2 Generation and population of OH

 In this section the generation of OH through photodissociation of water evaporated from grains in the radiation field of a new born star is investigated. Details about the evaporation process are given in Sect. 3.

Here we assume that grains are exposed to the heat flux from the star and will evaporate water. The term heat flux describes the total amount of energy from the star which heats the grains either by direct absorption or indirect absorption (e.g. photolytic heating). The evaporated water will be photolysed and yield OH with a peculiar nascent distribution over different quantum states. The continued formation and relaxation yields inversion between all those quantum states between which masing has been observed in star forming regions. The OH quantum state distribution presented here might also be affected by the absorption of FIR radiation from the surrounding dust and the reabsorption of the OH rotational lines which are emitted after photolysis of water because OH is formed in rotationally excited states. Because of the difficulties in quantifying these effects, they are excluded up to now. Nevertheless we think that the photodissociation and the subsequent IR relaxation will dominate the quantum state population at least under certain conditions, e.g. at high photodissociation rates. For the ground state masers the FIR pumping may play a more important role than for the excited state masers.

The evaporation from the grains is described by the evaporation rate $\Gamma$, i.e., the number of water molecules desorbed from grains per $\rm m^3$ and second. In the evaporation process water molecules go off the grains to the gas phase, which implies that the grain size decreases with time.

In the gas phase the water molecules are photolysed in the VUV radiation field with a photodissociation rate $\gamma_1$. The nascent OH is formed in different quantum states with a probability f_i for state i. The nascent quantum state population in OH relaxes subsequently by IR emission towards lower states with IR relaxation rates A_ik. There is an interesting competition between photodissociation and IR relaxation which depends on the photodissociation rate. The OH is also photolysed in the VUV radiation field with a photodissociation rate $\gamma_2$, which presents the only loss mechanism for ground state OH in the present model. The corresponding maser model is shown in Fig. 2.

Grains:

For a quantitative model, we consider a $\rm 1\,m^3$ volume of the maser with water containing grains of radius $r_{\rm gr}$. The effects of different grain radii are discussed in Sect. 3. A grain with radius $r_{\rm gr}$ has a surface area $A_{\rm gr}(r_{\rm gr})$ and a volume $V_{\rm gr}(r_{\rm gr})$. Although grains are expected to have a porous structure we assume spherical grains with $A_{\rm gr}=4\pi\cdot r_{\rm gr}^2$ and $V_{\rm gr}=4/3\pi\cdot r_{\rm gr}^3$ for simplicity.

The density of the grains with radius $r_{\rm gr}$ (number of grains of that radius per $\rm m^3$) is denoted by $n_{\rm gr}(r_{\rm gr})$. From the number $n_{\rm gr}(r_{\rm gr})$ of grains with radius $r_{\rm gr}$ and their surface $A_{\rm gr}(r_{\rm gr})$ we calculate the grain surface area contained in $\rm 1\,m^3$ of the maser volume by $O_{\rm gr}(r_{\rm gr}) = n_{\rm gr}(r_{\rm gr})\cdot A_{\rm gr}(r_{\rm gr})$.$O_{\rm gr}$ is thus the total surface (= the sum of the surfaces of all grains) contained in $\rm 1\,m^3$ with the unit $\rm m^2/m^3= m^{-1}$.

The number of molecules desorbed per area and second, i.e. the desorption flux density $j_{\rm des}$, multiplied with the surface area $O_{\rm gr}$ per $\rm m^3$ yields the evaporation rate $\Gamma$  

$$\Gamma=n_{\rm gr}\cdot A_{\rm gr} \cdot j_{\rm des}.$$ (1)

The desorption flux density $j_{\rm des}(T_{\rm gr})$ depends very sensitively on the grain temperature $T_{\rm gr}$ and increases steeply with increasing temperature. This will be seen to be important.

Photodissociation rates:

The density ${\rm [H_2O]}$ of gas phase water increases because of evaporation by $\Gamma$ and decreases because of photodissociation of water in the first absorption band by $\gamma_1 \cdot \rm [H_2O]$. This implies  

$$\frac{{\rm d}}{{\rm d} t} {\rm [H_2O]}=+\Gamma - \gamma_1 \cdot {\rm [H_2O]},$$ (2)

which can be simplified for the stationary case $\frac{{\rm d}}{{\rm d} t} {\rm [H_2O]}=0$ discussed here to:  

$$\Gamma = \gamma_1 \cdot {\rm [H_2O]}.$$ (3)

Because the first absorption band extends over a large wavelength range from 135 nm to 190 nm the photodissociation rate $\gamma_1$ is large and dominated by this band. Other losses, e.g. by photodissociation at shorter wavelength like Lyman $\alpha$, are neglected up to now. The photodissociation rate $\gamma_1$ is determined by the absorption cross section $\sigma_{\rm 1st}$ in the VUV and the spectral photon flux density $j_{\rm VUV}$ emitted by the star, more precisely:  

$$\gamma_1 =\int \limits_{\rm 1st} \sigma_{\rm 1st}(\nu) \cdot j_{\rm VUV}(\nu) {\rm d}\nu .$$ (4)

The spectrum emitted by a new born B/O-star peaks in the VUV region and has a very high luminosity L that may range from 10⁴-10⁶ solar luminosities (L₀). The short wavelength fraction of the emission below the Lyman $\alpha$ cut-off will be absorbed within the HII region whereas the light beyond the Lyman $\alpha$ cut-off will reach out to the border between the cold interstellar clouds and the HII region. The photodissociation rates vary strongly with the luminosity of the star and the distance between maser and star: for stars with L between 10⁴ to 10⁶ solar luminosities $(L_0 = 4\ 10^{26}\,{\rm W})$ and a temperature of 25000 K roughly 30% of the luminosity is emitted in the first absorption band. With an average photon energy of $10^{-18}\,{\rm J}$ this yields $1.2\ 10^{48} -1.2\ 10^{50}$ VUV- photons/s. At a distance of 1000 AU this implies $j_{\rm VUV} \approx 4\ 10^{18}-4\ 10^{20}\, {\rm m^{-2}\,s^{-1}}$ which, with an average absorption cross section of $\sigma_{\rm VUV}\approx 2\ 10^{-22}\,{\rm m}^{2}$ for the first absorption band, yields photodissociation rates between $\rm 10^{-3}-10^{-1}\,s^{-1}$. However at a distance of 200 AU, where the j=7/2 maser may be located, the VUV photon flux will be 25 times larger and vary between $2.5\ 10^{-2}$ and 2.5/s. These photodissociation rates are only valid if the VUV photon flux is not attenuated by other species. For the following results the photodissociation rates are varied from 10^-4/s (to account for possible attenuation) up to 1/s.

Each photodissociation event leads to an OH molecule. Thus the density [OH] of OH increases due to the photodissociation of water by $\gamma_1 {\rm [H_2O]}$. Simultaneously the OH density decreases due to the photodissociation of OH by $\gamma_2$[OH].  

$$\frac {{\rm d}}{{\rm d}t}{\rm [OH]}=+\gamma_1 {\rm [H_2O]} - \gamma_2{\rm [OH]}.$$ (5)

Under the stationary condition discussed later on the total OH density is constant ($\frac{{\rm d}}{{\rm d}t}{\rm [OH]}=0$) and is then given by (together with Eq. (3)):  

$${\rm [OH]}=\frac{\Gamma}{\gamma_2}\cdot$$ (6)

Because the absorption cross section of OH is almost identical to that of water ([Nee & Lee] 1984; [Watanabe & Zelikoff] 1952; [van Dishoeck & Dalgarno] 1984) we assume $\gamma_1= \gamma_2$ and will therefore often use $\gamma$ for both.

Now we show that the water evaporated from grains attenuates the VUV field not too much. The optical density for the absorption of VUV in the first absorption band of water is given by $\epsilon={\rm [H_2O]}\sigma_{\rm VUV}$, where the average absorption cross section is $\sigma_{\rm VUV}\approx 2\ 10^{-22}\,{\rm m}^{2}$. The density of water under stationary conditions is determined according to Eq. (2) by ${\rm [H_2O]}= \Gamma/\gamma$. With a high evaporation rate of $\Gamma=10^7\,\rm m^3\,s^{-1}$ and $\rm \gamma=10^{-1}/s$, which we will consider as an extreme case only for the j=7/2 masers, we obtain $\rm [H_2O]=10^8/m^3$. Even for this upper limit, the optical density is lower than $\rm 2\ 10^{-14}/m$ which implies that the VUV field can penetrate 1000 AUs deep into the maser volume. Close to the border of the HII region the attenuation of the VUV by evaporating water is therefore negligible.

2.1 Nascent OH population out of photodissociation of $\rm H_2O$

 

Notation of OH quantum states:

The OH from the photodissociation of water is formed in different quantum states numbered by the index i. The density of OH in state i is denoted by n_i. We use the labelling of states according to Fig. 1. The four hyperfine states in the $^2\Pi_{3/2}$ multiplet states in the j=3/2 OH quantum state are i=0-3, where 0, 1 correspond to the lower and 2, 3 to the upper $\Lambda$- doublet state. The hyperfine state with the lower F corresponds always to the lower number, i.e., here i=0, 2 for F=1 and i=1, 3 for F=2. The same notation is used for the upper states in $^2\Pi_{3/2}$ up to i=31. The states in $^2\Pi_{1/2}$ go from i=32, ..., 63, so that the 8 lowest rotational states with $N \le 7$ are included in the model. The population of states with larger N is negligible in the photodissociation of water.

 

Figure 1: Diagram of the OH energy levels up to N=7 with the notation used in this paper. The 32 lowest hyperfine states of the $^2\Pi_{3/2}$ ladder are denoted as n0 to n31 and the 32 lowest hyperfine states of the $^2\Pi_{1/2}$ ladder are denoted as n32 to n63. The observed maser transitions in W3(OH) are marked with arrows

The probability for the formation of OH after the photodissociation of water in quantum state i is denoted by f_i with $\sum_{i} f_i=1$. For the f_i we use the approach for the $\Lambda$-doublet population given in ([Andresen] 1985) which is roughly valid for the photodissociation of internally cold water. Because a large fraction of the water molecules will relax to lower quantum states before they are photolysed this is probably a good approximation. The nascent OH quantum state distribution is frequency dependent over the first absorption band, in particular for vibrational states. However, the distribution over rotational and $\Lambda$-doublet states depends only weakly on frequency. It should be mentioned that the population of the hyperfine states for $\Lambda$-doublet states was recently measured to be statistical ([Wurps] 1997; [Thissen et al.], submitted 1999).

A warning should be given here because the simplistic picture given above does not hold in detail. The state to state photodissociation yields in some cases even anti-inversion between the $\Lambda$-doublets. In addition vibrational excitation can not be neglected. An even more detailed study about the wavelength dependent nascent OH quantum state distribution from single $\rm H_2O$ quantum states is required to improve the present model.

 

Figure 2: The Maser Model in a schematic view. The quantum states shown here are the $\Lambda$-doublets of OH. Each hyperfine level (not shown here) is fed through photodissociation of water and depopulated via photolysis of OH. In their life time the OH molecules relax to lower states through IR relaxation and stimulated Microwave emission. These transitions are indicated be the solid arrows. The dotted arrows symbolise stimulated emission and absorbtion of FIR radiation, which is not yet included in the model

 

Figure 3: Nascent OH population out of photodissociation of water as a fraction of the total number of dissociated molecules. In equilibrium with OH photolysis the population is given as $n_i=\frac {\Gamma }{\gamma_2}\cdot f_i$ (Eq. (7))

The derived values of f_i are shown in Fig. 3. The nascent population can be calculated from the f_i values. In the stationary case Eqs. (2) and (5) vanish. This gives:  

$$n_i=\frac {\Gamma }{\gamma_2}\cdot f_i.$$ (7)

The population in each quantum state thus increases with the evaporation rate $\Gamma$ and decreases with the OH photolysis rate $\gamma_2$.

The reader may be reminded that the total OH density is simply the sum over the OH quantum state populations (compare Eq. (6)).

 

Figure 4: The effect of IR relaxation on OH quantum state population for two different photodissociation rates at an evaporation rate of $\Gamma=10^7/{\rm m^3\,s}$. For small rates $\gamma$, as in Fig. 4A, almost all OH molecules relax to the groundstate and yield a very high population of the groundstate. For high photodissociation rates the nascent OH population is much lower because OH is destroyed faster. The distribution over quantum states is less affected. There is still considerable population in the excited states

2.2 OH population including IR relaxation

 In this section we will demonstrate that the stationary OH quantum state population resulting from the above model has little to do with the nascent OH quantum state population that results directly from photolysis of water. We will demonstrate that the nascent distribution is considerably modified by an interesting competition between (a) photodissociation of water (b) subsequent IR relaxation and (c) photodissociation of OH. This competition yields very different OH quantum state distributions for different astrophysical conditions. It will be seen that for the small photodissociation rates that are expected for stars with lower luminosity or further away from a high luminosity star (1) the total OH density is large and (2) the OH quantum state distribution is shifted towards lower states with a very high population in the ground states. In contrast, for the high photodissociation that are expected close to stars with high luminosity (1) the total OH density is much smaller and (2) the OH quantum state distribution is less shifted to lower states and extends further out to higher states.

We will demonstrate that by far the largest population - and therefore the highest gain - is obtained in the ground states for small photodissociation rates. The result that the highest gain is obtained for low photodissociation rates may be surprising at the first glance but becomes obvious in the following discussion.

The density n_i in quantum state i of OH increases due to photodissociation of water by $f_i\gamma_1 {\rm [H_2O]}$ and decreases due to photodissociation of OH by $\gamma_ 2 n_i$. In addition, the quantum state density decreases because of IR relaxation out of state i to lower states by $\sum _{k} A_{ik} n_i$, where the sum goes over all final states that can be reached from state i and increases because of IR relaxation from higher states by $\sum _{k} A_{ki} n_k$. This implies  

$$\frac{{\rm d}}{{\rm d}t}n_i=+f_i\gamma_1 {\rm [H_2O]} - \gamma_2 n_i -\sum\limits_{k} A_{ik} n_i+\sum\limits_{k} A_{ki} n_k.$$ (8)

The IR relaxation rates are taken from [Goorvitch et al.] (1992), which includes all transitions possible according to the IR selection rules, e.g. also transitions with $\Delta N=0,-1,-2$ and $\Delta J=\Delta N-1$. The data is not resolved according to hyperfine states. The OH is destroyed with the same destruction rate $\gamma_2$ in all quantum states.

The equation system is solved numerically. For this we rewrite Eq. (8) as a matrix equation. We define a vector n by ${\vec n}=(n_0,...,n_{63})$ representing the density of OH in different quantum states and a vector f by ${\vec f} = (f_0,...,f_{63})$ containing the probabilities for the formation of OH in different quantum states as described in Sect. 2.1. We define further a diagonal matrix D with matrix elements (d_ij) with $d_{ij}= \delta_{ij} \sum _{k} A_{ik}$ which account for the IR relaxation out of state i and a matrix A with only off diagonal elements A_ki which accounts for the IR relaxation into state i from all states k that are above state i and are possible due to the IR selection rules. The matrices contain $64\times 64$ elements. With these definitions we rewrite Eq. (8):  

$$\frac{{\rm d}}{{\rm d}t} {\vec n}\!=\! {\vec f}\gamma_1 {\rm... ... {\vec f}\gamma_1 {\rm [H_2O]} \!+\! {\bf T(\gamma_2)}{\vec n}$$ (9)

where we introduce the matrix T by ${\bf T }(\gamma_2) = (-{\bf E}\gamma_2 - {\bf D} +{\bf A} )$.

For a stationary maser all time derivatives in Eqs. (2, 5, 8, 9, 12, 13) vanish. In particular we have $\frac{{\rm d}}{{\rm d}t} {\vec n}=0$ and obtain the quantum state densities from the solution of the matrix Eq. (15)  

$${\vec n}=-{\bf T}^{-1} {\vec f}\gamma_1 {\rm [H_2O]} =-{\bf T}^{-1} {\vec f}\Gamma.$$ (10)

For stationary conditions, we have $\Gamma=\gamma_1 {\rm [H_2O]}$ according to Eq. (2), which is used in Eq. (10). This implies that the density in every quantum state of OH increases linearly with the evaporation rate $\Gamma$ by the same amount. The importance of this linearity will be discussed later on.

Figure 4 shows how the OH quantum state populations are affected by the competition between photodissociation and IR relaxation. The population is given here only for the four lowest rotational states of the $^2\Pi_{3/2}$ ladder resolved according to hyperfine structure. An absolute value of $\Gamma=10^7/{\rm m^3\,s}$ is used here for the evaporation rate which determines the total OH density but has no effect on the OH quantum state distribution.

The nascent OH population resulting directly from photodissociation of $\rm H_2O$ is given in both plots (filled bars) to allow an easier comparison with the OH populations resulting from our maser model (open bars) which includes the competition between photodissociation and IR relaxation. The only difference between the two figures is that a $20\times$ smaller photodissociation rate is used in Fig. 4A. The actual value of the photodissociation rates is given in the corresponding figure.

The first and most obvious difference between Figs. 4A and 4B is that more than an order of magnitude larger total OH densities are obtained for the smaller photodissociation rate. The population in Fig. 4A goes up to more than $\rm 10^8/m^3$ whereas it stays at several times $\rm 10^6/m^3$ in Fig. 4B. This somewhat surprising result is easily understood in terms of Eq. (6), which predicts for a $20\times$ larger destruction rate of OH, $20\times$ less total OH. It may be emphasized again that the same evaporation rate is used in both cases. The very large population in the ground state of more than $\rm 10^8/m^3$ leads to large inversion and gain for the ground state masers for low photodissociation rates.

Large differences are also found for the relative distribution among the quantum states. Obviously the nascent distribution differs considerably from that of our maser model. Because IR relaxation transfers population from higher to lower states the distribution obtained from our model is always shifted towards lower states compared to the nascent distribution. This shift is more pronounced for small photodissociation rates because the OH molecules have more time to relax before they are destroyed by photodissociation.

For the small photodissociation rates almost all population is concentrated in the ground state with little population left in excited states. For the high photodissociation rate $\rm \gamma=0.2/s$ in Fig. 4B there is still more population in the ground state than in higher states. In comparison to the small photodissociation rate there is relatively more population in the first excited state (j=5/2) and also the j=7/2 state has a non-vanishing population. The excited states are populated more, because at higher dissociation rate (such as 0.2/s) the OH molecules have less time to relax before they are destroyed.

It may be interesting to note that the population of the first excited state (j=5/2) is enhanced compared to the nascent population. This is because the IR relaxation out of this state is slower than the IR relaxation into this state. Our results show an enhancement of the population in a given quantum state - relative to the nascent population - as long as the photodissociation rate $\gamma$ exceeds the IR relaxation rate $A_{\rm IR}(j)$ of this particular rotational state (e.g. $A_{\rm IR}(j=5/2)=0.136/{\rm s}$, $A_{\rm IR}(j=7/2)=0.514/{\rm s}$, $A_{\rm IR}(j=9/2)=1.25/{\rm s}$).

 

Figure 5: Population inversion (nu-nl) for the masers observed in W3(OH) calculated from the present model. The evaporation rate was chosen to be $\Gamma=9\ 10^6/{\rm m^3\,s}$. The population and the population inversion increase strongly with decreasing the dissociation rates. This is due to the fast relaxation from excited states. In contrast the inversions in the excited states are nearly constant over a large range of dissociation rates. For the inversion in the excited states there is a sharp bend off at the position when the dissociation rate becomes as large as the IR relaxation rate for this state ($A_{\rm IR}(j=5/2)=0.136\,{\rm s}^{-1}, A_{\rm IR}(j=7/2)=0.514\,{\rm s}^{-1}$)

In order to evaluate the influence of the photodissociation rate in a more detailed way the population inversion for some important OH maser transitions is plotted in Fig. 5. The population inversion PI_ul is defined as the difference between the population of the upper level (n_u) and the population of the lower level (n_l):

PI_ul=n_u-n_l.

In Fig. 5 the population inversion is plotted as a function of the photodissociation rate $\gamma$. The evaporation rate is chosen here to be $\Gamma=\rm 9\ 10^6/m^3\,s$.

The inversion in the excited states behaves in a very different manner than the inversion in the ground state. Whereas the inversion in the ground state increases dramatically with decreasing dissociation rates it remains almost constant for the excited states for small dissociation rates and drops off only at larger dissociation rates. This effect was already discussed above but becomes more obvious here. This effect is understood in terms of the different losses out of the ground and excited states: losses due to IR relaxation occur only out of the excited states whereas there is no such loss out of the ground state. Because the only loss in the ground state results from destruction of OH by photodissociation these losses decrease with decreasing dissociation rate.

For the excited states the loss rate is dominated by the losses due to IR relaxation, which is independent of $\gamma$. This leads to an almost constant population inversion for low dissociation rates, which is, however, much smaller than in the ground state. Only at dissociation rates higher than 0.1/s the losses due to photodissociation become comparable to the losses due to the IR relaxation. This causes the population and therefore the population inversion to go down. This bend-off is weaker for the higher excited states than for the lower lying excited states and the ground state.

It may be interesting to note that for higher $\gamma$'s ($\gt.3/{\rm s}$) the population inversion becomes higher in the excited states than for the ground states.

2.3 Collisional effects

Up to now we neglected collisional effects. Here we show why. To estimate the relative importance of collisions versus photodissociation, the OH collisional probability $p_{\rm c}$ (= probability for an OH collision per s) is compared with the OH photodissociation rates $\gamma_2$. We consider only collisions with $\rm H_2$ because this is expected to be the dominant species in the interstellar medium. This implies $p_{\rm c}={\rm [OH]}^{-1}\cdot \frac{{\rm d}}{{\rm d}t}{\rm [OH]}=k[\rm H_2]$. An order of magnitude estimate for the rate constant k for inelastic ${\rm OH{-}H_2}$ collisions is obtained from $k=\sigma_{\rm c} v$ with the cross section $\sigma_{\rm c}$ for inelastic collisions and the velocity v. For a typical cross section of $\sigma_{\rm c} \approx 10^{-15}\,{\rm cm}^2=10^{-19}\,{\rm m}^2$ and the nascent OH velocity from the photodissociation of water $v=1.3\ 10^3\,{\rm m/s}$ we obtain $k \approx 1.3\ 10^{-16}\,{\rm m^3/s}$.

The $\rm H_2$ densities are not expected to exceed $\rm 10^{14}\,m^{-3}$. This yields a collisional probability $p_{\rm c}= k\left[\rm H_2\right] \approx 10^{-2}/{\rm s}$. According to the discussion above, the photodissociation rates are expected to vary between $10^{-3}/{\rm s}$ and 1/s. This implies that collisions will affect the OH quantum state densities only for small photodissociation rates.

Because the IR relaxation from excited states is fast, the excited state OH masers have to be pumped fast to compete with IR relaxation. We assume that at least for the case of the 13441 MHz (j=7/2) with relaxation rates of $\rm 0.5\,s^{-1}$ collisions can be neglected safely. Even effects like FIR pumping will play a minor role so that the present model is expected to hold better for the excited state masers.

  

Table 1: Frequency and Einstein A-coefficients according to [Destombes et al.] (1977) and the calculated stimulation cross section (see Eq. (17))

2.4 Stimulated emission, gain and amplification

In this section we determined gain and amplification resulting from our model for different maser transitions under different astrophysical conditions. As a first step we operate only in the limit of the exponential small signal gain but give estimates for the onset of saturation.

If the lower and upper maser level are denoted by l and u, respectively, we obtain:  

$$\frac{{\rm d}}{{\rm d}t}n_l=+f_i\gamma_1 {\rm [H_2O]} - \gamma_2 n_l -\sum\limits_{k} A_{lk} n_l+S_{ul} n_l-S_{ul}n_u \\$$ (12)

 

$$\frac{{\rm d}}{{\rm d}t}n_u=+f_i\gamma_1 {\rm [H_2O]} - \gamma_2 n_i -\sum\limits_{k} A_{uk} n_u- S_{ul}n_u+S_{ul}n_l$$ (13)

S_ul is the stimulation rate. The stimulation rate is related to the maser photon flux $j_{\rm m}$ (photons per $\rm m^2$ and s) and the cross section $\sigma_{ul}$ for stimulated emission by  

$$S_{ul}=\sigma_{ul} j_{\rm m}.$$ (14)

The frequency integrated cross sections $\int \sigma_{ul}(\nu){\rm d}\nu$ for the stimulated emission are obtained from the Einstein A-coefficients from [Destombes et al.] (1977) via  

$$\int \sigma_{ul}(\nu){\rm d}\nu=\frac {A_{ul}c^2}{8\pi\nu^3}\cdot$$ (15)

The line width for the OH resulting from the photodissociation of water is given by the nascent velocity $v_{\rm OH}$ according to the Doppler effect by  

$$\Delta \nu =\frac {2\cdot v_{\rm OH}}{c}\cdot$$ (16)

With the nascent velocity of OH we have $v_{\rm OH} \approx 1.3\ 10^3\,{\rm m/s}$ and therefore $\Delta\nu\approx 10^{-5}\nu$. If we introduce the peak absorption cross section $\sigma^0_{ul}$ by $\int \sigma_{ul}(\nu){\rm d}\nu = \sigma^0_{ul} {\rm d}\nu = \sigma^0_{ul} 10^{-5}\cdot\nu$we obtain  

$$\sigma^0_{ul}=\frac {A_{ul}\ 10^5\cdot c^2}{8\pi\nu^3}\cdot$$ (17)

Equation (17) is used to calculate the (peak) cross section for stimulated emission for all observed maser frequencies. Table 1 gives the frequencies, Einstein A coefficients, the calculated $\sigma^0_{ul}$ and the microwave flux $I_{\rm sat}$ for the onset of saturation for all maser transitions. Here the onset of saturation is defined as

$$I_{\rm sat} \cdot \sigma_{ul}=5\%,$$ (18)

which means: The change in population due to the stimulated emission exceed 5 per cent of the population.

For the amplification of the photon flux along some propagation direction z we use the peak cross section for stimulated emission and obtain:  

$$\frac{{\rm d}}{{\rm d}z} j_{\rm m} = \sigma^0_{ul} (n_u-n_l) j_{\rm m}.$$ (19)

This implies a small signal gain  

$$g=\sigma^0_{ul} (n_u-n_l).$$ (20)

As seen in Sects. 2.1 and 2.2 the OH population is proportional to the evaporation rate $\Gamma$. Now we introduce a quantity called "specific gain" $g_{\rm s}$, which is defined as

$$g_{\rm s}=\frac {g}{\Gamma}=\frac {\sigma^0_{ul} (n_u-n_l)}{\Gamma}\cdot$$ (21)

The specific gain is the gain obtained if one water molecule is evaporated in $\rm 1\,m^3$ per second. The specific gain is used here because it is independent of the evaporation rate and describes the physics for the competition between photodissociation and IR relaxation. $g_{\rm s}$ is a characteristic feature of the proposed pumping scheme. The specific gain for the OH maser transitions observed in W3(OH) is displayed in Fig. 6.

 

Figure 6: The specific gain calculated from our model for all maser transitions observed in W3(OH). The absolute gain can be derived by multiplying the specific gain with the evaporation rate $\Gamma$

Figure 6A shows the specific gain for the maser transitions in the $^2\Pi_{3/2}$ ladder as a function of the photodissociation rate $\gamma$. In general, the results for the specific gain are similar to those for the population inversion that was shown in Fig. 5. This is not surprising because, apart from the evaporation rate, the only important difference between specific gain and population inversion is the different cross section for stimulated emission (compare Eq. (20)). The comparison between Figs. 5 and 6 thus shows the effects resulting from the different cross sections for stimulated emission. The general agreement in features, like increase of the specific gain in the ground state with decreasing $\gamma$ or the bend-over for the excited state at larger dissociation rates are therefore easily understood.

A large difference appears between specific gain and inversion for the main and satellite lines in the ground state. The specific gain for the 1612 MHz and 1720 MHz masers is much smaller than for the 1667 MHz and 1665 MHz masers although the inversion is similar. This is a result of the roughly $10\times$ smaller stimulation cross section $\sigma^0_{ul}$ for the satellite lines compared to the main lines.

For the excited states the specific gain is again almost constant over a wide range of $\gamma$'s and decreases when the photodissociation rate becomes as large as the IR relaxation rate. Because the stimulation rates are smaller for the excited states the specific gain - in contrast to the population inversion - is always smaller in the excited states than for the ground state main line transitions.

This seems to propose that ground state masers should dominate excited state masers even at high dissociation rates. Such conclusions cannot be drawn. First, it is not clear how the much larger spontaneous emission rates in the excited states compared to the ground state affect the appearance of a given maser transition.

Second, if the dissociation rates become larger than the IR relaxation rates, the population distribution for both the ground and excited states approaches the nascent population from direct photolysis. The nascent inversion in the ground state, however, depends strongly on the population of quantum states in the parent water molecule ([Häusler et al.] 1987). Photodissociation of the rotational ground state of water even yields anti-inversion for the ground state transitions in OH. For small dissociation rates this effect is less important because the high nascent inversion in the high lying rotational states is transferred to the ground state. This is because the IR relaxation conserves the inversion. Consequently the model given above is too simple for the ground state at large dissociation rates. A closer look at this problem reveals that for high $\gamma$'s the picture will change but remain more or less constant for low $\gamma$'s, because the population inversion is then more determined by IR relaxation from highly inverted upper rotational states.

Figure 6B shows the specific gain for the three transitions of the $^2\Pi_{1/2}(j=1/2)$ state. Photodissociation of $\rm H_2O$ yields anti-inversion between all $\Lambda$-doublets of the $^2\Pi_{1/2}$ multiplet and therefore yields a "negative specific gain" as seen for the 4750 MHz and the 4660 MHz transitions in Fig. 6B. For the 4765 MHz transition, however, a different behavior is observed. Here the specific gain is still negative for small dissociation rates below 0.05/s as expected. However, for $\gamma\gt.05/{\rm s}$ $g_{\rm s}$ the specific gain becomes positive with a maximum of $g_{\rm s}\approx 10^{-19}\,{\rm m^2\,s}$ at $\rm \gamma \approx 0.1/s$. Nevertheless the specific gain stays small and it may be hard to explain this maser without additional mechanisms like FIR pumping.

The gain g is obtained from the specific gain $g_{\rm s}$ by multiplication with $\Gamma$. It will be seen below that $\Gamma$ can vary strongly with the astrophysical conditions and depends sensitively on the density, sizes and temperatures of the grains. This implies in turn a large variation of maser phenomena with different astrophysical conditions.

The calculations given below neglect saturation effects which implies exponential growth of the maser intensity and yields for an amplification factor $V_{\rm m}$ 

$$V_{\rm m}=\frac {j_{\rm m}}{j_{\rm m_{in}}}{\rm e}^{gz}$$ (22)

where $j_{\rm m_{in}}$ is the incoming maser photon flux density and z is the length of the maser volume. Although the formalism presented above includes saturation effects these are excluded here and postponed to a future paper.

Some examples:

Now we give some specific examples for the gain g and the amplification factors $V_{\rm m}$ obtained for different maser transitions at different evaporation rates.

In Fig. 6 it can be seen that the specific gain in the j=3/2 maser can become very large, in particular for dissociation rates lower than $\rm 0.01\,s^{-1}$. Under these circumstances population and inversion are accumulated in the ground state due to IR-relaxation with little OH destroyed by photodissociation. Note that the nascent inversion in the ground state is lower than the inversion after relaxation.

This is illustrated by an example with a photodissociation rate of $\gamma_1 \approx 0.01\,\rm s^{- 1}$, where we obtain a specific gain of $2\ 10^{-16}\,{\rm m^2\,s}$ for the 1667 MHz maser. A moderate evaporation rate of $\Gamma \approx 3\ 10^3 \,\rm m^{-3}\,s^{-1}$ yields a gain of $6\ 10^{-13}\,{\rm m}^{-1}$, which implies under the assumption of exponential growth an amplification of $V_{\rm m} = 10^6$ for a tube length of roughly 150 AU. For the 6035 MHz maser the specific gain is $g_{\rm s} \approx 5\ 10^{- 18}\,{\rm m}^{-1}$ for $\gamma_1$ below 0.1/s. Nevertheless, a gain of $\approx 2.5\ 10^{-12}\,{\rm m}^{-1}$ can be obtained, with a much higher evaporation rate of $\Gamma=5\ 10^5\,\rm s^{-1}$, however. In this case an amplification of 10⁶ is obtained with a tube length of roughly 60 AU.

For the 13.44 GHz maser the specific gain is even smaller. In the range of low photodissociation rates we have $g_{\rm s} \approx 3.5\ 10^{-19}\,{\rm m^2\,s}$. To explain the gain of $\gamma= 2-3\ 10^{-12}\,{\rm m^{-1}}$, suggested by [Baudry & Diamond] (1998) for the single bright maser spot in W3(OH), an even higher evaporation rate of $\Gamma \approx 9\ 10^6 \,\rm m^{-3}\,s^{-1}$ is required. This gain yields an amplification of $V_{\rm m}=10^3$ for a tube length of 15 AU. It should be mentioned that there is a general background of 20 mJy at the maser frequency from all over the region of this maser. The amplification of this background by $V_{\rm m}=10^3$ yields exactly the 20 Jy observed for the single bright maser spot.

For the 4765 MHz maser in $^2\Pi_{1/2}(j=1/2)$ state gain is obtained only for photodissociation rates above 0.05/s. The maximum specific gain is obtained for $\gamma \approx 0.1\,\rm s^{-1}$ and is there around $g_{\rm s}=1\ 10^{-19}\,{\rm m^2\,s}$, the smallest specific gain from all observed masers. Even with $\Gamma=10^7\,\rm m^{-3}\,s^{-1}$ we obtain a gain of only $10^{-12}\,{\rm m}^{-1}$ and require a tube length of 50 AU for an amplification of 1000. This maser operates according to the present model only if the photodissociation and the evaporation rates are high.

To summarize the results from our gain considerations, we state that over a wide range of $\gamma$'s specific gain is obtained for all observed maser transitions that appear in regions of star formation. In contrast to the specific gain, the gain itself depends strongly upon the evaporation rates. For the same evaporation rate the relative gain between different states is determined by the relative specific gains. The specific gain for the excited state masers in $^2\Pi_{3/2}$ is much smaller than for the ground state masers. Small specific gain is obtained also for the 4765 MHz maser, however, only for $\gamma$ above 0.05/s. Much smaller specific gain is also obtained between states higher than j=7/2 in $^2\Pi_{3/2}$ where no masing is observed. However, the gain is so small for these states that masing is not expected from the present maser model, in agreement with observations.

For all states in $^2\Pi_{3/2}$ the gain for the F state with higher multiplicity dominates the gain with lower multiplicity. This predicts a preference for hyperfine transitions in the sense that the transition belonging to the larger F- value should be stronger. For the ground state masers the gain for the main lines dominates that for satellite lines. Because the specific gain can become very high in the ground state the satellite lines are also expected to yield maser radiation with a slight preference for the 1720 MHz transition.

2.5 Spontaneous emission and "self oscillation"

Here we discuss whether spontaneous emission from a tubular volume with cross sectional area ${\cal A}$ and length l can start the maser even without external background emission from e.g. grains. We use the quantum state populations given in Sect. 2.2. We consider an infinitesimal element ${\rm d}z$ along the maser's propagation direction z which has a volume ${\rm d}V={\cal A}{\rm d}z$ with a density n_u in the upper maser level. The number of spontaneous emission events from this volume is $A_{ul}n_u{\cal A}{\rm d}z$ per second. The fraction $\Delta \Omega/4\pi = {\cal A}/4\pi l^2$ of these spontaneous photons passes through the exit area of the maser tube. Only these spontaneous photons will propagate through the inverted maser volume, will be amplified on this path and contribute to the maser emission observed on earth. The maser photon flux density that exits the front area ${\cal A}$ of the maser tube is then $A_{ul}n_u{\cal A}{\rm d}z\, (\Delta \Omega/4\pi) /{\cal A}$. The maser flux density will give rise also to the stimulated emission $j(z) \sigma_{ul} (n_u - n_l ) {\cal A}{\rm d}z$ in the volume ${\cal A}{\rm d}z$. The increase ${\rm d}j$ of the maser flux density results therefore not only from the spontaneous emission $A_{ul}n_u{\rm d}z\, (\Delta \Omega/4\pi)$ but also from the stimulated emission $j(z) \sigma_{ul} (n_u - n_l ) {\cal A}{\rm d}z$ which implies

$$\frac{{\rm d}j}{{\rm d}z}=j(z) \sigma_{ul} (n_u - n_l ) + \frac {\Delta \Omega}{4\pi}\cdot A_{ul}n_u.$$ (23)

With the gain $g=\sigma_{ul} (n_u - n_l )=g_{\rm s}\Gamma$, this can be solved as long as the maser is not saturated by  

$$j(z)=\frac {\Delta \Omega}{4\pi} \frac{A_{ul}n_u}{g} ({\rm e}^{g_{\rm s}\Gamma \cdot z}-1).$$ (24)

Figure 7 shows results for the maser photon flux density obtained from spontaneous and stimulated emission for the maser transitions appearing in regions of star formation as a function of the maser tube length. The evaporation rate is taken to be $\Gamma=3\ 10^3/{\rm m^3\,s}$ and the photodissociation rate is $\gamma=0.01/{\rm s}$ as assumed for a distance of $\approx 1000$ AU from the central star. Under these condition only the ground state masers will start with spontaneous emission. In the present model only exponential growth and no saturation effect are encountered (see Eq. (24)). The emitted photon flux is largest for the main line ground state masers because their specific gains are largest at this particular dissociation rate. The 1667 MHz maser reaches the onset of saturation (see Table 1) at a tube length of $\approx 250$ AU.

 

Figure 7: Maser output as a function of the maser tube length as calculated from our model for $\Gamma=3\ 10^3/{\rm m^3\,s}$ and $\gamma=0.01\,{\rm s}^{-1}$ assumed for a distance of 1000 AU from the central star. It becomes clear that only the ground state maser can operate under these conditions. The excited state masers require higher evaporation rates and photodissociation rates. The dashed line marks the onset of saturation for the ground state main line transitions

In contrast, the ground state satellite line masers and the excited state maser show a much smaller growth under these conditions that are chosen to explain the main line ground state masers. This results from their much smaller specific gain which in turn comes from the small evaporation rate of only $\Gamma=3\ 10^3/{\rm m^3\,s}$. This implies the main line ground state masers will work even far away from the star, whereas the excited state masers will not operate there. According to the discussion below much higher evaporation rates may be expected closer to the star. There the specific gains of the different maser transitions becomes more similar at the expected higher dissociation rates (see Fig. 6). The competition between the masers under these conditions (closer to the star) have not been investigated yet.

Nevertheless the results show that with reasonable tube length the ground state maser may start with spontaneous emission without any amplification of background radiation from the surrounding dust shell. At a given maser tube length the output becomes higher with such additional microwave background radiation.