SCATTERING OF RADIATION Scattering depends completely on properties of incident radiation field, e.g intensity, frequency distribution (thermal emission instead does not - depends only on T). EXAMPLE: electron scattering e- +   e- +  Change of energy/momentum of electron and photon depend on initial energy/momentum of photon We consider ISOTROPIC COHERENT (ELASTIC, MONOCHROMATIC) SCATTERING = the photons absorbed/emitted (i.e scattered in/out) are distributed isotropically + overall distribution of frequency does not change for radiation field. NOTE: we talk about both “absorption” and “emission” even for scattered radiation Emission coefficient for coherent isotropic scattering j =  J power emitted = power “absorbed” (per unit volume and freq.) --  S = J and the RT equation becomes an integro-differential equation in I : dI /ds = -  (I – J ) Note: J is an integral of I

RANDOM WALK: ANOTHER APPROACH TO SCATTERING In the RT equation approach one deals with intensity of rays, i.e. with the collective behaviour of a large number of photons -  can be thought of as “average” behaviour for a single photon (probabilistic approach). EXAMPLE: Exp. decay of absorbed photon beam can mean that photon has a probability exp (-  ) to travel an optical depth  Now consider photon emitted in infinite homogeneous scattering region  -  photon undergoes several scatterings The typical distance travel by the photon l* is a multple of its mean free path l (which here means average distance covered between scattering events) For isotropic scattering: l* 2 = Nl 2 --  l*=N 1/2 l For a finite medium compare with characteristic size of medium L : the photon will continue to scatter until it escapes. Think of number of scatterers ~ number of “potential” absorbers, i.e. related to optical depth of medium For high optical depth l* ~ L because most photons will not escape  N ~ L 2 /l 2 -  N~  2 (  >>1). For low optical depth prob. of scattering ~ 1 – exp(-  )~  -  N ~  (  << 1)

COMBINED ABSORPTION AND SCATTERING Consider a thermal emitter whose photons can also undergo scattering. The full RT equation will be: dI /ds = (  +  )(I – S ) where S = (a B +  J )/(   +  ) = average of the “absorption” and “scattering” source functions averaged by the two coeffs. Generalized “optical depth” = extinction coefficient  d  = (  +  )ds Generalized “mean free path” -  l = (   ) -1 Considering the probability that a free path ends with absorption or scattering in an infinite medium one can defined also a generalized “mean displacement:” l* of a photon random walking. Random walks starts with thermal emission of a photon (creation), then many scatterings, and ends with true absorption (destruction). The “mean displacement” is also called thermalization length or effective mean path: l * ~ [    ) ] -1/2 can define also corresponding “effective optical depth”  For finite medium two simple cases,  > 1. In both cases emitted luminosity is that of a “modified blackbody” (for BB L =  B A)

COOLING PROCESSES IN THE ISM In general three cooling processes important: (1)Recombination: atom/ion/molecule captures electron and emits photon. Opposite process is photoionization: atom/ion/molecule absorbs photon and loses electron (2) Collisional de-excitation (heating via collisional excitation) e.g. free electron colliding with atom/ion/molecule (needs some ionization in gas to have free electrons) (3) Radiative cooling: bound electron in atom/ion/molecule decays to lower energy state emitting photon These processes can happen in all phases of ISM. In molecular phase (low T) molecules are more important (roto-vibrational as well as radiative transitions) Other cooling processes (within molecular clouds – see next lecture) are: (4) Cooling by dust (thermal emission) (5) Cooling via collisions between dust and gas (molecules), important at T < 100 K

Consider a partially ionized, hydrogen cloud (hydrogen most abundant element) that can cool via hydrogen recombination (WIM  WNM) In recombination ion captures electron and emits a photon of energy h equal to the kinetic energy lost by the electron. Recombinations to energy states higher than ground state cool the gas more efficiently because less energetic photon emitted in de-excitation has lower absorption probability by surrounding atoms (lower Einstein’s coeff. for absorption). In steady state photoionization (heating) rate=recombination (cooling) rate If no other cooling processes occur the equilibrium between the two rates will set the temperature of the cloud. The heating by photoionization H ion can be expressed as: H ion = n e n p  (T), h I being 13,6 ev for ground level of hydrogen and the mean energy is the result of weighting over the ambient photon flux and the (frequency dependent) ionization cross section. The cooling rate via recombination can be written as: C rec = n e n p  (T)k B T [ /k B T] where the mean is weighted by the Maxwell Boltzmann distributions for velocity of electrons Recombination coefficient

Setting H ion =C rec implies k B T gas ~ ~ h II ~ 10 ev in typical ISM conditions (smaller than h I of ground state because some hydrogen atoms will be in excited states).--  so T gas ~ 10 5 K >> T ISM (for WNM, WIM, CNM and molecular phase) Why is T WNM lower (~ 10 4 K)? Because (1) atoms can cool by emission of photons during spontaneous transition to state of lower energy (called radiative or line cooling) and (2) heavier elements such as carbon and oxygen, contribute a lot to such line cooling despite being much less abundant than hydrogen. Will show how... RADIATIVE and COLLISIONAL COOLING Consider partially ionized phase of ISM (e.g. WIM) - there are free electrons which collide with atoms (hydrogen or other elements) -  atom goes to higher energy state by absorbing kinetic energy from electron. It will then decay via collisional de-excitation, or spontaneous radiative decay. atom + e- --  atom* + e- --  atom + e- +  Collisional de-excitation depends on collision rate, which depends on ambient density. In typical ISM conditions densities are low enough that radiative emission dominates over collisions (opposite situation in the atmosphere of the Earth).