# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""A set of standard astronomical equivalencies."""
from __future__ import (absolute_import, division, print_function,
unicode_literals)
# THIRD-PARTY
import numpy as np
# LOCAL
from ..constants import si as _si
from . import si
from . import cgs
from . import astrophys
from . import dimensionless_unscaled
from .core import UnitsError
__all__ = ['parallax', 'spectral', 'spectral_density', 'doppler_radio',
'doppler_optical', 'doppler_relativistic', 'mass_energy',
'brightness_temperature', 'dimensionless_angles',
'logarithmic', 'temperature', 'temperature_energy']
[docs]def dimensionless_angles():
"""Allow angles to be equivalent to dimensionless (with 1 rad = 1 m/m = 1).
It is special compared to other equivalency pairs in that it
allows this independent of the power to which the angle is raised,
and independent of whether it is part of a more complicated unit.
"""
return [(si.radian, None)]
[docs]def logarithmic():
"""Allow logarithmic units to be converted to dimensionless fractions"""
return [
(dimensionless_unscaled, astrophys.dex,
lambda x: np.log10(x), lambda x: 10.**x)
]
[docs]def parallax():
"""
Returns a list of equivalence pairs that handle the conversion
between parallax angle and distance.
"""
return [
(si.arcsecond, astrophys.parsec, lambda x: 1. / x)
]
[docs]def spectral():
"""
Returns a list of equivalence pairs that handle spectral
wavelength, wave number, frequency, and energy equivalences.
Allows conversions between wavelength units, wave number units,
frequency units, and energy units as they relate to light.
There are two types of wave number:
* spectroscopic - :math:`1 / \\lambda` (per meter)
* angular - :math:`2 \\pi / \\lambda` (radian per meter)
"""
hc = _si.h.value * _si.c.value
two_pi = 2.0 * np.pi
inv_m_spec = si.m ** -1
inv_m_ang = si.radian / si.m
return [
(si.m, si.Hz, lambda x: _si.c.value / x),
(si.m, si.J, lambda x: hc / x),
(si.Hz, si.J, lambda x: _si.h.value * x, lambda x: x / _si.h.value),
(si.m, inv_m_spec, lambda x: 1.0 / x),
(si.Hz, inv_m_spec, lambda x: x / _si.c.value,
lambda x: _si.c.value * x),
(si.J, inv_m_spec, lambda x: x / hc, lambda x: hc * x),
(inv_m_spec, inv_m_ang, lambda x: x * two_pi, lambda x: x / two_pi),
(si.m, inv_m_ang, lambda x: two_pi / x),
(si.Hz, inv_m_ang, lambda x: two_pi * x / _si.c.value,
lambda x: _si.c.value * x / two_pi),
(si.J, inv_m_ang, lambda x: x * two_pi / hc, lambda x: hc * x / two_pi)
]
[docs]def spectral_density(wav, factor=None):
"""
Returns a list of equivalence pairs that handle spectral density
with regard to wavelength and frequency.
Parameters
----------
wav : `~astropy.units.Quantity`
`~astropy.units.Quantity` associated with values being converted
(e.g., wavelength or frequency).
Notes
-----
The ``factor`` argument is left for backward-compatibility with the syntax
``spectral_density(unit, factor)`` but users are encouraged to use
``spectral_density(factor * unit)`` instead.
"""
from .core import UnitBase
if isinstance(wav, UnitBase):
if factor is None:
raise ValueError(
'If `wav` is specified as a unit, `factor` should be set')
wav = factor * wav # Convert to Quantity
c_Aps = _si.c.to(si.AA / si.s).value # Angstrom/s
h_cgs = _si.h.cgs.value # erg * s
hc = c_Aps * h_cgs
fla = cgs.erg / si.angstrom / si.cm ** 2 / si.s
fnu = cgs.erg / si.Hz / si.cm ** 2 / si.s
nufnu = cgs.erg / si.cm ** 2 / si.s
lafla = nufnu
photlam = astrophys.photon / (si.cm ** 2 * si.s * si.AA)
photnu = astrophys.photon / (si.cm ** 2 * si.s * si.Hz)
def converter(x):
return x * (wav.to(si.AA, spectral()).value ** 2 / c_Aps)
def iconverter(x):
return x / (wav.to(si.AA, spectral()).value ** 2 / c_Aps)
def converter_fnu_nufnu(x):
return x * wav.to(si.Hz, spectral()).value
def iconverter_fnu_nufnu(x):
return x / wav.to(si.Hz, spectral()).value
def converter_fla_lafla(x):
return x * wav.to(si.AA, spectral()).value
def iconverter_fla_lafla(x):
return x / wav.to(si.AA, spectral()).value
def converter_photlam_fla(x):
return hc * x / wav.to(si.AA, spectral()).value
def iconverter_photlam_fla(x):
return x * wav.to(si.AA, spectral()).value / hc
def converter_photlam_fnu(x):
return h_cgs * x * wav.to(si.AA, spectral()).value
def iconverter_photlam_fnu(x):
return x / (wav.to(si.AA, spectral()).value * h_cgs)
def converter_photlam_photnu(x):
return x * wav.to(si.AA, spectral()).value ** 2 / c_Aps
def iconverter_photlam_photnu(x):
return c_Aps * x / wav.to(si.AA, spectral()).value ** 2
converter_photnu_fnu = converter_photlam_fla
iconverter_photnu_fnu = iconverter_photlam_fla
def converter_photnu_fla(x):
return x * hc * c_Aps / wav.to(si.AA, spectral()).value ** 3
def iconverter_photnu_fla(x):
return x * wav.to(si.AA, spectral()).value ** 3 / (hc * c_Aps)
return [
(fla, fnu, converter, iconverter),
(fnu, nufnu, converter_fnu_nufnu, iconverter_fnu_nufnu),
(fla, lafla, converter_fla_lafla, iconverter_fla_lafla),
(photlam, fla, converter_photlam_fla, iconverter_photlam_fla),
(photlam, fnu, converter_photlam_fnu, iconverter_photlam_fnu),
(photlam, photnu, converter_photlam_photnu, iconverter_photlam_photnu),
(photnu, fnu, converter_photnu_fnu, iconverter_photnu_fnu),
(photnu, fla, converter_photnu_fla, iconverter_photnu_fla)
]
[docs]def doppler_radio(rest):
r"""
Return the equivalency pairs for the radio convention for velocity.
The radio convention for the relation between velocity and frequency is:
:math:`V = c \frac{f_0 - f}{f_0} ; f(V) = f_0 ( 1 - V/c )`
Parameters
----------
rest : `~astropy.units.Quantity`
Any quantity supported by the standard spectral equivalencies
(wavelength, energy, frequency, wave number).
References
----------
`NRAO site defining the conventions <http://www.gb.nrao.edu/~fghigo/gbtdoc/doppler.html>`_
Examples
--------
>>> import astropy.units as u
>>> CO_restfreq = 115.27120*u.GHz # rest frequency of 12 CO 1-0 in GHz
>>> radio_CO_equiv = u.doppler_radio(CO_restfreq)
>>> measured_freq = 115.2832*u.GHz
>>> radio_velocity = measured_freq.to(u.km/u.s, equivalencies=radio_CO_equiv)
>>> radio_velocity # doctest: +FLOAT_CMP
<Quantity -31.209092088877583 km / s>
"""
assert_is_spectral_unit(rest)
ckms = _si.c.to('km/s').value
def to_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
return (restfreq-x) / (restfreq) * ckms
def from_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
voverc = x/ckms
return restfreq * (1-voverc)
def to_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
return (x-restwav) / (x) * ckms
def from_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
return restwav * ckms / (ckms-x)
def to_vel_en(x):
resten = rest.to(si.eV, equivalencies=spectral()).value
return (resten-x) / (resten) * ckms
def from_vel_en(x):
resten = rest.to(si.eV, equivalencies=spectral()).value
voverc = x/ckms
return resten * (1-voverc)
return [(si.Hz, si.km/si.s, to_vel_freq, from_vel_freq),
(si.AA, si.km/si.s, to_vel_wav, from_vel_wav),
(si.eV, si.km/si.s, to_vel_en, from_vel_en),
]
[docs]def doppler_optical(rest):
r"""
Return the equivalency pairs for the optical convention for velocity.
The optical convention for the relation between velocity and frequency is:
:math:`V = c \frac{f_0 - f}{f } ; f(V) = f_0 ( 1 + V/c )^{-1}`
Parameters
----------
rest : `~astropy.units.Quantity`
Any quantity supported by the standard spectral equivalencies
(wavelength, energy, frequency, wave number).
References
----------
`NRAO site defining the conventions <http://www.gb.nrao.edu/~fghigo/gbtdoc/doppler.html>`_
Examples
--------
>>> import astropy.units as u
>>> CO_restfreq = 115.27120*u.GHz # rest frequency of 12 CO 1-0 in GHz
>>> optical_CO_equiv = u.doppler_optical(CO_restfreq)
>>> measured_freq = 115.2832*u.GHz
>>> optical_velocity = measured_freq.to(u.km/u.s, equivalencies=optical_CO_equiv)
>>> optical_velocity # doctest: +FLOAT_CMP
<Quantity -31.20584348799674 km / s>
"""
assert_is_spectral_unit(rest)
ckms = _si.c.to('km/s').value
def to_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
return ckms * (restfreq-x) / x
def from_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
voverc = x/ckms
return restfreq / (1+voverc)
def to_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
return ckms * (x/restwav-1)
def from_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
voverc = x/ckms
return restwav * (1+voverc)
def to_vel_en(x):
resten = rest.to(si.eV, equivalencies=spectral()).value
return ckms * (resten-x) / x
def from_vel_en(x):
resten = rest.to(si.eV, equivalencies=spectral()).value
voverc = x/ckms
return resten / (1+voverc)
return [(si.Hz, si.km/si.s, to_vel_freq, from_vel_freq),
(si.AA, si.km/si.s, to_vel_wav, from_vel_wav),
(si.eV, si.km/si.s, to_vel_en, from_vel_en),
]
[docs]def doppler_relativistic(rest):
r"""
Return the equivalency pairs for the relativistic convention for velocity.
The full relativistic convention for the relation between velocity and frequency is:
:math:`V = c \frac{f_0^2 - f^2}{f_0^2 + f^2} ; f(V) = f_0 \frac{\left(1 - (V/c)^2\right)^{1/2}}{(1+V/c)}`
Parameters
----------
rest : `~astropy.units.Quantity`
Any quantity supported by the standard spectral equivalencies
(wavelength, energy, frequency, wave number).
References
----------
`NRAO site defining the conventions <http://www.gb.nrao.edu/~fghigo/gbtdoc/doppler.html>`_
Examples
--------
>>> import astropy.units as u
>>> CO_restfreq = 115.27120*u.GHz # rest frequency of 12 CO 1-0 in GHz
>>> relativistic_CO_equiv = u.doppler_relativistic(CO_restfreq)
>>> measured_freq = 115.2832*u.GHz
>>> relativistic_velocity = measured_freq.to(u.km/u.s, equivalencies=relativistic_CO_equiv)
>>> relativistic_velocity # doctest: +FLOAT_CMP
<Quantity -31.207467619351537 km / s>
>>> measured_velocity = 1250 * u.km/u.s
>>> relativistic_frequency = measured_velocity.to(u.GHz, equivalencies=relativistic_CO_equiv)
>>> relativistic_frequency # doctest: +FLOAT_CMP
<Quantity 114.79156866993588 GHz>
>>> relativistic_wavelength = measured_velocity.to(u.mm, equivalencies=relativistic_CO_equiv)
>>> relativistic_wavelength # doctest: +FLOAT_CMP
<Quantity 2.6116243681798923 mm>
"""
assert_is_spectral_unit(rest)
ckms = _si.c.to('km/s').value
def to_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
return (restfreq**2-x**2) / (restfreq**2+x**2) * ckms
def from_vel_freq(x):
restfreq = rest.to(si.Hz, equivalencies=spectral()).value
voverc = x/ckms
return restfreq * ((1-voverc) / (1+(voverc)))**0.5
def to_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
return (x**2-restwav**2) / (restwav**2+x**2) * ckms
def from_vel_wav(x):
restwav = rest.to(si.AA, spectral()).value
voverc = x/ckms
return restwav * ((1+voverc) / (1-voverc))**0.5
def to_vel_en(x):
resten = rest.to(si.eV, spectral()).value
return (resten**2-x**2) / (resten**2+x**2) * ckms
def from_vel_en(x):
resten = rest.to(si.eV, spectral()).value
voverc = x/ckms
return resten * ((1-voverc) / (1+(voverc)))**0.5
return [(si.Hz, si.km/si.s, to_vel_freq, from_vel_freq),
(si.AA, si.km/si.s, to_vel_wav, from_vel_wav),
(si.eV, si.km/si.s, to_vel_en, from_vel_en),
]
[docs]def mass_energy():
"""
Returns a list of equivalence pairs that handle the conversion
between mass and energy.
"""
return [(si.kg, si.J, lambda x: x * _si.c.value ** 2,
lambda x: x / _si.c.value ** 2),
(si.kg / si.m ** 2, si.J / si.m ** 2 ,
lambda x: x * _si.c.value ** 2,
lambda x: x / _si.c.value ** 2),
(si.kg / si.m ** 3, si.J / si.m ** 3 ,
lambda x: x * _si.c.value ** 2,
lambda x: x / _si.c.value ** 2),
(si.kg / si.s, si.J / si.s , lambda x: x * _si.c.value ** 2,
lambda x: x / _si.c.value ** 2),
]
[docs]def brightness_temperature(beam_area, disp):
"""
Defines the conversion between Jy/beam and "brightness temperature",
:math:`T_B`, in Kelvins. The brightness temperature is a unit very
commonly used in radio astronomy. See, e.g., "Tools of Radio Astronomy"
(Wilson 2009) eqn 8.16 and eqn 8.19 (these pages are available on `google
books
<http://books.google.com/books?id=9KHw6R8rQEMC&pg=PA179&source=gbs_toc_r&cad=4#v=onepage&q&f=false>`__).
:math:`T_B \equiv S_\\nu / \left(2 k \\nu^2 / c^2 \\right)`
However, the beam area is essential for this computation: the brightness
temperature is inversely proportional to the beam area
Parameters
----------
beam_area : Beam Area equivalent
Beam area in angular units, i.e. steradian equivalent
disp : `~astropy.units.Quantity` with spectral units
The observed `spectral` equivalent `~astropy.units.Unit` (e.g.,
frequency or wavelength)
Examples
--------
Arecibo C-band beam::
>>> import numpy as np
>>> from astropy import units as u
>>> beam_sigma = 50*u.arcsec
>>> beam_area = 2*np.pi*(beam_sigma)**2
>>> freq = 5*u.GHz
>>> equiv = u.brightness_temperature(beam_area, freq)
>>> u.Jy.to(u.K, equivalencies=equiv) # doctest: +FLOAT_CMP
3.526294429423223
>>> (1*u.Jy).to(u.K, equivalencies=equiv) # doctest: +FLOAT_CMP
<Quantity 3.526294429423223 K>
VLA synthetic beam::
>>> bmaj = 15*u.arcsec
>>> bmin = 15*u.arcsec
>>> fwhm_to_sigma = 1./(8*np.log(2))**0.5
>>> beam_area = 2.*np.pi*(bmaj*bmin/fwhm_to_sigma**2)
>>> freq = 5*u.GHz
>>> equiv = u.brightness_temperature(beam_area, freq)
>>> u.Jy.to(u.K, equivalencies=equiv) # doctest: +FLOAT_CMP
7.065788175060084
"""
beam = beam_area.to(si.sr).value
nu = disp.to(si.GHz, spectral())
def convert_Jy_to_K(x_jybm):
factor = (2 * _si.k_B * si.K * nu**2 / _si.c**2).to(astrophys.Jy).value
return (x_jybm / beam / factor)
def convert_K_to_Jy(x_K):
factor = (astrophys.Jy / (2 * _si.k_B * nu**2 / _si.c**2)).to(si.K).value
return (x_K * beam / factor)
return [(astrophys.Jy, si.K, convert_Jy_to_K, convert_K_to_Jy)]
[docs]def temperature():
"""Convert between Kelvin, Celsius, and Fahrenheit here because
Unit and CompositeUnit cannot do addition or subtraction properly.
"""
from .imperial import deg_F
return [
(si.K, si.deg_C, lambda x: x - 273.15, lambda x: x + 273.15),
(si.deg_C, deg_F, lambda x: x * 1.8 + 32.0, lambda x: (x - 32.0) / 1.8),
(si.K, deg_F, lambda x: (x - 273.15) * 1.8 + 32.0,
lambda x: ((x - 32.0) / 1.8) + 273.15)]
[docs]def temperature_energy():
"""Convert between Kelvin and keV(eV) to an equivalent amount."""
return [
(si.K, si.eV, lambda x: x / (_si.e.value / _si.k_B),
lambda x: x * (_si.e.value / _si.k_B))]
def assert_is_spectral_unit(value):
try:
value.to(si.Hz, spectral())
except (AttributeError, UnitsError) as ex:
raise UnitsError("The 'rest' value must be a spectral equivalent "
"(frequency, wavelength, or energy).")