analysis-and-interpretation-of-astronomical-sp

Analysis and Interpretation of Astronomical Spectra 51

ity class. Caveat: Dwarf or main sequence stars must not be confused with the White
Dwarfs. The latter are extremely dense, burned-out "stellar corpses" (sect. 14.3).

Examples: Sirius A: A1Vm, metal rich main sequence star (Dwarf), spectral class A1
Sirius B: DA2, White Dwarf (or „Degenerate“) spectral class A2
Omikron Andromedae: B6IIIep, Omikron Ceti (Mira): M7IIIe
Kapteyn‘s star: sdM1V

Suffix Sharp lines Prefixes
S Extraordinary sharp lines d Dwarf
c sd Sub Dwarf
Broad lines g Giant
B Normal lines
A Composite spectrum Special classes
comp Q Nova
H- emission lines by B- P Planetary Nebulae
e and O Stars D Dwarf, +additional letter for O, B,
He- and N- emission lines
F by O-stars A spectral class
Metallic emission lines W Wolf-Rayet Star + additional let-
Em
ter for C-, N- or O- lines
K Interstellar absorption S Stars with zirconium oxide ab-
M lines
n / nn Strong metal lines sorption bands
C Carbon stars
Wk Diffuse lines/strongly dif- L, Brown dwarfs
fuse lines T,
weak lines Y Theoretical class for brown

p, pec peculiar spectrum dwarfs <600 K

Sh Shell
V variation in spectrum
Fe, Mg… Excess or deficiency (–) of

the spec. element

Nowadays the special classes P (Planetary Nebulae) and Q (Novae) are barely in use!
The suffixes are not always applied consistently. We often see other versions. In the case of
shell stars e.g. pe, or shell is in use.

Analysis and Interpretation of Astronomical Spectra 52

13.8 The Rough Estimation of the Spectral Class

The rough, one-dimensional estimation of the spectral classes O, B, A, F, G, K, M, is easy
and even feasible for slightly advanced amateurs. The distinctive criteria are striking fea-
tures such as line- or band spectra, as well as absorption- or emission lines, which appear
prominently in certain spectral classes and vice versa are very weak or even completely ab-
sent in others. But the determination of the decimal subclasses and even more, the addi-
tional determination of the luminosity class (second dimension), require well-resolved spec-
tra with a large number of identified lines, as well as deeper theoretical knowledge. Possi-
ble distinctive criterions are eg the intensity ratio or the FWHM values of certain spectral
lines. On this topic exists an extensive literature eg [4]. Here follows just a brief introduc-
tion into the rough, one dimensional estimation of the spectral class. A further support is
the Spectroscopic Atlas for Amateur Astronomers [33]. The following figure shows super-

ifTmoAupnoFdsEeindL[a30n31d].lowly-resolÜvebdethreseicnthirte sSeqpueekncteraolfkthlaesspseecntral classes O - M, as it can be

Hε 3970
Hδ 4101
Hγ 4340
He I 4388
He I 4471
C III 4647/51
Hβ 4861
He I 4922
He I 5016
He I 50486
He II 5411

He I 5876
Na I 5890/95
Telluric O2
Hα 6562
He I 6678

O9.5 Alnitak
25‘000°K

B1 Spica
22‘000°K

B7 Regulus
A1 15‘000°K
A7
F0 Sirius
10‘000°K

Altair
7‘550°K
Adhafera
7‘030°K

F5 Procyon
6330°K

G2 Sonne
5‘700°K

G8 Vindemiatrix
4‘990°K

K1.5 Arcturus
4‘290°K

K5 Alterf
3‘950°K

M1.5 Antares Richard Walker 2010/05©
3‘600°K

M5 Rasalgethi
3‘300°K

KH TiO
Ca ll Mg l 5167-83
Ca I 4227 „Mg Triplet“ TiO TiO TiO
CH 4300
TiO
TiO

Analysis and Interpretation of Astronomical Spectra 53

What is already clearly noticeable here?

– In the upper third of the table (B2-A5), the strong lines of the H-Balmer series, i.e. Hα,
Hβ, Hγ, etc. They appear most pronounced in the class A2 and are weakening from here
towards earlier and later spectral classes.

– In the lower quarter of the table (K5-M5) the eye-catching shaded bands of molecular
absorption spectra, mainly due to titanium oxide (TiO).

– Just underneath the half of the table some spectra (F5-K0), showing only few prominent
features, but charged with a large number of fine metal lines. Striking features here are
only the Na I double line (Fraunhofer D1, 2) and in the “blue” part the impressive Fraun-
hofer lines of Ca II (K + H), gaining strength towards later spectral classes. Fraunhofer H
at λ 3968 starts around the early F-class to overprint the weakening Hε hydrogen line at
λ 3970. In addition, the H-Balmer series is further weakening towards later classes.

– Finally on the top of the table the extremely hot O-class with very few fine lines, mostly
ionised helium (He II) and multiply ionised metals. The H-Balmer series appears here
quite weak, as a result of the extremely high temperatures. The telluric H2O and O2 ab-
sorption bands are reaching high intensities here, because the strongest radiation of the
star takes place in the ultraviolet whereas the telluric absorption bands are located in
the undisturbed domain near the infrared part of the spectrum. By contrast the maximum
radiation of the late spectral classes takes place in the infrared part, enabling the stellar
TiO absorption bands to overprint here the telluric lines.

– In the spectra of hot stars (~ classes from early A – O) the double line of neutral sodium
Na I (Fraunhofer D1,2) must imperatively be of interstellar origin. Neutral sodium Na I has
a very low ionisation energy of just 5.1 eV and can therefore exist only in the atmos-
pheres of relatively cool stars. The wavelengths of the ionised Na II lie already in the ul-
traviolet range and are therefore not detectable by amateur equipment.

13.9 Diagrams for Estimation of the Spectral Class
Here follow two different flow diagrams, for the rough one-dimensional determination of
the spectral class: Sources: Lectures from University of Freiburg i.B. [56] and University of
Jena.

Version 1 Ca II and K? no O, B
A0–A5 <1 Ca II K/Hγ ~1 A5–A9
Applied lines:
– Fraunhofer K (Ca II K 3934Å) >1 F0–F3
– Fraunhofer H (Ca II H 3968Å)
– Fraunhofer G Band (CH molecular Ca II K/H no

4300–4310Å) ~1 G-Band
– Balmer line Hγ, 4340Å
– Mangan Mn 4031/4036Å >1 yes
– Titanoxide Bandhead: TiO 5168Å
G no Mn 4031, 4036 F3–F9
The intensity I of the compared lines (eg
Ca II K/Hγ) is calculated by the Peak in-
tensities (sect. 7.1).

yes

M <1 G–Band/TiO 5168 >1 K

Analysis and Interpretation of Astronomical Spectra 54

Version 2 Balmerlines no H and K no Spectral
visible? visible? type O
Applied lines: Spectral
– Fraunhofer K (Ca II K 3934Å) yes yes type K, M
– Fraunhofer H (Ca II H 3968Å)
– Balmer line Hγ, 4340Å H and K no Spectral
– Fe I, 4325Å visible? type B

yes

K/Hγ<1? no K/Hγ=1? no Spectral type

later A5

yes yes

Spectral Spectral no Hγ/Fe I<1?
type A0-A5 type A5
(4325)

no Hγ/Fe I=1? yes

(4325)

yes

Spectral Spectral Spectral type
type A6-F type G0-G4 later G5

13.10 Additional Criteria for Estimation of the Spectral Class

Here follow some additional indications (see also [3], [4], [33] and [80]).
Course of the Continuum curve:
Already by comparing of non-normalised raw spectra, eg an early B- star against a late K-
star, one can observe that the intensity maximum of the pseudo continuum is significantly
blue-shifted (Wien's Displacement Law). Extremely hot O- and early B- stars radiate mainly
in the ultraviolet (UV), cold M- stars chiefly in the infrared range (IR).
O- Class: singly ionised helium He II, also as emission line. Neutral He I, doubly ionised C III,
N III, O III, triply ionised Si IV. H- Balmer series only very weak. Maximum intensity of the
continuum is in the UV range.
Examples: Alnitak (Zeta Orionis): O9 Ib, Mintaka (Delta Orionis): O9.5 II,

B- Class: Neutral He I in absorption strongest at B2, Fraunhofer K- Line of Ca II becomes
faintly visible, further singly ionised OII, Si II, Mg II. H- Balmer series becomes stronger.
Examples: Spica: B1 III-IV, Regulus B7V, Alpheratz (Alpha Andromedae): B8 IVp Mn Hg Al-
gol (Beta Persei): B8V, as well as all bright stars of the Pleiades.

A- Klasse: H- Balmer lines are strongest at A2, Fraunhofer H+K lines Ca II become stronger,
neutral metal lines become visible, helium lines (He I) disappear.
Examples: Wega: A0 V, Sirius: A1 V m Castor A2 V m, Deneb: A2 Ia Denebola: A3 V,
Altair: A7V,

Analysis and Interpretation of Astronomical Spectra 55

F- Class: H- Balmer lines become weaker, H+K lines Ca II, neutral and singly ionised metal
lines become stronger (Fe I, Fe II, Cr II, Ti II). The striking “line double” of G-Band (CH mo-
lecular) and Hγ line can only be seen here and forms the unmistakable "Brand" of the mid-
dle F-class [33]!
Examples: Caph (Beta Cassiopeiae): F2III-IV, Mirphak (Alpha Persei): F5 Ib,
Polaris: F7 Ib-II, Sadr (Gamma Cygni): F8 Ib, Procyon: F5 IV-V

G- Class: Fraunhofer H+K lines Ca II very strong, H- Balmer lines get further weaker, Fraun-
hofer G- Band becomes stronger as well as many neutral metal lines eg Fe I, Fraunhofer D-
line (Na I).
Examples: Sun: G2V, the brighter component of Alpha Centauri G2V, Mufrid (Eta Bootis):
G0 IV, Capella G5IIIe + G0III (binary star composite spectrum).

K- Class: Is dominated by metal lines, H- Balmer lines get very weak,
Fraunhofer H+K Ca II are still strong, Ca I becomes strong now as well as the molecular
lines CH,CN. By the late K- types first appearance of TiO bands.
Examples: Pollux: K0IIIb, Arcturus: K1.5 III Fe, Hamal (Alpha Arietis): K2 III Ca, Alde-
baran: K5 III

M- Class: Molecular TiO- bands get increasingly dominant, many strong neutral metal lines,
eg Ca I. Maximum intensity of the continuum is in the IR range.
Examples: Mirach (Beta Andromedae): M0 IIIa, Betelgeuse: M1-2 Ia-Iab, Antares:
M1.5 Iab-b, Menkar (Alpha Ceti): M 1.5 IIIa, Scheat, (Beta Pegasi): M3 III Tejat Posterior
(mü Gemini): M3 III Ras Algheti: (Alpha Herculis): M5III

13.11 Appearance of Elements, Ions and Molecules in the Spectra

The following chart shows the relative change in the line intensity of characteristic spectral
lines as a function of the spectral type or the temperature. It was developed 1925 in a dis-
sertation by Cecilia Payne Gaposhkin (1900 – 1979).

This chart is not only of great value for determining the spectral class, but also prevents by
the line identification from large interpretation errors. Thus becomes immediately clear that
the photosphere of the Sun (spectral type G2V) is a few thousand degrees too cold to show
helium He l in a normal (photospheric) solar spectrum. He I is visible only during solar

eclipses as an emission line in the so called flash spectrum, which is produced mainly in
the much hotter solar chromosphere.

Temperature of the Photosphere (K)

50‘000 25‘000 10‘000 8‘000 6‘000 5‘000 4‘000 3‘000

Line Intensity EW H Ca II
He II He I
TiO
Mg II
Si IV Si III Si II Fe II Fe I Ca I

O5 B0 A0 F0 G0 K0 M0 M7

Spectral Type

Analysis and Interpretation of Astronomical Spectra 56

13.12 Effect of the Luminosity Class on the Line Width

Here we see within the same spectral class, how the line width increases with decreasing
luminosity. This happens primarily due to the so-called "pressure broadening", ie the broad-
ening of spectral lines due to increasing gas pressure. The main reason is the increasing
density of the stellar atmosphere with decreasing luminosity, ie the star becomes smaller,
less luminous and denser. Most densely are the atmospheres of the white dwarfs, class VII,
least densely among the Super Giants of class I. This effect is strongest by the H-Balmer
series of class A (below left). Already by the F-class (same wavelength domain, below
right), this effect is barely noticeable. This trend continues towards the later spectral
classes (excerpAtsusfrwomirk[u3n3g]).der Leuchtkraftklassen (LuminosAituysewffierkcut)nagudfeSrpLeekutrcahltkypraAftklasse (Lumin
Hγ 4340.47
DenDeAeb2neAlαab2CIαyagCyg Fe l 4326
RucRhubcahhbaδh CδaCsas Sc ll/Ti ll 4314
CH/Fe ll 4299-13
A5AI5II-IIIVI – IV
VegaVeαgaLyαr Lyr TAFCrELl/Ti ll224290
Fe l 4271-72
A0 VA0 V Zr ll 4258
Sc ll 4247
Fe ll 4231-33
Sr ll 4215.52
FFeel/lVl/lCl r4l2406266
Y ll/V ll/Fe ll 4177-79
FeFlel/Tlli4ll643147.62-73
ZFCFrerelllll44l41146446193279/1.99
HFSCFeiδerllll/4ll4lll141440525118881./83783/.40822
FSFeer llll44005875470.71
Fe l 4064
FFee lll44054250/23
MZTFFMTrieeinglllllll3444l4l900445092074037420811/01.-23576
TCFeialll/II4Y3l4l 943694883.74
Fe ll 4416.8
TCiallI4I 339953.30.466
Fe ll 4384-85
Fe ll 4366.17
Fe ll 4352
Hγ 4340.47
Sc ll/Ti ll 4314
Fe ll 4303.2
Fe l 4271-72

Fe ll 4233.17
Ca I 4226.73

Fe ll 4178.9
Fe l 4173.1

Si ll 4128/30
Hδ 4101.74
Sr ll 4077.71
Fe l 4067.6
Fe l 4045.82

Zr ll/Ti ll 4024/28
Fe l 4002/05

Hε 3970.07

Ca II 3933.66
Mirphak α Per
F2 Ib Ca I 4226.73
Ti ll 4154
Mirphak α Per
Mn l 4055
FC2aplbh β Cas
CaphF2β ICIIa–sIV

F2lll-lV

Porrima γ Vir
Porrima Fγ0VVir

F0 V

RuDDcVeehennbgeeaabbh
Vega

©Richard Walker 2010/05

13.13 Spectral Class and B-V Colour-Index

The following table shows the assignment of the unreddened star colours to the spectral
classes, according to Fitzgerald [421]. The figures, expressed in the photometric BV colour-
index, show here the index spread from the main sequence stars (V), up to the super giants
(I a), put here in parentheses. For the other luminosity classes see [421].

O5 –0.32 A0 –0.01 (+0.02) F8 +0.53 (0.55) K4 +1.00 (1.50)
O6 –0.32 A1 +0.02 (0.03) F9 +0.56 K5 +1.15 (1.60)
O7 –0.32 A2 +0.05 (0.05) G0 +0.60 (0.82) K7 +1.33 (1.62)
O8 –0.31 A3 +0.08 (0.06) G1 +0.62 (0.85) K9 +1.37 (1.64)
O9 –0.31 (–0.28) A4 +0.12 (0.08) G2 +0.63 (0.88) M0 +1.37 (1.65)
O9.5 –0.31 (–0.27) A5 +0.15 (0.10 G3 +0.65 (0.92) M1 +1.47 (1.65)
B0 –0.30 (–0.24) A7 +0.20 (0.13) G4 +0.66 M2 +1.47 (1.65)
B1 –0.26 (–0.19) A8 +0.27 (0.14) G5 +0.68 (1.00) M3 +1.47 (1.67)
B2 –0.24 (–0.17) A9 +0.30 (0.14) G6 +0.72 (1.04) M4 +1.52 (1.75)
B3 –0.20 (–0.13) F0 +0.32 (0.15) G7 +0.73 (1.10) M5 +1.61
B4 –0.18 (–0.11) F1 +0.34 (0.16) G8 +0.74 (1.14) M6 +1.64
B5 –0.16 (–0.09) F2 +0.35 (0.18) G9 +0.76 (1.16) M7 +1.68
B6 –0.14 (–0.07) F3 +0.41 K0 +0.81 (1.18) M8 +1.77
B7 –0.13 (–0.04) F4 +0.42 K1 +0.86 (1.20)
B8 –0.11 (–0.01) F5 +0.45 (0.26) K2 +0.92 (1.23)
B9 –0.07 (+0.00) F6 +0.48 K3 +0.95 (1.42)
B9.5 –0.04 (+0.01) F7 +0.50

Analysis and Interpretation of Astronomical Spectra 57

14 The Hertzsprung - Russell Diagram (HRD)

14.1 Introduction to the Basic Version

The HRD was developed in 1913 by Henry Russell, based on the work by Ejnar
Hertzsprung. It is probably the most fundamental and powerful illustrating tool in astro-
physics. On the topic of stellar evolution and HRD an extensive literature exists, which must
be studied anyway by the ambitious amateur. Here it is illustrated, how a wealth of informa-
tion can be gained about a star just with the help of the spectral class and the HRD. The fol-
lowing figure shows the basic version of the HRD with the luminosity (compared to the
sun), plotted against the spectral type. The luminosity classes Ia – VII are marked by lines
within the diagram. Further visible is the Main Sequence, identical to the line for luminosity
class V, and the two branches, where the Red Giants and White Dwarfs are gathering.

Luminosity compared to the Sun 106

IIab

105

Iab
104 Ib

II

103

102 III Red
Giant Branch
101 Main IV
Sequence
V
100 Sun

White Dwarfs VI
10-1 VII

10-2 O5 B0 B5 A0 A5 F0 F5 G0 G5 K0 K5 M0 M5

Spectral Type

The spectral class unambiguously determines the position of a star inside the diagram and
vice versa. The position of the sun, with the classification G2V is already marked (yellow
disk). With this diagram the luminosity of the spectral class, in comparison to the sun, can
already be determined – in the case of the sun
In the following it will be demonstrated, how further parameters of the star can be deter-
mined, just by modification of the HRD-scales.

Analysis and Interpretation of Astronomical Spectra 58

14.2 The Absolute Magnitude and Photospheric Temperature of the Star
The following version of the HRD shows on the horizontal axis – at the upper edge of the
diagram – the corresponding temperature of the stellar atmosphere, ie the photosphere of
the star, where the visible light is produced. The position of the Sun (G2V) is here also
marked with a yellow disc. At the top of the chart the so called effective temperature of
about 5,700 ° K can be read, on the left the absolute brightness (Absolute Magnitude) with
ca. 4M5. This corresponds to the apparent brightness, which a star generates in a normal-
ised distance of 10 parsecs, or some 32.6 light-years.

STELLAR TEMPERATURE 3,500K 2,400K
50,000K 25,000K 10,000K 7,500K 6,000K 4,900K

-8 Super Giants - Ia
68 Cygni Saiph Rigel Deneb
Betelgeuse
-6 Alnitak Super Giants - Ib Antares
Gacrux
-4 Adhara Mirfak Polaris

Spica Bright Giants - II

-2 Achernar

0 Regulus Algol Giants - III Capella Pollux AldebaranMira
Arcturus
Vega
2 SCiarisutsorA
ABSOLUTE MAGNITUDE
Altair Subgiants - IV
Procyon A
4 Sun

6 α Centauri B

8 61 Cygni A

61 Cygni B

10

12 Sirius B

14 Procyon B

16 05 05 05 05 Proxima Centauri
35 B
O AF G 0 5 05
SPECTRAL CLASS KM

Redrawn and supplemented following a graphics from: www.bdaugherty.tripod.com

The chart is further populated with a variety of common stars. The sun, along with Sirius,
Vega, Regulus, Spica, etc. is still a Dwarf star on the Main Sequence. Arcturus, Aldebaran,
Capella, Pollux, etc. have already left the main sequence and shining now on the Giant
Branch with the Luminosity Class III, Betelgeuse, Polaris, Rigel, Deneb, in the range of the
giants, with their respective classes Ia-Ib. On the branch of the White Dwarfs we see the
companion stars of Sirius and Procyon.

Analysis and Interpretation of Astronomical Spectra 59

Luminosity compared to the Sun14.3 The Evolution of the Sun in the HRD

The following simplified short description is based on [1]. It demonstrates how the spectral
class allows the determination of the stellar state of development. In the diagram below,
the evolutionary path of the sun is shown. By the contraction of a gas and dust cloud, at
first a protostar is formed, which subsequently moves within some million years on the
Main Sequence. Here, it stabilises at first the luminosity by about 70% of the today's value.
Within the next 9-10 billion years the luminosity increases to over 180%, whereby the
spectral class G2, eg the photospheric, or effective temperature, remains more of less con-
stant [1]. During this period as a Dwarf- or Main Sequence star, hydrogen is fused into he-
lium.

Towards the end of this stage the hydrogen is burning in a growing shell around the helium
core. The sun becomes now unstable and expands to a Red Giant of the M-class (luminosity
class ca. II). It moves now on the Red Giant Branch (RGB) to the top right of the HRD, where
after 12 billion years it comes to the ignition of the helium nucleus, the so called Helium
Flash. The photosphere of the Red Giant expands now almost to the Earth's orbit. By this
huge expansion the gravity acceleration within the stellar photosphere is reduced dramati-
cally and the star loses therefore at this stage about 30% of its mass [1].

Subsequently the giant moves with merging helium core on the Horizontal Branch (HB) to
the intermediate stage of a Yellow Giant of the K-class (lasting about 110M years) and then
via the reverse loop of the Asymptotic Giant Branch (AGB) to the upper left edge of the HRD
(details see [33]). The experts seem still be divided whether the sun is big enough to push
off at this stage the remaining shell as a visible Planetary Nebula [1], [2]. Assured however
is the final shrinkage process to an extremely dense White Dwarf of about Earth's size. Af-
ter further cooling, and a down move on the branch of the White Dwarfs, the sun will finally
get invisible as a Black Dwarf, and disappear from the HRD. During its live the Sun will pass
through a large part of the spectral classes, but with very different luminosities.

106

Ia
105 Planetary Nebula Iab

104 Ib

II AGB

103

102 III Giant Branch
Sun
101 today Proto Star

100

White Dwarf

10-1

10-2 O5 B0 B5 A0 A5 F0 F5 G0 G5 K0 K5 M0 M5

Spectral Type

Analysis and Interpretation of Astronomical Spectra 60

14.4 The Evolution of Massive Stars
In the next subsection it will be shown how the length of stay on the main sequence is
dramatically reduced with increasing stellar mass. Highly complex nuclear processes cause
complex pendulum like movements in the upper part of the HRD, showing various stages of
variables (more details see [33]). During this period also many of the heavy elements in the
periodic table are generated. Massive Stars about >8–10 solar masses, will not end as
White Dwarfs, but explode as Supernovae [2]. Depending on the mass of the star it finally
remains a Neutron Star, if the rest of the stellar mass is not greater than about 1.5–3 solar
masses (TOV limit: Tolman-Oppenheimer-Volkoff). Above this limit, eg stars with initial
>15–20 solar masses, will end in a Black Hole.

14.5 The Relation between Stellar Mass and Life Expectancy
The effect of the initial stellar mass is crucial for its further life. First, it decides which place
it occupies on the Main Sequence. The larger the mass the more left in the HRD or "earlier"
in respect of the spectral classification. Second, it has a dramatic impact on his entire life
expectancy, and the somewhat shorter period of time that he will spend on the Main Se-
quence. This ranges from roughly some million years for the early O- types up to >100 bil-
lion years for Red Dwarf stars of the M class. This is due to the fact that with increasing
mass the stars consume their "fuel" over proportionally faster. This relationship becomes
evident in the following table for Dwarf Stars on the Main Sequence (luminosity class V),
together with other parameters of interest. The values are taken from [53]. Mass, radius
and luminosity are given in relation to the values of the Sun ( ).

Spectral Mass Stay on Main Temperature Radius Luminosity
class, Main Sequence [Y] stellar atmos-
Sequence phere 9-15 90‘000 –800‘000
20 – 60 10 – 1M
O 3 – 18 400 – 10M >25 – 30‘000 K
B 2–3 3bn – 440M
A 1.1 – 1.6 7bn – 3M 10‘500–30‘000 K 3.0–8.4 95 – 52‘000
F 0.9–1.05 15bn – 8bn
G 0.6–0.8 7‘500 – 10‘000 K 1.7–2.7 8 – 55
K 0.08–0.5 >20bn
M 6‘000 – 7‘200 K 1.2–1.6 2.0 – 6.5

5‘500 – 6‘000 K 0.85–1.1 0.66 – 1.5

4‘000 – 5‘250 K 0.65–0.80 0.10 – 0.42

2‘600 – 3‘850 K 0.17–0.63 0.001 – 0.08

The length of stay of the K- and M-class Dwarf Stars differs, considerably depending on the
source. It has anyway rather theoretical importance, since these stars have a significantly
longer life expectancy than the current age of the universe from estimated 13.7bn years!

Analysis and Interpretation of Astronomical Spectra 61

14.6 Age Estimation of Star Clusters

The relationship between the spectral class and the according time, spent on the Main Se-
quence, allows the age estimation of star clusters – this under the assumption that such
clusters are formed within approximately the same period from a gas and dust cloud. If the
spectral classes of the cluster stars are transferred to the HRD, it gives the following inter-
esting picture: The older the cluster, the more right in the diagram (ie, "later") the distribu-
tion of stars turns off from the Main Sequence up to the realm of giants and Super Giants
(so called Turn off Point). M67 belongs with more than 3 billion years to the oldest open
clusters, ie the O-, B- and A-, as well as the early F-types have already left the Main Se-
quence, as shown on the chart. However all the bright stars of the Pleiades (M45), still be-

Alter eines Sternhaufenslong to the middle to late part of the B-class. This cluster must therefore necessarily be

younger than M67 (about 100M years). One can also say that the Main Sequence "burns
off" with increasing age of the cluster like a candle from top to down.

Source: [50] Lecture astrophysics, MPI

The horizontal axis of the HRD is divided here, instead of spectral types, with the equivalent
values of the Color-Magnitude Diagram (CMD). For the assignment see the according table
in sect. 13.13. This photometrically determined B – V color index is the brightness differ-
ence of the object spectrum (magnitudes) between the blue range (at 4,400 Å) and the
"visual" range at 5,500 Å (green). The difference = 0 corresponds to the spectral class A0
(standard star Vega). Earlier classes O, B have negative values, later classes are positive.
For the main sequence star Sun (G2), this value is + 0.63, for the Supergiant Betelgeuse
(M1) +1.85.

Analysis and Interpretation of Astronomical Spectra 62

15 The Measurement of the Radial Velocity

15.1 The Radial Velocity
The radial velocity is the motion-component of gases, solid substances or celestial bodies,
measured on the axis of our line of sight. In the following example, observer measures,
the radial velocity components of a passing star , which moves with the velocity . If the
vector is directed away from the observer , the observed wavelength appears stretched
and the spectrum therefore red-shifted. In the opposite case it appears compressed and the
spectrum blue-shifted. Perpendicular to the line of sight, the apparent Transverse velocity

of the star can be observed.

Vr Tv
SV

SV

SV Tv Vr = 0
Vr B

15.2 The Classical Doppler Effect

The astrophysics uses the classical Doppler Principle for the approximate determination of
low radial velocities (<500 km/s). This law is named after the Austrian physicist Christian
Doppler 1803 – 1853. It applies not only to sound-, but generally also to electromagnetic
waves, which can propagate even in vacuum. The classical explanation model is the
changing pitch, emitted by the siren of a passing emergency vehicle. The following graphic
shows the three basic cases, each with one oscillation period of a light signal, emitted by a
spaceship with the rest wavelength and the corresponding period . However, due to
the different radial velocities , observer accordingly sees different wavelengths and
period lengths . for all profiles.

λB = λ0 + Δλ

B

S S‘ + Vr

+ Δλ

λB = λ0

B

S Vr= 0

λB = λ0 – Δλ
B – Δλ – Vr

S‘ S

Analysis and Interpretation of Astronomical Spectra 63

1. In the upper, red profile the spaceship moves away by the small amount from to
, during the transmission of one entire oscillation period. Therefore the end of the

emitted oscillation period covers this additional way to the observer with a corre-

spondingly extended runtime . therefore measures a red shift, ie an extension of
the rest wavelength by . Here, the sign of the radial velocity is positive .

2. In the middle, black profile, observer measures the originally transmitted oscillation

period with the so-called rest wavelength , because the radial velocity of the space

ship is here . ,

3. In the lower, blue profile the spaceship approaches by the small amount from to
. Therefore the end of the emitted oscillation period covers this shortened way to the

observer with a correspondingly reduced runtime . therefore measures a blue-
shift, ie a reduction of the rest wavelength by . Here, the sign of the radial veloc-
ity is negative .

The amount of shift can easily be calculated from the total oscillation period [s] and
the radial velocity :

The oscillation period can be calculated from the rest wavelength and the speed of
light :

The Spectroscopic Doppler Formula with the radial velocity , can be obtained by insert of
in :

Measured wavelength of the considered spectral line
Measured shift in wavelength of a given spectral line
Rest wavelength of a spectral line, measured in a stationary system
Speed of light 300‘000 km/s

 If the spectrum is blue-shifted, the object is approaching us and becomes negative

 If the spectrum is red-shifted, the object is receding and becomes positive.

15.3 The z-Value - A Fundamental Measure of Modern Cosmology

In modern cosmology, the inconspicuous break in formula {16} has become the most

important measuring variable in extragalactic space. Now, due to the constant speed of
light, this so-called -value – applied at such extremely distant objects – became both, a

measure for the distance, as well as for the past. In a calibrated spectrum it can easily be

determined by the measured shift of a spectral line with the wavelength and their known

rest wavelength .

The -values remain also completely independent of the debated cosmological model pa-
rameters and thus are reliably comparable. According to {16}, the radial velocity of the clas-
sical Doppler Effect results to:

Analysis and Interpretation of Astronomical Spectra 64

15.4 The Relativistic Doppler Effect for Electromagnetic Waves

From the view of observer , at very high radial velocities an additional effect from the
Special Theory of Relativity STR becomes noticeable: detects that the system time of the
spaceship has significantly slowed down and therefore the period duration of the gener-
ated light signal is accordingly shortened from to . According to STR applies:

From the view of observer combines here the classical- with the relativistic Doppler-Effect:

1. Classical Doppler Effect: The observed duration of one oscillation period is the sum of
the period duration plus the extension, or shortening, of the run time due to the radial ve-
locity . (sect. 15.2).

2. Relativistic Doppler Effect: The combination with the relativistic Doppler Effect can be
obtained by replacing T in with T ' from

solved for

For reasons of proportionality and due to the following applies:

inserted into finally yields the well-known Relativistic Doppler Formula for cal-

culating the radial velocity .

This applies to kinematically induced radial velocities. Their extended application to the ap-
parent "cosmological escape velocity", due to the expansion of the "space-time lattice" is
under debate (see chap. 15.8).

15.5 The Measurement of the Doppler Shift

For radial velocity measurements ( ) in most cases, the Doppler shift is determined accord-
ing to sect. 15.3. The difference is calculated between a specific spectral line and their
well-known rest wavelength of a stationary laboratory spectrum.

The calculation of is finally enabled by formula {16} {18} or {20}. For the accurate deter-
mination of , a calibration lamp spectrum must be recorded, immediately before and/or
after the object spectrum with a totally unchanged spectrograph setup. The analysis soft-
ware (eg Vspec) allows in a calibrated spectral profile the -measurement with so called
Gaussian fits. For this purpose, clearly identified lines must be selected which are not too
strongly deformed by blends. A detailed description of the procedure can be found in [30].

In the red region of the spectrum and with high-resolution spectrographs, a very accurate
calibration can also be achieved by use of the atmospheric H2O lines [30]. In addition, this
is also enabled by the naturally- and artificially generated emissions of the night sky spec-
trum [33].

Analysis and Interpretation of Astronomical Spectra 65

15.6 Radial Velocities of Nearby Stars

The radial velocities of stars in the vicinity of the solar system reach for the most part only
one-or two-digit values in [km/s]. Examples: Aldebaran +54 km/s, Sirius –8.6 km/s, Betel-
geuse +21 km/s, Capella +22 km/s [100].

The corresponding shifts of are therefore very low, usually just a fraction of a 1Å. For
and based on the Hα line (6563 Å), corresponds to ~46 km/s (formula {16}).

Therefore highly-resolved and accurately calibrated spectra are necessary.

15.7 Relative Shift within a Spectrum caused by the Doppler Effect

Textbook examples for this effect are the so-called P Cygni profiles (sect. 5.6). For the de-
termination of the expansion velocity of the stellar envelope neither an absolutely wave-
length calibrated spectrum nor a heliocentric correction [30] is required. The measurement
of the relative shift between the absorption and the emission part of the P Cygni profile is
sufficient. For P Cygni this displacement within the Hα line amounts to some 4.4 Å, corre-
sponding to an expansion velocity of ~200 km/s [33].

15.8 Radial Velocities of Galaxies

In the area between the galaxies, which means outside of strongly gravitational acting sys-
tems, the cosmologically induced expansion of the so-called space-time lattice gets more
and more dominant and the kinematic peculiar motion of the galaxies increasingly negligi-
ble. This effect must always be taken into account by the interpretation of extragalactic
spectra. In the relative "near vicinity" up to some 100 Mega Parsec or 1 bn ly – which also
includes Messier's galaxies-world – it still needs to distinguish between the components of
the kinematic Doppler Effect, due to the peculiar motion, and the relativistic cosmological
redshift, caused by the expansion of the spacetime. This way, six of the 38 Messier-
galaxies tend to move – against the "cosmological trend" – ie with blue-shifted spectra, to-
wards our Milky Way! These include M31 (Andromeda) with about –300 km/s, and M33
(Triangulum) with some –179 km/s [101], see table sect. 15.10.

Georges Lemaître (1894 – 1966, picture Wikimedia) and Edwin Hub-
ble (1889-1953), are considered the discoverer of the expanding uni-
verse. Independent of each other they postulated, based on a statisti-
cally rather small basis of just 18 galaxies [431], in relation to the
photometrically determined distances D, an approximately propor-
tional increase of the redshift , and the associated radial velocity .

Today, the proportionality factor is called Hubble Parameter. Because nowadays it is

known that , considered over the time, doesn't remain a constant, this term has re-

placed the historically used term "Hubble Constant ". The current value for was first

determined in the so-called Key Project with the Hubble Space Telescope HST and

2012 improved with the Spitzer Space Telescope in the mid infrared. As the cause of the
progressive redshift, Hubble, in contrast to Lemaître, firstly still believed to the Doppler ef-
fect, as a result of a purely kinematic expansion. That's why, even today, the term "escape
velocity" is still in use. Today, however, it seems clear that, from a distance, longer than
some 100 mega parsecs, the cosmological spacetime expansion dominates and the phe-
nomenon of redshift has here nothing more to do with the Doppler effect! Due to the ex-
pansion- and curvature of space, in this extreme distance range, the classical notion of dis-
tance, measured in light years [ly] or parsec [pc], gets increasingly problematic and the
Hubble law as about >400 Mpc or , should no more be applied proportionally, ie

Analysis and Interpretation of Astronomical Spectra 66

without cosmological model parameters [431]. Here, therefore the -value has become es-
tablished, which can be measured directly in the spectrum (sect, 15.3 {17}) and remains
independent of the debated cosmological models. Therefore, the classical approach, to de-
termine the distance by the proportional Hubble law , remains restricted on the relative

"near vicinity" and is eg still applicable in the field of the Messier galaxies.

15.9 The Apparent Dilemma at

The range of the observed -values reaches today (2013) up to the Galaxy Abell 1835 with

. This was discovered in 2004 mainly by a French/Swiss team of researchers, with

the VLT of the ESO Southern Observatory! Anyway for values , applied in formulas

or , , at least an apparent dilemma shows up, because the radial velocity

, respectively the expansion of spacetime, seems to surpass the speed of light . In the

past, as a way out, sometimes the relativistic Doppler formula {20}, chap. 15.4, was pro-
posed. This procedure in fact prevents the exceeding of , but is nowadays no more ac-

cepted by most of the experts. So, for instance in the CDS database [100], the former rela-
tivistic - value for the quasar 3C273, still calculated on the basis of the STR , has re-

cently been deleted. However, the "classical" radial velocity, based on , is still to

find in this data set (2013).

Due to the increasing expansion of the space-time-lattice, at such extreme distances not
only the classical term for "distance" gets obviously senseless. This applies also to the
"classical" calculation of the apparent radial velocity by the Doppler principle, without
any consideration of cosmological models [431]. Accepted today, but still debated, are dif-
ferent cosmological models – most of them based on the General Theory of Relativity GTR.

15.10 Radial Velocity- and Cosmological Spacetime Expansion at Messier-Galaxies

The table on the following page shows, sorted by increasing distance , the measured he-
liocentric radial velocities for 38 Messier galaxies, according to NED NASA Extragalactic
Database [101]. The cosmological related spacetime expansion . is calculated accord-

ing to with distance values from CDS [100]. Positive values = red-shifted, negative val-
ues = blue-shifted. These figures clearly indicate, that in this “immediate neighbourhood”,
the kinematic peculiar motion of the galaxies still dominates. Nevertheless, at distances
about more than 50 million ly, the trend becomes already evident, that the measured radial
velocities tend, with increasing distance, more and more to the theoretical/cosmological
speed of the spacetime expansion . Anyhow in more than 50M ly distance still two gal-

axies can be found (M86 and M98) with relatively strong negative values (blue coloured
rows). This behaviour show 6 of 38, or about 16% of the Messier Galaxies. Most distant
galaxy is M109 with about 81M ly.

Based on the measured radial velocities , in the following table also the corresponding -

values can be found, calculated with formula . These modest amounts show

very impressively – as seen in the cosmic scale – how extremely small this area really is.

Analysis and Interpretation of Astronomical Spectra 67

Messier Galaxy Distance –value Radial velocity Cosmologic Spacetime
[Mpc / M ly] [km/s] expansion [km/s]
M31 Andromeda –0.0010
M33 Triangulum 0.79 / 2.6 –0.0006 –300 +58
M81 0.88 / 2.9 –0.0001 –179 +64
M82 3.7 / 12 +0.0007 –34 +270
M94 3.8 / 12 +0.0010 +203 +277
M64 5.1 / 17 +0.0014 +308 +372
M101 5.3 / 17 +0.0008 +408 +387
M102 6.9 / 22 +0.0008 +241 +503
M83 6.9 / 22 +0.0017 +241 +504
M106 7.0 / 23 +0.0015 +513 +511
M51 Whirlpool 7.4 / 24 +0.0020 +448 +540
M63 8.3 / 27 +0.0016 +600 +606
M74 8.3 / 27 +0.0022 +484 +606
M66 9.1 / 30 +0.0024 +657 +664
M95 10.0 / 32 +0.0026 +727 +730
M104 Sombrero 10.1 / 33 +0.0034 +778 +737
M105 10.4 / 34 +0.0030 +1024 +759
M96 10.4 / 34 +0.0030 +911 +759
M90 10.8 / 35 –0.0008 +897 +788
M65 12.3 / 40 +0.0027 –235 +898
M77 12.6 / 41 +0.0038 +807 +919
M108 13.5 / 44 +0.0023 +1137 +986
M99 14.3 / 47 +0.0080 +699 +1043
M89 15.4 / 50 +0.0011 +2407 +1124
M59 15.6 / 51 +0.0014 +340 +1138
M100 15.6 / 51 +0.0052 +410 +1138
M98 15.9 / 52 –0.0005 +1571 +1160
16.0 / 52 –142 +1168
M49 +0.0033
M86 16.0 / 52 –0.0008 +997 +1168
M91 16.2 / 53 +0.0016 –244 +1182
M60 16.2 / 53 +0.0037 +486 +1183
M61 16.3 / 53 +0.0052 +1117 +1189
M84 16.5 / 54 +0.0035 +1566 +1204
M87 16.8 / 55 +0.0044 +1060 +1226
M85 16.8 / 55 +0.0024 +1307 +1226
M88 17.0 / 55 +0.0076 +729 +1241
M58 18.9 / 62 +0.0051 +2281 +1380
M109 19.6 / 64 +0.0035 +1517 +1431
24.9 / 81 +1048 +1917

Analysis and Interpretation of Astronomical Spectra 68

15.11 The Redshift of the Quasar 3C273

The above listed, very low -values, clearly show, that Messier’s world of galaxies just be-
longs to our "backyard" of the universe. As a contrast the following graph shows the im-
pressively red-shifted H-emission lines of the apparently brightest quasar 3C273 in the
constellation Virgo. Thanks to advances, particularly in camera technology, also amateur
astronomers have nowadays the pleasure to deal with cosmologically relevant distance
ranges. We must therefore be aware of the effects of the STR, as well as of the cosmologi-
cal models, which are still under debate. 3C273 demonstrates, that the above presented
formulas have a practical use and are more than just “gimmicks at an academic level”.

uUsnetDQdilA)DurOetarSca:seGnranasttrminlyg3i2ss0Csu0iLco2mhn7mfg-31a,r5i,an0tμtiRmnosgeblistjd,er[esc4coht8srdi0efwdt]m.eoaNryef2o6gwt,he2a0ne1d2eaHrwyaisytlhlydiAttrirkceo3ag1gn4aLer+bdne-e10rdB°eCaac, 5solxmr1td2h0eee0drsdmLoimuncaehinsboeftttehrew(sitlhitlaeslosw

resoThleuitnidoicnatiosnliotf tshpe weacvterleonggthr,adpetherm–ineddewtiathiVlsspesceaet G[a3us5sfi]t.s, is provided red shifted on the original scale
The profile is normalised to the continuum Ic = 1, the intensity on the level of the wavelength axis is Ic = 0.6.

Hα 7580

Δλ≈1017Å

I Hβ 5632

Δλ≈771Å

Hγ 5023

Δλ≈683 Å

Hδ 4748

Δλ≈646 Å

1.0

0.6

Red shifted, calibrated original scale

©Richard Walker 2012/06

The splitting of the Hα line is caused here by the superposition with the intense Fraunhofer

A-band (O2). Therefore the evaluation of this line was omitted in the following table. The -

values have been measured at Gaussian fitted lines and calculated with formula {17}. The
individual amounts of shift are rounded to 1Å. They vary in size, but behave strictly propor-

tional to their according rest wavelengths . Therefore, the calculation of the -values for

all evaluated lines must yield almost the same amounts. This is also a good final test for the
quality of the wavelength calibration. The obtained values are consistent here up to almost
three decimal places with the accepted value , [100], [101].

5632 4861 771 0.1586
5023 4340 683 0.1574
4748 4102 646 0.1574

With the distance to 3C273 is already so large, that cosmological models and ap-

propriately chosen parameters should be applied. Nevertheless, with the measured -value

and applying a "classical" approach, the apparent radial velocity and the distance to

3C273, shall here roughly be estimated. Considering this enormous distance, the kinematic
peculiar motion of 3C273 can be neglected and therefore the measured apparent radial ve-

Analysis and Interpretation of Astronomical Spectra 69

locity be equalised with the expansion speed of the spacetime . Applying the "classi-

cal" Doppler formula {18} ), we obtain 47‘490 km s-1, i.e. almost 16% of

the speed of light! The corresponding value in the CDS Database [100] is .

With the Hubble Law {21} the Distance results to about 600 Mpc or 2 bn ly. The accepted

value is slightly higher ~2.5 bn ly.

The huge redshift of 3C273 also shows why the current and future space telescopes (eg
Herschel, James Webb) are more and more optimised for the infrared range. In addition to
such considerations we should always be aware that the light, which we analyse today
from 3C273, was “on the road”, since 2.5 bn years when our earth was still in the Precam-
brian geological age! But anyway, compared with for Abell 1835, this distance is

still relatively "close".

15.12 The Gravitational Redshift

Besides the classical Doppler effect and the relativistic spacetime expansion, it remains the
third option – the gravitational redshift. According to the General Theory of Relativity GTR,
light loses measurably energy as it propagates in an extreme gravitational field [431]. The
extreme case here are the black holes where the light is totally slowed down. In the spec-
troscopic amateur practice, however, already the much smaller gravitational field of white
dwarfs can massively distort the measurements of radial velocity! Anyway the average
value for this brake effect amounts in the spectral class DA to about 31 km/s [250]. The
corresponding, object-specific value must therefore be subtracted from the measured radial
velocity. For example, if for such a White Dwarf the radial velocity was measured to

, after the subtraction of gravitational redshift, it remains just
.

15.13 Short Excursus on "Hubble time" tH

In this case, it is worthwhile to take a small excursus, since the Hubble parameter allows in
a very simple way to estimate the approximate age of the universe! With the simplified as-
sumption of a constant expansion rate of the universe after the "Big Bang", by changing
formula {18}, it can be estimated, how long ago the entire matter was concentrated at “one
point”. This time span is also called Hubble time and is equal to the reciprocal of the
Hubble Parameter . This reciprocal value also corresponds to the Division of distance D by
the expansion velocity and thus the desired period of time !

To calculate the “Hubble time” we just need to put the units of the Hubble Parameter in to
equation {23} and to convert [ ] to [ ] and to [ ].

Analysis and Interpretation of Astronomical Spectra 70

16 The Measurement of the Rotation Velocity

16.1 Terms and Definitions

The Doppler shift , caused by the different radial

velocities of the eastern and western limb of a rotating
spherical celestial body, allows the determination of
the rotational surface speed. Here we limit ourselves
to the spectroscopically direct measurable portion of
the rotation speed, the so-called value, which

is projected into the visual line to Earth.

This term, read as a formula, allows calculating this
velocity share with given effective equatorial velocity

and the inclination angle between the rotation axis
and the visual line to Earth. The Wikipedia graph
shows here differently as .

If the rotation axis is perpendicular to the sight line to Earth then and s

Exclusively in this special case, we can exactly measure the equatorial velocity .

If , we look directly on a pole of the celestial body and thus the projected rotational

velocity becomes s

16.2 The Rotation Velocity of the Large Planets

If the slit of the spectrograph is aligned with the equator of a rotating planet, the absorption
lines of the reflected light appear slightly leaning. This is caused by the Doppler shift due to
the radial velocity difference between the eastern and western limb of the celestial
body. The rough alignment of the slit can be done by Jupiter with help of its moons and by
Saturn with its ring.

The radial velocity difference is calculated from the obliquity of the spectral lines. For
this purpose from a good quality spectrum two narrow strips, each on the lower and upper
edge are processed and calibrated in wavelength. The difference between the upper and
lower edge gives the Doppler shift and with formula {16} or {20} the velocity difference

can be obtained (detailed procedure, see [30]).

vsin i Δλ

vr
vr

-vr
-vr

-vsin i

Analysis and Interpretation of Astronomical Spectra 71

16.3 The Rotation Velocity of the Sun

This method allows also the estimation of the Sun’s rotational surface speed, of course
with an attached energy filter (!). This velocity however is so low (~1.9 km/s), that a high
resolution spectrograph is required. In this case, the picture of the Sun on the slit plate
would be too large, respectively the slit much too short to cover the entire solar equator.

Required in such cases is the recording of two separate wavelength-calibrated spectra,
each on the eastern- and western limb of the Sun. The following profiles were recorded

with the SQUES Echelle spectrograph [400] and show an averaged shift of

(blue: eastern limb, red: western limb). According to {15} and {25} results s .

The accepted value is ~ . This method was first practiced 1871 by Hermann Vogel.

Fe I 6252.561
Fe I 6254.262
Fe I 6256.37
Ti I 6258.103
Ti I 6258.706
Ti I 6261.101
Fe I 6265.14

Δλ

16.4 The Rotation Velocity of Galaxies

This method can also be applied to determine the ro-
tation velocity in the peripheral regions of galaxies.
This works only with objects, which we see nearly
“edge on” eg M31, M101 and M104 (Sombrero). For
Face On galaxies like M51 (Whirlpool), we can just
measure the radial velocity according to sect. 15.

16.5 Calculation of the Value with the Velocity Difference

Here two cases must be distinguished:

1. Light-Reflective Objects of the Solar System:

Analogously to the radar principle for light-reflecting ob- EaOsstt West
jects of our solar system, eg planets or moons, the Dop-
pler Effect, seen from earth, acts twice. A virtual ob-
server at the western limb of the planet sees the light of
the sun (yellow arrow) as already red-shifted by an
amount, corresponding to the radial velocity of this
point, relative to the Sun. This observer also notes that
this light is reflected unchanged towards Earth with the
same redshift (red arrow).

An observer on Earth measures the value of this reflected light, red-shifted by an addi-
tional amount, which is equal to the radial velocity of the western limb of the planet, rela-
tive to the Earth. This total amount is too high by the share, the virtual observer on the

western limb of the planet has already found, in the incoming sunlight. When the outer
planets are close to the opposition, both red-shifted amounts are virtually equal. A halving

Analysis and Interpretation of Astronomical Spectra 72

of the measured total amount therefore yields with formula {15} the velocity difference
between the eastern and western limb of the planet. This velocity difference must

now be halved again to obtain the desired, rotation velocity s . This corresponds finally
to ¼ of the originally measured redshift (½ x ½=¼).
Projected rotation velocity of light reflecting objects:

s

= Velocity difference calculated with the measured shift

2. Self-Luminous Celestial Bodies
For self-luminous celestial objects, e.g. the Sun or galaxies, only a halving of the measured
velocity difference is required:
Projected rotation velocity of self-luminous objects:

s

= Velocity difference calculated with the measured shift

Analysis and Interpretation of Astronomical Spectra 73

16.6 The Rotation Velocity of the Stars

Due to the huge distances, with few exceptions, even with large telescopes, stars can’t be
seen as discs, but just as small Airy disks due to diffraction effects in optics. In this case the
above presented method therefore fails and remains reserved for 2-dimensional appearing
objects. Today, numerous methods exist for determining the rotational speed of stars, eg
with photometrically detected brightness variations or by means of interferometry. The idea
to gain the projected rotational velocity s from the spectrum is almost as old as the
spectroscopy itself.

William Abney has proposed already in 1877 to determine s analysing the rotationally
induced broadening of spectral lines. This phenomenon is also caused by the Doppler Ef-
fect, because the spectrum is composed of the light from the entire stellar surface, facing
us. The broadening and flattening of the lines, is created by the rotationally caused different
radial velocities of the individual surface
points. This so-called "rotational broadening"
is not the only effect that influences the

of the spectral line (sect. 7.4). A suc-
cessful approach was to isolate the Doppler
Broadening-related share with various meth-
ods, eg by comparison with synthetically
modelled spectra or slowly rotating standard
stars. The graph on the right [52] shows this
influence on the shape of a Mg II line at
4481.2 Å, for a star of the spectral class A.

The numerous measured rotational velocities of main sequence stars show a remarkable
behaviour in terms of spectral classes. The
graph shows a decrease in speed from early
to late spectral types (Slettebak).

From spectral type G and later, s
amounts only to a few km/s (Sun ~2 km/s).
The entire speed range extends from
0 to >400 km/s. It has also shown that
stars, after leaving the Main Sequence on
their way to the Giant Branch in the HR dia-
gram (sect. 14), as expected greatly reduce
the rotation speed.

Since the s values get very low, from
the spectral class G on and later (so-called
slow rotators), the demand on the resolution
of the spectrograph is drastically increased. The focus of this method is therefore, particu-
larly for amateurs, on the early spectral classes O - B, which are dominated by the so-called
fast rotators. A typical example is Regulus, B7V, with s . The shape of this

star is thereby strongly flattened. But there are also outliers. Sirius for example, as a repre-

sentative of the early A-Class (A1V), is with 16 km/s a clear slow rotator [126].

An interesting case is Vega (A0V) with s , which for a long time has also

been considered as an outlier. But various studies, e.g. Y. Takeda et al. [120], have shown
that Vega is likely a fast rotator, where we look almost exactly on a pole ( ≈7°). This is sup-

ported by interferometrically detected, rotationally induced darkening effects (Peterson,

Aufdenberg et al. 2006). The spread of the newly estimated effective values for Vega is

broad and ranges in these studies of approximately 160 - 270 km/s. This example clearly
shows the difficulties, to determine the inclination and the associated effective rotation

Analysis and Interpretation of Astronomical Spectra 74

velocity , – this compared with the relatively simply determinable s value! Today the
- Method is complemented e.g. by a Fourier analysis of the line profile. The first

minimum point represents here the value for s with a resolution of some 2 km/s [125].
This method requires high-resolution spectra.

Various studies have shown, that the orientation of the stellar rotation axes is randomly dis-
tributed – in contrast to most of the planets in our solar system axes. Since the effective
equatorial velocity can be determined only in exceptional cases, the research is here limited
almost exclusively on statistical methods, based on extensive s - samples. Most ama-
teurs will probably limit themselves to reproduce well known literature values with high ro-
tation velocities, typically for the earlier spectral classes.

Empirical Formulas for s depending on ,s =

A considerable number of astrophysical publications in the SAO/NASA database (~1920 to
the presence) deal with the calibration of the rotational speed, relative to FWHM. Some of
the numerous "protagonists" are here A. Slettebak, O. Struve, G. Shajn, F. Royer and F. Fe-
kel. Probably the most cited standard work in this respect is the so called „New Slettebak

System“ from 1975. Anyway recent studies have shown that the provided s values are
systematically too low. That's why I present here more recent approaches. The variable in
most such formulas is , given in [Å] {3}. But in some formulas is also

expressed as a Doppler velocity [km/s]. In one case, also the Equivalent Width EW [Å] is in-
volved (sect. 7).

The Method according to F. Fekel

This well established method [122, 123] is based on two different calibration curves, each
one for the red and blue region of the spectrum. The two polynomial equations of the sec-
ond degree calibrate the "raw value" for s in [km/s], relating to the measured and ad-
justed {3} in [Å]. Below are the two calibration polynomials {26a, 27a} according

to Fekel, for the spectral ranges at 6430 Å and 4500 Å, followed by two further formulas
{26b, 27b}, which I have transformed from equations {26a, 27a} according to the explicitly
requested – Values:

The procedure for the determination of is described in detail and supported by an

example in [30]. Here follows a short overview:

First, several FWHM values [Å] are measured and averaged by weakly to moderately in-
tense and Gaussian fitted spectral lines (no H-Balmer lines). Further these values are
cleaned from the "instrumental broadening" ( ). Then the X values are calculated

by inserting the values in the above formulas {26b} or {27b}, according to the

wavelength ranges at 4500 Å or 6430 Å. These X-values must then be cleaned from the
line broadening due to the average macro turbulence velocity in the stellar atmosphere,

using the following formula. This will finally yield the desired value [km/s]

The variable depends on the spectral class. For the B- and A- class Fekel assumed

. For the early F- classes , Sun like Dwarfs , K- Dwarfs

Analysis and Interpretation of Astronomical Spectra 75

, early G- Giants , late G- und K- Giants and F – K
Sub Giants

Below, the commonly used spectral lines for determining the FWHM values are listed. The
lines, proposed by Fekel are in bold italics, such used by other authors only (eg Slettebak)
are written in italics. The supplement “(B)” means that the profile shape is blended with a
neighbouring line. “(S)” means a line deformation due to an electric field (Stark Effect):

– Late F–, G– and K– spectral classes: Analyse of lines preferably in the range at 6430Å:
eg Fe II 6432, Ca I 6455, Fe II 6456, Fe I 6469, Ca I 6471.

– Middle A–Classes and later: Analyse of moderately intense Fe I, Fe II and Ca I lines in
the range at 6430Å. For A3 – G0–Class Fe I 4071.8 (B) und 4072.5 (B).

– O–, B– and early A–classes: Analyse of lines in the range at 4500 Å:

– Middle B– to early F–classes: several Fe II and Ti II lines, as well as He I 4471 and
Mg II 4481.2.

– O–, early B– and Be– classes: He I 4026 (S), Si IV 4089, He I 4388,
He I 4470/71 (S,B), He II 4200 (S), He II 4542 (S), He II 4686, Al III, N II,

As an alternative to F. Fekel, A. Moskovitz [121] in connection with K-Giants, analysed ex-
clusively the relatively isolated Fe I line at 5434.5 with formula {26a}, respectively {27a}.

16.7 The Rotation Velocity of the Circumstellar Disks around Be Stars

Be stars form a large subgroup of spectral type B.
Gamma Cassiopeiae became the first Be star discov-
ered in 1868 by Father Secchi (sect. 13.3), who
wondered about the "bright lines" in this spectrum.
The lower case letter e (Be) already states that here
appear emission lines.

In contrast to the stars, which show P Cygni profiles
due to their expanding matter (sect. 17), it is here in
most cases a just temporarily formed, rotating cir-
University Western Ontario

cumstellar disk of gas in the equatorial plane of the Be star. Their formation mechanisms
are still not fully understood. This phenomenon is accompanied by hydrogen emission, and
strong infrared- as well as X-ray radiation. Outside of such episodic phases, the star leads a
seemingly normal life in the B-class. In research and monitoring programs of these objects,
many amateurs are involved with photometric and spectroscopic monitoring of the emis-
sion lines (particularly Hα). That's why I've gathered some information about this exciting
class, since here also applications for the EW and FWHM values can be presented. Here
follow some parameters of these objects, based on lectures given by Miroshnichenko
[140], [141], as well as publications by K. Robinson [5] and J. Kaler [3]:

– 25% of the 240 brightest Be stars have been identified as binary systems.

– Most Be stars are still on the Main Sequence of the HRD, spectral classes O1–A1 [141].
Other sources mention the range O7–F5 (F5 for shell stars).

– Be stars consistently show high rotational velocities, up to >400 km/s, in some cases
even close to the so-called "Break Up" limit. The main cause for the spread of these val-
ues should be the different inclination angles of the stellar rotation axes.

– The cause for the formation of the circumstellar disk and the associated mass loss is still
not fully understood. In addition to the strikingly high rotational speed [3], etc. it is also

Analysis and Interpretation of Astronomical Spectra 76

attributed to non-radial pulsations of the star or the passage of a close binary star com-
ponent in the periastron of the orbit.
– Such discs can arise within a short time but also disappear. This phenomenon may pass
through overall three stages: Common B star, Be–star and Be–shell star [33].
Classical example for that behaviour is Plejone (Plejades M45), which has passed all
three phases within a few decades [147]. Due to its viscosity, the disk material moves
outwards during the rotation [141].
– With increasing distance from the star, the thickness of the disk is growing and the den-
sity is fading.
– If the mass loss of the star exceeds that of the disc, the material is collected close to the
star. In the opposite case it can form a ring.
– The X-ray and infrared radiation is strongly increased.
If a B star mutates within a relatively short time to a Be star (eg scorpii δ), the Hα absorp-
tion from the stellar photosphere is changing to an emission line, now generated in the cir-
cumstellar disk. At the same time it becomes the most intense spectral feature (extensive
example see [30] sect. 22). It represents now the kinematic state of the ionised gas disk. It
is, similar to the absorption lines of ordinary stars, broadened by Doppler Effects, but here
due to the rotating disk of gas and additional, non-kinematic effects. Therefore the FWHM
value of the emission line is now a measure for the typical rotation velocity of the disc ma-
terial. For δ scorpii in [146] it is impressively demonstrated, that since about 2000 the
FWHM and EW values of the Hα emission line are subject to strong long-period fluctua-
tions. Between the first outbursts in 2000 to 2011, the Doppler velocity of the FWHM value
fluctuated between about 100–350 km/s and the EW value from –5 to –25 Å, indicating
highly dynamic processes in the disk formation process. Furthermore the chronological pro-
file of FWHM and EW values is strikingly phase-shifted.

Formula for the Rotation Velocity of the Disk Material:

Several formulas have been published, which allow to estimate the rotation speed of the
disc material from Be stars. In most cases is calculated with values at the

Hα line.

Following a formula by Dachs et al which Soria used in [145]: It expresses explicitly the
value, based on the [km/s] at the Hα emission line, combined with the

(negative) equivalent width EW [Å].

s

Example: The Hα Linie of Soria‘s Be–star yields , corresponding to a
.
Doppler Velocity of 278 km/s and . This results to

In [30] sect. 22.3 this formula is applied to a DADOS spectrum of δ scorpii. The accuracy of
this method is limited to ±30 km/s. Therefore "estimate", is here probably the better ex-
pression than "calculate".

Hanuschik [127] shows a simple linear formula, which expresses just with the

[km/s] of the Hα emission line. It corresponds to the median fit of a strongly

scattering sample with 115 Be stars, excluding those for which an underestimation of the
value was assumed; [127], {1b}.

With Soria’s (see above) results a different

Analysis and Interpretation of Astronomical Spectra 77

The following formula is applicable for the –values at the emission lines Hβ
(4861.3) and FE II (5317, 5169, 6384, 4584 Å).

,

The Distribution of the Rotation Velocity on the Disk R

Assuming that the disk rotation obeys the kinematic laws of Kepler, v <
the highest rotational velocity occurs on its inner edge – In many
cases, probably identical with the star equator. It decreases to- <
wards the outside (formula according to Robinson [5]).

The application of this formula is limited to high values ( or known inclina-
tion angle .

The Analysis of Double Peak Profiles ∆V peak

The emission lines of Be stars often show a double
peak. The gap between the two peaks is explained
with the Doppler-, self-absorption- and perspective
effects. Looking at the edge of the disk in the range R
of the symmetry axis the gas masses apparently Normalised Intensity V
move perpendicularly to our line of sight, ie the ra-
dial velocity there is . Indicated are important 1 Continuum level Ic=1

dimensions, which are used in the literature.

Ic

0 λHα

Wavelength λ

Distance between the Peaks

The chart on the right shows according to K. Robinson [5]
the modelled emission lines for different inclination angles
(sect. 16.1). It seems obvious that the distance
i=82°

increases with growing inclination . At the same time also 65°

the –values are increasing, if for all inclination an-

gles a similar effective equatorial velocity and a fixed disc 30°

radius are assumed. is expressed as a velocity

value according to the Doppler principle: according to 15°

{15}: . 0° Wavelength
λHα
Also according to Hanuschik [143] a rough correlation ex-
ists between the [km / s] at the Hα line and .

Analysis and Interpretation of Astronomical Spectra 78

According to Miroshnichenko [141] and Hanuschik [143] a decreasing disk radius R is also
associated with, an increasing . This statement is consistent with formula {33}

and {34}.

The outer disc radius Rs
r
The formula by Huang [143] allows the estimation of the outer disk
radius , expressed as [stellar radii r], based on and the

velocity at the inner edge of the disc, probably touching the star’s

equator in most of the cases ( ). <

The application of this formula is limited to high values
( or known inclination angle .

Peak Intensity Ratio (Violet/Red)

The V / R ratio is one of the main criteria for the description of the double peak emission of
Be stars. S. Stefl et al. [144] have supervised this ratio for about 10 years, at a sample of Be
stars. They have noted long-periodic variations of 5-10 years. One of the discussed hy-
potheses are oscillations in the inner part of the disk.

At δ scorpii the V / R ratio of the He I line (6678.15 Å) shows a strong variation since the
outbreak of 2000 [146]. See also example in [30] sect. 22.3.

According to Kaler [3] the V/R ratio also reflects the mass distribution within the disc and
may show a rather irregular course.

In Be-Binary systems a pattern of variation seems to occur, which is linked to the orbital pe-
riod of the system.

According to Hanuschik [143] asymmetries of the emission lines, eg , are related to
radial movements and darkening effects.

If no double peak is present in the emission line, according to [128] the asymmetry in the
steepness of the two edges can be analysed. V>R means the violet edge is steeper and vice
versa by R>V the red one.

Analysis and Interpretation of Astronomical Spectra 79

17 The Measurement of the Expansion Velocity

Different types of stars repel, at certain stages of evolution, more or less strongly matter.
The speed range reaches from relatively slow 20–30 km/s, typical for planetary nebulae,
up to several 1000 km/s for Novae and Supernovae (SNR). This process manifests itself in
different spectral symptoms, dependent mainly on the density of repelled material.

17.1 P Cygni Profiles
The P Cygni profiles have been already introduced in
sect. 5.6, as examples for mixed absorption- and
emission line spectra. They are a common spectral
phenomenon which occurs in all spectral classes,
and are a reliable sign for expanding star material.
The evaluation of this effect with the Doppler formula
is demonstrated here at the expansion velocity of the
stellar envelope of P Cygni.

As a rough approximation the offset [Å] is
measured between the peaks of the emission- and
the blue-shifted absorption part of the P Cygni
profiles [5]. Anyway, where exactly the offset should
be measured, depends also on the particular profile
shape. In the example for the Hα line of P Cygni
itself, the measured Peak-Peak difference yields:

(single measurement of 07.16.2009, 2200 UTC, DADOS and 900L/mm grating)

with , , yields:

The accepted values lie in the range of –185 bis – 205 km/s ±10km/s.The heliocentric
correction is not necessary here, since the Doppler shift is measured not absolutely but
relatively, as the difference, visible in the spectrum. P Cygni forms not a separate class
of stars. Such profiles are a common spectral phenomenon, which can be found in all
spectral classes and are a reliable indication for expanding stellar matter.

17.2 Inverse P Cygni Profiles

In contrast to the normal P Cygni profiles, the inverse ones are al-
ways a reliable sign for a contraction process. The absorption kink
of this feature is here shifted to the red side of the emission line.
Textbook example is the Protostar T Tauri which is formed by accre-
tion from a circumstellar gas and dust disk. The forbidden [O I] and
[S II] lines show here clearly inverse P Cygni profiles, indicating
large-scale contraction movements within the accretion disk,
headed towards the protostar. The Doppler analysis showed here
contraction velocities of some 600 km/s. The excerpt of the T Tauri
spectrum is from [33], Table 18.

Analysis and Interpretation of Astronomical Spectra 80

17.3 Broadening of the Emission Lines

P Cygni profiles are not only characteristic for stars with strong expansion movements, but
also for Novae and Supernovae. Anyway such extreme events show more often just a
strong broadening of the emission lines. This applies also to Wolf Rayet stars, even though
with significantly lower FWHM values (see [33]). According to [160] the expansion velocity

can be estimated, by putting in to the

conventional Doppler formula, instead of :

The scheme [160] shows for the Nova V475 Scuti
(2003) four developmental phases within 38 days. Here,
nearly the tenfold expansion velocity of the P Cygni oc-
curred, already an order of magnitude at which the rela-
tivistic Doppler formula should be applied {20}.

17.4 Splitting of the Emission Lines
In sect. 16.7 the determination of the rotation velocity by evaluating the double peak pro-
files in Be stars, was already presented. Split emission lines can also be observed in spectra
of relatively old, strongly expanded and thus optically transparent stellar envelops. Text-
book example is here SNR M1.

The chart at right explains the split up of the O III
emission lines due to the Doppler Effect.
The parts of the shell which move towards Direction
earth cause a blueshift of the lines and the earth
retreating ones are red-shifted. Thereby,
they are deformed to a so-called velocity el-
lipse. This effect is seen here at the noisy
[O III] lines of the M1 spectrum [33].

The chart at right shows the splitting of the Hα line in the central Hα SNR
area of the Crab Nebula M1 ([33] Table 85). Due to the transpar-
ency of the SNR, with the total expansion velocity is calculated, Δλ ≈ 31Å N

related to the diameter of the SNR, (here about 1800 km/s). The
Bradial velocity is finally obtained by halving this value. It yields
therefore just below 1000km/s 1

Olll 5006.84 An

Olll 4958.91
Hβ 4861.33

He II 4685

Analysis and Interpretation of Astronomical Spectra 81

18 The Measurement of the Stellar Photosphere Temperature

18.1 Introduction

Depending on the spectral class, stellar spectra and their measurable features reflect to a
different extent also the physical state of the photosphere. For the spectroscopic determi-
nation of the effective temperature (sect. 3.2), numerous methods exist with different

degrees of accuracy but also corresponding complexity.

18.2 Temperature Estimation of the Spectral Class

The spectral class reflects directly the sequence of the corresponding photospheric tem-
peratures (sect. 14.2). It is therefore the most direct, simple, but relatively imprecise way to
estimate the effective temperature . For main sequence stars of intermediate and late
spectral classes, the accuracy, realistically achievable by amateurs, should lie within some
100 K. In literature numerous tables can be found, assigning the effective temperatures to
the individual spectral classes. In addition, for the luminosity class III and V, separate and
significantly different values are presented. In the region of early spectral classes, even be-
tween renowned sources, differences may arise up to >1000 K. Thus, especially for the
early types of the spectral type O, for the same star often significantly different classifica-
tions are published. This shows that this method, at least for amateurs, is limited to the
main-sequence stars, since the determination of the luminosity class is quite difficult. As an
example, follows here a table with data, taken from a lecture at the University of Northern
Iowa: http://www.uni.edu/. At the early spectral classes, some of these values significantly
differ from those, shown in the table in sect. 14.5, or in [33].

Spectral Main Sequence Giants (III) Super Giants (I)
Type (K)
(V) (K) (K)

O5 54,000 21,000
O6 45,000 16,000
O7 43,300 14,000
O8 40,600 12,800
O9 37,800 11,500
B0 29,200 11,000
B1 23,000 10,500
B2 21,000 10,000
B3 17,600 9,700
B5 15,200 9,400
B6 14,300 9,100
B7 13,500 8,900
B8 12,300
B9 11,400 8,300
A0 9,600
A1 9,330 7,500
A2 9,040 7,200
A3 8,750
A4 8,480
A5 8,310
A7 7,920
F0 7,350
F2 7,050

Analysis and Interpretation of Astronomical Spectra 82

F3 6,850

F5 6,700 6,800

F6 6,550 6,150
5,800
F7 6,400
5,500
F8 6,300 5,100
5,050
G0 6,050 4,900
4,700
G1 5,930 4,500
4,300
G2 5,800 4,100
3,750
G5 5,660 5,010
3,660
G8 5,440 4,870 3,600
3,500
K0 5,240 4,720 3,300
3,100
K1 5,110 4,580 2,950

K2 4,960 4,460

K3 4,800 4,210

K4 4,600 4,010

K5 4,400 3,780

K7 4,000

M0 3,750 3,660

M1 3,700 3,600

M2 3,600 3,500

M3 3,500 3,300

M4 3,400 3,100

M5 3,200 2,950

M6 3,100 2,800

M7 2,900

M8 2,700

18.3 Temperature Estimation Applying Wien’s Displacement Law

A further approach is the estimation of with the principle of Wien's displacement law

(sect. 3.2). It is based on the assumption that the radiation characteristic of the star corre-
sponds approximately to that of a black body. Theoretically could be calculated, apply-

ing formula {2}, based on the wavelength , which has been measured at the maximum in-
tensity of the profile. This requires, however, a radiometrically corrected profile as de-

scribed in sect. 8.11. In sect. 3.3 it has already been demonstrated that the position of the
intensity maximum in the pseudo-continuum gives only a very rough indication for the tem-

perature of the radiator.

Further, the maximum intensity must lie within the recorded range – for a typical amateur
spectrograph about 3800 – 8000 Å. In the graphic below this criterion is met only by the
yellow graph for 6000 K. According to formula {2}, within this section, only profiles with

of about 7600 – 3600 K can be analysed by their maximum intensity. This corre-

sponds roughly to the spectral types M1 – F0.

Analysis and Interpretation of Astronomical Spectra 83

Inten-
sity

0 5000 10‘000 15‘000 20‘000

Wavelength [Å]

Therefore, the coverage of all spectral classes requires an adaptation of this method. One
possibility is enabled by the relationship between the continuum slope and .

I

λ

This method is provided by the Vspec software, applying the function Radiometry/Planck.

Thereby the radiometrically corrected profile is iteratively fitted by displayed continuum
curves, corresponding to entered values. This principle is demonstrated by the follow-

ing graph with a synthetically generated solar spectrum from the Vspec Library. Displayed
in red is the continuum curve, fitting to the Sun’s profile with 5800 K. The maximum inten-
sity of this curve lies at about 4900 Å, which, for this temperature, is also in accordance
with Wien’s displacement law (formula {1}). Further displayed for comparison is the black
curve for 10,000 K and the green one, belonging to 4,000 K.

Analysis and Interpretation of Astronomical Spectra 84

I

Synthetic
Solar Spectrum

λ

Of course, in practice, the temperature estimation of the investigated star is requested,
rather than the reproduction of an already known temperature of a synthetic profile. This
requests a radiometric correction of the obtained pseudo-continuum with a recorded stan-
dard star, according to sect. 8.11. Therefore, the accuracy of this measurement is also di-
rectly dependent on the quality of this relatively demanding correction procedure.

The following chart shows, in the context of the synthetic solar spectrum, the continua of
various effective temperatures according to Vspec Radiometry/Planck. In the range be-
tween 3,000 and 20,000 K, the increasingly closer following curves are displayed in 1,000
K steps. Above 20,000 K the intermediate spaces become so close, that here only the con-
tinua for 30,000 and 40,000 K are shown. It also shows that this estimation method is lim-
ited to the middle and late spectral types. In addition, with increasing temperature, the dif-
ferentiating influence of the spectral class to the radiometric correction is significantly de-
creasing (sect. 8.11).

I

7000 K Synthetic
Solar spectrum

λ

Analysis and Interpretation of Astronomical Spectra 85

18.4 Temperature Determination Based on Individual Lines

Generally these methods are based on the temperature dependency of the line intensity
. However, according to sect. 6.2, the intensity and shape of the spectral lines are

determined by many other variables, such as element-abundance, pressure, turbulence,

metallicity Fe/H and the rotation speed of the star. Similar to the determination of the rota-
tion velocity (sect. 16), all such methods must therefore be able to significantly hide such
interfering side influences or analyse lines, which are specifically sensitive to temperature
[11]. If the temperature should not only be "estimated" but rather accurately "determined",
therefore relatively sophisticated methods remain, based on high-resolution spectroscopy
and detailed analysis of especially temperature-sensitive, non ionised metal lines of the late
spectral classes K – M. A representative impression provides [190], [191], [191b]. Here,
ratios are calculated with the relative line depths of differently temperature-

sensitive metal absorptions [11]. These are subsequently calibrated in respect of known
values (LDR Line Depth Ratio). This way an accuracy of a few Kelvin can be achieved.

With a longer lasting temperature monitoring, eg the detection and even measurement of
giant dark sun spots is possible, typically observed at the late spectral type K [191b].

18.5 The “Balmer-Thermometer“ Hβ 4861
The temperature determination, based on Hβ 4861
the intensity of the H-Balmer lines, is often
called "Balmer-Thermometer". This method O9
is rather rudimentary, but it provides an in-
teresting experimental field. The H-lines are 25‘000 K
well suited, because the stellar photo-
spheres of most spectral classes consist to B1
>90% of hydrogen atoms. Only this element
can exclusively be detected and evaluated 22‘000 K
over almost the entire temperature se-
quence (classes O – M). B7
In contrast, ionised calcium Ca II appears
only within the spectral classes A – M. The 15‘000 K
often proposed sodium double line D1,2 is
analysable just at type ~F – M because Na I A1
becomes ionised at higher temperatures and
Na II absorptions appear exclusively in the 10‘000 K
UV range of the spectrum. In the earlier
spectral classes, Na I is therefore always of A7
interstellar origin and hence useless for this
purpose. 7‘550 K
The graphic on the right shows the intensity
profile of the Hβ line – a cutout of the over- F0
view to the spectral sequence in sect. 13.8
and [33]. From all Balmer lines, this absorp- 7‘030 K
tion can be observed within the largest
wavelength range. At this low resolution, it F5
remains analysable, even in the long wave-
length region, down to about K5. The maxi- 6‘330 K
mum intensity is reached at the spectral
class A1. The quantum-mechanical reasons G2
for this effect are discussed in sect. 9.2.
5‘700 K

G8

4‘990 K

K2

4‘290 K

K5

3‘950 K

Analysis and Interpretation of Astronomical Spectra 86

In the diagram below, the Hβ-equivalent widths (EW) of 24 Atlas stars [33] are plotted
against the effective temperatures . Due to the bell-shaped curve an ambigu-

ity arises, which requires a clarification by additional spectral information. A prominent fea-
ture, indicating the profile segment on the long wavelength side of 10,000 K, is the Fraun-
hofer K-line (Ca II) at 3934 Å – further details see [33]. The average temperature range is
here almost exclusively represented by main-sequence stars, the peripheral sections also
by giants of the luminosity class I – III.Despite of relatively few data points and a low reso-
lution (DADOS 200L/mm), the curve shape can clearly be recognised and is manually in-
serted here as an approximate "least square fit". It is just intended to demonstrate the prin-
ciple. For more accurate results the analysis of high-resolution spectra would be required
and further the separation between the luminosity classes.

α Gem
78 Virα CMa

α Aql

EW ζ Leo γ Vir γ Crv
Hβ α Cmi α Leo
[Å]

Sun α Vir 68 Cyg
η Boo δ Ori
α Ori
α Sco ε Vir ε Ori
61 Cyg ζ Ori

Effective temperature Teff [K]

Analysis and Interpretation of Astronomical Spectra 87

19 Spectroscopic Binary Stars

19.1 Terms and Definitions

> 50% of the stars in our galaxy are gravitationally connected components of double or
multiple systems. They concentrate primarily within the spectral classes A, F and G [170].
For the astrophysics these objects are also of special interest because they allow a deter-
mination of stellar masses independently of the spectral class. Soon after the invention of
the telescope, visual double stars were also observed by amateur astronomers. Today the
spectroscopy has opened for us also the field of spectroscopic binary stars.

An in-depth study of binary star orbits is demanding and requires eg celestial mechanics
skills. Here should only be indicated, what can be achieved by spectroscopic means. Scien-
tifically relevant results are usually only possible associated with long term astrometric and
photometric measurements. Spectroscopic binary stars are orbiting in such close distances
around a common gravity center, that they can’t be resolved even with the largest tele-
scopes in the world. They betray their binary nature just by the periodic change in spectral
characteristics. For such close orbits Kepler's laws require short orbital periods and high-
track velocities, significantly facilitating the spectroscopic observation of these objects.

In contrast to the complex behaviour of multiple systems, the motion of binary stars follows
the three simple Kepler’s laws. Its components rotate at variable velocities in elliptical

orbits around a common Barycenter B (center of gravity). The following sketch shows a fic-

tional binary star system with stars of the unequal sizes and . For simplicity their ellip-

tical orbits are running here exactly in the plane of the drawing as well as the sight line to
Earth which in addition runs parallel to the minor semi-axes. For this perspective special
case the orbital velocity at the Apastron (farthest orbital point) and Periastron (closest

orbital point) corresponds also to the observed radial velocity . The recorded maximum

values (amplitudes) are referred in the literature with . The following layout corresponds

to an orbit- inclination of (sect. 19.3).

VrM1 P= K1 VrM2 A

Apastron Periastron B Periastron Apastron

M1 M2 M1 Minor semi axis b Major semi axis a M2

VrM1 A VrM2 P= K2

Sight line to VrM1 A = Radial velocity M1 at Apastron
Earth VrM1 P = Radial velocity M1 at Periastron
VrM2 A = Radial velocity M2 at Apastron
VrM2 P = Radial velocity M2 at Periastron

Analysis and Interpretation of Astronomical Spectra 88

– Both orbital ellipse:
– must be in the same plane
– have different axis lengths, inversely proportional to their stellar masses

– must be similar to each other, i.e. have the same eccentricity

– The more massive star always runs on the smaller elliptical orbit and with the lower

velocity around the barycenter.

– The barycenter lies always in one of the two focal points

– und always run synchronously:

– During the entire orbit the connecting line between and runs permanently
through the barycenter

– and always reach the Apastron as well as the Periastron at the same time.

19.2 Effects of the Binary Orbit on the Spectrum

Due to the radial velocities the Doppler shift causes striking effects in the spectrum. The
above assumed perspective special case for the orbit orientation would maximize these
phenomena for a terrestrial observer (see below phase D). Generally, two different cases
can be distinguished [180].
1. Double stars with two components in the spectrum – SB2–systems

If the apparent brightness difference between the two components lies approximately at
, we can record the composite spectrum of both stars. The following phase diagrams

are based on the above assumptions and show these effects within one complete orbit:

Here the orbital velocities are di- ∆λ

rected perpendicular to the line of sight A
and thus the radial velocity with respect
to the Earth becomes The spec-

trum remains unchanged, i.e.

In the Apastron the orbital velocities ∆λA

are minimal. But they run now parallel to B

the line of sight, and correspond here to
the radial velocities, so . The

spectral line appears splitted:

Here the orbital velocities are di- ∆λ
∆λP
rected perpendicular to the line of sight C
and thus the radial velocity with respect D
to the Earth becomes The spec-

trum remains unchanged, i.e.

In the Periastron the orbital velocities

are maximal. They run now parallel to the

line of sight, and correspond here to the
radial velocities, so . The

spectral line appears here more splitted
compared to phase B.

Analysis and Interpretation of Astronomical Spectra 89

The above introduced formulas for the Doppler shift allow, after a simple change, to cal-
culate the sum of the radial velocities with the line splitting . For the general

case:

The Calculation of the Individual Radial Velocities and

If the mass difference is large enough, the splitting of the ∆λ1 ∆λ2
spectral line occurs asymmetrically with respect to the neu-
tral wavelength . With these uneven distances and

and by analogy to {39} the individual radial velocities
can be calculated separately [172].

As a result of the heliocentric radial movement of the entire λ1 λr0 λ2

star system [100], the wavelength of the "neutral",

unsplitted spectral line is shifted by the Doppler Effect from the stationary laboratory wave-
length to [170].

This is now the adjusted reference point for the measurement of the two distances , .
But first the , values must be heliocentrically corrected to , according to [30], sect.
10.3, step 7. Then it follows:

,,

If no asymmetry occurs of the splitted line with respect to , and are approximately
equal and the sum of the two radial velocities just needs to be halved.

2. Double stars with only one component in the spectrum – SB1–systems

In most cases the apparent brightness difference between the two components is signifi-
cantly . Here with amateur equipment, only the spectrum of the brighter star can be

recorded. Extreme cases are entirely invisible black holes as double star components, or
extrasolar planets, which shift the spectrum of the orbited star by just a few dozen me-
ters/sec! In such cases, a splitting of the line isn’t recognisable, but only the shift of to

the right or left in respect of the neutral position .

The following example, recorded with λr0 Adjusted reference point
DADOS 900L/mm shows this effect,
using the spectroscopic A components
within the quintuple system β Scorpii. 08.18.2009 22‘00 UTC 08.15.2009 22‘00 UTC
Impressively visible is here the Hα-shift ∆λ = –1.37Å = –63 km/s ∆λ = +1.71Å = +78 km/s

of the brighter component, within three
days. With this Vspec plot just this ef-
fect shall be demonstrated. A serious
investigation of the orbital parameters
would require the recording of multiple
orbits at least at daily intervals! The line
shifts are plotted here with respect to
. The X-axis is scaled here in Doppler

velocity, according to [30], sect. 19,
and allows a rough reading of the radial
velocity. The values , have been determined by Gaussian fit on the heliocentrically cor-
rected profiles. Detailed procedure see [30] sect. 24.2.

Analysis and Interpretation of Astronomical Spectra 90

Here are some orbital parameters of β Scorpii according to an earlier study by Peterson et
al. [177]. These values are based on the measured, maximum radial velocities und ,

obtained from the spectra of both components. The terms in the data list, and

for the masses, demonstrate that the inclination is unknown and therefore the

values are uncertain by this factor (for details see sect. 19.3 and 19.4).

- Spectral class of the brighter component: B0.5V, of the weaker component: unknown
- Orbital period:

- Stellar masses: , ss

- Mass Ratio:

- Max. recorded radial velocities: ,

- Major semi axes of the orbital ellipse: s s

These figures show that the mass and brightness difference is substantial. With the large
telescopes, involved in this study [177], the spectrum of the weaker component could still
be recognised, but analysed just with substantial difficulties.

19.3 The Perspectivic Influence from the Spatial Orientation of the Orbit

The orientation of the binary star orbital planes, with respect to our line of sight, shows a
random distribution. The angle between the axis, perpendicularly standing on the orbital
plane (normal vector), and our line of sight is called inclination [175]. Thus the definition

for the inclination of stellar and binary system rotation axes (sect. 16) is the same. Analogi-
cal, s is here the spectroscopically direct measurable share of the radial velocity ,
which is projected into the sight line to Earth. For we see the orbital ellipse exactly

"edge on" i.e. s .

– This elliptical orbit, with a given and fixed inclination , may be rotated freely around the
axis of the sight line, without any consequences for their apparent form.

– Thus for circular binary orbits the inclination fixes the only degree of freedom, which
affects the apparent shape of the orbit.

– For elliptical binary star orbits, the situation is more complex. In contrast to the circle,
the orientation of the ellipse axes in a given orbital plane forms an additional degree of
freedom, determining the apparent ellipse form.

vr

vr sin i i
vr sin i
Sight Line to Earth

vr

If remains unknown, the results can statistically only be evaluated, similar to the
s values of the stellar rotation (sect. 16.6).

Caveat:

There are also reputable sources that define as the angle between the line of sight and the

orbital plane, similar to the inclination angle convention between planetary orbits and the
ecliptic. Consequence: There must always be clarified which definition is used. The two
conventions can easily convert to each other as complementary angle .

Analysis and Interpretation of Astronomical Spectra 91

19.4 The Estimation of some Orbital Parameters

Based on purely spectroscopic observation some of the orbital parameters of the binary
system can be estimated. First, the measured radial velocities and are plotted as a

function of the time . The graph shows an orbital period of Mizar (ζ Ursae Majoris, A2V),

one of the showpieces for spectroscopic binaries with double lines (SB2 systems). Another
similar example is β Aurigae with an orbital period of about 4 days. The more these curves
show a sinusoidal shape, the lower is the eccentricity of the orbit ellipse [179].

K1

K2

Source: Uni Jena [170]

The Orbital Period

The orbital period T can directly be determined from the course of the velocity curves. As
the only variable it remains largely unaffected by perspective effects and is thus relatively
accurately determinable.

Simplifying to circular orbits

Since we are mostly confronted with randomly oriented, el- M1 rM1 B rM2 VM2
liptical orbits, a reasonably accurate determination of orbital M2
parameters is very complex. For this purpose, in addition to VM1
the spectroscopic, complementary astrometric measure-
ment data would be needed. Only for eclipsing binaries,
such as Algol, already a priori, a probable inclination of

can be assumed. For the rough estimation of other
parameters various sources suggest the simplification of the
elliptical to circular orbits. Thus, the radii and thus the
orbital velocities become constant. The mostly unknown
inclination is expressed in the formulas with the term .

The Orbital Velocity

To determine the orbital velocity we need from the velocity diagram, the maximum

values for both components. They correspond by definition, to the maximum amplitudes
and . For the circular orbit velocity follows:

s s
Determination of the orbital radii

With the orbital period and the velocity s for a circular orbit the corresponding

radius can be calculated. From geometric reasons follows generally:

s

Analysis and Interpretation of Astronomical Spectra 92

s

If both lines can be analysed, then with the obtained und the corresponding radii
and can be calculated separately.

Calculation of the Stellar Masses in SB2 Systems

If both lines can be analysed, then with the following formula the total mass of the system
, can be determined. This is obtained if {44}, changed as a sum of radii , is

used in the formula of the third Kepler's law, instead of the semi-major axis .

. with the partial radii

The partial masses can then be calculated by their total mass
and

For binary star systems is often expressed in solar masses and the distance in AU.
To convert:

Calculation of the Stellar Masses in SB1 Systems

The analysis of only one line has logically consequences on the information content and ac-
curacy of the system parameters to be determined. For SB1–systems in the spectrum, only
the radial velocity curve of the brighter component can be analysed, and therefore just

the amplitude can be determined. So with formula {44}, only the corresponding orbital

radius can be calculated. Unfortunately, neither their total mass nor their partial masses

, are directly determinable. Just the so called mass function , can be deter-

mined [51 sect. VI], [179]. This expression is not really demonstrative and means the cube
of the invisible partial mass in relation to the square of the total mass . For

a calculated example, see [171].

s
,

It can be roughly determined eg by estimation of the visible mass by the spectral type.
For a rough guide refer to the table in sect. 14.5. Such results still remain loaded with the
"uncertainty" of s .

Analysis and Interpretation of Astronomical Spectra 93

20 Balmer–Decrement

20.1 Introduction

In spectra where the H-Balmer series occurs in emission, the line intensity fades with

decreasing wavelength . This phenomenon is called the Balmer-Decrement D. The inten-

sity loss is reproducible by the laws of physics and is therefore a highly important indicator
for astrophysics. The hydrogen emission lines are formed by the electron transitions, which
end, as well known, in "downward direction" on the second-lowest energy level . The

probability, from which of the higher levels an electron comes, is determined for the indi-
vidual lines by quantum mechanical laws. From this it follows that the intensity is highest

at the line and gets continuously weaker at the shorter wavelength lines, , , ,

, etc. The extent of this decrease (decrement) is additionally, but only moderately, de-

pendent on the density and temperature of the electrons and .

20.2 Qualitative Analysis
For most amateurs probably qualitative applications of this effect are in the focus. On a low-
resolution spectrum of P Cygni (DADOS 200L), the intensity-decrease of the dominant hy-
drogen emission lines, normalised on a unified continuum intensity, is demonstrated.

Balmer – Decrement P Cygni Hα

Hβ

Hδ Hγ

Contrary to this trend Mira (o Ceti) shows only Hδ, and significantly attenuated, Hγ in emis-
sion. These impressive lines indicate what enormous intensities the Hα and Hβ-Balmer
emissions should really show, according to the Balmer-Decrement. But towards the long-
wave region, starting from ca. , these lines get, like almost all other discrete absorptions,
increasingly overprinted by titanium oxide vibration bands (TiO). This effect generates an
unusual, negative Balmer-Decrement, because TiO arises obviously in higher layers of the
stellar atmosphere, as the H emission lines, details see [33].

Balmer – Decrement Mira, o Ceti

Hδ Hγ TiO

TiO TiO TiO TiO
TiO
TiO
TiO

Analysis and Interpretation of Astronomical Spectra 94

20.3 Quantitative Analysis

Quantitatively analysed gets primarily the intensity deviation of the recorded hydrogen
emissions from the theoretically calculated Balmer-Decrement of the light-emitting source.
This effect is caused by the damping effects according to sect. 8.2, which selectively at-
tenuate the shortwave, blue part of the spectrum, more than the red region. The steeper the

decrement , compared to the theoretically determined values , the greater the absorb-

ance. The steeper the decrement runs, compared to theoretically calculated values ,

the stronger is the extinction of light. By convention, the intensity values are indicated in

relation to :

Inter alia, this allows the measurement of the "Interstellar Extinction" or the "Interstellar
Reddening". However the most important application, at least for amateurs, is probably the
reconstruction of the original intensity ratios of the recorded emission lines ( and -

values), which appear attenuated, even related to the continuum. This is performed using
the "template" of the theoretical Balmer-Decrement (see sect. 21.2). The proportional-
radiometric methods, as described in sect. 8.8, are not applicable here, because they can't
change the and -values and remain therefore reserved to the continuum-bound ab-
sorption lines.

The measurement of the Balmer-Decrement requires appropriate objects, radiating the
emission-lines of the H-Balmer series completely – for example most of the emission nebu-
lae, Be- and LBV stars. Representatives of the Mira Variables are unsuitable for the afore-
mentioned reasons. The following table shows by Brocklehurst [200] the theoretical dec-
rement values , quantum mechanically calculated for gases with a very low and high
electron density and electron temperatures of 10000K and 20000K.

Line Case A for Case B für
Hα thin gas (Ne = 102 cm-3) dense gas (Ne = 106 cm-3)

Te =10 000 K Te =20 000 K Te =10 000 K Te =20 000 K

2.85 2.8 2.74 2.72

Hβ 1 1 11

Hγ 0.47 0.47 0.48 0.48

Hδ 0.26 0.26 0.26 0.27

Hε 0.16 0.16 0.16 0.16

H8 0.11 0.11 0.11 0.11

In the specialist jargon the two density cases are called "Case A" and "Case B".

Case A: A very thin gas, which is permeable for Lyman photons so they are enabled to es-
cape the nebula.

Case B: A dense gas, which retains the short-wavelength photons, which
are therefore available for self-absorption processes.

The gas densities in emission nebulae are in the range between Case A and -B, by expand-
ing stellar envelopes (Be- and LBV stars, Planetary Nebulae) Case B [33]. The electron tem-
perature has in this range apparently only little influence on the decrement. The influ-
ence of the electron density is here noticeable only at the Hα line. Depending on the
source, these values may differ in some cases. Possible amateur applications are here
rough decrement comparisons between different objects, eg various planetary nebulae, as
well as Be- or LBV stars, located on different galactic latitudes.

Analysis and Interpretation of Astronomical Spectra 95

20.4 Quantitative Definition of the Balmer-Decrement is

For most astrophysical analysis the measured intensity ratio of the Hα and Hβ lines
required. This corresponds to the quantitative definition of the Balmer-Decrement:

{50}

20.5 Experiments with the Balmer-Decrement

The following graph shows the Balmer-Decrement of P Cygni, recorded with the DADOS
spectrograph. Once the decrements are referred to a radiometrically corrected profile (red),
which corresponds to the intensity profile of a B2 II star from the Vspec library, and once
based on a continuum with a normalised energy flow (blue). Due to the low spectral resolu-
tion the labelled decrements are intensity values , directly measured in the strongly
reddened P Cygni profile and subsequently divided by the corresponding Hβ intensities

.

Irel Hβ Hα

Hγ 1 1.8
Hε 5

0.5

1 λ
0.3

Flux normalised continuum

Within the intensity-normalised, blue profile, the roughly determined decrements match
pretty well to the spread of the accepted values –> for (Hα)/ (Hβ) between 4.3 – 5. The

red profile, represents approximately the intensity course of the original- and unattenuated
spectrum. It shows however drastically different and physically even impossible decrement
values. The defined Balmer-Decrement is here only 1.8, ie substantially

lower than !

The theoretical Balmer-Decrement is based on the probabilities for certain electron

transitions, determined by the laws of quantum physics. This statistical regularity (sect.
20.3) is overprinted here in the red-, radiometrically corrected profile by the blackbody-like

distributed and wavelength-dependent radiation intensity of the star, i.e. The

intensity of the individual emission lines depends in stellar atmospheres, inter alia on the
number of the occurring direct- or resonance absorptions and thus also on the photon den-

sity at the corresponding wavelength . In this respect, the blue, intensity-normalised pro-

file behaves here neutrally, due to

If a continuum is present, also in the professional sector, the Balmer-Decrement is deter-
mined by the EW-values and therefore also based on an intensity-normalised profile. At the
example of extra-galactic spectra this is demonstrated in The Balmer-Decrement of SDSS
Galaxies, by Brent Groves et al. [210].

Analysis and Interpretation of Astronomical Spectra 96

21 Spectroscopic Determination of Interstellar Extinction

21.1 Spectroscopic Definition of the Interstellar Extinction
With the Balmer-Decrement , the interstellar extinction can spectroscopically be deter-
mined. The extinction parameter characterises the entire extinction along the line of sight
between the object and the outer edge of Earth's atmosphere. It is defined as the logarith-
mic ratio between the theoretical (Th) and measured (obs) intensity of the line [8]:

, also called “logarithmic Balmer-Decrement” [201], is determined by the ratio be-

tween the measured and theoretical Balmer-Decrement and . The value –0.35 cor-

responds to the extinction factor at , according to the standard extinction curve

from Osterbrock (chart below) [201].

In the context of such calculations in the literature the value (Case B) has been

established for the theoretical Balmer-Decrement. inserted in yields:

21.2 Extinction Correction with the Measured Balmer-Decrement

The extinction is not constant but depends on the wavelength. With the correction function
[10], the emission lines are adapted relatively to (“Dereddening”).

is defined as follows, where is the extinction at and .

Thus, the measured intensities are reduced for and raised for (note

the sign of )! The value of is determined with an extinction curve, which exists in

different versions with slightly different values. For amateur applications intermediate val-

ues may be roughly interpolated. Bottom left is the galactic standard extinction curve from
Osterbrock (1989) with values [238]. The table values to the right are from Seaton

(1960).

–0.35 λ f(λ)
0.0 3500 +0.42
+0.14 4000 +0.24
4500 +0.10
Galactic Extinction Law from Osterbrock1989 4861
5000 0.00
6000 -0.04
7000 -0.26
8000 -0.45
-0.60

From Seatons 1960

Analysis and Interpretation of Astronomical Spectra 97
in [mag]
21.3 Balmer-Decrement and Color Excess

The measured Balmer-Decrement also determines the color excess
for the Balmer lines [10].

If follows: , in this special case, as expected, exists no
reddening.

The link to the "classical" photometry in the system provides the formula of C.S.
Reynolds [208]:

The associated parameters for:

Logarithmically transformed and inserted :

21.4 Balmer-Decrement and Extinction Correction in the Amateur Sector

Since amateurs (still) have no access to own space telescopes, here neither the exact de-
termination of the Interstellar extinction, nor of the reddened Balmer-Decrement is realistic
or useful. Furthermore the required, selective, accurate correction of atmospheric

and instrumental attenuation on the emission lines is not feasible with the here

presented methods. As the most important application remains the reconstruction of the
original intensity ratios for emission line objects. However, for this purpose, as already out-
lined, the radiometric methods according to sect. 8.10 - 8.11, totally fail. Fortunately most

of these objects produce H-emission lines. For example in emission nebulae, these are
generated far away from the star and mainly by the recombination of ionised H-atoms. Thus
their intensities correspond in the original spectrum almost to the unattenuated H-Balmer-
Decrement. Therefore, they can be applied – in relation to the measured decrement values
– as a kind of "correction template" even for all other species of emission lines.

The method according to sect. 21.2 is indeed provided for the partial correction of the in-

terstellar reddening . For amateur purpose formula {53} enables, within the relevant

range of Hα – Hβ, also for the other attenuating influences, and in a reasonable approxima-
tion, a very rough total intensity correction of the emission lines. The difference between
the original and pseudo-continuum is here, for the early and middle stellar spectral classes,
relatively low (see chart sect. 8.10). Further and show a quite similar char-

acteristic with an increasing attenuation towards shorter wavelengths. Anyway somewhat
different behaves as most of the today’s amateur cameras show a damping char-

acteristic, starting just from the green sector of the spectrum. Supplementary notes see
sect. 22.11.

In the professional sector, extinction corrections are performed with software support and
are indispensable by the analysis of extragalactic emission line objects. Even for objects

within our neighbour galaxy M31, usually does [201].

Analysis and Interpretation of Astronomical Spectra 98

22 Plasma Diagnostics for Emission Nebulae

22.1 Preliminary Remarks

In the “Spectroscopic Atlas for Amateur Astronomers” [33], a classification system is pre-
sented for the excitation classes of the ionised plasma in emission nebulae. Further this
process is practically demonstrated, based on several objects. Here, as a supplement, fur-
ther diagnostic possibilities are introduced, combined with the necessary physical back-
ground.

22.2 Overview of the Phenomenon “Emission Nebulae”

Reflection nebulae are interstellar gas and dust clouds which passively reflect the light of
the embedded stars. Emission nebulae however are shining actively. This process requires
that the atoms are first ionised by hot radiation sources with at least 25,000K. This re-
quires UV photons, above the so-called Lyman limit of 912 Å and corresponding to an ioni-
sation energy of >13.6 eV. This level is only achievable by very hot stars of the O- and early
B-Class generating this way a partially ionised plasma in the wider surroundings. By recom-
bination the ions recapture free electrons which subsequently cascade down to lower lev-
els, emitting photons of discrete frequencies , according to the energy difference

. Thus, this nebulae produce by the fluorescence effect, similar to gas discharge
lamps, mainly "quasi monochromatic" light, i.e. a limited number of discrete emission lines,
which, with exception of the Supernova Remnants (SNR), are superimposed to a very weak
emission-continuum. The energetic requirements are mainly met by H II regions (e.g. M42),
Planetary Nebulae PN (e.g. M57) and SNR (e.g. M1). Further mentionable are the Nuclei of
Active Galaxies (AGN) and T-Tauri stars (sect. 17.2). The matter of the Nebulae consists
mainly of hydrogen, helium, nitrogen, oxygen, carbon, sulphur, neon and dust (silicate,
graphite etc.). Besides the chemical composition, the energy of the UV radiation, and the
temperature as well as density of the free electrons characterise the local state of the
plasma. This manifests itself directly in the intensity of emission lines, which simply allows
a first rough estimation of important plasma -parameters.

22.3 Common Spectral Characteristics of Emission Nebulae Olll 5006.84 Hα 6562.82
In all types of emission nebulae, ionisation proc-
esses are active even if with very different exci-
tation energies. This explains the very similar
appearance of such spectra. The diagram shows
an excerpt from the spectrum of M42 with the
two most striking features:
Olll 4958.91
1. The intensity ratio of the brightest [O III] lines . Hβ 4861.33
is practically constant with

2. The difference between the measured and
theoretical course of the Balmer-Decrement

depends on the interstellar extinction as

well as and (sect.21).

22.4 Ionisation Processes in H II Emission Nebulae

These processes are first demonstrated schematically with a hydrogen atom. The high-
energy UV photons from the central star ionise the nebula atoms and are thus completely
absorbed, at latest at the end of the so-called Strömgren Sphere. Therefore here ends the
partially ionised plasma of the emission nebula. Since the observed intensity of spectral
lines barely varies, a permanent equilibrium between newly ionised and recombined ions
must exist. Rough indicators for the strength of the radiation field are the atomic species,

Analysis and Interpretation of Astronomical Spectra 99

the ionisation stage (sect. 11) and the abundance of the generated ions. The first two pa-
rameters can be directly read from the spectrum and compared with the required ionisation
energy (see table below and [33]).

Photon Elektron

Star

Y

Te Ne

High-energy Ionisation by UV- Electron gas
Radiator Teff>25‘000K Photons λ<912Å

The kinetic energy of the electrons, released by the ionisation process, heats the nebula

particles. corresponds to the surplus energy of the UV Photons, which remains after

photo ionisation and is fully transformed to kinetic energy of the free electrons. The elec-
tron temperature and -density affect the following recombination- and collision exci-

tation processes. is directly proportional to the average kinetic energy of the free elec-

trons (Boltzmann constant ).

Formula {55} yields in Joule with the electron mass and .
The short formula {55a} gives directly in electron volts [eV].

22.5 Recombination Process Electron

If an electron hits the ion centrally, it is captured and ends up first Photon
mostly on one of the upper excited levels (terms). The energy, gen-
erated this way is emitted as a photon . It corresponds to the +ΔEn
sum of the original kinetic energy of the electron and the dis-
crete energy difference due to the distance to the Ionisation
Limit. Since the share of the kinetic energy varies widely, from
the recombination process a broadband radiation is contributed to
the anyway weak continuum emission.

22.6 Line Emission by Electron Transition Recombination

After recombination the electron “falls” either directly or via sev-
eral intermediate levels (cascade), to the lowest energy ground

state . Such transitions generate discrete line emission, ac- Photon

cording to the energy difference . Most of these photons ΔEn

leave the nebula freely – including those which end in the pixel
field of our cameras! This process cools the nebula, because the
photons remove energy, providing thereby a thermal balance to n=1
the heating process by the free electrons. This regulates the elec-
tron temperature in the nebula in a range of ca.
5,000K < < 20,000K [237].

Electron Transition

Analysis and Interpretation of Astronomical Spectra 100

22.7 Line Emission by Collision Excitation

If an electron hits an ion, then in most cases not a recombination

but much more frequently a collision excitation occurs. If the impact Electron
energy is ≥ the electron is briefly raised to a higher level. By

allowed transitions it will immediately fall back to ground state and
radiate a photon of the discrete frequency , according to
the energy difference. Photon

Remark: Similarly, this process takes place in fluorescent lamps ΔEn

with low gas pressure. Due to the connected high tension, the elec- n=1
trons reach energies of several electron volts [eV], which subse-
quently excite mercury atoms to UV radiation. By contrast in dense
gases, the excitation occurs mainly by collisions between the ther-
mally excited atoms or molecules.
Collision Excitation

22.8 Line Emission by Permitted Transitions (Direct absorption)

In H II regions with O5 type central stars (eg M42) emission nebu-
lae have Strömgren spheres with diameters of several light years,
what extremely dilutes the radiation field. Thus, particularly in the
extreme outskirts of the nebula, the probability gets extremely
Photon Photon

small, that the energy of a photon exactly fits to the excitation
level of a hydrogen atom. Therefore the direct absorption of a pho-
ton doesn’t significantly contribute to the line emission. Further
the main part of the photons is radiated in the UV range. Conse-
quently many atoms are immediately ionised, once the energy of
the incident photons is above the ionisation limit. Therefore a sub-
stantial line emission of permitted transitions is only possible by
tahcetorrescoofmtbhienpateiromnipttreodcetrsasn. sTihtieonhsig, hisscpaeucsteradlbinyttehnesiatybuonfdHayndcreogwehnicahndisDHbireyeliscuetmvAe,brtahsleoormrpdateiiornsn
of magnitude higher than the remaining elements in the nebula. The frequency of a specific
electron transition also determines the relative intensity of the corresponding spectral line.

22.9 Line Emission by Forbidden Transitions

Emission nebulae contain various kinds of metal ions, most of them with several valence
electrons on the outer shell. These cause electric and magnetic interactions, which multi-
plies the possible energy states. Such term schemes (or Grotrian diagrams) are therefore
extremely complex and contain also so-called "Forbidden Transitions" (sect. 12). But the
extremely thin nebulae provide ideal conditions, because the highly impact-sensitive and
long-lasting metastable states become here very rarely untimely destroyed by impacts.

But first of all, these metal atoms must be ionised to the corresponding stage, which re-
quires high-energy UV-photons. The required energies are listed in the following table,
compared to hydrogen and helium [eV, λ]. The higher the required ionisation energy, the
closer to the star the ions are generated (so called “stratification”) [10] [201].

Ion [S II] [N II] [O III] [Ne III] [O II] H II He II He III

E [eV] 10.4 14.5 35.1 41.0 13.6 13.6 24.6 54.4

λ [Å] 1193 855 353 302 911 911 504 227

On the other hand The forbidden transitions of the metal ions need to populate the meta-
stable initial terms from the ground state just a few electron volts [eV]. This small amount
of energy is plentifully supplied by the frequent collision excitations of free electrons!