def lnlike(theta, fit_params, flux, err):
fit_params[3] = theta[0]
planetx, planety, z, rp, r, f, spotx, spoty, spotradius, spotcontrast = fit_params
fit = spotrod.integratetransit(
planetx, planety, z, rp, r, f, np.array([spotx]), np.array([spoty]),
np.array([spotradius]), np.array([spotcontrast]),
np.array(
[spotrod.circleangle(r, rp, z[i]) for i in xrange(z.shape[0])]))
residuals = flux - fit
return -0.5 * (np.nansum((residuals / err)**2.))
def lightcurve(t0,per,rp,inc,c1,c2):
f=2.0*quadraticlimbdarkening(r,c1,c2)
t_f=t-t0
smacm=(((per*24.*60.*60.)**2.*Grav*Ms)/(4*np.pi**2.))**(1./3.)
sma=smacm/Rs
MA=(2*np.pi/per)*t_f
eta=sma*np.cos(MA)
xi=sma*np.sin(MA)
b=sma*np.cos(inc*np.pi/180.)
planetx=b*eta/sma
planety=-xi
z=np.sqrt(planetx**2.+planety**2.)
planetangle=np.array([spotrod.circleangle(r,rp,z[i]) for i in xrange(z.shape[0])])
fitlightcurve=spotrod.integratetransit(planetx,planety,z,rp,r,f,np.array([spotx]),np.array([spoty]),np.array([spotradius]),np.array([spotcontrast]),planetangle)
return fitlightcurve
def lightcurve(fitparams, period, sma):
# Calculate orbital elements.
f, rp = fit_params
# eta, xi = spotrod.elements(t, period, semimajoraxis, k, h)
# if not np.isfinite(np.mean(eta)):
Mean_Anomaly = (2 * np.pi / period) * t
eta = semimajoraxis * np.cos(Mean_Anomaly)
xi = semimajoraxis * np.sin(Mean_Anomaly)
impactparam = semimajoraxis * np.cos(85.35 * np.pi / 180.)
planetx = impactparam * eta / semimajoraxis
planety = -xi
z = np.sqrt(np.power(planetx, 2) + np.power(planety, 2))
# Calculate planetangle array.
planetangle = np.array(
[spotrod.circleangle(r, rp, z[i]) for i in xrange(z.shape[0])])
fitlightcurve = spotrod.integratetransit(
planetx, planety, z, rp, r, f, np.array([spotx1, spotx2]),
np.array([spoty1, spoty2]), np.array([spotradius1, spotradius2]),
np.array([spotcontrast1, spotcontrast2]), planetangle)
return fitlightcurve
spoty=0.0
spotradius=0.0
spotcontrast=1.0
t_f=t-t0
f=2.0*quadraticlimbdarkening(r,u1,u2)
smacm=(((per*24.*60.*60.)**2.*Grav*Ms)/(4*np.pi**2.))**(1./3.)
sma=smacm/Rs
MA=(2*np.pi/per)*t_f
eta=sma*np.cos(MA)
xi=sma*np.sin(MA)
b=sma*np.cos(inc*np.pi/180.)
planetx=b*eta/sma
planety=-xi
z=np.sqrt(planetx**2.+planety**2.)
planetangle=np.array([spotrod.circleangle(r,rp,z[i]) for i in xrange(z.shape[0])])
def lightcurve(spotx,spoty,spotradius,spotcontrast):
fitlightcurve=spotrod.integratetransit(planetx,planety,z,rp,r,f,np.array([spotx]),np.array([spoty]),np.array([spotradius]),np.array([spotcontrast]),planetangle)
return fitlightcurve
####################################################
# Prior, Likelihood, Posterior Functions #
####################################################
def loggaus(mean,sig,param):
return -np.log(np.sqrt(2*np.pi*sig**2.))-((param-mean)**2.)/(2*sig**2.)
def lnprior(theta):
spotx,spoty,spotradius,spotcontrast=theta
if spotx<=1.0 and spotx>=-1.0 and spoty<=1.0 and spoty>-1.0 and spotx**2.+spoty**2.<1.0 and spotcontrast<=1.0 and spotcontrast>=0.0 and spotradius<=1.0 and spotradius>=0.0:
def lightcurve(rp,c1,c2,spotcontrast):
f=2.0*quadraticlimbdarkening(r,c1,c2)
planetangle=np.array([spotrod.circleangle(r,rp,z[i]) for i in xrange(z.shape[0])])
fitlightcurve=spotrod.integratetransit(planetx,planety,z,rp,r,f,np.array([spotx]),np.array([spoty]),np.array([spotradius]),np.array([spotcontrast]),planetangle)
return fitlightcurve
# Weights: 2.0 times limb darkening times width of integration annulii. f = 2.0 * quadraticlimbdarkening(r, u1, u2) / n; # Alternative: trapeziod rule. #r = numpy.linspace(0.0, 1.0, n); #f = 2.0 * quadraticlimbdarkening(r, u1, u2) * numpy.append(numpy.append([0.5], numpy.repeat(1.0, n-2)), [0.5]) / (n-1); # Calculate orbital elements. eta, xi = spotrod.elements(timebkjd-midtransit, period, semimajoraxis, k, h); # Planet coordinates in sky plane, in Rstar units. planetx = impactparam*eta/semimajoraxis; planety = -xi; # Distance from center, same as $z$ in Mandel, Agol 2002. z = numpy.sqrt(numpy.power(planetx,2) + numpy.power(planety,2)); # Calculate planetangle array. planetangle = numpy.array([spotrod.circleangle(r, rp, z[i]) for i in xrange(z.shape[0])]); # Prior for spot parameters: isotropic on the surface of the sphere. logp = lambda p: -0.5 * numpy.sum(numpy.log((1.0 - numpy.power(p[0::4], 2.0) - numpy.power(p[1::4], 2.0)))); # Likelihood for spot parameters. logl = lambda p: numpy.sum(numpy.power(spotrod.integratetransit(planetx, planety, z, rp, r, f, p[0::4], p[1::4], p[2::4], p[3::4], planetangle) - flux, 2.0) * minusoneovertwofluxerrsquared); # We have one spot, therefore phase space is 4D. ndim = 4; # Number of temperatures. ntemps = 10; # Number of parallel walkers at each temperature. nwalkers = 100; # Number of iterations. niter = 1000;
def spot_model(t, cube, nspot, texp=0.01881944, RESAMPLE=False, NRESAMPLE=20):
if RESAMPLE:
tin = np.array([])
for i in range(len(t)):
for j in range(NRESAMPLE):
jj = j + 1
tin = np.append(
tin,
t[i] + (jj - (NRESAMPLE + 1) / 2.) * (texp / NRESAMPLE))
else:
tin = t
# Extract parameters:
if nspot == 0:
rp, aR, inc, t0, q1, q2 = cube[0], cube[1], cube[2], cube[3], cube[
4], cube[5]
elif nspot == 1:
rp,aR,inc,t0,q1,q2,spotx,spoty,spotradius,spotcontrast = cube[0],cube[1],\
cube[2],cube[3],cube[4],cube[5],cube[6],cube[7],cube[8],cube[9]
elif nspot == 2:
rp,aR,inc,t0,q1,q2,spotx,spoty,spotradius,spotcontrast,spotx2,spoty2,spotradius2,\
spotcontrast2 = cube[0],cube[1],cube[2],cube[3],cube[4],cube[5],cube[6],cube[7],cube[8],\
cube[9],cube[10],cube[11],cube[12],cube[13]
# Generate model:
impactparam = aR * np.cos(inc * deg_to_rad)
phase = np.mod((tin - t0) / P + 0.5, 1.0) - 0.5
s1, s2 = reverse_ld_coeffs(ld_law, q1, q2)
# Initialize spotrod. Number of intergration rings:
n = 1500
# Integration annulii radii.
r = np.linspace(1.0 / (2 * n), 1.0 - 1.0 / (2 * n), n)
# Weights: 2.0 times limb darkening times width of integration annulii.
f = 2.0 * quadraticlimbdarkening(r, s1, s2) / n
# Calculate orbital elements.
if k == 0. and h == 0.:
# spotrod.elements (which uses the general equations of Pal, 2009MNRAS.396.1737P)
# fail to extract proper eta and xi for the case of circular orbits. It is however
# easy to obtain these following the paper (eqs. 6 and 7 on eqs 2 and 3):
M = (tin - t0) * (2 * np.pi) / P
lam = M + omega * deg_to_rad
eta0, xi0 = aR * np.cos(lam), aR * np.sin(lam)
eta,xi = eta0*np.cos(omega*deg_to_rad) - xi0*np.sin(omega*deg_to_rad),\
eta0*np.sin(omega*deg_to_rad) + xi0*np.cos(omega*deg_to_rad)
else:
eta, xi = spotrod.elements(tin - t0, P, aR, k, h)
planetx = impactparam * eta / aR
planety = -xi
z = np.sqrt(np.power(planetx, 2) + np.power(planety, 2))
planetangle = np.array(
[spotrod.circleangle(r, rp, z[i]) for i in xrange(z.shape[0])])
if nspot == 0:
spotx, spoty, spotradius, spotcontrast = 0., 0., 0., 1.
model = spotrod.integratetransit(planetx, planety, z, rp, r, f, np.array([spotx]),\
np.array([spoty]), np.array([spotradius]), np.array([spotcontrast]), planetangle)
elif nspot == 1:
model = spotrod.integratetransit(planetx, planety, z, rp, r, f, np.array([spotx]),\
np.array([spoty]), np.array([spotradius]), np.array([spotcontrast]), planetangle)
elif nspot == 2:
model = spotrod.integratetransit(planetx, planety, z, rp, r, f, np.array([spotx,spotx2]),\
np.array([spoty,spoty2]), np.array([spotradius,spotradius2]), np.array([spotcontrast,spotcontrast2]), planetangle)
if RESAMPLE:
model_out = np.zeros(len(t))
for i in range(len(t)):
model_out[i] = np.mean(model[i * NRESAMPLE:NRESAMPLE * (i + 1)])
return model_out
return model
r = numpy.linspace(1.0 / (2 * n), 1.0 - 1.0 / (2 * n), n)
# Weights: 2.0 times limb darkening times width of integration annulii.
f = 2.0 * quadraticlimbdarkening(r, u1, u2) / n
# Alternative: trapeziod rule.
#r = numpy.linspace(0.0, 1.0, n);
#f = 2.0 * quadraticlimbdarkening(r, u1, u2) * numpy.append(numpy.append([0.5], numpy.repeat(1.0, n-2)), [0.5]) / (n-1);
# Calculate orbital elements.
eta, xi = spotrod.elements(timebkjd - midtransit, period, semimajoraxis, k, h)
planetx = impactparam * eta / semimajoraxis
planety = -xi
z = numpy.sqrt(numpy.power(planetx, 2) + numpy.power(planety, 2))
# Calculate planetangle array.
planetangle = numpy.array(
[spotrod.circleangle(r, rp, z[i]) for i in xrange(z.shape[0])])
spotx = 0.204
spoty = 0.376
spotradius = 0.096
spotcontrast = 0.524
fitlightcurve = spotrod.integratetransit(planetx, planety, z, rp, r, f,
numpy.array([spotx]),
numpy.array([spoty]),
numpy.array([spotradius]),
numpy.array([spotcontrast]),
planetangle)
pyplot.plot(phase, flux, "b.")
pyplot.plot(phase, fitlightcurve, "k-")