funcs¶

Functions relating observable properties of binary stars and exoplanet systems to their fundamental properties, and vice versa. Also functions related to Keplerian orbits.

Parameters¶

Functions are defined in terms of the following parameters. [1]

a - orbital semi-major axis in solar radii = a_1 + a_2
P - orbital period in mean solar days
Mass - total system mass in solar masses, Mass = m_1 + m_2
ecc - orbital eccentricity
omdeg - longitude of periastron, omega, in _degrees_
sini - sine of the orbital inclination
K - 2.pi.a.sini/(P.sqrt(1-e^2)) = K_1 + K_2
K_1, K_2 - orbital semi-amplitudes in km/s
q - mass ratio = m_2/m_1 = K_1/K_2 = a_1/a_2
f_m - mass function = m_2^3.sini^3/(m_1+m_2)^2 in solar masses

= K_1^3.P/(2.pi.G).(1-e^2)^(3/2)
r_1 - radius of star 1 in units of the semi-major axis, r_1 = R_*/a
rho_1 - mean stellar density = 3.pi/(GP^2(1+q)r_1^3)
rstar - host star radius/semi-major axis, rstar = R_*/a
k - planet/star radius ratio, k = R_planet/R_star
tzero - time of mid-transit (minimum on-sky star-planet separation).
b - impact parameter, b = a.cos(i)/R_star

[1]	Hilditch, R.W., An Introduction to Close Binary Stars, CUP 2001.

Functions¶

pycheops.funcs.a_rsun(P, Mass)¶

Semi-major axis in solar radii

Parameters:	P – orbital period in mean solar days Mass – total mass in solar masses, M
Returns:	a = (G.M.P^2/(4.pi^2))^(1/3) in solar radii

pycheops.funcs.f_m(P, K, ecc=0)¶

Mass function in solar masses

Parameters:	P – orbital period in mean solar days K – semi-amplitude of the spectroscopic orbit in km/s ecc – orbital eccentricity
Returns:	f_m = m_2^3.sini^3/(m_1+m_2)^2 in solar masses

pycheops.funcs.m1sin3i(P, K_1, K_2, ecc=0)¶

Reduced mass of star 1 in solar masses

Parameters:	K_1 – semi-amplitude of star 1 in km/s K_2 – semi-amplitude of star 2 in km/s P – orbital period in mean solar days ecc – orbital eccentricity
Returns:	m_1.sini^3 in solar masses

pycheops.funcs.m2sin3i(P, K_1, K_2, ecc=0)¶

Reduced mass of star 2 in solar masses

Parameters:	K_1 – semi-amplitude of star 1 in km/s K_2 – semi-amplitude of star 2 in km/s P – orbital period in mean solar days ecc – orbital eccentricity
Returns:	m_2.sini^3 in solar masses

pycheops.funcs.asini(K, P, ecc=0)¶

a.sini in solar radii

Parameters:	K – semi-amplitude of the spectroscopic orbit in km/s P – orbital period in mean solar days
Returns:	a.sin(i) in solar radii

pycheops.funcs.rhostar(r_1, P, q=0)¶

Mean stellar density from scaled stellar radius.

Parameters:	r_1 – radius of star in units of the semi-major axis, r_1 = R_*/a P – orbital period in mean solar days q – mass ratio, m_2/m_1
Returns:	Mean stellar density in solar units

pycheops.funcs.K_kms(m_1, m_2, P, sini, ecc)¶

Semi-amplitudes of the spectroscopic orbits in km/s

K = 2.pi.a.sini/(P.sqrt(1-ecc^2))
K_1 = K * m_2/(m_1+m_2)
K_2 = K * m_1/(m_1+m_2)

Parameters:	m_1 – mass of star 1 in solar masses m_2 – mass of star 2 in solar masses P – orbital period in mean solar days sini – sine of the orbital inclination ecc – orbital eccentrcity
Returns:	K_1, K_2 – semi-amplitudes in km/s

pycheops.funcs.m_comp(f_m, m_1, sini)¶

Companion mass in solar masses given mass function and stellar mass

Parameters:	f_m – = K_1^3.P/(2.pi.G).(1-ecc^2)^(3/2) in solar masses m_1 – mass of star 1 in solar masses sini – sine of orbital inclination
Returns:	m_2 = mass of companion to star 1 in solar masses

pycheops.funcs.transit_width(r, k, b, P=1)¶

Total transit duration.

See equation (3) from Seager and Malen-Ornelas, 2003ApJ…585.1038S.

Parameters:	r – R_star/a k – R_planet/R_star b – impact parameter = a.cos(i)/R_star P – orbital period (optional, default P=1)
Returns:	Total transit duration in the same units as P.

pycheops.funcs.t2z(t, tzero, P, sini, rstar, ecc=0, omdeg=90, signFlag=False)¶

Calculate star-planet separation

Parameters:	t – time of observation (scalar or array) tzero – time of inferior conjunction, i.e., mid-transit P – orbital period sini – sine of orbital inclination rstar – scaled stellar radius, R_star/a ecc – eccentricity (optional, default=0) omdeg – longitude of periastron in degrees (optional, default=90) signFlag – set z negative if companion is further away than the star
Returns:	star-planet separation relative to scaled stellar radius
Example:

>>> from pycheops.funcs import t2z
>>> from numpy import linspace
>>> import matplotlib.pyplot as plt
>>> t = linspace(0,1,1000)
>>> sini = 0.999
>>> rstar = 0.1
>>> plt.plot(t, t2z(t,0,1,sini,rstar))
>>> plt.xlim(0,1)
>>> plt.ylim(0,12)
>>> ecc = 0.1
>>> for omdeg in (0, 90, 180, 270):
>>>     plt.plot(t, t2z(t,0,1,sini,rstar,ecc,omdeg))
>>> plt.show()

pycheops.funcs.tzero2tperi(tzero, P, sini, ecc, omdeg)¶

Calculate time of periastron from time of mid-eclipse

Uses the method by Lacy, 1992AJ….104.2213L

Parameters:	tzero – times of mid-eclipse P – orbital period sini – sine of orbital inclination ecc – eccentricity omdeg – longitude of periastron in degrees
Returns:	time of periastron prior to tzero
Example:	>>> tzero = 54321.6789 >>> P = 1.23456 >>> sini = 0.987 >>> ecc = 0.123 >>> omdeg = 89.01 >>> print(tzero2tperi(tzero,P,sini,ecc,omdeg)) 54321.6762764

pycheops.funcs.vrad(t, tzero, P, sini, K, ecc=0, omdeg=90)¶

Calculate radial velocity, V_r, for body in a Keplerian orbit

Parameters:	t – array of input times tzero – time of inferior conjunction, i.e., mid-transit P – orbital period sini – sine of the orbital inclination K – radial velocity semi-amplitude ecc – eccentricity (optional, default=0) omdeg – longitude of periastron in degrees (optional, default=90)
Returns:	V_r in same units as K relative to the barycentre of the binary

funcs¶

Parameters¶

Functions¶

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