3 Evaporation rates and life time of masers

 According to the present maser model, all masers observed at new born stars have gain and can yield maser radiation by photodissociation of the gaseous water evaporated from grains in the presence of VUV radiation. Because the gain increases linearly with the evaporation rate, the present maser model can yield any gain with sufficiently high evaporation rates. For the gains presented above, we simply postulated evaporation rates ranging from $\Gamma= 10^3 - 10^7\,\rm m^{-3}\,s^{-1}$ without any critical discussion of whether such rates are possible at all. In this section we try to establish order of magnitude estimates for the evaporation rates for the special astrophysical conditions at the border between a HII region and the surrounding cloud of grains.

It should be mentioned that the evaporation problem is closely related to an often discussed criticism of the photodissociation pump mechanism. The evaporation rate is responsible for the replacement of the large amount of gaseous water that is destroyed by photodissociation. Only with sufficient evaporation going on for an extended time the masers can operate without turning off. In the present stationary maser model the evaporation rate $\Gamma$ equals the number of water molecules destroyed per $\rm m^3$ and second (see Eq. (2)). An evaporation rate of $\approx 10^7 \, \rm m^{-3}\,s^{-1}$ is necessary if e.g. 10⁷ water molecules are destroyed per $\rm m^3$ and second! The question is whether such high evaporation rates are possible at all. The stationary maser model given here implies a highly dynamical equilibrium between evaporation and destruction in which the densities of OH and $\rm H_2O$ adjust to the given photodissociation rates. In this suggested mechanism the evaporated water is converted to translationally hot hydrogen atoms and relatively low oxygen atoms by the VUV radiation from the central star and heats the environment in areas where water is evaporated ("photolytic heating").

The region around the star that is exposed to the direct heat- and VUV- flux from the star will reach out only to a region where small grains are present in high abundance. With a grain density of $n_{\rm gr} \approx 1\,{\rm m}^{-3}$ for $1\,\mu {\rm m}$ grains, which is considered to be typical for cold interstellar clouds, an optical density of $\epsilon=n_{\rm gr}\sigma_{\rm gr} \approx3\ 10^{-12}\,{\rm m}^{-1}$ is obtained. This implies that the radiation will be attenuated to 1/e along a distance of only $\approx 2$ AU. This thin layer of a few AU thickness blocks the direct radiation. Although the radiation will diffuse further out, it is expected that there will be no VUV beyond that border. We will use the term "small grain border" for this thin layer below.

The small grains at this border are exposed to the direct heat flux from the star and will be evaporated with time leading to a growth of the HII region. The time until complete evaporation increases with the size of the grains. If the small $\rm 1 \,\mu m$ grains are already completely evaporated larger grains will remain at the inner side of the small grain border. Because their evaporation takes a longer time they will be a source of gas phase water molecules which are exposed to the VUV field from the central star. If we assume in addition that the grains move with their gravitational velocity towards the star large grains will penetrate deeper into the HII region than small grains. Because the optical density of the remaining large grains is much lower than that for the small grains at the border they are exposed to the direct heat- and VUV- flux from the star. This increases with decreasing distance to the star. Consequently evaporation and photolysis of water takes place simultaneously in this regions and yields the astrophysical situation where the present maser model applies. Below we will discuss the processes that occur to the larger grains at the inner side of the small gain border in more detail.

For a single grain the evaporation is determined mainly by its surface area and temperature. The surface temperature depends on the heat transfer from and to the surface. The heat transferred to the surface results from the direct heat flux from the star and from collisions with the surrounding gas, including e.g. the heat resulting from the adsorption of water or atoms that are generated by photolytic heating. The heat losses result from black body radiation, heat conduction to the bulk and evaporative cooling by desorption.

The evaporation of grains is extremely difficult to predict quantitatively because it depends on too many and mostly unknown parameters. Features like porosity or chemical composition of the surface strongly affect surface area and temperature leading to large variations in the evaporation rate. This problem is well known from comets (e.g. [Combi et al.] 1998; [Schultz et al.] 1992) where the identical problem is encountered. The astrophysical problem of the evaporation from grains near new born stars may also be compared with the evaporation of droplets in spray combustion where large difficulties are encountered in a quantitative treatment despite massive experimental and theoretical efforts. Because it is impossible to derive quantitative evaporation rates we restrict ourselves to a qualitative discussion of evaporation immediately from the analogy with spray combustion. The large droplets survive longer than small droplets in spray evaporation and penetrate deeper into the gas.

There are nevertheless some important qualitative conclusions which can be drawn. Here larger grains live longer when they are exposed to the direct heat flux from the star and penetrate the HII region deeper than the small grains. Closer to the star they are exposed to an increasing heat flux, acquire larger temperatures and may yield rather high evaporation rates in the presence of intense VUV radiation there.

To determine the evaporation rate $(\Gamma= O_{\rm gr} j_{\rm des} )$ both the desorption flux density $j_{\rm des}$ and the surface area of all grains in $\rm m^3$ maser volume have to be known. The desorption flux density is closely related to the vapour pressure of ice. Under thermal equilibrium conditions the same number of water molecules are adsorbed and desorbed on the surface, i.e., the desorption flux density $j_{\rm des}$ equals the adsorption flux density $j_{\rm ads}$. We approximate the adsorption flux density by $j_{\rm ads} = 1/6 \,{\rm n v}$ with the gas phase density n = p/kT and velocity $v = (2\,kT/m)^{0.5}$. This implies $j_{\rm des} = (18\, m kT)^{-0.5}\, p$ with the vapour pressure p of ice. For the vapour pressure p of ice we use the well known standard expression $p=A\exp(-B/T)$ ([Handb. of Chem. Phys.]). With $\alpha=(18\, m k)^{-0.5} A$ and $\beta= B$ this yields  

$$j_{\rm des}=\alpha \sqrt{T} {\rm e}^{-\beta/T}.$$ (25)

With $\beta=5.11\ 10^3$ K and $\alpha=3.32\ 10^{34}\frac{1}{sm^2 \sqrt{K}}$ this expression yields the correct vapour pressure of water at $100^\circ{\rm C}$ and $0^\circ{\rm C}$. The desorption flux density thus depends only on the surface temperature and increases exponentially with increasing temperature. For example, $j_{\rm des}$ increases from $j_{\rm des} \approx 10^{15}\,\rm m^{-2}\,s^{-1}$ at T = 120 K over $2\ 10^{22}\,\rm m^{-2}\,s^{-1}$ at 200 K to $j_{\rm des} \approx 2.8\ 10^{24}\,{\rm m^{-2}\,s}^{-1}$ at 250 K. Nevertheless, Eq. (25) is only used to obtain order of magnitude estimates for $j_{\rm des}$.

The evaporating grains have a finite life time that is determined by their surface temperature and their mass. In this qualitative discussion we neglect adsorption which would yield an even longer life time. Because $j_{\rm des} A_{\rm gr} {\rm d}t$ is the number ${\rm d}Z$ of water molecules desorbing from the grain in the time ${\rm d}t$, we have ${\rm d}Z/{\rm d}t = j_{\rm des} A_{\rm gr}$. With the density $\kappa \approx 3\ 10^{28}\,{\rm m}^{-3}$ of ice, the total number of water molecules on a grain is $Z=\kappa V_{\rm gr}= \kappa \frac {4}{3}\pi r_{\rm gr}^3$ under the assumption of spherical grains. Therefore ${\rm d}Z/{\rm d}t = {\rm d}/{\rm d}t(\kappa \frac {4}{3}\pi r_{\rm gr}^3) = \kappa \frac {4}{3}\pi r_{\rm gr}^2 {\rm d}r_{\rm gr}/{\rm d}t$ which yields $\kappa \frac {4}{3}\pi r_{\rm gr}^2 {\rm d}r_{\rm gr}/{\rm d}t = j_{\rm des} A_{\rm gr} = j_{\rm des} 4\pi r_{\rm gr}^2$. This simple model yields a decrease of the grain radius with time that is independent of the grain radius:

$$\frac{{\rm d}r_{\rm gr}}{{\rm d}t}=\frac{3}{\kappa}j_{\rm des}\cdot$$ (26)

The time $t_{\rm l}$ required to desorb all water molecules off the grain, i.e., the life time of a grain is

$$t_{\rm l}=\frac {\kappa}{3j_{\rm des}r_{\rm gr}}\cdot$$ (27)

If $r_{\rm gr}$ is measured in $\rm \mu m$ and $t_{\rm l}$ is given in years we obtain, for example, at a constant grain temperature of T=120 K and the desorption flux $j_{\rm des} \approx 10^{15}\,\rm m^{-2}\,s^{-1}$ corresponding to this temperature a grain life time of

$$t_1=0.317\cdot r_{\rm gr}.$$ (28)

The life of a grain depends linearly on its radius. Grains with $\rm 100\,\mu m$ radius last $\approx 30$ years, grains with $\rm 10\,\mu m$ for $\approx 3$ years and $\rm 1 \,\mu m$ only for $\approx 0.3$ years at 120 K. Large "grains" with radii of 10 km, like comets, will live 10⁸ times longer. As mentioned above, we assume that the grains move towards the central star with their gravitational velocity and will travel $\approx 1\, {\rm AU/year}$ for a velocity of e.g. $v \approx 5\ 10^3\,{\rm m/s}$. This implies that grains of increasing sizes will penetrate deeper and deeper beyond the border of the HII region. $\rm 100\,\mu m$ ($\rm 10\,\mu m$, $\rm 1 \,\mu m$) grains at T=120 K will travel roughly 30 AU (3 AU, 0.3 AU) towards the star before they are completely evaporated.

It is an obvious but important result that small grains are evaporated much faster than large grains and that large grains penetrate deeper. The same effect occurs as mentioned above in the evaporating sprays in Diesel engines where big droplets live much longer than small droplets and penetrate much deeper into the hot gas. This is well known to create one of the most important problems in Diesel combustion, i.e., soot formation which is avoided in modern engines today by the generation of finer droplets.

The importance of the above consideration should be stressed again: The larger water carrying grains penetrate the HII region and are evaporated in a region where they are exposed to the VUV radiation field. Large grains, like comets, may penetrate very deep in the HII region where the heat flux goes up and massive evaporation of water molecules sets in and intense VUV radiation fields are present for sure. This yields the astrophysical situation in which grains are exposed to the heat- and VUV- flux from the star, the situation in which our maser model applies.

Groundstate Masers:

We will now try to explain the ground state masers based on the astrophysical conditions met at the border of the HII region in a layer which may be a few tenths of AU thick, at the inner side of the small grain border where relatively small grains of $10-100\,\mu {\rm m}$ are still present and evaporate water. Then we will try to explain the excited state masers by the evaporation from much larger grains that penetrate deeper and may thus reach much higher temperatures there. Because these larger grains have finite life time the excited state masers should be transient phenomena.

First we treat the region close to the border of the small grain region. We calculate the grain density required for the evaporation rate of $\Gamma=3\ 10^3\,{\rm m^{-3}\,s^{-1}}$ that was used above to discuss j=3/2 masers. We assume a temperature of 120 K and obtain, according to Eq. (1), this rate $\Gamma$ with $O_{\rm gr}= n_{\rm gr} A_{\rm gr} = 3\ 10^{-12} \,{\rm m^2/m^3}$. This surface area (per $\rm m^3$ maser volume) originates from a grain size distribution that is not known. If we assume spherical $\rm 10\,\mu m$ $\rm (100\,\mu m)$ grains the required surface of area $3\ 10^{-12} \,{\rm m^2/m^3}$ is obtained with a grain density of $n_{\rm gr} \approx 24\ 10^{-3}\,{\rm m}^{-3}$ $(24\ 10^{-6} \,{\rm m}^{ -3})$. Much lower grain densities would result for highly porous grains. If we compare these grain densities to the often used grain density of $1/{\rm m}^3$ for $\rm 1 \,\mu m$ grain in cold clouds, this does not seem to be too exotic.

This implies that a gain of $g=10^{-12}\,{\rm m}^{-1}$ is obtained for the j=3/2 maser with e.g. a moderate grain density of only 24 grains with $\rm 100\,\mu m$ radius in a volume of $10^6\,{\rm m}^3$. These $\rm 100\,\mu m$ grains would live at T=120 K for roughly 30 years and penetrate the HII region 30 AU in this time. The continuous flow of grains towards the star could thus replaces the grains that are evaporated or the masers may simply move outwards with the expansion velocity of the HII region. This mechanism implies that the ground state masers can last as long as the cloud surrounding the HII region is not expanded so much that the evaporation and photodissociation rate become too small.

According to this picture the ground state OH masers could result from the evaporation of grains e.g. in the range of $\rm 10-100\,\mu m$ that lie in the circumstellar shell at the inner side of the small grain border. In this situation large maser tube lengths of several 100 AUs are possible tangentially to the border, i.e. large gain lengths are possible which in turn can yield intense ground state masers.

Excited State Masers:

The long lived larger grains suffer a different fate. On their continuing way towards the star they penetrate the HII region deeper and are exposed to an increasingly higher heat flux from the star. Due to the increasing heat flux with decreasing distance the surface temperature of the large grains will rise and the desorption flux as well as the evaporation rate will increase fast. Because the small grains are already completely evaporated deep inside the HII region the large grains there are exposed to the full direct heat and VUV flux. Then both water and OH are photolyzed efficiently to give H and O atoms. Because an energy of a few eV per photolyzed molecule goes to the gas surrounding the grain the gas heats up even more. This "photolytic heating" is well known for the comets approaching the sun. Consequently there is increasingly massive evaporation if a large grain approaches the star. Once the grain reaches temperatures above 180 K almost all energy will be used to desorb water because of the energy losses by black body radiation become small in comparison to the energy loss by desorption.

As mentioned above the high evaporation rate used above (e.g. $\Gamma \approx 9\ 10^6 \,\rm m^{-3}\,s^{-1}$) for the single intense maser spot given by [Baudry & Diamond] (1998) is much more difficult to explain: this evaporation rate is 3000 times higher than that assumed for the ground state masers. This high evaporation rate assumed for the 13.44 GHz maser can be explained qualitatively on the basis of the above discussion and the speculation that this maser is located very close to the star, e.g. at a distance of less than 200 AUs. This speculation is based on the location of the maximum of the 23.8 GHz continuum map, where according to [Baudry & Diamond] (1998) the star is suspected to be. It is interesting to note that with this location of the star most of the j=3/2 masers seem to be indeed at a distance of $\approx 1000$ AUs, where we assume the small grain border to be.

We assume, as before for the j=3/2 masers, that the j=7/2 maser is pumped by evaporation from water containing grains. In this case, however, the grains must have traveled from 1000 AU to 200 AU and still evaporate water. Therefore the "grains" must be very large because they survived the increasingly massive evaporation on their way towards the star. Because a time of roughly 800 years is required to travel the distance of 800 AU only very big "grains", like comets or planets, survive this journey. We assume an average temperature of 200 K for a "grain" with radius 20 km approaching the star to estimate its life time. With the desorption flux $j_{\rm des} = 2\ 10^{22}\,{\rm m^{-2}\,s^{-1}}$ at 200 K this yields a life time of $\approx 1600$ years and at 800 AUs the "grain" would still have a radius of 10 km. The situation described here is in closest analogy to comets that approach the sun and yield high water evaporation rates there.

At 200 AU, however, the temperature of this 10 km block can be considerably higher, in particular because the continuously increasing photolytic heating. Without any better knowledge we discuss an example with a temperature of 250 K for the 10 km block at 200 AU to explain the high evaporation rate of $\approx 10^7 \, \rm m^{-3}\,s^{-1}$. With the desorption flux $j_{\rm des} \approx 2.8\ 10^{24}\,{\rm m^{-2}\,s}^{-1}$ we obtain $O_{\rm gr} \approx 3\ 10^{-18}\,{\rm m}^2$ to explain this evaporation rate. Under the assumption of 10 km "grains" this requires a "grain" density $n_{\rm gr} \approx 2.4\ 10^{-27}\,{\rm m}^{-3}$.

This means that one block of 10 km radius is in a volume of $4\ 10^{26}\,{\rm m}^3$ or an average distance between these blocks of $\approx 10^9\,{\rm m}$. For this grain density a mass of $6.5\ 10^{23} \,{\rm kg}$ is contained in the tubular maser volume of $\rm 20\,{\rm AU}^3$ that was assumed above for the single intense 13.44 GHz maser spot by [Baudry & Diamond] (1998). With the mass $M \approx 2\ 10^{23}\,{\rm kg}$ of Earth this implies that one third of the mass of earth is contained in the maser volume. For "grains" with $100\times$ higher porosity much less mass is required in the maser volume. These porous grains may survive the passage because of poor heat conduction to the inner part of the grain. Because the j=7/2 masers are observed only once in the interstellar medium the somewhat exotic assumptions made here appear not to be too unreasonable.

One last remark is related to the probability for the appearance of a particular maser. This probability is determined not only by the gain but also by the probability for spontaneous emission, which clearly favours the excited state masers. On the other hand excited state masers require much larger evaporation rates than the ground state masers. Because these high evaporation rates are not available near the relatively cold border of the HII region the excited state masers should be much closer to the star where the heating of grains increases. The ground state masers on the other hand require neither high evaporation nor high photodissociation rates. If they are located at the border of the HII region they have long tube length and can nevertheless start even by spontaneous emission. Closer to the star, where the evaporation is high, the excited state masers may operate as amplifiers with e.g. the 20 mJy background radiation observed for the 13441 MHz maser.