# 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. 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 it 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:

```
>>> from astropy import units as u
>>> 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:

```
>>> import numpy as np
>>> 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:

```
>>> import numpy as np
>>> 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
>>> freq = 5 * u.GHz
>>> (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:

```
>>> import numpy as np
>>> 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
>>> freq = 5 * u.GHz
>>> (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`

:

```
>>> import numpy as np
>>> 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¶

This 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:

```
>>> import astropy.units as u
>>> 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:

```
>>> import astropy.units as u
>>> 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¶

This 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:

```
>>> import astropy.units as u
>>> import astropy.constants as const
>>> 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:

```
>>> import astropy.units as u
>>> 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:

```
>>> import astropy.units as u
>>> 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:

```
>>> import astropy.units as u
>>> 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¶

This 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)>
```

### Reduced Hubble Constant and “little-h” Equivalency¶

The dimensionless version of the Hubble constant — often known as “little h” —
is a frequently used quantity in extragalactic astrophysics. It is also widely
known as the bane of beginners’ existence in such fields (See e.g., the title of
this paper, which also provides
valuable advice on the use of little h). `astropy`

provides an equivalency
that helps keep this straight in at least some of these cases, by providing a
way to convert to/from physical to “little h” units.

#### Examples¶

To convert to or from physical to “little h” units:

```
>>> import astropy.units as u
>>> H0_70 = 70 * u.km/u.s / u.Mpc
>>> distance = 70 * (u.Mpc/u.littleh)
>>> distance.to(u.Mpc, u.with_H0(H0_70))
<Quantity 100.0 Mpc>
>>> luminosity = 0.49 * u.Lsun * u.littleh**-2
>>> luminosity.to(u.Lsun, u.with_H0(H0_70))
<Quantity 1.0 solLum>
```

Note the unit name `littleh`

: while this unit is usually expressed in the
literature as just `h`

, here it is `littleh`

to avoid confusion with
“hours.”

If no argument is given (or the argument is `None`

), this equivalency assumes
the `H0`

from the current default cosmology:

```
>>> distance = 100 * (u.Mpc/u.littleh)
>>> distance.to(u.Mpc, u.with_H0())
<Quantity 147.79781259 Mpc>
```

This equivalency also allows a common magnitude formulation of little h scaling:

```
>>> mag_quantity = 12 * (u.mag - u.MagUnit(u.littleh**2))
>>> mag_quantity
<Magnitude 12. mag(1 / littleh2)>
>>> mag_quantity.to(u.mag, u.with_H0(H0_70))
<Quantity 11.2254902 mag>
```

### Temperature Equivalency¶

The `temperature()`

equivalency allows conversion
between the Celsius, Fahrenheit, Rankine and Kelvin.

#### Example¶

To convert between temperature scales:

```
>>> import astropy.units as u
>>> 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, mass and energy can be equivalent units. For instance:

```
>>> import astropy.units as u
>>> (1 * u.g).to(u.eV, u.mass_energy())
<Quantity 5.60958865e+32 eV>
```

## 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 such an 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:

```
>>> from astropy.units import imperial
>>> 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 ,
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:

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

Here, any list of equivalencies could be used, or you could add, for example,
`spectral()`

and
`spectral_density()`

(since these return
lists, they should indeed be combined by adding them together).

The disadvantage of the above approach is that you may forget to turn the default off (done by giving an empty argument). To automate this, a context manager is provided:

```
>>> import astropy.units as u
>>> with u.set_enabled_equivalencies(u.dimensionless_angles()):
... phase = 0.5 * u.cycle
... c = np.exp(1j*phase)
>>> c
<Quantity (-1+1.2246063538223773e-16j) >
```