Obtain from theory/experiment a value for the absorbtion cross-section. Calculate the number of atoms/ions/molecules giving rise to the line.

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1 Line Emission Observations of spectral lines offer the possibility of determining a wealth of information about an astronomical source. A usual plan of attack is :- Identify the line. Obtain from theory/experiment a value for the absorbtion cross-section. Calculate the number of atoms/ions/molecules giving rise to the line. Infer information about the population of states. Determine temperature and density. The position of the lines gives information on the velocity of the emitters along the line of sight or the cosmological redshift.

2 Einstein Coefficients n E=hν 1 B 1 B 1 A 1 1 n 1 Spontaneous emission rate = Stimulated emission rate = Absorbtion rate = (NB sometimes and and ). defined so that rates are Line profile normalised so that If only radiative transitions link the two states then! #%$'& " ( $'& ) & ( &*$ ) $ ( $+&,

3 X W ; F F F 4 F? Finding -/.% :7 54 are properties of the emitters, so if we can determine them in any specific case then we can apply them in any general case. Consider the case where the emitters and the radiation are in thermal equilibrium ; <>=? 7 =A@CB D (Planck function). Also occupancy of the two states given by Boltzmann function E 4 E 5? ; <=? S W*V S V+W%X 5HG MONPRQSTB D U%V+W S V+W V G DK ? ; MONA`? a [Z =]\ ^ V G DK. So if we can find one Einstein coefficient then we can calculate the other two. Can use Fermi s Golden Rule to find 7 54.

4 i k m i k k k l Radiative transfer: b_c and d c e The equation for radiative transfer is e q i fhgi fj k l m iygnipo, the spontaneous emission coefficient is the energy emitted per time per volume per solid angle per unit frequency. e m qri stusvxwzy{r ~} s9{r ƒ z s9{ ˆŠ ƒ }Œ tuwƒ s9{nž vxs> st y k cy šƒ qri œ cž is the absobtion coefficient and includes the effect of stimulated emission. It is the fractional loss of intensity per unit length. fhgi fj k s]t svÿwƒy{ ˆ~ ƒ } t wƒ s {nžcvÿsn st y{r z ƒ u s9{r ~} s k cy šƒ œ cž _ gni gi ž fhg i g i fj { c œ cž+ _ šz l _ œ cž+ _ ª «c l _

5 Ö Ú µ ² ½ µ µ ³ Excitation Temperature Define the excitation temperature, for the two levels through ± ² ±_³ µ ³/ º¹L» ¼O½¾Y hà %Á Â If the emitters are in local thermodynamic equilibrium then Á Ã. The level is said to be thermalised. If not in thermal equilibrium then in general Ä. The ratio of how much stimulated emission there is compared to absorbtion is Å~Æ~Ç È É ÊŒË ÆÍÌnÎ ÌÈ Ç ÅxÅ~Ç ÏƒÐ Ë ÑÅ~ÏƒÒŸÑ Æ~Ç ÏƒÐ ± ² ±_³ ² Ó Ô¹L» ¼O½¾Y Õ Á Â Stimulated emission is very important at radio frequencies where ¼O½¾Y Õ Ä Ä so that ØÙ Ã ¹½ Â ±_³[Û ²ÜªÝ Ü Þ ß ² ³ à á ² ¼O½ Õ Á

6 â ë ê ì ì è Collisions n E=hν 1 C 1 C 1 1 n 1 â Atoms can become excited or de-excited though collisions. â The rate of collisional excitation is ãåäxã æ>ç äè. (ã æ is the density of colliders). â The rate of collisional de-excitation is ãºè ã æçéè ä. â Consider a system where the particles are in thermal equilibrium and where radiative transitions are negligable. ç äè çéè ä ã_ä[ç äè ê ã è>çéè ä äîí ïð ñlò óoôõyö _ø ù ç äè and ç è ä are properties of the particles so the above is always true. â The ç s can be related to a collisional cross-section and calculated.

7 ú ú ú Critical Density for Collisional De-excitation ú Assume ûnü is low so absorbtion and stimulated emission are negligable. ý þ ÿ þ þ þ þ ÿ ú Substituting for and gives þ þ "!$#&%(' ) * +-, +/. þ is the critical density for collisional de-excitation. þ ÿ 7 7 þ10 8 collisional + de-excitation much faster than spontaneous emission; + approaches its equilibrium value and the line is thermalised. ú þ ÿ 9 9 þ108 spontaneous emission much faster than collisional de-excitation; each time a collision excites ý þ: the 9 high 9 energy state it decays through emitting a photon equilibrium value sub-thermal excitation. If we relax the assumption that ûrü is low the effect is to modify þ10 8 þ10<; 3 6 = 5 6.

8 Example: Cooling rate of gaseous nebulae > Cooling occurs when collisions excite gas into high energy state which then radiates a photon; this leaves the nebula and the cloud cools. > For most collisions which excites a gas particle are de-excited by collisions too. Number of excited gas particles B? C rate of emission B? C rate of cooling B?. > For low density? D D? A most collisions which excites a gas particle results in emission of a photon (because collisional de-excitation negligable). Rate of collisions B?:E C rate of cooling B? E. Example: CO F G H I J rotational line > R S K L M N O P For the CO rotational line at 115 GHz, EQ L M OUTWVYXT Q[Z \ EQ L M OT Q]_^ ` XT Q. C? A L R O/Oba ^ T ` > K L Below this density the number of CO molecules in the M level is lower than one would expect if the system was in C thermal equilibrium weak emission the line is subthermally excited. So generally only see CO from regions where O/O a ^ T `

9 o k d Line Shape c Spectral lines always have a width 3 contributions to this width. c Natural line shape:- dfeg hi gives the mean occupation time in the upper state; Heisenberg s uncertainty relationship gives the corresponding energy width of the line. jkml"nyo p1q shape. k6l t l$u[n r s h<v p hqxw Lorentzian line c Pressure broadening:- collisions while emitting disrupt the radiation train y line width. Lorentzian line shape. c Doppler broadening:- velocity of emitter along the line of sight results in a Doppler shift. Many emitters with Maxwellian random velocities result in Gaussian line shape, width dependant on temperature or bulk motion of the gas. c In most cases the natural line shape and pressure broadening can be ignored and we just consider Doppler broadening. c The shape will also depend on the optical depth. The brightness temperature z {~}6 is a function of frequency given by z { }6 z t ƒ k(t nˆn

11 Calculating fš Œ Consider an emitter moving along the line of sight at speed Ž. The emitted light at frequency / in the emitteds rest frame is Doppler shifted to freqency given by $ Ž Ž Œ If the velocity distribution is Maxwellian then the velocity distribution is given by Žfš Ž œf 1žŸ Ž œ žÿ Ž Œ Equating š and Žfš Ž$š gives š œf 1žŸ Œ Note that since ª«includes a š term, then the optical $ š œ žÿ depth of a line is also proportional to š.

12 Figure 1: Low optical depth cloud Figure : High optical depth cloud

13 Equivalent Width The actual profile of an absobtion line may be difficult to measure precisely. Measure the area in the line and calculate the equivalent width,. ± ²³ ²µ ˆ ± ¹º»1 ±b¼ ³¾½ˆ½ Can then estimate À I 0 I I ν Á Ä Ä Ä Ä Ä Ä Ä Ä Ä Å Å Å Å Å Å Å Å Å Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Å Å Å Å Å Å Å Å Å Å Å Å Å Å Ä Ä Ä Ä Ä Å Å Å Å Å Ä Ä Ä Ä Ä Ä Å Å Å Å Å Å along the line of sight. Equal Areas W Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã Ã ν

14 Æ à Example: H 1 cm line Æ Magnetic ÇÉÈ Ê dipole transition between nuclear hyperfine levels Ë F = 0, 1; Degeneracies 1,3 ÌÎÍ Ï Ð ÑÓÒÕÔ&Ö ÍÏ Ð ÖÒÙØ/Ú Û Ñ ÜÝ Í Þbß Ý Í à GHz; occu- á Ñ Ü¾â pancy time years. Æ Ñ Ü Í Ç ã Ý ä For a cloud at temperature 100 K and density the time between H atom collisions is 100 years levels are set å:æçá Ñ Übè Ý ä à by collisions. collisional de-excitation much faster than spontaneous emission and the line is thermalised. Æ The spin temperature, é ê Ð ë Ì ÍÏ[ì$íî Ð ÜïÒÙÜñð K. å Í å Ï Ð òóõôö ø ë Ì ì$í î é êµù Ð ò à å Ï Ð å Ô Æ An optically thick line will give an estimate of é ê. Æ An optically thin line will give an estimate of ú åûü.

15 Summary ý The equation of radiative transfer for emission lines can be written in terms of the transitions Einstein coefficients þ ÿ, ÿ and ÿ. ý Fermi s Golden Rule can be used to find ÿ ; the other two coefficients can be calculated from this. ý Collisions can also excite/de-excite ÿ atomic transitions with ÿ rates governed by and. ý There is a critical density above which the rate of cooling of a gas varies linearly with density and below which it varies quadratically. ý Emission line widths are determined by their natural line shapes (uncertainty principle), pressure broadening and Doppler broadening.

The incident energy per unit area on the electron is given by the Poynting vector, '

The incident energy per unit area on the electron is given by the Poynting vector, ' ' ' # Thompson Scattering Consider a beam of radiation moving in the direction, and being scattered by an electron through an angle in the plane. The electron experiences the following electric fields