Transcript Document

DEPARTMENT OF PHYSICS AND ASTRONOMY
LIFECYCLES OF STARS
Option 2601
M.R. Burleigh 2601/Unit 2
Stellar Physics
 Unit 1 - Observational properties of
stars
 Unit 2 - Stellar Spectra
 Unit 3 - The Sun
 Unit 4 - Stellar Structure
 Unit 5 - Stellar Evolution
 Unit 6 - Stars of particular interest
M.R. Burleigh 2601/Unit 2
DEPARTMENT OF PHYSICS AND ASTRONOMY
Unit 2
Stellar Spectra
M.R. Burleigh 2601/Unit 2
Unit 1 Slides and Notes
 Reminder, can be found at…
– www.star.le.ac.uk/~mbu/lectures.html
 In case of problems see me in lectures
or email me… [email protected]
M.R. Burleigh 2601/Unit 2
Book Chapters
 Zeilik and Gregory
– Part II, Chapters 8,10-13,
– Part III, Chapters 15-18
 Phillips
– Chapters 1-6
M.R. Burleigh 2601/Unit 2
Stellar Spectra
 Review of atomic physics
 Absorption and emission processes
 Qualitative treatment of spectral line
formation
 Atmospheric opacity
 Spectral classification of stars
 Hertzsprung-Russell diagram
 Atmosphere models
M.R. Burleigh 2601/Unit 2
Basic Atomic Physics
Bohr atom – quantized orbits
æ h ö
Bohr postulate: mvr = n ç
÷
è 2p ø
2p 2 me 4 Z 2
Energy of orbits: E ( n) = n2h2
As n  , E  0
M.R. Burleigh 2601/Unit 2
NB. It is –ve
i.e. bound
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Quantized Radiation
Electron transition between orbits
Emission:
E = h
E na   E nb   h
If na > nb
Absorption:
E nb   h  E na 

E na   E nb 
Frequency of photon:  ab 
h
M.R. Burleigh 2601/Unit 2
Quantized Radiation
 Emission – transition from higher to
lower orbit
 Absorption – transition from lower to
higher orbit
 1 quantum emitted or absorbed
 electron can jump over several levels
 Can cascade to lower orbit emitting
several photons of intermediate energy
M.R. Burleigh 2601/Unit 2
Example for hydrogen
 2 2 me 4  1
1



E n   
  R'  2 
2
2

n 
 h
n
1
ab
1 1
R'  1 1 



  2  2   R 2  2 
c ch  nb na 
 nb na 
 ab
The Rydberg constant
(10.96776m-1)
Example: Lyman series
Lyman :
1 1
 R  

1 4 
M.R. Burleigh 2601/Unit 2
1
  = 1216Å (121.6nm)
Important Terms
 Bound electrons – in orbits around
atoms
 Free electrons – not in orbits associated
with individual atoms
M.R. Burleigh 2601/Unit 2
Excitation
Atoms can be excited (increase in energy)
 Radiatively – by absorption of a photon
 Collisional – by a free particle
(electron/atom)...
– Returns by emitting a photon
 Line formation – decay of radiatively
excited states
M.R. Burleigh 2601/Unit 2
De-excitation
 Atoms remain excited for very short
times (~10-8 seconds)
 Atoms always interacting, cause excited
atom to jump spontaneously to lower
level
– Radiative de-excitation – emission of
photon
– Collisional de-excitation – colliding particle
gains kinetic energy
M.R. Burleigh 2601/Unit 2
Ionization
Liberation of an electron:  + energy  + + eEnergy required = ionisation potential
e.g. for hydrogen
13.6eV for the ground state:
13.6
IPn   E    E n   2 eV
n
M.R. Burleigh 2601/Unit 2
Ion notation
 Chemical notation - + or ++ etc.
– but ++++++++ would be silly!
 Spectroscopic notation - (I), (II) etc.
– e.g. neutral atoms… HI, HeI, CI
– Singly ionized… HII (H+), HeII (He+)
– Doubly ionized… CIII (C++), NIII
M.R. Burleigh 2601/Unit 2
Spectra
 Bound transitions  absorption at
discrete wavelengths  series limit
– e.g. Lyman (n=1), Balmer (n=2), Paschen
(n=3), Brackett (n=4), Ffund (n=5)
– Lyman limit at 13.6eV = 91.2nm
M.R. Burleigh 2601/Unit 2
M.R. Burleigh 2601/Unit 2
Spectra of atoms/ions
 Very similar except for effects of charge
 Transitions give rise to emission or
absorption features in spectra
Wave number 
1
ab
2
1 1

 RZ  2  2 
 nb na 
Z = value of the ionisation state
M.R. Burleigh 2601/Unit 2
Spectra of molecules
Spectra can arise from
1. Electronic energy states from
combined electron cloud
2. Internuclear distances quantised into
“vibrational” energy states
3. Quantised rotational energy
Appear as bands in spectra
M.R. Burleigh 2601/Unit 2
Spectral Lines
Spectral line intensities – equivalent width
Line strength  area of the line in the plot (absorption)
This can be represented by ‘equivalent width’
Pressure
Doppler effects
in gas
0
M.R. Burleigh 2601/Unit 2
Equal
areas
Equivalent
width
Excitation equilibrium
No of transitions depends on population of energy state
From which the transition occurs
Level populations depend upon temperature
Mean kinetic energy of a gas particle:
1 2 3
mv  kT
2
2
Thermal equilibrium  mean no of atoms in given states constant
Boltzmann’s equation: N B   g B  exp  E A  EB 
NB / NA = excitation ratio
 kT
 gA 
g = multiplicity
N = number density of state
E = energy of level
NA
M.R. Burleigh 2601/Unit 2

Ionization equilibrium
Population of ions also depends on temperature
Saha equation:
3


N i 1  AkT 2 
 i 

exp   
Ni
 Ne 
 kT 


Ni+1 = higher ion number density
Ni = lower ion number density
A = constant incorporating atomic data
i = ionisation potential of ion i
Ne = electron density
M.R. Burleigh 2601/Unit 2
Local thermodynamic equilibrium
 Combination of Boltzmann & Saha eqns
specify state of gas completely
 Iteration for each state and level
 Plasma where all populations specified
by T and Ne is said to be in Local
Thermodynamic Equilibrium (LTE)
 Often assumed as an approximation in
atmosphere modelling
M.R. Burleigh 2601/Unit 2
Spectral Classification
Division of stars into groups depending upon
features in their spectra
 Angelo Secchi (1863) found different types,
but ordering difficult
 Annie J. Cannon (1910) developed Harvard
scheme  H Balmer strengths
 Later re-arranged in order of decreasing
temperature (see Saha & Boltzman eqns)
M.R. Burleigh 2601/Unit 2
Harvard scheme
 Seven letters – O B A F G K M (L T)
 Each subdivided from 0 to 9
 e.g. Sun has spectral type G2
Mnemonic – Only Bold Astronomers
Forge Great Knowledgeable Minds
or the 1950s/Katy Perry version
- Oh Be A Fine Girl Kiss Me
M.R. Burleigh 2601/Unit 2
Harvard Scheme
M.R. Burleigh 2601/Unit 2
Harvard spectral classifications
Type
Colour
Approximate
surface
temperature (K)
Main characteristics
Examples
O
Blue
> 25,000
Singly ionised helium lines either in
emission or absorption. Strong ultraviolet
continuum.
10 Lacertra
B
Blue
11,000 – 25,000
Neutral helium lines in absorption.
Rigel, Spica
A
Blue
7,500 – 11,000
Hydrogen lines at maximum strength for
A0 stars, decreasing thereafter.
Sirius, Vega
F
Blue to
white
6,000 – 7,500
Metallic lines become noticeable.
Canopus, Procyon
G
White
to
yellow
5,000 – 6,000
Solar-type spectra. Absorption lines of
neutral metallic atoms and ions (e.g. onceionised calcium) grow in strength.
Sun, Capella
K
Orange
to red
3,500 – 5,000
Metallic lines dominate. Weak blue
continuum.
Arcturus,
Aldebaran
M
Red
< 3,500
Molecular bands of titanium oxide
noticeable.
Betelgeuse,
Antares
M.R. Burleigh 2601/Unit 2
Absorption spectra
O
B
A
F
G
K
M
M.R. Burleigh 2601/Unit 2
Stellar spectra
M.R. Burleigh 2601/Unit 2
Stellar
Spectra
M.R. Burleigh 2601/Unit 2
Spectral
Type
The Sun
Vega
M.R. Burleigh 2601/Unit 2
Luminosity Classification
Observers noted differences in spectral
line shapes
 Narrow lines  star more luminous
 Morgan & Keenan  6 luminosity
classes
 e.g. Sun is a G2 V star
M.R. Burleigh 2601/Unit 2
Morgan-Keenan luminosity classes
Ia
Most luminous supergiants.
Ib
Less luminous supergiants.
II
Luminous giants.
III
Normal giants.
IV
Subgiants.
V
Main sequence stars
(dwarfs).
M.R. Burleigh 2601/Unit 2
Stellar
Spectra
M.R. Burleigh 2601/Unit 2
Luminosity
Class
Colour/Magnitude diagram
Hertzsprung-Russell (H-R) diagram
1. Plot luminosity vs. spectral type
2. Plot magnitude vs. colour… same idea
but different parameters
–
Colour measures changes in spectral
shape
M.R. Burleigh 2601/Unit 2
H-R diagram
M.R. Burleigh 2601/Unit 2
Important equations
Bohr postulate:
h
mvr  n
2
Energy of orbits:
2 2 me 4 Z 2
E n   
n2h2
Transition wavelength:
1
ab
n = 1, 2, 3
1 1
 R 2  2 
 nb na 
R = Rydberg constant = 10.96776m-1
M.R. Burleigh 2601/Unit 2
Boltzmann’s equation:
N = number density of state
g = multiplicity
NB  gB 
  E A  E B 
   exp 

NA  gA 
 kT
E = energy of level
3


N i 1  AkT 2 
 i 

exp   
Ni
 Ne 
 kT 
Saha equation:


Ni+1 = number density of the higher ion
Ni = number density of the lower ion
A = constant incorporating atomic data
i = ionisation potential of ion I
Ne = electron density
M.R. Burleigh 2601/Unit 2
Atmosphere Models
Flux is constant:
F
4
 Teff
Scale height of the atmosphere is << R*, so we can represent
the atmosphere as a plane parallel layer of infinite extent
Equation of radiative transfer:
64
2 3 dT
Lr   
r T
3 r  r 
dr
 = Rosseland
mean opacity
M.R. Burleigh 2601/Unit 2
Flux equation:
dP r 
4
c
 T
d
dt = -k dh
 = optical depth
h>0
=0
d > 0
h=0
M.R. Burleigh 2601/Unit 2
>0
Flux is constant so we can integrate:
 4
Pr   Teff   q 
c
Constant
Calculate q from the boundary conditions:
P(r) = P(r = surface) at  = 0
c
q
P

surface

4
Teff
M.R. Burleigh 2601/Unit 2
Assume that locally the radiation field is a Planck
function. At the stellar surface, radiation outflow
is in one direction – outwards.  Surface
radiation pressure is half that given by the Planck
formula.
2 4
2
Psurface  
Teff  q 
3c
3
and:
3 4
2
T  Teff   
4 
3
4
1st simple model
equation
This gives T as a function of  (Rosseland mean optical depth)
Note:
1) Teff is T at  = 2/3
and
2) T(0) = Teff / 21/4 = 0.841 Teff
Surface
M.R. Burleigh 2601/Unit 2
To complete the model add hydrostatic
equilibrium to find pressure and density
distribution:
dP
GM r 


2
dh
r
Variation in h is small compared to R
Matm << M  M(r) = M and r = R
dP MG
   2    g
dh
R
And dividing by  gives:
M.R. Burleigh 2601/Unit 2
dP g

d 
Surface gravity
Schematic model atmosphere calculation
INITIAL MODEL e.g. Grey approximation
T,  structure
CALCULATE ION AND LEVEL POPULATIONS
i.e. solve Saha-Boltzmann equations
CALCULATE RADIATIVE TRANSFER
DETERMINE NEW TEMPERATURE
STRUCTURE
SOLVE EQUATION OF HYDROSTATIC
EQUILIBRIUM
COMPARE NEW MODEL WITH OLD
If differences are small
END
M.R. Burleigh 2601/Unit 2
LOOP
BACK
If
differences
are large
i.e. > some
limit
Stellar Spectra
 Review of atomic physics
 Absorption and emission processes
 Qualitative treatment of spectral line
formation
 Atmospheric opacity
 Spectral classification of stars
 Hertzsprung-Russell diagram
 Atmosphere models
M.R. Burleigh 2601/Unit 2
DEPARTMENT OF PHYSICS AND ASTRONOMY
Unit 2
Stellar Spectra
M.R. Burleigh 2601/Unit 2
DEPARTMENT OF PHYSICS AND ASTRONOMY
LIFECYCLES OF STARS
Option 2601
M.R. Burleigh 2601/Unit 2