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8. The Classification of Stellar Spectra
Goals:
1. Gain a working familiarity with the primary
spectral types and luminosity classes of stars
with regard to strengths of specific spectral
lines in stellar spectra.
2. Become familiar with the Boltzmann and Saha
equations and how they are used with stellar
spectra to establish surface temperatures and
luminosities for individual stars.
3. Link a knowledge of stellar temperatures and
luminosities to the location of stars in the
Hertzsprung-Russell diagram.
History
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 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.
A major accomplishment in
subsequent years was the
classification of all stars brighter
than ~10th magnitude photographic
by Annie Jump Cannon from
objective prism plates taken at
observatories operated by the
Harvard College Observatory. The
resulting Henry Draper Catalogue
and Extension (HD and HDE) is still
used today.
Special designations were also developed for stars having peculiar
spectral properties, namely:
c
g
d
pec
k
pq
e
=
=
=
=
=
=
=
ev
v
n
=
=
=
nn
s
=
=
narrow lines (typical of supergiant stars)
giant spectrum
dwarf (main-sequence) spectrum
peculiar spectrum (also abbreviated to p)
interstellar (sharp) Ca II K-line present
nova-like spectrum (broad emission blends) .
emission lines present (the letter "f" was used to designate
emission for O stars)
variable emission
variable spectrum (as, for example, in a pulsating star)
wide and diffuse (nebulous) spectral lines (now attributed
to rapid rotation)
very diffuse spectral lines (for very rapid rotation)
sharp lines (generally resulting from a very low projected
rotational velocity)
Morgan-Keenan (MK) System
The MK classification scheme is a
refinement of 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 (1943, called
the MKK scheme), and was later
revised by Morgan and Keenan (see
Johnson & Morgan 1953, ApJ, 117,
313) to the present MK scheme.
Later modifications include more
recent MK dagger types, denoted
MK†, and spectral subtypes and
designations added by Nolan
William Wilson Morgan
Walborn, Morgan's last graduate
1906-1995
student.
The MK scheme includes the temperature subtypes of the
Harvard scheme, with additions to fill out the temperature
subclasses detectable at the dispersion (~100 Å/mm) used for MK
classification, e.g. B0, B0.5, B1… B9, B9.5, A0, A … etc. The Otype stars were reclassified numerically, the original temperature
ordering becoming O6, O7, O8, O9, O9.5, B0. Luminosity classes
were added using the Saha ionization law, although the original
scheme was an empirical system based upon the observable
spectral features of similar-temperature stars having known
absolute magnitudes (measurable parallaxes and members of
clusters and associations). Temperature subtypes and luminosity
classes were established from specific line ratios determined from
stellar spectra. The luminosity types used are:
Ia = luminous supergiants (0 designates hypergiants)
Ib = less luminous supergiants
II = bright giants
III = giants
IV = subgiants
V = dwarfs (main-sequence stars)
Interpolated luminosity classes are also used, 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 metalpoor 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.
The original MK scheme has been improved upon by 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 O3, O4, and O5, and
a luminosity classification for O-type stars 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 showing anomalous intensities of
Mg, Si, Eu, and the rare earths Sr, Cr, etc.) and Am stars (metallic
line strong for the strength of Ca K), and such stars are now
recognized as being representative of the need for a third
parameter, such as chemical composition, in the system.
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
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:
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



i
N total N neutral  N ion
Nn Nn  N 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 II λ4063 weakens at lower Pe
(supergiants) where presumably most of the elements are in the
same or lower ionization states.
The situation for G8 stars of different luminosity. Note the strong
absorption by CN bands in luminous stars.
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.
A Comment About Spectral Types in the Literature
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 also highly desirable.
The Hertzsprung-Russell Diagram
The original “H-R
Diagrams” were graphs
by Ejnar Hertzsprung
(Denmark) and Henry
Norris Russell
(Princeton) that plotted
the absolute magnitudes
of stars as a function of
their spectral types,
which are closely tied to
temperature. The main
features are well known
to astronomy students.
Methods of Plotting the H-R Diagram
Original Version: MV versus Spectral Type (OBAFGKM)
Open Cluster Version: MV or V versus (B–V)0 (intrinsic colour
index)
Theoretician’s Version: log L versus log Teff (effective
temperature)
Tables correlate spectral type with (B–V)0, log Teff, and bolometric
correction (BC) as functions of luminosity class. Recall that:
Mbol = MV + BC
Mbol() = +4.79
Mbol – Mbol() = –2.5 log (L/L)
log (L/L) = 0.4 [4.79 – MV – BC] = 1.916 – 0.4 MV – 0.4 BC
L = 4πR2σTeff4
log R/R = 8.4817 – 0.2 (MV + BC) – 2 log Teff .
The open cluster
version of the H-R
diagram is usually
referred to as a colourmagnitude diagram.
It can be plotted using
either “observed”
colours and magnitudes
or “unreddened”
colours and magnitudes
in which the effects of
interstellar reddening
and extinction have
been removed.
An “unreddened” colourmagnitude diagram for the
open cluster IC 1590, from
Guetter & Turner 1997, AJ,
113, 2116. The cluster sits in
the H II region NGC 281
and is obscured by varying
amounts of interstellar dust
along the line of sight to
individual stars.
The curved line is the ZAMS
adjusted to the distance of
the cluster, while dotted lines
represent model isochrones
for pre-main-sequence stars
with ages of 106 years
(upper) and 3.5 × 106 years
(lower).
Fundamental Information from H-R Diagrams
The position of a star on the main sequence is a function of the
mass and chemical composition of the star. The abundances of H,
He, and Z (elements of atomic number > 2) in a star directly affect
the main-sequence location of the star.
Such a result is generally referred to as the Vogt–Russell Theorem.
The zero-age main sequence (ZAMS) is the locus of stars at the
initiation of hydrogen burning. If a star burns X% of its mass
before leaving the ZAMS, then the main-sequence lifetime, τms,
can be expressed by:
X M
X M
X 1
1



100 L 100 M 4 100 M 3 M 3
where M and L are the mass and luminosity of the star. The
fraction X depends on the mass of the star for high mass stars,
since it essentially specifies the core mass of the star (which
increases with increasing mass). For the Sun X ≈ 0.10.
 ms 
The original equation can be used to obtain crude estimates for
main-sequence lifetimes for stars of different mass, e.g.:
 ms 25 M Sun  13
1
 3
 ms 1 M Sun  25 16,000
Which gives τms(25 M) ≈ 5 × 105 years. Because X increases with
increasing stellar mass, however, the value of τms(25 M) is
actually larger than the estimate given here, i.e. τms(25 M) ≈ 2 ×
106 years (for an 09 star, roughly).
Another feature of the theoretical H-R diagram is that lines of
constant radius are defined by:
log (L/L) = 2 log R/R + 4 log Teff/Teff()
and run diagonally across the diagram. See following.
Note how lines of constant radius run diagonally across the
diagram from upper left to lower right. They have a slope of –4 in
terms of the logarithmic parameters.
Lines of constant radius in the colour-magnitude version of the
H-R diagram follow lines of constant surface brightness, which
are curved. Numbers denote logarithms of the radii in solar units.
The functional dependence of –BC (negative bolometric
correction) as a function of B–V colour, as established for stars
with well-established dimension.