Atoms and spectral lines
Atoms: small dense nucleus consisting of protons and neutrons, usually with:
p ≈ n.
Light electrons orbit the nucleus like planets around a star, but more complicated
orbits!
In Bohr’s model electron orbits are quantized. That is, electrons stay in discrete,
well-defined ‘shells.’ More precisely,….
1. Orbits (shells) balance Coulomb attraction and centripetal force,
mv 2 (Ze)e
= 2 .
r
r
The discreteness comes from the assumption that orbital
angular €
momentum is quantized mvr = nh/(2π)
2. Note especially the existence of a lowest or ground state
orbit with n=1.
This doesn’t exist classically, it is an intrinsically quantum
phenom.
3. The spacing between the orbits gets smaller for large n.
Proof:
2
m  nh  (nh /(2π )) 2 Ze 2
= 2

 =
3
r  2πmr 
mr
r
where we've substituted for v.
(nh /(2π )) 2
∴ rn =
.
2
mZe
 (h /2π) 2 
2
2
Then, Δr = rn +1 − rn = 
(n
+
1)
−
n
,

[
]
2
 mZe 
€
€
Δr 2n + 1 2
=
≈ , for large n.
2
r
n
n
In terms of orbital energy:
2
4 2
Ze
me
Z
E n = 12 mv 2 −
=− 2
r
2n (h /2π) 2
me 4 Z 2  1
1
so, ΔE = E n +1 − E n = −
− 2 .
2
2
2(h /2π)  (n + 1)
n 
Energy levels
€
0
-E1
4. There exists a continuum, E ≥ 0, where electrons are no longer bound.
5. The structure of orbits is different for each element. The orbits are drawn
closer in for elements with more protons, larger A, Z.
6. Within shells there can be subshells (quantum numbers l, m, …). The number
of electrons in the outer shell largely determines the chemistry.
7. The Pauli exclusion principle states that no more than 2 electrons can occupy
a given orbital.
Line Radiation
Absorption:
e- lifted from the ground state Eg to excited state Ee by absorbing a photon of
energy: ε = hν = Ee - Eg
More generally, hνnm = En - Em,
for n > m.
Ee
Eg
Emission:
e- spontaneously drops to lower orbits, until it reaches the ground state.
Photon emitted in each transition. How?
“After the fifty years of concious brooding have brought me no closer to the answer to
the question, ‘What are light quanta?’ Of course today every rascal thinks he knows the
answer, but he is deluding himself.”
(A. Einstein, 1951)
The most important example of emission & absorption lines in astronomy is the
spectrum of the hydrogen atom.
Paschen series
Hα
656.3
Hβ
486.1
Balmer series
Ly α
121.6
nm
Lyman series
Excitation and Ionization
At low temperatures and with no external disturbances atoms like to live in the
ground state. Two ways to excite electrons to higher energy levels.
1. Collisional excitation
e-
Atom
Some of the relative kinetic energy of the collision is used to boost the
electron.
Given a Maxwellian velocity distribution…
Low T <--> exponentially few fast collisions <--> little excitation.
High T <--> many fast collisions <--> lots of excitation.
2. Collisional ionization
If enough energy is imparted to the electron in the collision it may be knocked free.
e-
3. Photoabsorption
Electron absorbs photon with energy equal to the difference between levels.
For photoionization need Ephoton > |Ebind|.
Atom
Kirchhoff’s laws redux: we can now understand them from a microscopic
point of view.
2. Low density, non-opaque gas that is hot
enough to excite n = 2,3,… levels effciently
---> emission lines.
1. Very hot, opaque gas, with much ionization,
excitation and recombination
---> continuum emission.
3. Cold gas: photoabsorption
---> reemission into random directions
+ absorption lines in forward direction.
Kirchhoff’s laws and stellar spectra
1. Approximate thermal continua come from radiation originating in deep
layers, with many scatterings on the way up.
2. Massive stars have very hot outer layers ---> emission lines superimposed
on the continuum.
I
λ −−>
3. Less massive stars have cooler outer layers ---> absorption lines.
I
λ −−>
Spectral classification: arranging the stars
Stars can be classified by the relative strengths of important lines in their spectra
(as well as by color). The development of the Henry Draper system
(OBAFGKM) was an important accomplishment during the early 20th century.
Originally, it was a sequence A-P based on the strength of the Balmer lines, but
as it evolved some letters were dropped, others rearranged.
Recall: the optical Balmer line series goes from level n to level 2 in hydrogen.
In reality, need to use lines from other elements too.
Spectral Classification Criteria (from text)
Type
Characteristics
O (1,2…, 9)
Strong HeII (He+) lines, in absorption or emission. Strong
UV continuum. Weak HeI. Prominent H lines. SiIV, OIII, NIII,
CIII.
B
Strong HeI in absorp. H lines stronger. MgII, SiII.
A
H lines at max in A0. FeII, SiII, MgII at max. Some CaII. Weak
neutral metals.
F
H lines weaker. CaII H&K stronger. Neutral metals stronger.
G
CaII lines dominate. H very weak. FeI, MnI, CaI become stronger.
CH G-band strong.
K
Neutral metals strong. Begion TiO bands.
M
Neutral metals and molecules (CH, TiO, etc.).
Luminosity Class: the 2nd dimension.
Spectral classification has a 2nd dimension. It was discovered empirically that
intrinsically brighter stars have narrower lines. This led to the Morgan-Keenan
luminosity classification.
MV (near G0)
I
(a, ab, b)
Supergiant
-5, -8
II
“
Bright giant
-2, -4
III
“
Giant
0, -2
IV
Subgiant
1, 2
V
Dwarf (main sequence) 3, 4
VI
Subdwarf
The complete spectral type (e.g., G2V) nearly uniquely places a star in the HR
diagram (later).
Physics of luminosity class
= pressure broadening due to surface gravity
E.g., compare a giant and dwarf at the same surface temp.
P ~ ρT is lower in the giant, so the thermal Doppler line width is lower.
Dependence of L.C. on abundances
Classification schemes assume solar abundances. Slight changes are needed for
metal-poor stars.
OBAFGKM and more. Each spectral type is divided into 10 subclasses, A0, A1, A2, ...A9 etc. The spectral
types and sub-classes represent a temperature sequence, from hotter (O stars) to cooler (M stars), and from
hotter (subclass 0) to cooler (subclass 9). The temperature defines the star's "color" and surface brightness.
Spectral Type
O
B
A
F
G
K
M
C
Surface Temperature
> 25,000K
10,000-25,000K
7,500-10,000K
6,000-7,500K
5,000-6,000K
3,500-5,000K
< 3,500K
< 3,500K
Distinguishing Features
H; HeI; HeII
H; HeI; HeII absent
H; CaII; HeI and HeII absent
H; metals (CaII, Fe, etc)
H; metals; some molecular species
metals; some molecular species
metals; molecular species (TiO!)
metals; molecular species (C2!)
Stars are also classified by luminosity class. Luminosity classes are determined from spectral features and
photometric measurements, coupled with information regarding the distance to the star and theamount of
extinction of the starlight from interstellar material. The luminosity class designation describes the size
(gravitational acceleration in photosphere) of a star from the atmospheric pressure. For larger stars of a given
spectral type, the surface gravity decreases relative to what it was on the main sequence, and this decreases
the equivalent widths of the absorption lines.
Luminosity Class
0
Ia
Ib
II
III
IV
V
sd
D
Description
Hypergiants
Supergiants!
Supergiants!
Bright Giants
Giants
Sub-Giants
Dwarfs
Sub-Dwarfs
White Dwarfs
Comments
extreme
large and luminous
less luminous than Ia
Main Sequence
From cfa-www.harvard.edu/~pberlind/atlas/htmls/note.html
Sample stellar spectra
A0V
B1V
O6V
F6V
G0V
K3V
M2V
N-type
AGB