14.4: Doppler Broadening of Spectral Lines

The atoms that make up the gas of the stellar atmosphere are constantly in motion, and this motion shifts the wavelengths, seen by an observer, at which the atoms can absorb radiation. This motion may be only the thermal motion of the gas, or it may include the larger-scale motions of turbulence or rotation. Whatever the combination, the shifting of the rest wavelengths by varying amounts for different populations of atoms will usually result in the observed line's being broadened by an amount significantly greater than the natural width determined by atomic properties.

The shifting of the rest wavelength caused by the motion of the atoms not only produces a change as seen by the observer, but also may expose the atom to a somewhat different radiation field. This will be true if the motion is locally random so that the motion of each atom is uncorrelated with that of its neighbors. However, should the motions be large-scale, then entire collections of atoms will have their rest wavelengths shifted by the same amount with respect to the observer and the star. If these collections of atoms constitute an optically thick ensemble, then the radiation field of the ensemble will be shifted along with the rest wavelength. To atoms within such a "cloud" there will be no effect of the motion on the atoms themselves. It will be as if a "mini-atmosphere" was moving, and no additional photons will be absorbed as a result of the motion. Such motions will not affect the equivalent widths of lines but may change the profiles considerably.

Contrast this with the situation resulting from an atom whose motion is uncorrelated with that of its neighbors. Imagine a line with an arbitrarily sharp atomic absorption coefficient [that is, \(\boldsymbol{S}_\nu=\delta(\nu-\nu_0)\)]. If there were no motion in the atmosphere, the lowest-lying atoms would absorb all the photons at frequency \(v_0\), leaving none to be absorbed by the overlying atoms. Such a line is said to be saturated because the addition of absorbing material will make no change in the line profile or equivalent width. But, allow some motion, and the rest frequency of these atoms is changed slightly from \(\nu_0\). Now these atoms will be capable of absorbing photons at the neighboring frequencies, and the line will appear wider and stronger. Its equivalent width will be increased simply as a result of the Doppler shifts experienced by some atoms. Thus, if the motion consists of collections of atoms that are optically thin, we can expect changes in the line strengths as well as in the profiles. However, if those collections of atoms are large enough to be optically thick, then no change in the equivalent width will occur in spite of marked changes in the line profile. We refer to the motions of the first case as microscopic motions so as to contrast them with the second case of macroscopic motion.

a. Microscopic Doppler Broadening

Again, it is useful to make a further subdivision of the classes of microscopic motions based on the nature of those motions. In the case of thermal motions, we may make plausible assumptions regarding the velocity field of the atoms.

Thermal Doppler Broadening The assumption of LTE from Section 9.1b stated that the particles that make up the gas obeyed Maxwell-Boltzmann statistics appropriate for the local values of temperature and density. For establishing the Saha-Boltzmann ionization and excitation formulas, it was really only necessary that the electrons dominating the collision spectrum exhibit a maxwellian energy spectrum. However, we will now insist that the ions also obey Maxwell-Boltzmann statistics so that we may specify the velocity field for the atoms. With this assumption, we may write \[\frac{d N(v)}{N}=\frac{1}{\sqrt{\pi}} e^{-v^2 / v_0^2} \frac{d v}{v_0}\label{14.3.1}\]

where dN/N is just the fraction of particles having a speed lying between v and v + dv and so it is a probability density function of the particle energy distribution. It is properly normalized since the integrals of both sides of equation \ref{14.3.1} are unity. The second moment of this energy distribution gives \[\left\langle v^2\right\rangle=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} v^2 e^{-\left(v / v_0\right)^2} \frac{d v}{v_0}=\frac{1}{2} v_0^2\label{14.3.2}\]

which we may relate to the kinetic energy of the gas.

Now we wish to pick the speed used in equations \ref{14.3.1}, and \ref{14.3.2} to be the radial or line of sight velocity. Since there is no preferred frame of reference for the random velocities of thermal motion, this choice is as good as any other. However, the mean square velocity \(<\mathrm{v}^2>\) calculated in equation \ref{14.3.2} is then only averaged over line-of-sight or radial motions and thus represents only 1 degree of freedom for the particles of the gas. So the energy associated with that motion is equal to ½kT for a monatomic gas, and \[v_0^2=\frac{2 k T}{m}\label{14.3.3}\]

With the aid of the first-order (classical) Doppler shift, we define the Doppler width of a line in terms of v₀ as \[\frac{\Delta \lambda_d}{\lambda_0}=\frac{\Delta \nu_d}{\nu_0}=\frac{v_0}{c}\label{14.3.4}\]

Using equation \ref{14.3.4}, we may rewrite the particle distribution function for velocity as one for the fraction of atoms capable of absorbing at a frequency shift \(\Delta v\) (or wavelength shift \(\Delta\lambda\)). \[\frac{d N(\Delta \lambda)}{N}=\frac{1}{\sqrt{\pi}} e^{-\left(\Delta \lambda / \Delta \lambda_d\right)^2} \frac{d \lambda}{\Delta \lambda_d}=\frac{1}{\sqrt{\pi}} e^{-\left(\Delta \nu / \Delta \nu_d\right)^2} \frac{d \nu}{\Delta \nu_d}=\frac{d N(\Delta \nu)}{N}\label{14.3.5}\]

Since the atomic line absorption coefficient is basically the probability of an atom's absorbing a photon at a given frequency, that probability should be proportional to the number of atoms capable of absorbing at that frequency. Thus, \[S_\nu d \nu=\frac{A}{\sqrt{\pi}} e^{-\left(\Delta \nu / \Delta \nu_d\right)^2} \frac{d \nu}{\Delta \nu_d}\label{14.3.6}\]

where \(A\) is simply a constant of proportionality. This constant can be related to the Einstein coefficient by equations \ref{14.1.3}, and \ref{14.1.4}, with the result that \[S_\nu d \nu=\frac{h \nu_0 B_{i k}}{4 \pi \sqrt{\pi} \Delta \nu_d} e^{-\left(\Delta \nu / \Delta \nu_d\right)^2} d \nu=\frac{\sqrt{\pi} e^2 f_{i k}}{m_e c \Delta \nu_d} e^{-\left(\Delta \nu / \Delta \nu_d\right)^2} d \nu\label{14.3.7}\]

To get the result on the far right-hand side, we used the relationship between the \(f\) value for a particular transition and the Einstein coefficient given by equation \ref{14.2.21}. This is the expression for the atomic line absorption coefficient for thermal Doppler broadening. It differs significantly from the Lorentz profile of radiation damping by exhibiting much stronger frequency dependence. A spectral line where both broadening mechanisms are present will possess a line core that is dominated by Doppler broadening while the far wings of the line will be dominated by the damping profile as the gaussian profile of the Doppler core rapidly goes to zero.

Microturbulent Broadening In addition to the thermal velocity field, the atoms in the atmospheres of many stars experience motion due to turbulence. Unfortunately, the theory of turbulent flow is insufficiently developed to enable us to make specific predictions concerning the velocity distribution function of the turbulent elements. So, for simplicity, we assume that they also exhibit a maxwellian velocity distribution, but one having a characteristic velocity different from the thermal velocity. Thus, the form of the probability density distribution function for turbulent elements is the same as equation \ref{14.3.1} except that the velocity is the radial velocity of the turbulent cell: \[\frac{d N}{N}=\frac{1}{\sqrt{\pi}} e^{-\left(\upsilon / \upsilon_0\right)^2} \frac{d \upsilon}{\upsilon_0}\label{14.3.8}\]

If there were no other processes to consider, the atomic absorption coefficient for a turbulently broadened line would have the same form as one that is thermally broadened except for a minor change in the interpretation of the Doppler half-width. However, we are interested in the combined effects of thermal and turbulent broadening, and so we consider how this combination may be carried out.

Since equation \ref{14.3.1} represents the fraction of particles with a thermal velocity within a particular range, we may write the probability that a given atom will have a thermal velocity lying between v and v + dv as \[P_{\mathrm{th}}(v)=\frac{N}{v_0 \sqrt{\pi}} e^{-\left(v / v_0\right)^2}\label{14.3.9}\]

The probability that this same atom will reside in a particular turbulent element having a turbulent velocity lying between \(\upsilon+\mathrm{d}\upsilon\) can be obtained, in a similar manner, from equation \ref{14.3.8} and is \[p_{\text {turb }}(\upsilon)=\frac{1}{\upsilon_0 \sqrt{\pi}} e^{-\left(\upsilon / \upsilon_0\right)^2}\label{14.3.10}\]

However, the observer does not regard these velocities as being independent since she or he is interested only in those combinations of velocities that add to produce a particular radial velocity v which yields a Doppler-shifted line. So we must regard the thermal and turbulent velocities to be constrained by \[\mathrm{v}=v+\upsilon\label{14.3.11}\]

Now the joint probability that an atom will have a velocity v lying between \(\mathrm{v}\) and \(\mathrm{v+dv}\) resulting from specific thermal and turbulent velocities \(\mathrm{v}\) and \(\upsilon\), respectively, is given by the product of equations \ref{14.3.9} and \ref{14.3.10}. But we are not interested in just the probability that a thermal velocity \(\mathrm{v}\) and a turbulent velocity \(\upsilon\) will yield an observed velocity \(\mathrm{v}\); rather we are interested in all combinations of \(\mathrm{v}\) and \(\upsilon\) that will yield \(\mathrm{v}\). Thus we must sum the product probability over all combinations of \(\mathrm{v}\) and \(\upsilon\) subject to the constraint given by equation \ref{14.3.11}. With this in mind, we can write the combined probability that a given atom will have combined thermal and turbulent velocities that yield a specific observed radial velocity as \[\begin{aligned}
p(\mathrm{v})=\frac{d N}{d v} & =\int_{-\infty}^{+\infty} P_{\mathrm{th}}(v) p_{\mathrm{turb}}(\mathrm{v}-v) d v \\
& =\frac{N}{\pi v_0 \upsilon_0} \int_{-\infty}^{+\infty} e^{-\left(v / v_0\right)^2-\left[(\mathrm{v}-v) / \upsilon_0\right]^2} d v
\end{aligned}\label{14.3.12}\]

Since the velocities involved in equation \ref{14.3.12} are radial velocities, they may take on both positive and negative values. Thus the range of integration must run from -∞ to +∞. After some algebra, equation \ref{14.3.13} yields the fraction of atoms with a combined velocity v to be \[\frac{d N}{N}=\frac{e^{-\left(\mathrm{v} / \mathrm{v}_0\right)^2}}{\mathrm{v}_0 \sqrt{\pi}} d \mathrm{v}\label{14.3.13}\]

where \[\mathrm{v}_0^2=v_0^2+\upsilon_0^2\label{14.3.14}\]

The similarity of the form of equation \ref{14.3.13} to that of equations \ref{14.3.1}, and \ref{14.3.8} is no accident. The integral in equation \ref{14.3.12} is known as a convolution integral. The combined probability of p(a) and p(b) involves taking the product p(a) × p(b). If, in addition, one has a constraint q(c) = q(a,b), then he must consider all combinations of a and b that yield c and sum over them. That is, one wants the probability of (a₁,b₁) or (a₂,b₂) etc. that yields c. Combining probabilities of A or B involves summing those probabilities. So, in general, if one wishes to find the combined probability of two processes subject to an additional constraint, one "convolves" the two probabilities. It is a general property of convolution integrals where the probability distributions have the same form that the resultant probability will also have the same form with a variance that is just the sum of the variances of the two initial probability distribution functions. Thus the convolution of any two Gaussian distribution functions will itself be a Gaussian distribution function having a variance that is just the sum of the two initial variances. This explains the form of equation \ref{14.3.14}. As a result, we may immediately write the atomic absorption coefficient for the combined effects of thermal and turbulent Doppler broadening as \[S_\nu=\frac{\sqrt{\pi} e^2 f_{i k}}{m_e c \Delta \nu_d} e^{-\left(\Delta \nu / \Delta \nu_d\right)^2}\label{14.3.15}\]

where \[\Delta \nu_d=\frac{\nu_0 \mathrm{v}_0}{c}\label{14.3.16}\]

and \[\mathrm{v}_0^2=\frac{2 k T}{m}+\upsilon_0^2\label{14.3.17}\]

It is now clear why we assumed the turbulent broadening to have a Maxwellian velocity distribution. If this were not the case, the convolution integral would be more complicated. If the turbulent velocity distribution function had the form \[p_{\text {turb }}(\upsilon)=\Phi(\upsilon)\label{14.3.18}\]

then the convolution integral with thermal broadening would become \[p(\mathrm{v})=N \int_{-\infty}^{+\infty} \frac{\Phi(x)}{\sqrt{\pi} v_0} e^{-(x-\mathrm{v})^2 / v_0^2} d x\label{14.3.19}\]

If the function Φ(x) is sufficiently simple, the integral may be expressed in terms of analytic functions. If not, then the integral must be evaluated numerically as part of the larger calculation for finding the line profile.

Combination of Doppler Broadening and Radiation Damping Any spectral line will be subject to the effects of radiation damping or some other intrinsic broadening mechanism as well as the broadening introduced by Doppler motions. So to get a reasonably complete description of the atomic absorption coefficient, we have to convolve the Doppler profile with the classical damping profile given by equation \ref{14.2.19}. However, since the atomic absorption coefficient is expressed in terms of frequency, the constraint on the independent variables of velocity and frequency must contain the Doppler effect of that velocity on the observed frequency. Thus the frequency \(\nu’\) at which the atom will absorb in terms of the rest frequency \(\nu_0\) is \[\nu^{\prime}=\nu_0+\frac{\nu_0 \mathrm{v}}{c}\label{14.3.20}\]

For an atom moving with a line-of-sight velocity v, the atomic absorption for radiation damping is \[S_\nu=\frac{\pi e^2 f_{i k} \Gamma_{i k}}{4 \pi^2 m_e c} \frac{1}{\left(\nu_0+\nu_0 \mathrm{v} / c-\nu\right)^2+\left[\Gamma_{i k} /(4 \pi)\right]^2}\label{14.3.21}\]

This atomic absorption coefficient is essentially the probability that an atom having velocity v will absorb a photon at frequency \(v\). To get the total absorption coefficient, we must multiply by the probability that the atom will have the velocity v [equation \ref{14.3.13}] and sum all possible velocities that can result in an absorption at \(\nu\). Thus, \[S_\nu=\frac{\pi e^2 f_{i k} \Gamma_{i k}}{4 \pi m_e c} \int_{-\infty}^{+\infty} \frac{1}{\mathrm{v}_0 \sqrt{\pi}} \frac{e^{-\left(\mathrm{v} / \mathrm{v}_0\right)^2} d \mathrm{v}}{\left(\nu_0+\nu_0 \mathrm{v} / c-\nu\right)^2+\left[\Gamma_{i k} /(4 \pi)\right]^2}\label{14.3.22}\]

The convolution integral represented by equation \ref{14.3.22} is clearly not a simple one. When one is faced with a difficult integral, it is advisable to change variables so that the integrand is made up of dimensionless quantities. This fact will remove all the physical parameters to the front of the integral, clarifying their role in the result, and reduce the integral to a dimensionless weighting factor. This also facilitates the numerical evaluation of the integral since the relative values of all the parameters of the integrand are clear. With this in mind we introduce the following traditional dimensionless variables: \[\begin{aligned}
u & \equiv \frac{c\left(\nu-\nu_0\right)}{\nu_0 \mathrm{v}_0} \\
y & \equiv \frac{\mathrm{v}}{\mathrm{v}_0} \\
a & \equiv \frac{c \Gamma_{i k}}{4 \pi \nu_0 \mathrm{v}_0}=\frac{\Gamma_{i k}}{4 \pi \Delta v_d}
\end{aligned}\label{14.3.23}\]

Substituting these into equation \ref{14.3.22}, we get \[S_\nu(u)=\frac{e^2 f_{i k} a}{m_e \nu_0 \mathrm{v}_0 \sqrt{\pi}} \int_{-\infty}^{+\infty} \frac{e^{-y^2} d y}{a^2+(u-y)^2}\label{14.3.24}\]

It is common to absorb all the physical parameters on the right-hand side of equation \ref{14.3.24} into a single constant that has the units of an absorption coefficient so that \[S_0 \equiv \frac{\sqrt{\pi} e^2 f_{i k}}{m_e \nu_0 \mathrm{v}_0}\label{14.3.25}\]

The remaining dimensionless function can be written as \[H(a, u) \equiv \frac{a}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-y^2} d y}{a^2+(u-y)^2}\label{14.3.26}\]

This is known as the Voigt function, and it allows us to write the atomic absorption coefficient in the following simple way: \[S_\omega(u)=S_0 H(a, u)\label{14.3.27}\]

For small values of the damping constant (a < 0.2), the Voigt function is near unity at the line center (that is, u = 0) and falls off rapidly for increasing values of │u│. For values of │u│near zero the Voigt function is dominated by the exponential that corresponds to the Doppler core of the line. However, at larger values of │u│, the denominator dominates the value of the integral. This corresponds to the damping wings of the line profile.

Considerable effort has gone into the evaluation of the Voigt function because it plays a central role in the calculation of the atomic line absorption coefficient. One of the earliest attempts involved expressing the Voigt function as \[H(a, u)=\sum_{i=0}^{\infty} a^i \mathrm{H}_i(u)\label{14.3.28}\]

where the functions H_i(u) are known as the Harris functions⁶. More commonly one finds the alternative function \[V(a, u)=\frac{H(a, u)}{\sqrt{\pi}}\label{14.3.29}\]

whose integral over all u is unity. This function is known as the normalized Voigt function. Extensive tables of this function were calculated by D.Hummer⁷ and a reasonably efficient computing scheme has been given by G.Finn and D.Mugglestone⁸. However, with the advent of fast computers emphasis has been put on finding a fast and accurate computational algorithm for the Voigt function. The best to date is that given by J.Humí ek⁹. This has been expanded by McKenna¹⁰ to include functions closely related to the Voigt function. All this effort has made it possible to obtain accurate values for the Voigt function with great speed, making the inclusion of this function in computer codes little more difficult than including trigonometric functions.

b. Macroscopic Doppler Broadening

The fact that each atom was subject to all the broadening mechanisms described above caused most of the problems in calculating the atomic absorption coefficient through the introduction of a convolution integral. This approach assumed that each atom could "see" other atoms subject to the different velocity sources. However, if the turbulent elements were sufficiently large that they themselves were optically thick, then each element would optically behave independently of the others. The line profiles of each would be similar, but shifted relative to the others by an amount determined by the turbulent velocity of the element. Indeed, this would be the case if any motions involving optically thick sections of the atmosphere were present.

The proper approach to this problem involves finding the locally emitted specific intensity, convolving it with the velocity distribution function and integrating the result over the visible surface of the star to obtain the integrated flux. This flux can then be normalized to produce the traditional line profile. However, since the macroscopic motions can affect the structure of the atmosphere, the problem can become exceedingly difficult and solvable only with the aid of large computers. In spite of this, much can be learned about the qualitative behavior of these broadening mechanisms from considering some greatly simplified examples. We discuss just two, the first involves motions of large sections of the atmosphere in a presumably uncorrelated fashion, and the second involves the correlated motion of the entire star.

Broadening by Macroturbulence It would be a mistake to assume that turbulent elements only come in sizes that are either optically thick or thin. However, to gain some insight into the degree to which turbulence can affect a line profile, we divide the phenomena into these two cases. We have already discussed the effects that small turbulent elements have on the resulting atomic line absorption coefficient (i.e., microturbulence), and we have seen that they lead to an increase in value of that parameter for all frequencies. Such is not the case for macroturbulence. The motion of optically thick elements cannot change the value of the atomic line absorption coefficient because the environment of a particular atom concealed within the turbulent element is unaffected by the motion of that element. Thus, each element behaves as a separate "atmosphere", producing its own line profile, which contributes to the stellar profile by an amount proportional to the ratio of the visible area of the element to that of the apparent disk of the star. Thus, the combining (or convolution) of line profiles occurs not on the atomic level as with microturbulent Doppler broadening, but after the radiative transfer has been locally solved to yield a local line profile. This requires that we make assumptions that apply globally to the entire star in order to relate one turbulent element to another.

To demonstrate the nature of this effect, we consider a particularly simple situation where there is no limb-darkening in or out of the line. In addition, we assume that the local line profile is given by a Dirac delta function of frequency and that the macroturbulent motion is purely radial with a velocity \(\forall \mathrm{v}_m\). Under these conditions, zones of constant radial velocity will appear as concentric circles on the apparent disk (see Figure 14.3).

Since the intrinsic line profile is a delta function of frequency, the line profile originating at a ring of constant radial velocity located at an angle θ measured from the center of the disk will be Doppler shifted by an amount \[\Delta \nu=\nu-\nu_0= \pm \frac{v_m \nu_0 \mu}{c}\label{14.3.30}\]

where, as usual, \[\frac{F_c-F_\nu(\text { line })}{F_c}=1-r_\nu \propto 1-\mu^2=1-\left(\frac{\Delta \nu c}{\nu_0 v_m}\right)^2\label{14.3.31}\]

The amount of energy removed from the total continuum flux by the local line absorption will simply be proportional to the area of the differential annulus located at the particular value of µ corresponding to \(\Delta\nu\), Thus, \[\frac{F_c-F_\nu(\text { line })}{F_c}=1-r_\nu \propto\left[1-\mu^2\right]^{1 / 2}=1-\left(\frac{\Delta \nu c}{\nu_0 v_m}\right)^214.3.32\]

Therefore, the line profile would be given by \[r_\nu=1-\left(1-r_0\right)\left[1-\left(\frac{\Delta \nu c}{v_m \nu_0}\right)^2\right]^{1 / 2} \quad \Delta \nu \leq \frac{\nu_0 v_m}{c}\label{14.3.33}\]

This line profile is dish-shaped and is characteristic of this type of mass atmospheric motion. Since the equivalent width remains constant for macroscopic broadening, the central depth of the line will decrease for increasing \(\mathrm{v_m}\).

Figure 14.3 schematically indicates the apparent disks of two idealized stars. Panel (\(a\)) depicts the lines of constant line-of-sight velocity for a macroturbulent stellar atmosphere where the turbulent motion is assumed to be along the stellar radius and of a fixed magnitude \(\mathrm{v_m}\). Panel (\(b\)) also indicates the lines of constant radial velocity for a spherical star that is spinning rigidly.

Clearly a real situation replete with limb-darkening, a velocity dispersion of the turbulent elements, an anisotropic velocity field, along with a spectrum of sizes for the turbulent eddies, would make the problem significantly more difficult. A great deal of work has been done to treat the problem of turbulence in a more complete manner, but the results are neither simple to discuss nor easy to review. D.Mihalas¹¹ gives an introduction and excellent references to this problem.

Broadening by Stellar Rotation As we saw in Chapter 7, rapid rotation of the entire star will lead to significant distortion of the star and a wide variation of the parameters that define a stellar atmosphere over its surface. In such a situation, most of the assumptions we have made for the purpose of modeling the atmosphere no longer apply and recourse must be made to a more numerical approach (see G.Collins¹² and J.Cassinelli¹³). However, as with macroturbulence, some insight may be gained by considering the effects of rotation on the line profile of a slowly rotating star. Such a model is originally due to G.Shajn and O.Struve¹³ and is now commonly referred to as the Struve model.

Consider a uniformly bright spherical star which is rotating as a solid body. Except for the rotation, this is essentially the same model as that used for the discussion of macroturbulence (see Figure 14.3). If we define \(\theta\) and \(\phi\), respectively, to be the polar and azimuthal angles of a spherical coordinate system with its polar axis aligned with the rotation axis of the star, then the velocity toward the observer's line of sight is \[v_r=v_{\mathrm{eq}} \sin \theta \sin \phi \sin i\label{14.3.34}\]

where \(\mathrm{v_{eq}}\) is the equatorial velocity of the star and \(i\) is the angle between the line-of-sight and the rotation axis, called the inclination. An inspection of Figure 14.3 and some geometry leads one to the conclusion that for spherical stars the product \(\sin i\sin\theta\) is constant on the stellar surface along any plane parallel to the meridian plane. Thus, any chord on the apparent disk that is parallel to the central meridian is a locus of constant radial velocity (see Figure 14.3). Any profile formed along this cord will be displaced in frequency by an amount \[\Delta \nu=\frac{v_{\mathrm{eq}} \nu_0}{c}\left(1-\mu^2\right)^{1 / 2} \sin i=a\left(1-\mu^2\right)^{1 / 2}\label{14.3.35}\]

For a sphere of unit radius, the length of the chord is \(2\mu\). If we make the same assumptions about the intrinsic line profile as were made for the case of macroturbulence (i.e., it can be locally represented by a delta function), then the amount of flux removed from the continuum intensity by any profile located on one of these chords will just be proportional to the length of the chord. Therefore, \[\frac{F_c-F_\nu(\text { line })}{F_c}=1-r_\nu \propto 2 \mu=2\left(1-\frac{\Delta \nu^2}{a^2}\right)^{1 / 2}\label{14.3.36}\]

which leads to a profile of the form \[r_\nu=1-\left(1-r_0\right)\left(1-\frac{\Delta \nu^2 c^2}{\nu_0^2 v_{\mathrm{eq}}^2 \sin ^2 i}\right)^{1 / 2} \quad \Delta \nu \leq \frac{\nu_0 v_{\mathrm{eq}} \sin i}{c}\label{14.3.37}\]

Except for the replacement of the turbulent velocity by the equatorial velocity, the rotational profile has the same form as the profile for macroturbulence [equation \ref{14.3.33}]. This points out a fundamental problem of Doppler broadening by mass motions. In general, it is not possible to infer the velocity field from the line profile alone. To be sure, the presence of limb-darkening would affect these two cases differently, as would the introduction of gravity darkening for the case of rotation. But the non-uniqueness remains for the general case, and any determination of the velocity field from the analysis of line profiles is strongly model-dependent and usually relies on some assumed symmetry.

Many of the simplifying assumptions of these models for macroturbulence and rotation can be removed for a modest increase in complexity. In the case of rotation, if the local line profile were not given by a delta function but had an intrinsic shape r'(x) where \[x=\frac{\Delta \nu c}{\nu_0 v_{\mathrm{eq}}}\label{14.3.38}\]

then the observed line profile would be given by the convolution integral \[1-r(x)=\int_{-1}^{+1}\left[1-r^{\prime}(x-y)\right] Q(y) d y\label{14.3.39}\]

Here Q(y) is known as the rotational broadening function which, if limb-darkening is included, is given by A.Unsöld¹⁵ as \[Q(y)=\frac{3}{3+2 \beta}\left[\frac{2}{\pi}\left(1-y^2\right)^{1 / 2}+\frac{\beta}{2}\left(1-y^2\right)\right]\label{14.3.40}\]

The parameter β is the first-order limb-darkening coefficient. Consider the case for β = 0 and that the intrinsic line profile is a delta function. It is clear that equations \ref{14.3.39} and \ref{14.3.40} will yield equation \ref{14.3.37} as long as the integral of Q(y) is normalized to unity. Integration of equation \ref{14.3.40} will satisfy the skeptic that this is indeed the case. It is also clear that the general effects of rotation are not qualitatively very different from those implied by equation \ref{14.3.37}. While quantitative comparison with observation will clearly be affected by such things as the intrinsic line profile and limb-darkening, a truly useful comparison will have to go even further and include the effects of the variation of the atmospheric structure over the surface on the line profile.

While macroturbulence and rotation constitute the most important forms of macroscopic broadening, there are others. The presence of magnetic fields can split atomic lines through the Zeeman effect. In some instances, this can lead to anomalously broad spectral lines and subsequent errors in the abundances derived from these lines. In some instances, the broadening is sufficiently large to allow the estimation of the magnetic field itself. Fortunately, strong magnetic fields appear to be sufficiently rare among normal stellar atmospheres to allow us to ignore their effects most of the time. However, we should be ever mindful of the possibility of their existence and of the effects that they can introduce in the shaping of spectral lines.