An adequate explanation of spectral line formation requires the ability to track the path that a photon will take in the stellar atmosphere. This is achieved using the equation of radiative
14 Chapter1. Introduction
Figure 1.4: This is the geometry used to solve the transfer equation. It shows the relation between the radiation leaving the surface at an angleθto the normal and the optical depthτν,s along s.
1.5.1
Equation of radiative transfer
The equation of radiative transfer is most easily understood under the assumption of a purely absorbing and thermally radiating gas. Under the assumption that absorption occurs through- out the atmosphere of a star, and ignoring scattering processes, one can model the basic physics of stellar line formation. To begin, assume a plane parallel atmosphere (a good approximation for a star where the depth of the atmosphere is a tiny fraction of the stellar radius). Figure 1.4 outlines the geometry in the atmosphere of a star. Consider radiation emitted from the star, passing through the atmosphere at an angle θ. The intensity of radiation Iν(θ) at a given fre- quencyν passes along the vector s. As per the equation of radiative transfer, some radiation will be absorbed and emitted along path s, but not lost due to scattering processes:
dIν(θ)
ds =−κνIν(θ)+ǫν. (1.10)
Absorption of radiation along path s is taken into account by the opacity coefficient,κν(cm−1), where the emissivity,ǫν, is the amount of radiation being emitted (ergs cm−3s−1Hz−1ster−1). In principle, if dIν/ds > 0 the intensity of radiation increases along the vector s withǫν > κνIν. This produces an emission spectrum. If dIν/ds < 0, meaning thatǫν < κνIν, there is a loss of intensity along the path s, which produces absorption features in a spectrum.
1.5. Spectral line formation 15
of the equation of radiative transfer is
cos(θ)dIν(θ)
dτν
= Iν(θ)−Sν, (1.11) where Sν is the source function and is defined as ǫν/κν (the emissivity divided by the opacity coefficient). Solving this differential equation, one can obtain the surface intensity and find that
Iν(0, θ)=
Z ∞
0
Sν(τν)e−τνsecθd(τνsecθ). (1.12)
This equation can be easily understood. The optical depth along s is d(τνsecθ) and
Sνe−τνsecθd(τνsecθ) is the fraction of radiation that reaches the surface of the star. One can obtain an explicit solution by assuming a linear source function,
Sν =aν+bντcont, (1.13)
where aν and bν are taken as constants (that in principle would depend upon the choice of frequency) andτcont is the continuum optical depth. Further, local thermodynamic equilibrium
(LTE) is assumed so that Sν = Bν, where Bν is the Planck function (black body radiation). The assumption of LTE is a good approximation, regardless of the fact that energy is lost at the surface of the star. This is due to the fact that the mean free path of the radiation is small compared to the size scale of the region under consideration. Definingµ= cosθ, So = aν, and β=bν/aνand solving Equation 1.12 one finds that
Icont(θ)= So(1+βµ)=Scont(τcont =µ). (1.14)
This is a very powerful equation. For a given source function, one has the emerging specific intensity at varying inclinationsµto the vertical. Notice that the emergent specific intensity depends upon the choice of frequency and the angle. As a simple example, consider limb darkening in the Sun. At the centre of the Sun, whereθ = 0, Icont is maximum whereas at the
edge of the Sun (θ =90◦) Icont is minimum.
1.5.2
Spectral Lines
We know that κν = κcont + κline, where the total opacity is the sum of the continuum opacity
and the opacity in the spectral line. Definingην to beκline/κcont =constant one can express the
opacity at a given frequency as,
16 Chapter1. Introduction
Here,ην describes the line profile. Forκline << κcont, the opacity in the line is small compared
to that of the continuum and only weak lines (if any) are observed. Conversely, forκline> κcont
the absorption features are substantially stronger than that of the continuum; stronger lines are observed. Assuming thatην is independent of depth in the atmosphere (the Milne-Eddington approximation), one can rewrite Equation 1.15, recalling thatτν =κνs:
τν = (1+ην)τcont, (1.16)
withην now equallyτline/τcont. Modifying Equation 1.13, the source function now becomes
Sν(τν)=So(1+ βτν 1+ην
). (1.17)
Simply solving Equation 1.12 for this new source function yields a new expression for the surface intensity:
Iν(0)= So(1+ βµ 1+ην
). (1.18)
This expression (which applies locally to a point on the stellar surface) describes the specific intensity that is emergent throughout a spectral line. At line centre, notice that ην >> 1 (ie. τline >> τcont); the intensity in the line decreases to create a larger absorption feature because
of the larger line opacities. Nearer the wings, the opacity drops offand the intensity increases in the line.
Absorption Features
The creation of absorption lines requires a temperature gradient. In stars, this gradient is cre- ated with the cooler atmosphere. Specifically, absorption features are due to bound electrons in a certain energy state absorbing a photon of energy hνand jumping to a higher energy level. The absorption feature will occur at frequencyν, but can only occur if there is an energy level to jump to such that Ef = Ei+hν. However, the physics is more complicated: What broadens a spectral line? What determines line depth?
Photons coming into the atmosphere encounter atoms moving at different velocities. These thermal motions allow photons of slightly different energies to be absorbed and contribute locally to the line. Away from line centre, a photon that is slightly shifted in wavelength (λ±∆λ) will see an optically thin medium for a longer distance in the line-forming region than it would otherwise without the thermal motions of the atom, which means that the photon will not be absorbed immediately. This serves to broaden the spectral line around the wavelength,λ. This is referred to as thermal broadening. Other local broadening effects include microturbulence
1.5. Spectral line formation 17
and magnetic fields. Microturbulence is characterised by small-scale motions in the stellar atmosphere that are characterised by a Gaussian velocity distribution. Analogous to thermal broadening, the microturbulent velocities produce Doppler shifts that broaden spectral lines. The broadening of spectral lines due to magnetic fields can be seen following the discussion of Section 1.4 (see Equation 1.8). Stellar rotation is a non-localised effect that broadens spectral lines. The rotation of the star creates blue and red-shifted components (as portions of the star moves toward and away from the observer, respectively). The spectral lines necessarily broaden proportional to the rotation velocity of the star. This is often the dominant form of spectral line broadening in a star.
The strength of a spectral line is a function of the depth in the atmosphere from which the line was formed. Stronger lines form at shallower depths whereas weaker lines form at deeper depths. This can easily be seen from Equation 1.18. The emergent specific intensity is described completely byην. Forην << 1 (optically thin case) the line depth is proportional to the total number of absorbing atoms. This creates weak spectral lines. Asην approaches 1 or becomes much greater than 1 (optically thick case) the line strengths necessarily increase. As per Equation 1.18, the line centre has the lowest intensity whenην >> 1 where Iν(0) = So = Sν(0) = Bν(0) for LTE. In the wings, the line is still optically thin. The line depth increases withην, which increases the line width causing the line profile to broaden: with increasingην, the profile is first Gaussian, becoming more rectangular and eventually Lorentzian in shape.
1.5.3
Polarisation and the equations of radiative transfer
In the presence of polarised light, the equation of radiative transfer has to be modified from the above discussion. We describe polarised light using the Stokes’ parameters: I, Q, U and
V. In brief, I is the total specific intensity, V is the circular polarisation (see Section 1.4)
and linear polarisation is described by Q and U. Specifically, V = Iright − Ileft, measuring
the intensity of light beams through two circular polarisers, one of which allows only right circularly polarised light and the other only left circularly polarised light. Similarly, linear polarisation is measured using a linear polariser that only allows light linearly polarised at a certain angle to be transmitted, while blocking the orthogonal polarisation. Specifically,
Q= I0−I90(comparing light linearly polarised at 0◦to 90◦), whereas U = I45−I135(comparing
linearly polarised light at 45◦ to 135◦). In general, all of the Stokes quantities are functions of frequency.
18 Chapter1. Introduction
Stenflo, 1994, for a more complete discussion): µdI dτν = ηI(I −Bν)+ηQQ+ηVV µdQ dτν = ηQ(I −Bν)+ηIQ+ρRU µdU dτν = ρRQ+ηIU−ρWV µdV dτν = ηV(I−Bν)+ρWU+ηIV. (1.19) In the above equations, ηQ and ηV are the ratios of the Zeeman opacities, which allow us to follow the linear and circular polarisation in a given light ray. The variables ρR and ρW describe the anomalous dispersion. The equations of radiative transfer are now much more complicated and involve solving four coupled, linear, first order differential equations (see Martin & Wickramasinghe, 1979, for solution methods).