Observe a phase curve of a spotted star.

This example demonstrates stellar contamination of a phase curve.

A phase curve with a long enough baseline can be contaminated by stellar variability. We take the phase curve of GJ1214 b, analyzed by Kempton et al. [2023] using JWST MIRI-LRS, as an example.

from pathlib import Path
import numpy as np
import matplotlib.pyplot as plt
from astropy import units as u
from cartopy import crs as ccrs
import pypsg
from pypsg.globes import PyGCM

from VSPEC import ObservationModel,PhaseAnalyzer
from VSPEC import params
from VSPEC.config import MSH

SEED = 1214
pypsg.docker.set_url_and_run()

Saved settings to /home/ted/.pypsg/settings.json
Reloading settings...

Create the configurations

# Instrument
inst = params.InstrumentParameters.miri_lrs()

# Observation
observation = params.ObservationParameters(
    observation_time=41.0*u.hr,
    integration_time=15*u.min
)

# PSG
psg_params = params.psgParameters(
    gcm_binning=200,
    phase_binning=1,
    nmax=0,
    lmax=0,
    continuum=['Rayleigh', 'Refraction', 'CIA_all'],
    use_molecular_signatures=True
)

# Star and Planet
star_teff = 3250*u.K
star_rad = 0.215*u.R_sun
planet_rad = 2.742*u.R_earth
orbit_rad = 0.01490*u.AU
orbit_period = 1.58040433*u.day
planet_rot_period = orbit_period
star_rot_period = 120 * u.day
planet_mass = 8.17*u.M_earth
star_mass = 0.178*u.M_sun
inclination = 88.7*u.deg

start_time_before_eclipse = 2*u.hr
angle_before_eclipse = (2*np.pi*u.rad * start_time_before_eclipse/orbit_period).to(u.deg)
initial_phase = 0*u.deg - angle_before_eclipse

planet_params = params.PlanetParameters(
    name='GJ1214b',
    radius=planet_rad,
    gravity=params.GravityParameters('kg',planet_mass),
    semimajor_axis=orbit_rad,
    orbit_period=orbit_period,
    rotation_period=planet_rot_period,
    eccentricity=0,
    obliquity=0*u.deg,
    obliquity_direction=0*u.deg,
    init_phase=initial_phase,
    init_substellar_lon=0*u.deg
)

system_params = params.SystemParameters(
    distance=14.6427*u.pc,
    inclination=inclination,
    phase_of_periasteron=0*u.deg
)

star_dict = {
    'teff': star_teff,
    'radius': star_rad
}
planet_dict = {'semimajor_axis': orbit_rad}

gcm_dict = {
    'nlayer': 30,
    'nlon': 30,
    'nlat': 15,
    'epsilon': 6,
    'albedo': 0.3,
    'emissivity': 1.0,
    'lat_redistribution': 0.1,
    'gamma': 1.4,
    'psurf': 1*u.bar,
    'ptop': 1e-5*u.bar,
    'wind': {'U': '0 m/s','V':'0 m/s'},
    'molecules':{'CO2':0.99}
}

gcm = params.gcmParameters.from_dict({
    'star':star_dict,
    'planet':planet_dict,
    'gcm':{'vspec':gcm_dict,'mean_molec_weight':44}
})

star_kwargs = dict(
    psg_star_template='M',
    teff=star_teff,
    mass=star_mass,
    radius=star_rad,
    period=star_rot_period,
    misalignment=0*u.deg,
    misalignment_dir=0*u.deg,
    ld=params.LimbDarkeningParameters.proxima(),
    faculae=params.FaculaParameters.none(),
    flares=params.FlareParameters.none(),
    granulation=params.GranulationParameters.none(),
    grid_params=(500,1000),
    spectral_grid='default'
)

quiet_star = params.StarParameters(
    spots=params.SpotParameters.none(),
    **star_kwargs
)
spotted_star = params.StarParameters(
    spots=params.SpotParameters(
        distribution='iso',
        initial_coverage=0.2,
        area_mean=300*MSH,
        area_logsigma=0.2,
        teff_umbra=2700*u.K,
        teff_penumbra=2700*u.K,
        equillibrium_coverage=0.2,
        burn_in=0*u.s,
        growth_rate=0.0/u.day,
        decay_rate=0*MSH/u.day,
        initial_area=10*MSH
    ),
    **star_kwargs
)

# Set parameters for simulation
header_kwargs = dict(
    teff_min=2300*u.K,teff_max=3400*u.K,
    seed = SEED,
    verbose = 0
)
internal_params_kwargs = dict(
    planet=planet_params,
    system=system_params,
    obs=observation,
    gcm=gcm,
    psg=psg_params,
    inst=inst
)

params_quiet = params.InternalParameters(
    header=params.Header(data_path=Path('.vspec/gj1214_quiet'),**header_kwargs),
    star = quiet_star,
    **internal_params_kwargs
)

params_spotted = params.InternalParameters(
    header=params.Header(data_path=Path('.vspec/gj1214_spotted'),**header_kwargs),
    star = spotted_star,
    **internal_params_kwargs
)

Map the planetary surface

Before we run VSPEC, let’s look at the planet.

gcm_data:PyGCM = gcm.get_gcm()

tsurf = gcm_data.tsurf.dat.to_value(u.K)

fig = plt.figure()
proj = ccrs.Robinson(central_longitude=0)
ax = fig.add_subplot(projection=proj)

lats = gcm_data.lat_start + np.arange(gcm_data.shape[2]+1) * gcm_data.dlat.to_value(u.deg)
lons = gcm_data.lon_start + np.arange(gcm_data.shape[1]+1) * gcm_data.dlon.to_value(u.deg)

im = ax.pcolor(lons,lats,tsurf.T,cmap='gist_heat',transform=ccrs.PlateCarree())
gl = ax.gridlines(crs=ccrs.PlateCarree(),draw_labels=True,
    color='grey', alpha=0.8, linestyle='--')
gl.top_xlabels = False
gl.right_ylabels = False


_=fig.colorbar(im,ax=ax,label='Surface Temperature (K)')

/home/ted/github/VSPEC/VSPEC/gcm/heat_transfer.py:538: RuntimeWarning: Energy balance off by -1.5 %. eps = 6.0, T0 = 0.1
  warnings.warn(msg, RuntimeWarning)

Run the spotless model

model_quiet = ObservationModel(params=params_quiet)
model_quiet.build_planet()
model_quiet.build_spectra()

/home/ted/github/VSPEC/VSPEC/gcm/heat_transfer.py:538: RuntimeWarning: Energy balance off by -1.5 %. eps = 6.0, T0 = 0.1
  warnings.warn(msg, RuntimeWarning)
/home/ted/github/VSPEC/VSPEC/gcm/heat_transfer.py:538: RuntimeWarning: Energy balance off by -1.5 %. eps = 6.0, T0 = 0.1
  warnings.warn(msg, RuntimeWarning)

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Plot the lightcurve

data_quiet = PhaseAnalyzer(model_quiet.directories['all_model'])
flux_unit = u.Unit('W m-2 um-1')
def get_star(data:PhaseAnalyzer):
    i_eclipse1 = np.argmin(data.lightcurve('total',(0,-1))[:data.N_images//4])
    i_eclipse2 = np.argmin(data.lightcurve('total',(0,-1))[3*data.N_images//4:]) + 3*data.N_images//4
    time = (data.time-data.time[0]).to_value(u.hr)
    star_spec1 = data.spectrum('total',i_eclipse1).to_value(flux_unit)
    star_spec2 = data.spectrum('total',i_eclipse2).to_value(flux_unit)

    def func(t:float):
        m = (star_spec2 - star_spec1)/(time[i_eclipse2]-time[i_eclipse1])
        x = t-time[i_eclipse1]
        b = star_spec1
        y = m * x + b
        return y

    return func


def plot_lc(data:PhaseAnalyzer):
    fig,axes = plt.subplots(2,1,tight_layout=True)

    axes[0].scatter((data.time-data.time[0]).to(u.hr),
        data.lightcurve('total',(0,-1)),label='white light',s=5,c='k')
    axes[0].set_xlabel('Time since start of observation (hour)')
    axes[0].set_ylabel('Flux (W m-2 um-1)')
    axes[0].legend()
    first_four = data.time-data.time[0] <= 4*u.hour
    axins = axes[0].inset_axes([0.08, 0.15, 0.35, 0.5])
    axins.scatter((data.time-data.time[0]).to(u.hr)[first_four],
        data.lightcurve('total',(0,-1))[first_four],label='white light',s=5,c='k')
    axes[0].indicate_inset_zoom(axins)

    interp = get_star(data)
    t = (data.time-data.time[0]).to_value(u.hr)

    n_steps = 10
    colors = plt.get_cmap('viridis')
    indices = np.arange(start=0,stop=data.N_images,step=data.N_images//n_steps)

    for index in indices:
        star_spec = interp(t[index])
        pl_spec = data.spectrum('total',index).to_value(flux_unit) - star_spec
        axes[1].plot(data.wavelength,1e6*pl_spec/star_spec,c=colors(index/data.N_images))
    axes[1].set_xlabel('Wavelength (um)')
    axes[1].set_ylabel('Planet flux (ppm)')
    return fig

plot_lc(data_quiet).show()

Plot the spectroscopic phase curve

We can throw out the transit points while we’re at it.

def get_phase_map(data:PhaseAnalyzer):
    white_light_curve = data_quiet.lightcurve('total',(0,-1),normalize=0)
    points_to_use = white_light_curve > 0.5*(np.median(white_light_curve)+ np.min(white_light_curve))

    interp = get_star(data)
    ts = (data.time-data.time[0]).to_value(u.hr)

    # get the planet flux, except plance nan during transit
    star_im = np.array([interp(t) for t in ts]).T
    total_im = data.total.to_value(flux_unit)
    pl_im = np.where(
        points_to_use,
        total_im-star_im,
        np.nan
    )
    return pl_im,star_im


def plot_phasecurve(data:PhaseAnalyzer):
    pl_im,star_im = get_phase_map(data)

    fig,ax = plt.subplots(1,1)
    im = ax.pcolormesh(
        (data.time-data.time[0]).to_value(u.hr),
        data.wavelength.to_value(u.um),
        pl_im/star_im*1e6,
        cmap='viridis'
    )
    fig.colorbar(im,ax=ax,label='Planet flux (ppm)')
    ax.set_xlabel('Time since start of observation (hour)')
    ax.set_ylabel('Wavelength (um)')
    return fig

plot_phasecurve(data_quiet).show()

We can easily make out the phase curve because the star is static.

Run the spotted model

Because we are using the same planet parameters, we won’t rerun PSG for this. Instead, we will just rerun the stellar part of the code. In a way this is cheating but it will save time. Be careful because we are overwriting our old data.

model_spotted = ObservationModel(params_spotted)
model_spotted.build_planet()
model_spotted.build_spectra()

data_spotted = PhaseAnalyzer(model_spotted.directories['all_model'])

/home/ted/github/VSPEC/VSPEC/gcm/heat_transfer.py:538: RuntimeWarning: Energy balance off by -1.5 %. eps = 6.0, T0 = 0.1
  warnings.warn(msg, RuntimeWarning)
/home/ted/github/VSPEC/VSPEC/gcm/heat_transfer.py:538: RuntimeWarning: Energy balance off by -1.5 %. eps = 6.0, T0 = 0.1
  warnings.warn(msg, RuntimeWarning)
Generated 124 mature spots

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Plot the lightcurve, again

We redo our earlier analysis

plot_lc(data_spotted).show()

And the phase curve

plot_phasecurve(data_spotted).show()

Compare phase curves

Kempton et al. [2023] break the spectrum up into \(0.5 \mu m\) bins to analyze the phase curve. We will do the same to observe the effects of stellar contamination.

def get_lc(data:PhaseAnalyzer,w1:u.Quantity,w2:u.Quantity):
    """Get the lightcurve given a bandpass"""
    wl = data.wavelength
    i_left = int(np.argwhere(wl > w1)[0])
    try:
        i_right = int(np.argwhere(wl > w2)[0])
    except IndexError:
        i_right = -1
    interp = get_star(data)
    ts = (data.time-data.time[0]).to_value(u.hr)
    star_im = np.array([interp(t) for t in ts]).T
    total_im = data.total.to_value(flux_unit)
    pl_im = total_im-star_im
    lc = 1e6*pl_im[i_left:i_right,:]/star_im[i_left:i_right,:]
    return lc.mean(axis=0)

bin_edges = np.arange(5.0,12.0,0.5)
n_ax = len(bin_edges)
fig,axes = plt.subplots(n_ax,1,figsize=(7,10),sharex=True)
for edge,ax in zip(bin_edges,axes):
    w1,w2 = edge*u.um,(edge+0.5)*u.um
    quiet_lc = get_lc(data_quiet,w1,w2)
    spotted_lc = get_lc(data_spotted,w1,w2)
    time = (data_quiet.time - data_quiet.time[0]).to(u.hr)
    ax.plot(time,(quiet_lc),c='xkcd:azure',label='No Spots')
    ax.plot(time,(spotted_lc),c='xkcd:lavender',label='Spotted')
    ax.text(0.7,0.7,f'{w1:.1f} - {w2:.1f}',transform=ax.transAxes)
    ax.set_ylim(-100,700)
fig.subplots_adjust(hspace=0,wspace=0)
axes[0].legend()
axes[-1].set_xlabel('Time (hour)')
_ = axes[n_ax//2].set_ylabel('Planet Flux (ppm)')

/home/ted/github/VSPEC/examples/end_to_end/plot_gj1214b_kempton23.py:354: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
  i_left = int(np.argwhere(wl > w1)[0])
/home/ted/github/VSPEC/examples/end_to_end/plot_gj1214b_kempton23.py:356: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
  i_right = int(np.argwhere(wl > w2)[0])

2D residuals

Let’s take a look at how much of the planet flux (from the spotted model) is actaully due to spots.

pl_quiet,star_quiet = get_phase_map(data_quiet)
pl_spotted,_ = get_phase_map(data_spotted)

contribution_from_spots = pl_spotted-pl_quiet
contrast = (contribution_from_spots/star_quiet*1e6)
t = (data_quiet.time - data_quiet.time[0]).to_value(u.hr)
wl = data_quiet.wavelength.to_value(u.um)

fig,ax = plt.subplots(1,1)
im = ax.pcolormesh(t,wl,contrast)
fig.colorbar(im,ax=ax,label='False planet flux (ppm)')
ax.set_xlabel('Time since start of observation (hour)')
_=ax.set_ylabel('Wavelength (um)')