an order of magnitude criterion.

There is one case for whichwe can simplify Eq. (3.2.9) without losing its exactness: the case where

maser radiation is generated at the \optimum" density, i.e., the density that provides the maximum

(unsaturated) value of jk j for the line in question. As was discussed in Section 3.1, the optimum

net

density for maser ampli cation is much higher than the density of maximum inversion, and, as Figs. 5

and 6 show, this optimum density is actually rather close to the density of thermalization. At such high

densities, A in Eq. (3.2.9) can always b e ignored as compared with (C + C ). For 5

t t 21

T =10 K, we use the following approximation:

9 5

(C + C ) 3 10 n N ; (3:2:10)

t 21 e

which is a simpli ed version based on Gee's et al. (1976) analytical approximations for collision rates

max

and which is exact to 5%. Substituting for N the approximation (3.1.6) for N , and taking A

e 21

5 3

9 1 15 1

6:3 10 n (s ) and 6:6 10 n (s ) | common approximations valid for -lines with n 1

1 1

| and nally neglecting A in Eq. (3.2.9) as compared with (C + C ), the equation reduces to

t t 21

2 4:7 2

J D n S 10 cm

: (3:2:11) 0:2

J kp c 10 Jy l

The previous assumption, that the observed masing lines arise in regions of \optimum" density,is

of course an assumption which can b e wrong. One reason may b e the interaction of the adjacent masing

transitions due to saturation. Supp ose some line 2 ! 1 mases at its optimum density and reaches strong

saturation. We can ask how such saturation might a ect the trends of masing in the next transition

higher, 3 ! 2. If the 2 ! 1 line were not saturated, the 3 ! 2 line would b e if not thermalized, at this

high density, at least amplifying less strongly than its own optimum (lower) densitywould predict (see

Fig.5 for an illustration). According to Eq. (3.2.1), however, saturation of the 2 ! 1 transition means

that N decreases due to the transfer of some p opulation from level 2 to level 1, and implies that the

p opulation of the lower level of the 3 ! 2 transition decreases. In other words, saturation of the transition

2 ! 1 should increase the inversion of the transition 3 ! 2. In thermo dynamic language, saturation of

the transition 2 ! 1 op ens a new channel of sink to the maser on the transition 3 ! 2, thereby increasing

the eciency of its pumping. As a result, the 3 ! 2 line will b e ampli ed to a higher intensity. Ifitin

turn saturates, it will increase in the same manner the inversion of the next higher transition, etc. The

eciency of this upward di usion of the reenforced pump sink decreases, however, at every step, until

eventually, at some higher value of n, the steeply increasing relative rate of thermalizing collisions will

remove the p opulation inversion.

Thus, a saturated high frequency masing line can \attract" an adjacent group of lower frequency

lines to mase in a region of density higher than their unsaturated \optimum" densities. We will apply

this mechanism, in Pap er 3, to interpreting some observed prop erties of the multi-line hydrogen maser in

MWC349. In any particular case, however, this qualitative idea requires a more exact numerical check|

a calculation of level p opulations including induced radiative transitions in the lines which are supp osed

to b e saturated masers.

4. HYDROGEN \DASARS"

The e ect of overco oling, demonstrated for hydrogen Rydb erg levels in Fig. 4a and 5 and discussed

in Section 2.3, is actually a well known phenomenon for radio lines of some interstellar molecules (H CO,

OH). Observationally it app ears as an enhancement of a line seen in absorption. The observational e ect of

overco oling was called \DASAR" (\Darkness Ampli cation by the Stimulated Absorption of Radiation")

byC.Townes and A. Shawlow. Overco oling is sometimes called \anti-inversion," and the corresp onding

enhanced absorption | \anti-maser." The analysis of non-LTE states of a quantum transition given in 12

Section 2.2 shows, however, that overco oling is really more an analogue of overheating than of inversion

(see more in Strelnitski, 1983).

We rst show that for an homogeneous, ionized cloud with no external continuum background, the

observation of a line in absorption, i.e., r<0, requires > 1{overco oling of the corresp onding

transition.

From Eq. (2.1.6), r<0 corresp onds to

1exp ( )

< : (4:1:1)

1 exp ( )

c l

It is easily seen that the right-hand side of Eq. (4.1.1) is always < 1, whatever the value (or sign) of ,

and hence <1, which is only p ossible according to Eq. (2.1.2) if > 1.

Thus, in the framework of this simple mo del, the very observation of a recombination line in absorp-

tion on the prop er free-free continuum of the cloud is a direct indication that the transition is overco oled.

More complicated circumstances, however | a hotter external continuum background for example, or

some p eculiar morphology of the cloud, like p erhaps if the kinetic temp erature decreases radially out-

ward | can pro duce an absorption recombination line without overco oling. Usually such cases can b e

identi ed by a detailed analysis of the source.

Unfortunately, the reverse statement, namely that > 1(overco oling) always results in r<0(a

line seen in absorption), is not true. To see when overco oling do es pro duce an absorption line in our

mo del, we consider three particular cases:

(1) 1; 1;

c l

(2) 1; 1;

c l

(3) 1.

In the rst case, expanding the exp onents in Eq. (2.1.6) to rst order terms and using Eqs. (2.1.2)

and (2.1.4), wehave:

net

r b : (4:1:2) 1=

Since all the terms on the right-hand side are p ositive, r can only b e p ositive in this case. In other

words, only emission recombination lines can b e observed from an optically thin cloud with no external

continuum background.

0 0

Case (2) can o ccur for high frequency recombination lines (n 30), for which k k (see Fig.10).

Ignoring k in Eq. (2.1.2) and expanding the denominator on the right-hand side of Eq. (2.1.6) to rst

order, wehave, as the condition of r<0 for this case:

1 exp ( )

: (4:1:3) >

Thus, for high frequency lines overco oling can pro duce an absorption line even in a cloud optically thin

in the continuum, however, the line optical depth of the cloud needs to b e 1 and its continuum optical

depth should satisfy the condition ab ove (4.1.3).

Finally,we consider case (3), rst in its asymptotic form, when 1. Eq. (2.1.6) reduces then to

r 1, and the condition r<0 b ecomes <1 which, according to Eq. (2.1.2), is identically satis ed

if > 1. Thus, for a homogeneous cloud optically thick in continuum an overco oled recombination line

will always b e observed in absorption (whatever the value of ).

Nowwe show that this is true even for a moderately optically thick cloud ( 1). In this case b oth

numerator and denominator in Eq. (2.1.6) are less than 1 and greater than (1 e ) 0:63. Thus

the condition r<0 in this case b ecomes <0:63. After some algebra on equation (2.1.2) for , and

rememb ering that b =b < 1 when >1, the r<0 condition b ecomes:

2 1

k b 0:6

1 >

: (4:1:4)

k 1:6

0 0

Taking (at its maximum) from Fig. 5, b from Fig. 2, and k ;k from Fig. 10, it is easy to see that

12 1

Eq. (4.1.4) is satis ed for all the transitions which can b e overco oled in principle, n 20 up to at least

n = 56, seen in Fig. 5. Thus, for all these transitions (and in fact for higher n transitions as well)

overco oled lines will b e observed in absorption, if 1.

c 13

Observation of an absorption line on the prop er free-free continuum of an HI I region is the most

direct way to reveal overco oling, but it is not the only p ossible way.Atlow optical depths (Case 1)

there is another obvious way: to observe more than one line in the interval of n where overco oling is

theoretically predicted and to determine the decrementofrwith resp ect to n. According to Eq. (4.1.2),

in the optically thin case r / b , hence, the observed sign of the decrementofrshould show whether or

not overco oling takes place.

5. PROSPECTS

In this Section we address the observational prosp ects for HRL masers, lasers, and dasars. Our anal-

ysis of masing shows that there are at least two preliminary questions to b e addressed when considering

an ob ject, or a class of ob jects, as a p otential candidate: (1) what are the densities of ionized hydrogen in

the source? and (2) what is the linear scale, or, more exactly, the column density p er thermal line width,

at each density? The answer to the rst question shows us which lines mightbeinverted or overco oled in

the source, the answer to the second | whether or not these non-LTE lines can b ecome high gain masers

or detectable dasars.

We examine now, with these two test questions, several classes of ob jects. This list of candidates

should not b e regarded as a complete one: we only give several examples. We do not discuss here the

ob jects similar to MWC349 | hot emission line stars with neutral disks ionized on their surface. Some

observational prosp ects regarding this class of sources will b e presented in Pap er 3.

5.1. Compact HII Regions

Simple estimates show that among the compact HI I regions asso ciated with the sites of massive star

formation, only the densest , \ultracompact," ones can have optical depths in HRL close or surpassing

unityby absolute value as required for masing. Typical electron densities in ultracompact HI I regions

4 5 3

are N 10 {10 cm , but in some of them an average density is estimated to b e as muchasN

e e

5 3

310 cm (Wo o d and Churchwell 1989; Kurtz et al. 1993). Figure 8 shows that the lines in a broad

17 1

interval, from 35 to 60 ,have comparably high jk j at this density, whichmay reach 10 cm ,

net

if the temp erature is somewhat lower than 10,000 K. Since many of these ultracompact HI I regions have

dimensions L 10 cm, the maser gain jk j L may surpass unity in these lines, even if the velo city

net

disp ersion reduces the e ectivevalue of jk j by a factor of 1.5{2. Thus, some increase of intensity and

net

line narrowing due to masing is anticipated in this interval of -lines from the most compact HI I regions.

Some of these regions are highly inhomogeneous. Since chances of high-gain masing increase, in

general, with the increasing density, masing in lower n lines is also quite p ossible from lo cal density

enhancements, even if such zones of higher density are considerably smaller than the dimensions corre-

sp onding to the average density of the ob ject.

HI I regions asso ciated with evolved ob jects, for example planetary nebulae, have in general lower

densities. Nevertheless studies of forbidden line transitions (e.g., Osterbro ck, 1989) show that in some

6 3

cases like IC4997 there are regions with densities up to 10 cm .

E ects of masing can b e studied by comparing the intensities and widths of several -lines over a

broad range of n, and also by comparison with the -lines from nearby n (see Pap er 3). The analysis

would help sp ecify more exactly the physical parameters of the HI I region.

5.2. Wolf-Rayet Stars

The idea to search for HRL line masers in Wolf-Rayet (WR) stars originated with H.E. Matthews

(1992; private communication). These stars p ossess strong winds and expanding ionized circumstell ar

disks which might promote p opulation inversions and maser ampli cation similar to those in the disk

of MWC349. However, the rst attempts to nd mm and submm hydrogen masers in WR stars gave

negative results (Matthews 1994; private communication). 14

The most probable reason is insucient optical depth in the lines (30 -26 ) studied. They require

7 8 3

N 10 {10 cm (see Table 1) for e ective masing | but these densities are not sustained in large

enough column densities in a WR disk out ow. If we approximate the ionized envelop e of a WR star as

1 2

1000 km s constantvelo city (thus N / R ) out ow with a mass loss rate a spherically symmetric,

4 1 < <

2 10 M yr , it is easy to show that the linear scale of the region of density N will b e R(N ) of

e e

1=2

18 max 8 3

10 N cm. At the optimum density of the H26 line, for example, where N 1 10 cm ,

14 max 14 1 max

2 10 cm. Table 1 gives k 10 cm for this line, but, in calculating we nd R(N )

net e

the optical thickness wemust reduce this value by the ratio of the radial velo city disp ersion in a WR

3 1 1

envelop e, V 10 km s , to the thermal line width, v 20 km s (see Sect. 3.1). As a result, the

max 2

net optical depth in the line reduces to k R(N )(v =V ) 10 . In the same waywe can

net net th

show that j j1 for other mm and submm lines. Though j j do es increase with decreasing n,it

net net

< >

10. Thus, high gain lasers in hydrogen recombination -lines from the expanding 1 only at n b ecomes

10. shells of WR stars are most promising for the IR lines with n

At this high a frequency a di erent source of line emission can b e even more e ective at pro ducing

laser lines in WR stars: the atmosphere. In the standard mo del of a WR star the non-expanding

11 13 3 11

atmosphere consists of a turbulentlayer where N 10 {10 cm and whichis510 cm thick.

7 lines (see Table 1). Multiplying the width of this layer Such densities are optimum for lasing in n

max 12 3

by our calculated values of k ,we nd very large optical depths; with Ne 10 cm , for example,

net

the optical depth of B should b e 200! Even assuming the lower limit to the density in this layer

11 3

(deduced by Aller from optical sp ectroscopy) of N 10 cm ,we still have optical depths of 0:2,

3, and 2 for the B ,H6 , and H7 lines resp ectivel y, with B b eing the most promising line if the

density is higher than this lower limit.

The observability of an HRL laser is, however, strongly contingent up on its geometry. If the solid

angle of lasing is not small enough, this emission tends to b e lost on the stronger, sp ontaneous emission

background. This problem is discussed in more detail in Pap er 3.

5.3. Be Stars

Smith et al. (1979) suggested that Be stars mightbelikely candidates for lasing in the infrared

hydrogen lines, although at that time the lack of theoretical high density p opulation data precluded a

quantitative discussion. These complex ob jects have ionized Keplerian disks (Slettebak et al. 1992), high

mass loss rates, photometric variability, and, most imp ortant for our considerations, inferred densities

11 13 3 12 13

in the region emitting Balmer lines of N 10 {10 cm ,over dimensions 10 {10 cm. These

parameters are similar to those in the turbulentlayer of a WR atmosphere, and similar considerations

hold here to o, suggesting they are promising candidates for lasing in the near IR recombination -lines

(n 10).

5.4. Active Galactic Nuclei and Starburst Galaxies

Enormous masses of ionized hydrogen are present in the central parts of galaxies with activenuclei

(AGN), while starburst galaxies, whichmay sometimes b e the result of two galaxies in collision, often

have bright compact HI I regions outside the nucleus. In the case of AGN's we consider the p ossibiliti es of

maser action in twotyp es of regions (e.g., Osterbro ck 1989). In the so-called "Broad Line Regions" (BLR)

8 10 3

N 10 {10 cm , and may range b etween L 0:01{0:1 p c (Seifert 1 galaxies) to 1 p c (quasars). In

2 4 3

the "Narrow Line Regions" (NLR) N 10 {10 cm ;L 1000 p c. However, the densities ab ove do not

apply to these linear scales but ruther to much smaller \clumps" or cloudlets| the lling factor should

be much less than one.

At the densities typical for BLR, H -lines with n 10 25 have maximum lo cal maser gain (Fig. 8).

max

Taking the values of k for these lines from Fig. 6 or Table 1, we see that linear scales of only 10 AU

net

(25 )to 0:01 AU (10 ) are needed to make these lines optically thick, if line broadening is thermal.

If the individual cloudlets in BLR are of these dimensions or bigger, then laser emission in these submm

and IR lines is anticipated, and would b e seen as relatively strong narrow features centered at the radial

velo cities of individual cloudlets. It seems quite p ossible. For example, in the Scoville and Norman 15

(1988) mo del of the broad emission line clouds as photoionized mass-loss envelop es of giant stars, the

Stromgrem sphere reaches down to a radius of 10 cm in the stellar envelop e. With the density there

9 3

b eing 10 cm , the 10 -15 lines should have high unsaturated laser gains | on the order of tens.

The NLR gas seems to b e a less likely prosp ect for high-gain masing in HRLs. The densities of the

4 3 >

50. Densities 10 cm have linear scales NLR gas are optimum for masing in high n -lines | n

up to 1pcinAGNs (viz., the ionized accretion disk \halo" p ostulated by Scoville and Norman, 1995).

According to Fig. 8, the 50 {80 lines have the highest jk j at this density, but it is easy to see that

net

2 3

0.1 are achievable. The regions with N 10 cm can b e much larger, up to maser gains of only

1 kp c, but considering the velo city disp ersion in such a large region, wehave to reduce the optical

depth at each frequency by a factor of V=v 50, and turns then out to b e < 1 in this case to o.

th net

Maser e ects in mm HRLs are anticipated in the starburst galaxies whose archetyp e, M82, was in

fact the rst external galaxy where a radio recombination line (H166 ) had b een detected (Shaver et al.

7 3

1977). Kronb erg et al. (1985) p ostulate dense (N 10 cm ) ionized shells surrounding sup ernovae

which had explo ded inside dense molecular clouds of the M82-like starburst galaxies. According to

Fig. 8, -lines with n 20{35 have the highest maser gains at this density. With the typical linear scale

for the shells b eing 3 10 cm, Fig. 8 shows that the gain can surpass unity, if large-scale motions do

not broaden the HRL to o much, as compared with the thermal width, and if T is not much higher than

10,000 K. Though the b oth latter assumptions can b e wrong, some indications of narrow, presumably

weakly masing comp onents in the 36 line from M82 has recently b een detected by Smith et al. (1995).

5.4. Hydrogen Dasars

From an observational p oint of view the ma jor diculty with the overco oling phenomenon in HRLs,

5 3

as compared with inversion, is the low upp er limit for N :overco oling disapp ears at N 10 cm (see

e e

Sect. 4). At this limiting densityhydrogen -lines with n 20{25 are overco oled, and the Case 2 (cf.,

Sect. 4) applies. Since is only slightly greater than 1 for these lines, Eq. (4.1.3) shows that must b e

12 l

> 1 to observe an absorption line on the prop er continuum of an HI I region. The typical value of k for

5 3 20 1

these lines, at N 10 cm ,is k 10 cm (Fig. 10), and thus very large HI I region dimensions,

e l

of > 300 p c, are needed for > 1. HI I regions with such parameters are unknown.

Atlower densities, which might make it easier to nd larger regions, the maximum of overco oling

shifts to the higher n lines. The values of b , and the oscillator strengths of the lines all increase, but

n 12

the N factor decreases faster, so the required linear scale increases. Of known ob jects, we b elieve only

AGN's can b e considered to day as p ossible candidates for direct observations of hydrogen dasars. If the

4 3

regions of intermediate density, N 10 cm ,have dimensions comparable with NLR's (up to 1000

p c), -lines from n 30{35 may acquire 1 and app ear as dasars.

The most realistic way to detect the e ect of overco oling in hydrogen, however, is to observe more

than one line in the n range where overco oling is predicted at a given density, and to determine the

decrementofrwith n, as describ ed in Sect. 4. This way of inferring overco oling is applicable to any

galactic HI I region if the observations are done in the prop er frequency range.

6. CONCLUDING REMARKS

Hydrogen, the simplest and the most abundant atom in the Universe, is also the rst known atomic

astrophysical maser. Perhaps the most striking feature of this maser, as compared with molecular astro-

physical masers, is its obstinate uniqueness: six years after its discovery in MWC349 there is still only one

known strong hydrogen maser source, despite intense attempts to nd another. In contrast, practically

every newly discovered molecular maser line was quickly detected in a multitude of similar sources.

Any useful theory of the hydrogen maser phenomenon must directly address this striking fact.

In this pap er wehave attempted to show that, although hydrogen may b e common, the physical

conditions necessary for creating a high gain HRL line maser are rather sp ecial: for every line (or narrow

group of lines) only a limited range of density is able to pro ducing the high negativevalues of the lo cal

absorption co ecients required. Furthermore the region with this sp ecial densitymust b e adequately

extended, and homogeneous enough in its radial velo city, to create an optical depth < 1. The velo city 16

disp ersion parameter is an esp ecially imp ortant one. One of the advantages of molecular masers is that

they are lo cated in relatively cold, neutral gas clouds and are pro duced by relatively heavy particles, so

the width of the lo cal thermal velo city pro le is, in general, only 1kms .Together with the high

density of these cold clouds, such a small velo city disp ersion can pro duce over even a short distance a

high negative optical depth at the line center, and thus very strong masing. A HRL maser, on the other

hand, is generated in a hot ionized gas and is radiated by the lightest particle, whose thermal velo city

disp ersion is therefore very high, 20 km s . As a result the lo cal absorption co ecient is relatively

low, regardless of the degree of inversion, and so long distances are required to create <1. But

longer distances bring an additional velo city disp ersion to the picture due to their large scale motions |

expansion, contraction, and the like, which broaden the frequency pro le of the ampli ed radiation even

more, thus hindering strong masing.

In such situations chance may b ecome a decisive factor for creating an observable maser. In the case

of MWC349, there are at least two lucky coincidences: (1) a very high UV- luminous a star, p erhaps

in a particular out ow phase of its life, surrounded by a dense, massive neutral disk, resulting in an

unusually dense, extended HI I region on the disk's surface; and (2), a fortuitous pro jection of the disk to

our line-of-sight as edge-on. A Keplerian velo city eld seen edge-on is a b etter \organizer" of the atoms'

radial velo cities than an out ow | that is whywe observe stronger mm and submm masers from the

disk of MWC349 than from its out ow (cf. Pap er 3).

Whatever the particulars of HRL line masers and lasers, nature is fabulously inventive. Conditions

favorable to the creation of rather strong hydrogen masers and lasers, or even abnormally strong ab-

sorb ers (dasars), might o ccur in several classes of astrophysical ob jects, b oth galactic and extra-galactic.

The strong hydrogen maser in MWC349 was discovered accidentally, during a routine search for more

recombination lines in this well-known emission line star, at a time when the paradigm of \decrease of

stimulated emission e ects toward higher frequencies" was strongly held. Now, thanks to the example of

MWC349 coupled with a thorough analysis of high densityhydrogen level p opulations, we can b e much

more optimistic ab out nding this intrinsicall y interesting and informative phenomenon elsewhere in the

Universe.

The authors are grateful to D.G. Hummer and P.J. Storey for the p ossibility of using their machine-

readable hydrogen p opulation calculation les prior to their publication. V.P. thanks the partial supp ort

of this researchby the Tomalla Foundation Fellowship and the International Center for Fundamental

Physics in Moscow for its help in obtaining this Fellowship. VSS and VOP thank the Smithsonian

Institution, National Air and Space Museum for a senior fellowship and visiting fellowship, resp ectivel y,

to work in the Lab oratory for Astrophysics on this program and related observations. Together with

HAS they acknowledge nancial supp ort from the Institution's Scholarly Studies Program. HAS also

acknowledges partial supp ort from NASA grantNAGW-1711. 17

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FIGURE CAPTIONS

Fig. 1. To the classi cation of non-LTE states of a quantum transition.

4 3

Fig. 2. Solid lines: b (n)atT =10 K and at di erent electron numb er densities N (cm ),

n e e

indicated near the curves. Numerical data from Storey and Hummer (1995). Broken line: b (n)at

N = 0 (\radiative limit"), calculated with the program of one of us (V.P.) mentioned in text.

Fig. 3. To the explanation of the overco oling mechanism: b (l ) for n =3045, at T =10 K

nl e

3 3

and N =10 cm . Note the rearrangement of the b factors in favor of the high-l sublevels with

e nl

the decrease of n, due to the increasing unblurring of the l -structure. Numerical data from Storey and

Hummer (1995).

4 3 7 3

Fig. 4. a | (n)atT =10 K and at N =10 10 cm (numb ers near the curves).

n;n+1 e e

8 12 3

b | The same for N =10 10 cm . Numerical data from Storey and Hummer (1995).

Fig. 5. (N )atT =10 K for di erent n lines (n is indicated near the curves). Numerical

n;n+1 e e

data from Storey and Hummer (1995). The bars, in the lower part of the gure, show the lo cation and

the width (FWHM) of the k (N ) and k (N ) pro les for the 36 line, according to our calculations

l e net e

(compare with Fig. 6b).

4 4

Fig. 6. a{i | k (N ) for some -lines at T =10 K. j | k (N )atT =10 K for some -lines.

net e e net e e

Fig. 7. Solide lines: log jk j (log N ) for higher n -lines (n is indicated near the curves), for

net e

4 4 3

T =10 K. Broken lines { lo ci of the maximal jk j for individual -lines, for T =10 and 5 10 K.

e net e

Dotted line { the envelop e of the jk j (N ) plots for T =510 K (the plots are not shown).

net e e

4 3

Fig. 8 Log jk j (n) for di erent electron densities, for T =10 K (solide lines) and 5 10 K (broken

net e

lines).

max

Fig. 9. Dots: Graphical representation of the N (n) dep endence, with the numerical data from

Table 1 for T =10 K. Dashed line: approximation (3.1.2) for 5 n 40.

0 0

Fig. 10. The LTE line and continuum absorption co ecients, k and k , as functions of n, for

4 3

T =10 and 5 10 K.

TABLE CAPTION

max max

Table 1. Optimum densities of unsaturated maser ampli cation, N , and the values of k for

e net

hydrogen recombination - and -lines at T = 10,000 and 5,000K.

e 19

VLADIMIR S. STRELNITSKI: Lab oratory for Astrophysics, MRC 321, National Air and Space

Museum, Smithsonian Institution, Washington, DC 20560. e-mail: [email protected]

HOWARD A. SMITH: Lab oratory for Astrophysics, MRC 321, National Air and Space Museum,

Smithsonian Institution, Washington, DC 20560. e-mail: [email protected]

VICTOR O. PONOMAREV: Astro-Space Center of P.N. Leb edev Physical Institute, Leninsky

prosp ect, 53, Moscow 117924. e-mail: p onomarev@ras an.serpukhov.su 20