Lecture 2 : Radiation Processes

Lecture 2 : Radiation Processes

Virtually everything we know about the physical universe outside the Earth comes from electromagnetic radiation.

Before the 1950s, the electromagnetic spectrum was restricted to optical radiation (circa 3000 to 7000 Angstroms). From the 1950s radio wavelengths were opened up. Most of the rest of the electromagnetic spectrum is blocked by the Earth's atmosphere, and so it was not until pretty recently that observations could be made from above it, in near space. Since the 1970s huge advances have been made from space, where the remaining regions gamma, X-ray, utraviolet (UV), infra-red (IR) and microwave are observed.

Electromagnetic radiation is characterised by a fundamental energy, E and wavelength, l, or frequency n, where


E = h n       and         l = c/n.

The Planck constant has the value


h = 6.625 ×10-27 erg s.

A temperature T can be associated with the the wavelength via the Boltzmann constant


k = 1.38 ×10-16 erg K-1.

Radiation is divided broadly into gamma, X-ray, UV, optical (or visible), IR, microwave and radio, as shown in the table.

Table 1: Electromagnetic Radiation
Region Wavelength (Å) Wavelength (cm) Frequency (Hz) Energy (eV)
Radio > 109 > 10 < 3 ×109 < 10-5
Microwave 109 - 106 10 - 0.01 3 ×109 - 3 ×1012 10-5 - 0.01
Infrared 106 - 7000 0.01 - 7 ×10-5 3 ×1012 - 4.3 ×1014 0.01 - 2
Visible 7000 - 4000 7 ×10-5 - 4 ×10-5 4.3 ×1014 - 7.5 ×1014 2 - 3
Ultraviolet 4000 - 10 4 ×10-5 - 10-7 7.5 ×1014 - 3 ×1017 3 - 103
X-Ray 10 - 0.1 10-7 - 10-9 3 ×1017 - 3 ×1019 103 - 105
Gamma Ray < 0.1 < 10-9 > 3 ×1019 > 105
Source: http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html

Table 2: Some radiation sources
Region Wavelength (Å) Main Sources
Radio > 109 Supernova remnants, galaxies, galactic center, HII regions, quasars
Microwave 109 - 106 Sun, Crab nebula, microwave background, Milky Way, radio galaxies, quasars
Infrared 106 - 7000 Sun, planets, Galactic center, nebulae, dust, quasars, active galaxies
Visible 7000 - 4000 Sun, planets, stars, galaxies, quasars
Ultraviolet 4000 - 10 Sun, hot stars, active galaxies, quasars
X-Ray 10 - 0.1 Sun, compact binaries, black holes, hot gas in galaxy clusters, AGN, quasars
Gamma Ray < 0.1 crab pulsar, vela pulsar, galactic disk, matter anihilation, extragalactic sources(?)
Source: ``Cosmology'', M. Rowan-Robinson, 2003, Oxford University Press

1  Astronomy from Space

A brief guide to the wavelength regions and missions currently or recently in space, and some notes on Tuorla Observatory's involvements with these missions:

Table 3: Ongoing or upcoming missions at various wavelengths
Region Missions
Radio VSOP/HALCA: http://www.vsop.isas.ac.jp/
Microwave Planck: http://astro.estec.esa.nl/SA-general/Projects/Planck/
Infrared Herschel/FIRST: http://astro.esa.int/herschel/
ISO: http://www.iso.vilspa.esa.es/
Visible HST: http://oposite.stsci.edu/pubinfo/
X-Ray http://imagine.gsfc.nasa.gov/docs/sats_n_data/xray_missions.html
Chandra: http://chandra.harvard.edu/
Newton: http://sci.esa.int/xmm/
Gamma Ray http://imagine.gsfc.nasa.gov/docs/sats_n_data/gamma_missions.html
AGILEhttp://agile.mi.iasf.cnr.it/Homepage/index.shtml

The flagships of X-ray astronomy right now are the Chandra mission flown by NASA and the XMM or Newton mission, flown by the European Space Agency. At Tuorla Observatory, Chandra will be used for studying quasars, active galactic nuclei and black hole candidates. In the future, plans are afoot for using the Italian AGILE mission for studying active galaxies.

In the radio region the Japanese VSOP/HALCA satellite was recently in operation: researchers at Tuorla are working with the data from this mission, which provides very high angular resolution radio images.

At Tuorla we are preparing for the ESA mission Planck, which is scheduled for launch in 2007 and will be much more sensitive than MAP. Both missions will answer very basic questions about the structure of the Universe, the cosmological parameters and the formation place and time of galaxies.

In the infra-red, Tuorla is participating in the ESA FIRST mission (now renamed to the Herschel Space Observatory), which will do imaging and spectroscopy in the far infra-red (60-670 m m) range. The test mirror, made from a special material SiC (silicon carbide) was made in the optical labs at Tuorla. A recently completed infra-red mission was ESA's ISO (Infra-red Space Observatory) mission. At Tuorla studies of active galaxies have been made with this satellite.

The flagship in the optical region is the Hubble Space Telescope. At Tuorla extensive use has been made of the Hubble, among them studies of the amount of dark matter in the Galaxy.

2  Radiation emission processes

Radiation is produced in astrophysical sources by many processes: blackbody emission, bremsstrahlung, synchrotron, compton scattering, as well as the more familiar line emission from excitation in atoms and molecules.

3  Blackbody Emission

If a radiation field is on ``thermal balance'', which means that its temperature is constant, then it emits a characteristic ``blackbody'' spectrum. The spectrum is very often called a ``thermal spectrum'' and the emission called ``thermal''. These all refer to bodies which have a uniform or near-uniform temperature.

The spectrum emitted is called the Planck function. The form was calculated by Planck in the 19th century, and is a consequence of assuming that light has a quantum nature. The Planck function is


In =  2hn3

c2
 1

ehn/kT-1
(1)

where h = 6.626 ×10-27 erg s is the Planck constant and k = 1.38 ×10-16 erg K-1 is the Boltzmann constant.





Problem 2.1 Show that, in units of wavelength l = c/n, the Planck function is


Il =  2hc2

l5
 1

ehc/lkT-1
.
(2)

What are the units of Il?





For hn/kT << 1, the photons behave like classical particles. In this case


ehn/kT » 1 +  hn

kT
+  ¼
(3)

and hence to first order


In =  2n2

c2
k T          hn << kT
(4)

This is called the Rayleigh-Jeans part of the black-body spectrum. In astrophysical sources, it generally applies at low frequencies (e.g. the radio region). It is the straightline part of the curve shown in figure 2.1.

Extending this law into the non-classical regime (hn > kT) leads to infinite total emitted energy! It is this non-physical behaviour which was termed the ``ultraviolet catastrophy'' and was a key problem which lead to the discovery of the quantum nature of light.

For hn/kT >> 1, the quantum nature of the photons comes into play. We have


ehn/kT - 1 » ehn/kT
(5)

and thus


In =  2hn3

c2
e-hn/kT          hn >> kT.
(6)

This part of the black-body spectrum is called the Wien Law, and is seen as a rapid fall-off in the intensity going toward higher frequencies.

The microwave background radiation, thought to be a relic of the Big Bang, is almost exactly black-body.

The energy density un for black-body radiation is given by


un =  4p

c
In
(7)

and the pressure Pn of the field (due to photon momentum) is


Pn =  4p

3c
In.
(8)

3.1  Total flux

The total flux integrated over all frequencies is


I = ó
õ
¥

0 
In dn =  2h

c2
æ
è
 kT

h
ö
ø
4

 
ó
õ
¥

0 
 x3 dx

ex-1
(9)

where x = hn/kT. Performing the integral leads to


I =  2p4k4

15c2h3
T4 =  s

p
T4
(10)

and thus the important result emerges that the luminosity L of a body in thermal equilibrium (i.e. at constant temperature) is


L µ T4.
(11)

The constant of proportionality is called the Stefan-Boltzmann constant, s = 2p5k4/15c2h3 and is 5.67 ×10-5 erg cm-2 K-4 s-1 in cgs units.

CMB_intensity.gif
Figure 1: Brightness Ln versus frequency n of the Cosmic background radiation. The spectrum is very close to a perfect black-body, and has a temperature of circa 2.73 Kelvin. Data points from the COBE FIRAS experiment match the theoretical curve so closely that the differences do not show up at this scale. Source : http://spectrum.lbl.gov/www/cobe/cobe.html





Problem 2.2 Use Gnuplot to make a plot of the Planck spectrum for temperatures T of 10, 102, 104, 106 and 108 degrees Kelvin. Use log Intensity (In) versus log frequency (n) for the axes. Mark on the curves the Rayleigh-Jeans and Wien parts of the spectrum. Identify what type of radiation is associated with the peak of each spectrum (i.e. is it in the optical, X-ray, gamma-ray, radio, IR or UV region?)





3.2  Monotonicity with temperature

For any two blackbody curves, the one with higher temperature has greater intensity than the cooler one for all wavelengths: i.e. a greater flux is emitted at all wavelengths.

Also, as T ® 0, the total flux ® 0,

and, as T ® ¥, the total flux ® ¥.

3.3  Wien displacement law

At what n and l are In and Il maximum?





Problem 2.3 Find the peak luminosity for the functions In and Il, by solving


 d In

d n
= 0    and     d Il

dl
= 0

Useful substitutions to make are x = [(hn)/kT] and y = [ hc/(lkT)].

(Hint: the first case is equivalent to solving x = 3(1-e-x) and the second case to solving y = 5(1-e-y)).





The problem above shows that the maximum emission in the black body curve occurs at different places in l and n. Let's denote these lmax and nmax.

At temperature T, the following relations hold:


nmax / T = 5.88 ×1010 Hz K-1

and


lmax T = 0.290 cm K.

In other words nmax is a linear function of temperature, T. This simple relationship is called the Wien Law.





Problem 2.4 Check from your plots that the peak of the blackbody spectrum shifts linearly with temperature, i.e. that it follows the Wien Law.





3.4  Some characteristic temperatures

3.4.1  Brightness temperature

If we know the true brightness In of a source of radiation at some frequency, n, then it is possible to fit a unique blackbody curve of temperature T through the point. This establishes a ``brightness temperature'' Tb. Note that to know the true brightness one must know the distance to the source.





Problem 2.5 In the Rayleigh-Jeans part of the spectrum, show the brightness temperature is given by


TB =  c2 In

2 n2 k
.





For a thermal (i.e. blackbody) source, the brightness temperature is of course the same at all frequencies. For non-thermal sources it will change with frequency.

3.4.2  Colour temperature

If, as is more often the case, the distance to a source is not known, then a measurement of the flux at two different frequencies can be used to fit a blackbody curve, establishing a ``colour temperature'' for the source.

This is what one commonly does for stars: measuring the amount of light FB and FV in two bands (or colours) B and V, allows one to define a colour


B-V = -2.5 log10  FB

FV
.

Because stars have spectra which are quite close to blackbodies, this colour is a measure of temperature.

stellarspectra.gif
Figure 2: Intensity spectra for stars of different spectral types. The flux is in arbitrary units, and all spectra are normalised at 5500 Angstroms. Hot stars are blue, cool stars are red - as can be seen in the amount of blue versus red light put out by stars of different spectra type. The peak in the energy distribution of O + B stars is well into the ultra-violet, while for M stars it is well into the infra-red (off to the left and right respectively on the lower scale). For G type stars like the Sun it peaks just about where our eyes are most sensitive to photons. Note that the spectral distributions shown above are for a star observed outside of the Earth's atmosphere - the atmosphere can change the distribution markedly in the UV and IR. Source : http://www.ifa.hawaii.edu/ pickles/AJP/hilib.html and A.J. Pickles, 1998, Pub. Astr. Soc. Pac., vol 110, page 863

If one observes the peak in the spectrum, one can fit a very good colour temperature via the Wien displacement law.


Tcol =  h nmax

2.82 k

3.4.3  Effective temperature

For sources which are emitting radiation which is close to a black-body, it is very useful to define the effective temperature Teff, where


F = ó
õ
¥

0 
cosqIn dndW = sTeff4.
(12)

One can think of Teff as a ``best fitting'' black body curve to the actual spectrum.





Problem 2.6 Estimate the wavelength where the peak emission occurs for the three spectra in figure 2.3. What temperature does each source have? Make an estimate of the ``spectral type'' of the stars (i.e. OBAFGK or M) from its temperature (a table of spectral types and temperatures is in the previous lecture).

problemspectra.gif
Figure 3: Three stellar spectra, in the optical region.








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On 6 Feb 2005, 20:53.