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occultquad.py
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from numpy import size,zeros,where,arccos,sqrt,pi,log
# Computes Hasting's polynomial approximation for the complete
# elliptic integral of the first (ek) and second (kk) kind
def ellke(k):
m1=1.-k**2
logm1 = log(m1)
a1=0.44325141463
a2=0.06260601220
a3=0.04757383546
a4=0.01736506451
b1=0.24998368310
b2=0.09200180037
b3=0.04069697526
b4=0.00526449639
ee1=1.+m1*(a1+m1*(a2+m1*(a3+m1*a4)))
ee2=m1*(b1+m1*(b2+m1*(b3+m1*b4)))*(-logm1)
ek = ee1+ee2
a0=1.38629436112
a1=0.09666344259
a2=0.03590092383
a3=0.03742563713
a4=0.01451196212
b0=0.5
b1=0.12498593597
b2=0.06880248576
b3=0.03328355346
b4=0.00441787012
ek1=a0+m1*(a1+m1*(a2+m1*(a3+m1*a4)))
ek2=(b0+m1*(b1+m1*(b2+m1*(b3+m1*b4))))*logm1
kk = ek1-ek2
return [ek,kk]
# Computes the complete elliptical integral of the third kind using
# the algorithm of Bulirsch (1965):
def ellpic_bulirsch(n,k):
kc=sqrt(1.-k**2); p=n+1.
if(min(p) < 0.):
print('Negative p')
m0=1.; c=1.; p=sqrt(p); d=1./p; e=kc
while 1:
f = c; c = d/p+c; g = e/p; d = 2.*(f*g+d)
p = g + p; g = m0; m0 = kc + m0
if max(abs(1.-kc/g)) > 1.e-8:
kc = 2*sqrt(e); e=kc*m0
else:
return 0.5*pi*(c*m0+d)/(m0*(m0+p))
# Python translation of IDL code.
# This routine computes the lightcurve for occultation of a
# quadratically limb-darkened source without microlensing. Please
# cite Mandel & Agol (2002) and Eastman et al (2012) if you make use
# of this routine in your research. Please report errors or bugs to
def occultquad(z,u1,u2,p0):
nz = size(z)
lambdad = zeros(nz)
etad = zeros(nz)
lambdae = zeros(nz)
omega=1.-u1/3.-u2/6.
## tolerance for double precision equalities
## special case integrations
tol = 1e-14
p = abs(p0)
z = where(abs(p-z) < tol,p,z)
z = where(abs((p-1)-z) < tol,p-1.,z)
z = where(abs((1-p)-z) < tol,1.-p,z)
z = where(z < tol,0.,z)
x1=(p-z)**2.
x2=(p+z)**2.
x3=p**2.-z**2.
## trivial case of no planet
if p <= 0.:
muo1 = zeros(nz) + 1.
mu0 = zeros(nz) + 1.
return [muo1,mu0]
## Case 1 - the star is unocculted:
## only consider points with z lt 1+p
notusedyet = where( z < (1. + p) )
notusedyet = notusedyet[0]
if size(notusedyet) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*(lambdad+2./3.*(p > z))+ \
u2*etad)/omega
mu0=1.-lambdae
return muo1
# Case 11 - the source is completely occulted:
if p >= 1.:
occulted = where(z[notusedyet] <= p-1.)#,complement=notused2)
if size(occulted) != 0:
ndxuse = notusedyet[occulted]
etad[ndxuse] = 0.5 # corrected typo in paper
lambdae[ndxuse] = 1.
# lambdad = 0 already
notused2 = where(z[notusedyet] > p-1)
if size(notused2) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*(lambdad+2./3.* \
(p > z))+u2*etad)/omega
mu0=1.-lambdae
return muo1
notusedyet = notusedyet[notused2]
# Case 2, 7, 8 - ingress/egress (uniform disk only)
inegressuni = where((z[notusedyet] >= abs(1.-p)) & (z[notusedyet] < 1.+p))
if size(inegressuni) != 0:
ndxuse = notusedyet[inegressuni]
tmp = (1.-p**2.+z[ndxuse]**2.)/2./z[ndxuse]
tmp = where(tmp > 1.,1.,tmp)
tmp = where(tmp < -1.,-1.,tmp)
kap1 = arccos(tmp)
tmp = (p**2.+z[ndxuse]**2-1.)/2./p/z[ndxuse]
tmp = where(tmp > 1.,1.,tmp)
tmp = where(tmp < -1.,-1.,tmp)
kap0 = arccos(tmp)
tmp = 4.*z[ndxuse]**2-(1.+z[ndxuse]**2-p**2)**2
tmp = where(tmp < 0,0,tmp)
lambdae[ndxuse] = (p**2*kap0+kap1 - 0.5*sqrt(tmp))/pi
# eta_1
etad[ndxuse] = 1./2./pi*(kap1+p**2*(p**2+2.*z[ndxuse]**2)*kap0- \
(1.+5.*p**2+z[ndxuse]**2)/4.*sqrt((1.-x1[ndxuse])*(x2[ndxuse]-1.)))
# Case 5, 6, 7 - the edge of planet lies at origin of star
ocltor = where(z[notusedyet] == p)#, complement=notused3)
t = where(z[notusedyet] == p)
if size(ocltor) != 0:
ndxuse = notusedyet[ocltor]
if p < 0.5:
# Case 5
q=2.*p # corrected typo in paper (2k -> 2p)
Ek,Kk = ellke(q)
# lambda_4
lambdad[ndxuse] = 1./3.+2./9./pi*(4.*(2.*p**2-1.)*Ek+\
(1.-4.*p**2)*Kk)
# eta_2
etad[ndxuse] = p**2/2.*(p**2+2.*z[ndxuse]**2)
lambdae[ndxuse] = p**2 # uniform disk
elif p > 0.5:
# Case 7
q=0.5/p # corrected typo in paper (1/2k -> 1/2p)
Ek,Kk = ellke(q)
# lambda_3
lambdad[ndxuse] = 1./3.+16.*p/9./pi*(2.*p**2-1.)*Ek-\
(32.*p**4-20.*p**2+3.)/9./pi/p*Kk
# etad = eta_1 already
else:
# Case 6
lambdad[ndxuse] = 1./3.-4./pi/9.
etad[ndxuse] = 3./32.
notused3 = where(z[notusedyet] != p)
if size(notused3) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*\
(lambdad+2./3.*(p > z))+u2*etad)/omega
mu0=1.-lambdae
return muo1
notusedyet = notusedyet[notused3]
# Case 2, Case 8 - ingress/egress (with limb darkening)
inegress = where( ((z[notusedyet] > 0.5+abs(p-0.5)) & \
(z[notusedyet] < 1.+p)) | \
( (p > 0.5) & (z[notusedyet] > abs(1.-p)) & \
(z[notusedyet] < p)) )#, complement=notused4)
if size(inegress) != 0:
ndxuse = notusedyet[inegress]
q=sqrt((1.-x1[ndxuse])/(x2[ndxuse]-x1[ndxuse]))
Ek,Kk = ellke(q)
n=1./x1[ndxuse]-1.
# lambda_1:
lambdad[ndxuse]=2./9./pi/sqrt(x2[ndxuse]-x1[ndxuse])*\
(((1.-x2[ndxuse])*(2.*x2[ndxuse]+x1[ndxuse]-3.)-\
3.*x3[ndxuse]*(x2[ndxuse]-2.))*Kk+(x2[ndxuse]-\
x1[ndxuse])*(z[ndxuse]**2+7.*p**2-4.)*Ek-\
3.*x3[ndxuse]/x1[ndxuse]*ellpic_bulirsch(n,q))
notused4 = where( ( (z[notusedyet] <= 0.5+abs(p-0.5)) | \
(z[notusedyet] >= 1.+p) ) & ( (p <= 0.5) | \
(z[notusedyet] <= abs(1.-p)) | \
(z[notusedyet] >= p) ))
if size(notused4) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*(lambdad+2./3.*\
(p > z))+u2*etad)/omega
mu0=1.-lambdae
return muo1
notusedyet = notusedyet[notused4]
# Case 3, 4, 9, 10 - planet completely inside star
if p < 1.:
inside = where(z[notusedyet] <= (1.-p))#, complement=notused5)
if size(inside) != 0:
ndxuse = notusedyet[inside]
## eta_2
etad[ndxuse] = p**2/2.*(p**2+2.*z[ndxuse]**2)
## uniform disk
lambdae[ndxuse] = p**2
## Case 4 - edge of planet hits edge of star
edge = where(z[ndxuse] == 1.-p)#, complement=notused6)
if size(edge[0]) != 0:
## lambda_5
lambdad[ndxuse[edge]] = 2./3./pi*arccos(1.-2.*p)-\
4./9./pi*sqrt(p*(1.-p))*(3.+2.*p-8.*p**2)
if p > 0.5:
lambdad[ndxuse[edge]] -= 2./3.
notused6 = where(z[ndxuse] != 1.-p)
if size(notused6) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*\
(lambdad+2./3.*(p > z))+u2*etad)/omega
mu0=1.-lambdae
return muo1
ndxuse = ndxuse[notused6[0]]
## Case 10 - origin of planet hits origin of star
origin = where(z[ndxuse] == 0)#, complement=notused7)
if size(origin) != 0:
## lambda_6
lambdad[ndxuse[origin]] = -2./3.*(1.-p**2)**1.5
notused7 = where(z[ndxuse] != 0)
if size(notused7) == 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*\
(lambdad+2./3.*(p > z))+u2*etad)/omega
mu0=1.-lambdae
return muo1
ndxuse = ndxuse[notused7[0]]
q=sqrt((x2[ndxuse]-x1[ndxuse])/(1.-x1[ndxuse]))
n=x2[ndxuse]/x1[ndxuse]-1.
Ek,Kk = ellke(q)
## Case 3, Case 9 - anywhere in between
## lambda_2
lambdad[ndxuse] = 2./9./pi/sqrt(1.-x1[ndxuse])*\
((1.-5.*z[ndxuse]**2+p**2+x3[ndxuse]**2)*Kk+\
(1.-x1[ndxuse])*(z[ndxuse]**2+7.*p**2-4.)*Ek-\
3.*x3[ndxuse]/x1[ndxuse]*ellpic_bulirsch(n,q))
## if there are still unused elements, there's a bug in the code
## (please report it)
notused5 = where(z[notusedyet] > (1.-p))
if notused5[0] != 0:
print("ERROR: the following values of z didn't fit into a case:")
return [-1,-1]
if p0 > 0:
muo1 =1.-((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*(lambdad+2./3.*\
(p > z))+u2*etad)/omega
mu0=1.-lambdae
else:
muo1 =1.+((1.-u1-2.*u2)*lambdae+(u1+2.*u2)*(lambdad+2./3.*\
(p > z))+u2*etad)/omega
mu0=1.+lambdae
return muo1