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- 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()` or `Unit.to()` 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 given 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 `parallax()` function handles conversions between parallax angles and length.

In general, you should not be able to change units of length into angles or vice versa, so `to()` raises an exception:

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

To trigger the conversion between parallax angle and distance, provide `parallax()` as the optional keyword argument (`equivalencies=`) to the `to()` method.

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

### 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.) The function `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

Normally the following would raise exceptions:

    >>> u.degree.to('')
    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)
    Traceback (most recent call last):
      ...
    UnitConversionError: 'cycle2 kg m2 / s2' and 'J' (energy) are not convertible

But when passing the proper conversion function, `dimensionless_angles()`, it works.

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

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 set default equivalencies.

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 \(\omega\) in radians per second to the corresponding frequency \(f\) in hertz (i.e., to implement \(f=\omega/2\pi\)). However, attempting this yields:

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

Here, we might have expected ~0.159 Hz in the first example and 1 Hz in the second. However, `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)])
    <Quantity 0.15915494309189535 Hz>
    >>> (1*u.cy/u.s).to(u.Hz, equivalencies=[(u.cy/u.s, u.Hz)])
    <Quantity 1. Hz>

### Spectral Units

`spectral()` is a function that returns an equivalency list to handle conversions between wavelength, frequency, energy, and wave number.

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

    >>> ([1000, 2000] * u.nm).to(u.Hz, equivalencies=u.spectral())
    <Quantity [2.99792458e+14, 1.49896229e+14] Hz>
    >>> ([1000, 2000] * u.nm).to(u.eV, equivalencies=u.spectral())
    <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())
    1.869180759162485e-27

### 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. In these conventions \(f_0\) is the rest frequency, \(f\) is the observed frequency, \(V\) is the velocity, and \(c\) is the speed of light:

> - Radio \(V = c \frac{f_0 - f}{f_0} ; f(V) = f_0 ( 1 - V/c )\)

> - Optical \(V = c \frac{f_0 - f}{f } ; f(V) = f_0 ( 1 + V/c )^{-1}\)

> - Relativistic \(V = c \frac{f_0^2 - f^2}{f_0^2 + f^2} ; f(V) = f_0 \frac{\left(1 - (V/c)^2\right)^{1/2}}{(1+V/c)}\)

These three conventions are implemented in `astropy.units.equivalencies` as `doppler_optical()`, `doppler_radio()`, and `doppler_relativistic()`.

#### Example

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)
    <Quantity -1895.4321928669085 km / s>

### 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 `spectral_density()`. This function takes as its arguments the `Quantity` for the spectral location.

#### Example

To perform unit conversions with `spectral_density()`:

    >>> (1.5 * u.Jy).to(u.photon / u.cm**2 / u.s / u.Hz,
    ...                 equivalencies=u.spectral_density(3500 * u.AA))
    <Quantity 2.6429114293019694e-12 ph / (cm2 Hz s)>
    >>> (1.5 * u.Jy).to(u.photon / u.cm**2 / u.s / u.micron,
    ...                 equivalencies=u.spectral_density(3500 * u.AA))
    <Quantity 6467.9584789120845 ph / (cm2 micron s)>
    >>> a = 1. * (u.photon / u.s / u.angstrom)
    >>> a.to(u.erg / u.s / u.Hz,
    ...      equivalencies=u.spectral_density(5500 * u.AA))
    <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
    <Quantity 2.51705828e+11 ph / (cm2 s)>
    >>> b.to(a.unit, u.spectral_density(w))
    <Quantity 1. erg / (cm2 s)>

### 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 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

The `brightness_temperature()` equivalency requires the beam area and frequency as arguments. Recalling that the area of a 2D Gaussian is \(2 \pi \sigma^2\) (see wikipedia), 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))
    <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))
    <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))
    <Quantity 19.553932298231704 K>

### 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 `beam_angular_area()` equivalency.

#### Example

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))
    <Quantity 15.019166691021288 MJy / sr>

Note that the radio_beam package deals with beam input/output and various operations more directly.

### Temperature Energy Equivalency

The `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

To convert between temperature and its equivalent in energy:

    >>> t_k = 1e6 * u.K
    >>> t_k.to(u.eV, equivalencies=u.temperature_energy())
    <Quantity 86.17332384960955 eV>

### Thermodynamic Temperature Equivalency

This `thermodynamic_temperature()` equivalency allows conversion between `Jy/beam` and “thermodynamic temperature”, \(T_{CMB}\), in Kelvins.

#### Examples

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))
    <Quantity 1. MJy / sr>

By default, this will use the \(T_{CMB}\) value for the default cosmology in `astropy`, but it is possible to specify a custom \(T_{CMB}\) 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))
    <Quantity 0.99982392 MJy / sr>

### Molar Mass AMU Equivalency

The `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.

#### Example

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())
    <Quantity 1.0 u>
    >>> y.to(u.g/u.mol, equivalencies=u.molar_mass_amu())
    <Quantity 1.0 g / mol>

### 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

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)
    <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 `plate_scale()`:

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

The `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)
    <Quantity 26.0096 cm>

### Photometric Zero Point Equivalency

The `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 Magnitudes and Other Logarithmic Units.

#### Example

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))
    <Magnitude 22.30195136 mag(AB)>

### Temperature Equivalency

The `temperature()` equivalency allows conversion between the Celsius, Fahrenheit, Rankine and Kelvin.

#### Example

To convert between temperature scales:

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

Note

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

### Mass-Energy Equivalency

In a special relativity context it can be convenient to use the `mass_energy()` equivalency. For instance:

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

### Doppler Redshift Equivalency

Conversion between Doppler redshift and radial velocity can be done with the `doppler_redshift()` equivalency.

#### Example

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

    >>> z = 0.1 * u.dimensionless_unscaled
    >>> z.to(u.km / u.s, u.doppler_redshift())
    <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 for more information.

## Writing New Equivalencies

An equivalence list is a `list` of tuples, where each `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 A More Complex Example: Spectral Doppler Equivalencies for more details.

### Examples

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)
    8.345404463333525

And it also works in the other direction:

    >>> imperial.lb.to(imperial.pint, 1, equivalencies=liters_water)
    0.9586114172355459

### A More Complex Example: Spectral Doppler Equivalencies

We show how to define an equivalency using the radio convention for CO 1-0. This function is already defined in `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)
    -1895.4321928669262
    >>> (116e9 * u.Hz).to(u.km / u.s, equivalencies=freq_to_vel)
    <Quantity -1895.4321928669262 km / s>

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 `Constant`.

## Displaying Available Equivalencies

The `find_equivalent_units()` method also understands equivalencies.

### Example

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          ,
      Bq           | 1 / s                  | becquerel             ,
      Ci           | 3.7e+10 / s            | curie                 ,
      Hz           | 1 / s                  | Hertz, hertz          ,
      J            | kg m2 / s2             | Joule, joule          ,
      Ry           | 2.17987e-18 kg m2 / s2 | rydberg               ,
      cm           | 0.01 m                 | centimeter            ,
      eV           | 1.60218e-19 kg m2 / s2 | electronvolt          ,
      earthRad     | 6.3781e+06 m           | R_earth, Rearth       ,
      erg          | 1e-07 kg m2 / s2       |                       ,
      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           ,
    ]

## 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 enable other units.

### Examples

To enable radians to be treated as a dimensionless unit use `set_enabled_equivalencies()` as a context manager:

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

To permanently and globally enable radians to be treated as a dimensionless unit use `set_enabled_equivalencies()` not as a context manager:

    >>> u.set_enabled_equivalencies(u.dimensionless_angles())
    <astropy.units.core._UnitContext object at ...>
    >>> u.deg.to('')
    0.017453292519943295

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

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