.. _unit_equivalencies:

Equivalencies
*************

The unit module has machinery for supporting equivalences between
different units in certain contexts, namely when equations can
uniquely relate a value in one unit to a different unit. A good
example is the equivalence between wavelength, frequency, and energy
for specifying a wavelength of radiation. Normally these units are not
convertible, but when understood as representing light, they are
convertible in certain contexts. Here we describe how to use the
equivalencies included in `astropy.units` and how to
define new equivalencies.

Equivalencies are used by passing a list of equivalency pairs to the
``equivalencies`` keyword argument of `Quantity.to()
<astropy.units.quantity.Quantity.to>` `Unit.to()
<astropy.units.core.UnitBase.to>` or `Unit.get_converter()
<astropy.units.core.UnitBase.get_converter>` methods.
The list can be supplied directly,
but ``astropy`` contains several functions that return appropriate lists so
constructing them is often not necessary. Alternatively, if a larger piece of
code needs the same equivalencies, you can set them for a :ref:`given context
<equivalency-context>`.

Built-In Equivalencies
======================

How to Convert Parallax to Distance
-----------------------------------

The length unit *parsec* is defined such that a star one parsec away
will exhibit a 1-arcsecond parallax. (Think of the name as a contraction
between *parallax* and *arcsecond*.)

The :func:`~astropy.units.parallax` function handles
conversions between parallax angles and length.

.. EXAMPLE START: Converting Parallax to Distance

In general, you should not be able to change units of length into
angles or vice versa, so :meth:`~astropy.units.core.UnitBase.to`
raises an exception::

  >>> from astropy import units as u
  >>> (0.8 * u.arcsec).to(u.parsec)  # doctest: +IGNORE_EXCEPTION_DETAIL
  Traceback (most recent call last):
    ...
  UnitConversionError: 'arcsec' (angle) and 'pc' (length) are not convertible

To trigger the conversion between parallax angle and distance, provide
:func:`~astropy.units.parallax` as the optional keyword
argument (``equivalencies=``) to the
:meth:`~astropy.units.core.UnitBase.to` method.

    >>> (0.8 * u.arcsec).to(u.parsec, equivalencies=u.parallax())
    <Quantity 1.25 pc>

.. EXAMPLE END

Angles as Dimensionless Units
-----------------------------

Angles are treated as a physically distinct type, which usually helps to avoid
mistakes. However, this is not very handy when working with units related to
rotational energy or the small angle approximation. (Indeed, this
double-sidedness underlies why radians went from a `supplementary to derived unit
<https://www.bipm.org/en/committees/cg/cgpm/20-1995/resolution-8>`__.) The function
:func:`~astropy.units.dimensionless_angles` provides the required
equivalency list that helps convert between angles and dimensionless units. It
is somewhat different from all others in that it allows an arbitrary change in
the number of powers to which radians is raised (i.e., including zero and
thus dimensionless).

Examples
^^^^^^^^

.. EXAMPLE START: Angles as Dimensionless Units

Normally the following would raise exceptions::

  >>> u.degree.to('')  # doctest: +IGNORE_EXCEPTION_DETAIL
  Traceback (most recent call last):
    ...
  UnitConversionError: 'deg' (angle) and '' (dimensionless) are not convertible
  >>> (u.kg * u.m**2 * (u.cycle / u.s)**2).to(u.J)  # doctest: +IGNORE_EXCEPTION_DETAIL
  Traceback (most recent call last):
    ...
  UnitConversionError: 'cycle2 kg m2 / s2' and 'J' (energy) are not convertible

But when passing the proper conversion function,
:func:`~astropy.units.dimensionless_angles`, it works.

  >>> u.deg.to('', equivalencies=u.dimensionless_angles())  # doctest: +FLOAT_CMP
  0.017453292519943295
  >>> (0.5e38 * u.kg * u.m**2 * (u.cycle / u.s)**2).to(u.J,
  ...                            equivalencies=u.dimensionless_angles())  # doctest: +FLOAT_CMP
  <Quantity 1.9739208802178715e+39 J>
  >>> import numpy as np
  >>> np.exp((1j*0.125*u.cycle).to('', equivalencies=u.dimensionless_angles())) # doctest: +FLOAT_CMP
  <Quantity  0.70710678+0.70710678j>

.. EXAMPLE END

In an example with complex numbers you may well be doing a fair
number of similar calculations. For such situations, there is the
option to :ref:`set default equivalencies <equivalency-context>`.

In some situations, this equivalency may behave differently than
anticipated. For instance, it might at first seem reasonable to use it
for converting from an angular velocity :math:`\omega` in radians per
second to the corresponding frequency :math:`f` in hertz (i.e., to
implement :math:`f=\omega/2\pi`). However, attempting this yields:

  >>> (1*u.rad/u.s).to(u.Hz, equivalencies=u.dimensionless_angles())  # doctest: +FLOAT_CMP
  <Quantity 1. Hz>
  >>> (1*u.cycle/u.s).to(u.Hz, equivalencies=u.dimensionless_angles())  # doctest: +FLOAT_CMP
  <Quantity 6.283185307179586 Hz>

Here, we might have expected ~0.159 Hz in the first example and 1 Hz in
the second. However, :func:`~astropy.units.dimensionless_angles`
converts to radians per second and then drops radians as a unit. The
implicit mistake made in these examples is that the unit Hz is taken to be
equivalent to cycles per second, which it is not (it is just "per second").
This realization also leads to the solution: to use an explicit equivalency
between cycles per second and hertz:

  >>> (1*u.rad/u.s).to(u.Hz, equivalencies=[(u.cy/u.s, u.Hz)])  # doctest: +FLOAT_CMP
  <Quantity 0.15915494309189535 Hz>
  >>> (1*u.cy/u.s).to(u.Hz, equivalencies=[(u.cy/u.s, u.Hz)])  # doctest: +FLOAT_CMP
  <Quantity 1. Hz>

.. _astropy-units-spectral-equivalency:

Spectral Units
--------------

:func:`~astropy.units.spectral` is a function that returns
an equivalency list to handle conversions between wavelength,
frequency, energy, and wave number.

.. EXAMPLE START: Using Spectral Units for Conversions

As mentioned with parallax units, we pass a list of equivalencies (in this case,
the result of :func:`~astropy.units.spectral`) as the second
argument to the :meth:`~astropy.units.quantity.Quantity.to` method and
wavelength, and then frequency and energy can be converted.

  >>> ([1000, 2000] * u.nm).to(u.Hz, equivalencies=u.spectral())  # doctest: +FLOAT_CMP
  <Quantity [2.99792458e+14, 1.49896229e+14] Hz>
  >>> ([1000, 2000] * u.nm).to(u.eV, equivalencies=u.spectral())  # doctest: +FLOAT_CMP
  <Quantity [1.23984193, 0.61992096] eV>

These equivalencies even work with non-base units::

  >>> # Inches to calories
  >>> from astropy.units import imperial
  >>> imperial.inch.to(imperial.Cal, equivalencies=u.spectral())  # doctest: +FLOAT_CMP
  np.float64(1.869180759162485e-27)

.. EXAMPLE END

.. _astropy-units-doppler-equivalencies:

Spectral (Doppler) Equivalencies
--------------------------------

Spectral equivalencies allow you to convert between wavelength,
frequency, energy, and wave number, but not to velocity, which is
frequently the quantity of interest.

It is fairly convenient to define the equivalency, but note that there are
different `conventions <https://www.gb.nrao.edu/~fghigo/gbtdoc/doppler.html>`__.
In these conventions :math:`f_0` is the rest frequency, :math:`f` is the
observed frequency, :math:`V` is the velocity, and :math:`c` is the speed of
light:

    * Radio         :math:`V = c \frac{f_0 - f}{f_0}  ;  f(V) = f_0 ( 1 - V/c )`
    * Optical       :math:`V = c \frac{f_0 - f}{f  }  ;  f(V) = f_0 ( 1 + V/c )^{-1}`
    * Relativistic  :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)}`

These three conventions are implemented as
:func:`~astropy.units.doppler_optical`, :func:`~astropy.units.doppler_radio`,
and :func:`~astropy.units.doppler_relativistic`.

Example
^^^^^^^

.. EXAMPLE START: Using Spectral (Doppler) Equivalencies

To define an equivalency::

    >>> restfreq = 115.27120 * u.GHz  # rest frequency of 12 CO 1-0 in GHz
    >>> freq_to_vel = u.doppler_radio(restfreq)
    >>> (116e9 * u.Hz).to(u.km / u.s, equivalencies=freq_to_vel)  # doctest: +FLOAT_CMP
    <Quantity -1895.4321928669085 km / s>

.. EXAMPLE END

Spectral Flux and Luminosity Density Units
------------------------------------------

There is also support for spectral flux and luminosity density units,
their equivalent surface brightness units, and integrated flux units. Their use
is more complex, since it is necessary to also supply the location in the
spectrum for which the conversions will be done, and the units of those spectral
locations. The function that handles these unit conversions is
:func:`~astropy.units.spectral_density`. This function takes as
its arguments the |Quantity| for the spectral location.

Example
^^^^^^^

.. EXAMPLE START: Converting Spectral Flux and Luminosity Density Units

To perform unit conversions with :func:`~astropy.units.spectral_density`::

    >>> (1.5 * u.Jy).to(u.photon / u.cm**2 / u.s / u.Hz,
    ...                 equivalencies=u.spectral_density(3500 * u.AA)) # doctest: +FLOAT_CMP
    <Quantity 2.6429112e-12 ph / (Hz s cm2)>
    >>> (1.5 * u.Jy).to(u.photon / u.cm**2 / u.s / u.micron,
    ...                 equivalencies=u.spectral_density(3500 * u.AA))  # doctest: +FLOAT_CMP
    <Quantity 6467.95791275 ph / (micron s cm2)>
    >>> a = 1. * (u.photon / u.s / u.angstrom)
    >>> a.to(u.erg / u.s / u.Hz,
    ...      equivalencies=u.spectral_density(5500 * u.AA))  # doctest: +FLOAT_CMP
    <Quantity 3.6443382634999996e-23 erg / (Hz s)>
    >>> w = 5000 * u.AA
    >>> a = 1. * (u.erg / u.cm**2 / u.s)
    >>> b = a.to(u.photon / u.cm**2 / u.s, u.spectral_density(w))
    >>> b  # doctest: +FLOAT_CMP
    <Quantity 2.51705828e+11 ph / (s cm2)>
    >>> b.to(a.unit, u.spectral_density(w))  # doctest: +FLOAT_CMP
    <Quantity 1. erg / (s cm2)>

.. EXAMPLE END

Brightness Temperature and Surface Brightness Equivalency
---------------------------------------------------------

There is an equivalency between surface brightness (flux density per area) and
brightness temperature. This equivalency is often referred to as "Antenna Gain"
since, at a given frequency, telescope brightness sensitivity is unrelated to
aperture size, but flux density sensitivity is, so this equivalency is only
dependent on the aperture size. See `Tools of Radio Astronomy
<https://books.google.com/books?id=9KHw6R8rQEMC&pg=PA179&source=gbs_toc_r&cad=4#v=onepage&q&f=false>`_
for details.

.. note:: The brightness temperature mentioned here is the Rayleigh-Jeans
          equivalent temperature, which results in a linear relation between
          flux and temperature. This is the convention that is most often used
          in relation to observations, but if you are interested in computing
          the *exact* temperature of a blackbody function that would produce a
          given flux, you should not use this equivalency.

Examples
^^^^^^^^

.. EXAMPLE START: Converting Brightness Temperature and Surface Brightness
   Equivalency

The :func:`~astropy.units.brightness_temperature` equivalency
requires the beam area and frequency as arguments. Recalling that the area of a
2D Gaussian is :math:`2 \pi \sigma^2` (see `wikipedia
<https://en.wikipedia.org/wiki/Gaussian_function#Two-dimensional_Gaussian_function>`_),
here is an example::

    >>> beam_sigma = 50*u.arcsec
    >>> omega_B = 2 * np.pi * beam_sigma**2
    >>> freq = 5 * u.GHz
    >>> (1*u.Jy/omega_B).to(u.K, equivalencies=u.brightness_temperature(freq))  # doctest: +FLOAT_CMP
    <Quantity 3.526295144567176 K>

If you have beam full-width half-maxima (FWHM), which are often quoted and are
the values stored in the FITS header keywords BMAJ and BMIN, a more appropriate
example converts the FWHM to sigma::

    >>> beam_fwhm = 50*u.arcsec
    >>> fwhm_to_sigma = 1. / (8 * np.log(2))**0.5
    >>> beam_sigma = beam_fwhm * fwhm_to_sigma
    >>> omega_B = 2 * np.pi * beam_sigma**2
    >>> (1*u.Jy/omega_B).to(u.K, equivalencies=u.brightness_temperature(freq))  # doctest: +FLOAT_CMP
    <Quantity 19.553932298231704 K>

You can also convert between ``Jy/beam`` and ``K`` by specifying the beam area::

    >>> (1*u.Jy/u.beam).to(u.K, u.brightness_temperature(freq, beam_area=omega_B))  # doctest: +FLOAT_CMP
    <Quantity 19.553932298231704 K>

.. EXAMPLE END

Beam Equivalency
----------------

Radio data, especially from interferometers, is often produced in units of
``Jy/beam``. Converting this number to a beam-independent value (e.g.,
``Jy/sr``), can be done with the
:func:`~astropy.units.beam_angular_area` equivalency.

Example
^^^^^^^

.. EXAMPLE START: Converting Radio Data to a Beam-Independent Value

To convert units of ``Jy/beam`` to ``Jy/sr``::

    >>> beam_fwhm = 50*u.arcsec
    >>> fwhm_to_sigma = 1. / (8 * np.log(2))**0.5
    >>> beam_sigma = beam_fwhm * fwhm_to_sigma
    >>> omega_B = 2 * np.pi * beam_sigma**2
    >>> (1*u.Jy/u.beam).to(u.MJy/u.sr, equivalencies=u.beam_angular_area(omega_B))  # doctest: +FLOAT_CMP
    <Quantity 15.019166691021288 MJy / sr>


Note that the `radio_beam <https://github.com/radio-astro-tools/radio-beam>`_
package deals with beam input/output and various operations more directly.

.. EXAMPLE END

Temperature Energy Equivalency
------------------------------

The :func:`~astropy.units.temperature_energy` equivalency allows
conversion between temperature and its equivalent in energy (i.e., the
temperature multiplied by the Boltzmann constant), usually expressed in
electronvolts. This is used frequently for observations at high-energy, be it
for solar or X-ray astronomy.

Example
^^^^^^^

.. EXAMPLE START: Temperature Energy Equivalency

To convert between temperature and its equivalent in energy::

    >>> t_k = 1e6 * u.K
    >>> t_k.to(u.eV, equivalencies=u.temperature_energy())  # doctest: +FLOAT_CMP
    <Quantity 86.17332384960955 eV>

.. EXAMPLE END

.. _tcmb-equivalency:

Thermodynamic Temperature Equivalency
-------------------------------------

This :func:`~astropy.units.thermodynamic_temperature`
equivalency allows conversion between ``Jy/beam`` and "thermodynamic
temperature", :math:`T_{CMB}`, in Kelvins.

Examples
^^^^^^^^

.. EXAMPLE START: Thermodynamic Temperature Equivalency

To convert between ``Jy/beam`` and thermodynamic temperature::

    >>> nu = 143 * u.GHz
    >>> t_k = 0.002632051878 * u.K
    >>> t_k.to(u.MJy / u.sr, equivalencies=u.thermodynamic_temperature(nu))  # doctest: +FLOAT_CMP
    <Quantity 1. MJy / sr>

By default, this will use the :math:`T_{CMB}` value for the default
:ref:`cosmology <astropy-cosmology>` in ``astropy``, but it is possible to
specify a custom :math:`T_{CMB}` value for a specific cosmology as the second
argument to the equivalency::

    >>> from astropy.cosmology import WMAP9
    >>> t_k.to(u.MJy / u.sr, equivalencies=u.thermodynamic_temperature(nu, T_cmb=WMAP9.Tcmb0))  # doctest: +FLOAT_CMP
    <Quantity 0.99982392 MJy / sr>

.. EXAMPLE END

Molar Mass AMU Equivalency
--------------------------

The :func:`~astropy.units.molar_mass_amu` equivalency allows
conversion between the atomic mass unit and the equivalent g/mol. For context,
refer to the `NIST definition of SI Base Units
<https://www.nist.gov/si-redefinition/definitions-si-base-units>`_.

Example
^^^^^^^

.. EXAMPLE START: Molar Mass AMU Equivalency

To convert between atomic mass unit and the equivalent g/mol::

    >>> x = 1 * (u.g / u.mol)
    >>> y = 1 * u.u
    >>> x.to(u.u, equivalencies=u.molar_mass_amu()) # doctest: +FLOAT_CMP
    <Quantity 1.0 u>
    >>> y.to(u.g/u.mol, equivalencies=u.molar_mass_amu()) # doctest: +FLOAT_CMP
    <Quantity 1.0 g / mol>

.. EXAMPLE END

Pixel and Plate Scale Equivalencies
-----------------------------------

These equivalencies are for converting between angular scales and either linear
scales in the focal plane or distances in units of the number of pixels.

Examples
^^^^^^^^

.. EXAMPLE START: Pixel and Plate Scale Equivalencies

Suppose you are working with cutouts from the Sloan Digital Sky Survey,
which defaults to a pixel scale of 0.4 arcseconds per pixel, and want to know
the true size of something that you measure to be 240 pixels across in the
cutout image::

    >>> sdss_pixelscale = u.pixel_scale(0.4*u.arcsec/u.pixel)
    >>> (240*u.pixel).to(u.arcmin, sdss_pixelscale)  # doctest: +FLOAT_CMP
    <Quantity 1.6 arcmin>

Or maybe you are designing an instrument for a telescope that someone told you
has an inverse plate scale of 7.8 meters per radian (for your desired focus),
and you want to know how big your pixels need to be to cover half an arcsecond.
Using :func:`~astropy.units.plate_scale`::

    >>> tel_platescale = u.plate_scale(7.8*u.m/u.radian)
    >>> (0.5*u.arcsec).to(u.micron, tel_platescale)  # doctest: +FLOAT_CMP
    <Quantity 18.9077335632719 micron>

The :func:`~astropy.units.pixel_scale` equivalency can also work
in more general context, where the scale is specified as any quantity that is
reducible to ``<composite unit>/u.pix`` or ``u.pix/<composite unit>`` (that is,
the dimensionality of ``u.pix`` is 1 or -1). For instance, you may define the
dots per inch (DPI) for a digital image to calculate its physical size::

    >>> dpi = u.pixel_scale(100 * u.pix / u.imperial.inch)
    >>> (1024 * u.pix).to(u.cm, dpi)  # doctest: +FLOAT_CMP
    <Quantity 26.0096 cm>

.. EXAMPLE END

Photometric Zero Point Equivalency
----------------------------------

The :func:`~astropy.units.zero_point_flux` equivalency provides a way to move
between photometric systems (i.e., those defined relative to a particular
zero-point flux) and absolute fluxes. This is most useful in conjunction with
support for :ref:`logarithmic_units`.

Example
^^^^^^^

.. EXAMPLE START: Photometric Zero Point Equivalency

Suppose you are observing a target with a filter with a reported standard zero
point of 3631.1 Jy::

    >>> target_flux = 1.2 * u.nanomaggy
    >>> zero_point_star_equiv = u.zero_point_flux(3631.1 * u.Jy)
    >>> u.Magnitude(target_flux.to(u.AB, zero_point_star_equiv))  # doctest: +FLOAT_CMP
    <Magnitude 22.30195136 mag(AB)>

.. EXAMPLE END

Temperature Equivalency
-----------------------

Conversions between kelvins and degrees Rankine do not require using any
equivalencies because the Kelvin scale and the Rankine scale have the same zero
point::

   >>> temp_K = 100 * u.K
   >>> temp_R = temp_K.to(u.imperial.deg_R)
   >>> temp_R
   <Quantity 180. deg_R>
   >>> temp_R.to(u.K)
   <Quantity 100. K>

.. EXAMPLE START: Using the Temperature Equivalency

The Celsius and Fahrenheit scales have different zero points, which
neccessitates the use of the :func:`~astropy.units.temperature` equivalency for
unit conversions involving degrees Celsius or degrees Fahrenheit::

    >>> temp_C = 0 * u.Celsius
    >>> temp_Kelvin = temp_C.to(u.K, equivalencies=u.temperature())
    >>> temp_Kelvin  # doctest: +FLOAT_CMP
    <Quantity 273.15 K>
    >>> temp_F = temp_C.to(u.imperial.deg_F, equivalencies=u.temperature())
    >>> temp_F  # doctest: +FLOAT_CMP
    <Quantity 32. deg_F>
    >>> temp_R = temp_C.to(u.imperial.deg_R, equivalencies=u.temperature())
    >>> temp_R  # doctest: +FLOAT_CMP
    <Quantity 491.67 deg_R>

.. note:: You can also use ``u.deg_C`` instead of ``u.Celsius``.

.. EXAMPLE END

Mass-Energy Equivalency
-----------------------

.. EXAMPLE START: Using the Mass-Energy Equivalency

In a special relativity context it can be convenient to use the
:func:`~astropy.units.mass_energy` equivalency. For instance::

    >>> (1 * u.g).to(u.eV, u.mass_energy())  # doctest: +FLOAT_CMP
    <Quantity 5.60958865e+32 eV>

.. EXAMPLE END

Doppler Redshift Equivalency
----------------------------

Conversion between Doppler redshift and radial velocity can be done with the
:func:`~astropy.units.doppler_redshift` equivalency.

Example
^^^^^^^

.. EXAMPLE START: Converting Doppler redshift to radial velocity

To convert Doppler redshift (unitless) to ``km/s``::

    >>> z = 0.1 * u.dimensionless_unscaled
    >>> z.to(u.km / u.s, u.doppler_redshift())  # doctest: +FLOAT_CMP
    <Quantity 28487.0661448 km / s>

However, it cannot take the cosmological redshift unit from `astropy.cosmology.units`
because the latter should not be interpreted the same since the recessional
velocity from the expansion of space can exceed the speed of light; see
`Hubble's law: Redshift velocity and recessional velocity <https://en.wikipedia.org/wiki/Hubble%27s_law#Redshift_velocity_and_recessional_velocity>`_
for more information.

.. EXAMPLE END

Magnetic Flux Density and Field Strength Equivalency
----------------------------------------------------

The :func:`~astropy.units.magnetic_flux_field` equivalency allows
conversion between Magnetic Flux Density (B) and its equivalent Magnetic Field
Strength (H), governed by the equation

.. math::

        \mathbf{B} = \mu_r \mu_0 \mathbf{H}.

Where :math:`\mu_0` is the vacuum permeability and :math:`\mu_r` is the
relative permeability of the medium. For a vacuum, :math:`\mu_r=1`.

Example
^^^^^^^

.. EXAMPLE START: Magnetic Flux Density Magnetic Field Strength Equivalency

To convert between Magnetic Flux Density (B) and its equivalent Magnetic Field
Strength (H) in a vacuum.

    >>> H = 1 * u.Oe
    >>> H.to(u.G, u.magnetic_flux_field())  # doctest: +FLOAT_CMP
    <Quantity 1. G>
    >>> H.to(u.T, u.magnetic_flux_field())  # doctest: +FLOAT_CMP
    <Quantity 0.0001 T>
    >>> B = 1 * u.T
    >>> B.to(u.A / u.m, equivalencies=u.magnetic_flux_field())  # doctest: +FLOAT_CMP
    <Quantity 795774.71502628 A / m>

Conversion in a medium with :math:`\mu_r=0.9`::

    >>> H.to(u.G, u.magnetic_flux_field(mu_r=0.9))  # doctest: +FLOAT_CMP
    <Quantity 0.9 G>
    >>> B.to(u.A / u.m, equivalencies=u.magnetic_flux_field(mu_r=0.9))  # doctest: +FLOAT_CMP
    <Quantity 884194.12780697 A / m>

.. EXAMPLE END

Writing New Equivalencies
=========================

An equivalence list is a :class:`list` of tuples, where each :class:`tuple` has
four elements::

  (from_unit, to_unit, forward, backward)

``from_unit`` and ``to_unit`` are the equivalent units. ``forward`` and
``backward`` are functions that convert values between those units. ``forward``
and ``backward`` are optional, and if omitted then the equivalency declares
that the two units should be taken as equivalent. The functions must take and
return non-|Quantity| objects to avoid infinite recursion; See
:ref:`complicated-equiv-example` for more details.

Examples
--------

.. EXAMPLE START: Writing New Equivalencies

Until 1964, the metric liter was defined as the volume of 1kg of water at 4°C at
760mm mercury pressure. Volumes and masses are not normally directly
convertible, but if we hold the constants in the 1964 definition of the liter as
true, we could build an equivalency for them::

  >>> liters_water = [
  ...    (u.l, u.g, lambda x: 1000.0 * x, lambda x: x / 1000.0)
  ... ]
  >>> u.l.to(u.kg, 1, equivalencies=liters_water)
  1.0

Note that the equivalency can be used with any other compatible unit::

  >>> imperial.gallon.to(imperial.pound, 1, equivalencies=liters_water)  # doctest: +FLOAT_CMP
  8.345404463333525

And it also works in the other direction::

  >>> imperial.lb.to(imperial.pint, 1, equivalencies=liters_water)  # doctest: +FLOAT_CMP
  0.9586114172355459

.. EXAMPLE END

.. _complicated-equiv-example:

A More Complex Example: Spectral Doppler Equivalencies
------------------------------------------------------

.. EXAMPLE START: Writing Spectral Doppler Equivalencies

We show how to define an equivalency using the radio convention for CO 1-0.
This function is already defined in
:func:`~astropy.units.doppler_radio`, but this example is
illustrative::

    >>> from astropy.constants import si
    >>> restfreq = 115.27120  # rest frequency of 12 CO 1-0 in GHz
    >>> freq_to_vel = [(u.GHz, u.km/u.s,
    ... lambda x: (restfreq-x) / restfreq * si.c.to_value('km/s'),
    ... lambda x: (1-x/si.c.to_value('km/s')) * restfreq )]
    >>> u.Hz.to(u.km / u.s, 116e9, equivalencies=freq_to_vel)  # doctest: +FLOAT_CMP
    np.float64(-1895.4321928669262)
    >>> (116e9 * u.Hz).to(u.km / u.s, equivalencies=freq_to_vel)  # doctest: +FLOAT_CMP
    <Quantity -1895.4321928669262 km / s>

.. EXAMPLE END

Note that once this is defined for GHz and km/s, it will work for all other
units of frequency and velocity. ``x`` is converted from the input frequency
unit (e.g., Hz) to GHz before being passed to ``lambda x:``. Similarly, the
return value is assumed to be in units of ``km/s``, which is why the ``value``
of ``c`` is used instead of the :class:`~astropy.constants.Constant`.

Displaying Available Equivalencies
==================================

The :meth:`~astropy.units.core.UnitBase.find_equivalent_units` method also
understands equivalencies.

Example
-------

.. EXAMPLE START: Displaying Available Equivalencies

Without passing equivalencies, there are three compatible units for ``Hz`` in
the standard set::

  >>> u.Hz.find_equivalent_units()
    Primary name | Unit definition | Aliases
  [
    Bq           | 1 / s           | becquerel    ,
    Ci           | 3.7e+10 / s    | curie        ,
    Hz           | 1 / s           | Hertz, hertz ,
  ]

However, when passing the spectral equivalency, you can see there are
all kinds of things that ``Hz`` can be converted to::

  >>> u.Hz.find_equivalent_units(equivalencies=u.spectral())
  Primary name | Unit definition        | Aliases
  [
    AU           | 1.49598e+11 m          | au, astronomical_unit            ,
    Angstrom     | 1e-10 m                | AA, angstrom, Å                  ,
    Bq           | 1 / s                  | becquerel                        ,
    Ci           | 3.7e+10 / s            | curie                            ,
    Hz           | 1 / s                  | Hertz, hertz                     ,
    J            | m2 kg / s2             | Joule, joule                     ,
    Ry           | 2.17987e-18 m2 kg / s2 | rydberg                          ,
    cm           | 0.01 m                 | centimeter                       ,
    eV           | 1.60218e-19 m2 kg / s2 | electronvolt                     ,
    earthRad     | 6.3781e+06 m           | R_earth, Rearth                  ,
    erg          | 1e-07 m2 kg / s2       |                                  ,
    foe          | 1e+44 m2 kg / s2       | Bethe, bethe                     ,
    jupiterRad   | 7.1492e+07 m           | R_jup, Rjup, R_jupiter, Rjupiter ,
    k            | 100 / m                | Kayser, kayser                   ,
    lsec         | 2.99792e+08 m          | lightsecond                      ,
    lyr          | 9.46073e+15 m          | lightyear                        ,
    m            | irreducible            | meter                            ,
    micron       | 1e-06 m                |                                  ,
    pc           | 3.08568e+16 m          | parsec                           ,
    solRad       | 6.957e+08 m            | R_sun, Rsun                      ,
  ]

.. EXAMPLE END

.. _equivalency-context:

Using Equivalencies in Larger Pieces of Code
============================================

Sometimes you may have an involved calculation where you are regularly switching
back and forth between equivalent units. For these cases, you can set
equivalencies that will by default be used, in a way similar to how you can
:ref:`enable other units <enabling-other-units>`.

Examples
--------

.. EXAMPLE START: Using Equivalencies in Larger Pieces of Code

To enable radians to be treated as a dimensionless unit use
:func:`~astropy.units.set_enabled_equivalencies` as a `context manager
<https://docs.python.org/3/reference/datamodel.html#context-managers>`_::

  >>> with u.set_enabled_equivalencies(u.dimensionless_angles()):
  ...    phase = 0.5 * u.cycle
  ...    c = np.exp(1j*phase)
  >>> c  # doctest: +FLOAT_CMP
  <Quantity -1.+1.2246468e-16j>

To permanently and globally enable radians to be treated as a dimensionless
unit use :func:`~astropy.units.set_enabled_equivalencies` not as a context
manager:

.. doctest-skip::

  >>> u.set_enabled_equivalencies(u.dimensionless_angles())
  <astropy.units.core._UnitContext object at ...>
  >>> u.deg.to('')  # doctest: +FLOAT_CMP
  0.017453292519943295

The disadvantage of the above approach is that you may forget to turn the
default off (done by giving an empty argument).

:func:`~astropy.units.set_enabled_equivalencies` accepts any list of
equivalencies, so you could add, for example,
:func:`~astropy.units.spectral` and
:func:`~astropy.units.spectral_density` (since these return
lists, they should indeed be combined by adding them together).

.. EXAMPLE END
