9.1  Optically thin emission lines

In the `standard model' for interpreting line intensities there are three fundamental assumptions that serve to simplify the problem considerably (these are within the CHIANTI models but also normally assumed):

1. the plasma is in a steady state;
2. atomic processes affecting the ionisation state of an element can be separated from those affecting the level balance within an ion;
3. all lines are optically thin.

The last one normally holds for the EUV lines that CDS observes (with the exception of the helium lines), while the major uncertainties reside in the unknown ionization state. This casts serious doubts on results that can be obtained with spectroscopic methods.

The intensity I(l_ij), of an optically thin spectral line (having filling factor=1) of wavelength l_ij (frequency n_ij = [ c/(h l_ij)]) is

I(lij)=  h nij 4 p    ó õ Nj   Aji  dh    [ergs  cm-2  s-1  sr-1]

(4)

where i, j are the lower and upper levels, A_ji is the spontaneous transition probability, N_j is the number density of the upper level j of the emitting ion and h is the line of sight through the emitting plasma.

In low density plasmas the collisional excitation processes are generally faster than ionization and recombination timescales, therefore the collisional excitation is dominant over ionization and recombination in populating the excited states.

This allows the low-lying level populations to be treated separately from the ionization and recombination processes.

For allowed transitions we have N_j(X^+m) A_ji ~ N_e. The population of the level j can be expressed as:

Nj(X+m) =  Nj(X+m) N(X+m)     N(X+m) N(X)     N(X) N(H)  N(H) Ne   Ne

(5)

• N(X^+m)/N(X) is the ionization ratio of the ion X^+m relative to the total number density of element X (contained in the files in the ioneq/ directory);
• Ab(X)=N(X)/N(H) is the elemental abundance relative to Hydrogen (contained in the files in the abundance/ directory);
• N(H)/N_e is the Hydrogen density relative to the free electron density. Often assumed to be equal to 0.83, as hydrogen and helium are usually completely ionised for hot optically thin plasmas.

  See the CHIANTI routine PROTON_DENS for details on how to calculate N(H)/N_e.

• The fraction n_i = [(N_j(X^+m))/(N(X^+m))] of ions X^+m lying in the state j is determined within CHIANTI by solving the statistical equilibrium equations for a number of low-lying levels of the ion including all the important collisional and radiative excitation and de-excitation mechanisms.

For the basic CHIANTI model these processes are simply electron and proton excitation and de-excitation, and the generalised radiative decay:

aij = Ne Cije + Np Cijp + Aij

(6)

where C_ij^e is the electron excitation-de-excitation rate, C_ij^p is the proton excitation-de-excitation rate, N_p is the proton density, A_ij is the generalized radiative decay rate, that includes A_ij, the radiative decay rate which is zero for i < j (the `A-values' are contained in the CHIANTI .wgfa files), and the photoexcitation and stimulated emission.

C_ij^e is given by:

Cije = Ne qij     i < j

(7)

Cije = Ne  wj wi exp æ è  DE kT ö ø qji     i > j

(8)

where w_i is the statistical weight of level i, k is Boltzmann's constant, T the electron temperature, and q_ij the electron excitation rate coefficient which is given by:

qij=2.172×10-8 æ è  I¥ kT ö ø 1/2   exp æ è -   DE kT ö ø  Uij wi        [cm3 s-1]

(9)

where I_¥ is the Rydberg energy (13.61 eV), and U_ij is the thermally-averaged collision strength for the i ® j excitation. The U_ij are derived from the scaled data in the CHIANTI .splups files.

Within CHIANTI, we presently model the Photoexcitation and Stimulated Emission by assuming a a blackbody radiation field of temperature T_*. The generalized photon rate coefficient in this case is:

Aij = ì ï ï ï í ï ï ï î W(R) Aji  wj wi  1 exp(DE/kT*) -1     i < j Aji é ë 1 + W(R)  1 exp(DE/kT*) -1 ù û     i > j

(10)

where A_ji is the radiative decay rate and W(R) is the radiation dilution factor which accounts for the weakening of the radiation field at distances R from the source center.

We also assume an uniform (no limb brightening/darkening) spherical source with radius R_*:

W =  1 2 é ë 1 - æ è 1 -  1 r2 ö ø 1/2   ù û

(11)

where

r =   R R*

(12)

The solution of Eq.  is performed by the CHIANTI routine pop_solver.pro, which gives the fraction of ions in the state i.

The level populations for a given ion can be calculated and displayed with plot_populations.pro (but also see pop_plot.pro).

We rewrite the intensity as:

I(lij) = ó õ Ab(X) C(T,lij,Ne) Ne NH dh

(13)

where the function

C(T,lij,Ne) =  h nij 4 p     Aji Ne     Nj(X+m) N(X+m)     N(X+m) N(X)   [ergs   cm+3   s-1],

(14)

called the contribution function, contains all of the relevant atomic physics parameters and is strongly peaked in temperature.

gofnt.pro calculates these contribution functions.

Please note that in the literature there are various definitions of contribution functions. Aside from having values in in either photons or ergs, sometime the factor [  1/(4p)] is not included. Sometimes a value of 0.83 for N(H)/N_e is assumed and included. Sometimes the element abundance factor is also included. Any of the above (or any other) variations also affect the definition of a line intensity in terms of the contribution function and the DEM. In the following we will refer to the functions C(T,l_ij,N_e) and G(T,l_ij,Ab(X),N_e) = Ab(X)  C(T,l_ij,N_e) ( i.e. the contribution function that contains the abundance factor ).

If we define, assuming that is a single-value function of the temperature, the differential emission measure DEM (T) function as

DEM (T) =  Ne NH  dh dT   [cm-5 K-1]

(15)

the intensity can be rewritten, assuming that the abundance is constant along the line of sight:

I(lij) = Ab(X) ó õ T   C(T,lij,Ne)  DEM (T)   dT    [ergs   cm-2   s-1  sr-1]

(16)

The DEM gives an indication of the amount of plasma along the line of sight that is emitting the radiation observed and has a temperature between T and T+dT.

The CHIANTI routine chianti_dem.pro calculates the Differential Emission Measure DEM(T) using the CHIANTI database, from a given set of observed lines.

9.1.1  Isothermal approximation

In the isothermal approximation, all plasma is assumed to be at a single temperature (T_o) and the intensity becomes:

I(lij) = C(To, lij, Ne) Ab(X) EMh

(17)

where we have defined the column emission measure

EMh = ó õ Ne NH dh     [cm-5]

(18)

Please note that in the literature many different definitions of Differential Emission Measures, Emission Measures and approximations can be found (see Del Zanna et al., 2002 for some clarifications).