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									   PH507        Astrophysics      Professor Glenn White                 1

The planet inj the news:

Lecture 9: Radiation processes

Almost all astronomical information from beyond the Solar System comes to us
from some form of electromagnetic radiation (EMR). We can now detect and
study EMR over a range of wavelength or, equivalently, photon energy,
covering a range of at least 1016- from short wavelength, high photon energy
gamma rays to long wavelength low energy radio photons. Out of all this vast
range of wavelengths, our eyes are sensitive to a tiny slice of wavelengths-
roughly from 4500 to 6500 Å. The range of wavelengths our eyes are sensitive to
is called the visible wavelength range. We will define a wavelength region
reaching somewhat shorter (to about 3200 Å) to somewhat longer (about 10,000
Å) than the visible as the optical part of the spectrum. (Note: Physicists measure
optical wavelengths in nanometers (nm). Astronomers tend to use _Angstroms.
1 Å = 10-10 m = 0.1 nm. Thus, a physicist would say the optical region extends
from 320 to 1000 nm.)

All EMR comes in discrete lumps called photons. A photon has a definite
energy and frequency or wavelength. The relation between photon energy (Eph)
and photon frequency  is given by:

                                    Eph = h

or, since c = 
   PH507        Astrophysics        Professor Glenn White                 2

                                      E ph 

where h is Planck’s constant and  is the wavelength, and c is the speed of
light. The energy of visible photons is around a few eV (electron volts). (An
electron volt is a non- metric unit of energy that is a good size for measuring
energies associated with changes of electron levels in atoms, and also for
measuring energy of visible light photons. 1 eV = 1.602 x 10-19 Joules.)

In purely astronomical terms, the optical portion of the spectrum is important
because most stars and galaxies emit a significant fraction of their energy in this
part of the spectrum. (This is not true for objects significantly colder than stars -
e.g. planets, interstellar dust and molecular clouds, which emit in the infrared
or at longer wavelengths - or significantly hotter- e.g. ionised gas clouds,
neutron stars, which emit in the ultraviolet and x-ray regions of the spectrum.
Another reason the optical region is important is that many molecules and
atoms have electronic transitions in the optical wavelength region.

Blackbody Radiation

Where then does a thermal continuous spectrum come from? Such a continuous
spectrum comes from a blackbody whose spectrum depends only upon the absolute
temperature. A blackbody is so named because it absorbs all electromagnetic energy
incident upon it - it is completely black. To be in perfect thermal equilibrium,
however, such a body must radiate energy at exactly the same rate that it absorbs
energy; otherwise, the body will heat up or cool down (its temperature will change).
Ideally, a blackbody is a perfectly insulated enclosure within which radiation has come
into thermal equilibrium with the walls of the enclosure. Practically, blackbody
radiation may be sampled by observing the enclosure through a tiny
pinhole in one of the walls. The gases in the interior of a star are opaque (highly ab-
sorbent) to all radiation (otherwise, we would see the stellar interior at some
wavelength!); hence, the radiation there is blackbody in character. We sample this
radiation as it slowly leaks from the surface of the star - to a rough approximation, the
continuum radiation from some stars is blackbody in nature.

We will define the regions of the Electromagnetic Spectrum to have wavelengthds
as follows:
      Gamma-rays: < 0.1Å, highest frequency, shortest wavelength, highest

      X-Rays: 0.1Å -- 100Å

      Ultraviolet light: 100Å -- 3000Å

      Visible light: 3000Å -- 10000Å = 1µm (micrometer or micron)
   PH507        Astrophysics          Professor Glenn White              3

      Infrared Light: 1µm -- 1mm

      Radio waves: >1mm, lowest frequency, longest wavelength, lowest

Planck’s Radiation Law

After Maxwell's theory of electromagnetism appeared in 1864, many attempts were
made to understand blackbody radiation theoretically. None succeeded until, in 1900,
Max K. E. L. Planck (1858-1947) postulated that electromagnetic energy can propagate
only in discrete quanta, or photons, each of energy E = hv. He then derived the
spectral intensity relationship, or Planck blackbody radiation law:

                                       
                     2h 3  1      
           I( )d   2  h 
                      c   k T 
                                e  1
                                     

where I(v)dv is the intensity (J/m2 . s . sr) of radiation from a blackbody at temperature
T in the frequency range between v and v + dv, h is Planck's constant, c is the
speed of light, and k is Boltzmann's constant. Note the exponential in the denomi-

Because the frequency v and wavelength of electromagnetic radiation are related by
v = c, we may also express Planck's formula in terms of the intensity emitted per
unit wavelength interval:

This is illustrated for several values of T:
   PH507       Astrophysics       Professor Glenn White                 4

Note that both I() and I(v) increase as the blackbody temperature increases - the
blackbody becomes brighter. This effect is easily interpreted when we note that I(v)∆v
is directly proportional to the number of photons emitted per second near the energy
hv. The Planck function is special enough so that its given its own symbol, B() or B(v),
for intensity.

Wien’s Law

A blackbody emits at a peak intensity that shifts to shorter wavelengths as its
temperature increases.
   PH507       Astrophysics       Professor Glenn White                5

      Wilhelm Wien (1864-1928) expressed the wavelength at which the
       maximum intensity of blackbody radiation is emitted - the peak (that
       wavelength for which dI()/d = 0) of the Planck curve (found from
       taking the first derivative of Planck's law) - by Wien's displacement law:

                            max = 2.898 x 10-3 / T

 where max is in metres when T is in Kelvin. Note that because maxT =
constant, increasing one proportionally decreases the other.

For example, the continuum spectrum from our Sun is approximately
blackbody, peaking at max ≈ 500 nm. Therefore, the surface temperature is
near 5800 K.

The Law of Stefan and Boltzmann

The area under the Planck curve (integrating the Planck function) represents the
total energy flux, F (W/m2), emitted by a blackbody when we sum over all
wavelengths and solid angles:
   PH507        Astrophysics        Professor Glenn White                  6

where = 5.669 x 10-8 W/m2 . K4. The strong temperature dependence of this
formula was first deduced from thermodynamics in 1879 by Josef Stefan (1835-
1893) and was derived from statistical mechanics in 1884 by Boltzmann.
Therefore we call the expression the Stefan-Boltzmann law. The brightness of
a blackbody increases as the fourth power of its temperature. If we
approximate a star by a blackbody, the total energy output per unit time of the
star (its power or luminosity in watts) is just L = 4R2T4 since the surface area
of a sphere of radius R is 4R2

To summarise: A blackbody radiator has a number of special characteristics.
One, a blackbody emits some energy at all wavelengths. Two, a hotter
blackbody emits more energy per unit area and time at all wavelengths than
does a cooler one. Three, a hotter blackbody emits a greater proportion of its
radiation at shorter wavelengths than does a cooler one. Four, the amount of
radiation emitted per second by a unit surface area of a blackbody depends on
the fourth power of its temperature.

Stellar Material

Our Sun is the only star for which I( has been accurately observed. Indeed, Ibol is
related to the solar constant: the total solar radiative flux received at the Earth’s orbit
outside our atmosphere (1370 W/m2). The solar luminosity L (3.86 x 1026 W) is
calculated from the solar constant in the following manner. Using the inverse-square
law, we find the radiative flux at the Sun’s surface R. Then Lis just  times this flux.
The solar energy distribution curve may be approximated by a Planck blackbody
curve at the effective temperature Teff, defined as the temperature of a blackbody that
would emit the same total energy as an emitting body, such as the Sun or a star. Then
the Stefan-Boltzmann law implies

                                L = 4π R2 T4eff     J s-1

where is the Stefan-Boltzmann constant.

Stellar Atmospheres
   PH507        Astrophysics        Professor Glenn White                 7

The spectral energy distribution of starlight is determined in a star’s atmosphere, the
region from which radiation can freely escape. To understand stellar spectra, we first
discuss a model stellar atmosphere and investigate the characteristics that determine
the spectral features.

Physical Characteristics

The stellar photosphere, a thin, gaseous layer, shields the stellar interior from view.
The photosphere is thin relative to the stellar radius, and so we regard it as a uniform
shell of gas. The physical properties of this shell may be approximately specified by
the average values of its pressure P, temperature T, and chemical composition µ
(chemical abundances).

We also assume that the gas obeys the perfect-gas law:

                                        P =nkT

where k is Boltzmann’s constant. This relationship is also known as Boyle’s Law.

An important result that follows from it is that the kinetic energy of a particle, or
assemblage of particles, is given by the relationship;

                                        KE  kT

Thus temperature is just a measure of the kinetic energy of a gas, or an assemblage of
particles. This equation applies equally well to a star as a whole, as to a single particle,
and later we will look at the comparison between a star’s kinetic and gravitational
(potential) energies.

The kinetic energy is also a measure of the velocity that atoms or molecules are
moving about at - the hotter they are, the faster they move. Thus, for a cloud of gas
surrounding a hot star of temperature T = 15,000 K, which consists of hydrogen atoms
(mass = 1.67 10-27 kg);

                                    3          1
                                      kT  KE  mv 2
                                    2          2
                           v        19 km s 1  50,000 mph

The particle number density is related to both the mass density (kg/m3) and the
composition (or mean molecular weight) µ by the following definition of µ:

                                         1       mH n
                                                 
   PH507         Astrophysics      Professor Glenn White                 8

where mH = 1.67 x 10-27 kg is the mass of a hydrogen atom. For a star of pure atomic
hydrogen, µ = 1. If the hydrogen is completely ionised, µ = 1/2 because electrons and
protons (hydrogen nuclei) are equal in number and electrons are far less massive than
protons. In general, stellar interior gases are ionised and

                                         3   1
                                       2X  Y  Z
                                           4   2

where X is the mass fraction of hydrogen, Y is that of helium, and Z is that of all
heavier elements. The mass fraction is the percentage by mass of one species relative
to the total. Thus, for a pure hydrogen star (X=1.0, Y = 0.0, Z = 0.0),  ~ 0.5, and for a
white dwarf star (X = 0.0, Y = 1.0, Z = 0.0)  ~ 1.33.


The continuous spectrum, or continuum, from a star may be approximated by the
Planck blackbody spectral-energy distribution. For a given star, the continuum
defines a colour temperature by fitting the appropriate Planck curve. We can also
define the temperature from Wien’s displacement law: maxT = 2.898 x 10-3 m . K
which states that the peak intensity of the Planck curve occurs at a wavelength max
that varies inversely with the Planck temperature T. The value of max then defines a
temperature. Also note here that the hotter a star is, the greater will be its luminous
flux (in W/m2), in accordance with the Stefan-Boltzmann law: F = T4 where = 5.67
x 10-8 W/m2 . K4. Then the relation

L = 4πR2T4eff

defines the effective temperature of the photosphere.

A word of caution: the effective temperature of a star is usually not identical to its
excitation (Boltzmann eqn) or ionisation temperature (Saha eqn) because spectral-
line formation redistributes radiation from the continuum. This effect is called line
blanketing and becomes important when the numbers and strengths of spectral lines
are large.
   PH507       Astrophysics        Professor Glenn White               9

When spectral features are not numerous, we can detect the continuum between them
and obtain a reasonably accurate value for the star’s effective surface temperature.
The line blanketing alters the atmosphere’s blackbody character.


The goal of the observational astronomer to to make measurements of the EMR
from celestial objects with as much detail, or the finest resolution, possible.
There are of course different types of detail that we want to observe. These
include angular detail, wavelength detail, and time detail. The perfect
astronomical observing system would tell us the amount of radiation, as a
function of wavelength, from the entire sky in arbitrarily small angular slices.
Such a system does not exist!

We are always limited in angular and wavelength coverage, and limited in
resolution in angle and wavelength. If we want good information about the
wavelength distribution of EMR from an object (spectroscopy or
spectrophotometry) we have to give up angular detail. If we want good angular
resolution over a wide area of sky (imaging) we usually have to give up
wavelength resolution or coverage.

The ideal goal of spectrophotometry is to obtain the spectral energy
distribution (SED) of celestial objects, or how the energy from the object is
distributed in wavelength. We want to measure the amount of energy received
by an observer outside the Earth's atmosphere, per second, per unit area, per
unit wavelength or frequency interval. Units of spectral flux (in cgs) look like:

                              f  = ergs s-1 cm-2 Å -1

if we measure per unit wavelength interval, or

                              f = ergs s-1 cm-2 Hz -1
   PH507       Astrophysics       Professor Glenn White                10

(pronounced f nu if we measure per unit frequency interval.

Classifying Stellar Spectra


A single stellar spectrum is produced when starlight is focused by a telescope onto a
spectrometer or spectrograph, where it is dispersed (spread out) in wavelength and
recorded photographically or electronically. If the star is bright, we may obtain a
high-dispersion spectrum, that is, a few mÅ per millimetre on the spectrogram,
because there is enough radiation to be spread broadly and thinly. At high dispersion,
a wealth of detail appears in the spectrum, but the method is slow (only one stellar
spectrum at a time) and limited to fairly bright stars. Dispersion is the key to
unlocking the information in starlight.

The Spectral-Line Sequence

At first glance, the spectra of different stars seem to bear no relationship to one
another. In 1863, however, Angelo Secchi found that he could crudely order the
spectra and define different spectral types. Alternative ordering schemes appeared in
the ensuing years, but the system developed at the Harvard Observatory by Annie J.
Cannon and her colleagues was internationally adopted in 1910. This sequence, the
Harvard spectral classification system, is still used today. (About 400,000 stars were
classified by Cannon and published in various volumes of the Henry Draper Catalogue,
1910-1924, and its Extension, 1949.

At first, the Harvard scheme was based upon the strengths of the hydrogen Balmer
absorption lines in stellar spectra, and the spectral ordering was alphabetical (A
through to P). Some letters were eventually dropped, and the ordering was rearranged
to correspond to a sequence of decreasing temperatures (see the effects of the
Boltzmann and Saha equations): OBAFGKMRNS. Stars nearer the beginning of the
spectral sequence (closer to O) are sometimes called early-type stars, and those closer
to the M end are referred to as late-type. Each spectral type is divided into ten parts
from 0 (early) to 9 (late); for example, . . . F8 F9 G0 G1 G2 . . . G9 K0 . . . . In this
scheme, our Sun is spectral type G2. In 1922, the International Astronomical Union
(IAU) adopted the Harvard system (with some modifications) as the international

Many mnemonics have been devised to help students retain the spectral sequence. A
variation of the traditional one is “Oh, Be a Fine Girl, Kiss Me Right Now, Smack.”

The next Figure shows exemplary stellar spectra arranged in order; note how the con-
spicuous spectral features strengthen and diminish in a characteristic way through the
spectral types.
   PH507         Astrophysics          Professor Glenn White                    11

   Comparison of spectra observed for seven different stars having a range of surface
 temperatures. The hottest stars, at the top, show lines of helium and multiply-ionised
heavy elements. In the coolest stars, at the bottom, helium lines are not seen, but lines of
  neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen
   lines are strongest. The actual compositions of all seven stars are about the same.
   PH507       Astrophysics      Professor Glenn White              12

The Temperature Sequence

The spectral sequence is a temperature sequence, but we must carefully qualify this
statement. There are many different kinds of temperatures and many ways to deter-
mine them.

Theoretically, the temperature should correlate with spectral type and so with the
star’s colour. From the spectra of intermediate-type stars (A to K), we find that the
(continuum) colour temperature does so, but difficulties occur at both ends of the
sequence. For O and B stars, the continuum peaks in the far ultraviolet, where it is
undetectable by ground-based observations. Through satellite observations in the far
ultraviolet, we are beginning to understand the ultraviolet spectra of O and B stars.
For the cool M stars, not only does the Planck curve peak in the infrared, but
numerous molecular bands also blanket the spectra of these low-temperature stars.
PH507   Astrophysics   Professor Glenn White   13
   PH507        Astrophysics        Professor Glenn White                14

When the strengths of various spectral features are plotted against excitation-
ionisation (or Boltzmann-Saha) temperature; the spectral sequence does correlate with
this temperature as seen below;

In practice, we measure a star’s colour index, CI = B - V, to determine the effective
stellar temperature. If the stellar continuum is Planckian and contains no spectral
lines, this procedure clearly gives a unique temperature, but observational
uncertainties and physical effects do lead to problems: (a) for the very hot O and B
stars, CI varies slowly with Teff and small uncertainties in its value lead to very large
uncertainties in T; (b) for the very cool M stars, CI is large and positive, but these faint
stars have not been adequately observed and so CI is not well determined for them; (c)
any instrumental deficiencies, calibration errors, or unknown blanketing in the B or V
bands affect the value of CI - and thus the deduced T. Hence, it is best to define the CI
versus T relation observationally.


• Last year discussed stellar spectra and classification on an empirical basis:
  Spectral sequence      O B A F G K M
  Temperature        ~40,000 K      ---->    2500 K
  Classification based on relative line strengths of He, H, Ca, metal, molecular
• We will now look a little deeper at stellar spectra and what they tell us about
  stellar atmospheres.
   PH507         Astrophysics       Professor Glenn White                  15

Radiative Transfer Equation
• Imagine a beam of radiation of intensity I passing through a layer of gas:

Power passing into volume                         Area
                                                   dA          Power passing out
                                                               of volume
      
E = I d dA d
                                                               E  + dE
where I  = intensity into
solid angle element d

                                 path length ds

NB in all these equations subscripts  can be replaced by 

In the volume of gas there is:
   ABSORPTION - Power is reduced by amount

          dE = -   E ds = -   I d dA d ds

 = the cross-section for absorption of radiation of wavelength  (frequency )
  per unit mass of gas. Units of are m2 kg-1
  The quantity  is the fraction of power in a beam of radiation of wavelength
   absorbed by unit depth of gas. It has units of m-1. (NB in many texts  is
simply given the symbol  in the equations given here - beware!)

EMISSION - Power is increased by amount

                      dE = j  d dA d ds                                       (1)

where j = EMISSION COEFFICIENT = amount of energy emitted per second
  per unit mass per unit wavelength into unit solid angle.
            Units of j (j) are W kg-1 µm-1 sr-1 (W kg-1 Hz-1 sr-1) or m s-3 sr-1
      (NB power production per unit volume per unit wavelength into unit solid
           angle is  =j More confusion is possible here, since is also the
       symbol used for total power output of a gas, units are W kg-1, - Beware!)
  So total change in power is
              dE = dI d dA d = -   I d dA d ds + j  d dA d ds
  which reduces to

                       dI = -   I ds + j  ds                                (2)
   PH507          Astrophysics                       Professor Glenn White   16

                  = - I + j 
  This is a form of the radiative transfer equation in the plane parallel case.

Optical depth
• Take a volume of gas which only absorbs radiation (j = 0) at  :
                   dI = -   I ds
  For a depth of gas s, the fractional change in intensity is given by
          I (s)         s
               I
                   =
                          -  ds
         I (0)        0
                                             I (s)
Integrating ==>
                                      ln (       
                                             I (0)
                                                     ) = -
                                                               ds

                                         -    ds
==>              I (s) = I (0) e
                             
  We define Optical depth 


              ds
                         
                                                                                 (4)

  So              I (s) = I (0) e                                                 (5)
                                 
• Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its original value if optical
  depth = 1.
• Optical depth is not a physical depth. A large optical depth can occur in a
  short physical distance if the absorption coefficient  is large, or a large
  physical distance if  is small.

Full Radiative Transfer Equation again
               = - I + j 

divide by  
   PH507            Astrophysics                       Professor Glenn White                      17

                dI                             j
                            = -I +             
             ds                            
                                                  
                                                                             = -I + S
                                                                     d                                      (6)
  As ds --> 0,  is constant over ds.                                  

  This is the RADIATIVE TRANSFER EQUATION in the plane parallel case.
                    S =         or     j =  S
                                            
  where                     

  and S is the SOURCE FUNCTION.

Radiative transfer in a blackbody
• Remember definition of a blackbody as a perfect absorber and emitter of
  radiation. Matter and radiation are in THERMODYNAMIC EQUILIBRIUM,
  ie gross properties do not change with time. Therefore a beam of radiation in
  a blackbody is constant:
                 = 0 = - I + j 

                                                        from definition of source function, j =  S
==> 0 =  (I - S),
  i.e. I = S.
  but for a blackbody I = B                                                 the PLANCK FUNCTION
                             2                                            3
                    2hc                   1                        2h             1
           B =                                              B =
                                                                   2
                                hc/kT
                                 (e           - 1)                  c          hkT
                                                                              (e       - 1)
  Summary: in complete thermodynamic equilibrium the source function equals the
  Planck function,
  i.e.    j =  B                                                                                     (7)         (K
• In studies of stellar atmospheres we make the assumption of LOCAL
  THERMODYNAMIC EQUILIBRIUM (LTE), i.e. thermodynamic equilibrium
  for each particular layer of a star.

• Note that if incoming radiation at a particular wavelength (e.g. in a spectral
  line) enters a blackbody gas it is absorbed, but emission is distributed over all
  wavelengths according to the Planck function. All information about the
  original energy distribution of the radiation is lost. This is what happens in
  interior layers of a star where the density is high and photons of any
  wavelength are absorbed in a very short distance. Such a gas is said to be
  optically thick (see below).

Emission and Absorption lines
   PH507              Astrophysics                          Professor Glenn White             18

•  the absorption coefficient describes the efficiency of absorption of material
  in the volume of gas. In a low density gas, photons can generally pass
  through without interaction with atoms unless they have an energy
  corresponding to a particular transition (electron energy level transition, or
  vibrational/rotational state transition in molecules). At this particular
  energy/frequency/wavelength the absorption coefficient  is large.

• Let's imagine the volume of gas shown earlier with both absorption and

                                                    I  (0)                                           I

                                                                            path length s
                     = S - I
                                       

Multiply both sides by e and re-arrange
                      dI                   
                              e + I e = S e
                                                           

                               d               
  ==>                              (I e ) = S e 
                                                       
                                            integrate over whole volume, i.e. from 0 to s, or 0 to 
                                                                   
                                                                   
                                                             
                                   I e             =       S e
                                                              
                                                0                   0

                                                                        assuming S = constant along path
  ==>        I e - I(0) = S e - S

==>     I        =  I(0) e- +                       S (1 - e- )                                (8)
                   radiation left                              light from radiation
                  over from light                                 emitted in the
                    entering box.                                      box.

• Case 1       >> 1: OPTICALLY THICK CASE
  If  >> 1, then e- --> 0, and eqn (8) becomes I = S                                                  (9)
  In LTE                                        S = B, the Planck function.
   PH507        Astrophysics        Professor Glenn White                    19

  So for an optically thick gas, the emergent spectrum is the Planck function,
  independent of composition or input intensity distribution. True for stellar
  photosphere (the visible "surface" of a star).

• Case 2      << 1 OPTICALLY THIN CASE
                                                        If  << 1, then e- ≈ 1 - 
(first two terms of Taylor series expansion)
   eqn (8) becomes                        I =       I(0) (1 - )   +   S (1 - 1 + )
   ==>I = I(0) +  ( S - I(0) )                                                        (10)

• If I(0) = 0 : no radiation entering the box (from direction of interest):
From eqn (8)                       I =  S           (=  B in LTE)
  Since  = ∫  , then          I =   s S
                If  is large (true at wavelength of spectral lines) then I is large,
                we see EMISSION LINES. This happens for example in gaseous
                               nebulae or the solar corona when the sun is eclipsed.

• If I(0) ≠ 0 , let's examine eqn (8)
                  I = I(0) +  ( S - I(0) )

  If S > I(0) then right hand term is +ve
  when  is large (ie  is large) we see higher intensity than I(0)
  If S < I(0) then right hand term is -ve
  when  is large (ie  is large) we see lower intensity than I(0)
  For stars we see absorption lines. This means I(0) > S,
         i.e. (intensity from deeper layers) > (source function for the top layers

  Assuming LTE (S = B) the source function increases as temperature
  increases:                   I(0) = B(Tdeep layer) > S = B(Touter layer).
  Therefore temperature must be increasing as we go into the star for
  absorption lines to be observed.

• To summarise
  - We see CONTINUUM RADIATION for an optically thick gas
            (= PLANCK FUNCTION assuming LTE).
  - We see EMISSION LINES for an optically thin gas.
  - We see ABSORPTION LINES + CONTINUUM for an optically thick gas
          overlaid by optically thin gas with temperature decreasing outwards.
  - We see EMISSION LINES + CONTINUUM for an optically thick gas
  overlaid by an optically thin gas with temperature increasing outwards.
PH507   Astrophysics   Professor Glenn White   20

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