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4. Stellar Parallaxes
Stellar 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 companions
πabs = absolute parallax, is the true parallax of a star, or
what is measured for it
In 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 SemiAutomatic 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 mas
McCormick ±18 mas Van Vleck
±10 mas
Yale
±15 mas Herstmonceux
±8 mas
Cape
±17 mas Lick
±8 mas
Greenwich ±18 mas USNO (average)
±4 mas
Mt. Wilson ±20 mas USNO (fine-grained emulsions) ±2 mas
Hipparcos
±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 Velocities
Radial 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 artificiallygenerated 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 Classification
Stellar 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 photographicallyrecorded 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 spectrum
d = dwarf (main-sequence) spectrum
pec = peculiar spectrum (also abbreviated to p)
k = interstellar (sharp) Ca II K-line present
pq = nova-like spectrum (broad emission blends)
e = emission lines present (the letter “f” was used to
designate emission for O stars)
ev = variable emission
v = 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 Morgan
1906-1995
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 supergiants
II = bright giants
III = giants
IV = subgiants
V = 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 (latetype) 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:
OBAFGKMLT
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 collisiondominated, 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:
3
 m  2
Nv
 v e
 4 
N
 2 k T 
2
 mv 2
2k T
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:
vmp   in figure  
while the average velocity occurs at:
vavg  v in figure  
 2k T 


 m 
 8k T 


m
~ v2
~ 1/exp(v2)
The root-mean-square velocity occurs at:
vrms  u in figure  
 3k T 


 m 
In a high temperature gas consisting mostly of atoms and ions, the
atomic absorption and emission mechanisms are as summarized
below:
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:
Kinetic Energy  e
E
2k T
referred to as the Boltzmann factor.
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:
Psm  e Em kT
  En kT  e Em  En  kT
Psn  e
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:
P  Em  g m e

P  En  g n e  E n
 Em kT
kT
gm  Em  En  kT

e
gn
The result is an equation, the Boltzmann Equation, that expresses
the proportion of atoms in two atomic energy levels as:
N m g m  Emn  kT g m mn  kT

e

e
N n gn
gn
where ξmn = Emn is the energy difference between levels m and n. It
is generally easier to express the ratio logarithmically as:
Nm
gm
log
   mn  log
Nn
gn
where:
log 10 e 5040
units of K / eV , best value 5039.8


kT
T
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:
Nm
gm
log
   m  log
N
u T 
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.
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:
N2
5040
8
10.196  log
log

N total
T
2
 log 0.5
5040 10.196  57,000 K
So T 
log 4  log 0.5
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?
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:
2ui 1 T   2 me kT  2   i
N


 e
i
2
N
N e ui T  
h

i 1
3
kT
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:
 2ui 1 T 
N i 1
log i  2.5 log T   i  log Pe  0.4771  log 



N
u
T
 i

where Pe = NekT is the electron pressure (dynes/cm2) in a perfect
gas.
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:
N2
N2
N2 Nn
N2 Nn



N total N neutral  N ion
Nn Nn  N i Nn
1  N i Nn

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 Spectra
The 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:
N i 1
Pe   T 
i
N
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:

l

Where lλ is the line opacity function and κλ is the continuous
opacity function.
Balmer Lines of Hydrogen
The 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:
f  , T  N H N e
H 

 NH

Ne
l
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.
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:
f  , T  N H N e
H 

 Ne

NH
l
which is dependent on gravity, g, which determines Ne. The H
lines in B and A-type stars are therefore strongly gravitydependent 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.
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:
f  , T  N H N e
H 

 f  , T 

NH Ne
l
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.
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 Ratios
In 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:
i 1
N
Pe N Pe
i
l  N 

 Pe
 T 
 T 
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:
i
N
 T  N  T  1
i 1
l  N 


Pe
Pe
Pe
For early-type stars of type B and A, κλ ~ NH (the abundance of
hydrogen), so:
l
Pe


 Pe
 NH
for Case 1
l
N


 constant terms
 NH
l
1
1



  N H Pe Pe
for Case 2
for Case 3
The obvious conclusion from such an analysis is that luminositysensitive (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.
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:

l


Pe
N H Pe
l
N
l
1
 constant terms
1



  N H Pe Pe
1


 2
2
  N H Pe
Pe
for Case 1
for Case 2
for Case 3
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.
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:
Sr II  4215
Fe I  4260
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.
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 signalto-noise ratio spectra (as obtained, for example, using
CCD detectors) are highly desirable in the present era.
How would you classify the following stars?
1.6
V439 Cyg
1.4
1.2
Flux
1
0.8
0.6
0.4
0.2
0
3725
3925
4125
4325
4525
4725
4925
5125
Flux
Wavelength
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
3725
Polaris B
3925
4125
4325
4525
Wavelength
4725
4925
5125