Download PY3A06 Astronomical On-line Notes

On-line Notes
• R.J. Ruten
PY3A06
Astronomical
Spectroscopy
Dr. Brian Espey
SNIAM 1.04
[email protected]
PY3A06 Astronomical Spectroscopy
• Lecturer: Dr. Brian Espey, SNIAM 1.04
• email: [email protected]
• web info: TBA
• Lectures:
– “Radiative Transfer in Stellar Atmospheres”
– http://esmn.astro.uu.nl
• E.H. Avrett
– “Lecture Notes: Introduction to Non-LTE Radiative
Transfer and Atmospheric Modelling”
– https://www.cfa.harvard.edu/~avrett/nonltenotes.pdf
• G.W. Collins II
– The Fundamentals of Stellar Astrophysics,”
http://ads.harvard.edu/books/1989fsa..book/
• I. Hubeny
– “Stellar Atmospheres Theory: Introduction”
– http://aegis.as.arizona.edu/~hubeny/ASTR545/eadn.pdf
Course Text- Pradhan & Nahar
• Hamilton, Lending
• S-LEN 523.01 (3 copies)
• ISBN 9780521825368
– SNIAM
• Lecture Theatre Wednesdays at 10:00,
weeks 1,3-6,8-10  UPDATED!
• Lecture Room Tuesdays at 14:00, weeks
6,8-12
• Lecture Room Tuesday at 12:00, week 5
(tutorial)
 Consider tie-ins with what
you learn in the
Thermodynamics portion of
this module, and also with
the Atomic & Molecular
Spectroscopy Module
Overview
•
•
•
•
1
2
3
4
Intro, Boltzmann / Saha
Spectral Classification
Radiative Transfer
Optical Depth
• 5 Spectral Line Formation
• 6 Line Shapes
• 7 Emission Line Processes/Diagnostics
• Including: Tutorials – Practical, real
data & some exam-type Problems
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Introduction
“Of all objects, the planets (or stars) …we
can never known anything of their chemical
or mineralogical structure……”
Auguste Comte, The Positive Philosophy,
Book II, Chapter 1 (1842)
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Spectroscopy
The em Spectrum
• Diagnosis of plasmas (partially neutral
combinations of electrons and ions)
• Plasmas constitute >99% of observed
material in the Universe
• The ultimate in remote sensing!
– e.g., study of Cosmic Microwave Background
spectrum from the early stages of the Big Bang
• Complete understanding requires wide
range of energies/frequencies/wavelengths
Will concentrate on:
• UV – many diagnostic lines of resonance lines from
common elements
• Optical – (most observations) also used to classify
stars
Spectra
• Spectra provide an elemental
fingerprint for individual species, as
well as providing information on an
object’s:
– temperature
– density
Photometry – a basic form
of Spectroscopy
• With photometry can
infer continuum
shape
– can estimate
blackbody continuum
temperature
• To examine
emission/absorption
details need spectra
– pressure
– distance
– motion
The ultimate in remote sensing!
Diagnostic Range
Intergalactic medium 
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Example objects and
spectra
2
Representative Objects
“Dumbell”
Planetary Nebula
• Different types of astronomical
objects:
– photoionised nebulae
– shock-excited nebulae
Types of object
• Note that the type of spectrum
can vary over time, or have
one or more of the following
components:
– continuum emission/absorption
– combination objects
“Veil” Supernova
remnant
• Ionisation may vary over time
• Densities may vary
SN1987A ejecta
– low density
– ~10-2 – 106 cm -3 (~104 – 1012 m–3)
– line emission/absorption
• Note also that
spectra/material is broken
into two main groups:
– Those with emission similar to a
blackbody (thermal spectra),
such as emission from stars
– Those with non-blackbody spectra
(non-thermal ), such as
synchrotron emission
– high density
– ~ 106 – 1012 cm -3(~1012 – 1018 m–3)
Supernova in 1987 (SN1987A) history
Photo- / Collisional ionisation
SN1987A observed with HST
Final ring
~1 light
year in
diameter
Note that the spectrum from the supernova shock will also
evolve over time
Photoionisation
optical
x-ray
radio
Representative
Spectra
Ionisation in
same object
can vary in
both space
and time
e.g. SN1987A
Collisional
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3
Electrons & Ions
• Universe fractional abundance (by mass):
H ~ 0.70; He ~ 0.28; “Metals”: 0.02
Basic Definitions:Classical Physics
Y
X
Z
• Generally, astronomical objects are neutral
• However, electrons are the active players...
– since me ~ mp /1836, we find ve = 42.85 vp
(and ions will be even slower...)
• Metals contribute:
– bulk of spectral features, and opacity (more later)
due to many energy levels/lines
– good proportion of free electrons in cool objects
(many outer electrons & binding energy low)
Important Parameters
• Astrophysics tends to use old-fashioned
(cgs) units, or non-SI values:
• Temperature
– Kelvin
– also eV (1.602 x 10–19 J)
• Density-
– particles per cubic cm (unit: cm–3)
• Energy-
– erg (or K)
Electron energy
• For metals we are used to the idea of a work
function, ϕ :
½ m v2 = h ν – ϕ
(Einstein 1905)
• For single atoms, this work function for a single
electron is called the ionisation energy Ei and is
usually given in eV
• Energy may be supplied through direct transfer
of particle energy (jiggling by other atoms/
ions/e- either due to individual thermal motions
or collective motion such as shocks), called
collisional ionisation or via. the photoelectric
effect, called photoionisation
Electron energy
Energy, Speed etc.
• Energy in excess of the ionisation energy goes to
k.e. and results in further heating
• For a particle ensemble, average particle energy is:
• Recombination of electrons with ions results in
the emission of photons.
• In thermal equilibrium,
particles will have a Maxwellian
of speeds:
• Depending on the opacity (more later), these
photons may be– absorbed in the same medium (and lead to heating
elsewhere)
or
– may escape the medium altogether, and thus serve to
cool the medium
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distribution
• Solving, we get for a single particle in 3-D:
where kB is Boltzmann‟s constant =1.380x10–16 erg K–1
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Mean particle speed in 3-D
• Using this energy,
, we can
determine the average k.e. of a single particle
• The particles in question are usually electrons, so
we can define the temperature of a plasma (the
ensemble distribution of particles) in terms of
either the electron temperature (Te) or the ion
temperature (Ti )
• The ion temperature Ti or some equivalent may
be more relevant, depending on the situation,
but usually the distributions are charge-coupled
since the plasma is neutral overall and we
consider the electron temperature for most work.
Electron energy
• The collective distribution of particle energies can
be used to define the medium‟s temperature by
comparing it with the theoretical Maxwellian
distribution.
• Note that the temperature of the radiation (e.g.,
from a nearby star) may be very different from
that of the particles. For example:
– The plasma may be partially transparent, so that more
energetic radiation passes straight through
– Particle radiation can also cause ionisation
– The energy of the ionisation potential needs to be taken
into account.
Energy-Temperature Equivalence
Energy-Temperature Equivalence
• Equivalently, we can define the temperature in
terms of the electron energy, usually expressed
in electron-volts (eV)
• Note that we will come across instances of both
continuum (quasi-blackbody) emission, and also
emission which is well away from equilibrium,
when the background radiation field can lead to
ionisation and non-thermal electron velocities
• Using the appropriate values, we get the
equivalence 1 eV = 11,604.45K
• We also have the regular relationship: E = h υ so
we can also describe photon energies in eV
Blackbody radiation
Basic Definitions:Quantum Physics
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• If a material is very opaque to radiation
(optically thick – more on this later), photons
will scatter many times before being emitted,
and the photons and particles come into
thermodynamic equilibrium (TE).
• Under these circumstances, the particles of
the emitting medium and the emitted photons
come to equilibrium, and the average particle
energy will equal the average photon energy.
Under these circumstances, a unique
temperature can describe the material and its
radiation.
5
Blackbody radiation
• For non-relativistic monoatomic particles, the
energy sharing between particles and photons
results in the equality:
The best blackbody that we know:- the
Cosmic Background Radiation
• At its earliest stages the Universe was both
hot and dense, so blackbody conditions were
met:
• To within a factor of a few, we find:
• Expressing in eV units, we obtain:
Blackbody radiation
• The total radiation per unit area emitted by a
perfect radiator in thermal equilibrium is given
by the Stefan-Boltzmann Law:
The Planck Function
• For photons, the equivalent to the particle M-B
distribution is the Planck function:
• A perfect radiator (also a perfect absorber) is
called a black body, and emits is the most
efficient emitter at all wavelengths, with a
continuum emissivity given by the Planck Law
• (Kirchoff’s Law relates the emissivity and the
absorptivity – a less efficient radiator is also a
less efficient absorber)
The Planck Function
Planck Function and Luminosity
• The temperature in this case is the radiation
temperature of the object – e.g., 2.7K for the
CMBR
• Note that the peak of the radiation scales
inversely with the temperature- this is the
Wien Law
– Peak wavelength = 2.8978 x 107 Å / T
where T is in Kelvin
– Where does the peak occur for the Sun?
(T = 5770K)?
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6
Planck Function and Luminosity
• The total luminosity over all wavelengths (=
bolometric luminosity)of a spherical blackbody is
given by:
where σ is Stefan-Boltzmann‟s constant
• In practice, stars are not perfect blackbodies, but
we can relate the total (bolometric) luminosity of
a star to that of a perfect blackbody:
where Teff is called the star‟s effective temperature
Thermodynamic Equilibrium
• If TE holds, the system in question is easily
described, so it is a powerful concept.
• In reality, observable astrophysical systems
are not in true equilibrium*, but equilibrium
can apply locally (Local Thermodynamic
Equilibrium, or LTE).
• If TE (closely) applies, then we can also
describe the relative electron populations of
the molecules, atoms or ions, and ionisation
state of the medium
* Why is this?
Maxwell-Boltzmann statistics
• For quantised (bound) and free non-degenerate
particles in thermodynamic equilibrium, there are
three basic rules:
1. All quantum states of equal energy have
equal probability of being populated
2. The probability of populating a state with an
energy, E, at a kinetic temperature, T, is
equal to exp(-E/kT)
3. No more than one electron may occupy a
given quantum state [we include spin states
here]
Electronic Level Populations
• For gas in (close to) thermodynamic
equilibrium, we can use M-B
statistics to determine the relative
level populations:
where:
• nj , ni are the number of atoms in the lower, upper states,
respectively,
• gj and gi represent the statistical weights of the two levels
(g= 2J+1),
• Eji is the excitation energy between the lower and upper
states,
•
The internal Partition Function
• The number in any level relative to the total
number of electrons is:
• We define the internal partition function, U, for
the atom or ion as the denominator of the
above equation, viz:
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and k and T have their usual meaning
Maxwell-Boltzmann statistics
• The partition function represents the
level-by-level population of occupied
levels in an atom or ion of a plasma in
thermal equilibrium which is
characterised by a (local) temperature, T
• We can then represent the level
population more compactly as:
7
Boltzmann statistics
• Note that the partition function, U, is formally a
divergent sum over an infinite number of levels,
with statistical weights increasing as:
• However, in general, a natural truncation of the
partition function sum occurs due to the
interaction and perturbation of the upper atomic
levels with neighbouring atoms or ions, since
the size of the orbitals also increases roughly as
The free electron Partition Function
• For (non-degenerate) free electrons, the
energy is given by the kinetic energy, and the
wavenumber is given by:
• The number of electron states per unit energy
per unit volume (their statistical weight) is
given by:
where the leading factor of 2 accounts for the
electron spin
• Truncation also occurs at higher plasma
densities due to increased inter-ion interaction
The free electron Partition Function
The free electron Partition Function
• From the 3 basic rules given earlier, we can
define:
• For a Maxwellian velocity distribution, we have:
• With the normalisation:
• The free electron partition function, Ue , is given
by:
• and the fractional population of states is given
by:
• This equation holds for non-degenerate
electrons where the number of available states
is much greater than the number of electrons
 true for most locations outside high density /
high temperature stellar cores
Stellar Atmospheres
Stellar atmospheres are the main connecting
link between observations and the rest of stellar
astrophysics – observables!
•
Stellar Atmospheres…
…some general points…
• By definition, most of stellar photons we
receive are from „photosphere‟ ( optical depth
2/3 at 500nm)
• Need to model in order to compare with
observations
• Initial model constructed on the basis of
observations & known physical laws.
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8
Stellar Atmospheres
Initial model is modified and improved iteratively
until good match achieved. Can then infer certain
properties of a star: temperature, surface gravity,
radius, chemical composition, rate of rotation, etc. as
well as the thermodynamic properties of the
atmosphere itself.
•
Stellar Model Fits to Data
• Metallicity (top) and temperature (bottom) for
two spectral regions in a star:
•Model fits
to the Sun
and α Boo
Stellar Atmospheres
A number of simplifications usually necessary!:
Plane-parallel geometry (more 3-D models coming)
 making all physical variables a function of only one space
coordinate
Hydrostatic Equilibrium
 no large scale accelerations in photosphere, comparable to
surface gravity, no dynamical significant mass loss
No fine structures
 such as granulation, starspots
Magnetic fields are (generally) excluded
Black Body Stars & Thermal Equilibrium
•Basic condition for the BB as emitting source
 negligible fraction of radiation escapes!
•Optical depth to the surface is high for lower
photosphere, so most photons do not escape.
 reabsorbed close to emission site, so
thermodynamic equilibrium - & radiation laws of BB
apply.
•However, a star cannot be in perfect thermodynamic
equilibrium! Net outflow of energy!
Black Body Stars & Thermal Equilibrium?
Black Body Stars & Thermal Equilibrium?
However, a star cannot be in perfect thermodynamic
equilibrium as there must be a nett outflow of energy!
•(L)TE means atoms, electrons & photons interact
enough that the energy is distributed equally among
all possible forms (kinetic, radiant, excitation etc),
and the following theoretical distributions can be
used to understand physical processes:
•Higher layers deviate increasingly from BB as leakage
becomes more significant.
• TE can be applied to relatively small volumes of the
model photosphere – (volumes with dimensions of order
unity in optical depth – more on this later)  Local
Thermodynamic Equilibrium, or LTE
 continuous transition from near-perfect TE deep in
the star to LTE deep in the photosphere to complete
non-equilibrium (non-LTE) high in the atmosphere.
•photon energies: Planck Law (Black-Body
Relationship)
• kinetic energies: Maxwell-Boltzmann Relationship
• excitation level populations: Boltzmann Equation
• ionization state populations: Saha Equation
So one temperature can be used to describe the gas locally!
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Black Body Stars & Thermal Equilibrium?
•LTE usually assumed
 works for non-extreme conditions
•LTE poorly describes:
• very hot stars (strong radiation field)
• very extended stars (low densities, i.e., red giants)
•LTE works for some spectral features, but not for
other features in the same star
(different lines form in different photospheric regions)
•Generally models can reproduce stellar spectra
extremely well….
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