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ma02.py
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"""------------------
The Mandel & Agol (2002) transit light curve equations.
------------------
:FUNCTIONS:
:func:`occultuniform` -- uniform-disk transit light curve
:func:`occultquad` -- quadratic limb-darkening
:func:`occultnonlin` -- full (4-parameter) nonlinear limb-darkening
:func:`occultnonlin_small` -- small-planet approximation with full
nonlinear limb-darkening.
:func:`t2z` -- convert dates to transiting z-parameter for circular
orbits.
:REQUIREMENTS:
`numpy <http://www.numpy.org/>`_
`scipy.special <http://www.scipy.org/>`_
:NOTES:
Certain values of p (<0.09, >0.5) cause some routines to hang;
your mileage may vary. If you find out why, please let me know!
Cursory testing suggests that the Python routines contained within
are slower than the corresponding IDL code by a factor of 5-10.
For :func:`occultquad` I relied heavily on the IDL code of E. Agol
and J. Eastman.
Function :func:`appellf1` comes from the mpmath compilation, and
is adopted (with modification) for use herein in compliance with
its BSD license (see function documentation for more details).
:REFERENCE:
The main reference is that seminal work by `Mandel and Agol (2002)
<http://adsabs.harvard.edu/abs/2002ApJ...580L.171M>`_.
:LICENSE:
Created by `Ian Crossfield <http://www.astro.ucla.edu/~ianc/>`_ at
UCLA. The code contained herein may be reused, adapted, or
modified so long as proper attribution is made to the original
authors.
:REVISIONS:
2011-04-22 11:08 IJMC: Finished, renamed occultation functions.
Cleaned up documentation. Published to
website.
2011-04-25 17:32 IJMC: Fixed bug in :func:`ellpic_bulirsch`.
2012-03-09 08:38 IJMC: Several major bugs fixed, courtest of
S. Aigrain at Oxford U.
-----------------
"""
import numpy as np
from scipy import special, misc
import pdb
eps = np.finfo(float).eps
zeroval = eps*1e6
def appelf1_ac(a, b1, b2, c, z1, z2, **kwargs):
"""Analytic continuations of the Appell hypergeometric function of 2 variables.
:REFERENCE:
Olsson 1964, Colavecchia et al. 2001
"""
# 2012-03-09 12:05 IJMC: Created
def appellf1(a,b1,b2,c,z1,z2,**kwargs):
"""Give the Appell hypergeometric function of two variables.
:INPUTS:
six parameters, all scalars.
:OPTIONS:
eps -- scalar, machine tolerance precision. Defaults to 1e-12.
:NOTES:
Adapted from the `mpmath <http://code.google.com/p/mpmath/>`_
module, but using the scipy (instead of mpmath) Gauss
hypergeometric function speeds things up.
:LICENSE:
MPMATH Copyright (c) 2005-2010 Fredrik Johansson and mpmath
contributors. All rights reserved.
Redistribution and use in source and binary forms, with or
without modification, are permitted provided that the following
conditions are met:
a. Redistributions of source code must retain the above
copyright notice, this list of conditions and the following
disclaimer.
b. Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following
disclaimer in the documentation and/or other materials
provided with the distribution.
c. Neither the name of mpmath nor the names of its contributors
may be used to endorse or promote products derived from this
software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND
CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,
INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING
IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
THE POSSIBILITY OF SUCH DAMAGE.
"""
#2011-04-22 10:15 IJMC: Adapted from mpmath, but using scipy Gauss
#hypergeo. function
if kwargs.has_key('eps'):
eps = kwargs['eps']
else:
eps = 1e-12
# Assume z1 smaller
# We will use z1 for the outer loop
if abs(z1) > abs(z2):
z1, z2 = z2, z1
b1, b2 = b2, b1
def ok(x):
return abs(x) < 0.99
# IJMC: Ignore the finite cases for now....
## Finite cases
#if ctx.isnpint(a):
# pass
#elif ctx.isnpint(b1):
# pass
#elif ctx.isnpint(b2):
# z1, z2, b1, b2 = z2, z1, b2, b1
#else:
# #print z1, z2
# # Note: ok if |z2| > 1, because
# # 2F1 implements analytic continuation
if not ok(z1):
u1 = (z1-z2)/(z1-1)
if not ok(u1):
raise ValueError("Analytic continuation not implemented")
#print "Using analytic continuation"
return (1-z1)**(-b1)*(1-z2)**(c-a-b2)*\
appellf1(c-a,b1,c-b1-b2,c,u1,z2,**kwargs)
#print "inner is", a, b2, c
##one = ctx.one
s = 0
t = 1
k = 0
while 1:
#h = ctx.hyp2f1(a,b2,c,z2,zeroprec=ctx.prec,**kwargs)
#print a.__class__, b2.__class__, c.__class__, z2.__class__
h = special.hyp2f1(float(a), float(b2), float(c), float(z2))
term = t * h
if abs(term) < eps and abs(h) > 10*eps:
break
s += term
k += 1
t = (t*a*b1*z1) / (c*k)
c += 1 # one
a += 1 # one
b1 += 1 # one
return s
def ellke2(k, tol=100*eps, maxiter=100):
"""Compute complete elliptic integrals of the first kind (K) and
second kind (E) using the series expansions."""
# 2011-04-24 21:14 IJMC: Created
k = np.array(k)
ksum = 0*k
kprevsum = ksum.copy()
kresidual = ksum + 1
#esum = 0*k
#eprevsum = esum.copy()
#eresidual = esum + 1
n = 0
sqrtpi = np.sqrt(np.pi)
while (np.abs(kresidual) > tol).any() and n <= maxiter:
ksum += ((misc.factorial2(2*n - 1)/misc.factorial2(2*n))**2) * k**(2*n)
#ksum += (special.gamma(n + 0.5)/special.gamma(n + 1) / sqrtpi) * k**(2*n)
kresidual = ksum - kprevsum
kprevsum = ksum.copy()
n += 1
#print n, kresidual
return ksum * (np.pi/2.)
def ellke(k):
"""Compute Hasting's polynomial approximation for the complete
elliptic integral of the first (ek) and second (kk) kind.
:INPUTS:
k -- scalar or Numpy array
:OUTPUTS:
ek, kk
:NOTES:
Adapted from the IDL function of the same name by J. Eastman (OSU).
"""
# 2011-04-19 09:15 IJC: Adapted from J. Eastman's IDL code.
m1 = 1. - k**2
logm1 = np.log(m1)
# First kind:
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)
# Second kind:
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
return ee1 + ee2, ek1 - ek2
def ellpic_bulirsch(n, k, tol=100*eps, maxiter=1e4):
"""Compute the complete elliptical integral of the third kind
using the algorithm of Bulirsch (1965).
:INPUTS:
n -- scalar or Numpy array
k-- scalar or Numpy array
:NOTES:
Adapted from the IDL function of the same name by J. Eastman (OSU).
"""
# 2011-04-19 09:15 IJMC: Adapted from J. Eastman's IDL code.
# 2011-04-25 11:40 IJMC: Set a more stringent tolerance (from 1e-8
# to 1e-14), and fixed tolerance flag to the
# maximum of all residuals.
# Make p, k into vectors:
#if not hasattr(n, '__iter__'):
# n = array([n])
#if not hasattr(k, '__iter__'):
# k = array([k])
if not hasattr(n,'__iter__'):
n = np.array([n])
if not hasattr(k,'__iter__'):
k = np.array([k])
if len(n)==0 or len(k)==0:
return np.array([])
kc = np.sqrt(1. - k**2)
p = n + 1.
if min(p) < 0:
print ("Negative p")
# Initialize:
m0 = np.array(1.)
c = np.array(1.)
p = np.sqrt(p)
d = 1./p
e = kc.copy()
outsideTolerance = True
iter = 0
while outsideTolerance and iter<maxiter:
f = c.copy()
c = d/p + c
g = e/p
d = 2. * (f*g + d)
p = g + p
g = m0.copy()
m0 = kc + m0
if max(np.abs(1. - kc/g)) > tol:
kc = 2. * np.sqrt(e)
e = kc * m0
else:
outsideTolerance = False
#if (iter/10.) == (iter/10):
# print iter, (np.abs(1. - kc/g))
#pdb.set_trace()
iter += 1
## For debugging:
#print min(np.abs(1. - kc/g)) > tol
#print 'tolerance>>', tol
#print 'minimum>> ', min(np.abs(1. - kc/g))
#print 'maximum>> ', max(np.abs(1. - kc/g)) #, (np.abs(1. - kc/g))
return .5 * np.pi * (c*m0 + d) / (m0 * (m0 + p))
def z2dt_circular(per, inc, ars, z):
""" Convert transit crossing parameter z to a time offset for circular orbits.
:INPUTS:
per -- scalar. planetary orbital period
inc -- scalar. orbital inclination (in degrees)
ars -- scalar. ratio a/Rs, orbital semimajor axis over stellar radius
z -- scalar or array; transit crossing parameter z.
:RETURNS:
|dt| -- magnitude of the time offset from transit center at
which specified z occurs.
"""
# 2011-06-14 11:26 IJMC: Created.
numer = (z / ars)**2 - 1.
denom = np.cos(inc*np.pi/180.)**2 - 1.
dt = (per / (2*np.pi)) * np.arccos(np.sqrt(numer / denom))
return dt
def t2z(tt, per, inc, hjd, ars, ecc=0, longperi=0):
"""Convert HJD (time) to transit crossing parameter z.
:INPUTS:
tt -- scalar. transit ephemeris
per -- scalar. planetary orbital period
inc -- scalar. orbital inclination (in degrees)
hjd -- scalar or array of times, typically heliocentric or
barycentric julian date.
ars -- scalar. ratio a/Rs, orbital semimajor axis over stellar radius
ecc -- scalar. orbital eccentricity.
longperi=0 scalar. longitude of periapse (in radians)
:ALGORITHM:
At zero eccentricity, z relates to physical quantities by:
z = (a/Rs) * sqrt(sin[w*(t-t0)]**2+[cos(i)*cos(w*[t-t0])]**2)
"""
# 2010-01-11 18:18 IJC: Created
# 2011-04-19 15:20 IJMC: Updated documentation.
# 2011-04-22 11:27 IJMC: Updated to avoid reliance on planet objects.
# 2011-05-22 16:51 IJMC: Temporarily removed eccentricity
# dependence... I'll deal with that later.
#if not p.transit:
# print "Must use a transiting exoplanet!"
# return False
import analysis as an
if ecc <> 0:
ecc = 0
print "WARNING: setting ecc=0 for now until I get this function working"
if ecc==0:
omega_orb = 2*np.pi/per
z = ars * np.sqrt(np.sin(omega_orb*(hjd-tt))**2 + \
(np.cos(inc*np.pi/180.)*np.cos(omega_orb*(hjd-tt)))**2)
else:
if longperi is None:
longperi = 180.
f = an.trueanomaly(ecc, (2*np.pi/per) * (hjd - tt))
z = ars * (1. - ecc**2) * np.sqrt(1. - (np.sin(longperi + f) * np.sin(inc))**2) / \
(1. + ecc * np.cos(f))
return z
def uniform(*arg, **kw):
"""Placeholder for my old code; the new function is called
:func:`occultuniform`.
"""
# 2011-04-19 15:06 IJMC: Created
print "The function 'transit.uniform()' is deprecated."
print "Please use transit.occultuniform() in the future."
return occultuniform(*arg, **kw)
def occultuniform(z, p, complement=False):
"""Uniform-disk transit light curve (i.e., no limb darkening).
:INPUTS:
z -- scalar or sequence; positional offset values of planet in
units of the stellar radius.
p -- scalar; planet/star radius ratio.
complement : bool
If True, return (1 - occultuniform(z, p))
:SEE ALSO: :func:`t2z`, :func:`occultquad`, :func:`occultnonlin_small`
"""
# 2011-04-15 16:56 IJC: Added a tad of documentation
# 2011-04-19 15:21 IJMC: Cleaned up documentation.
# 2011-04-25 11:07 IJMC: Can now handle scalar z input.
# 2011-05-15 10:20 IJMC: Fixed indexing check (size, not len)
# 2012-03-09 08:30 IJMC: Added "complement" argument for backwards
# compatibility, and fixed arccos error at
# 1st/4th contact point (credit to
# S. Aigrain @ Oxford)
z = np.abs(np.array(z,copy=True))
fsecondary = np.zeros(z.shape,float)
if p < 0:
pneg = True
p = np.abs(p)
else:
pneg = False
p2 = p**2
if len(z.shape)>0: # array entered
i1 = (1+p)<z
i2 = (np.abs(1-p) < z) * (z<= (1+p))
i3 = z<= (1-p)
i4 = z<=(p-1)
#print i1.sum(),i2.sum(),i3.sum(),i4.sum()
z2 = z[i2]**2
acosarg1 = (p2+z2-1)/(2.*p*z[i2])
acosarg2 = (1-p2+z2)/(2*z[i2])
acosarg1[acosarg1 > 1] = 1. # quick fix for numerical precision errors
acosarg2[acosarg2 > 1] = 1. # quick fix for numerical precision errors
k0 = np.arccos(acosarg1)
k1 = np.arccos(acosarg2)
k2 = 0.5*np.sqrt(4*z2-(1+z2-p2)**2)
fsecondary[i1] = 0.
fsecondary[i2] = (1./np.pi)*(p2*k0 + k1 - k2)
fsecondary[i3] = p2
fsecondary[i4] = 1.
if not (i1+i2+i3+i4).all():
print("warning -- some input values not indexed!")
if (i1.sum()+i2.sum()+i3.sum()+i4.sum() <> z.size):
print("warning -- indexing didn't get the right number of values")
else: # scalar entered
if (1+p)<=z:
fsecondary = 0.
elif (np.abs(1-p) < z) * (z<= (1+p)):
z2 = z**2
k0 = np.arccos((p2+z2-1)/(2.*p*z))
k1 = np.arccos((1-p2+z2)/(2*z))
k2 = 0.5*np.sqrt(4*z2-(1+z2-p2)**2)
fsecondary = (1./np.pi)*(p2*k0 + k1 - k2)
elif z<= (1-p):
fsecondary = p2
elif z<=(p-1):
fsecondary = 1.
if pneg:
fsecondary *= -1
if complement:
return fsecondary
else:
return 1. - fsecondary
def depthchisq(z, planet, data, ddepth=[-.1,.1], ndepth=20, w=None):
#z = transit.t2z(planet, planet.i, hjd, 0.211)
nobs = z.size
depths = np.linspace(ddepth[0],ddepth[1], ndepth)
print depths
chisq = np.zeros(ndepth, float)
for ii in range(ndepth):
tr = -(transit.occultuniform(z, np.sqrt(planet.depth))/depths[ii])
if w is None:
w = np.ones(nobs,float)/data[tr==0].std()
print 'w>>',w[0]
baseline = np.ones(nobs,float) * an.wmean(data[tr==0], w[tr==0])
print 'b>>',baseline[0]
print 'd[ii]>>',depths[ii]
model = baseline + tr*depths[ii]
plot(model)
chisq[ii] = (w*(model-data)**2).sum()
return depths, chisq
def integral_smallplanet_nonlinear(z, p, cn, lower, upper):
"""Return the integral in I*(z) in Eqn. 8 of Mandel & Agol (2002).
-- Int[I(r) 2r dr]_{z-p}^{1}, where:
:INPUTS:
z = scalar or array. Distance between center of star &
planet, normalized by the stellar radius.
p = scalar. Planet/star radius ratio.
cn = 4-sequence. Nonlinear limb-darkening coefficients,
e.g. from Claret 2000.
lower, upper -- floats. Limits of integration in units of mu
:RETURNS:
value of the integral at specified z.
"""
# 2010-11-06 14:12 IJC: Created
# 2012-03-09 08:54 IJMC: Added a cheat for z very close to zero
#import pdb
z = np.array(z, copy=True)
z[z==0] = zeroval
lower = np.array(lower, copy=True)
upper = np.array(upper, copy=True)
a = (z - p)**2
def eval_int_at_limit(limit, cn):
"""Evaluate the integral at a specified limit (upper or lower)"""
term1 = cn[0] * (1. - 0.8 * np.sqrt(limit))
term2 = cn[1] * (1. - (2./3.) * limit)
term3 = cn[2] * (1. - (4./7.) * limit**1.5)
term4 = cn[3] * (1. - 0.5 * limit**2)
return -(limit**2) * (1. - term1 - term2 - term3 - term4)
ret = eval_int_at_limit(upper, cn) - eval_int_at_limit(lower, cn)
return ret
def smallplanet_nonlinear(*arg, **kw):
"""Placeholder for backwards compatibility with my old code. The
function is now called :func:`occultnonlin_small`.
"""
# 2011-04-19 15:10 IJMC: Created
print("The function 'transit.smallplanet_nonlinear()' is deprecated.")
print("Please use transit.occultnonlin_small() in the future.")
return occultnonlin_small(*arg, **kw)
def occultnonlin_small(z,p, cn):
"""Nonlinear limb-darkening light curve in the small-planet
approximation (section 5 of Mandel & Agol 2002).
:INPUTS:
z -- sequence of positional offset values
p -- planet/star radius ratio
cn -- four-sequence nonlinear limb darkening coefficients. If
a shorter sequence is entered, the later values will be
set to zero.
:NOTE:
I had to divide the effect at the near-edge of the light curve
by pi for consistency; this factor was not in Mandel & Agol, so
I may have coded something incorrectly (or there was a typo).
:EXAMPLE:
::
# Reproduce Figure 2 of Mandel & Agol (2002):
from pylab import *
import transit
z = linspace(0, 1.2, 100)
cns = vstack((zeros(4), eye(4)))
figure()
for coef in cns:
f = transit.occultnonlin_small(z, 0.1, coef)
plot(z, f, '--')
:SEE ALSO:
:func:`t2z`
"""
# 2010-11-06 14:23 IJC: Created
# 2011-04-19 15:22 IJMC: Updated documentation. Renamed.
# 2011-05-24 14:00 IJMC: Now check the size of cn.
# 2012-03-09 08:54 IJMC: Added a cheat for z very close to zero
cn = np.array([cn], copy=True).ravel()
if cn.size < 4:
cn = np.concatenate((cn, [0.]*(4-cn.size)))
z = np.array(z, copy=True)
F = np.ones(z.shape, float)
z[z==0] = zeroval # cheat!
a = (z - p)**2
b = (z + p)**2
c0 = 1. - np.sum(cn)
Omega = 0.25 * c0 + np.sum( cn / np.arange(5., 9.) )
ind1 = ((1. - p) < z) * ((1. + p) > z)
ind2 = z <= (1. - p)
# Need to specify limits of integration in terms of mu (not r)
Istar_edge = integral_smallplanet_nonlinear(z[ind1], p, cn, \
np.sqrt(1. - a[ind1]), 0.) / \
(1. - a[ind1])
Istar_inside = integral_smallplanet_nonlinear(z[ind2], p, cn, \
np.sqrt(1. - a[ind2]), \
np.sqrt(1. - b[ind2])) / \
(4. * z[ind2] * p)
term1 = 0.25 * Istar_edge / (np.pi * Omega)
term2 = p**2 * np.arccos((z[ind1] - 1.) / p)
term3 = (z[ind1] - 1) * np.sqrt(p**2 - (z[ind1] - 1)**2)
term4 = 0.25 * p**2 * Istar_inside / Omega
F[ind1] = 1. - term1 * (term2 - term3)
F[ind2] = 1. - term4
return F
def occultquad(z,p0, gamma, retall=False, verbose=False):
"""Quadratic limb-darkening light curve; cf. Section 4 of Mandel & Agol (2002).
:INPUTS:
z -- sequence of positional offset values
p0 -- planet/star radius ratio
gamma -- two-sequence.
quadratic limb darkening coefficients. (c1=c3=0; c2 =
gamma[0] + 2*gamma[1], c4 = -gamma[1]). If only a single
gamma is used, then you're assuming linear limb-darkening.
:OPTIONS:
retall -- bool.
If True, in addition to the light curve return the
uniform-disk light curve, lambda^d, and eta^d parameters.
Using these quantities allows for quicker model generation
with new limb-darkening coefficients -- the speed boost is
roughly a factor of 50. See the second example below.
:EXAMPLE:
::
# Reproduce Figure 2 of Mandel & Agol (2002):
from pylab import *
import transit
z = linspace(0, 1.2, 100)
gammavals = [[0., 0.], [1., 0.], [2., -1.]]
figure()
for gammas in gammavals:
f = transit.occultquad(z, 0.1, gammas)
plot(z, f)
::
# Calculate the same geometric transit with two different
# sets of limb darkening coefficients:
from pylab import *
import transit
p, b = 0.1, 0.5
x = (arange(300.)/299. - 0.5)*2.
z = sqrt(x**2 + b**2)
gammas = [.25, .75]
F1, Funi, lambdad, etad = transit.occultquad(z, p, gammas, retall=True)
gammas = [.35, .55]
F2 = 1. - ((1. - gammas[0] - 2.*gammas[1])*(1. - F1) +
(gammas[0] + 2.*gammas[1])*(lambdad + 2./3.*(p > z)) + gammas[1]*etad) /
(1. - gammas[0]/3. - gammas[1]/6.)
figure()
plot(x, F1, x, F2)
legend(['F1', 'F2'])
:SEE ALSO:
:func:`t2z`, :func:`occultnonlin_small`, :func:`occultuniform`
:NOTES:
In writing this I relied heavily on the occultquad IDL routine
by E. Agol and J. Eastman, especially for efficient computation
of elliptical integrals and for identification of several
apparent typographic errors in the 2002 paper (see comments in
the source code).
From some cursory testing, this routine appears about 9 times
slower than the IDL version. The difference drops only
slightly when using precomputed quantities (i.e., retall=True).
"""
# 2011-04-15 15:58 IJC: Created; forking from smallplanet_nonlinear
# 2011-05-14 22:03 IJMC: Now linear-limb-darkening is allowed with
# a single parameter passed in.
# Initialize:
gamma = np.array(gamma, copy=True)
if gamma.size < 2: # Linear limb-darkening
gamma = np.array([gamma.ravel(), [0.]])
z = np.array(z, copy=True)
lambdad = np.zeros(z.shape, float)
etad = np.zeros(z.shape, float)
F = np.ones(z.shape, float)
p = np.abs(p0) # Save the original input
# Define limb-darkening coefficients:
c2 = gamma[0] + 2 * gamma[1]
c4 = -gamma[1]
# Test the simplest case (a zero-sized planet):
if p==0:
if retall:
ret = np.ones(z.shape, float), np.ones(z.shape, float), \
np.zeros(z.shape, float), np.zeros(z.shape, float)
else:
ret = np.ones(z.shape, float)
return ret
# Define useful constants:
fourOmega = 1. - gamma[0]/3. - gamma[1]/6. # Actually 4*Omega
a = (z - p)**2
b = (z + p)**2
k = 0.5 * np.sqrt((1. - a) / (z * p))
p2 = p**2
z2 = z**2
# Define the many necessary indices for the different cases:
i01 = (p > 0) * (z >= (1. + p))
i02 = (p > 0) * (z > (.5 + np.abs(p - 0.5))) * (z < (1. + p))
i03 = (p > 0) * (p < 0.5) * (z > p) * (z < (1. - p))
i04 = (p > 0) * (p < 0.5) * (z == (1. - p))
i05 = (p > 0) * (p < 0.5) * (z == p)
i06 = (p == 0.5) * (z == 0.5)
i07 = (p > 0.5) * (z == p)
i08 = (p > 0.5) * (z >= np.abs(1. - p)) * (z < p)
i09 = (p > 0) * (p < 1) * (z > 0) * (z < (0.5 - np.abs(p - 0.5)))
i10 = (p > 0) * (p < 1) * (z == 0)
i11 = (p > 1) * (z >= 0.) * (z < (p - 1.))
if verbose:
allind = i01 + i02 + i03 + i04 + i05 + i06 + i07 + i08 + i09 + i10 + i11
nused = (i01.sum() + i02.sum() + i03.sum() + i04.sum() + \
i05.sum() + i06.sum() + i07.sum() + i08.sum() + \
i09.sum() + i10.sum() + i11.sum())
print "%i/%i indices used" % (nused, i01.size)
if not allind.all():
print "Some indices not used!"
#pdb.set_trace()
# Lambda^e is easy:
lambdae = 1. - occultuniform(z, p)
# Lambda^e and eta^d are more tricky:
# Simple cases:
lambdad[i01] = 0.
etad[i01] = 0.
lambdad[i06] = 1./3. - 4./9./np.pi
etad[i06] = 3./32.
lambdad[i11] = 1.
# etad[i11] = 1. # This is what the paper says
etad[i11] = 0.5 # Typo in paper (according to J. Eastman)
# Lambda_1:
ilam1 = i02 + i08
q1 = p2 - z2[ilam1]
## This is what the paper says:
#ellippi = ellpic_bulirsch(1. - 1./a[ilam1], k[ilam1])
# ellipe, ellipk = ellke(k[ilam1])
# This is what J. Eastman's code has:
# 2011-04-24 20:32 IJMC: The following codes act funny when
# sqrt((1-a)/(b-a)) approaches unity.
qq = np.sqrt((1. - a[ilam1]) / (b[ilam1] - a[ilam1]))
ellippi = ellpic_bulirsch(1./a[ilam1] - 1., qq)
ellipe, ellipk = ellke(qq)
lambdad[i02 + i08] = (1./ (9.*np.pi*np.sqrt(p*z[ilam1]))) * \
( ((1. - b[ilam1])*(2*b[ilam1] + a[ilam1] - 3) - \
3*q1*(b[ilam1] - 2.)) * ellipk + \
4*p*z[ilam1]*(z2[ilam1] + 7*p2 - 4.) * ellipe - \
3*(q1/a[ilam1])*ellippi)
# Lambda_2:
ilam2 = i03 + i09
q2 = p2 - z2[ilam2]
## This is what the paper says:
#ellippi = ellpic_bulirsch(1. - b[ilam2]/a[ilam2], 1./k[ilam2])
# ellipe, ellipk = ellke(1./k[ilam2])
# This is what J. Eastman's code has:
ellippi = ellpic_bulirsch(b[ilam2]/a[ilam2] - 1, np.sqrt((b[ilam2] - a[ilam2])/(1. - a[ilam2])))
ellipe, ellipk = ellke(np.sqrt((b[ilam2] - a[ilam2])/(1. - a[ilam2])))
lambdad[ilam2] = (2. / (9*np.pi*np.sqrt(1.-a[ilam2]))) * \
((1. - 5*z2[ilam2] + p2 + q2**2) * ellipk + \
(1. - a[ilam2])*(z2[ilam2] + 7*p2 - 4.) * ellipe - \
3*(q2/a[ilam2])*ellippi)
# Lambda_3:
#ellipe, ellipk = ellke(0.5/ k) # This is what the paper says
ellipe, ellipk = ellke(0.5/ p) # Corrected typo (1/2k -> 1/2p), according to J. Eastman
lambdad[i07] = 1./3. + (16.*p*(2*p2 - 1.)*ellipe -
(1. - 4*p2)*(3. - 8*p2)*ellipk / p) / (9*np.pi)
# Lambda_4
#ellipe, ellipk = ellke(2. * k) # This is what the paper says
ellipe, ellipk = ellke(2. * p) # Corrected typo (2k -> 2p), according to J. Eastman
lambdad[i05] = 1./3. + (2./(9*np.pi)) * (4*(2*p2 - 1.)*ellipe + (1. - 4*p2)*ellipk)
# Lambda_5
## The following line is what the 2002 paper says:
#lambdad[i04] = (2./(3*np.pi)) * (np.arccos(1 - 2*p) - (2./3.) * (3. + 2*p - 8*p2))
# The following line is what J. Eastman's code says:
lambdad[i04] = (2./3.) * (np.arccos(1. - 2*p)/np.pi - \
(2./(3*np.pi)) * np.sqrt(p * (1.-p)) * \
(3. + 2*p - 8*p2) - \
float(p > 0.5))
# Lambda_6
lambdad[i10] = -(2./3.) * (1. - p2)**1.5
# Eta_1:
kappa0 = np.arccos((p2+z2[i02 + i07 + i08]-1)/(2.*p*z[i02 + i07 + i08]))
kappa1 = np.arccos((1-p2+z2[i02 + i07 + i08])/(2*z[i02 + i07 + i08]))
etad[i02 + i07 + i08] = \
(0.5/np.pi) * (kappa1 + kappa0*p2*(p2 + 2*z2[i02 + i07 + i08]) - \
0.25*(1. + 5*p2 + z2[i02 + i07 + i08]) * \
np.sqrt((1. - a[i02 + i07 + i08]) * (b[i02 + i07 + i08] - 1.)))
# Eta_2:
etad[i03 + i04 + i05 + i09 + i10] = 0.5 * p2 * (p2 + 2. * z2[i03 + i04 + i05 + i09 + i10])
# We're done!
## The following are handy for debugging:
#term1 = (1. - c2) * lambdae
#term2 = c2*lambdad
#term3 = c2*(2./3.) * (p>z).astype(float)
#term4 = c4 * etad
F = 1. - ((1. - c2) * lambdae + \
c2 * (lambdad + (2./3.) * (p > z).astype(float)) - \
c4 * etad) / fourOmega
#pdb.set_trace()
if retall:
ret = F, lambdae, lambdad, etad
else:
ret = F
#pdb.set_trace()
return ret
def occultnonlin(z,p0, cn):
"""Nonlinear limb-darkening light curve; cf. Section 3 of Mandel & Agol (2002).
:INPUTS:
z -- sequence of positional offset values
p0 -- planet/star radius ratio
cn -- four-sequence. nonlinear limb darkening coefficients
:EXAMPLE:
::
# Reproduce Figure 2 of Mandel & Agol (2002):
from pylab import *
import transit
z = linspace(0, 1.2, 50)
cns = vstack((zeros(4), eye(4)))
figure()
for coef in cns:
f = transit.occultnonlin(z, 0.1, coef)
plot(z, f)
:SEE ALSO:
:func:`t2z`, :func:`occultnonlin_small`, :func:`occultuniform`, :func:`occultquad`
:NOTES:
Scipy is much faster than mpmath for computing the Beta and
Gauss hypergeometric functions. However, Scipy does not have
the Appell hypergeometric function -- the current version is
not vectorized.
"""
# 2011-04-15 15:58 IJC: Created; forking from occultquad
#import pdb
# Initialize:
cn0 = np.array(cn, copy=True)
z = np.array(z, copy=True)
F = np.ones(z.shape, float)
p = np.abs(p0) # Save the original input
# Test the simplest case (a zero-sized planet):
if p==0:
ret = np.ones(z.shape, float)
return ret
# Define useful constants:
c0 = 1. - np.sum(cn0)
# Row vectors:
c = np.concatenate(([c0], cn0))
n = np.arange(5, dtype=float)
# Column vectors:
cc = c.reshape(5, 1)
nn = n.reshape(5,1)
np4 = n + 4.
nd4 = n / 4.
twoOmega = 0.5*c[0] + 0.4*c[1] + c[2]/3. + 2.*c[3]/7. + 0.25*c[4]
a = (z - p)**2
b = (z + p)**2
am1 = a - 1.
bma = b - a
k = 0.5 * np.sqrt(-am1 / (z * p))
p2 = p**2
z2 = z**2
# Define the many necessary indices for the different cases:
i01 = (p > 0) * (z >= (1. + p))
i02 = (p > 0) * (z > (.5 + np.abs(p - 0.5))) * (z < (1. + p))
i03 = (p > 0) * (p < 0.5) * (z > p) * (z <= (1. - p)) # also contains Case 4
#i04 = (z==(1. - p))
i05 = (p > 0) * (p < 0.5) * (z == p)
i06 = (p == 0.5) * (z == 0.5)
i07 = (p > 0.5) * (z == p)
i08 = (p > 0.5) * (z >= np.abs(1. - p)) * (z < p)
i08a = (p == 1) * (z == 0)
i09 = (p > 0) * (p < 1) * (z > 0) * (z < (0.5 - np.abs(p - 0.5)))
i10 = (p > 0) * (p < 1) * (z == 0)
i11 = (p > 1) * (z >= 0.) * (z < (p - 1.))
iN = i02 + i08
iM = i03 + i09
# Compute N and M for the appropriate indices:
# (Use the slow, non-vectorized appellf1 function:)
myappellf1 = np.frompyfunc(appellf1, 6, 1)
N = np.zeros((5, z.size), float)
M = np.zeros((3, z.size), float)
#pdb.set_trace()
termN = myappellf1(0.5, 1., 0.5, 0.25*nn + 2.5, am1[iN]/a[iN], -am1[iN]/bma[iN])
#pdb.set_trace()
termM = myappellf1(0.5, -0.25*nn[1:4] - 1., 1., 1., -bma[iM]/am1[iM], -bma[iM]/a[iM])
N[:, iN] = ((-am1[iN])**(0.25*nn + 1.5)) / np.sqrt(bma[iN]) * \
special.beta(0.25*nn + 2., 0.5) * \
(((z2[iN] - p2) / a[iN]) * termN - \
special.hyp2f1(0.5, 0.5, 0.25*nn + 2.5, -am1[iN]/bma[iN]))
M[:, iM] = ((-am1[iM])**(0.25*nn[1:4] + 1.)) * \
(((z2[iM] - p2)/a[iM]) * termM - \
special.hyp2f1(-0.25*nn[1:4] - 1., 0.5, 1., -bma[iM]/am1[iM]))
# Begin going through all the cases: