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lcspot2.py
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lcspot2.py
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# port of lcspot.pro
# Analytic model of intensity variations, due to a circular spot on a
# rotating stars, based on equations in Eker (1994, ApJ, 420, 373).
#
# Inputs:
# phase (array[nphi]) rotational phases for output light curve, where
# phase=0.0 is when stellar longitude 0 crosses disk center.
# par (array[7]) stellar and spot parameters
# par[0:6] = [limb1, limb2, inc_deg, lon_deg, lat_deg, rad_deg, iratio]
# limb1: linear term in limb-darkening law
# I[mu] = I[0] .* (1 - limb1.*(1-mu) - limb2.*(1-mu.^2)), where mu=cos(theta)
# Form used by Claret (2004)# Eker (1994) uses u1=limb1, u2=-limb2.
# Limb-darkening coefficients depend on filter bandpass.
# limb2: quadratic term in limb-darkening law [see limb1]
# inc_deg: inclination of stellar rotation axis (in degrees)
# lon_deg: longitude of spot center (in degrees)
# lat_deg: latitude of spot center (in degrees)
# rad_deg: angle radius of spot (in degrees)
# iratio: spot intensity divided by non-spot intensity
# 0:black, <1:dark, 1:photosphere, >1:bright
#
from numpy import *
import itertools
def lcspot(time, params):
#Internal constants
d2r = pi / 180
#Impose Boundaries
bps = boundparams(params)
#Extract stellar and spot properties from parameter vector
inc_deg = bps[0]
lon_deg = bps[1]
lat_deg = bps[2]
rad_deg = bps[3]
period = bps[4]
limb1 = 0.45 #linear coefficient in limb-darkening law
limb2 = 0.3 #quadratic coefficient in limb-darkening law
iratio = 0.67 #intensity in spot / intensity out of spot
#Convert input angles from degrees to radians
inc = inc_deg * d2r #stellar inclination
lam = lon_deg * d2r #spot longitude
bet = lat_deg * d2r #spot latitude
rad = rad_deg * d2r #spot radius
#Useful scalar quantities
cosrad = cos(rad)
sinrad = sin(rad)
#Calculate a vector of rotational phases (could be passed as an argument)
phase = time / period
phi = 2.0 * pi * phase
nphi = len(phi)
#Calculate angle "theta" between two vector originating at spot center:
# Vector 1) normal to stellar surface directed away from center of star
# Vector 2) directed towards the observer.
#See equation 20 of Eker (1994).
costhe0 = cos(inc) * sin(bet) + sin(inc) * cos(bet) * cos(phi-lam)
#Useful quantities
sinthe0 = sqrt(1.0 - costhe0**2)
the0 = arccos(costhe0)
#Find rotational phases when spot is full, gibbous, crescent, occulted
jf = flatnonzero(the0 <= pi/2-rad)
nf = len(jf)
jg = flatnonzero(logical_and(the0 > pi/2-rad, the0 <= pi/2))
ng = len(jg)
jc = flatnonzero(logical_and(the0 > pi/2, the0 <= pi/2+rad))
nc = len(jc)
jo = flatnonzero(the0 > pi/2+rad)
no = len(jo)
#Allocate vectors for intensity integrals.
ic = zeros(nphi) #constant intensity term
il = zeros(nphi) #linear intensity term
iq = zeros(nphi) #quadratic intensity term
#
# Rotational phases when spot is full (entirely visible)
#
#Useful quantities for full phases.
if nf >= 1:
#unused: the0_f = the0[jf] #angle between spot and LOS
costhe0_f = costhe0[jf]
sinthe0_f = sinthe0[jf]
#Calculate intensity integrals for phases when spot is fully visible
#See equations 13a, 13b, and 13c of Eker (1994)
ic[jf] = pi * sin(rad)**2 * costhe0_f
il[jf] = 2*pi/3 * (1 - cosrad**3) - pi * cosrad * sinrad**2 * sinthe0_f**2
iq[jf] = pi/2 * (1 - cosrad**4) * costhe0_f**3 + 3*pi/4 * sinrad**4 * costhe0_f * sinthe0_f**2
#
# Rotational phases when spot is gibbous (more than half visible).
#
#Useful quantities for gibbous phases.
if ng >= 1:
the0_g = the0[jg] #angle between spot and LOS
costhe0_g = costhe0[jg]
sinthe0_g = sinthe0[jg]
#Calculate integration limits for partially visible spot.
#See equation 16 of Eker (1994).
cosphi0_g = - 1.0 / ( tan(the0_g) * tan(rad) )
rad0_g = abs( the0_g - pi/2 )
#Useful quantities.
#if any(abs(cosphi0_g) > 1.0): print 'IVW @ 114 & 115',params
phi0_g = arccos(cosphi0_g)
sinphi0_g = sqrt(1.0 - cosphi0_g**2)
cosrad0_g = cos(rad0_g)
sinrad0_g = sin(rad0_g)
#Auxiliary quantities for gibbous phases.
#See unnumbered equations that follow equation 18b in Eker (1994).
k1_g = ((pi - phi0_g) / 4) * (cosrad0_g**4 - cosrad**4)
k2_g = (sinphi0_g / 8) * ( rad0_g - rad + 0.5 * ( sin(2*rad) * cos(2*rad) - sin(2*rad0_g) * cos(2*rad0_g) ) )
k3_g = (1.0 / 8) * (pi - phi0_g - sinphi0_g * cosphi0_g) * (sinrad**4 - sinrad0_g**4)
k4_g = - (sinphi0_g - sinphi0_g**3 / 3) * ( (3.0 / 8) * (rad - rad0_g) + (1.0 / 16) * ( sin(2*rad) * (cos(2*rad) - 4) - sin(2*rad0_g) * (cos(2*rad0_g) - 4) ) )
#Corrections to intensity integrals for gibbous phases.
#See equations 18a and 18b in Eker (1994).
cl_g = ((pi - phi0_g) / 3) * (cosrad**3 - cosrad0_g**3) * (1 - 3*costhe0_g**2) - (pi - phi0_g - sinphi0_g * cosphi0_g) * (cosrad - cosrad0_g) * sinthe0_g**2 - (4.0 / 3) * sinphi0_g * (sinrad**3 - sinrad0_g**3) * sinthe0_g * costhe0_g - (1.0 / 3) * sinphi0_g * cosphi0_g * (cosrad**3 - cosrad0_g**3) * sinthe0_g**2
cq_g = 2 * costhe0_g**3 * k1_g + 6 * costhe0_g**2 * sinthe0_g * k2_g + 6 * costhe0_g * sinthe0_g**2 * k3_g + 2 * sinthe0_g**3 * k4_g
#Calculate intensity integrals for phases when spot is fully visible.
#Constant intensity integral. See equation 17 of Eker (1994)
#if any(abs(cosrad / sinthe0_g) > 1.0): print 'IVW @ 134',params
ic[jg] = phi0_g * costhe0_g * sinrad**2 - arcsin(cosrad / sinthe0_g) - 0.5 * sinthe0_g * sinphi0_g * sin(2*rad) + pi/2
#Apply corrections to linear and quadratic intensity integrals.
il[jg] = 2*pi/3 * (1 - cosrad**3) - pi * cosrad * sinrad**2 * sinthe0_g**2 - cl_g
iq[jg] = pi/2 * (1 - cosrad**4) * costhe0_g**3 + 3*pi/4 * sinrad**4 * costhe0_g * sinthe0_g**2 - cq_g
#
# Rotational phases when spot is crescent (less than half visible).
#
#Useful quantities for crescent phases.
if nc >= 1:
the0_c = the0[jc] #angle between spot and LOS
costhe0_c = costhe0[jc]
sinthe0_c = sinthe0[jc]
#Calculate integration limits for partially visible spot.
#See equation 16 of Eker (1994).
cosphi0_c = - 1.0 / ( tan(the0_c) * tan(rad) )
rad0_c = abs( the0_c - pi/2 )
#Useful quantities.
#if any(abs(cosphi0_c) > 1.0): print 'IVW @ 157 & 158',params
phi0_c = arccos(cosphi0_c)
sinphi0_c = sqrt(1.0 - cosphi0_c**2)
cosrad0_c = cos(rad0_c)
sinrad0_c = sin(rad0_c)
#Auxiliary quantities for crescent phases.
#See unnumbered equations that follow equation 18b in Eker (1994).
k1_c = (phi0_c / 4) * (cosrad0_c**4 - cosrad**4)
k2_c = - (sinphi0_c / 8) * ( rad0_c - rad + 0.5 * ( sin(2*rad) * cos(2*rad) - sin(2*rad0_c) * cos(2*rad0_c) ) )
k3_c = (1.0 / 8) * (phi0_c + sinphi0_c * cosphi0_c) * (sinrad**4 - sinrad0_c**4)
k4_c = (sinphi0_c - sinphi0_c**3 / 3) * ( (3.0 / 8) * (rad - rad0_c) + (1.0 / 16) * ( sin(2*rad) * (cos(2*rad) - 4) - sin(2*rad0_c) * (cos(2*rad0_c) - 4) ) )
#Corrections to intensity integrals for crescent phases.
#See equations 18a and 18b in Eker (1994).
#unused: cl_c = ((pi - phi0_c) / 3) * (cosrad**3 - cosrad0_c**3) * (1 - 3*costhe0_c**2) - (pi - phi0_c - sinphi0_c * cosphi0_c) * (cosrad - cosrad0_c) * sinthe0_c**2 - (4.0 / 3) * sinphi0_c * (sinrad**3 - sinrad0_c**3) * sinthe0_c * costhe0_c - (1.0 / 3) * sinphi0_c * cosphi0_c * (cosrad**3 - cosrad0_c**3) * sinthe0**2
cq_c = 2 * costhe0_c**3 * k1_c + 6 * costhe0_c**2 * sinthe0_c * k2_c + 6 * costhe0_c * sinthe0_c**2 * k3_c + 2 * sinthe0_c**3 * k4_c
#Calculate intensity integrals for phases when spot is fully visible.
#Constant intensity integral. See equation 17 of Eker (1994)
#if any(abs(cosrad / sinthe0_c) > 1.0): print 'IVW @ 177',params
ic[jc] = phi0_c * costhe0_c * sinrad**2 - arcsin(cosrad / sinthe0_c) - 0.5 * sinthe0_c * sinphi0_c * sin(2*rad) + pi/2
#Linear intensity integral. See equation 19a of Eker (1994)
il[jc] = (phi0_c / 3) * (cosrad**3 - cosrad0_c**3) * (1 - 3 * costhe0_c**2) - (phi0_c + sinphi0_c * cosphi0_c) * (cosrad - cosrad0_c) * sinthe0_c**2 + (4.0 / 3) * sinphi0_c * (sinrad**3 - sinrad0_c**3) * sinthe0_c * costhe0_c + (1.0 / 3) * sinphi0_c * cosphi0_c * (cosrad**3 - cosrad0_c**3) * sinthe0_c**2
#Apply corrections to linear and quadratic intensity integrals.
iq[jc] = cq_c
#
# Rotational phases when spot is completely occulted (back of star).
#
if no >=1:
ic[jo] = 0.0
il[jo] = 0.0
iq[jo] = 0.0
#
# Calculate light curve. Equation 12c from Eker (1994).
#
lc = 1.0 + (iratio - 1.0) / (pi * (1.0 - limb1/3.0 + limb2/6.0)) * ((1.0 - limb1 + limb2)*ic + (limb1 - 2.0 * limb2)*il + limb2*iq)
return lc
### Single Spot Helper Functions ###
def lcspotdiffs(params, time, data):
return data - lcspot(time, params)
def lcspotdiffs3(params, time, data):
diffs = data - lcspot(time, params)
penalty = 0.0
if params[0] < 0.0:
penalty = penalty + ((-params[0]/45.0)**2)/len(diffs)
if params[0] > 90.0:
penalty = penalty + (((params[0] - 90.0)/45.0)**2)/len(diffs)
if params[1] < 0.0:
penalty = penalty + ((-params[1]/180.0)**2)/len(diffs)
if params[1] > 360.0:
penalty = penalty + (((params[1] - 360.0)/180.0)**2)/len(diffs)
if params[2] < -90.0:
penalty = penalty + (((-params[2] - 90.0)/90.0)**2)/len(diffs)
if params[2] > 90.0:
penalty = penalty + (((params[2] - 90.0)/90.0)**2)/len(diffs)
if params[3] < 0.01:
penalty = penalty + (((-params[3] + 0.01)/20.0)**2)/len(diffs)
return sqrt(diffs**2 + penalty)
def lcspotsse(params, time, data):
return sum(lcspotdiffs(params, time, data)**2)
def lcspotsse3(params, time, data):
return sum(lcspotdiffs3(params, time, data)**2)
### Fixed Inclination Helper Functions ###
def lcspotfi(time, inc, sparams):
return lcspot(time, [inc] + list(sparams))
def lcspotdiffsfi(params, time, data, inc):
return lcspotdiffs3(array([inc] + list(params)), time, data)
def lcspotssefi(params, time, data, inc):
return lcspotsse(array([inc] + list(params)), time, data)
def lcspotssefi3(params, time, data, inc):
return sum(lcspotdiffsfi(params, time, data, inc)**2)
def lcspotfstar(time, inc, teq, alpha, sparams):
lat = sparams[1]
period = teq / (1 - alpha * sin(lat)**2)
return lcspot(time, [inc] + list(sparams) + [period])
def lcspotdiffsfstar(params, time, data, inc, teq, alpha):
lat = params[1]
period = teq / (1 - alpha * sin(lat)**2)
return lcspotdiffs3(array([inc] + list(params) + [period]), time, data)
def lcspotssefstar(params, time, data, inc, teq, alpha):
return sum(data - lcspotfstar(time, inc, teq, alpha, params))**2
### Multi-Spot Helper Functions ###
def lccomb(lc1, lc2):
return lc1 + lc2 - 1
def lcsep(combined, component):
return combined - component + 1
def lcmultispot(time, pset):
inc = pset[0]
teq = pset[1]
alpha = pset[2]
spots = pset[3:]
intensity = 1
for spot in spots:
tspot = teq / (1.0 - alpha * sin(spot[1] * pi / 180.0)**2)
intensity = lccomb(intensity, lcspot(time,[inc]+list(spot)+[tspot]))
return intensity
def lcmultispotdiffs(params,time,data):
inc = params[0]
teq = params[1]
alpha = params[2]
nspots = (len(params)-3)/3
spots = [[params[3+3*i],params[3+3*i+1],params[3+3*i+2]] for i in range(nspots)]
pset = [inc,teq,alpha]+spots
diffs = abs(data - lcmultispot(time, pset))
if inc < 0.0:
diffs = diffs + ((inc/-45.0)**2)/len(diffs)
if inc > 90.0:
diffs = diffs + (((inc-90.0)/45.0)**2)/len(diffs)
for s in spots:
if s[0] < 0.0:
diffs = diffs + ((s[0]/-180.0)**2)/len(diffs)
if s[0] > 360.0:
diffs = diffs + (((s[0]-360.0)/180.0)**2)/len(diffs)
if s[1] < -90.0:
diffs = diffs + (((s[1]+90.0)/-90.0)**2)/len(diffs)
if s[1] > 90.0:
diffs = diffs + (((s[1]-90.0)/90.0)**2)/len(diffs)
if s[2] < 0.0:
diffs = diffs + ((s[2]/-20.0)**2)/len(diffs)
return diffs
def lcmultispotsse(params, time, data):
inc = params[0]
teq = params[1]
alpha = params[2]
nspots = (len(params)-3)/3
spots = [[params[3+3*i],params[3+3*i+1],params[3+3*i+2]] for i in range(nspots)]
pset = [inc,teq,alpha]+spots
diffs = abs(data - lcmultispot(time, pset))
return sum(diffs**2)
### Other Helper Functions ###
def paramdist(fps, tps, scalevals=array([45.0, 180.0, 90.0, 17.5, 25.0])):
return sqrt(sum(((array(fps) - array(tps))/array(scalevals))**2))
def spotparamdist(fps, tps, scalevals=array([180.0, 90.0, 17.5])):
return paramdist(fps, tps, scalevals)
def multispotparamdist(fps, tps, scalevals=array([180.0, 90.0, 17.5])):
return sqrt(sum(array([paramdist(fps[i], tps[i], scalevals) for i in range(len(fps))])**2))
def spotmatch(fspots,tspots,scalevals=array([180.0, 90.0, 17.5])):
nspots = len(fspots)
if nspots == 1:
return fspots, tspots, spotparamdist(fspots[0],tspots[0],scalevals)
else:
mindist = double('inf')
for r in range(nspots):
rfspots, rtspots, rdist = spotmatch(fspots[1:], tspots[:r]+tspots[r+1:], scalevals)
totaldist = spotparamdist(fspots[0],tspots[r],scalevals)
if totaldist < mindist:
mindist = totaldist
nfspots = fspots[0:1] + rfspots
ntspots = tspots[r:r+1] + rtspots
return nfspots, ntspots, mindist
def fpl2gpl(fpl):
return [fpl[0], fpl[1], fpl[2]] + [[fpl[3+3*i],fpl[3+3*i+1],fpl[3+3*i+2]] for i in range((len(fpl)-3)/3)]
def gpl2fpl(gpl):
return [gpl[0], gpl[1], gpl[2]] + [sp for sps in gpl[3:] for sp in sps]
def spacedvals(min, max, nvals):
spacing = (max - min)/double(nvals)
return arange(min+(spacing/2.0),max,spacing)
def spacedparams(nvals, minr=10, maxr=25, mint=1.0, maxt=45.0):
incvals = spacedvals(0,90,nvals)
lonvals = spacedvals(0,360,nvals)
latvals = spacedvals(-90,90,nvals)
radvals = spacedvals(minr,maxr,nvals)
pervals = spacedvals(mint,maxt,nvals)
return [list(x) for x in itertools.product(incvals,lonvals,latvals,radvals,pervals)]
def spacedparamsfi(nvals, minr=10, maxr=25, mint=1.0, maxt=45.0):
lonvals = spacedvals(0,360,nvals)
latvals = spacedvals(-90,90,nvals)
radvals = spacedvals(minr,maxr,nvals)
pervals = spacedvals(mint,maxt,nvals)
return [list(x) for x in itertools.product(lonvals,latvals,radvals,pervals)]
def spacedparamsspot(nvals, minr=10, maxr=25):
lonvals = spacedvals(0,360,nvals)
latvals = spacedvals(-90,90,nvals)
radvals = spacedvals(minr,maxr,nvals)
return [list(x) for x in itertools.product(lonvals,latvals,radvals)]
def boundparams(params):
cps = copy(params)
if cps[0] < 0.0: cps[0] = 0.0
if cps[0] > 90.0: cps[0] = 90.0
if cps[1] < 0.0: cps[1] = 0.0
if cps[1] > 360.0: cps[1] = 360.0
if cps[2] < -90.0: cps[2] = -90.0
if cps[2] > 90.0: cps[2] = 90.0
if cps[3] < 0.01: cps[3] = 0.01
if cps[4] < 0.01: cps[4] = 0.01
return cps
def boundparamsfi(params):
cps = copy(params)
if cps[0] < 0.0: cps[0] = 0.0
if cps[0] > 360.0: cps[0] = 360.0
if cps[1] < -90.0: cps[1] = -90.0
if cps[1] > 90.0: cps[1] = 90.0
if cps[2] < 0.01: cps[2] = 0.01
if cps[3] < 0.01: cps[3] = 0.01
return cps
def boundparamsfstar(params):
cps = copy(params)
if cps[0] < 0.0: cps[0] = 0.0
if cps[0] > 360.0: cps[0] = 360.0
if cps[1] < -90.0: cps[1] = -90.0
if cps[1] > 90.0: cps[1] = 90.0
if cps[2] < 0.01: cps[2] = 0.01
return cps