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Astron. Astrophys. 337, 85-95 (1998)

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2. Quantum heating

Standard interstellar dust grains (radii [FORMULA] Å) are large enough to be in thermal equilibrium with an outer radiation field. Therefore, an equilibrium temperature [FORMULA] can be obtained using the balance equation


where [FORMULA] is the mean intensity of the incoming radiation field at the frequency [FORMULA], [FORMULA] the Planck function, [FORMULA] the absorption cross section, and [FORMULA] the temperature of the dust grain.

The heat capacity of smaller dust grains, however, is so low that even the absorption of one UV/visible photon can increase the heat content of the grain substantially. As a result of this, small dust grains undergo temperature fluctuations if they are exposed to UV radiation. Therefore, instead of an equilibrium temperature, a temperature distribution is required. In this paper, we will refer to this effect, which is often called transient heating , as to quantum heating because the effect is related to the absorption of individual photons.

Two methods published in previous papers for calculating the quantum heating process seem to be applicable for radiative transfer calculations. The first one we refer to (Sect. 2.1) is a general method for calculating the quantum heating process of small particles and was developed by Guhathakurta & Draine (1989). The second one (Sect. 2.2) was mainly developed by L[FORMULA]ger et al. 1989 to calculate the emission of PAHs, and was first used in radiative transfer calculations by Siebenmorgen & Krügel (1992).

2.1. Emission of small dust grains

To calculate the emission of small dust grains (radii [FORMULA] 100 Å), we use an extension of the method developed by Guhathakurta & Draine (1989), hereafter GD89. Knowing the range of the temperature fluctuations [FORMULA], a grid for temperatures [FORMULA] and corresponding enthalpies [FORMULA] is created. For these (artificial) energy levels, transition rates per unit time ([FORMULA]) for heating ([FORMULA]) and cooling ([FORMULA]) can be calculated. After solving the stationary matrix equation for the probabilities [FORMULA] of finding the grain in the [FORMULA]th bin (in the temperature interval [FORMULA]),


the monochromatic energy per unit time [FORMULA] emitted by the grain can be obtained by


In the following section we present details on calculating the transition matrix and solving Eq. (2) as well as some results for carbon grains.

This method, starting from the definition of the temperature grid up to the calculation of the emitted intensity, is implemented as the inner loop in the algorithm summarized in Fig. 1.

[FIGURE] Fig. 1. Algorithm used to calculate the emission of small dust grains due to quantum heating.

2.1.1. Transition matrix

Following the formalism of GD89, the matrix elements for cooling from state i to state f are defined by


where [FORMULA] and [FORMULA] are the values and widths of the enthalpy bins. The frequencies [FORMULA] and [FORMULA] are defined by [FORMULA] and [FORMULA].

To solve Eq. (2) for the complete matrix (cooling according to Eq.(4)), we used the bi-conjugate gradient method BiCGStab of Sleijpen & Van der Vorst (1995), where the solution of the approximate matrix (cooling after Eq. (5)) is used for preconditioning. Due to the nature of the matrix, standard matrix solvers like LU-decomposition or the Gauß algorithm (Press et al. 1986) failed to give reliable results.

Because solving Eq. (2) for such a matrix can be very time-consuming, GD89 used a more simple approach to calculate the cooling rates. In their approximation, cooling from level i occurs only to the next lower level [FORMULA]. Therefore, cooling rates are given by,


A matrix defined in such a way is a lower triangular matrix with only one additional upper side diagonal and, therefore, Eq. (2) can now be solved directly, what makes the procedure very fast. There is no physical reason for this approximative treatment of cooling because the temperature and enthalpy levels are artificial, whereas the physical states are expected to form a quasi-continuum.

This procedure can only be understood in terms of numerics. If "coarse" grids are used, the widths of the enthalpy bins often increase to such an extent that the integration limits numerically approach [FORMULA] in Eq. (4). This effectively transforms Eq. (4) into Eq. (5).

In Fig. 2, we show the emission of a 10 Å graphite grain located in the vicinity of a B-type star, and obtained by the use of both kinds of transition matrices. For wavelengths up to 30 µm both methods produce almost identical emission. However, in case of longer wavelengths, using cooling according to Eq. (5) systematically underestimates the emission up to an order of magnitude. These differences are due to the larger number of transition channels which are available if the exact cooling (Eq. (4)) is used.

[FIGURE] Fig. 2. Emission of a 10 Å graphite grain located at a distance of 1pc from a B-type star ([FORMULA], [FORMULA]K). The solid line represents the emission calculated using the complete cooling equation (Eq. (4)) and the dashed line is emission for approximative cooling after Eq. (5). The "+" signs indicate the "classical" emission after Eq. (1), which is shown to illustrate the effect of quantum heating. Note that for both kinds of matrix definitions the frequency-integrated emission is the same.

To test our implementation of the quantum heating algorithm, we compared our results with those published by Siebenmorgen et al. (1992). To illustrate the effect of quantum heating, the emission of the 10 and 40 Å grains are compared with the emission obtained under the assumption of thermal equilibrium (Eq. (1)). In Fig 3a, one can nicely see the effect of grain size on the probability density. The probability density broadens with decreasing grain size. In the "classical limit" where the grain reaches thermal equilibrium, the distribution becomes almost a [FORMULA]-function. In Fig. 3b, the main effect of quantum heating, namely the increase of emission shortwards of about [FORMULA]m wavelength, is clearly observable. For the environment used here, [FORMULA] sized grains are almost in thermal equilibrium. As one can see from Fig. 3a, the resulting probability densities are quite similar to the densities obtained by Siebenmorgen et al. (1992).

[FIGURE] Fig. 3. a Probability densities for graphite grains (after Eq. 5) located at a distance of [FORMULA]cm from a B-type star ([FORMULA], [FORMULA]K). The dots represent the results of Siebenmorgen et al. (1992). b Emission of 10 and 40 Å graphite grains due to quantum heating. The "+" signs indicates the emission obtained when assuming thermal equilibrium.

2.1.2. Adaptive temperature grid

For the calculations shown in Fig. 3, we used a fixed temperature grid. But as illustrated in Fig. 4a-f, the probability distribution is not calculated with sufficient accuracy if the grid is too coarse. Not only the shape of the probability distribution may be wrong, also its maximum may be located at the wrong position. A good indicator for the reliability of the solution of Eq. (2) is the conservation of energy. As shown in Table 1, the quantity [FORMULA] decreases with increasing number of temperature grid points. Here, the deviation from the conservation of energy [FORMULA] and the absorbed and emitted energies per unit time are defined by


To calculate the transition matrix and to solve Eq. (2) may take several seconds on a DEC-Alpha 3000/500 workstations if more than 200 grid points are used. Depending on the size of the dust particle and on the spectral energy distribution of the outer radiation field, much more grid points may be required to obtain reliable results (Table 1).

[FIGURE] Fig. 4a-f. Probability densities and spectral deviation [FORMULA] for 40 and 10 Å graphite grains near a B-type star ([FORMULA], [FORMULA]K). Left row: results for a 40 Å grain at [FORMULA] distance using a fixed temperature interval of [FORMULA] K and different numbers of grid points. Middle row: like the left row but using the iterative scheme from Fig. 1 with adaptive temperature grids, see Table 1 for details on temperature grids. Right row: results for a 10 Å grain at [FORMULA] distance using a fixed temperature interval of [FORMULA] K. The equilibrium temperatures (after Eq. (1)) are 605 K for the 40 Å grain, and 1084 K for the 10 Å grain.


Table 1. Conservation of energy due to quantum heating calculation for the plots presented in Fig. 4a-f

If the quantum heating of small grains is considered in 2D radiative transfer calculations it is necessary to apply the procedure described above up to several thousand times during one run of the code. Therefore, even gaining only one second CPU time for each quantum heating calculation results in a significant acceleration of the combined code. From this point of view, it does not seem to be appropriate to use always several hundreds of grid points to ensure reliable results for the quantum heating calculations. Hence, we developed the algorithm shown in Fig. 1.

The basic idea of this algorithm is to calculate the quantum heating via an iterative scheme. For each iteration, a new temperature interval [FORMULA] is chosen, based on an analysis of the probability densities obtained from the previous iteration. The new interval excludes temperatures which are not necessary for the quantum heating calculation (adaptive temperature grid) and is defined by


where [FORMULA]. Additionally, to further increase the accuracy of the calculations, the number of temperature grid points is increased by 50% for every new iteration.

We found that another good indicator to check the convergence of the quantum heating algorithm, apart from the conservation of energy, is the maximum spectral deviation [FORMULA] defined by


where the emitted mean intensity [FORMULA], which is defined by


was multiplied by the factor [FORMULA] to correct for remaining numerically-caused deviations from conservation of energy. Apart from indicating convergence of the algorithm, [FORMULA] provides information about the importance of the quantum heating for the analyzed grain (under the actual radiative conditions). A small spectral deviation indicates that quantum heating can be neglected for all larger grains exposed to the same radiation field. Their emission can be calculated using Eq. (1) whereas high levels of spectral deviation, like those shown in Fig. 4a-f f, are clear indicators for the necessity of using the quantum heating algorithm.

In the current implementation of the algorithm (Fig. 1), we use [FORMULA] (limit for [FORMULA]) and [FORMULA] (limit for [FORMULA]). We also tested our code by using smaller values for [FORMULA] and [FORMULA] but this did not significantly influence the results. The deviations around 1µm in the cases shown in Fig. 4a-f d-f, together with the criteria we just introduced, demonstrate that quantum heating has to be taken into account for these grain sizes.

The effect of iterating can be seen by comparing the results of Fig. 4a-f a with those of Fig. 4a-f b. Already with an adapted 252-point grid, the position and shape of the probability density curve are similar to those obtained by a 400-point grid without iterating (for energy conservation, see Table 1). Additionally, with the 252-point grid (Fig. 4a-f e), almost the same low levels in spectral deviation are reached as for the 400-point grid without iterating (Fig. 4a-f d). For the sake of comparison, the resulting probability density and spectral deviation for a 10 Å grain located in the same environment as the 40 Å grain are presented in Fig. 4a-f c,f. Here, the results are much less sensitive to the number of temperature grid points for a simple reason. For a given temperature grid, the width of the enthalpy bins is proportional to the number of atoms in the grain. Therefore, they are proportional to the cube of the grain radius. This results in a much finer enthalpy grid if the grain size decreases.

Our calculations clearly show that the use of fixed temperature grids, even if they contain about 400 grid points, cannot always ensure reliable results for the quantum heating of small interstellar dust grains. We demonstrated that iterative schemes with adaptive grids are a good approach to obtain a sufficient and controlled accuracy, for all grain sizes and environments of astrophysical interest. Using the algorithms presented above (Fig. 1), enables us to consider the emission from small grains in a 2D radiative transfer code.

2.2. Heating of PAHs

In general, we use the same quantum heating algorithm for the PAHs as for the small dust grains, assuming that the absorption cross section of PAHs is independent of temperature. But as PAHs may consist of only some dozens of carbon atoms, the time between the absorption of two UV/visible photons (Eq. (10)) can be so long that the PAH can cool down via IR emission, before the next high-energy photon impinges. In this case, a much faster method is used to calculate the emission, based on the results of Léger et al. (1989). It was first used in radiative transfer calculations (for spherical geometry) by Siebenmorgen et al. (1992):

First, the mean time [FORMULA] for absorbing a UV/visible photon is determined,


where [FORMULA] denotes the low-frequency cut-off in the UV/visible absorption cross section of the PAHs. Knowing the energy absorbed during this time interval, the peak temperature and corresponding enthalpy [FORMULA] can be calculated. Then, the PAH cools down via IR-emission, according to the energy-time dependence found by Allamandola et al. (1989), and Léger et al. (1989)


Here we assume that this cooling behaviour found for chrysene (24 C-atoms) is also valid for all larger compact PAHs. For each time step [FORMULA] during the cooling (Eq. (12)) conservation of energy is considered,


with [FORMULA] and [FORMULA]. Finally, the emission is again obtained by a superposition of blackbody curves, similar to Eq. (3), with probabilities [FORMULA] given by


In Fig. 5 we compare the emission for a PAH obtained using both kinds of calculation methods, the matrix method of GD89 (Eqs. (2) - (5)) and the cooling curve method (Eqs. (10) - (13)).

[FIGURE] Fig. 5. Emission of a compact PAH consisting of 30 carbon atoms, located at a distance of [FORMULA]cm from a B-type star ([FORMULA], [FORMULA]K). The solid line is for the emission calculated using the method of GD89, whereas the diamonds indicate the results obtained using the cooling curve (Eq. (11)). The absorption cross section of the PAH was taken from Schutte et al. (1993).

We conclude in agreement with Siebenmorgen et al. (1992) that the cooling curve method gives reliable results for large absorption times [FORMULA]. Actually, we demand that [FORMULA] minutes, which is much larger than the typical "cooling time scale" of about 10 seconds (Allamandola et al. 1989).

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998