# Source code for transit

```
#! /usr/bin/env python
# -*- coding: utf-8 -*-
"""
This module is used to compute a grid containing drawing of disks of stars,
planets and atmospheres.
"""
import numpy as np
from scipy.interpolate import interp1d
from scipy.special import voigt_profile
from scipy.integrate import simps
from flatstar import draw, utils
__all__ = ["draw_transit", "radiative_transfer_2d", "profile_los",
"optical_depth_2d"]
# Draw a grid
[docs]def draw_transit(planet_to_star_ratio, planet_physical_radius,
impact_parameter=0.0, phase=0.0, grid_size=100,
supersampling=None, resample_method=None,
limb_darkening_law=None, ld_coefficient=None):
"""
Calculate a normalized transit map. Additionally, calculate two-dimensional
arrays of density and line-of-sight radial velocities around the planet.
Parameters
----------
planet_to_star_ratio (``float``):
Ratio between the radii of the planet and the star.
planet_physical_radius (``float``):
Physical radius of the planet in whatever unit you want to work with.
impact_parameter (``float``, optional):
Transit impact parameter of the planet (not the atmosphere). Default is
0.0.
phase (``float``, optional):
Phase of the transit. -0.5, 0.0, and +0.5 correspond to the center of
planet being located at, respectively, the left limb of the star, the
center, and the right limb. Default is 0.0.
grid_size (``int``, optional):
Size of the transit grid. Default is 100.
supersampling (``float`` or ``None``, optional):
In order to avoid pixels with hard edges, it is useful to first compute
the transit grid at a high resolution and then downscale it to a
manageable grid size. Supersampling is the factor by which to increase
the grid size at first and then downscale to the requested grid size. If
``None``, no supersampling is applied. Default is ``None``.
resample_method (``str`` or ``None``, optional):
Method by which to resample the image if supersampling is used. If
``None``, then fallback to a "box" method. Available methods are
``"nearest"``, ``"box"``, ``"bilinear"``, ``"hamming"`` and
``"lanczos"`` (the last two are not recommended for research-grade
results. Default is ``None``.
limb_darkening_law (``None`` or ``str``, optional):
String with the name of the limb-darkening law. The options are the same
currently implemented in the code ``flatstar``: ``'linear'``,
``'quadratic'``, ``'square-root'``, ``'log'``, ``'exp'``, ``'s3'``,
``'c4'``, or ``None`` (no limb-darkening). Default is ``None``.
ld_coefficient (``float`` or ``array-like``):
In case of a linear limb-darkening law, the value of the coefficient
should be a float. In all other options it should be array-like. Default
is ``None``.
Returns
-------
normalized_intensity_map (``numpy.ndarray``):
2-D map of intensity normalized in such a way that the sum of the array
will be 1.0 if the planet is not transiting.
transit_depth (``float``):
Absorption caused by the opaque disk of the planet in the specified
transit configuration.
r_from_planet (``numpy.ndarray``):
2-D map of radial distances of each pixel from the center of the planet
in the same unit as ``planet_physical_radius``.
"""
if supersampling is not None:
effective_grid_size = int(round(grid_size * supersampling))
rescaling = 1 / supersampling
else:
effective_grid_size = grid_size
rescaling = None
star_grid = draw.star(effective_grid_size,
limb_darkening_law=limb_darkening_law,
ld_coefficient=ld_coefficient)
transit_grid = draw.planet_transit(star_grid, planet_to_star_ratio,
impact_parameter, phase,
rescaling_factor=rescaling,
resample_method=resample_method)
# We will calculate the densities and wind radial velocities of the
# upper atmosphere. We need to know the matrix r_p containing distances from
# planet center when we draw the extended atmosphere
pl_ref = transit_grid.planet_px_coordinates
planet_centric_r = utils.cylindrical_r(transit_grid.intensity, pl_ref)
# We also need to know the physical size of the pixel in the grid
planet_radius = transit_grid.planet_radius_px
px_size = planet_physical_radius / planet_radius
r_from_planet = planet_centric_r * px_size
# Finally
normalized_intensity_map = transit_grid.intensity
transit_depth = transit_grid.transit_depth
return normalized_intensity_map, transit_depth, r_from_planet
# Calculate the radiative transfer
[docs]def radiative_transfer_2d(intensity_0, r_from_planet, radius_profile,
density_profile, velocity_profile, central_wavelength,
oscillator_strength, einstein_coefficient,
wavelength_grid, gas_temperature, particle_mass,
bulk_los_velocity=0.0, planet_radial_velocity=0.0,
wind_broadening_method='average',
turbulence_broadening=False):
"""
Calculate the absorbed intensity profile in a wavelength grid.
Parameters
----------
intensity_0 (``float`` or ``numpy.ndarray``):
Original flux intensity originating from a background illuminating
source. If ``numpy.ndarray``, must have the same shape as
``r_from_planet``.
r_from_planet (``numpy.ndarray``):
2-D map of radial distances of each pixel from the center of the planet.
The unit has to be consistent with the other input parameters. E.g., if
you use unit of meters, all other input parameters with length in the
unit also have to be in meters.
radius_profile (``numpy.ndarray``):
1-D profile of radii in which the densities are sampled. Unit has to be
the same as ``r_from_planet``.
density_profile (``numpy.ndarray``):
1-D profile of volumetric number densities in function of radius. Unit
has to be 1 / length ** 3, where length is the same unit as
``r_from_planet``.
velocity_profile (``numpy.ndarray``):
1-D profile of velocities in function of radius. Unit has to be
m / s.
central_wavelength (``float`` or ``array_like``):
Central wavelength of the transition in unit of m. If more than one line
is to be calculated, then input this parameter as an array-like object.
oscillator_strength (``float``):
Oscillator strength of the transition. The format or shape of this input
parameter needs to be consistent with that of ``central_wavelength``.
einstein_coefficient (``float``):
Einstein coefficient of the transition in 1 / s. The format or shape of
this input parameter needs to be consistent with that of
``central_wavelength``.
wavelength_grid (``numpy.ndarray``):
Wavelengths to calculate the profile in unit of m.
gas_temperature (``float``):
Gas temperature in K.
particle_mass (``float``):
Mass of the particle corresponding to the transition in unit of kg.
bulk_los_velocity (``float``, optional):
Bulk velocity of the gas cell in the line of sight in unit of m / s.
Default is 0.0.
planet_radial_velocity (``float``, optional):
Radial velocity (i.e., the component in the line of sight) of the planet
in relation to the rest frame of the host star and in unit of m / s.
Default is 0.0.
wind_broadening_method (``str``, optional):
Method of calculation for the wind broadening. There are two options:
1) ``'formal'``: the formal definition of radiative transfer taking into
account the full dimensionality of the wind (slower);
2) ``'average'``: assumes the Parker wind broadening contributes to the
Gaussian term of the Voigt profile with an additive factor proportional
to the square of the density-averaged, line-of-sight velocity (faster).
turbulence_broadening (``bool``, optional):
If ``True``, adds a turbulence broadening, defined as in Lampón et al.
2020, to the Gaussian term of the Voigt profile. Default is ``False``.
Returns
-------
intensity (``numpy.ndarray``):
Absorbed intensity profile in function of ``wavelength_grid``.
"""
# First, calculate the optical depth in function of radial distance from the
# planet and the wavelength
optical_depth_profile = optical_depth_2d(
radius_profile, density_profile, velocity_profile, central_wavelength,
oscillator_strength, einstein_coefficient, wavelength_grid,
gas_temperature, particle_mass, bulk_los_velocity,
planet_radial_velocity, wind_broadening_method, turbulence_broadening
)
# Now we interpolate the optical depths to each radius from the planet
f = interp1d(radius_profile, optical_depth_profile, axis=0,
bounds_error=False, fill_value=0.0)
optical_depth_array = f(r_from_planet)
# A bit of array-manipulation magic here to allow us to properly broadcast
intensity_0 = np.reshape(intensity_0, np.shape(intensity_0) + (1,))
# Finally calculate the radiative transfer
intensity = np.sum(intensity_0 * np.exp(-optical_depth_array), axis=(0, 1))
return intensity
# Density and velocity profiles in the line of sight
[docs]def profile_los(radius_profile, density_profile, velocity_profile,
z_grid_size=200):
"""
Calculate the profiles of radius and line-of-sight velocities in function of
sky-projected radial distance from the planet (axis 0) and the line of sight
direction (axis 1).
Parameters
----------
radius_profile (``numpy.ndarray``):
1-D profile of radii in which the densities are sampled, in whatever
unit you want to work with.
density_profile (``numpy.ndarray``):
1-D profile of volumetric number densities in function of radius, in
whatever unit you want to work with.
velocity_profile (``numpy.ndarray``):
1-D profile of velocities in function of radius, in whatever unit you
want to work with.
Returns
-------
los_density_r_z (``numpy.ndarray``):
2-D map of densities in the x- and z-axis distances from the center of
the planet.
los_velocity_r_z (``numpy.ndarray``):
2-D map of line-of-sight velocities in the x- and z-axis distances from
the center of the planet.
"""
# Create a two-dimensional array that measures the distance from the planet
# The second dimension is the line-of-sight direction.
r_top_atm = radius_profile[-1]
los_z = np.linspace(-r_top_atm, r_top_atm, z_grid_size)
# Maybe in the future implement z in log-space instead of linear space
coordinates = np.array(np.meshgrid(radius_profile, los_z))
distances = np.sum(coordinates ** 2, axis=0) ** 0.5
# The line-of-sight wind velocity is given by Eq. 6 in Seidel et al.
# (2020)
# (https://ui.adsabs.harvard.edu/abs/2020A%26A...633A..86S/abstract)
f_v = interp1d(radius_profile, velocity_profile, bounds_error=False,
fill_value=0.0)
v_vertical = f_v(distances)
# We need to add one dimension to `z` to allow for proper array broadcasting
z_expanded = np.expand_dims(los_z, axis=1)
los_velocity_r_z = v_vertical * z_expanded / \
(z_expanded ** 2 + radius_profile ** 2) ** 0.5
# Calculate the line-of-sight density
d_v = interp1d(radius_profile, density_profile, bounds_error=False,
fill_value=0.0)
los_density_r_z = d_v(distances)
return los_density_r_z, los_velocity_r_z, los_z
# Optical depth in function of cylindrical radius from the planet and
# wavelength. Hold on to your hat because this code is very complex.
[docs]def optical_depth_2d(radius_profile, density_profile, velocity_profile,
central_wavelength, oscillator_strength,
einstein_coefficient, wavelength_grid, gas_temperature,
particle_mass, bulk_los_velocity=0.0,
planet_radial_velocity=0.0,
wind_broadening_method='average',
turbulence_broadening=False):
"""
Calculate the optical depth in function of cylindrical radius from the
planet and the wavelength.
Parameters
----------
radius_profile (``numpy.ndarray``):
1-D profile of radii in which the densities are sampled. Unit has to be
consistent with the other input parameters involving lenghts and
densities.
density_profile (``numpy.ndarray``):
1-D profile of volumetric number densities in function of radius. Unit
has to be 1 / length ** 3, where length is the same unit as
``radius_profile``.
velocity_profile (``numpy.ndarray``):
1-D profile of velocities in function of radius. Unit has to be
m / s.
central_wavelength (``float`` or ``array_like``):
Central wavelength of the transition in unit of m. If more than one line
is to be calculated, then input this parameter as an array-like object.
oscillator_strength (``float``):
Oscillator strength of the transition. The format or shape of this input
parameter needs to be consistent with that of ``central_wavelength``.
einstein_coefficient (``float``):
Einstein coefficient of the transition in 1 / s. The format or shape of
this input parameter needs to be consistent with that of
``central_wavelength``.
wavelength_grid (``numpy.ndarray``):
Wavelengths to calculate the profile in unit of m.
gas_temperature (``float``):
Gas temperature in K.
particle_mass (``float``):
Mass of the particle corresponding to the transition in unit of kg.
bulk_los_velocity (``float``, optional):
Bulk velocity of the gas cell in the line of sight in unit of m / s.
Default is 0.0.
planet_radial_velocity (``float``, optional):
Radial velocity (i.e., the component in the line of sight) of the planet
in relation to the rest frame of the host star and in unit of m / s.
Default is 0.0.
wind_broadening_method (``str``, optional):
Method of calculation for the wind broadening. There are two options:
1) ``'formal'``: the formal definition of radiative transfer taking into
account the full dimensionality of the wind (slower);
2) ``'average'``: assumes the Parker wind broadening contributes to the
Gaussian term of the Voigt profile with an additive factor proportional
to the square of the density-averaged, line-of-sight velocity (faster).
turbulence_broadening (``bool``, optional):
If ``True``, adds a turbulence broadening, defined as in Lampón et al.
2020, to the Gaussian term of the Voigt profile. Only used if
``wind_broadening_method`` is set to ``'average'``. Default is ``False``.
Returns
-------
optical_depth_array (``numpy.ndarray``):
Optical depth in function of radial distance from the center of the
planet (axis 0) and in function of wavelength (axis 1).
"""
# Calculate density and velocity profiles in the line of sight
density_los, velocity_los, z_los = \
profile_los(radius_profile, density_profile, velocity_profile)
spatial_shape = np.shape(density_los)
# Spectral line properties
w0 = central_wavelength # Reference wavelength in m
f = oscillator_strength # Unit-less
a_ij = einstein_coefficient # In unit of 1 / s
# Transform `w0`, `f` and `a_ij` into arrays if they are not already arrays
if isinstance(w0, float):
w0 = np.array([w0, ])
f = np.array([f, ])
a_ij = np.array([a_ij, ])
elif isinstance(w0, list):
w0 = np.array(w0)
f = np.array(f)
a_ij = np.array(a_ij)
elif isinstance(w0, np.ndarray):
pass
else:
raise ValueError('The spectral line properties must be float, list or'
' numpy.ndarray, and their format must be consistent'
' with each other.')
# Some necessary book-keeping here
n_lines = len(w0)
wl_grid = wavelength_grid
c_speed = 2.99792458e+08 # Speed of light in m / s
k_b = 1.380649e-23 # Boltzmann's constant in J / K
nu0 = c_speed / w0 # Reference frequency in Hz
nu_grid_rest = c_speed / wl_grid
v_bulk = bulk_los_velocity
rv_planet = planet_radial_velocity
temp = gas_temperature
mass = particle_mass
# Change the shape of `nu_grid_rest` to allow for proper array broadcasting
nu_grid_rest = np.reshape(nu_grid_rest, (len(nu_grid_rest), 1))
# For each line we calculate the grid of frequency shifts from the central
# frequency
delta_nu = (nu_grid_rest - nu0).T
# Cross-section
k = 2.654008854574474e-06 # Physical constant in units of m ** 2 * Hz
sigma = k * f
# This is convoluted, but we create a nested function to calculate the
# optical depth divided by the cross-section (i.e. normalized optical depth)
def _normalized_optical_depth(delta_nu_grid, nu0_k, a_ij_k,
_method=wind_broadening_method):
# Calculate the Lorentzian width of the Voigt profile
gamma = a_ij_k / 4 / np.pi
# Formal calculation of optical depth (slower)
if _method == 'formal':
# Calculate Doppler width (standard deviation) of the Voigt profile
alpha_nu = \
nu0_k / c_speed * (2 * k_b * temp / mass) ** 0.5
# Calculate the frequency shifts due to wind and bulk motion
delta_nu_wind = (velocity_los + v_bulk +
rv_planet) / c_speed * nu0_k
delta_nu_add = np.reshape(delta_nu_wind, spatial_shape + (1,))
# Faster calculation of optical depth: We assume that the Parker wind
# broadening has a Gaussian shape. To this end, we take a wind velocity
# and add it quadratically to the Gaussian broadening term of the Voigt
# profile.
elif _method == 'average':
# Calculate the wind broadening velocity as the density-averaged
# line-of-sight velocity of the Parker wind
weights = density_los
wind_broadening_velocity = \
(np.sum(velocity_los ** 2 * weights) /
np.sum(weights)) ** 0.5
if turbulence_broadening is True:
# Similar to Lampon et al. (2020), calculate the broadening
# as the turbulent velocity = sqrt(5/3 * kT / m)
turbulence_velocity = np.sqrt(5 / 3 * k_b * temp / mass)
else:
turbulence_velocity = 0.0
# Calculate Doppler width of the Voigt profile
alpha_nu = nu0_k / c_speed * (2 * k_b * temp / mass +
wind_broadening_velocity ** 2 +
turbulence_velocity ** 2) ** 0.5
# Frequency shift due to bulk line-of-sight velocity (not to be
# confused with the Parker wind velocity).
delta_nu_add = (v_bulk + rv_planet) / c_speed * nu0_k
else:
raise ValueError('The chosen ``wind_broadening_method`` is not '
'implemented.')
# Finally calculate the Voigt profiles
profiles = voigt_profile(delta_nu_grid + delta_nu_add, alpha_nu,
gamma)
# Calculate the optical depths divided by the cross section
density_expanded = np.expand_dims(density_los, axis=-1)
opt_depth_over_cross_section_r_nu = \
simps(profiles * density_expanded, z_los, axis=0)
return opt_depth_over_cross_section_r_nu
# Calculate the optical depth for each line
optical_depth_list = []
for i in range(n_lines):
norm_opt_depth = _normalized_optical_depth(
delta_nu_grid=delta_nu[i],
nu0_k=nu0[i],
a_ij_k=a_ij[i]
)
optical_depth_i = norm_opt_depth * sigma[i]
optical_depth_list.append(optical_depth_i)
optical_depth_array = np.array(optical_depth_list)
optical_depth_array = np.sum(optical_depth_array, axis=0)
return optical_depth_array
```