Astron. Astrophys. 332, 732-738 (1998)

2. Theory

The rate of a resonance scattering process where photons of frequency and polarization are absorbed, while photons of frequency and polarization are emitted, is given by

where and are the initial, intermediate, and final atomic states. Eq. (1) is a special case of the Kramers-Heisenberg formula of dispersion (Louisell 1973).

For an atom moving relative to the emitting light source the center of mass of the atom will become time-dependent. Denoting the velocity of the atom relative to the source and the observer by , the effect of a moving center of mass position is equivalent to making the following two substitutions in Eq. (1)

These substitutions give the Doppler shift in the absorbed and emitted light frequencies due to the relative motion of the atom.

The response of an atom to an incoherent light source, containing a distribution of different photon frequencies, is obtained by summing Eq. (1) over all incoming frequencies. If we denote the flux of photons in mode by then due to the fact that the modes are very closely spaced can be replaced by a continuous distribution ( has the units m^-2 s^-1 Hz^-1). Integrating over all incoming frequencies the total transition rate is

In Eq. (3) we have only retained the Doppler shift in the argument of and in the resonance profile, as there small shifts in frequency can become critical. The finite frequency resolution of the photon detector that measures the scattered light, is taken into account by using the density of photon states: . Eq. (3) then becomes

where we have introduced the fine structure constant and denoted the shifted argument of by . Expression (4) gives the probability per unit time that a photon having a frequency within the interval is scattered into the solid angle surrounding .

2.1. LOS integral

Eq. (4) refers to the scattering from a single atom. In any real situation there is allways a large number of scattering atoms present. In what follows we will assume that a photon is only scattered once on its way to the observer (the thin medium approximation). Referring to Fig. 1 the number of atoms at a distance s from the observer within the solid angle equals , where is the number density at point s. The dependence of the photon flux on the distance to the light source is in the absence of absorption , where is the flux at the position of the observer and where we assumed a point source. As the scattered intensity falls off as , the term in the expression for N will cancel. We take the velocities of atoms at to be distributed according to the normalized velocity distribution :

where f is the solution to the appropriate Boltzmann equation for the gas. From f we can also obtain the density n as

If we take the fraction of the atoms in the ground state to be , then by multiplying Eq. (4) by N and and integrating over all velocities and along the LOS, we obtain

I() gives the number of photons counted in a detector of angular opening , area dA, and frequency resolution . It is to be noted that the dipole matrix elements in Eq. (7) contain an implicit s dependence due to the orthogonality relation between the polarization vectors and the photon momentum. Eq. (7) is our main result and one would in general have to resort to numerical calculations to evaluate the integrals. In the next section we will, however, show that for an important special case an analytical solution is possible.

Fig. 1. Definition of the quantities used in deriving the LOS integral.

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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