Planetary Eccentricity vs Eclipse Location

I think that STARRY might not be computing the eclipse location relative to eccentricity of the planetary orbit correctly. I may be attributing eccentricity or Omega incorrectly; please let me know if my code snippet below is correct.

Here is a plot of four different starry phase curves with different combinations of eccentricity and Omega (see legend)

This second figure below is identical to the one above; but zoomed in on the eclipse location, which has not changed; but was expected to.

The vertical dashed line shows the location of phase = 0.5. The shape of the eclipse bottom changes; it begins to look like a grazing eclipse at some versions of eccentricity and omega.

My understanding is that if the eccentricity is > 0.0 and the Omega (“The longitude of ascending node in degrees”) is > 0.0, then the eclipse location should change (+/-) by up to several tenths in phase (0.3 - 0.7)

As described here:

def deltaphase_eclipse(ecc, omega):
    ''' Compute the phase location of the eclipse for a given orbit
        
        Parameters
        -----------
        ecc (float): eccentricity of the orbit; ranging from 0.0 - 1.0
        omega (float): argument of the ascending node in degrees; ranging 0-360
        
        Returns
        --------
        phase_shift (float): shift in phase for the center location of the eclipse
    '''
    phase_shift = 0.5*( 1 + (4. / np.pi) * ecc * np.cos(omega * 180/np.pi))
    
    return phase_shift

This script returns the following phase locations for the values of eccentricity and omega in the script below and figure above

print(deltaphase_eclipse(0.3,0)))
# 0.691

print(deltaphase_eclipse(0.3,90))
# 0.443

print(deltaphase_eclipse(0.3,180))
# 0.343

print(deltaphase_eclipse(0.3,270))
# 0.651

print(deltaphase_eclipse(0.3,360))
# 0.566

---

Code Snippet

from starry import kepler

from pylab import *; ion()
import numpy as np
try:
    import exomast_api # pip install git+https://github.com/exowanderer/exoMAST_API
except:
    !pip install git+https://github.com/exowanderer/exoMAST_API
    import exomast_api

def update_starry(system, times, model_params):
    ''' Update planet - system parameters '''
    planet = system.secondaries[0]
    
    edepth = model_params['edepth']
    day2night = model_params['day2night']
    phase_offset = model_params['phase_offset']
    
    planet.lambda0 = model_params['lambda0'] # Mean longitude in degrees at reference time
    planet.r = model_params['Rp_Rs'] # planetary radius in stellar radius
    planet.inc = model_params['inclination'] # orbital inclination 
    planet.a = model_params['a_Rs'] # orbital distance in stellar radius
    planet.prot = model_params['orbital_period'] # synchronous rotation
    planet.porb = model_params['orbital_period'] # synchronous rotation
    planet.tref = model_params['transit_time'] # MJD for transit center time
    
    planet.ecc = model_params['eccentricity'] # eccentricity of orbit
    planet.Omega = model_params['omega'] # argument of the ascending node
    
    max_day2night = np.sqrt(3)/2
    Y_1_0_base = day2night * max_day2night if edepth > 0 else 0
    
    # cos(x + x0) = Y_1_0*cos(x0) - Y_1n1*sin(x0)
    Y_1_0 = Y_1_0_base*cos(phase_offset/180*pi)
    Y_1n1 = -Y_1_0_base*sin(phase_offset/180*pi)
    Y_1p1 = 0.0
    
    planet[1,0] = Y_1_0
    planet[1, -1] = Y_1n1
    planet[1, 1] = Y_1p1
    planet.L = edepth / planet.flux()
    
    system.compute(times)

def plot_starry_phase_curve_phys(system, times, model_params, fig=None, ax=None, label='', figsize=(15,10)):
    
    if None in [fig, ax]:
        fig = plt.figure(figsize=figsize)
        ax = fig.add_subplot(111)
    
    update_starry(system, times, model_params)
    
    edepth = model_params['edepth']
    day2night = model_params['day2night']
    phase_offset = model_params['phase_offset']
    
    phase = (times - system.secondaries[0].tref) / system.secondaries[0].porb
    ax.plot(phase,system.lightcurve, label=label)
    
    # Universal
    if edepth > 0: ax.set_ylim(1-.1*edepth,1 + 1.1*edepth)
    
    ax.set_xlabel('Time [day]', fontsize=30)
    ax.set_ylabel('normalized flux', fontsize=30)
    
    for tick in ax.xaxis.get_major_ticks():
        tick.label.set_fontsize(20)
    
    for tick in ax.yaxis.get_major_ticks():
        tick.label.set_fontsize(20)


planet_info = exomast_api.exoMAST_API(planet_name='HD 189733 b')

lmax  = 1

phase = np.linspace(0, 1.0, 1000)
times = phase*planet_info.orbital_period + planet_info.transit_time

''' Instantiate Kepler STARRY model; taken from HD 189733b example'''
# Star
star = kepler.Primary()

# Planet
lambda0 = 90.0

planet = kepler.Secondary(lmax=lmax)
planet.lambda0 = lambda0 # Mean longitude in degrees at reference time
planet.r = planet_info.Rp_Rs # planetary radius in stellar radius
planet.L = 0.0 # flux from planet relative to star
planet.inc = planet_info.inclination # orbital inclination 
planet.a = planet_info.a_Rs # orbital distance in stellar radius
planet.prot = planet_info.orbital_period # synchronous rotation
planet.porb = planet_info.orbital_period # synchronous rotation
planet.tref = planet_info.transit_time # MJD for transit center time

planet.ecc = planet_info.eccentricity # eccentricity of orbit
planet.Omega = planet_info.omega # argument of the ascending node

# Instantiate the system
system = kepler.System(star, planet)

# Specific plottings
fpfs = 1000/1e6

### Run Plots


fig = plt.figure(figsize=(20,6))
ax = fig.add_subplot(111)

''' Instantiate Kepler STARRY model; taken from HD 189733b example'''
lambda0 = 90.0

model_params = {}
model_params['lambda0'] = lambda0 # Mean longitude in degrees at reference time
model_params['Rp_Rs'] = planet_info.Rp_Rs # planetary radius in stellar radius
model_params['inclination'] = planet_info.inclination # orbital inclination 
model_params['a_Rs'] = planet_info.a_Rs # orbital distance in stellar radius
model_params['orbital_period'] = planet_info.orbital_period # synchronous rotation
model_params['transit_time'] = planet_info.transit_time # MJD for transit center time

model_params['eccentricity'] = planet_info.eccentricity # eccentricity of orbit
model_params['omega'] = planet_info.omega # argument of the ascending node

model_params['edepth'] = fpfs
model_params['day2night'] = 1.0
model_params['phase_offset'] = 0

model_params['eccentricity'] = 0.0
model_params['omega'] = 0.0
label='eccentricity={}; omega={}'.format(model_params['eccentricity'], model_params['omega'])
plot_starry_phase_curve_phys(system, times, model_params, fig=fig, ax=ax, label=label)

model_params['eccentricity'] = 0.3
model_params['omega'] = 0.0
label='eccentricity={}; omega={}'.format(model_params['eccentricity'], model_params['omega'])

plot_starry_phase_curve_phys(system, times, model_params, fig=fig, ax=ax, label=label)

model_params['eccentricity'] = 0.3
model_params['omega'] = 90.0
label='eccentricity={}; omega={}'.format(model_params['eccentricity'], model_params['omega'])

plot_starry_phase_curve_phys(system, times, model_params, fig=fig, ax=ax, label=label)

model_params['omega'] = 180.0
label='eccentricity={}; omega={}'.format(model_params['eccentricity'], model_params['omega'])

plot_starry_phase_curve_phys(system, times, model_params, fig=fig, ax=ax, label=label)

model_params['omega'] = 270.0
label='eccentricity={}; omega={}'.format(model_params['eccentricity'], model_params['omega'])

plot_starry_phase_curve_phys(system, times, model_params, fig=fig, ax=ax, label=label)

ax.legend(loc=0, fontsize=15)

plt.axvline(0.5, ls='--');
plt.axhline(1.0, ls='--');

[rodluger]

@exowanderer I think you want to change w (the longitude of pericenter, which rotates the ellipse of the orbit about the orbital plane) instead of Omega (the argument of ascending node, which rotates the orbit on the plane of the sky, assuming it is viewed ~edge on). Changing Omega should have no observational effect for a single-planet system, since it just rotates everything on the plane of the sky.

[rodluger]

@exowanderer Thanks, that’s a better way to think about it. I’ll implement the general conversion between t0 and tref, lambda0 so the user can provide either one or the other.