be assumed within a few nucleus radii of the surface once production rates exceeded
20–40 kg s
À1 (Fig. 3.19). Beyond this and in the outer coma, LTE cannot be
assumed and the source function must be found by solving the radiative transfer
equation together with a statistical equilibrium equation that gives the populations of
electrons in each energy level of the molecule and the associated emission rates. The
calculation needs to take into account molecular collisions, electron-molecule collisions, fluorescence, and IR pumping.
Zakharov et al. (2007) compared two approaches to addressing non-LTE. The
escape probability method (Sobolev’s method) converts the global problem of
solving the radiative transfer equation to a local one by assuming that each point
within the gas field is coupled radiatively with the region surrounding it. A local
region can be defined such that a photon has a finite probability of being absorbed
along a line within the region and beyond which there is a vanishing probability of
absorption. For a spherically symmetric medium expanding at constant velocity
(which approximates cometary atmospheres as in Eq. 3.1) the escape probabilities
are given by Litvak and Kuiper (1982) and these are required to compute the average
intensity received from all angles within the local region (see also Rybicki 1984;
Bockelée-Morvan 1987). Unfortunately, realistic comae present radial and azimuthal variations in gas density and velocity and this is especially the case when
studying the innermost coma. In such cases, a Monte Carlo approach is necessary
(Zakharov et al. 2007) at the cost of a major increase in computing time.
In the Monte Carlo approach, a domain is divided into a large number of cells
with constant physical properties (density, velocity, temperature, etc.). It is assumed
that the molecular excitation in the cell is uniform. For each cell, the average
intensity received is approximated by the summation over a random set of rays
from elsewhere which enter the cell and contribute to the radiation field within
it. Knowing this average intensity and, assuming statistical equilibrium, the level
populations can be determined from classical balance equations (Zakharov et al.
2007).
Yamada et al. (2018) have recently published a new open source code for the
investigation of rotational line emission in non-LTE systems which implements a
deterministic method for the solution of the multi-level non-LTE problem in a 1D
spherically symmetric atmosphere and is able to treat optically thick transitions in
both static and expanding atmospheres accurately.
Non-LTE is particularly important for infrared emissions as we shall now illustrate using the H 2 O molecule. Figure 3.10 shows the fine structure of the
ro-vibrational band of H 2
16 O at 2.69 microns. The plot gives the spectral line
intensity, S s (T,p). This calculation (produced using the HITRAN database) assumes
that the ground-state populations are in LTE and follow a Boltzmann distribution at
the specified temperature. The population is given by.
P l ¼
w l
Z T
e
À
E l
kT
ð3:36Þ
where w l is the statistical weight and Z T is the partition function given by
198
3 Gas Emissions Near the Nucleus
20–40 kg s
À1 (Fig. 3.19). Beyond this and in the outer coma, LTE cannot be
assumed and the source function must be found by solving the radiative transfer
equation together with a statistical equilibrium equation that gives the populations of
electrons in each energy level of the molecule and the associated emission rates. The
calculation needs to take into account molecular collisions, electron-molecule collisions, fluorescence, and IR pumping.
Zakharov et al. (2007) compared two approaches to addressing non-LTE. The
escape probability method (Sobolev’s method) converts the global problem of
solving the radiative transfer equation to a local one by assuming that each point
within the gas field is coupled radiatively with the region surrounding it. A local
region can be defined such that a photon has a finite probability of being absorbed
along a line within the region and beyond which there is a vanishing probability of
absorption. For a spherically symmetric medium expanding at constant velocity
(which approximates cometary atmospheres as in Eq. 3.1) the escape probabilities
are given by Litvak and Kuiper (1982) and these are required to compute the average
intensity received from all angles within the local region (see also Rybicki 1984;
Bockelée-Morvan 1987). Unfortunately, realistic comae present radial and azimuthal variations in gas density and velocity and this is especially the case when
studying the innermost coma. In such cases, a Monte Carlo approach is necessary
(Zakharov et al. 2007) at the cost of a major increase in computing time.
In the Monte Carlo approach, a domain is divided into a large number of cells
with constant physical properties (density, velocity, temperature, etc.). It is assumed
that the molecular excitation in the cell is uniform. For each cell, the average
intensity received is approximated by the summation over a random set of rays
from elsewhere which enter the cell and contribute to the radiation field within
it. Knowing this average intensity and, assuming statistical equilibrium, the level
populations can be determined from classical balance equations (Zakharov et al.
2007).
Yamada et al. (2018) have recently published a new open source code for the
investigation of rotational line emission in non-LTE systems which implements a
deterministic method for the solution of the multi-level non-LTE problem in a 1D
spherically symmetric atmosphere and is able to treat optically thick transitions in
both static and expanding atmospheres accurately.
Non-LTE is particularly important for infrared emissions as we shall now illustrate using the H 2 O molecule. Figure 3.10 shows the fine structure of the
ro-vibrational band of H 2
16 O at 2.69 microns. The plot gives the spectral line
intensity, S s (T,p). This calculation (produced using the HITRAN database) assumes
that the ground-state populations are in LTE and follow a Boltzmann distribution at
the specified temperature. The population is given by.
P l ¼
w l
Z T
e
À
E l
kT
ð3:36Þ
where w l is the statistical weight and Z T is the partition function given by
198
3 Gas Emissions Near the Nucleus