4201 Essential Physics: Radiation

Monica Ladner-Gabney: solar spectrum from bus-shelter

The revolutions in astronomy

• Systematic observations (needed writing, Babylon 1500 BC)
• Model for the universe (needed observations, Ptolemy 100 AD)
• Collecting light (needed clear glass, telescope, Galileo, 1605 AD)
• Predictive Model (required calculus : Newton,. 1666)
• Saving light (needed chemistry, photography, 1860)
• Extending the spectrum (needed radar: Jansky radio-telescopes, 1955)
• Escaping the atmosphere (needed satellites, 1958)
• Synthesising the ideas (needed computers, 1960)
• Note that most of these had nothing to do with astronomy.
• Note that 4 require better understanding of light.

Electromagnetic Spectrum

99.9999% of our information about the universe comes as EM radiation:

But light is part of the whole electromagnetic spectrum: wavelengths we can detect go from 10-16 m to 104m

We "see" only one octave. Compare to sound, we hear (well, some of us!) 10 octaves (20Hz to 20000 Hz: middle C is at 252 Hz.)

Why do we see so little of the spectrum? Answer lies in the transparency of the atmosphere. This ignores extra problems like clouds and "twinkle".

Electromagnetic Spectrum

In more detail: electromagnetic radiation tells us

• Luminosity
• Temperature
• Velocity of object
• Chemical composition of star
• Pressure
• We had better understand
  • Wavelength, frequency, energy
  • Detection
  • Production (i.e. spectra)
  • Modification (i.e. Doppler shift, Zeeman effect)
• Basic relations:
  
  \color{red}{ \begin{array}{l} \nu \lambda = c \\ E = h\nu \\ \end{array}}

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  Spectra

  Remarks apply to any part of the spectrum. Simplest spectrum is black-body: produced by any object above 0 K Note: we will always use Kelvin temp scale: 0 K ∼ -273°C

  • Energy emitted by hot body increases with temp.

    Stefan-Boltzmann Law:

    \color{red}{ F = \sigma T^4 ,\sigma = 5.67 \times 10^{ - 8} Wm^{ - 2} K^{ - 4} }
  • Peak wavelength gets shorter as we heat an object:

    black -> red -> orange -> yellow -> white -> blue

    0°K -> 900°K -> 2000°K -> 5000°K -> 20000°K -> ... Wien's Displacement Law:

    \color{red}{ \lambda _{\max } = \frac{b}{T},b = 2.9 \times 10^{ - 3} mK}

  Combined in the Einstein-Planck radiation law:Intensity at a given wavelength
  

  \color{red}{ I\left( \lambda \right) = \frac{{2hc^2 }}{{\lambda ^5 }}\frac{1}{{e^{ - hx/\lambda kT} - 1}}}

  h = Planck's constant, c = speed of light

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  Graph shows the distribution of photon intensities due to a radiating block body at the temperatures of 2500K, 4000K, and 5000K. Note that as the temperature increases the peak moves to lower wavelengths (higher energies) the intensity increases

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  Line spectra:

  Radiation from atoms and molecules. All atoms show same general features:

  Emission spectrum consists of bright lines on a dark background: e.g. sodium (Na). This gives the characteristic bright yellow colour of sodium lights (there are some other weaker lines)
  Usually in stars we see Absorption Spectra: Dark lines on a rainbow continuum Note that lines occur at same place as in emission.

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  e.g. Hydrogen: Easiest to understand theoretically and most important (H is 90% of universe by number). Line at 656 nm is responsible for red colour of (e.g.) nebulae in Orion .
  Line at 656 nm is responsible for red colour of (e.g.) nebulae in Orion. (note the star colours in this picture are B-B colours!)

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  When do we see spectral lines?

  • Obviously we need a hot gas, but how hot? Recall Ideal Gas:
    \color{red}{ PV = nRT = NkT}
    • P = Pressure,
    • V = Volume,
    • n = # moles,
    • R = gas constant = 8.31Joules/mol.K
    • T = temperature in Kelvins,
    • N = no. of atoms present,
    • k = Boltzmann's constant = 1.38x10^-23 JK^-1
  • ............................................However......this is not very useful, since we really want to relate the temperature to the energy of the photons, not the pressure and volume. Average energy of gas molecules ∝temp
    \color{red}{ \frac{1}{2}m\bar v^2 = \frac{3}{2}kT}
    Energy per degree of freedom \color{red}{E = \frac{1}{2}kT}

  • Boltzmann's constant: k = 1.38x10^-23J/K = 8.625x10^-5 eV K^-1

    i.e. a gas molecule at room temp has an average energy of

    \color{red}{ E \sim kT \sim 0.0026eV \sim \frac{1}{{40}}eV}

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  This is described by Maxwell-Boltzmann equation:

  \color{red}{ N = \frac{{2N_0 mv^2 }}{{\pi kT}}e^{ - \frac{{mv^2 }}{{2kT}}} }

  What does this look like? The Maxwell-Boltzmann distribution is shown here for three different peak temperatures, those being 650K, 800K, and 1000K. Note that as the temperature increases, the distribution becomes more broad, and the peak of the distribution declines in magnitude.

  A more complicated molecule has more degrees of freedom.

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  Boltzmann Equation and Collisional Excitation

  (...just what we need to bring us back to emission spectra...).

  A useful rule of thumb: At the atomic level, nothing happens until the energy required is comparable to the average energy: say (say 1/10 E) e.g.:

  • a hot gas won't emit yellow light until the average energy is ∼ 0.2eV ∼ 2000 K (T ∼ E/k)
  • nuclear processes involve 100's of keV's to MeV's of energy, so won't happen until temperatures get to > 10⁸ K
  • For an atom to radiate, it must be in higher energy state: this requires that it has gained energy via a collision with another atom.
  • Suppose we have a very simple atom with only two levels: A and B.The ratio of number of states in the two levels is:
    \color{red}{ \frac{{N_B }}{{N_A }} = \frac{{g_B }}{{g_A }}e^{\frac{{E_A - E_B }}{{kT}}} }

    The g's are "spectroscopic factors": can usually assume they are 1

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  Hence probability of first excited state of hydrogen (ΔE = -13.6+3.4 = 10.2 eV) and second state (ΔE = -13.6+1.51 = 12.1 eV). The population probabilities are shown for the n=2 and n=3 (lower curve). Hence very few H atoms are excited to the lowest levels until T ∼ 10,000 K). Note that there are always some: hence you can see H lines even at 2000 K

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  Absorption lines:

  Emission lines arise from an electron in an excited state decaying to one in a lower state
  Absorption lines occur when all wavelengths are incident on an atom, and an electron can make a jump to a higher level. Hence H gas will absorb light from a continuum spectrum

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  Hence absorption occurs by removing a few wavelengths from continuum.

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  Need considerable optical depth of material, so difficult to show.

  Note (important)
  • Most astro objects are black bodies shining through gas:
  • i.e. we see BB spectrum + absorption lines
  • Very rarely do we see emission lines

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  Saha equation

  Now all this is complicated by ionization. At stellar temperatures, gas will be partly split up into ions. Also an equilibrium process, described by the Saha equation
  \color{red}{ N_ + N_e = \frac{{2N_0 }}{{h^3 }}\left( {2\pi mkT} \right)^{3/2} e^{\frac{\chi }{{kT}}} }

  where χ is the ionisation energy, m is the electron mass.

  What this means is that as the temperature increases, the electron is more and more likely to be removed from the atom.

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  Boltzmann-Saha combined

  At low T, no excitation, no ionization, so no H lines Medium T, excitation, no ionization, so strong Balmer lines High T: high ionization, so no atoms left to be excited, so no Balmer line

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  Ionic Spectra

  But there is one more complication (you were surprised?). Ions can emit their own characteristic radiation, which has no relation to atomic spectrum. Also different ions of a given element exist at different temperatures. Nomenclature:

  • H_I is unionised hydrogen

    Note: pronounced this un-ionised, not union-ised!

  • H_II is ionised hydrogen (i.e. a proton)
  • Mg_IV is three times ionised magnesium, i.e. Mg⁺⁺⁺

  Since considerable energy is required to ionise an atom many times, ionic spectra show high temperatures.

• e.g. ...He_II
• To ionise He takes ∼ 24.6eV (say 100000 K, hotter than almost any star). Then ion is like H_I, except energy levels are shifted
  \color{red}{ E_m = - \frac{{13.6Z^2 }}{{m^2 }},Z = 2}

  i.e. red Balmer line show up at ¹/₄ wavelength of H.

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  Molecular spectra:

  A typical diatomic molecule shows a rotational spectrum.

  Binding energy small (H₂ => 2H at 4eV), so molecules don't exist in stars (except for TiO) but do exist in dust clouds. Spectrum mainly in IR. Final line spectrum is

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  21 cm line

  .

  Energy difference arises due to magnetic interaction of electron and proton (hyperfine interaction).

  \color{red}{ \Delta E \approx 6 \times 10^{ - 6} eV}
  • A rule of thumb is that lifetime of excited states ∝ 1/ΔE³ :
  • Typical lifetimes are ∼ 10^-8 --10^-12 s.
  • This one has lifetime ∼ 10⁶ yrs.
  • H atoms collide and can exchange electrons, so para-state is continuously refilled.

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  Doppler Effect

  During the time taken to emit one wavelength, the emitter moves away a distance v ΔT, i.e. λ' = λ + v ΔT. Time taken to emit one λ is ΔT = λ/c. Hence$$ \color{red}{ \lambda ' = \lambda \left( {1 + \frac{v}{c}} \right),\nu ' = \frac{v}{{\left( {1 + \frac{v}{c}} \right)}}} $$

  This is red-shift: v >0, λ' > λ and ν' < ν. This is only true if v << c: we might need the relativistic formula:

  $$ \color{red}{ \nu ' = v\sqrt {\frac{{1 - \frac{v}{c}}}{{1 + \frac{v}{c}}}} } $$

  In general, define red-shift, z, via

  $\color{red}{\lambda ' = \lambda \left( {1 + z} \right)}$ so $z = \frac{v}{c}$ for non-relativistic shifts, relativistically $$ \color{red}{ z = \sqrt {\frac{{1 + \frac{v}{c}}}{{1 - \frac{v}{c}}}} - 1} $$ Typically
  • stars have velocities -10 km s^-1 < v< 100 km s^-1
  • galaxies have velocities 0 < v< 0.99c

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  Other Effects

  • Another effect of Doppler is that hot stars have broadened lines
  • Natural widths of spectral lines result from Heisenberg Uncertainty:
    \color{red}{ \Delta E > \frac{\hbar }{{\Delta t}}}
  • Pressure broadening: If object is at very high pressure, an excited atom can collide with another before it has had time to radiate ⇒ uncertainty in the energy.

  • Zeeman effect: Atomic levels become split in a mag field, since Electron in orbital acts like magnetic dipole

  • Forbidden lines: Pressure can be so low that some lines which are never seen on earth can be seen in space.

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  Spectra in summary:

  We can use spectral info to measure
  • Temp (BB peak, strength of lines, Doppler broadening, ionisation)
  • Elements and concentrations in stars (relative strengths of lines)
  • Pressure (broadening)
  • Velocity of emitter (Doppler)
  • Magnetic Field of emitter(Zeeman)
  • We cannot
  • learn anything useful about solids (all give a roughly BB spectrum with no lines)
  • see cold gas (except for H)
  • deduce what molecules are present unless we have a reference spectrum from either the lab or (now) calculation. There are still a large number of un-identified lines
  • Now how do we manipulate E. M. radiation?