5 NLTE Models: PRO2

The program PRO2 (Werner 1986; Werner et al. 2003; Rauch & Deetjen 2003; Werner et al. 2012) calculates plane parallel NLTE model atmospheres in radiative and hydrostatic equilibrium. It is based on the “Accelerated Lambda Iteration (ALI) method (Werner & Husfeld 1985). It consists mainly out of two iteration cycles. The “outer” cycle is called “Scharmer iteration” (this name is taken from Scharmer 1981 who worked on solutions of radiative transfer problems using approximate lambda operators) and iterates the radiation field according to

The source function S of the actual iteration n is depending on the radiation field J which is to be calculated. To calculate S, the non-linear statistical equations have to be solved under consideration of the radiative and hydrostatic equilibrium. This is done in the “inner” iteration cycle by a [quasi-] Newton-Raphson iteration. In the following this procedure is called “linearizations”.

5.2 The PARAMETERs

For the compilation of PRO2 the files PARA.INC, PARA1.INC, and PARA3.INC have to be adjusted to the used atomic data file ATOMS (Sect. 2) and the frequency grid FGRID.

	PARAMETER(	NLMAX=	229,	NIONMAX=	79,	NATOMAX=	17,
>		NFMAX=	7999,	LTEMAX =	931,	NLATOM =	229	)

	PARAMETER(	NRBBMAX=	249,	NRBFMAX=	229,	NCBBMAX=	599,	NCBFMAX=	229,
>		NRFFMAX=	61,	NCBXMAX=	4999,	NRLLMAX=	1,	NRLUMAX=	1,
>		NRBBMAD=	999,	NRBFMAD=	6,	NCBBMAD=	6,	NCBFMAD=	2,
>		NRFFMAD=	2,	NCBXMAD=	8,	NRLLMAD=	1,	NRLUMAD=	1	)

	PARAMETER(	NRDIMAX=	1,	NRDIMAD=	1,	NSIG=	599999	)

The PARAMETER denote maximum numbers of e.g. elements etc. (see below) which can be treated with PRO2 if the files are used for its compilation, respectively.

parameter	maximum numbers of

NLMAX	NLTE levels
NIONMAX	ions
NATOMAX	elements
NFMAX	frequency points
LTEMAX	LTE levels
NLATOM	NLTE levels (block-matrix method, for one element, set value equal to NLMAX)
NRBBMAX	radiative bound-bound transitions
NRBFMAX	radiative bound-free transitions
NCBBMAX	collisional bound-bound transitions
NCBFMAX	collisional bound-free transitions
NRFFMAX	radiative free-free transitions
NCBXMAX	collisional bound-bound transitions (to LTE levels)
NRLLMAX	radiative transitions within a band of a complex ion
NRLUMAX	radiative transitions lowup of complex ions
NRBBMAD	data points in formula used for RBB
NRBFMAD	data points in formula used for RBF
NCBBMAD	data points in formula used for CBB
NCBFMAD	data points in formula used for CBF
NRFFMAD	data points in formula used for RFF
NCBXMAD	data points in formula used for CBX
NRLLMAD	data points in formula used for RLL
NRLUMAD	data points in formula used for RLU
NRDIMAX	radiative di-electronic transitions
NRDIMAD	data points in formula used for RDI
NSIG	size of array SIGBF (NRBF× frequency points)

NSIG uses the number of frequency points between the thresholds and the maximum energy. This reduces the size of he array SIGBF and saves core memory.

These PARAMETERS are given in the output of ATOMS2 and SETF2 (please use e.g. grep para to extract them for this output). The PARAMETER NSIG is named there NRBFMAW. In the PARAMETER files used for the compilation of LINE1, NSIG has to be eliminated from PARA3.INC and NRBFMAW has to be added in PARA1.INC accordingly.

5.3 Input and Output Files

The program PRO2 expects the following input files (only those which are marked with “^*” are necessary):

DATEN*
contains all input cards, used as a here document for STDIN (file name can be freely chosen)

RBF_CUTOFF
optional input file, that can be used to set specific bound-free absorption cross-section equal to 0, starting at a given frequency.
Example:

cat > RBF_CUTOFF << eos
LOW |~|

CUTOFF-FREQUENCY
C432S |~|

C522S

1.0E16
HE28 |~|

HE31

1.2E17

NEGDENS
Indicates that negative density values exist in MODOUT. MODOUT can be read by LINE1 and transformed into a formatted version. This model may be edited then.

STOP
Indicates that a model converged. PRO2 will stop immediately in case that this file exists in the working directory because it assumes convergence then (cf. Sect. 10). This can be used to stop PRO2 calculations (e.g. to restart the jobs with a changed input file DATEN.

5.4 Input Options

The program PRO2 can be controlled by a number of input options. Some of them are initialized with a default value (indicated by *) and can be omitted if no other values shall be requested. Indispensable blanks are indicated by |~|

. Numerical data can be inserted format free.

ABUNDANCE

AA x
introduces the abundance of the element AA (flushed to the left). The input x is the number ratio relative to hydrogen. If the start model contains already the element AA, this card is disregarded.

ABUNDANCE

MASS

FRACTION
introduces the abundance of the element AA (flushed to the left). The input x is the mass fraction. If the start model contains already the element AA, this card is disregarded.

ACCELERATION |~|

CONVERGENCE |~|

FROM

ITERATION 999*
the Ng method for acceleration of convergence is used, starting from iteration 999. After every 4th iteration, all photospheric parameters are extrapolated from the previous iterations.

BROYDEN=0*, |~|

SWITCH-LIMITS |~|

0-->1,

1-->2,

2-->1

: 1.E-33* 1.E-33* 1.E+33*
This option invokes a quasi-Newton method (“Broyden method”) instead of the Newton-Raphson iteration of the linearized equations.

While in the case of diagonal operator the switches occur locally following local criteria, the tridiagonal operators require global judgement and used the highest relative correction found in the whole atmosphere. Informations about the switches can be printed to STDOUT (see below).

CHANGE

EFFECTIVE TEMPERATURE |~|

x
generally, PRO2 takes the information about Teff from the start model (Sect. 4). With this card, the another Teff can be chosen. This works in small steps and takes much less time than the calculation of a new model with almost the same parameters.

CHANGE

LOGG

x
like CHANGE EFFECTIVE TEMPERATURE, but for a new value of the surface gravity g

CHANGE

ABUNDANCE

MASS

FRACTION
like CHANGE EFFECTIVE TEMPERATURE, see ABUNDANCE for MASS FRACTION.

COMMENT:
With this card any comment can be printed at the beginning of the output.

DEPTH

DEPENDENT

LINE

PROFILES,

LINEARIZATION: |~|

XXXXX
This option XXXXX = FIRST or EACH |~|

is indispensable if depth dependent line profiles shall be calculated. FIRST saves a lot of computational time.

ERRSCH=1.E-04*
limit for the relative corrections to stop the Scharmer iteration

FORMAL

SOLUTION
PRO2 carries out two formal solutions (with/without line opacities) and saves the line profile in the file LINES.

STEP

F-VALUES:

*****-START: |~|

IIII

x₁ x₂ x₃
The oscillator strengths of all lines of a selected ion IIII are reduced by a factor of x₁ at the beginning of the iterations and stepped up by a factor of x₂ every x₃ Scharmer iterations.

For ***** the two possibilities INPUT or MODEL can be requested. With INPUT, PRO2 uses the values of x₁, x₂, and x₃ as requested in the card; with MODEL, PRO2 reads these values from the start model (Sect. 4) if a previous PRO2 has not reached unity for all step-up values and thus, written them to the output model.

STEP

F-VALUES:

KANTOROVICH |~|

RESET

AFTER

STEP

UP
In general, after each increase of f-values, the Broyden/Kantorovich method is reset. This card suppresses this reset.

STEP

F-VALUES:

LIMIT

FOR

ABS.

REL.

CORRECTION |~|

x
In case that the absolute relative total density correction exceeds x (default value: 0.1), the next step up is skipped.

GAMMA=1*
The value defines the optical depth

which separates the region of the line wing for which the core and the wing approximations shall be used.

IGNORE

CPU

TIME

LIMIT
set the security time (necessary to complete the job) equal 0. This can only be used if it is sure that the requested number of iterations can be completed within the jobs run time limit.

INCREASE

COLLISIONAL |~|

RATES

FACTOR

x
The collisional rates can be increased in order to simulate a LTE model atmosphere. The factor x has to be > 10⁺³⁰, in order to guarantee that collisions dominate the photosphere ...

JACOBI

FRESH-UP,INTERVAL 5*
calculation of the Jacobian every 5^th Scharmer iteration. Only valid in case of Broyden or Kantorovich method.

KANTOROVICH=2, |~|

SWITCH LIMITS |~|

0-->1,

1-->2,

2-->1

: 1.E33* 0.1* 1.0*
like the BROYDEN card.

LINEARIZE

HYDROSTATIC |~|

EQUATION
With this option the hydrostatic equation is solved simultaneously with the statistical equation and not (default) solved subsequently.

RENORMALIZATION |~|

COMPLEX

LINE

CROSS

SECTIONS
undoes a previously made re-normalization of the sample cross-sections for iron group elements

TEMPERATURE |~|

CORRECTION
the radiative equilibrium is omitted from the linearization

T-CORRECTION |~|

ONLY

LOG

INTERVAL:

m_min m_max
the temperature correction is restricted to the selected log m interval.

T-CORRECTION |~|

PROFILE:

X
the temperature correction is damped by simple functions (ND = total number of depth points)

X

1	depth point / ND
2	(depth point / ND)²
3

Λ = 3	diagonal operator by Olson & Kunasz
Λ = 4	tridiagonal operator by Olson & Kunasz

Both operators are generally parameter-free. In practice, it is less time consuming to do a Λ iteration in the optical thin case. This can be done by requesting a special GAMMA parameter (see above).

In order to accelerate the convergence, both operator switch to core saturation if

> 100.

LINEARIZATION |~|

MODE:

BLOCK-MATRIX-ITERATION, |~|

XX YY ZZ
with this option, only the occupation numbers of the given elements (here XX YY ZZ) are iterated. In combination with the card SOLVE STATISTICAL EQUATIONS ONLY, a “classical” line formation iteration is done. In all depth points, XX is Newton-Raphson iterated (inclusive constraint equations) first, then YY, and at least ZZ. Attention: this option alone does not switch off the constraint equations!

LINEARIZATION |~|

MODE:

BLOCK-MATRIX-ITERATION, |~|

ALL
like LINEARIZATION MODE: BLOCK-MATRIX-ITERATION, XX YY ZZ. All elements are iterated in order of their appearance in the atomic data file ATOMS.

MICROTURBULENCE |~|

[KM/S] 0.0*
with this option, the microturbulence pressure is considered in the hydrostatic equation but not in the calculation of the line profiles!

OCCUPATION |~|

PROBABILITY |~|

FORMALISM

FOR

AA
The Hummer-Mihalas (HM) formalism is used to calculate the level dissolution for AA = H1 or AA = <ion>.

OPACITY

PROJECT

RBF

DATA

: START

EDGE
PRO2 can calculate bound-free cross-section using data from the Opacity Project (OP). Here the OP data set is used and the cross-sections start at the threshold energy of the respective level.

OPACITY

PROJECT

RBF

DATA:

FULL

DATA

SET
see above, the complete OP data set is used (which may start at lower energy the the level energy in order to simulate the transition from the line absorption to the continuous absorption at the series limit.

OPACITY

PROJECT

RBF

DATA:

ONLY

EDGE

VALUE
see above, a mean value of the OP data at the threshold energy is calculated, then the absorption cross-sections are calculated using the Seaton formula in hydrogenic approximation.

OPACITY

PROJECT

RBF

DATA:

MISSING

HYDROGENIC
see above, for those levels which are not found in the OP data set hydrogen-like threshold cross-sections are calculated. Then, the Seaton formula is used.

OPACITY

PROJECT

RBF

DATA:

HYDROGENIC
see above, all bound-free transitions, which are found in the atomic data file ATOMS and request explicitly the use of OP data are calculated with hydrogen-like threshold cross-sections and the Seaton formula.

RADIATIVE

EQUILIBRIUM: |~|

DIFFERENTIAL/INTEGRAL |~|

FORM
The radiative equilibrium is calculated using a combination of the differential and the integral method.

READ

NEW

TEMPERATURE |~|

STRUCTURE
A temperature structure is read in from file TEMPERATURE.IN. This temperature may be interpolated from other models in order to speed up the calculation of “in-between” models. TEMPERATURE.IN is an ASCII table with log m and T, however, the log m values are ignored.

REDUCE

LOG

CVEC

x
This options scales the inhomogeneity vector of the linearized equations. This is useful to overcome numerical instabilities and to avoid too-large corrections in the beginning of the iterations. This card has to be inactivated to get a finally converged model.

SAVING

MODEL:

EACH

ITERATION
a temporary model (MODTMP) is saved in the working directory $TMPDIR. An example how to save this temporary model automatically in case of a failed model calculation is given in Sect. D.8.5. An

before the start of the model calculation, where ${name} is the assigned name of the output model helps to identify the model.

SET

TEMPERATURE t log m₁ log m₂ [scaling factor]
within log m₁ < log m < log m₂, the model temperature is set to

SHIFT

EDGE

BBBBBBBBBBFFFFFFFFFF x
The absorption threshold of the level BBBBBBBBBBFFFFFFFFFF given in the TMAP code (Sect. 2.1) is artificially shifted to the frequency point x. Attention: The given frequency x has to fit exactly a point within the frequency grid FGRID! All shifts are printed to STDOUT.

SKIP

ATOM

BELOW x
the element AA (flushed to the left) with occupation numbers less than x are eliminated from the statistical equations

SKIP

ION

III

BELOW x
ion III (flushed to the left) with occupation numbers less than x are eliminated from the statistical equations

SKIP

LEVELS

BELOW x
levels with occupation numbers less than x are eliminated from the statistical equations

SKIP

OCCDRVF

AFTER

1ST

LINEARIZATION
The derivations of the source functions are only calculated once (in the 1st linearization) and are then kept fixed. Attention: This saves an enormous amount of computational time in case of many NLTE levels and many frequency points but it works well only in the case of almost converged models.

SOLVE

STATISTICAL |~|

EQUATIONS

ONLY
switches off the constraint equations for n_e, n_H, n_g, T

SOLVE

STATISTICAL |~|

EQUATIONS

ONLY

RE-SOLVE PARTICLE CONSERVATION
switches off the constraint equations for n_e, n_H, n_g, T within the Newton-Raphson iteration. The hydrostatic and particle conservation equations are then solved subsequently.

SWITCH

OFF

LINES
All line transitions are ignored. Attention: sample lines of iron-group elements can not be switched off.

TIME

LIMIT 2000*
cpu time for the job to calculate. Only valid on non-CRAY machines

TRANSITION |~|

DETAILED

RADIATIVE

BALANCE

XXXXXXXXXXYYYYYYYYYY
With this option, the radiative transition XXXXXXXXXXYYYYYYYYYY given in the TMAP code (Sect. 2.1) is calculated in detailed radiative equilibrium. Thus, this transition is eliminated from the statistical equations but not from the calculation of the opacities and emissivities.

UNSOELD-LUCY |~|

TEMPERATURE |~|

CORRECTION |~|

DAMP= x
With this option, the radiative equilibrium is eliminated from the linearization. The temperature structure is calculated after the linearizations with the Unsöld-Lucy temperature correction method. The damping factor x is used to avoid over-corrections and numerical instabilities.

UPPER

AND

LOWER

LIMITS

FOR

RELATIVE

T-CORRECTION |~|

t₁ t₂
The relative temperature corrections (iteration i) are limited to the range T_i ^. t₁ < T_i+1 < T_i ^. t₂ at all depth points.

5.5 Output Options

The following PRINT and PLOT cards create output on STDOUTt and plot data files, respectively. The plot files are written in WRPLOT readable format. The option should be easily understandable ... Generally, for XXXX EACH or LAST can be inserted. ii indicates an I2 format specifier (FORTRAN).

5.5.1 Output to STDOUT

ABUNDANCES

PRINT |~|

BROYDEN

INFORMATIONS

PRINT |~|

CORRECTIONS |~|

LAST

LINEARIZATION,ITERATION:LAST,DEPTH |~|

INCREMENT: i

PRINT |~|

CORRECTIONS |~|

TOTAL

DENSITIES,ITERATION:LAST,DEPTH |~|

INCREMENT: i
in the case of non-linearization of the hydrostatic equation (default, see above)

CP-TIME/ITERATION,XXXX

PRINT |~|

DATA

FOR

DIELECTRONIC |~|

RECOMBINATIONS
prints especially the frequencies which are selected by PRO2.

DEPARTURE

COEFFICIENTS,ITERATION:XXXX,DEPTH |~|

INCREMENT:15

PRINT |~|

EMERGENT

FLUX,ITERATION:XXXX
prints also Teff, which is an sensitive control parameter

FREQUENCY

GRID

PRINT |~|

INFORMATION |~|

ABOUT

INPUT

SAMPLE

CROSS

SECTIONS

PRINT |~|

INPUT

MODEL,DEPTH |~|

INCREMENT:ii

PRINT |~|

INPUT

MODEL,DEPTH |~|

INCREMENT:ii |~|

(STRUCTURE |~|

ONLY)

PRINT |~|

INTEGRATED |~|

SURFACE

FLUX,ITERATION:XXXX

PRINT |~|

KANTOROVICH |~|

INFORMATIONS

PRINT |~|

LEVELS
prints level names, energies, and statistical weights of all LTE / NLTE levels

LEVEL

SKIP

INFORMATION

PRINT |~|

MAX.

REL.

CORRECTIONS |~|

EVERY

ITERATIONS

PRINT |~|

OCCUPATION |~|

PROBABILITIES,ITERATION:XXXX,DEPTH |~|

INCREMENT:ii

PRINT |~|

OPACITY

PROJECT

INFORMATION

PRINT |~|

OUTPUT

MODEL,ITERATION:XXXX,DEPTH |~|

INCREMENT:ii
prints temperature and density structure, and all occupation numbers of the NLTE levels

OUTPUT

MODEL,ITERATION:XXXX,DEPTH |~|

INCREMENT:ii |~|

(STRUCTURE |~|

ONLY)
prints temperature and density structure

MODEL

ATOMS

(OVERVIEW)

PRINT |~|

NUMBER

FUV

PHOTONS

PRINT |~|

OPTIONS
prints the most important input parameters

PARAMETER

NSIG
for the optimization of the PARAMETER NSIG, NRLUMAD, and NRLLMAD

RADIATION

FIELD,ITERATION:XXXX,DEPTH |~|

INCREMENT:ii

PRINT |~|

RBF

CUTOFF

FREQUENCIES
prints the frequency where the RBF cross-section of a level is set equal 0

ROSS,ITERATION:LAST
print the value of ROSS for the inner boundary condition

TAUSCALES,ITERATION:XXX,DEPTH |~|

INCREMENT:ii

PRINT |~|

WARNINGS

LP-PLOT |~|

OPTICALLY

THICK/THIN,ITERATION:XXXX
A plot of the

= 1 limit in the atmosphere is printed and saved in a file (PLLP).

5.5.2 Output into Plot Data Files

PLOT

EMERGENT

FLUX,ITERATION:XXXX
Same like PRINT card, output is written into file PRFLUX.

PLOT

IONIZATION |~|

FRACTIONS AA xmin xmax ymin ymax
Plot data is written into file IONPLOT.

LP-PLOT

OPTICALLY

THICK/THIN,ITERATION:XXXX,FILE |~|

ONLY
A plot of the

= 1 limit in the atmosphere is saved in the file PLLP only.

PLOT

RBF

CROSS

SECTIONS XXXXXXXXXXYYYYYYYYYY 0 0 0 0
The RBF cross section of level XXXXXXXXXXYYYYYYYYYY is saved in the file PLRBF.