4. Stellar Parallaxes4. Stellar ParallaxesStellar parallax is the displacement in a star’s position in the sky with respect to the stellar background arising from the orbital motion of the Earth about the Sun. Denoted by the angle π, it is defined to be the angle subtended by 1 A.U. (the semi-major axis of the Earth’s orbit) at the distance of the star. In practice one can observe the annual displacement of a star resulting from Earth’s orbit about the Sun as 2π. In the skinny triangle approximation, 1 A.U. is the chord length subtended at the star by the angle π, measured in radians. In this case, the chord length ≈ the arc length subtended at the star = dπ, where d is the distance to the star. Since π < 1" for all stars, the equation can be written as an equality, i.e. 1 A.U. = dπ, or:

In order to take advantage of trigonometric stellar parallaxes measured in arcseconds, it is useful to define a unit of distance corresponding to that angle. Thus, the parsec is the distance to an object when 1 astronomical unit (A.U.) subtends an angle of 1 arcsecond:

Since all stars should exhibit parallax, measured values (trigonometric parallaxes) are of two types:πrel = relative parallax, is the annual displacement of a star measured relative to its nearby companionsπabs = absolute parallax, is the true parallax of a star, or what is measured for itIn the past, all parallaxes were relative parallaxes, and were adjusted to absolute via:πabs = πrel + correction

The definition of parallax and parsec = distance at which one Astronomical Unit (A.U.) subtends an angle of 1 arcsecond.

Note, by definition:

1 pc = 206265 A.U.

How parallax is measured.

The complication of the parallactic ellipse. In practice all parallaxes are measured using only points near maximum parallax displacement.

The concept of relative parallax is also illustrated.

The estimated frequency of an average 11th magnitude star (upper) and an average 16th magnitude star (lower) as a function of distance (visual magnitudes dashed line, photographic magnitudes solid line). 11th magnitude stars peak for a distance of ~250 pc, corresponding to a correction factor of 0".004. 16th magnitude stars peak for a distance of ~800 pc, corresponding to a correction factor of 0".00125.

The distance to any star or object with a measured absolute parallax is given by:

The relative uncertainty in distance is given by:

Typical corrections to absolute are +0".003 to +0".005 for the old refractor parallaxes, but are more like +0".001 for more recent reflector parallaxes from the U.S. Naval Observatory. Space-based parallaxes from the Hipparcos mission are all absolute parallaxes; they were measured relative to all other stars observed by the satellite. They have typical uncertainties of less than 1 mas (milliarcsecond), i.e. <0".001, although systematic errors of order 0".001 or more are suspected in many cases.

Example: What is the distance to the star Spica (α Virginis), which has a measured parallax according to Hipparcos of πabs = 12.44 ±0.86 mas?

Solution.

The distance to Spica is given by the parallax equation, i.e.

The uncertainty is:

The distance to Spica is 80.39 ±5.56 parsecs.

In general practice, parallax images at an observatory are taken ~6 months apart during evening and morning twilight when the program stars lie close to the meridian. That assures the parallax factor has been maximized (the star lies closest to the extremes of the parallactic ellipse) while the effects of atmospheric refraction are kept to a minimum. Image series (mainly plate series in the past) were continued for several years to allow removal of proper motion from the observed displacements. Other effects, aberration, refraction, etc., are common to all stars in the same field, and are usually negligible.The large proportion of older measured trigonometric parallaxes for stars were obtained photographically using long focal length refractors. Such photographic parallax programs originated with Schlesinger in 1903 with the Yerkes 40-inch refractor. Schlesinger later started the parallax programs at the Allegheny Observatory (1914) and at the Yale Southern Station in Johannesburg, South Africa (1925).

Colleagues of Schlesinger were responsible for continuing the program at Yerkes, and for developing the programs at Van Vleck and Leander McCormick Observatories. Programs at the Greenwich Observatory, Cape Observatory, and Sproul Observatory were initiated in the same mode as Schlesinger’s precepts. Van Maanen’s parallax program on the 60-inch reflector at Mount Wilson Observatory used different techniques. Since the era of the ‘60s, the U.S. Naval Observatory (USNO) has used the 60-inch reflector at Flagstaff to take parallax plates (or, more recently, CCD images) for relatively faint stars. All of the photographic images in the series were measured automatically using the Strand Semi-Automatic Measuring Machine (SAMM) in Washington, D.C. The new series makes use of a CCD detector to record even fainter stars.

The standard sources of older trigonometric parallax data are Jenkins’ General Catalogue of Trigonometric Parallaxes (1952) and its Supplement (1963), published by the Yale Observatory. The catalogues contain ~7000 stars with older measured parallaxes, all reduced to a common system using a detailed intercomparison of internal and external observatory errors known as the Yale precepts. A new, updated, version of the catalogue, which incorporates newer results, was published by Art Upgren of the Yale Observatory in 1995, but without the useful intercomparison of observatory results in the original Jenkins catalogue.Errors in trigonometric parallaxes arise from several different sources, including optical instabilities in the telescopes (particularly severe for the old refractors), errors in measurement, and the problems of establishing a proper reference frame of field stars. (Air pollution etching of Lick Observatory’s refractor may even account for some problems with Lick parallaxes!)

The main sources of error are: (i) internal errors (hopefully random) from all of the above problems, and (ii) external errors (systematic) arising from various sources, such as the use of variable-width sectors at Allegheny for measuring bright stars, and corrections from relative to absolute parallax. According to studies into such problems: (i) the USNO parallaxes appear to be free of external errors, so that their internal error estimates represent the true parallax errors, and (ii) the old Allegheny parallaxes for bright stars contain a magnitude dependent error. Thus, πrel (Allegheny) = πrel (published) + 0".003 (7 – B), for blue magnitude B, such that 3 < B < 7.The systematic error in the Allegheny parallaxes is unfortunate, since all other tests indicate that the Allegheny parallaxes have the smallest internal and external errors of all published, older, refractor parallaxes.

McCormick parallaxes have the largest errors of the older series, and their internal error estimates appear to be unrelated to the true parallax errors for the stars. Currently, the USNO parallaxes are the most accurate available. The total parallax errors, Δπ, for trigonometric parallaxes measured at various observatories (old and new) have been estimated to be:Allegheny ±9 mas Cambridge ±23 masMcCormick ±18 mas Van Vleck ±10 masYale ±15 mas Herstmonceux ±8 masCape ±17 mas Lick ±8 masGreenwich ±18 mas USNO (average) ±4 masMt. Wilson ±20 mas USNO (fine-grained emulsions) ±2 masHipparcos ±1 mas USNO (CCD detector) ±1 mas

Weights are assigned to measured parallaxes from different sources based upon the estimated errors. Thus,

The mean trigonometric parallax resulting from the combination of several estimates is found using:

It has long been recognized that there are problems with the older parallaxes collected in the GCTSP, since its π values are all πrel reduced to the system of the Allegheny Observatory with a correction of +0."003 added to bring them to absolute. The complicated Yale precepts end up making all Observatory corrections negative except those for the Allegheny Observatory. Some of the published πrel values depend on the order of publication for various imaging series. For example, remeasured McCormick parallaxes at one period were in better agreement with Allegheny parallaxes published after the publication of their preliminary value than with their own original estimate! That appears to be a case of trying to match the results from a recognized higher quality institution.

Also complicating the issue is the impossibility of retaining a refractor’s imaging characteristics after it has been disassembled for cleaning. Computers allow more recent parallax series to be reduced using much larger reference frames (12-16 stars or so) than the original 3-4 star reference frame used in Schlesinger’s era. That reduced the random errors in parallax measurements considerably, although uncertain external observatory errors remain. The parallaxes generated using the U.S. Naval Observatory reflector are the best available from ground; tests indicate that they are free of external error. They also make use of faint stars for their reference frames, so their estimated internal errors of ±0".002 to ±0".004 are smaller than the corrections to absolute. The initiation of parallax measurements using CCD detectors has generated parallaxes (1990) with quoted uncertainties of only ±0".001, comparable to those generated by the Hipparcos satellite without concerns about systematic errors.

Calibration of Luminosities Using Parallax Data.Various problems are encountered when using parallax measurements to calibrate luminosities of different objects. One is the tendency for any survey to sample preferentially the most luminous objects of its type when it is limited by apparent magnitude, for example, the determination of MV for B5 V stars. That is known as Malmquist bias, for which suitable corrections exist. The presence of measuring errors in trigonometric parallaxes also makes their use in luminosity calibrations susceptible to bias. Lutz and Kelker (PASP, 85, 573, 1973) first drew attention to the problem by noting the difference between the observed absolute parallax for a star, π0, and its true absolute parallax, πabs. The observed parallax has an associated uncertainty, ±σ, such that π0–σ < πabs < π0+σ. It means that the true distance to the star, d*, must lie in the spherical-volume shell defined by d±Δd, where d = 1/π0. For a uniform spatial distribution of stars, more stars lie beyond the shell and are scattered into it than vice versa.

In essence, the distance to a parallax star is statistically likely to be larger than d given by the value of 1/π0. Corrections can be estimated for the amount of bias based upon the known values for π0 and ±σ. A significant improvement upon the general Lutz/Kelker corrections was made by Hanson (MNRAS, 186, 875, 1979). His technique makes use of proper motion data for the sample of parallax stars in order to estimate their actual space distribution. Typical samples of parallax stars lack the degree of completeness necessary to consider them to be drawn from a uniformly distributed spatial selection of stars. As demonstrated by Hanson, different bias corrections are required for non-uniform statistical samples than is the case for the uniform-density samples considered by Lutz and Kelker. Further studies of this type of bias were published by Lutz (MNRAS, 189, 273, 1979; IAU Colloq., 76, 41, 1983) and Smith (A&A, 188, 233, 1987). But see Smith (MNRAS, 338, 891, 2003) for a dissenting view of such parallax adjustments.

Examples of Hanson’s method as applied to cumulative distributions for the first 500 stars in Luyten’s (1955) LFT catalogue (left), and Luyten’s LHS catalogue (right). The distribution at left follows a N(μ) ~ μ–2.3 distribution.

Hanson’s method applied to parallaxes in the U.S. Naval Observatory catalogues.

A calibration of stellar luminosities as a function of MK spectral type, from Sharpless, Basic Astronomical Data.

An independent calibration of stellar luminosities as a function of MK spectral type, from Blaauw, Basic Astronomical Data.

5. Stellar Radial Velocities5. Stellar Radial VelocitiesRadial velocities of stars are determined from a comparison of the observed wavelengths for identified spectral lines λ with the corresponding rest wavelengths λ0:

where c is the speed of light. Note that there is no need of relativistic corrections for most stellar velocities, although observed velocities do need to be corrected for the rotation (0.47 km/s at the equator) and orbital motion (29.8 km/s) of the Earth in order that they are heliocentric (or, more properly, barycentric). In stellar spectroscopy, the stellar spectrum is usually exposed along with a suitable bracketing emission spectrum from a comparison source, normally Fe arc, hollow cathode Fe comparison source, He-Ar lamp, or Ne lamp, depending upon the wavelength region of interest.

Fe lines are most plentiful in the blue-green spectral region. The rest wavelengths of the comparison lines are well known from laboratory measurements. The velocity of the star is determined by first establishing a relationship between plate of CCD position and wavelength from measurement of the comparison arc lines, and then comparing the similarly-measured positions of the stellar lines (converted to wavelengths) with their rest positions using the above equation. Each spectral line gives an estimate of VR for the star, and the individual values for the “best” lines are normally averaged to obtain a more accurate mean value. With modern detectors, cross-correlation techniques are more typically used. It should be obvious that the procedure is subject to some uncertainties. Early-type stars, for example, have very few spectral lines, and those that are found in their spectra are often quite broad and difficult to measure accurately.

The prominent lines are frequently also blends of several separate lines, and their “effective wavelengths” may need to be calibrated using observations of stars of known velocity. The typical uncertainties in radial velocity measurements for such stars are about ±2 to ±3 km/s. Late-type stars exhibit many more spectral lines, most of which are quite sharp and easy to measure. The effects of line blending are usually better understood, although still a problem. Typical uncertainties in radial velocity measurements for such stars are about ±1 km/s or less. In general, the uncertainties in radial velocity measurements vary as the inverse of the spectrograph dispersion, i.e. σ ~ 1/dispersion.Many measurements of radial velocity for late-type stars are currently done using radial velocity spectrometers (or speedometers), which use a comparison star mask (typically a spectrum of Arcturus or an artificially-generated spectrum) to scan repeatedly across the spectrum of the program star.

The stellar spectrum signal transmitted through a mask is recorded as a function of scanning x-position of the mask using a photomultiplier tube (or other device), and the position of minimum throughput is used to specify the star’s velocity. The spectrometer is calibrated in various ways using a reference source, the sky spectrum (during twilight hours), and the solar spectrum reflected from planetary satellites. Such devices are somewhat limited in their spectral coverage by the mask employed (usually they are restricted to spectral types F or later), but are nevertheless easier to use and faster than photographic plates, reach fainter stars, and have high internal accuracy. Typical uncertainties in single radial velocity measurements with such devices (Griffin-type Radial Velocity Scanners, CORAVEL, etc.) are about ±0.5 km/s. The measurements for early-type stars can often be obtained using CCD detectors, with cross-correlation techniques and synthetically-generated comparison spectra used to optimize the results.

Precision radial velocities are calibrated using HF tubes or night sky lines for calibration purposes, and the wavelength coverage is normally in the near infrared.The major difficulties encountered in determining reliable radial velocities for stars are:(i) Variability. Stellar pulsation and orbital motion about an unseen companion produce temporal variations in the observed radial velocities for some types of stars.(ii) Blending. Double-lined spectroscopic binaries can have blended spectral lines at some or all phases that make it difficult to derive accurate radial velocities for the systems.(iii) Rapid Rotation. Rapidly-rotating stars viewed nearly equator-on have very broad and diffuse spectral lines that are extremely difficult to measure for radial velocities. Line profile fitting software is extremely valuable in such instances.

(iv) Emission Lines. Emission components of spectral lines usually distort the line profiles in such a manner as to make it impossible to obtain the true radial velocities of the stars. The emission produced by a diffuse shell surrounding a star typically produces a P Cygni–type of profile consisting of a sharp blue absorption feature on an broad emission line. The determination of realistic systemic velocities for such stars is an ongoing problem in astronomy.The accuracy of radial velocity measurements depends mainly upon the spectroscopic dispersion used to obtain the stellar spectra. The higher the dispersion (10 Å/mm vs 50 Å/mm), the more accurate is VR. Spectrographs mounted on telescopes are also susceptible to flexure, bad optics, and thermal effects, which can produce systematic effects on the derived radial velocities. Radial velocity spectrometers are typically mounted at the Coudé focus of telescopes to minimize such problems, and produce more reliable results than do Cassegrain spectrographs.

6. Spectral Classification6. Spectral ClassificationStellar spectra were observed and recorded long before the field of spectroscopy had fully developed. Prior to the laboratory identification of spectral lines at specific wavelengths with certain elements, some method of classifying stellar spectra was desirable. The hydrogen Balmer line sequence was recognized from the earliest such studies, and so the earliest classification scheme of any duration was a Harvard scheme developed by Pickering and Fleming based upon photographically-recorded blue-green spectra in the λ3900–5000 Å region that designated stars according to a letter sequence A, B, C, D,... based upon the decreasing strength of the hydrogen Balmer lines visible (type A having the strongest lines). Pickering and Annie Jump Cannon later revised the scheme according to information gleaned from atomic physics, which allowed one to establish element identification and degree of excitation or ionization for specific elements.

The revised scheme eliminated certain redundant types and reordered the types in a logical sequence: O, B, A, F, G, K, M being a sequence in order of decreasing surface temperature.

The arrangement of spectral types in the Harvard scheme was subdivided numerically into finer temperature subtypes, such as B0, B2, B3, B5… A0, A2, A3, A5… etc. Note that not all spectral subtypes were used; types B1 and B4, among others, simply did not exist. The O-type stars were subdivided differently, i.e. Oa, Ob, Oc, Od, and Oe, in order to avoid confusion with an alternate classification scheme of that era.

Special designations were also developed for stars having peculiar spectral properties, namely:c = narrow lines (typical of supergiant stars)g = giant spectrumd = dwarf (main-sequence) spectrumpec = peculiar spectrum (also abbreviated to p)k = interstellar (sharp) Ca II K-line presentpq = nova-like spectrum (broad emission blends)e = emission lines present (the letter “f” was used to designate emission for O stars)ev = variable emissionv = variable spectrum (as, for example, in a pulsating star)n = wide and diffuse (nebulous) spectral lines (now recognized as due to rapid rotation)nn = very diffuse spectral lines (for very rapid rotation)s = sharp lines (generally resulting from a very low projected rotational velocity)

The MK classification scheme is a refinement to the original Harvard system of stellar classification that includes a designation for the star’s luminosity as well as its temperature. The scheme went through an initial stage with a paper by Morgan, Keenan, and Kellerman (called the MKK scheme), but was later revised by Morgan and Keenan to the present MK scheme. Later modifications include more recent MK dagger types, denoted MK†, and spectral subtypes and designations added by Nolan Walborn, a student of Morgan’s (his last!).

William Wilson Morgan1906-1995

The MK scheme includes the basic temperature subtypes of the Harvard scheme, with additions to fill out the temperature subclasses that can be distinguished at the dispersion (~100 Å/mm) used for MK classification, e.g. B0, B0.5, B1,... B9, B9.5, A0, A1,...etc. The O-type stars were reclassified numerically, and the original temperature ordering went O6, O7, O8, O9, O9.5, B0. Walborn has recently extended it to O2.

Nolan Walborn

Luminosity classes were added on the basis of the Saha ionization law, although the scheme was originally established as an empirical system based upon the observable spectral features of similar-temperature stars having known absolute magnitudes (stars with measurable parallaxes and stars in clusters and associations). In all cases, temperature subtypes and luminosity classes are established on the basis of specific line ratios which can be determined from stellar spectra.The luminosity types used are: Ia = luminous supergiants (class 0 is used for supersupergiants = hypergiants)Ib = less luminous supergiantsII = bright giantsIII = giantsIV = subgiantsV = dwarfs (main-sequence stars)

Interpolated luminosity classes are also possible, such as IV-V, II-III, etc. Provisions were also made in the original scheme for subdwarfs (= VI) and white dwarfs (= VIII), but those classes never became popular.Subdwarfs are now recognized as metal-poor stars that are difficult to classify in any case, and white dwarfs are degenerate stars (D) that have since been given their own classification scheme for two recognized sequences — the hydrogen sequence (DO = He II strong, He I or H present, DA = only Balmer lines, no He I or metals present, and DA–F) and the helium-carbon (He-C) sequence (DC = continuous spectrum, no lines deeper than 5%, DB = He I lines, no H or metals present, DZ = metal lines only, no H or He, and DQ = carbon features, molecular or atomic). No luminosity classification is possible for white dwarfs which all have similar surface gravities (log g ≈ 8), but their surface temperatures vary from ~5000K to ~100,000K, specified in a temperature index = 10 θeff = 50,400/Teff.

Thus, Teff = 15,000K, index = 3, Teff ≤ 5500K, index = 9 (can be increased if lots of cool degenerates are found), Teff = 50,000K, index = 1, and Teff > 100,000K, index = 0]. Additional designations are used for: P = polarized magnetic stars, H = magnetic stars with no polarization, Z = peculiar or unclassifiable spectra, and V = optional for VV Ceti variables (pulsating DAs). In the new scheme (Sion et al. 1983, ApJ, 269, 253), D = degenerate star, A,B,C,O,Z & Q = dominant spectroscopic type, A,B,C,O,Z & Q = secondary spectroscopic type (if needed), and 0,1,2,3,...9 = number designating Teff (from colour). The complete designations for white dwarfs can therefore be relatively simple or moderately complex, e.g. DA1, DB3, DBZ4, DXH3, DOZ1.The original MK scheme has been improved upon by the work of Morgan’s students, such as Nolan Walborn, and by students of their students (e.g. Richard Gray).

The result has been the extension of the temperature sequence for O stars to subtypes O2, O3, O4, and O5, and a luminosity classification for O-type stars that is based upon their degree of “Of-ness,” i.e. type f are class I, type (f) are class III, and type ((f)) are classes IV and V. Abt has done a lot of work classifying the Ap stars (A stars display anomalous intensities of Mg, Si, Eu, and the rare earths Sr, Cr, etc.) and Am stars (metallic lines strong for the strength of Ca K), and the stars are now recognized as being representative of the need for a third parameter in the system, such as chemical composition.Spectral classification has always been recognized as an essential means of identifying important properties of stars such as: (i) temperature, (ii) luminosity, (iii) chemical composition, (iv) rotation rate, (v) companions, etc. Spectral classification itself is typically performed with the aid of spectra (originally prism spectra were used, but now grating spectra are used exclusively) of stars at a specific dispersion.

Classical MK dispersion is around 100-120 Å/mm, although some classifications are done at dispersions near ~60 Å/mm. The classification of hot (early-type) stars is done exclusively using blue spectra covering the interval 3900-4900 Å, but redder spectral regions are often used in the classification of red M-type stars and other cool (late-type) objects. Extensions of the original schemes to the ultraviolet and infrared regions have been done in recent years. Standard practice for photographic spectra was to widen the spectra by ~1/3 of the separation of Hγ and Hδ. Since that corresponds to 4340Å – 4101Å = 239Å, at a dispersion of ~100 Å/mm the required widening was ~1/3 (239/100) mm ≈ 0.8 mm.The characteristics of stars of different Teff and luminosity can be explained using the Boltzmann and Saha equations. They specify the energy level populations for atoms and ions, as well as the population of different ionization states for different species of the same element.

The principal temperature sensitive criteria are:O stars: the He II/He I ratio is extremely sensitive to T.B stars: the ratios of Si III/Si IV and Si II/He I are quite sensitive to T. He I lines peak in strength at B2–B3, the H lines strengthen towards A0, as do Mg II λ4481 and Ca II K λ3933 later than B5.A stars: all metal lines increase in strength and H lines decrease in strength as T decreases. Balmer lines reach maximum strength at A0–A2.F stars: the G band (of CH) increases in strength, the H lines decrease in strength, and Ca II H&K increase in strength as T decreases.G stars: Ca II H&K, the G band, Ca I λ4226, etc. all increase in strength as T decreases.K stars: Ca I increases in strength with decreasing T, and the lines of Fe I reach maximum strength.M stars: molecular bands (such as TiO) are prominent.

Anomalies are of different types. The most important represent stars in which the products of nuclear energy generation at the stellar core are seen at the stellar surface. Those include Wolf-Rayet stars (WR), which were first discovered by Wolf and Rayet in 1867. In such stars the products of H-burning via the CN0 cycle (excess N and He — the WN stars), He-burning (excess C and He — the WC stars), and C-burning (excess O — the WO stars) are seen at the surface of OB stars (of ~20 M) that apparently have very high mass loss rates, or to a minor degree in the spectra of the central stars of some planetary nebulae, which are evolved Population II or old disk stars (of ~1 M) in which the outer layers have been stripped away by envelope helium flash events prior to the planetary nebula stage (all such objects are WC or WO types).

Also exhibiting compositional anomalies are carbon stars, which are late-type stars exhibiting excessive amounts of C in their spectra. The older types R, N, the carbon star equivalents of types K and M, have now been replaced by a newer designation C#,#, where the first number is the temperature type and the second is the abundance excess e.g. C1,0; C7,4; etc. (see Keenan’s chapter in Basic Astronomical Data). Such stars are all giants that have lost their outer layers, or stars in which deep convection has dredged up nuclear-processed material from the cores. The S stars are equivalent to M-type stars, but the monoxides TiO, VO, ScO, AlO, etc. of light metals have been replaced by oxides of heavy metals, e.g. ZrO, La0, YO, etc. Comparable objects are the barium stars. The S stars represent objects in which nuclear-processed material by the S (slow) process has been brought to the stellar surface. They therefore represent advanced evolutionary stages of low mass stars.

The peculiar A stars, designated Ap, are stars with anomalously intense lines of rare earth elements (La, Ce, Sm, Eu, Yb, etc.) and Mn, Si, Cr, and Sr, of differing intensity. They are subdivided (Mn, Si, Cr-Eu, Sr groups) on the basis of which spectral lines dominate. They represent different temperature subclasses in the B5 to F0 spectral interval, and appear to be slowly-rotating stars in which gravitational settling and magnetic effects have stratified and enhanced the surface compositions of some elements in the outer layers. The metallic-lined stars, designated Am, are otherwise normal A dwarfs in which the metallic lines appear anomalously strong in comparison with the lines of Ca II H&K. Generally of spectral types A1–A6 on the basis of their H&K lines, their metallic line strengths typically are those of cooler objects. Present notation is to estimate their temperature class using the strength of Ca II H&K, with a separate designation preceded by “m” to denote their temperature class based on the metal line strengths, e.g. AlmA6.

Low metallicity can make normal spectral classification particularly difficult, as is the case for Population II stars of all metallicities (see Keenan’s chapter in Basic Astronomical Data). The need for a third dimension (metallicity) in the classification of such stars has made reliable classification a daunting task.

MK Classification Criteria

New sequence:O B A F G K M L T

Atlas spectra shown on subsequent pages as they appear on photographic spectrograms.

Walborn’s development of luminosity criteria for O-type stars.

Digital Spectra

Physical Basis for Spectroscopic Parallaxes

It is important to consider where stellar spectra originate, namely in the hot, gaseous atmospheres that constitute the outermost thin layers of all stars. Visible light penetrates not very deeply into stellar atmospheres, but goes deep enough to pass through the cool surface layers at the top of the atmospheres into deeper regions where the local temperatures are usually at least twice as high. Stellar atmospheres are therefore not in thermodynamic equilibrium since there is a constant flow of heat energy through them. However, any one point in the atmosphere has local conditions such that the gas atoms are dominated by collisions with one another and the temperature does not vary. Such a condition is termed local thermodynamic equilibrium, or LTE. LTE does not exist when the energy levels in the gas atoms are not collision-dominated, as, for example, in rarefied regimes such as gaseous nebulae and the outer atmospheres of luminous stars.

When LTE exists, the laws of statistical mechanics can be applied to describe the parameters of the local gas. An important formula describing the equilibrium velocity distribution for particles of mass m is Maxwell’s Law for Speeds:

where Nv is the number of particles with velocity v relative to the total number of particles N, T is the temperature, and k is the Boltzmann constant. In such a distribution, the peak, or most probable velocity, occurs at:

while the average velocity occurs at:

Tkmv

v evTk

m

N

N 2222

3

24

m

Tkv

2figureinmp

m

Tkvv

8

figureinavg

The root-mean-square velocity occurs at:

m

Tkuv

3figureinrms

~ v2

~ 1/exp(v2)

For high densities LTE applies and level populations are determined by collisions between atoms. The population of different atomic energy levels is then governed by the energy of colliding atoms, which depend directly on:

referred to as the Boltzmann factor.

In a high temperature gas consisting mostly of atoms and ions, the atomic absorption and emission mechanisms are as summarized below:

TkE

e 2EnergyKinetic

The expected number of atoms in excited energy levels m and n should therefore vary according to the probabilities of populating those energy levels, which for states sm and sn vary as:

Atomic energy levels are degenerate, however, which means that more than one electron can be in a given level provided that its quantum properties (spin, orbital angular momentum) are not shared by other electrons. The result is that the probability factors must also include statistical weights, gm and gn, denoting the possible ways that an energy level can be filled. The resulting probability ratio is then expressed in terms of energy states by:

kTEEkTE

kTE

n

m nm

n

m

ee

e

sP

sP

kTEE

n

mkTE

n

kTEm

n

m nm

n

m

eg

g

eg

eg

EP

EP

The result is an equation, the Boltzmann Equation, that expresses the proportion of atoms in two atomic energy levels as:

where ξmn = Emn is the energy difference between levels m and n. It is generally easier to express the ratio logarithmically as:

where:

kT

n

mkTE

n

m

n

m mnmn eg

ge

g

g

N

N

n

mmn

n

m

g

g

N

Nloglog

8.5039valuebest,/ofunits5040log10 eVKTTk

e

An alternate form of the Boltzmann Equation expresses the proportion of atoms in a specific atomic energy level relative to those in all possible atomic energy levels, namely:

where u(T) is the partition function, which expresses how the various atomic energy levels are populated at a specific temperature T. In most cases, except for high temperatures or low atomic excitation potentials, 99% or more of the atoms are populated only in the ground state, the lowest electronic energy level. A simple approximation is therefore u(T) = g1.

In any event, there are tables that can be consulted to establish u(T) for a particular situation.

Tug

N

N mm

m loglog

Example:For a gas of neutral hydrogen, at what temperature are half of the atoms in the first excited level? The excitation potential for n = 2 is ξ2 = 10.196 eV, g2 = 8, and u(T) ≈ g1 = 2 can be assumed.

Solution:From the Boltzmann Equation:

But stars are rarely this hot, so why are Balmer lines, which originate from the n = 2 level of hydrogen, so strong in stellar spectra?

K000,57

5.0log4log

196.105040So

5.0log

2

8log196.10

5040log

total

2

T

TN

N

The answer lies in the Saha Equation, an extension of the Boltzmann Equation that accounts for ionization of atoms. The equation is formulated exactly like the Boltzmann equation, but includes a term in electron numbers, Ne, to account for the ionization of one species to become an ion and an electron. The resulting equation is:

where Ni+1 is the number of ions in the (i+1)th state, Ni is the number in the ith state, and χi is the ionization energy from the ith state. As usual, it is much easier to express the Saha equation in a logarithmic form:

where Pe = NekT is the electron pressure (dynes/cm2) in a perfect gas.

kTe

ie

ii

iie

h

kTm

TuN

Tu

N

N

23

21

1 22

Tu

TuPT

N

N

i

ieii

i1

1 2log4771.0loglog5.2log

For hydrogen, χi = 13.595 eV, u1(T) ≈ g1 = 2 (as before), and u2(T) = 1 (there is only one quantum state for a free electron). For Pe (or Ne) it is necessary to have a formula that is appropriate for the atmospheres of typical stars. Normally it is possible to find an approximation formula suitable for a typical group of stars, specified in terms of temperature T. Typical results are shown graphically below. Note that hydrogen becomes mostly ionized above T = 10,000 K.

It is now possible to tackle the question of where specific spectral lines should reach maximum strength in stellar spectra as a function of temperature T. The hydrogen Balmer lines are the easiest to address, since they originate from the first excited level (n = 2) and hydrogen has only two ionization states. Thus:

n

n2

nnn

n2

ionneutral

2

total

2

1 NN

NN

NNNN

NN

NN

N

N

Nii

where the numerator involves the Boltzmann equation and the denominator the Saha equation. When the expression is evaluated with a suitable equation for Pe(T), one obtains the results depicted. Note that maximum strength for the hydrogen Balmer lines is expected for T ≈ 10,000 K.

An alternate view of the same results, where the number ratio is plotted logarithmically, which is usually much easier to interpret. Maximum in this diagram is around T = 9500 K.

Subsequent application of the same technique to other atomic species is more involved, since there are more ionization states possible. But it is still possible to establish trends as a function of temperature or spectral type, as indicated below:

Note how the dominant lines of calcium (Ca) vary with temperature according to which ionized species is most abundant.

The spectral sequence is indeed

a temperature sequence ― the

variation of excitation and

ionization potentials for

dominant spectral lines as a function of spectral type.

The hydrogen Balmer lines appear to reach their greatest strength in dwarfs (luminosity class V, also known as main-sequence stars) around spectral types A0–A2, so the effective temperatures of such stars must be ~9500 K.

The effective temperature of the Sun (G2 V) is 5779 K, and its spectrum is dominated by the H and K lines of Ca II (singly ionized calcium).

Other spectral types are treated as lab exercises.

Luminosity Effects in Stellar SpectraThe problem of how to recognize the effects of differing luminosities among stars is addressed in simple fashion, but it helps to recognize the Saha Equation in its simplest possible form, namely:

where φ(T) is the complicated function of temperature that includes all of the other terms. It is also standard practice to express the strength of any spectral line as a signal to noise ratio, η, namely:

Where lλ is the line opacity function and κλ is the continuous opacity function.

TPN

Nei

i

1

l

Balmer Lines of HydrogenThe strength of the hydrogen lines in stellar spectra is governed by the abundance of hydrogen, by the effects of Stark broadening on the hydrogen spectral lines (caused by the charge effects of passing electrons), and by the continuous opacity source in the stellar atmosphere. The line opacity function can therefore be expressed in simple fashion as lλ(H) = f(λ,T) NH Ne , but the continuous opacity function κλ depends directly on the type of star considered.

For hot O-type stars the dominant continuous opacity source in stellar atmospheres is electron scattering, for which κλ ~ Ne. The hydrogen line strength is therefore given by:

which is independent of gravity, g, which determines Ne. The H lines in hot O-type stars are therefore gravity-independent. Recall example presented earlier.

H

e

eH NN

NNTfl

,H

Note how the hydrogen Balmer lines, the prominent series of absorption lines, remain relatively unchanged in appearance for dwarfs (class V) and supergiants (classes Ia and Ib).

For B and A-type stars the dominant continuous opacity source in stellar atmospheres is atomic hydrogen, for which κλ ~ NH. The hydrogen line strength is therefore given by:

which is dependent on gravity, g, which determines Ne. The H lines in B and A-type stars are therefore strongly gravity-dependent through the Stark effect. That can be seen for both B3 stars and A0 stars, for which spectra are illustrated for stars of different luminosity classes, always with supergiants (class Ia) at the top and dwarfs (class V) at the bottom.

e

H

eH NN

NNTfl

,H

Spectra for stars of spectral type B2.

B2 Ia

B2 Ib

B2 III

B2 IV

B2 V

B2 Vh

Spectra for stars of spectral type A0.

A0 Ia

A0 Ib

A0 III

A0 Va

A0 Vb

For cool F, G, and K-type stars, the dominant continuous opacity source in stellar atmospheres is the negative hydrogen ion (H–), for which κλ ~ NH Ne. The hydrogen line strength is therefore given by:

which is independent of gravity, g, specified by Ne. The H lines in F, G, and K-type stars are therefore gravity-independent. That can be seen for F8 stars, for which spectra are illustrated for stars of different luminosity classes. The strongest lines in F8 stars are the H and K lines of Ca II, but the H lines are also relatively strong. They are not affected by changes in gravity, however.

TfNN

NNTfl

eH

eH ,,

H

Spectra for stars of spectral type F8. The hydrogen Balmer lines are indicated as Hγ, Hδ, Hε, H8.

F8 Ia

F8 Ib

F8 III

F8 III-IV

F8 V

Gravity Dependence of Metal Line RatiosIn standard MK spectral classification, it is line ratios, rather than just line strengths, which are important. The method of examining this type of dependence is to consider the following different cases:

Case 1. The lines of an element in one ionization state, where most of the atoms of that element are in the next higher ionization state,

Case 2. The lines of an element in one ionization state, where most of the atoms of that element are in the same ionization state, and

Case 3. The lines of an element in one ionization state, where most of the atoms of that element are in the next lower ionization state.

Ionic broadening plays only a minor role in the strength of metal lines. The strength of most is governed primarily by the abundance of the species responsible for the lines. Here we can use the simple form of the Saha equation to deduce the results.

For Case 1, the line opacity coefficient is lλ ~ Ni, while the abundance of the element can be approximated by N ≈ Ni+1, so:

For Case 2, the line opacity coefficient is lλ ~ Ni ≈ N (the abundance of the element), so is independent of electron pressure.

For Case 3, the line opacity coefficient is lλ ~ Ni+1, while the abundance of the element can be approximated by N ≈ Ni, so:

eee

ii P

T

PN

T

PNNl

1

eee

ii

PP

TN

P

TNNl

11

For early-type stars of type B and A, κλ ~ NH (the abundance of hydrogen), so:

The obvious conclusion from such an analysis is that luminosity-sensitive (Pe–dependent) metal lines for early-type stars are those arising from elements in ionization states other than those where the element is most abundant. Good examples are the O II lines and N II (e.g. λ3995) lines in early B stars, where most of the species are O III and N III, respectively.

3Casefor11

2Casefortermsconstant

1Casefor

eeH

H

eH

e

PPN

l

N

Nl

PN

Pl

At spectral type B5 the spectral lines of Mg II and Si II are stronger in supergiants (low Pe) where presumably most of the elements are in higher ionization states.

For late-type stars of types F, G, and K, κλ ~ NHNe = NHPe (the abundance of H– ions depends upon the abundance of electrons and hydrogen atoms), so:

In this instance, highly-ionized species are very sensitive luminosity indicators in late-type stars, where most of the species are either neutral or in a lower ionization state. Some specific examples can be found in the spectra of late-type stars.

3Casefor11

2Casefor1

1Casefortermsconstant

22eeH

eeH

eH

e

PPN

l

PPN

Nl

PN

Pl

In F, G, and K-type stars, for instance, the lines of Fe I (neutral iron, e.g. λ4045) are not gravity dependent, whereas the lines of Fe II (singly ionized iron, e.g. λ4233) are gravity dependent. In G and K-type stars, the lines of Sr II (singly ionized strontium, e.g. λ4215) are also gravity dependent. Actually, the specific luminosity indicators in MK classification are line ratios. Thus, for late-type stars, the ratio of line strengths for:

is an excellent indicator of luminosity class for the stars (most Sr and Fe is neutral), and is relatively independent of any variations in the abundance of strontium or iron.

Another sensitive indicator of high luminosity is the band head of CN (cyanogen), which is also used for galaxy redshifts.

See examples.

4260IF

4215IISr

e

At spectral type K0 the spectral line of Sr II λ4077 strengthen while the spectral line of Fe I λ4063 weakens at lower Pe (supergiants) where presumably most of the elements are in the lower or the same ionization states, respectively.

The situation for G8 stars of different luminosity. Note the strong absorption by CN bands in luminous stars.

The spectroscopic temperature class criteria for M stars according to Keenan (unpublished).

Textbook Example:The solar atmosphere contains ~500,000 hydrogen atoms for every calcium atom. Why then are the H and K lines of Ca II so much stronger than the Balmer lines of H I in the solar spectrum?

Solution (see textbook):From the Saha and Boltzmann equations:

N2(H I)/Ntotal (H) = 4.90 × 10–9 for hydrogen (the Balmer lines originate from the n =2 level)

N1(Ca II)/Ntotal(Ca) = 0.995 for calcium (the calcium H and K lines originate from the n =1 level, termed resonance lines)

So N2(H I)/N1(Ca II) = 500,000 × 4.90 × 10–9 ≈ 0.0025 = 1/400

So there are ~400 times more atoms in the solar atmosphere capable of producing the Ca II H and K lines than there are atoms capable of producing the H I Balmer lines. The great difference in their lines strengths in the solar spectrum is therefore a temperature effect, not an abundance effect.

The main problems with spectral classification are inexperience and the use of observational data of questionable quality. Novices invariably attempt to match the overall appearance of stellar spectra with those of established spectroscopic standards, and it takes some practice to develop the techniques of using observable line ratios to classify stellar spectra. The best spectral classifiers were invariably taught by Morgan or his students. Automated techniques are somewhat poorly designed for the examination of spectral line ratios in comparison with eye estimates, although such techniques are gradually being introduced successfully. High signal-to-noise ratio spectra (as obtained, for example, using CCD detectors) are highly desirable in the present era.

How would you classify the following stars?

V439 Cyg

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3725 3925 4125 4325 4525 4725 4925 5125

Wavelength

Flu

x

Polaris B

0

0.2

0.4

0.6

0.8

11.2

1.4

1.6

1.8

2

3725 3925 4125 4325 4525 4725 4925 5125

Wavelength

Flu

x