# Presentation: 937344

## Emission mechanisms. II 

***                      ******Giorgio****** ******Matt***
***(Dipartimento di Fisica, Università Roma Tre, Italy)***
***Reference: Rybicki & Lightman, “Radiative processes in astrophysics”, Wiley***
***                   ******Kahn, in SAAS-Fee 2000, Springer***

## Outline of the lecture 

*** ******Basics on atomic transitions***
*** ******Collisional equilibrium***
*** ******Photoionization equilibrium***
*** ******Line diagnostics***

## Einstein’s Coefficients (for atoms)

** ****Spontaneous emission****. The system is in an excited level 2 at energy **
***    ******E+h******ν******0***** ****and  drops to a lower level 1 (energy E) by emitting a **
**    ****photon of energy *****h******ν******0***** **
**    *****A******21*****: transition probability per unit time of spontaneous emission**
** ****Absorption****. The system, at level 1 with energy *****E*****, absorbs a **
**    ****photon of energy h****ν****0**** and reach the level 2 at energy *****E+h******ν******0*****. The **
**    ****transition probability depends on the radiation field.**
**    *****B******12******J*****: transition probability per unit time of absorption**
** ****Stimulated emission****. The system goes from level 2 to level 1 **
**    ****stimulated by the presence of a radiation field.**
**     *****B******21******J*****: transition probability per unit time of stimulated emission**
**At the equilibrium, the rate of emission must be equal to the rate **
**of absorption:                     **

**    *****n******1******B******12******J = n******2 ******(A******21******+B******21 ******J)***

##  

**At thermodynamic equilibrium:    **
**                                               *****n******1 ******/n******2 ******=(g******1******/g******2******)e ******h******ν******/kT******     J=B(T) ***
**and therefore:**

**called “detailed balance relations” and valid universally **
**(not only for  thermodynamic equilibrium).**

****** **

**Not all atomic transitions are allowed. Selection rules are such that**
***Δ******S=0,  ******Δ******L=0,±1,  ******Δ******J=0,±1***** (but *****J=0 → 0 strictly forbidden*****). **
**However, selection rules may be violated because they are derived in an approximated way. In practice, strictly forbidden means very low probability of occurrance.                                                                     **

## Line profiles 

**Let us call *****φ******(******ν******)***** the probability that the transition occurs by emitting or absorbing a photon with energy *****h******ν****** ******(emission or absorption line ******( ∫******φ******(******ν******)d******ν****** ******≡****** 1)  *****An unavoidable source of broadening is due to the uncertainty principle -- *****dEdt ~ h/2******π*****, dt being the timescale of decay -- this natural broadening has the form of a  Lorentzian function (****γ**** is the decay rate):                                                                            **

**The combination of the two gives rise to the *****Voigt profile,   *****composed by a ****Doppler core**** and ****Lorentzian wings**

**Further broadening is due to the thermal motion of atoms: **

**Forbidden lines are narrower than **
**resonant lines**

## Photon Excitation/de-excitation 

**A photon can be absorbed by an electron in an atom, which jumps to a higher level (*****excitation*****). The probability of absorption depends on the *****oscillator strenght f *****(related to the Einstein coefficients). *****f *****is large for *****resonant lines*****, low for *****forbidden lines. ***

**Line absorption from a population of atoms is measured in terms of the **
***Equivalent Width (EW). ***** **
***I******ν******(c)***** intensity of the continuum without absorption**
***I******ν******(l)*****   actual intensity of the continuum.**
**It corresponds to the area in the spectrum removed by the absorption, and depends on the probability of the transition and the amount of matter. **

## Optically thick in the wings   

**If matter is optically thin even at the line center, the line profile is unsaturated at any frequency. Increasing the optical depth, the line saturates, first in the Doppler core and then in the Lorentzian wings.**
***Curve of growth***

***Optically thick in the wings   *********
***Optically thick in the core     *********
***Optically thin                          *********

***N******H******=σ******T******R***

## The inverse process is de-excitation, when an electron in an excited atom falls into a lower level by emitting one (or more, if the de-excitation occurs as a cascade) photon.  Also for the emission 

**The inverse process is de-excitation, when an electron in an excited atom falls into a lower level by emitting one (or more, if the de-excitation occurs as a cascade) photon.  Also for the emission **
**lines can be defined an equivalent width:**

**If line emission occurs via the exact inverse transition with respect to absorption, the process is called *****resonant scattering*****.  Resonant scattering is important for resonant lines, both because absorption is more likely (larger oscillator strengths) and because forbidden de-excitation occurs on long timescale (and therefore something different is likely to occur in the meantime).**

## Collisional Excitation/de-excitation 

**An atom can be excited by interacting with another atom or a free electron.**
**The inverse process is collisional de-excitation, when an electron in an excited atom falls to a lower level ceding the energy to the passing electron. **

## Ionization/recombination 

**Process ****(ionization)****Collisional ionization**
**Photoionization**
**(Photoelectric absorption)**
**Autoionization **
**(Auger effect)**

**Inverse process **
**(recombination)**
**3-body recombination**
**Radiative recombination**
**Dielectronic recombination**

## Collisional ionization: similar to collisional excitation, but the excited electron ends up in a continuum, rather than bound, state.

***Collisional ionization*****: similar to collisional excitation, but the excited electron ends up in a continuum, rather than bound, state.**
***3-body recombination*****: 2 free electrons interact with an ion. One of them gets captured, the other one remains free **
**carrying out the excess energy **

***Autoionization*****:  an excited atom decays by ejecting an electron from an outer levels. **
***Dielectronic recombination*****: capture of a free electron, with the excess energy used to excite the atom. The excited atom may then decay radiatively**

## Photoelectric absorption 

**A bound electron is expelled from the atom by the absorption of a photon with *****E ≥ E******th***** with *****E******th****** *****the ionization potential.**
**Above the threshold, the cross section is:**

**Given the *****E******-3.5***** dependence, the absorption is dominated by photons just above threshold.**

**Summing over all shells and convolving with cosmic element abundances, **
**  ****the total cross section  can be  derived.**
**Photoionization is very important in the UV and soft X-ray band**

## Radiative recombination 

**Radiative recombination (i.e. the capture of an electron by an atom with release of one or more photons) can occur either via**
**a recombination cascade or directly to the ground state.**

**In the latter case, a pseudo continuum is created, as**
**the photon carries out the ionization potential plus the kinetic energy of the electron.**
*********    ******Radiative Recombination Continuum*****  ****                **

**The recombination rate decreases with the electron velocity (temperature)**

## Fluorescent emission 

**If ionization occurs in an inner shell, the atom is not only ionized but also excited. De-excitation can**
**occur via Auger effect (double ionization) or radiatively via emission of a *****fluorescent photon*****.**
**The probability of a radiative de-excitation is called *****fluorescent yield***

**If the ionization is in the K shell,  fluorescence may occur via a ****L→K (K****α**** photon), M→K (K****β**** photon),**** etc. **
**K****α**** transition is the most probable (9/10 for iron)**

## Collisional  equilibrium 

**Let us assume matter in thermal equilibrium. Let us also assume that the radiation field is negligible. At equilibrium, ionization and recombination rates must be equal.**

**By solving this system of equations the ionization equilibrium **
**(i.e. the fraction of each ion of each element) can be obtained as **
**a function of temperature.**

## Line cooling

**In collisionally ionized plasma, cooling by line emission may be important. Once solved for the ionization structure, and summing up the emissivity due to all ions, the total emissivity is: **

**The main continuum emission process in a plasma is thermal bremsstrahlung, for which: **

**For cosmic solar abundances, bremsstrahlung dominates **
**above ****~2x10****7**** K (i.e. about 2 keV),**** line cooling below.**

**When in the line cooling regime, **
**cooling becomes very fast**

## Spectra from collisionally ionized plasma 

**kT=0.3 keV – *****line emission  dominates***
**kT=1 keV – *****line and continuum emission both important***

**kT=7 keV – *****continuum (brems.) emission dominates***

## Example: Clusters of Galaxies

**IGM (Inter Galactic medium) is indeed in collisional equilibrium!**

## Photoionization  equilibrium 

**Let us now assume that matter is in equilibrium with the radiation field. Photoabsorption may now be the main ionization process. Again, at equilibrium ionization and recombination rates must be equal. Assuming that the recombination time scale, *****1/******α******(X******i+1******)n******e *****, is short:**

**The ionization rate depends on the ionizing photon flux, the recombination rate on the matter density. The ionization structure is therefore governed by the so called **
***ionization parameter U***

## Photoionization  equilibrium 

**Temperature does not change much with the ionization parameter until**
**the matter  is completely ionized. At  that point, photons can no longer be used for ionization, and the main interaction becomes Compton scattering.  **

**The *****Compton temperature***** is **
**then reached:**

## Example: warm reflectors in AGN 

***Urry & Padovani (1995)***

***NGC 5738 - HST***

## Example: warm reflectors in AGN 

***Bianchi et al. 2006***

## Example: warm reflectors in AGN 

**NGC 1068**** *****(Kinkhabwala****** et al. 2002)***

**While in collisionally ionized plasmas lines tend to concentrate at energies around kT, in photoionized plasmas lines are more spread over the spectrum**
**(depending on the ionizing spectral distribution)**

## Line diagnostics

**Apart from the broad band spectral fitting, other tools to distinguish between collisionally and photoionized plasma are: **

**Line ratios in He-like elements**
**(z=forbidden,**
**w=resonant,**
**x,y=intercombination)**
**Also density diagnostic**

***Porquet & Dubau 2000***

**Radiative recombination preferentially **
**occurs there**

## Line diagnostics

**The presence of a prominent RRC also indicates photoionized plasma (in collisionally ionized plasma it would be very broad and hard to detect).**

## Compton Reflection

**A rather common astrophysical situation is when X-rays illuminates `cold’ matter.**
**It produces the so called *****Compton***
***reflection continuum***

**The shape of the continuum is **
**due to the competition between ****photoabsorption**** and ****Compton**
**scattering****. ****Fluorescent lines**** are also produced, ****Fe K****α**** being the most prominent.**

***(Reynolds et al. 1995)***

## Iron line spectroscopy and GR

**Iron line can be used to probe General Relativity effects around black holes in ****Active Galactic Nuclei**** and **
**Galactic Black Hole systems **

## Black Holes 

**  ****The mass ****M  **
**The angular momentum ****J **** **
**The electric charge ****Q **

**A Black Holeis fully described by three quantities:    **

**If ****Q=0**** (as usually assumed), the space-time is described **
**by the **** Kerr ****metric**
**If also ****J=0  ****(i.e. spherical symmetry), the (much simpler) ****Schwarzschild  ****metric can be used**
**  ****r****g****=GM/c****2**** is the gravitational radius. In the following, all distances will be given in units of r****g****  **
**a=Jc/GM****2****  ****is the adimensional angular momentum per unit mass, ‘spin’ thereafter **

## Accretion discs

**We can assume that the inner disc radius corresponds to the innermost stable circular orbit (****ISCO****)**
**The ****ISCO**** depends on the BH spin and on whether the disc is co- or counter-rotating with the BH**

**Let us assume a geometrically thin, optically thick accretion disc. ****Matter rotates in (quasi) circular orbits (i.e. ****Vφ >> Vr**** ) with Keplerian velocities.**

## Accretion discs

**The Keplerian velocity (in the Locally Non-Rotating Frame) **
**is given by:   **
** **
**Vφ/c = (r****2**** – 2ar****1/2**** + a****2 ****)**
**            ****/(r****2**** + a****2**** – 2r)****1/2 ****(r****3/2**** + a)**
**which, for small r, can be a significant fraction of *****c***

## Photon trajectories

**In GR, photon geodesics are no longer straight lines (****light bending****)**
** **
**In Schwarzschild metric **
**the trajectories are **
**two-dimensional, in Kerr metric they are fully **
**three-dimensional **

***Luminet ***
***1979***

## Photon shifts

**Photons emitted in the accretion disc appear to the distant observer as ****redshifted  ****because of the Gravitational redshift and the Doppler transverse effect, and  ****blueshifted**** /****redshifted**** **
**by the Doppler effect when the matter is ****approaching****/****receding**

**i=30                     75                         90**

**a=0**
**a=0.5**
**a=0.998**

***(Dabrowski 1998)***

## Doppler boosting

**i=30                     75                         90**

**a=0**
**a=0.5**
**a=0.998**

***(Dabrowski 1998)***

**The quantity **
**I****υ****/υ****3**** **
**is a Lorentz invariant.**
** **
**Therefore, the ****blueshifted**** radiation is brighter **
***(***_***Doppler boosting***_***),***** the ****redshifted**** **
**is fainter.**

## Iron Lines

***(Fabian et al. 2000)***

**The abovementioned**
**SR and GR effects**
**modify the line profile**
**in a characteristic and **
**well-recognizable way**

## Iron Lines

***i=20***
***i=50***
***        ***
***   ******i=87***

***(Matt et al. 1992)***

**a=1**

**a=0**

***Fabian et al. (2000)***

**Isotropic**
**illumination**
***(e.g. George & Fabian 1991, ***
***Matt et al.***
***1991, 1992)***

## Observations

**MCG-6-30-15**

**ASCA***** (Tanaka et al. 1995)***

**BeppoSAX***** (Guainazzi et al. 1999)***

**XMM-Newton *****(Wilms et al. 2001)***

**Rin <6, a>0 !!!**

**Rin > 6**