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PY3A06 Astronomical Spectroscopy

Dr. Brian Espey

SNIAM 1.04

Brian.Espey@tcd.ie

On-line Notes • R.J. Ruten

– “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

PY3A06 Astronomical Spectroscopy • Lecturer: Dr. Brian Espey, SNIAM 1.04

• email: Brian.Espey@tcd.ie

• web info: TBA

• Lectures:

–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)

Course Text- Pradhan & Nahar

• Hamilton, Lending

• S-LEN 523.01 (3 copies)

• ISBN 9780521825368

Consider tie-ins with what you learn in the Thermodynamics portion of this module, and also with the Atomic & Molecular Spectroscopy Module

Overview • 1 Intro, Boltzmann / Saha

• 2 Spectral Classification

• 3 Radiative Transfer

• 4 Optical Depth

• 5 Spectral Line Formation

• 6 Line Shapes

• 7 Emission Line Processes/Diagnostics

• Including: Tutorials – Practical, real data & some exam-type Problems

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

• 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

The em Spectrum

• Spectra provide an elemental

fingerprint for individual species, as

well as providing information on an

object’s:

– temperature

– density

– pressure

– distance

– motion

The ultimate in remote sensing!

Spectra Photometry – a basic form

of Spectroscopy

• With photometry can infer continuum shape

– can estimate blackbody continuum temperature

• To examine emission/absorption details need spectra

Diagnostic Range

Intergalactic medium

Example objects and

spectra

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• Different types of astronomical objects:

– photoionised nebulae

– shock-excited nebulae

– combination objects

• Ionisation may vary over time

• Densities may vary

– low density

– ~10-2 – 106 cm -3 (~104 – 1012 m–3)

– high density

– ~ 106 – 1012 cm -3(~1012 – 1018 m–3)

Representative Objects “Dumbell”

Planetary Nebula

“Veil” Supernova remnant

SN1987A ejecta

• Note that the type of spectrum can vary over time, or have one or more of the following components:

– continuum emission/absorption

– 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

Types of object

Final ring ~1 light year in

diameter

Supernova in 1987 (SN1987A) history

SN1987A observed with HST

Photo- / Collisional ionisation

Note that the spectrum from the supernova shock will also evolve over time

e.g. SN1987A

Ionisation in same object can vary in both space and time

optical x-ray radio Photoionisation

Collisional

Representative

Spectra

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Basic Definitions:-

Classical Physics

Electrons & Ions

• 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)

X Y Z

• Universe fractional abundance (by mass):

H ~ 0.70; He ~ 0.28; “Metals”: 0.02

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 in excess of the ionisation energy goes to

k.e. and results in further heating

• Recombination of electrons with ions results in the emission of photons.

• 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

Energy, Speed etc. • For a particle ensemble, average particle energy is:

• In thermal equilibrium, particles will have a Maxwellian distribution of speeds:

• 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

• Equivalently, we can define the temperature in terms of the electron energy, usually expressed

in electron-volts (eV)

• 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

Energy-Temperature Equivalence

• 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

Basic Definitions:-

Quantum Physics

Blackbody radiation

• 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.

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Blackbody radiation

• For non-relativistic monoatomic particles, the energy sharing between particles and photons results in the equality:

• To within a factor of a few, we find:

• Expressing in eV units, we obtain:

• At its earliest stages the Universe was both hot and dense, so blackbody conditions were met:

The best blackbody that we know:- the Cosmic Background Radiation

Blackbody radiation

• The total radiation per unit area emitted by a perfect radiator in thermal equilibrium is given by the Stefan-Boltzmann Law:

• 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 • For photons, the equivalent to the particle M-B

distribution is the Planck function:

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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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,

• and k and T have their usual meaning

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:

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:

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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

• Truncation also occurs at higher plasma densities due to increased inter-ion interaction

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

The free electron Partition Function

• From the 3 basic rules given earlier, we can define:

• With the normalisation:

• The free electron partition function, Ue , is given by:

The free electron Partition Function

• For a Maxwellian velocity distribution, we have:

• 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…

…some general points…

Stellar Atmospheres • Stellar atmospheres are the main connecting link between observations and the rest of stellar astrophysics – observables!

• 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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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. •Model fits to the Sun

and α Boo

Stellar Model Fits to Data • Metallicity (top) and temperature (bottom) for

two spectral regions in a star:

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? However, a star cannot be in perfect thermodynamic equilibrium as there must be a nett outflow of energy!

•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.

•(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:

•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!

Black Body Stars & Thermal Equilibrium?

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•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….

Black Body Stars & Thermal Equilibrium?