def transitDuration_Rs(sma, rprs, inc, period, exact=False):
"""
Calculate transit duration
Invokes :py:func:`transitDuration` after parameter transformation.
Parameters
----------
sma : float
Semi-major axis [stellar radii]
rprs : float
Planet-to-star radius ratio (Rp/Rs)
inc : float
Orbital inclination [deg]
period : float
Orbital period
exact : boolean, optional
If True, a slightly more accurate expression is evaluated.
The default is False.
Returns
-------
Transit duration : float
The duration of the transit (same units as `period`).
"""
from PyAstronomy import constants as PC
c = PC.PyAConstants()
rs = c.RSun
return transitDuration(sma*rs/c.AU, rprs*rs/c.RJ, 1., inc, period, exact)
def transitDuration(sma, rp, rs, inc, period, exact=False):
"""
Calculate the transit duration.
The routine calculates the transit duration assuming
a spherical star and planet and a circular planetary orbit.
It evaluates the expression
.. math::
T_D = \\frac{P}{\\pi} \\arcsin\\left(\\frac{\\sqrt{(R_s+R_p)^2 - b^2}}{a} \\right)
where P is the orbital period, b the impact parameter (:math:`b=a \\cos(i)`),
and a the semi-major axis.
If the `exact` flag is set True, the slightly more accurate expression
.. math::
T_D = \\frac{P}{\\pi} \\arcsin\\left(\\frac{\\sqrt{(R_s+R_p)^2 - b^2}}{a \\sin(i)} \\right)
is evaluated, where the additional sine term accounts for orbit curvature.
.. note:: The units of the transit duration are the same as the units
of the input orbital period.
Parameters
----------
sma : float
The semi-major axis in AU.
rp : float
The planetary radius in Jovian radii.
rs : float
The stellar radius in solar radii.
inc : float
The orbital inclination in degrees.
period : float
The orbital period.
exact : boolean, optional
If True, a slightly more accurate expression is evaluated.
The default is False.
Returns
-------
Transit duration : float
The duration of the transit (same units as `period`).
"""
from PyAstronomy import constants as PC
c = PC.PyAConstants()
# Calculate the impact parameter
impact = (sma * c.AU) * np.cos(inc / 180. * np.pi)
# Correction factor (inclination dependent factor)
cfac = 1.0 if not exact else np.sin(inc / 180. * np.pi)
# Calculate the geometric transit duration
dur = (period / np.pi) * \
np.arcsin(np.sqrt((rs * c.RSun + rp * c.RJ)
** 2 - impact**2) / (sma * c.AU * cfac))
return dur
def ingressDuration_Rs(sma, rprs, inc, period):
"""
Calculate transit duration
Invokes :py:func:`transitDuration` after parameter transformation.
Parameters
----------
sma : float
Semi-major axis [stellar radii]
rprs : float
Planet-to-star radius ratio (Rp/Rs)
inc : float
Orbital inclination [deg]
period : float
Orbital period
Returns
-------
Transit duration : float
The duration of the transit (same units as `period`).
"""
from PyAstronomy import constants as PC
c = PC.PyAConstants()
rs = c.RSun
return ingressDuration(sma*rs/c.AU, rprs*rs/c.RJ, 1., inc, period)
def planck(T, lam=None, nu=None):
"""
Evaluate Planck's radiation law.
Depending on whether wavelength or frequency is specified as input, the
function evaluates:
.. math::
B_{\\nu} = \\frac{2\\pi h \\nu^3}{c^2} \\frac{1}{e^{\\frac{h\\nu}{kT}} - 1}
or
.. math::
B_{\\lambda} = \\frac{2\\pi h c^2}{\\lambda^5} \\frac{1}{e^{\\frac{h c}{\\lambda kT}} - 1} \\; .
If lambda is given (in meters), the output units are W/(m^2 m). To convert into erg/(cm^2 A s),
the output has to be multiplied by a factor of 1e-7.
Parameters
----------
T : float
Temperature in Kelvin.
lam : float or array, optional
Wavelength in meters.
nu : float or array, optional
Frequency in Hz.
Returns
-------
Spectral radiance : float or array
Depending on whether `lam` or `nu` were specified, returns the
spectral radiance per area and wavelength or frequency. The unit (SI)
will be W/(m^2 m) if `lam` was given and W/(m^2 Hz) if `nu` was
specified.
"""
if _ic.check["quantities"]:
from PyAstronomy import constants
c = constants.PyAConstants(unitSystem="SI")
else:
raise (PE.PyARequiredImport(
"You need to install 'quantities' to use this function.\n 'quantities' can be obtained from 'http://pypi.python.org/pypi/quantities'."
))
return None
if (lam is not None) and (nu is not None):
raise (PE.PyAParameterConflict(
"Specify either 'lam' OR 'nu', but not both."))
if (lam is None) and (nu is None):
raise (PE.PyAParameterConflict("Specify either 'lam' OR 'nu'."))
# Parameters have been specified properly
if lam is not None:
result = 2. * np.pi * c.h * (c.c**2) / (lam**5)
result /= (np.exp(c.h * c.c / (lam * c.k * T)) - 1.0)
return result
elif nu is not None:
result = 2. * np.pi * c.h * (nu**3) / (c.c**2)
result /= (np.exp(c.h * nu / (c.k * T)) - 1.0)
return result
def thermalBroadeningWidth(lam0, T, m=None, fwhm=True):
"""
Calculate the width for thermal broadening.
Thermal motion of particles causes a Doppler
broadening of the line profile. The resulting
line profile is Gaussian with FWHM given by
.. math::
fwhm = \\lambda_0\\sqrt{\\frac{8k_B T\\ln(2)}{m c^2}}
See, e.g.,
http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/broaden.html
Parameters
----------
lam0 : float
Wavelength at which to calculate the width.
T : float
Temperature [K].
m : float, optional
Mass of the particles [kg]. If not specified,
the proton mass is assumed.
fwhm : boolean, optional
If True (default), the FWHM of the Gaussian
broadening kernel will be returned. Otherwise,
the standard deviation is returned.
Returns
-------
Width : float
The width of the Gaussian broadening kernel.
By default, the FWHM is returned. To obtain
the standard deviation, set the `fwhm` flag to
False.
"""
from PyAstronomy import constants as PC
pc = PC.PyAConstants()
pc.setSystem("SI")
if m is None:
# Use proton mass if not specified otherwise
m = pc.mp
result = lam0 * np.sqrt(8.0 * pc.k * T * np.log(2.0) / (m * (pc.c)**2))
if not fwhm:
return result / (2.0 * np.sqrt(2.0 * np.log(2.0)))
else:
return result
def ingressDuration(sma, rp, rs, inc, period):
"""
Calculate the ingress (and egress) duration of the transit.
The routine calculates the transit in- and egress duration assuming
a spherical star and planet and a circular planetary orbit.
It evaluates the expression
.. math::
T_I = \\frac{P}{2 \\pi} \\left( \\arcsin\\left(\\frac{\\sqrt{(R_s+R_p)^2 - b^2}}{a \\sin(i)} \\right) - \\arcsin\\left(\\frac{\\sqrt{(R_s-R_p)^2 - b^2}}{a \\sin(i)} \\right) \\right)
is evaluated, where b is the impact parameter (:math:`b=a \\cos(i)`).
.. note:: The units of the ingress duration are the same as the units
of the input orbital period.
Parameters
----------
sma : float
The semi-major axis in AU.
rp : float
The planetary radius in Jovian radii.
rs : float
The stellar radius in solar radii.
inc : float
The orbital inclination in degrees.
period : float
The orbital period.
Returns
-------
Ingress duration : float
The duration of in- and egress (same units as `period`).
"""
from PyAstronomy import constants as PC
c = PC.PyAConstants()
# Calculate the impact parameter
impact = (sma * c.AU) * np.cos(inc / 180. * np.pi)
imsin = (sma * c.AU) * np.sin(inc / 180. * np.pi)
# Calculate the geometric transit duration
idur = (period / (2.*np.pi)) * ( \
np.arcsin(np.sqrt((rs * c.RSun + rp * c.RJ)** 2 - impact**2) / imsin) - \
np.arcsin(np.sqrt((rs * c.RSun - rp * c.RJ)** 2 - impact**2) / imsin) \
)
return idur
def transitDuration(sma, rp, rs, inc, period):
"""
Calculate the transit duration.
The routine calculates the transit duration assuming
a circular star and planet and a circular planetary orbit.
In particular, it evaluates:
.. math::
T_D = \\frac{P}{\\pi} \\arcsin\\left(\\frac{\\sqrt{(R_s+R_p)^2 - b^2}}{a} \\right)
where P is the orbital period, b the impact parameter (:math:`b=a/R_s\\cos(i)`),
and a the semi-major axis.
.. note:: The units of the transit duration are the same as the units
of the input orbital period.
Parameters
----------
sma : float
The semi-major axis in AU.
rp : float
The planetary radius in Jovian radii.
rs : float
The stellar radius in solar radii.
inc : float
The orbital inclination in degrees.
period : float
The orbital period.
Returns
-------
Transit duration : float
The duration of the transit (same units as
`period`).
"""
from PyAstronomy import constants as PC
c = PC.PyAConstants()
# Calculate the impact parameter
impact = (sma * c.AU) * np.cos(inc / 180. * np.pi)
# Calculate the geometric transit duration
dur = (period / np.pi) * \
np.arcsin(np.sqrt((rs * c.RSun + rp * c.RJ)
** 2 - impact**2) / (sma * c.AU))
return dur
def tempFromthermalBroadeningWidth(lam0, width, m=None, awidth=0.0):
"""
Calculate the temperature required to obtain thermal broadening width.
Thermal motion of particles causes Doppler
broadening of the line profile. The resulting
line profile is Gaussian with FWHM given by
.. math::
fwhm = \\lambda_0\\sqrt{\\frac{8k_B T\\ln(2)}{m c^2}}
See, e.g.,
http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/broaden.html
Here, the relation is reversed, so that the temperature, T, can be
obtained from the width. The relation is only strictly valid in the case
of Gaussian lines.
.. note:: Units of lam0, width, and awidth have to be consistent.
Parameters
----------
lam0 : float
Wavelength at which to width (FWHM) was measured.
width : float
Measured width of the feature (FWHM)
m : float, optional
Mass of the particles [kg]. If not specified,
the proton mass is assumed.
awidth : float
Additional width contributed by other effects such as
instrumental resolution. The thermal width, ft, is obtained by
quadratic subtraction, i.e., ft**2 = width**2 - awidth**2.
Default is zero.
Returns
-------
Temperature : float
The temperature required to achieve the specified broadening
width.
Thermal width : float
The thermal width used in the calculations (may be modified
if `awidth` has been specified.
"""
from PyAstronomy import constants as PC
pc = PC.PyAConstants()
pc.setSystem("SI")
if m is None:
# Use proton mass if not specified otherwise
m = pc.mp
if awidth > width:
raise(PE.PyAValError("The additional width (awidth) is larger than the specified width of the feature.", \
where="tempFromthermalBroadeningWidth", \
solution="Check width specification."))
# Take into account additional width
width = np.sqrt(width**2 - awidth**2)
result = width**2 / lam0**2 * (m * (pc.c)**2) / (8.0 * pc.k * np.log(2.0))
return result
def __init__(self, ccfs="rutten", afc="middelkoop", rphot="noyes"):
"""
Converting Mount-Wilson S-index into RHK index.
The Mount-Wilson S-index is a measure of the emission-line cores
of the Ca II H and K lines at about 3933 A and 3968 A in two
narrow bands normalized by two adjacent continuum bands.
The activity index RHK is closely related to the S-index. In
particular, it gives the emission in the narrow bands normalized
by the bolometric brightness of the star
.. math::
R_{HK} = \\frac{4\\pi R_s^2 (F_H + F_K)}{4\\pi R_s^2\\sigma T_{eff}^4} = \\frac{F_H+F_K}{\\sigma T_{eff}^4} \\; .
The stellar surface flux in "arbitrary units" in the narrow
H and K bands (fH + fK) is related to the Mount-Wilson S-index
through the relation
.. math::
f_H + f_K = S C_{cf} T_{eff}^4 10^{-14} \\; ,
where Ccf is a conversion factor, which can be parameterized in terms of
the B-V color and the luminosity class. The conversion between
arbitrary units and physical units, needed to derive the true surface
flux and the RHK index, has been derived by several authors
starting with Middelkoop 1982.
Their factor was also used by Noyes et al. 1984---in particular in their
appendix a, where is appears implicitly. Later, the value of the conversion
factor has been revised by several authors, e.g., Oranje 1983 and Rutten 1984,
who estimated a value about 70% larger than previously proposed. Hall et al.
2007 derive a value 40% larger than that of Middelkoop 1982 and provide
a thorough discussion on the differences between the individual approaches
and results given in the literature.
Finally, the RHK index thus derived still covers a photospheric
contribution, which is always present and not related to the
chromosphere. To obtain the purely chromospheric, primed RHK index,
an estimate of the photospheric surface flux in the H and K pass-bands
has to be subtracted. For active stars, the photospheric correction
is usually quite irrelevant. For inactive, quiet stars, it can,
however, be important.
The issue of the Mount-Wilson S-index conversion has been revisited
by Mittag et al. 2013, who provide an alternative conversion procedure
and revised photospheric corrections for various luminosity classes.
.. note:: In the default configuration, the conversion of the S-index
into RHK is identical to the relation stated by Noyes et al. 1984
in their appendix (a)
.. math::
R_{HK} = 1.340 \\times 10^{-4} C_{cf} S
where the factor 1.34e-4 is a combination of the conversion from
arbitrary to physical units, 1e-14, and the Stefan-Boltzmann
constant, in particular 1.34e-4 = 7.6e5*1e-14/5.67e-5. The Ccf factor
is, however, calculated according to Rutten 1984.
The relations and coefficients used here are taken from the
following publications (and references therein):
- Middelkoop 1982, A&A 107, 31
- Oranje 1983, A&A 124, 43
- Noyes et al. 1984, A&A 279, 763
- Rutten 1984, A&A 130, 353
- Hall et al. 2007, AJ 133, 862
- Mittag et al. 2013, A&A 549, 117
Parameters
----------
ccfs : string, {rutten, noyes}, optional
Source of the conversion factor between S-index and RHK.
afc : string, {rutten, oranje, middelkoop, hall}, optional
Source of conversion factor between "arbitrary units"
and physical units of surface flux.
rphot : string, {noyes}
The source for the photospheric correction for the RHK
index.
"""
if ccfs == "rutten":
self._log10ccf = self.log10ccfRutten
elif ccfs == "noyes":
self._log10ccf = self.log10ccfNoyes
else:
raise(PE.PyAValError("No such Ccf source: " + str(ccfs), \
solution="Use 'rutten' or 'noyes'."))
self._ccfs = ccfs
if rphot == "noyes":
self._logrphot = self.logRphotNoyes
else:
raise(PE.PyAValError("No such source for the photospheric correction: " + str(rphot), \
solution="Use 'noyes'."))
self._rphots = rphot
self._afc = afc
# The conversion factor from "arbitrary units" to physical units.
self._absCal = {
"rutten": 1.29e6,
"oranje": 1.21e6,
"middelkoop": 7.6e5,
"hall": 1.07e6
}
if not self._afc in self._absCal:
raise(PE.PyAValError("No such source for the conversion from arbitrary to physical units: " + str(self._afc), \
solution="Use either of: " + ', '.join(self._absCal.keys()) ))
from PyAstronomy.pyasl import _ic
if not _ic.check["quantities"]:
raise(PE.PyARequiredImport("The 'quantities' package is not installed, which is required to use 'SMW_RHK'.", \
where="SMW_RHK", \
solution="Install quantities (https://pypi.python.org/pypi/quantities/0.10.1)."))
from PyAstronomy import constants as PC
self._pc = PC.PyAConstants()
self._pc.setSystem("cgs")
