helium

helium.radiative_processes(spectrum_at_planet)[source]

Calculate the photoionization rate of helium at null optical depth based on the EUV spectrum arriving at the planet.

Parameters
spectrum_at_planet (``dict``):

Spectrum of the host star arriving at the planet covering fluxes at least up to the wavelength corresponding to the energy to ionize helium (4.8 eV, or 2593 Angstrom).

Returns
phi_1 (float):

Ionization rate of helium singlet at null optical depth in unit of 1 / s.

phi_3 (float):

Ionization rate of helium triplet at null optical depth in unit of 1 / s.

a_1 (float):

Flux-averaged photoionization cross-section of helium singlet in unit of cm ** 2.

a_3 (float):

Flux-averaged photoionization cross-section of helium triplet in unit of cm ** 2.

a_h_1 (float):

Flux-averaged photoionization cross-section of hydrogen in the range absorbed by helium singlet in unit of cm ** 2.

a_h_3 (float):

Flux-averaged photoionization cross-section of hydrogen in the range absorbed by helium triplet in unit of cm ** 2.

helium.radiative_processes_mono(flux_euv, flux_fuv, average_euv_photon_wavelength=242.0, average_fuv_photon_wavelength=2348.0)[source]

Calculate the photoionization rate of helium at null optical depth based on the EUV spectrum arriving at the planet.

Parameters
flux_euv (``float``):

Monochromatic extreme-ultraviolet (0 - 504 Angstrom) flux arriving at the planet in units of erg / s / cm ** 2. Attention: notice that this flux_euv is different from the one used for hydrogen, since helium ionization happens at a shorter wavelength.

flux_fuv (``float``):

Monochromatic far- to middle-ultraviolet (911 - 2593 Angstrom) flux arriving at the planet in units of erg / s / cm ** 2.

average_euv_photon_wavelength (``float``):

Average wavelength of EUV photons ionizing the He singlet state, in unit of Angstrom. Default value is 242 Angstrom. The default value is based on a flux-weighted average of the solar spectrum between 0 and 504 Angstrom.

average_fuv_photon_wavelength (``float``):

Average wavelength of FUV-NUV photons ionizing the He triplet state, in unit of Angstrom. Default value is 2348 Angstrom. The default value is based on a flux-weighted average of the solar spectrum between 911 and 2593 Angstrom.

Returns
phi_1 (float):

Ionization rate of helium singlet at null optical depth in unit of 1 / s.

phi_3 (float):

Ionization rate of helium triplet at null optical depth in unit of 1 / s.

a_1 (float):

Flux-averaged photoionization cross-section of helium singlet in unit of cm ** 2.

a_3 (float):

Flux-averaged photoionization cross-section of helium triplet in unit of cm ** 2.

a_h_1 (float):

Flux-averaged photoionization cross-section of hydrogen in the range absorbed by helium singlet in unit of cm ** 2.

a_h_3 (float):

Flux-averaged photoionization cross-section of hydrogen in the range absorbed by helium triplet in unit of cm ** 2.

helium.recombination(temperature)[source]

Calculates the helium singlet and triplet recombination rates for a gas at a certain temperature.

Parameters
temperature (``float``):

Isothermal temperature of the upper atmosphere in unit of Kelvin.

Returns
alpha_rec_1 (float):

Recombination rate of helium singlet in units of cm ** 3 / s.

alpha_rec_3 (float):

Recombination rate of helium triplet in units of cm ** 3 / s.

helium.collision(temperature)[source]

Calculates the helium singlet and triplet collisional population rates for a gas at a certain temperature.

Parameters
temperature (``float``):

Isothermal temperature of the upper atmosphere in unit of Kelvin.

Returns
q_13 (float):

Rate of helium transition from singlet (1^1S) to triplet (2^3S) due to collisions with free electrons in units of cm ** 3 / s.

q_31a (float):

Rate of helium transition from triplet (2^3S) to 2^1S due to collisions with free electrons in units of cm ** 3 / s.

q_31b (float):

Rate of helium transition from triplet (2^3S) to 2^1P due to collisions with free electrons in units of cm ** 3 / s.

big_q_he (float):

Rate of charge exchange between helium singlet and ionized hydrogen in units of cm ** 3 / s.

big_q_he_plus (float):

Rate of charge exchange between ionized helium and atomic hydrogen in units of cm ** 3 / s.

helium.population_fraction(radius_profile, velocity, density, hydrogen_ion_fraction, planet_radius, temperature, h_fraction, speed_sonic_point, radius_sonic_point, density_sonic_point, spectrum_at_planet=None, flux_euv=None, flux_fuv=None, initial_state=array([0.5, 0.5]), relax_solution=False, convergence=0.01, max_n_relax=10, method='odeint', **options_solve_ivp)[source]

Calculate the fraction of helium in singlet and triplet state in the upper atmosphere in function of the radius in unit of planetary radius.

Parameters
radius_profile (``numpy.ndarray``):

Radius in unit of planetary radii.

velocity (``numpy.ndarray``):

Velocities sampled at the values of radius_profile in units of sound speed. Similar to the output of parker.structure().

density (``numpy.ndarray``):

Densities sampled at the values of radius_profile in units of density at the sonic point. Similar to the output of parker.structure().

hydrogen_ion_fraction (``numpy.ndarray``):

Number fraction of H ion over total H in the upper atmosphere in function of radius. Similar to the output of hydrogen.ion_fraction().

planet_radius (``float``):

Planetary radius in unit of Jupiter radius.

temperature (``float``):

Isothermal temperature of the upper atmosphere in unit of Kelvin.

h_fraction (``float``):

Total (ion + neutral) H number fraction of the atmosphere.

speed_sonic_point (``float``):

Speed of sound in the outflow in units of km / s.

radius_sonic_point (``float``):

Radius of the sonic point in unit of Jupiter radius.

density_sonic_point (``float``):

Density at the sonic point in units of g / cm ** 3.

spectrum_at_planet (``dict``, optional):

Spectrum of the host star arriving at the planet covering fluxes at least up to the wavelength corresponding to the energy to populate the helium states (4.8 eV, or 2593 Angstrom). Can be generated using tools.make_spectrum_dict. If None, then flux_euv and flux_fuv must be provided instead. Default is None.

flux_euv (``float``, optional):

Monochromatic extreme-ultraviolet (0 - 1200 Angstrom) flux arriving at the planet in units of erg / s / cm ** 2. If None, then spectrum_at_planet must be provided instead. Default is None.

flux_fuv (``float``, optional):

Monochromatic far- to middle-ultraviolet (1200 - 2600 Angstrom) flux arriving at the planet in units of erg / s / cm ** 2. If None, then spectrum_at_planet must be provided instead. Default is None.

initial_state (``numpy.ndarray``, optional):

The initial state is the y0 of the differential equation to be solved. This array has two items: the initial value of the fractions of singlet and triplet state in the inner layer of the atmosphere. The default value for this parameter is numpy.array([0.5, 0.5]), i.e., fully neutral at the inner layer with 50% in singlet and 50% in triplet states.

relax_solution (``bool``, optional):

The first solution is calculating by initially assuming the entire atmosphere is in neutral state. If True, the solution will be re-calculated in a loop until it converges to a delta_f of 1%, or for a maximum of 10 loops (default parameters). Default is False.

convergence (``float``, optional):

Value of delta_f at which to stop the relaxation of the solution for f_r. Default is 0.01.

max_n_relax (``int``, optional):

Maximum number of loops to perform the relaxation of the solution for f_r. Default is 10.

method (``str``, optional):

If method is 'odeint', then scipy.integrate.odeint() is used instead of scipy.integrate.solve_ivp() to calculate the steady-state distribution of helium. The first seems to be at least twice faster than the second in some situations. Any other method will fallback to an option of solve_ivp() methods. For example, if method is set to 'Radau', then use solve_ivp(method='Radau'). Default is 'odeint'.

**options_solve_ivp:

Options to be passed to the scipy.integrate.solve_ivp() solver. You may want to change the options atol (absolute tolerance; default is 1E-6) or rtol (relative tolerance; default is 1E-3). If you are having numerical issues, you may want to decrease the tolerance by a factor of 10 or 100, or 1000 in extreme cases.

Returns
f_1_r (numpy.ndarray):

Fraction of helium in singlet state in function of radius.

f_3_r (numpy.ndarray):

Fraction of helium in triplet state in function of radius.