----- Documentation for Model\abs_eff.pro -----
 NAME:
	 ABS_EFF

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE THE WAVELENGTH DISTRIBUTION OF THE ABSORPTION EFFICIENCY
	(THIN DUSTY SHELL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	ABS_EFF,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + IMPACT PARAMETER RELATIVE TO STELLAR DIAM (STRUCTURE).
	A = PARAMETER.

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH WAVELENGTH.


 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING THE PARTIAL DERIVATIVE. PD_F(I)
            = DERIVATIVE AT ITH WAVELENGTH W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = X^(-A)
	WITH:	X = XX.WVL = WAVELENGTH
		A = SPECTRAL INDEX
	PD_F = -LN(X)*F	           ;PARTIAL DERIVATIVE W/RESPECT TO A

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/08
	last modification by pcr 2008/02/12

----- Documentation for Model\central_depth.pro -----
 NAME:
       CENTRAL_DEPTH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE OPTICAL DEPTH RADIAL DISTRIBUTION IN CENTRAL DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	CENTRAL_DEPTH,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + RELATIVE IMPACT PARAMETERS (STRUCTURE).
	A = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH WAVELENGTH AND SIGHT ANGLE.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = ARRAY CONTAINING PARTIAL DERIVATIVE.
       PD_F(I,J,K) = DERIVATIVE AT ITH WAVELENGTH AND JTH SIGHT ANGLE
                     W/RESPECT TO KTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
       F = NB_DENS(A1,A2,A4)*ABS_EFF(A3)*CENT_PATH(A0,A1,A2)/A5
	WITH: 	NB_DENS = RELATIVE NUMBER DENSITY AT OUTER BOUNDARY = (A2/A1)^A4
		ABS_EFF = WAVELENGTH DISTRIBUTION OF ABSORPTION EFFICIENCY = (1/X)^A3
		CENT_PATH = ANGULAR OPTICAL PATH IN CENTRAL DIRECTION = A0*(SQRT((A2/A0)^2-R^2)-SQRT((A1/A0)^2-R^2))/2
  		X = XX.WVL = WAVELENGTH
		R = XX.RIP = RELATIVE IMPACT PARAMETER
		A0 = STELLAR DIAMETER
		A1 = SHELL INNER DIAMETER
		A2 = SHELL OUTER DIAMETER
		A3 = SPECTRAL INDEX
		A4 = POWER-LAW DENSITY INDEX
		A5 = REFERENCE GEOMETRICAL PATH
	PD_F_0 = F*PD_CENT_PATH_0/CENT_PATH						;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = F*(PD_NB_DENS_1/NB_DENS+PD_CENT_PATH_1/CENT_PATH)			;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = F*(PD_NB_DENS_2/NB_DENS+PD_CENT_PATH_2/CENT_PATH)			;PARTIAL DERIVATIVE W/RESPECT TO A2
	PD_F_3 = F*PD_ABS_EFF_3/ABS_EFF							;PARTIAL DERIVATIVE W/RESPECT TO A3
	PD_F_4 = F*PD_NB_DENS_4/NB_DENS							;PARTIAL DERIVATIVE W/RESPECT TO A4
	PD_F_5 = -F/A5								 	;PARTIAL DERIVATIVE W/RESPECT TO A5

 LOCAL PROCEDURE CALLED:
	ABS_EFF
	CENT_PATH
	NB_DENS

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/08
       last modification by pcr 2009/03/18

----- Documentation for Model\cent_path.pro -----
 NAME:
       CENT_PATH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE GEOMETRICAL-THICKNESS RADIAL DISTRIBUTION IN CENTRAL DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	CENT_PATH,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + RELATIVE IMPACT PARAMETERS (STRUCTURE).
	A = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF THICKNESS AT EACH SIGHT ANGLE.


 OPTIONAL OUTPUT PARAMETERS:
	PD_F = ARRAY CONTAINING PARTIAL DERIVATIVE.
       PD_F(I,J) = DERIVATIVE AT ITH SIGHT ANGLE W/RESPECT TO JTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = A0*(R2-R1)/2
	WITH: 	R = XX.RIP = RELATIVE IMPACT PARAMETER
		A0 = STELLAR DIAMETER
		A1 = SHELL INNER DIAMETER
		A2 = SHELL OUTER DIAMETER
		AA1 = A1/A0
		AA2 = A2/A0
		R1 = SQRT(AA1^2-R^2)
		R2 = SQRT(AA2^2-R^2)
	PD_F_0 = (AA1^2/R1-AA2^2/R2+R2-R1)/2 	;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = -AA1/R1/2		 	;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = AA2/R2/2			;PARTIAL DERIVATIVE W/RESPECT TO A2

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/08
	last modification by pcr 2008/12/28

----- Documentation for Model\core_depth.pro -----
 NAME:
	CORE_DEPTH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE OPTICAL DEPTH RADIAL DISTRIBUTION IN CORE DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	CORE_DEPTH,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + IMPACT PARAMETER RELATIVE TO INNER DIAM (STRUCTURE).
	A  = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH WAVELENGTH AND SIGHT ANGLE.


 OPTIONAL OUTPUT PARAMETERS:
	PD_F = ARRAY CONTAINING PARTIAL DERIVATIVE.
       PD_F(I,J,K) = DERIVATIVE AT ITH WAVELENGTH AND JTH SIGHT ANGLE
                     W/RESPECT TO KTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = NB_DENS(A0,A1,A3)*ABS_EFF(A2)*CORE_PATH(A0,A1,A2)/A4
	WITH: 	NB_DENS = RELATIVE NUMBER DENSITY AT OUTER BOUNDARY = (A1/A0)^A3
		ABS_EFF = WAVELENGTH DISTRIBUTION OF ABSORPTION EFFICIENCY = X^(-A2)
		CORE_PATH = ANGULAR OPTICAL PATH IN CORE DIRECTION = 2*(SQRT(A1^2-R^2)-SQRT(A0^2-R^2))
  		X = XX.WVL = WAVELENGTH
		R = A0/2*XX.RIP = IMPACT PARAMETER
		A0 = SHELL INNER DIAMETER
		A1 = SHELL OUTER DIAMETER
		A2 = SPECTRAL INDEX
		A3 = POWER-LAW DENSITY INDEX
		A4 = REFERENCE GEOMETRICAL DEPTH
	PD_F_0 = F*(PD_NB_DENS_0/NB_DENS+PD_CENT_PATH_0/CENT_PATH)			;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = F*(PD_NB_DENS_1/NB_DENS+PD_CENT_PATH_1/CENT_PATH)			;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = F*PD_ABS_EFF_2/ABS_EFF							;PARTIAL DERIVATIVE W/RESPECT TO A2
	PD_F_3 = F*PD_NB_DENS_3/NB_DENS							;PARTIAL DERIVATIVE W/RESPECT TO A3
	PD_F_4 = -F/A4								 	;PARTIAL DERIVATIVE W/RESPECT TO A4

 LOCAL PROCEDURE CALLED:
	ABS_EFF
	CORE_PATH
	NB_DENS

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/10
	last modification by pcr 2009/03/18

----- Documentation for Model\core_path.pro -----
 NAME:
	CORE_PATH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE GEOMETRICAL-THICKNESS RADIAL DISTRIBUTION IN CORE DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	CORE_PATH,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + IMPACT PARAMETER RELATIVE TO INNER DIAM (STRUCTURE).
	A  = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF THICKNESS AT EACH SIGHT ANGLE.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = ARRAY CONTAINING PARTIAL DERIVATIVE.
       PD_F(I,J) = DERIVATIVE AT ITH SIGHT ANGLE W/RESPECT TO JTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = A0*(R1-R0)
	WITH: 	R = XX.RIP = RELATIVE IMPACT PARAMETER
		A0 = SHELL INNER DIAMETER
		A1 = SHELL OUTER DIAMETER
		AA = A1/A0
		R0 = SQRT(1-R^2)
		R1 = SQRT(AA^2-R^2)
	PD_F_0 = R1-R0-AA^2/R1 			;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = AA/R1 				;PARTIAL DERIVATIVE W/RESPECT TO A1

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/10
	last modification by pcr 2008/12/28

----- Documentation for Model\delta_ring.pro -----
 NAME:
	DELTA_RING

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE "DELTA RING" FUNCTION
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	DELTA_RING,X,A,F,PD_F

 INPUTS:.
	X = VALUES OF INDEPENDENT VARIABLES (SPATIAL FREQUENCIES).
	A = PARAMETER OF EQUATION DESCRIBED BELOW (RING DIAMETER IN MAS).

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH X(I).

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_F(I) = DERIVATIVE AT ITH POINT W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = A*J0(PI*A*X)
       PD_F = J0(PI*A*X)-(PI*A*X)*J1(PI*A*X)

 LOCAL PROCEDURE CALLED: 
	NONE 

 LOCAL FUNCTION USED: 
	NONE 

 LOCAL SYSTEM VARIABLE USED: 
	MAS2RADIAN 
     
 REVISION HISTORY:
       Written by pcr 2006/09/18
	last modification by pcr 2008/01/23

----- Documentation for Model\engelke.pro -----
 NAME:
	ENGELKE

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE "ENGELKE" (=MODIFIED PLANCK) SPECTRAL RADIANT EXITANCE
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE
	(ENGELKE, AJ 104, 1248 (1992)).

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	ENGELKE,X,A,F,PDER_F

 INPUTS:.
	X = ROW VALUES OF INDEPENDENT VARIABLES (WAVELENGTHS IN MU).
	A = PARAMETER OF EQUATION DESCRIBED BELOW (TEMPERATURE IN KELVIN).

 OUTPUTS:
	F = ROW VALUES OF FUNCTION (in W.m^-2.mu^-1).


 OPTIONAL OUTPUT PARAMETERS:
	PDER_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_F(I) = DERIVATIVE AT ITH POINT W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F=PI*C1/(X^5)/(EXP(C2/(X*B))-1)
       PDER_F=F*(C2/(X*B))/B/(1-EXP(-C2/(X*B)))*B/A*(1-C5*(C4/(X*A))/(1+(C4/(X*A))))
 where	B=(C3*A*(1+C4/(A*X))^C5

 LOCAL PROCEDURE CALLED:
 	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
 	NONE

 REVISION HISTORY:
       Written by pcr 2007/01/03
	last modification by pcr 2008/02/12

----- Documentation for Model\gaussian_disk.pro -----
 NAME:
	GAUSSIAN_DISK

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE "GAUSSIAN_DISK" FUNCTION
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	GAUSSIAN_DISK,X,A,F,PD_F

 INPUTS:.
	X = VALUES OF INDEPENDENT VARIABLES (SPATIAL FREQUENCIES).
	A = PARAMETER OF EQUATION DESCRIBED BELOW (DISK FWHM IN MAS).

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH X(I).

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_F(I) = DERIVATIVE AT ITH POINT W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = (A^2)/4/LN(2)*EXP(-((PI*A*X)^2)/4/LN(2))
       PD_F = F*2/A*(1-((PI*A*X)^2)/4/LN(2))

 LOCAL PROCEDURE CALLED: 
	NONE 

 LOCAL FUNCTION USED: 
	GAUSSIAN

 LOCAL SYSTEM VARIABLE USED:
	MAS2RADIAN

 REVISION HISTORY:
       Written by pcr 2006/09/18
	last modification by pcr 2009/01/28

----- Documentation for Model\inner_temp.pro -----
 NAME:
	INNER_TEMP

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE DUST EFFECTIVE TEMPERATURE AT INNER BOUNDARY
	(THIN DUSTY SHELL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	INNER_TEMP,A,F,PD_F

 INPUTS:
	A = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF FUNCTION.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_F(K) = DERIVATIVE W/RESPECT TO KTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = A0*(1-MU)^(1/(A3+4))
	WITH: 	A0 = STELLAR EFFECTIVE TEMPERATURE 
		A1 = STELLAR DIAMETER
		A2 = SHELL INNER DIAMETER
		A3 = SPECTRAL INDEX
		MU = SQRT(1-(A1/A2)^2)
	PD_F_0 = F/A0					;PARTIAL DERIVATIVE W/RESPECT TO A0					
	PD_F_1 = F/(A3+4)*A1/A2^2/MU/(1-MU)		;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = -F/(A3+4)*A1^2/A2^3/MU/(1-MU)		;PARTIAL DERIVATIVE W/RESPECT TO A2
	PD_F_3 = -F/(A3+4)^2*LN(1-MU)			;PARTIAL DERIVATIVE W/RESPECT TO A3	

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/10
	last modficiation by pcr 2008/02/12

----- Documentation for Model\marcs_disk.pro -----
 NAME:
	MARCS_DISK

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE TRAPEZOIDAL TRANSFER FUNCTION OF MARCS INTENSITY
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	MARCS_DISK,XX,A,FORI,F,PD_F

 INPUTS:.
	XX = ROW VECTOR OF SPATIAL FREQUENCIES (FOR A GIVEN BASELINE)
           OR FREQUENCIES + RELATIVE IMPACT PARAMETERS (STRUCTURE).
	A = PARAMETER VECTOR (WITH 1 ELEMENT = STELLAR DISK).
	FORI = FLUX OR INTENSITY VALUES (SAME DIMENSION AS XX). 

 OUTPUTS:
	FF = VALUE OF FUNCTION AT EACH FREQUENCY.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_FF(I,K) = DERIVATIVE AT ITH SPATIAL FREQUENCY 
                    W/RESPECT TO STELLAR DIAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
   IF FORI=INTENSITY (AND FREQ=XX.WVL AND RIP=XX.RIP):
	FF = TRAPEZ_HANKEL(FREQ,A,RIP,FORI) 
       PD_F = PD_TRAPEZ_HANKEL(FREQ,A,RIP,FORI)
   IF FORI=FLUX (AND FREQ=XX):
	FF = FORI*UNIFORM_DISK(FREQ,A) 
       PD_F = FORI*PD_UNIFORM_DISK(FREQ,A)

 LOCAL PROCEDURE CALLED:
	TRAPEZ_HANKEL 
	UNIFORM_DISK 

 LOCAL FUNCTION USED:
	NONE 

 LOCAL SYSTEM VARIABLE USED:
	NONE 
     
 REVISION HISTORY:
       Written by pcr 2007/12/03
	last modification by pcr 2008/01/23

----- Documentation for Model\model_data_funct.pro -----
 NAME:
	MODEL_DATA_FUNCT

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE ANALYTICAL MODEL DATA FUNCTION (FLUX/VISI/BISP)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.
	NORMALLY, PROCEDURE USED BY LM_FIT TO FIT MODEL 
       ON INTERFEROMETRIC CALIBRATED DATA.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	MODEL_DATA_FUNCT,X,B,BWARR,PAR,MODEL,VIS,TEFF,REF,DID,FCT,PDER

 INPUTS:
	X = ROW VECTOR OR STRUCTURE OF INDEPENDENT VARIABLES 
	     (WAVELENGTH OR WAVELENGTH AND RELATIVE IMPACT PARAMETER).
       B = ROW COMPLEX VECTOR OF BASELINES (LENGTH/ANGLE).
	BWARR = 2-D ARRAY WITH # OF BASELENGTHS AND # OF WAVELENGTHS
               FOR EACH PART (DEFINED BY P2VM SETTING).
	        BWARR(I,0) = # OF BASELENGTHS FOR PART I
               BWARR(I,1) = # OF WAVELENGTHS FOR PART I
	PAR = ROW VECTOR OF PARAMETERS OF EQUATION DESCRIBED BELOW.
       IDXF = ROW VECTOR OF UNFREEZE-PARAMETER INDEXES.
       MODEL = MODEL NUMBER FOR CALCULATION OF FCTION (AND PARTIAL DERIVATIVES).
       VIS = FLAG FOR VISIBILITY AND PARTIAL DERIVATIVES CALCULATION (0:FLUX/1:VISI),
             FOR BISPECTRUM/TRIPLE PRODUCT CALCULATION VIS=+/-TABLE.
	TEFF = EFFECTIVE TEMPERATURE USED WITH MARCS TABULATED FLUX/INTENSITY.
       REF = ROW VECTOR OF NB_LAMBDA REFERENCE FLUXES
             OR MATRIX OF NB_LAMBDA*(NB_LUMIN+1) INTENSITIES AND ROSS2LD RATIOS.
 OUTPUTS:
	PAR = ROW VECTOR OF PARAMETERS.
       DID = 2-DIM COMPLEX ARRAY WITH OPTICAL DEPTH AND INTENSITY
             OR FLAG (=1/0) FOR OPT.DEPTH BASED MODELS.
	FCT = ROW REAL(COMPLEX) VECTOR OF NB_FREQ=NB_LAMBDA*NB_BASE FCTION VALUES
	(OF NB_BISP=NB_LAMBDA*NB_BASE^3 BISP VALUES)

 OPTIONAL OUTPUT PARAMETERS:
	PDER = 2-D REAL(COMPLEX) ARRAY CONTAINING THE
		PARTIAL DERIVATIVES.  PDER(I,J) = DERIVATIVE
		AT ITH SPATIAL FREQUENCY (OR ITH FREQ. TRIPLET)
               W/RESPECT TO JTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	X = WAVELENGTH VECTOR OR WAVELENGTH AND RELATIVE IMPACT PARAMETER STRUCTURE.
	B = COMPLEX BASELINE VECTOR (LENGTH/ANGLE).
	REF = REFERENCE FLUX VECTOR OR INTENSITY + DIAM RATIO ARRAY.
	IF VIS=0:
		FCT = ABS(SPEC).
		PDER = SIGN(SPEC)*PDSPEC.
 	IF VIS=1:
    		FCT = ABS(SPEC/SPEC0).
    		PDER = SIGN(SPEC)*(PDSPEC/SPEC0-SPEC/SPEC0*PDSPEC0/SPEC0).
 	IF VIS=TAB:
    		FCT = SPEC1*CONJ(SPEC2)*SPEC3.
    		PDER = PDSPEC1*CONJ(SPEC2)*SPEC3+SPEC1*CONJ(PDSPEC2)*SPEC3
                      +SPEC1*SPEC2*PDSPEC3.
 	IF VIS=(-1)*TAB:
    		FCT = (SPEC1/SPEC0)*CONJ(SPEC2/SPEC0)*(SPEC3/SPEC0).
    		PDER = (PDSPEC1/SPEC0-SPEC1/SPEC0*PDSPEC0/SPEC0)*CONJ(SPEC2/SPEC0)*(SPEC3/SPEC0)
    		     + (SPEC1/SPEC0)*CONJ(PDSPEC2/SPEC0-SPEC2/SPEC0*PDSPEC0/SPEC0)*(SPEC3/SPEC0)
    		     + (SPEC1/SPEC0)*CONJ(SPEC2/SPEC0)*(PDSPEC3/SPEC0-SPEC3/SPEC0*PDSPEC0/SPEC0).
	WHERE:	SPEC,SPEC0,SPEC1,SPEC2,SPEC3,
		AND PDSPEC,PDSPEC0,PDSPEC1,PDSPEC2,PDSPEC2,
		GIVEN BY MODEL_SPECTRUM
		AND INDEX OF TAB GIVEN BY MAKE_TAB3.

 COMMENTS:
	BASED ON FCT.PRO

 LOCAL PROCEDURE CALLED:
	MODEL_SPECTRUM

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLES USED:
 	NB_BASE_STEP
       NB_RAD_MAX

 REVISION HISTORY:
       Written by pcr 2006/09/25
	last modification by pcr 2010/08/25

----- Documentation for Model\model_data_funct_proj.pro -----
 NAME:
	MODEL_DATA_FUNCT_PROJ

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE ANALYTICAL MODEL PROJECTED DATA FUNCTION
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.
	NORMALLY, FUNCTION USED BY LM_FIT TO FIT
	MODEL ON INTERFEROMETRIC CALIBRATED DATA.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	MODEL_DATA_FUNCT_PROJ,XX,BASE,BW_ARR,PARAM,INDEX_FREE,
             MODEL_NUMBER,VISI_FLAG,TEMP,REF_DATA,DEPTH_INTENS_DATA,
             PROJ_FUNCT,PROJ_PDER

 INPUTS:
	XX = ROW VECTOR OR STRUCTURE OF INDEPENDENT VARIABLES
            (WAVELENGTH OR WAVELENGTH AND RELATIVE IMPACT PARAMETER).
       BASE = ROW COMPLEX VECTOR OF BASELINES (LENGTH/ANGLE).
	BW_ARR = 2-D ARRAY WITH # OF BASELENGTHS AND # OF WAVELENGTHS
                FOR EACH PART (DEFINED BY P2VM SETTING).
		BW_ARR(I,0) = # OF BASELENGTHS FOR PART I
		BW_ARR(I,1) = # OF WAVELENGTHS FOR PART I
	PARAM = ROW VECTOR OF PARAMETERS OF EQUATION DESCRIBED BELOW.
       INDEX_FREE = ROW VECTOR OF FREE-PARAMETER INDEXES.
       MODEL_NUMBER = NBER OF MODEL USED FOR CALCULATION OF FUNCTION
                     (AND PARTIAL DERIVATIVES).
       VISI_FLAG = FLAG FOR VISIBILITY AND PARTIAL DERIVATIVES CALCULATION
                  (0:FLUX/1:VISI),
		    FOR BISPECTRUM/TRIPLE PRODUCT CALCULATION VISI_FLAG=+/-TAB3.
	TEMP = EFFECTIVE TEMPERATURE USED WITH MARCS TABULATED FLUX/INTENSITY.
       REF_DATA = ROW VECTOR OF LENGTH(XX) REFERENCE FLUXES
		   OR MATRIX OF LENGTH(XX.WVL)*(NB_LUMIN+1) REFERENCE INTENSITIES
                  AND ROSS2LD RATIO.

 OUTPUTS:
	PARAM = ROW VECTOR OF PARAMETERS.
       DEPTH_INTENS_DATA = 2-DIM COMPLEX ARRAY WITH OPTICAL DEPTH AND INTENSITY
                           OR FLAG (=1/0) FOR OPT.DEPTH BASED MODELS.
	PROJ_FUNCT = ROW VECTOR OF PROJECTED FUNCTION VALUES AT EACH
                    PROJECTED FREQUENCY (OR PROJECTED WAVELENGTH IF NULL BASE)

 OPTIONAL OUTPUT PARAMETERS:
	PROJ_PDER = 2-D ARRAY CONTAINING PROJECTED PARTIAL DERIVATIVES
       PROJ_PDER(I,J) = PROJECTED DERIVATIVE AT ITH PROJECTED FREQUENCY/WAVELENGTH
                        W/RESPECT TO JTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	PROJ_FUNCT = FUNCTPROJ(FUNCT) OR BISP_PROJ(FUNCT)
	PROJ_PDER = FUNCTPROJ(PDER) OR BISP_PROJ(PDER)
	WHERE:	FUNCT AND PDER ARE GIVEN BY MODEL_DATA_FUNCT

 COMMENTS:
	BASED ON FUNCT.PRO

 LOCAL PROCEDURE CALLED:
	MODEL_DATA_FUNCT

 LOCAL FUNCTION USED:
	FUNCTPROJ
	BISP_PROJ

 LOCAL SYSTEM VARIABLE USED:
       EPSILON
	NB_BASE_STEP
       NB_DATA_MAX
	NB_RES_STEP

 MODIFICATION HISTORY:
       Written by pcr 2007/11/30
	last modification by pcr 2010/08/25

----- Documentation for Model\model_spectrum.pro -----
 NAME:
	MODEL_SPECTRUM

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
       PRINT PARAMETER SIGNIFICANCE AND VALUES, RETURN FLUX,
	CREATE PARAMETER SETS FOR TEST MODEL, AND CALCULATE OPTICAL 
       DEPTH AND INTENSITY
	OR
	EVALUATE MODEL COMPLEX SPATIAL SPECTRUM FOR GIVEN COMPLEX BASELINE
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.
	PROCEDURE USED BY LM_FIT TO FIT MODEL TO INTERFEROMETRIC CALIBRATED DATA.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	MODEL_SPECTRUM,A,MODEL,FLUX,ODEPTH,PRANGE,DIAM_INDEX,PTIT,
           XX,BASE,EM,SPEC,PDER_SPEC

 INPUTS:
	A = ROW VECTOR OF MODEL PARAMETERS OF EQUATION DESCRIBED BELOW.
       MODEL = USED-MODEL NUMBER.
	XX = VALUES OF INDEPENDENT VARIABLES = WAVELENGTHS (VECTOR, IN MU)
            OR WAVELENGTHS + RELATIVE IMPACT PARAMETERS (STRUCTURE).
       BASE = COMPLEX BASELINE (LENGTH IN M / ANGLE IN DEG).
       EM = SPECTRAL EXITANCE OR SPECTRAL RADIANCE.

 OUTPUTS:
	A = ROW VECTOR OF MODEL PARAMETERS.
	FLUX = STELLAR FLUX FLAG (1 FOR BLACKBODY, 0 FOR ENGELKE, -1 FOR MARCS).
	ODEPTH = 2-DIM COMPLEX ARRAY WITH OPTICAL DEPTH AND INTENSITY
               OR FLAG (=1/0) FOR OPT.DEPTH BASED MODELS.
	PRANGE = 2-DIM ARRAY OF PARAMETER MIN AND MAX VALUES (SIZE=2*LENGTH(A)),
                USED WITH TEST_MODEL AND CREATE_DATA,
		OR EFFECTIVE TEMPERATURE USED WITH MARCS FLUX/INTENSITY TABULATED DATA.
	DIAM_INDEX = INDEX OF PARAMETER USED AS CENTRAL DIAMETER,
                    USED WITH TEST_MODEL, CREATE_DATA AND FIT_MODEL.
	PSTRING = 2-DIM ARRAY OF PARAMETER NAME AND UNIT STRINGS (SIZE=2*LENGTH(A)),
                FOR PLOT OF PARAMETER AXIS TITLE.
	SPEC = COMPLEX VALUE OF SPATIAL SPECTRUM AT EACH SPATIAL FREQUENCY B/X
              (X = WAVELENGTH VECTOR).

 OPTIONAL OUTPUT PARAMETERS:
	PDER_SPEC = (NLAMBDA,NPARAM) COMPLEX ARRAY CONTAINING PARTIAL DERIVATIVES.
	SPEC_PDER(I,J) = DERIVATIVE AT ITH WAVELENGTH W/RESPECT TO JTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:

    CENTRAL_DEPTH = optical depth in the central direction (thin-shell model)
    CORE_DEPTH = optical depth in the core direction (thin-shell model)
    DELTA_RING(D) = delta ring Hankel transform of diameter D
    GAUSSIAN_DISK(W) = gaussian disk Hankel transform of full width
                       at half maximum W
    ENGELKE(T) = Engelke exitance (modified Planck) with temperature T
    INNER_TEMP = dust effective temperature at the inner boundary (thin-shell model)
    PLANCK(T)  = blackbody exitance (Planck) with surface temperature T
    SHELL_DEPTH = optical depth in the shell direction (thin-shell model)
    TRAPEZ_HANKEL(I) = trapezoidal Hankel transform of intensity I
    UNIFORM_DISK(D) = Hankel transform of uniform disk of diameter D

 *****************************************
  SINGLE COMPONENT 1-D MODELS (MODEL<100):
 *****************************************

 MODEL=1 (stellar disk with tabulated exitance/radiance)
    if P0 = spectral radiance:
	SPEC = TRAPEZ_HANKEL(A0,P0)
    if P0 = spectral exitance:
	SPEC = P0*UNIFORM_DISK(A0)
    where A0 = disk diameter
 MODEL=10 (single ud bb model)
	SPEC = PLANCK(A0)*UNIFORM_DISK(A1)
    where A0 = effective temperature
	   A1 = disk diameter
 MODEL=11 (single gd bb)
	SPEC = PLANCK(A0)*GAUSSIAN_DISK(A1)
    where A0 = effective temperature
	   A1 = disk fwhm
 MODEL=12 (single dr bb)
	SPEC = PLANCK(A0)*DELTA_RING(A1)
 	where A0 = effective temperature
             A1 = ring diameter
 MODEL=13 (single ud Engelke model)
	SPEC = ENGELKE(A0)*UNIFORM_DISK(A1)
 	where A0 = effective temperature
             A1 = disk diameter
 MODEL=20 (single ur bb)
	SPEC =  PLANCK(A0)*(UNIFORM_DISK(A2)-UNIFORM_DISK(A1))
 	where A0 = effective temperature
 	      A1 = outer ring internal diameter
 	      A2 = outer ring external diameter
 MODEL=21 (single ud bb model, increas pow law of diam vs lambda)
	SPEC = PLANCK(A0)*UNIFORM_DISK(A1*X^A2)
 	where A0 = effective temperature
             A1 = disk diameter @ 1 mu
             A2 = spectral index
 MODEL=51 (radiative-transfer analytical model, central direction)
  	TRM = EXP(-CENTRAL_DEPTH(A)) = central transmittance
	if P0 = spectral radiance:
		SPEC = TRAPEZ_HANKEL(P0*TRM)
	if P0 = spectral exitance:
		SPEC = P0*TRAPEZ_HANKEL(TRM)
	where A0 = central source diameter
 	      A1 = shell inner diameter
 	      A2 = shell outer diameter
 	      A3 = spectral index
 	      A4 = grain density index
 	      A5 = reference geometrical thickness
 MODEL=52 (radiative-transfer analytical model, core direction)
  	TRM = 1-EXP(-CORE_DEPTH(B)) = core transmittance
	SPEC = PLANCK(A0)*TRAPEZ_HANKEL(TRM)
	where A0 = dust temperature
 	      A1 = B0 = shell inner diameter
 	      A2 = B1 = shell outer diameter
 	      A3 = B2 = spectral index
 	      A4 = B3 = grain density index
 	      A5 = B4 = reference geometrical thickness
 MODEL=53 (radiative-transfer analytical model, shell direction)
  	TRM = 1-EXP(-SHELL_DEPTH(B)) = shell transmittance
	SPEC = PLANCK(A0)*TRAPEZ_HANKEL(TRM)
	where A0 = dust temperature
 	      A1 = B0 = shell inner diameter
 	      A2 = B1 = shell outer diameter
 	      A3 = B2 = spectral index
 	      A4 = B3 = grain density index
 	      A5 = B4 = reference geometrical thickness
 MODEL=54 (radiative-transfer analytical model, core+shell directions)
  	TRM1 = 1-EXP(-CORE_DEPTH(B)) = core transmittance
  	TRM2 = 1-EXP(-SHELL_DEPTH_1(B)) = shell transmittance, inner direction (until inner diam A1)
  	TRM3 = 1-EXP(-SHELL_DEPTH_2(B)) = shell transmittance, outer direction (until outer diam A2)
	SPEC = PLANCK(A0)*(TRAPEZ_HANKEL(TRM2)-TRAPEZ_HANKEL(TRM1)+TRAPEZ_HANKEL(TRM3))
	where A0 = dust temperature
 	      A1 = B0 = shell inner diameter
 	      A2 = B1 = shell outer diameter
 	      A3 = B2 = spectral index
 	      A4 = B3 = grain density index
 	      A5 = B4 = reference geometrical thickness


 *********************************
  COMPOSITE 1-D MODELS (MODEL>99):
 *********************************

 MODEL=130 (stellar disk with tabulated exitance/radiance + 1 ud bb)
    if P0 = spectral radiance:
	SPEC = TRAPEZ_HANKEL(A0,P0)*EXP(-A3)+PLANCK(A1)*UNIFORM_DISK(A2)*(1-EXP(-A3))
    if P0 = spectral exitance:
	SPEC = P0*UNIFORM_DISK(A0)*EXP(-A3)+PLANCK(A1)*UNIFORM_DISK(A2)*(1-EXP(-A3))
       where A0 = disk diameter
             A1 = outer effective temperature
 	      A2 = outer diameter
 	      A3 = global optical depth
 MODEL=140 (2 ud bbs)
	SPEC = PLANCK(A0)*UNIFORM_DISK(A1)*EXP(-A4)+PLANCK(A2)*UNIFORM_DISK(A3)*(1-EXP(-A4))
 	where A0 = inner effective temperature
 	      A1 = inner diameter
             A2 = outer effective temperature
 	      A3 = outer diameter
 	      A4 = global optical depth
 MODEL=141 (1 ud bb + 1 gd bb)
	SPEC = PLANCK(A0)*UNIFORM_DISK(A1)*EXP(-A4))+PLANCK(A2)*GAUSSIAN_DISK(A3)*(1-EXP(-A4))
 	where A0 = inner effective temperature
 	      A1 = inner diameter
             A2 = outer effective temperature
 	      A3 = outer fwhm
 	      A4 = global optical depth
 MODEL=142 (stellar disk with tabulated exitance/radiance + 1 ud bb, decreas pow law of opt depth vs lambda)
    if P0 = spectral radiance:
	SPEC = TRAPEZ_HANKEL(A0,P0)*EXP(-(A3*X^(-A4)))+PLANCK(A1)*UNIFORM_DISK(A2)*(1-EXP(-(A3*X^(-A4))))
    if P0 = spectral exitance:
	SPEC = P0*UNIFORM_DISK(A0)*EXP(-(A3*X^(-A4)))+PLANCK(A1)*UNIFORM_DISK(A2)*(1-EXP(-(A3*X^(-A4))))
       where A0 = disk diameter
             A1 = outer effective temperature
 	      A2 = outer diameter
 	      A3 = global optical depth @ 1 mum
 	      A4 = spectral index
 MODEL=150 (1 ud bb + 1 ur bb)
	SPEC =  PLANCK(A0)*UNIFORM_DISK(A1)*EXP(-A5)+PLANCK(A2)*(UNIFORM_DISK(A4)-UNIFORM_DISK(A3))*(1-EXP(-A5))
 	where A0 = inner effective temperature
 	      A1 = inner disk diameter
             A2 = outer effective temperature
 	      A3 = outer ring internal diameter
 	      A4 = outer ring external diameter
 	      A5 = global optical depth
 MODEL=151 (2 ud bbs, decreas pow law of opt depth vs lambda)
	SPEC = PLANCK(A0)*UNIFORM_DISK(A1)*EXP(-(A4*X^(-A5)))+PLANCK(A2)*UNIFORM_DISK(A3)*(1-EXP(-(A4*X^(-A5))))
 	where A0 = inner effective temperature
 	      A1 = inner diameter
             A2 = outer effective temperature
 	      A3 = outer diameter
 	      A4 = global optical depth @ 1 mum
             A5 = spectral index
 MODEL=152 (MARCS disk + dust thin shell = radiative-transfer full analytical model)
	T1 = INNER_TEMP(AA)
  	TRM0 = EXP(-CENTRAL_DEPTH(A)) = central transmittance
  	TRM1 = 1-EXP(-CORE_DEPTH_0(B)) = core transmittance, central direction (until stellar diam A0)
  	TRM2 = 1-EXP(-CORE_DEPTH_1(B)) = core transmittance, inner direction (until inner diam A1)
  	TRM3 = 1-EXP(-SHELL_DEPTH_1(B)) = shell transmittance, inner direction (until inner diam A1)
  	TRM4 = 1-EXP(-SHELL_DEPTH_2(B)) = shell transmittance, outer direction (until outer diam A2)
	if P0 = spectral radiance:
		SPEC0 = TRAPEZ_HANKEL(P0*TRM0)
	if P0 = spectral exitance:
		SPEC0 = P0*TRAPEZ_HANKEL(TRM0)
	SPEC1 = TRAPEZ_HANKEL(TRM1)
	SPEC2 = TRAPEZ_HANKEL(TRM2)
	SPEC3 = TRAPEZ_HANKEL(TRM3)
	SPEC4 = TRAPEZ_HANKEL(TRM4)
	SPEC = SPEC0+PLANCK(T1)*(SPEC2-SPEC1+SPEC4-SPEC3)
   where T0 = AA0 = central source temperature (fixed, not a model parameter)
         A0 = AA1 = central source diameter
         A1 = AA2 = B0 = shell inner diameter
 	  A2 = B1 = shell outer diameter
 	  A3 = AA3 = B2 = spectral index
 	  A4 = B3 = grain density index
 	  A5 = B4 = reference geometrical thickness

 LOCAL PROCEDURE CALLED:
	CENTRAL_DEPTH
	CORE_DEPTH
	DELTA_RING
	ENGELKE
	GAUSSIAN_DISK
	INNER_TEMP
	MARCS_DISK
	PLANCK
	SHELL_DEPTH
	UNIFORM_DISK
 LOCAL FUNCTION USED:
	COSD
       SIND
 LOCAL SYSTEM VARIABLE USED:
       EPSILON
	MAS2RADIAN
	NB_CORE
	NB_MAX_PARAM
	NB_RES_STEP
	NB_SHELL
	NB_STAR
	OBS_PATH
       PARAM_REL_DIST

 REVISION HISTORY:
       Written by pcr 2006/09/16
	last modification by pcr 2011/01/18

----- Documentation for Model\nb_dens.pro -----
 NAME:
	NB_DENS

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE RELATIVE NUMBER DENSITY AT OUTER BOUNDARY
	(THIN DUSTY SHELL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	NB_DENS,A,F,PD_F

 INPUTS:.
	A = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF FUNCTION.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE
       PD_F(K) = DERIVATIVE W/RESPECT TO KTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = (A1/A0)^A2
	WITH: 	A0 = SHELL INNER DIAMETER
		A1 = SHELL OUTER DIAMETER
		A2 = POWER-LAW DENSITY INDEX
	PD_F_0 = -A2/A0*F		;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = A2/A1*F		;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = LN(A1/A0)*F		;PARTIAL DERIVATIVE W/RESPECT TO A2

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED: 
	NONE 

 LOCAL SYSTEM VARIABLE USED: 
	NONE 
     
 REVISION HISTORY:
       Written by pcr 2008/01/08
	last modification by pcr 2008/02/12

----- Documentation for Model\planck.pro -----
 NAME:
	PLANCK

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE "PLANCK" (BLACK BODY) SPECTRAL RADIANT EXITANCE
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	PLANCK,X,A,F,PDER_F

 INPUTS:.
	X = ROW VALUES OF INDEPENDENT VARIABLES (WAVELENGTHS IN MU).
	A = PARAMETER OF EQUATION DESCRIBED BELOW (TEMPERATURE IN KELVIN).

 OUTPUTS:
	F = ROW VALUES OF FUNCTION (IN W.m^-2.mu^-1).

 OPTIONAL OUTPUT PARAMETERS:
	PDER_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PDER_F(I) = DERIVATIVE AT ITH POINT W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F=PI*C1/(X^5)/(EXP(C2/(X*A))-1)
       PDER_F=F*(C2/(X*A))/A/(1-EXP(-C2/(X*A)))

 LOCAL PROCEDURE CALLED:
 	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
 	NONE
     
 REVISION HISTORY:
       Written by pcr 2006/08/20
	last modification by pcr 2008/02/12

----- Documentation for Model\shell_depth.pro -----
 NAME:
	SHELL_DEPTH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE OPTICAL DEPTH RADIAL DISTRIBUTION IN SHELL DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	SHELL_DEPTH,XX,A,F,PD_F

 INPUTS:.
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + IMPACT PARAMETER RELATIVE TO OUTER DIAM (STRUCTURE).
	A  = ROW VECTOR OF PARAMETERS.

 OUTPUTS:
	F = VALUE OF FUNCTION AT EACH WAVELENGTH AND SIGHT ANGLE.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = ARRAY CONTAINING PARTIAL DERIVATIVE.
       PD_F(I,J,K) = DERIVATIVE AT ITH WAVELENGTH AND JTH SIGHT ANGLE
                     W/RESPECT TO KTH PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = NB_DENS(A0,A1,A3)*ABS_EFF(A2)*SHELL_PATH(A1)/A4
	WITH: 	NB_DENS = RELATIVE NUMBER DENSITY AT OUTER BOUNDARY = (A1/A0)^A3
		ABS_EFF = WAVELENGTH DISTRIBUTION OF ABSORPTION EFFICIENCY = (1/X)^A2
		SHELL_PATH = ANGULAR OPTICAL PATH IN SHELL DIRECTION = A1*SQRT(1-R^2)
  		X = XX.WVL = WAVELENGTH
		R = XX.RIP = RELATIVE IMPACT PARAMETER
		A0 = SHELL INNER DIAMETER
		A1 = SHELL OUTER DIAMETER
		A2 = SPECTRAL INDEX
		A3 = POWER-LAW DENSITY INDEX
		A4 = REFERENCE GEOMETRICAL PATH
	PD_F_0 = F*PD_NB_DENS_0/NB_DENS							;PARTIAL DERIVATIVE W/RESPECT TO A0
	PD_F_1 = F*(PD_NB_DENS_1/NB_DENS+PD_SHELL_PATH_1/SHELL_PATH)			;PARTIAL DERIVATIVE W/RESPECT TO A1
	PD_F_2 = F*PD_ABS_EFF_2/ABS_EFF							;PARTIAL DERIVATIVE W/RESPECT TO A2
	PD_F_3 = F*PD_NB_DENS_3/NB_DENS							;PARTIAL DERIVATIVE W/RESPECT TO A3
	PD_F_4 = -F/A4	                                                                ;PARTIAL DERIVATIVE W/RESPECT TO A4

 LOCAL PROCEDURE CALLED:
	ABS_EFF
	NB_DENS
	SHELL_PATH

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 MODIFICATION HISTORY:
       Written by pcr 2008/01/10
	last modification by pcr 2009/03/18

----- Documentation for Model\shell_path.pro -----
 NAME:
	SHELL_PATH

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE GEOMETRICAL-THICKNESS RADIAL DISTRIBUTION IN SHELL DIRECTION
	(RADIATIVE-TRANSFER ANALYTICAL MODEL)
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVES.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	SHELL_PATH,XX,A,F,PD_F

 INPUTS:
	XX = VALUES OF INDEPENDENT VARIABLES
          = WAVELENGTHS + IMPACT PARAMETER RELATIVE TO OUTER DIAM (STRUCTURE).
	A  = PARAMETER.

 OUTPUTS:
	F = VALUE OF THICKNESS AT EACH SIGHT ANGLE.

 OPTIONAL OUTPUT PARAMETERS:
	PD_F = VECTOR CONTAINING PARTIAL DERIVATIVE.
       PD_F(I) = DERIVATIVE AT ITH SIGHT ANGLE W/RESPECT TO PARAMETER.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	F = A*SQRT(1-R^2)
	WITH: 	R = XX.RIP = RELATIVE IMPACT PARAMETER
		A = SHELL OUTER DIAMETER
	PD_F = SQRT(1-R^2)		 	;PARTIAL DERIVATIVE W/RESPECT TO A

 LOCAL PROCEDURE CALLED:
	NONE

 LOCAL FUNCTION USED:
	NONE

 LOCAL SYSTEM VARIABLE USED:
	NONE

 REVISION HISTORY:
       Written by pcr 2008/01/10
	last modification by pcr 2008/12/28

----- Documentation for Model\uniform_disk.pro -----
 NAME:
	UNIFORM_DISK

 AUTHOR:
       pierre.cruzalebes@oca.eu

 PURPOSE:
	EVALUATE "UNIFORM_DISK" FUNCTION
	AND OPTIONALLY RETURN VALUE OF PARTIAL DERIVATIVE.

 CATEGORY:
	MODELLING.

 CALLING SEQUENCE:
	UNIFORM_DISK,X,A,FCT,PD_FCT

 INPUTS:.
	X = VALUES OF INDEPENDENT VARIABLES (SPATIAL FREQUENCIES, IN RAD^-1).
	A = PARAMETER OF EQUATION DESCRIBED BELOW (DISK DIAMETER, IN MAS).

 OUTPUTS:
	FCT = VALUE OF FUNCTION AT EACH X(I).

 OPTIONAL OUTPUT PARAMETERS:
	PD_FCT = VECTOR CONTAINING THE PARTIAL DERIVATIVE.
       PD_F(I) = DERIVATIVE AT ITH POINT W/RESPECT TO A.

 COMMON BLOCKS:
	NONE.

 SIDE EFFECTS:
	NONE.

 RESTRICTIONS:
	NONE.

 PROCEDURE:
	FCT = (A^2)/4*2*J1(PI*A*X)/(PI*A*X)
       PD_FCT = (A/2)*J0(PI*A*X)

 LOCAL PROCEDURE CALLED:
	NONE 

 LOCAL FUNCTION USED: 
	NONE

 LOCAL SYSTEM VARIABLE USED:
       EPSILON
	MAS2RADIAN 
     
 REVISION HISTORY:
       Written by pcr 2006/09/18
	last modification by pcr 2011/01/13