Bell_EBM package

Submodules

Bell_EBM.H2_Dissociation_Routines module

Bell_EBM.H2_Dissociation_Routines.cp_H2(T)

Get the isobaric specific heat capacity of H2 as a function of temperature.

Parameters:T (ndarray) – The temperature. Returns:The isobaric specific heat capacity of H2. Return type:ndarray

Bell_EBM.H2_Dissociation_Routines.dDissFracApprox(T, mu=3320.680532597579, std=471.38088012739126)

Calculate the derivative in the dissociation fraction of H2 using an erf approximation.

Parameters:

• T (ndarray) – The temperature.
• mu (float, optional) – The mean for the Gaussian function.
• std (float, optional) – The standard deviation for the Gaussian function.

Returns:

The derivative in the dissociation fraction of H2.

Return type:

ndarray

Bell_EBM.H2_Dissociation_Routines.dDissFracSaha(T, P)

Calculate the derivative of the dissociation fraction of H2 using the Saha Equation.

Parameters:

• T (ndarray) – The temperature.
• P (ndarray) – The pressure

Returns:

The derivative of the dissociation fraction of H2.

Return type:

ndarray

Bell_EBM.H2_Dissociation_Routines.delta_cp_H2(T)

Get the derivative of the isobaric specific heat capacity of H2 as a function of temperature.

Pretty sure cp_H2 should already include this factor…

Parameters:T (ndarray) – The temperature. Returns:The derivative of theisobaric specific heat capacity of H2. Return type:ndarray

Bell_EBM.H2_Dissociation_Routines.dissFracApprox(T, mu=3320.680532597579, std=471.38088012739126)

Calculate the dissociation fraction of H2 using an erf approximation.

Parameters:

• T (ndarray) – The temperature.
• mu (float, optional) – The mean for the error function.
• std (float, optional) – The standard deviation for the error function.

Returns:

The dissociation fraction of H2.

Return type:

ndarray

Bell_EBM.H2_Dissociation_Routines.dissFracSaha(T, P)

Calculate the dissociation fraction of H2 using the Saha Equation.

Parameters:

• T (ndarray) – The temperature.
• P (ndarray) – The pressure

Returns:

The dissociation fraction of H2.

Return type:

ndarray

Bell_EBM.H2_Dissociation_Routines.getSahaApproxParams(P=10132.5)

Get the Gaussian and erf parameters used to approximate the Saha equation.

Parameters:P (ndarray) – The pressure. Returns:2 floats containing the mean and the standard deviatio for the Gaussian/erf functions. Return type:list

Bell_EBM.H2_Dissociation_Routines.nQ(mu, T)

Calculate the quantum concentration.

Parameters:

• mu (ndarray) – The mean molecular weight in units of u.
• T (ndarray) – The temperature.

Returns:

The quantum concentration.

Return type:

ndarray

Bell_EBM.H2_Dissociation_Routines.true_cp(T, mu=3320.680532597579, std=471.38088012739126)

Get the isobaric specific heat capacity of an LTE mix of H2+H as a function of temperature.

Accounts for the energy of H2 dissociation/recombination.

Parameters:

• T (ndarray) – The temperature.
• mu (float) – The mean for the Gaussian/erf approximations to the Saha equation.
• std (float) – The standard deviation for the Gaussian/erf approximations to the Saha equation.

Returns:

The isobaric specific heat capacity of an LTE mix of H2+H.

Return type:

ndarray

Bell_EBM.KeplerOrbit module

class Bell_EBM.KeplerOrbit.KeplerOrbit(a=149597870700.0, Porb=None, inc=90.0, t0=0.0, e=0.0, Omega=270.0, argp=90.0, obliq=0.0, argobliq=0.0, Prot=None, wWind=0.0, m1=1.9884754153381438e+30, m2=0.0)

Bases: object

A Keplerian orbit.

a

float – The semi-major axis in m.

inc

float – The orbial inclination (in degrees above face-on)

t0

float – The linear ephemeris in days.

e

float – The orbital eccentricity.

Omega

float – The longitude of ascending node (in degrees CCW from line-of-sight).

argp

float – The argument of periastron (in degrees CCW from Omega).

obliq

float, optional – The obliquity (axial tilt) of body 2 (in degrees toward body 1).

argobliq

float, optional – The reference orbital angle used for obliq (in degrees from inferior conjunction).

wWind

float, optional – Body 2’s wind angular velocity in revolutions/s.

t_peri

float – Time of body 2’s closest approach to body 1.

t_ecl

float – Time of body 2’s eclipse by body 1.

mean_motion

float – The mean motion in radians.

FSSI(Y, x, f, fP)

Fast Switch and Spline Inversion method from Tommasini+2018.

Parameters:

• Y (ndarray) – The f(x) values to invert.
• x (ndarray) – x values spanning the domain (more values for higher precision).
• f (callable) – The function f.
• fP (callable) – The first derivative of the function f with respect to x.

Returns:

The numerical approximation of f^-(y).

Return type:

ndarray

FSSI_Eccentric_Inverse(M, xtol=1e-10)

Convert mean anomaly to eccentric anomaly using FSSI (Tommasini+2018).

Parameters:

• M (ndarray) – The mean anomaly in radians.
• xtol (float) – tolarance on error in eccentric anomaly.

Returns:

The eccentric anomaly in radians.

Return type:

ndarray

Newton_Eccentric_Inverse(M, xtol=1e-10)

Convert mean anomaly to eccentric anomaly using Newton.

Parameters:

• M (ndarray) – The mean anomaly in radians.
• xtol (float) – tolarance on error in eccentric anomaly.

Returns:

The eccentric anomaly in radians.

Return type:

ndarray

Porb

float – Body 2’s orbital period in days.

Changing this will update Prot if none was provided when the orbit was initialized.

Prot

float – Body 2’s rotational period in days.

distance(t=None, TA=None, xtol=1e-10)

Find the separation between the two bodies.

Parameters:

• t (ndarray) – The time in days.
• TA (ndarray) – The true anomaly in radians (if t and TA are given, only TA will be used).
• xtol (float) – tolarance on error in eccentric anomaly (calculated along the way).

Returns:

The separation between the two bodies.

Return type:

ndarray

ea_to_ma(ea)

Convert eccentric anomaly to mean anomaly.

Parameters:ea (ndarray) – The eccentric anomaly in radians. Returns:The mean anomaly in radians. Return type:ndarray

eccentric_anomaly(t, useFSSI=None, xtol=1e-10)

Convert time to eccentric anomaly, numerically.

Parameters:

• t (ndarray) – The time in days.
• useFSSI (bool) – Whether or not to use FSSI to invert Kepler’s equation.
• xtol (float) – tolarance on error in eccentric anomaly.

Returns:

The eccentric anomaly in radians.

Return type:

ndarray

get_phase(t, TA=None)

Get the orbital phase.

Parameters:t (ndarray) – The time in days. Returns:The orbital phase. Return type:ndarray

get_sop(t)

Calculate the sub-observer longitude and latitude.

Parameters:t (ndarray) – The time in days. Returns:A list of 2 ndarrays containing the sub-observer longitude and latitude.

Each ndarray is in the same shape as t.

Return type:list

get_ssp(t, TA=None)

Calculate the sub-stellar longitude and latitude.

Parameters:t (ndarray) – The time in days. Returns:A list of 2 ndarrays containing the sub-stellar longitude and latitude.

Each ndarray is in the same shape as t.

Return type:list

m1

float – Body 1’s mass in kg.

If no period was provided when the orbit was initialized, changing this will update the period.

m2

float – Body 2’s mass in kg.

If no period was provided when the orbit was initialized, changing this will update the period.

mean_anomaly(t)

Convert time to mean anomaly.

Parameters:t (ndarray) – The time in days. Returns:The mean anomaly in radians. Return type:ndarray

phase_eclipse

float – The orbital phase of eclipse.

Read-only.

phase_periastron

float – The orbital phase of periastron.

Read-only.

phase_transit

float – The orbital phase of transit.

Read-only.

plot_orbit()

A convenience routine to visualize the orbit

Returns:The figure containing the plot. Return type:figure

solve_period()

Find the Keplerian orbital period.

Returns:The Keplerian orbital period. Return type:float

t_trans

float – Time of body 1’s eclipse by body 2.

Read-only.

ta_to_ea(ta)

Convert true anomaly to eccentric anomaly.

Parameters:ta (ndarray) – The true anomaly in radians. Returns:The eccentric anomaly in radians. Return type:ndarray

ta_to_ma(ta)

Convert true anomaly to mean anomaly.

Parameters:ta (ndarray) – The true anomaly in radians. Returns:The mean anomaly in radians. Return type:ndarray

true_anomaly(t, xtol=1e-10)

Convert time to true anomaly, numerically.

Parameters:

• t (ndarray) – The time in days.
• xtol (float) – tolarance on error in eccentric anomaly (calculated along the way).

Returns:

The true anomaly in radians.

Return type:

ndarray

xyz(t, xtol=1e-10)

Find the coordinates of body 2 with respect to body 1.

Parameters:

• t (ndarray) – The time in days.
• xtol (float) – tolarance on error in eccentric anomaly (calculated along the way).

Returns:

A list of 3 ndarrays containing the x,y,z coordinate of body 2 with respect to body 1.

The x coordinate is along the line-of-sight. The y coordinate is perpendicular to the line-of-sight and in the orbital plane. The z coordinate is perpendicular to the line-of-sight and above the orbital plane

Return type:

list

Bell_EBM.Map module

class Bell_EBM.Map.Map(values=None, time=0.0, nlat=16, nlon=None, useHealpix=False, nside=7)

Bases: object

A map.

lat

ndarray, optional – The unique latitude values in degrees.

latGrid

ndarray – The latitude grid in degrees.

lon

ndarray, optional – The unique longitude values in degrees.

lonGrid

ndarray – The longitude grid in degrees.

nlat

int, optional – The number of latitudinal cells to use for rectangular maps.

nlon

int, optional – The number of longitudinal cells to use for rectangular maps.

nside

int, optional – A parameter that sets the resolution of healpy maps.

pixArea

ndarray – The area of each pixel.

time

float – Time of map in days.

useHealpix

bool – Whether the planet’s map uses a healpix grid.

values

ndarray – The temperature map values.

load_custom_map(values, time=None, lat=None, lon=None, latGrid=None, lonGrid=None, pixArea=None)

Set the whole map object.

Parameters:

• values (ndarray) – The map temperatures (in K) with a size of self.npix.
• time (float, optional) – Time of map in days.
• lat (ndarray, optional) – The unique latitude values in degrees.
• lon (ndarray, optional) – The unique longitude values in degrees.
• latGrid (ndarray, optional) – The latitude of every pixel in degrees.
• lonGrid (ndarray, optional) – The longitude of every pixel in degrees.
• pixArea (ndarray, optional) – The angular area of each pixel in steradians.

plot_H2_dissociation(refLon=None)

A convenience routine to plot the H2 dissociation map.

Parameters:refLon (float, optional) – The sub-stellar longitude used to de-rotate the map. Returns:The figure containing the plot. Return type:figure

plot_map(refLon=None)

A convenience routine to plot the temperature map

Parameters:refLon (float, optional) – The sub-stellar longitude used to de-rotate the map. Returns:The figure containing the plot. Return type:figure

set_values(values, time=None)

Set the temperature map.

Parameters:

• values (ndarray) – The map temperatures (in K) with a size of self.npix.
• time (float, optional) – Time of map in days.

Bell_EBM.Planet module

class Bell_EBM.Planet.Planet(plType='gas', rad=71492000.0, mass=1.8981871658715508e+27, a=4487936121.0, Porb=None, Prot=None, inc=90.0, t0=0.0, e=0.0, Omega=270.0, argp=90, obliq=0.0, argobliq=0.0, vWind=0.0, albedo=0.0, cp=None, cpParams=None, mlDepth=None, mlDensity=None, T_exponent=4.0, emissivity=1.0, trasmissivity=0.0, nlat=16, nlon=None, useHealpix=False, nside=7)

Bases: object

A planet.

albedo

float – The planet’s Bond albedo.

cpParams

iterable, optional – Any parameters to be passed to cp if using the bell2018 LTE H2+H mix cp

C

float, optional – The planet’s heat capacity in J/m^2/K.

emissivity

float – The emissivity of the emitting layer (between 0 and 1).

g

float – The planet’s surface gravity in m/s^2.

map

Bell_EBM.Map – The planet’s temperature map.

orbit

Bell_EBM.KeplerOrbit – The planet’s orbit.

plType

str – The planet’s composition.

trasmissivity

float – The trasmissivity of the emitting layer (between 0 and 1).

T_exponent

float – The exponent which determinges the rate at which the planet cools (4 for blackbody cooling, 1 for Newtonian cooling) when calculating Fout with bolo=True.

useHealpix

bool – Whether the planet’s map uses a healpix grid.

Fout(T=None, bolo=True, wav=1e-06)

Calculate the instantaneous outgoing flux.

Parameters:

• T (ndarray) – The temperature (if None, use self.map.values).
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• wav (float, optional) – The wavelength to use if bolo==False.

Returns:

The emitted flux in the same shape as T.

Return type:

ndarray

Fp_vis(t, T=None, bolo=True, wav=4.5e-06)

Calculate apparent outgoing planetary flux (used for making phasecurves).

Weight flux by visibility/illumination kernel, assuming the star/observer are infinitely far away for now.

Parameters:

• t (ndarray) – The time in days.
• T (ndarray) – The temperature (if None, use self.map.values).
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• wav (float, optional) – The wavelength to use if bolo==False

Returns:

The apparent emitted flux. Has shape (t.size, self.map.npix).

Return type:

ndarray

Porb

float – The planet’s orbital period in days.

Prot

float – The planet’s rotational period in days.

absorptivity

float – The fraction of flux absorbed by the planet.

Read-only.

cp

float or callable – The planet’s isobaric specific heat capacity in J/kg/K.

Changing the planet’s cp may update planet.C

mass

float – The planet’s mass in kg.

Changing the planet’s mass will update planet.g and possibly planet.mlDensity and planet.C

mlDensity

float – The density of the planet’s mixed layer

In kg/m^3 if rock/water models or the inverse of the planet’s surface gravity (in s^2/m) for gas/bell2018 models.

Changing the planet’s mlDensity may update planet.C

mlDepth

float – The depth of the planet’s mixed layer.

In units of m for rock/water models, or Pa for gas/bell2018 models.

Changing the planet’s mlDepth may update planet.C

plot_H2_dissociation(tempMap=None, time=None)

A convenience routine to plot the planet’s H2 dissociation map.

Parameters:

• tempMap (ndarray, optional) – The temperature map (if None, use self.map.values).
• time (float, optional) – The time corresponding to the map used to de-rotate the map.

Returns:

The figure containing the plot.

Return type:

figure

plot_map(tempMap=None, time=None)

A convenience routine to plot the planet’s temperature map.

Parameters:

• tempMap (ndarray) – The temperature map (if None, use self.map.values).
• time (float, optional) – The time corresponding to the map used to de-rotate the map.

Returns:

The figure containing the plot.

Return type:

figure

rad

float – The planet’s radius if m.

Changing the planet’s radius will update planet.g and possibly planet.mlDensity and planet.C

t0

float – The planet’s linear ephemeris in days.

weight(t, TA=None, refPos='SSP')

Calculate the weighting of map pixels.

Weight flux by visibility/illumination kernel, assuming the star/observer are infinitely far away for now.

Parameters:

• t (ndarray) – The time in days.
• EA (ndarray, optional) – The eccentric anomaly in radians.
• refPos (str, optional) – The reference position; SSP (sub-stellar point) or SOP (sub-observer point).

Returns:

The weighting of map mixels at time t. Has shape (t.size, self.map.npix).

Return type:

ndarray

Bell_EBM.Star module

class Bell_EBM.Star.Star(teff=5778.0, rad=1.0, mass=1.0)

Bases: object

A star.

teff

float – The star’s effective temperature in K.

rad

float – The star’s radius in solar radii.

mass

float – The star’s mass in solar masses.

Fstar(bolo=True, tBright=None, wav=4.5e-06)

Calculate the stellar flux for lightcurve normalization purposes.

Parameters:

• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tBright (ndarray) – The brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.

Returns:

The emitted flux in the same shape as T.

Return type:

ndarray

Bell_EBM.StarPlanetSystem module

class Bell_EBM.StarPlanetSystem.System(star=None, planet=None)

Bases: object

A Star+Planet System.

star

Bell_EBM.Star – The host star.

planet

Bell_EBM.Planet – The planet.

Fin(t=0, TA=None, bolo=True, tStarBright=None, wav=4.5e-06)

Calculate the instantaneous incident flux.

Parameters:

• t (ndarray, optional) – The time in days.
• TA (ndarray, optional) – The true anomaly in radians.
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tStarBright (ndarray) – The stellar brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.

Returns:

The instantaneous incident flux.

Return type:

ndarray

Firr(t=0.0, TA=None, bolo=True, tStarBright=None, wav=4.5e-06)

Calculate the instantaneous irradiation.

Parameters:

• t (ndarray, optional) – The time in days.
• TA (ndarray, optional) – The true anomaly in radians.
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tStarBright (ndarray) – The stellar brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.

Returns:

The instantaneous irradiation.

Return type:

ndarray

ODE(t, T, TA=None)

The derivative in temperature with respect to time.

Used by scipy.integrate.ode to update the map

Parameters:

• t (ndarray) – The time in days.
• T (ndarray) – The temperature map with shape (self.planet.map.npix).
• TA (ndarray, optional) – The true anomaly in radians (much faster to compute if provided).

Returns:

The derivative in temperature with respect to time.

Return type:

ndarray

get_phase(t)

Get the orbital phase.

Parameters:t (ndarray) – The time in days. Returns:The orbital phase. Return type:ndarray

get_phase_eclipse()

Get the orbital phase of eclipse.

Returns:The orbital phase of eclipse. Return type:float

get_phase_periastron()

Get the orbital phase of periastron.

Returns:The orbital phase of periastron. Return type:float

get_phase_transit()

Get the orbital phase of transit.

Returns:The orbital phase of transit. Return type:float

get_teq(t=0)

Get the planet’s equilibrium temperature.

Parameters:t (ndarray, optional) – The time in days. Returns:The planet’s equilibrium temperature at time(s) t. Return type:ndarray

get_tirr(t=0.0)

Get the planet’s irradiation temperature.

Parameters:t (ndarray, optional) – The time in days. Returns:The planet’s irradiation temperature at time(s) t. Return type:ndarray

invert_lc(fp_fstar, bolo=True, tStarBright=None, wav=4.5e-06)

Invert the fp/fstar phasecurve into an apparent temperature phasecurve.

Parameters:

• fp_fstar (ndarray) – The observed planetary flux normalized by the stellar flux.
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tBright (ndarray) – The brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.

Returns:

The apparent, disk-integrated temperature.

Return type:

ndarray

lightcurve(t=None, T=None, bolo=True, tStarBright=None, wav=4.5e-06, allowReflect=True, allowThermal=True)

Calculate the planet’s lightcurve (ignoring any occultations).

Parameters:

• t (ndarray, optional) – The time in days. If None, will use 1000 time steps around orbit.
• T (ndarray, optional) – The temperature map (either shape (1, self.planet.map.npix) and constant over time or shape is (t.shape, self.planet.map.npix). If None, use self.planet.map.values instead (default).
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tStarBright (ndarray) – The stellar brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.
• allowReflect (bool, optional) – Account for the contribution from reflected light.
• allowThermal (bool, optional) – Account for the contribution from thermal emission.

Returns:

The observed planetary flux normalized by the stellar flux.

Return type:

ndarray

plot_lightcurve(t=None, T=None, bolo=True, tStarBright=None, wav=4.5e-06, allowReflect=True, allowThermal=True)

A convenience plotting routine to show the planet’s phasecurve.

Parameters:

• t (ndarray, optional) – The time in days with shape (t.size,1). If None, will use 1000 time steps around orbit.
• T (ndarray, optional) – The temperature map in K with shape (1, self.planet.map.npix) if the map is constant or (t.size,self.planet.map.npix). If None, use self.planet.map.values instead.
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tBright (ndarray) – The brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.
• allowReflect (bool, optional) – Account for the contribution from reflected light.
• allowThermal (bool, optional) – Account for the contribution from thermal emission.

Returns:

The figure containing the plot.

Return type:

figure

plot_tempcurve(t=None, T=None, bolo=True, tStarBright=None, wav=4.5e-06, allowReflect=False, allowThermal=True)

A convenience plotting routine to show the planet’s phasecurve in units of temperature.

Parameters:

• t (ndarray, optional) – The time in days with shape (t.size,1). If None, will use 1000 time steps around orbit.
• T (ndarray, optional) – The temperature map in K with shape (1, self.planet.map.npix) if the map is constant or (t.size,self.planet.map.npix). If None, use self.planet.map.values instead.
• bolo (bool, optional) – Determines whether computed flux is bolometric (True, default) or wavelength dependent (False).
• tBright (ndarray) – The brightness temperature to use if bolo==False.
• wav (float, optional) – The wavelength to use if bolo==False.
• allowReflect (bool, optional) – Account for the contribution from reflected light.
• allowThermal (bool, optional) – Account for the contribution from thermal emission.

Returns:

The figure containing the plot.

Return type:

figure

run_model(T0=None, t0=0.0, t1=None, dt=None, verbose=True, intermediates=False)

Evolve the planet’s temperature map with time.

Parameters:

• T0 (ndarray) – The initial temperature map with shape (self.planet.map.npix). If None, use self.planet.map.values instead (default).
• t0 (float, optional) – The time corresponding to T0 (default is 0).
• t1 (float, optional) – The end point of the run (default is 1 orbital period later).
• dt (float, optional) – The time step used to evolve the map (default is 1/100 of the orbital period).
• verbose (bool, optional) – Output comments of the progress of the run (default = False)?
• intermediates (bool, optional) – Output the map from every time step? Otherwise just returns the last step.

Returns:

A list of 2 ndarrays containing the time and map of each time step.

Return type:

list

Module contents