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class EclipsingStarModel( NonLinearModel )	[source]

Model for the light curve of an eclipsing double star, as a function of time, t.

par	symbol	name	description	limits	comment
p0	e	eccentricity	of the elliptic orbit	0<e<1	0 is circular
p1	P	period	of the velocity variation	P>0
p2	T	phase	phase until t = 0	0<T<2π
p3	i	inclination	of orbit (close to 90)	0<i<π
p4	ω	longitude	from north to periastron	0<ω<2π
p5	r1	radius1	radius of star 1	0<r1<1
p6	r2	radius2	radius of star 2	0<r2<1
p7	f1	lumen1	luminosity of star 1
p8	f2	lumen2	luminosity of star 2
p9	fs	spot	spot illumination	fs >= 0	0 is no spot
p10	m1	mass1	relative mass of star 1	0 < m1 < 1	m2 = 1 - m1

The amplitude (semimajor axis) is set to 1.0. Both stellar radii are given as a fraction of the amplitude. The mass of star 2 is ( 1 - m1 ). The mass ratio is used in calculating the tidal distortion.

When the sum of the radii, p₅ + p₆, is larger than the periastron distance, 1 - p₀, the stars clash.

When the model is constructed as circular, the parameter p₀ becomes 0 by definition and p₄ looses its meaning. They are removed from the model.

This class uses StellarOrbitModel to find the true orbit

The parameters are initialized at
     [0.0, 1.0, 0.0, pi/2, 0.0, 0.1, 0.1, 1.0, 1.0, 0.0, 0.5]. It is a non-linear model.

For the mathematics of this model and further explanation see ../EclipsingStars.md

(Partial) derivatives were obtained with the help of
https://www.derivative-calculator.net Muchas Gracias

Attributes

• storbit : StellarOrbitModel
       to calculate the true stellar orbit
• circular : bool
       whether the orbit is circular (eccentricity == phase == longitude == 0)
• spot : bool
       apply spot illumination and tidal distortion
• tides : bool
       apply tidal distortion
• occultation : bool
       True: eclipses stricty enforced
• fixpar : lambda function
       to provide parameters for StellarOrbitModel

Examples

esm = EclipsingStarModel( spot=True, tides=True )
print( esm.npars )
10

EclipsingStarModel( circular=False, spot=False, tides=False, occultation=True, copy=None, **kwargs )	[source]

Radial velocity model.

Number of parameters depends on the settings of ( False or True ) of
  circular spot tides ( 9 or 6 ) + ( 0 or 1 ) + ( 0 or 1 )

Parameters

• circular : bool
       stellar orbit is circular: eccentricity = 0 ==> lonod = 0
• spot : bool or number
       apply spot illumination; more pronounced when spot > 1
• tides : bool
       apply tidal distortion
• occultation : bool (True)
       eclipses are stricty enforced
• copy : EclipsingStarModel
       model to copy

copy( )	[source]

Copy method.

distanceConstraint( logL, problem, allpars, lowLhood )	[source]

Constrain the sizes of the stars for use in NestedSampling to avoid collapse. and the inclination to ensure eclipses

Parameters

• logL : float
       log Likelihood obtained with allpars
• problem : Problem
       to solve
• allpars : array
       all parameters involved in the problem
• lowLhood : float
       present value of the likelihood constraint

logCombiPrior( allpars )	[source]

Get extra prior for this combination of parameters.

Parameters

• allpars : array
       all parameters involved in the problem

baseResult( xdata, params )	[source]

Returns the result of the model function.

Parameters

• xdata : array_like
       values at which to calculate the result
• params : array_like
       values for the parameters.

lightCurve( xy, z, params, debug=0 )	[source]

Return the visible area of two eclipsing stars, multiplied by their luminocities.

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       of the model
• debug : int
       debug partial derivativesin stages

visibleFraction( xy, z, r1, r2 )	[source]

Calculate the fraction of visibility for both stars.

If z is positive, star 2 is in front of star 1
If z is negative, star 2 is behind star 1

Parameters

• xy : array
       distance between overlapping circles
• z : array
       distance of star 2 to the sky plane
• r1 : float or array
       radius of star 1
• r2 : float or array
       radius of star 2

VFderivative( xy, z, r1, r2 )	[source]

Calculate the derivatives for the visible fraction to xy and z

Parameters

• xy : array
       distance between overlapping circles
• z : array
       distance of star 2 to the sky plane
• r1 : float or array
       radius of star 1
• r2 : float or array
       radius of star 2

Returns
( dV1dx, dV2dx, dV1dz, dV2dz )
     derivatives of the visibility to xy and z

VFpartial( xy, z, r1, r2 )	[source]

Calculate the partial derivatives for the visible fraction to r1 and r2.

Parameters

• xy : array
       distance between overlapping circles
• z : array
       distance of star 2 to the sky plane
• r1 : float or array
       radius of star 1
• r2 : float or array
       radius of star 2

Returns
( dV1dr1, dV2dr1, dV1dr2, dV2dr2 )
     partials of the visibility to r1 and r2

overlap( xy, r1, r2 )	[source]

Calculate the overlap area between two partially overlapping circles

https://scipython.com/books/book2/chapter-8-scipy/problems/overlapping-circles/

Parameters

• xy : array
       distance between overlapping circles
• r1 : float or array
       radius of circle 1
• r2 : float or array
       radius of circle 2

spotIllumination( xy, z, params )	[source]

Illumination of the stars on each other. Depending on distance between stars and aspect angle

The algoritm is taken from DOI: 10.5817/OEJV2025-0258

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model

tidalDistortion( xy, z, params )	[source]

Calculate the tidal distortion of the stars in a binary system as elongated, prolate spheroids (cigars).

See equation. (1.468) from https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node69.html
  (for as long as it lasts)

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model

Returns
( a1, b1, a2, b2 )
     apparent stretch and squeeze of both stars

basePartial( xdata, params, parlist=None )	[source]

Returns the partials at the input value.

Parameters

• xdata : array
       values at which to calculate the result
• params : array
       values for the parameters.
• parlist : array
       list of indices active parameters (or None for all)

baseDerivative( xdata, params )	[source]

Returns the derivative of f to t (dfdt) at the input values.

Parameters

• xdata : array
       times at which to calculate the result
• params : array
       values for the parameters.

OVpartial( xy, r1, r2 )	[source]

calculate the partials of overlap to r1 and r2.

Parameters

• xy : array
       projected distance between stars
• r1 : array
       radius of star 1
• r2 : array
       radius of star 2

SIpartial( xy, z, params, surface=(1,1) )	[source]

Returns partials of the SpotIllumination. Specificly of the modified f1 and f2 to params[-5:] (radius₁, radius₂, lumen₁, lumen₂, spot)

If no spot illumination is requested, the dF*dfs are returned as zero.

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       values for the parameters.
• surface : tuple of arrays
       normalized surface areas of the stars

Returns
( dF1dr1, dF1dr2, dF1df1, dF1df2, dF1dfs,
   dF2dr1, dF2dr2, dF2df1, dF2df2, dF2dfs )

LCpartial( xy, z, params )	[source]

Return partial derivatives of LightCurve to each of the parameters

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model

TDpartial( xy, z, params, TDresult=None )	[source]

Calculate partials of the tidal distortion to the distortion parameter

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model
• TDresult : None or tuple
       result of a call to tidalDistortion

Returns

• ( da1dm, db1dm, da2dm, db2dm, da1dr, db1dr, da2dr, db2dr ) : 8 arrays
       Partials of the normalized projected axes to the parameters

OVderivative( xy, r1, r2 )	[source]

calculate the derivative of overlap to xy

Parameters

• xy : array
       projected distance between stars
• r1 : array
       radius of star 1
• r2 : array
       radius of star 2

LCderivative( xy, z, params )	[source]

Return derivative of LightCurve to xy and z, as

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model

SIderivative( xy, z, params, surface=(1,1) )	[source]

Returns derivatives of the SpotIllimunation. Specificly of the modified f1 and f2 to xy and z

Parameters

• xy : float
       distance between the stars in the sky plane
• z : float
       distance of star 2 to the sky plane
• params : array_like
       values for the parameters.
• surface : tuple of 2 array
       normalized surface areas of the stars

Returns
( dF1dx, dF2dx, dF1dz, dF2dz )

TDderivative( xy, z, params, TDresult=None )	[source]

Calculate the derivative of the tidal distortion to xy and z

Parameters

• xy : array
       distance between the stars in the sky plane
• z : array
       distance of star 2 to the sky plane
• params : array
       parameters of the model
• TDresult : tuple
       result of call tp tidalDostortion

Returns
( da1dx, db1dx, da2dx, db2dx, da1dz, db1dz, da2dz, db2dz )
     derivatives of apparent major and minor axes to xy and z

baseName( )	[source]

Returns a string representation of the model.

reportParameters( param, stdevs=None, toMags=False )	[source]

Print parameters and stdevs (if present)

Parameters

• param : array
       to be converted and printed
• stdevs : array
       to be converted and printed
• toMags : bool or float (False)
       true convert luminosities to magnitudes
       float true and use number as scaling factor

Methods inherited from NonLinearModel

• setMixedModel( lindex )
• isMixed( )
• getNonLinearIndex( )
• partial( xdata, param=None, useNum=False )

Methods inherited from Model

• chainLength( )
• isNullModel( )
• isolateModel( k )
• addModel( model )
• subtractModel( model )
• multiplyModel( model )
• divideModel( model )
• pipeModel( model )
• appendModel( model, operation )
• correctParameters( params )
• result( xdata, param=None )
• operate( res, pars, next )
• derivative( xdata, param, useNum=False )
• selectPipe( ndim, ninter, ndout )
• pipe_0( dGd, dHdG )
• pipe_1( dGd, dHdG )
• pipe_2( dGd, dHdG )
• pipe_3( dGd, dHdG )
• pipe_4( dGdx, dHdG )
• pipe_5( dGdx, dHdG )
• pipe_6( dGdx, dHdG )
• pipe_7( dGdx, dHdG )
• pipe_8( dGdx, dHdG )
• pipe_9( dGdx, dHdG )
• shortName( )
• getNumberOfParameters( )
• numDerivative( xdata, param )
• numPartial( xdata, param )
• isDynamic( )
• hasPriors( isBound=True )
• getPrior( kpar )
• setPrior( kpar, prior=None, **kwargs )
• getParameterName( kpar )
• getParameterUnit( kpar )
• getIntegralUnit( )
• setLimits( lowLimits=None, highLimits=None )
• getLimits( )
• hasLimits( fitindex=None )
• unit2Domain( uvalue, kpar=None )
• domain2Unit( dvalue, kpar=None )
• partialDomain2Unit( dvalue )
• nextPrior( )
• getLinearIndex( )
• testPartial( xdata, params, silent=True )
• strictNumericPartial( xdata, params, parlist=None )
• assignDF1( partial, i, dpi )
• assignDF2( partial, i, dpi )
• strictNumericDerivative( xdata, param )

Methods inherited from FixedModel

• select( params )
• selectNames( names )
• expand( xdata, param )

Methods inherited from BaseModel

• checkParameter( param )
• checkPositive( param )
• checkZeroParameter( param )
• isModifiable( )
• basePrior( kpar )
• getParameterIndex( parname )
• getParameterValue( param, name, default=None )
• baseParameterName( kpar )
• baseParameterUnit( kpar )