Spectral lines

Spectral lines provide powerful diagnostics of the conditions in the emitting region of a celestial source. Normal stars exhibit absorption lines due to decreasing temperature (with altitude) in the photosphere (Fig. 10a) while ejected gas near a star or an active corona can result in emission lines (Fig. 10b). Here we discuss the several types of spectral lines and their measurable characteristics. In the following section, we present the physics of radiation propagation that creates the lines.

Absorption and emission lines

Spectral lines arise from atoms or molecules undergoing transitions between two energy states differing in energy by A E. Such transitions in the hydrogen atom are shown as arrows in Fig. 10.1. If the atom is going from a high (excited) energy state to a lower energy state, the excess energy is emitted as a photon of energy AE = hv. If many atoms do this, many photons with the same energy are emitted giving rise to an emission line (Fig. 1a), provided the photons can emerge without further scatters.

On the other hand, if these atoms are being excited to a higher energy state through the absorption of photons, only those photons of the correct energy, AE = h v will be absorbed. If the absorbing atoms are between us and the source of the original photons (say, a hot star), a deficiency of photons at that frequency, an absorption line, will be observed.

Each type of atom or molecule emits or absorbs radiation at frequencies characteristic of that atom; the observed frequencies therefore indicate the type of atom involved. The sodium doublet (X ~ 589 nm) is one example, and the Ha line of hydrogen at X = 656.2 nm is another.

The emission and absorption processes are a function not only of the kinds of atoms that are present but also of the conditions of temperature and pressure in which the atoms find themselves. For instance, the hotter stars do not show hydrogen absorption because the hydrogen is entirely ionized. Thus the conditions in the stellar atmosphere are directly indicated by the presence or absence of certain lines. See for example our discussion of the Saha equation in Section 9.4.

Origin of spectral lines

Figure 11 shows how the emission and absorption lines arise. A hot incandescent lamp emitting a continuum spectrum illuminates a cool cloud containing sodium (Na) atoms. Three observers analyze the light with a prism; each has a different perspective, and each sees a different spectrum. Each can choose to observe the light emerging from the prism directly by eye (observers A,B) or with the aid of a lens and piece of film (observer C).

Observer A studies the light coming directly from the lamp and sees the continuum spectrum. Observer B studies the light from the cloud and observes the Na doublet in emission. The emission-line spectrum arises from the re-emission of the radiation initially absorbed by the Na atoms. If the gas is sufficiently hot, collisions of the gas atoms will also excite the atoms to produce the lines of interest;

Figure 11.11. Origins of the spectral lines. A continuum source of light illuminates a cloud of sodium (Na) atoms. Three observers, A, B, and C see respectively, the continuum spectrum, emission lines, and the continuum with absorption lines.

the illuminating source need not be present, though it could be the source of the heating. In either case, the characteristic photons are emitted into all directions. Observer C studies the light coming from the direction of the lamp, such that the light has passed through the cloud. The continuum spectrum of the lamp is observed but the sodium atoms have removed many of the photons at the two frequencies of the Na doublet. Thus the doublet is observed in absorption.

In the laboratory there are two classic cases that produce lines. Emission lines are observed from a heated gas such as a Bunsen burner flame, and absorption lines are observed when a cool gas is placed in front of a hot source. The former case is a variant of observer B's situation. The latter is the case of observer C. It is also characteristic of stellar atmospheres which typically exhibit absorption lines.

For the case of observer C, both absorption and emission processes along the path from the lamp to observer C must be taken into account. In each differential layer of gas, radiation from the lamp is absorbed at the particular frequencies characteristic of the gas atoms and is diminished as it does so. The atoms in the layer are also emitting radiation characteristic of its cooler temperature. The net spectrum seen by observer C is obtained by integrating these effects layer by layer through the cloud, for each frequency element of the spectrum. The formalism for this radiative transfer calculation is presented in the next section.

Figure 11.12. Light from a star exhibiting absorption (observer C') and emission lines (B', B''). The view of the stellar disk is blocked for B'' as in a solar eclipse because the chromospheric emission lines seen by B'' are weak compared to the light from the disk. A normal star viewed with no such aid usually exhibits only absorption lines formed in the photosphere, unless large amounts of ejected hot or fluorescing gas are in its vicinity.

Figure 11.12. Light from a star exhibiting absorption (observer C') and emission lines (B', B''). The view of the stellar disk is blocked for B'' as in a solar eclipse because the chromospheric emission lines seen by B'' are weak compared to the light from the disk. A normal star viewed with no such aid usually exhibits only absorption lines formed in the photosphere, unless large amounts of ejected hot or fluorescing gas are in its vicinity.

Stars and nebulae

Radiation from stars exhibits both absorption and emission lines (Fig. 12). When the star is viewed directly, the decreasing temperature with increasing radius in the photosphere results in the production of absorption lines (observer C'). These are known as Fraunhofer lines after the discoverer of such lines in the solar spectrum.

Consider now that a large volume of gas is ejected from the star, possibly because the star is rotating so rapidly (see arrow) that centrifugal force ejects part of the atmosphere in the equatorial regions. Photons emitted from the ejected cloud yield emission lines (observer B'). Of course, the star is so distant that an observer on the earth can not distinguish the different parts of the cloud; all the light (and the spectral lines) appear to come from the same place in the sky. Thus, the emission lines must compete with the continuum in the detected spectrum.

Most stars do not have large quantities of ejected gas and thus exhibit only absorption lines (Fig. 10a). A tangential line of sight through the outer layers of a star's atmosphere (chromosphere) does give rise to emission lines, but they are not strong enough to overcome the light from the stellar disk with its absorption lines at the same frequencies. In this case, the emission lines serve to fill in the absorption lines only slightly. Emission lines from the transparent outer layers of the sun (chromosphere) may be observed by blocking the light from the solar disk, either artificially or with the moon during a solar eclipse (Fig. 12, observer B").

Stars that do in fact exhibit emission lines must have large amounts of gas that are strongly illuminated by the star or are hot in their own right. Examples are centrifugally ejected gas, an accretion disk, an intense stellar wind, and a very active corona. Massive and hence highly luminous stars, called luminous blue variables, can produce major ejections of gas that then exhibit emission lines to distant observers. A portion of the spectrum from such a star, n Carinae, is in Fig. 10b.

Another important example of emission lines in optical astronomy is the Balmer radiation emitted from HII regions (Section 10.2). As noted, the prominent hydrogen Ha emission line gives rise to the red regions in photographs of these nebulae.

Permitted and forbidden lines Emission lines that arise from allowed transitions are called permitted lines. The selection rules of quantum mechanics allow these transitions to occur rapidly. The emitted radiation is electric-dipole radiation. If the atom is in an upper state of a permitted transition, the transition will occur after a very short time, ~10-8 s.

In contrast, the emission of electric-dipole radiation between certain states of the atom is forbidden by the selection rules of quantum mechanics (angular momentum and parity). These transitions in fact can take place, but with a much lower probability of occurrence. The observed lines are called forbidden lines. In such cases, the atom will remain in the upper state for a mean lifetime of order 1 s. The actual mean lifetime depends upon the particular atomic states involved; it is a value between 0.01 s and 100 s for most forbidden transitions. Such a long-lived upper state is called a metastable state.

In laboratory situations, even in an excellent vacuum, collision times between atoms will be much shorter than the ~1-s decay time. Thus the metastable states will be collisionally de-excited long before they get around to freely radiating a photon. Thus one rarely sees forbidden spectral lines in earth laboratories.

In contrast, the densities within a celestial emission nebula are exceptionally low, ~108 m-3, compared to an excellent laboratory vacuum of ~1014 m-3 (10-9 torr), and collisions occur infrequently. Hence, the atoms in space will often decay via forbidden transitions. (They are excited to the upper states by infrequent collisions with electrons.) In some regions such as the Orion nebula, Fig. 1.7, green light is prominently seen; this is due to emission from two forbidden lines of doubly ionized oxygen, O III, at X = 496 nm and X = 501 nm. Forbidden lines are often indicated with brackets, e.g., the latter line would be designated "[OIII] 500.7 nm". The most prominent lines from the Orion nebula are listed in Table 1.

Table 11.1. Prominent emission lines in Orion nebula

Wavelength (nm)a Name/color

Two ultraviolet lines Hp line, blue Green pair Ha line, red Infrared line a Brackets indicate forbidden lines

Spectral lines at non-optical frequencies Spectral lines are studied in all bands from the radio through gamma-ray. In radio astronomy, the study of line emission and the Doppler shifts in frequency of these lines has provided valuable information about the existence of molecules in space (Fig. 13) which are the building blocks of life. The Doppler shifts of "21-cm" spectral lines from the hyperfine (spin flip) transition of hydrogen (Fig. 10.1) reveal the revolution of stars and gas about the center of the Galaxy.

In x-ray astronomy, as noted in the previous section, high-resolution spectra of supernova remnants reveal the heavy elements ejected in the explosion (Fig. 5a). Observations of stellar coronae similarly reveal the elements therein (Fig, 5b). In general, x-ray line studies of Fe (iron) and other elements provide diagnostics of temperatures and densities of the hot (T ~ 107 K) plasmas in the source regions.

Line strengths and shapes

Equivalent width

A typical spectral line will have a profile that may be more or less Gaussian in shape and which can be severely distorted under certain conditions (Fig. 14). The total (integrated) area of an absorption or emission line in the observed spectrum is the measure of its strength or total power. The irregular shape of some lines makes it useful to define a quantity that is a measure of this integrated power relative to the other light from the star.

This quantity is called the equivalent width (EW). The EW is defined as the width (in wavelength or frequency) of the nearby continuum flux that contains the same power (same area on the plot) as the real line, that is, the dark shaded areas in Fig. 14. A large EW means the line is quite pronounced compared to the continuum flux. In

Figure 11.13. Sample radio spectra showing molecular lines from different sources. The objects are (a) late-type carbon star, (b) active molecular cloud core region, (c) a supernova remnant, (d) dark cloud core, (e) diffuse cloud, (f) giant molecular cloud. The observed lines are identified as being specific lines in SiS, CN, etc., so their rest frequencies are known. The abscissa indicates the radial Doppler velocity that is associated with a given observed frequency for the specified line. These plots thus give directly the radial velocities of the emitting clouds. [From B. Turner & M. Ziurys, in Galactic and Extragalactic Radio Astronomy, eds. G. Verschuur and K. Kellermann, 2nd Ed. Springer, 1988, p. 210]

Figure 11.13. Sample radio spectra showing molecular lines from different sources. The objects are (a) late-type carbon star, (b) active molecular cloud core region, (c) a supernova remnant, (d) dark cloud core, (e) diffuse cloud, (f) giant molecular cloud. The observed lines are identified as being specific lines in SiS, CN, etc., so their rest frequencies are known. The abscissa indicates the radial Doppler velocity that is associated with a given observed frequency for the specified line. These plots thus give directly the radial velocities of the emitting clouds. [From B. Turner & M. Ziurys, in Galactic and Extragalactic Radio Astronomy, eds. G. Verschuur and K. Kellermann, 2nd Ed. Springer, 1988, p. 210]

the case of absorption (Fig. 14a,c), this hypothetical (dark-shaded) line has total absorption over the equivalent width. For emission (Fig. 14b,d), the equivalent width can be extremely large if the line is intense and the continuum is very small; it will be infinite if there is no continuum flux!

width width

Figure 11.14. Rough sketches of hypothetical spectra for weak lines (a,b), moderately strong lines (c,d), and intense lines (e,f). The concept of equivalent width EW is illustrated in (a-d). The weak wings due to pressure broadening become prominent for the more intense lines. The emitting cloud can begin to absorb its own radiation if it is sufficiently dense (f).

Figure 11.14. Rough sketches of hypothetical spectra for weak lines (a,b), moderately strong lines (c,d), and intense lines (e,f). The concept of equivalent width EW is illustrated in (a-d). The weak wings due to pressure broadening become prominent for the more intense lines. The emitting cloud can begin to absorb its own radiation if it is sufficiently dense (f).

Damping and thermal profiles

The profile of a spectral line from a sample of unperturbed gas has a shape governed primarily by two factors. First, the atoms in the gas will have a thermal spectrum governed by the Maxwell-Boltzmann (M-B) velocity distribution. The atoms are receding from and approaching the observer; radiation emitted or absorbed by them will be shifted in frequency due to the Doppler shift. This broadens the lines because both red and blue shifts are present. The Doppler shift of the frequency is proportional to the velocity component along the line of sight. The M-B distribution gives the number of atoms at each speed. Thus the line shape is governed by the M-B distribution; this is called thermal Doppler broadening, or simply, thermal broadening.

The M-B distribution is a Gaussian (or normal) distribution of speeds (6.3). The speeds translate directly to a Gaussian distribution for the line shape, that is, the line amplitude ki(v) vs. frequency v. For an emission line, (v - vo)2 n

K1,0

2a 2

(Emission line shape; (11.28)

Doppler broadening)

where k10 is the amplitude at v = v0, where v0 is the central frequency of the line, and where a is the standard deviation (characteristic width) of the Gaussian (6.3). The parameter k1 represents the frequency dependent opacity (m2/kg). We explore the connection between spectral lines and opacity in the next section.

The second factor that broadens the spectral line is known as the damping profile. Classically, this is the damping term of the classical oscillator. Friction or radiation shortens the life of the oscillator and broadens the resonance curve (amplitude vs. frequency). In the quantum-mechanical interpretation, the limited lifetime At of the initial state leads to an uncertainty AE in the energy of that state, according to the Heisenberg uncertainty principle, h

principle)

where At is the time the atom is in the upper state and h is the Planck constant. The longer the atom is in the state, the more precisely its energy can be measured. A large transition probability leads to a short life in the state and a large energy uncertainty. The emitted (or absorbed) photons thus yield a broadened line shape.

The expected damping line profile obeys the classical formula,

K2 Y

where y is the full width at half the maximum height of the line profile and k2 0 is an adjustable amplitude factor. It turns out that Y is proportional to the transition probability (not shown here). A large transition probability, i.e., a shorter lifetime, thus corresponds to a wider profile, or energy uncertainty, as expected.

The line shapes for the two effects (damping and thermal broadening) are shown in Fig. 15 for the case where the width at half maximum of the damping curve is half that of the thermal curve. In stellar atmospheres the central portion of the damping profile is quite narrow. Thus one might expect that the damping would not be important. But the exponential in (28) drives the thermal term strongly toward zero as one moves off center, and the damping curve dominates in the wings.

The combined thermal/damping line profile can be visualized in the following manner. Consider that each moving atom emits photons with the narrow damping profile centered on the Doppler-shifted line frequency v' of that particular atom,

Figure 11.15. Line profiles due separately to the thermal and the damping terms, from (28) and (30) respectively. The opacity kis plotted against frequency offset from the line center (v - v0). Both curves are normalized to unity at v = v0, and the widths at half maximum are set to 1.0 and 0.5 for the thermal and damping curves, respectively. Note how the wings of the damping curve dominate at large offsets from the line center.

Figure 11.15. Line profiles due separately to the thermal and the damping terms, from (28) and (30) respectively. The opacity kis plotted against frequency offset from the line center (v - v0). Both curves are normalized to unity at v = v0, and the widths at half maximum are set to 1.0 and 0.5 for the thermal and damping curves, respectively. Note how the wings of the damping curve dominate at large offsets from the line center.

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