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, one can set them for a given context.

Built-in equivalencies

Parallax Units

parallax() is a function that returns an equivalency list to handle conversions between angles and length.

Length and angles are not normally convertible, so to() raises an exception:

>>> from astropy import units as u
>>> (8.0 * u.arcsec).to(u.parsec)  
Traceback (most recent call last):
  ...
UnitConversionError: 'arcsec' (angle) and 'pc' (length) are not convertible

However, when passing the result of parallax() as the third argument to the to() method, angles can be converted into units of length (and vice versa).

>>> (8.0 * u.arcsec).to(u.parsec, equivalencies=u.parallax())
<Quantity 0.125 pc>
>>> u.AU.to(u.arcminute, equivalencies=u.parallax())
3437.7467707580054

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 radian went from 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 radian is raised (i.e., including zero and thus dimensionless). For instance, normally the following 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>

The example with complex numbers is also one 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 above with parallax units, we simply pass a list of equivalencies (in this case, the result of spectral()) as the third argument to the to() method and wavelength, 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 straightforward 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 use:

>>> 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 / Luminosity Density Units

There is also support for spectral flux and luminosity density 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. For example:

>>> (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)>

Brightness Temperature / 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 planck function that would produce a given flux, you should not use this equivalency.

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>

Finally, there is an equivalency that allows you to convert from Jansky to Kelvin. In this case, the Jansky unit is implicitly Jansky/beam. Because of the implicit assumed per beam unit, this approach is deprecated.:

>>> 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
>>> # DEPRECATED
>>> (1*u.Jy).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:

>>> 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:

>>> 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. Example:

>>> 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 reference to why this was added, refer to NIST Mole Reference The following is an example of it’s usage:

>>> 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. For example, 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 a (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>

Photometric Zero Point Equivalency

This equivalency provides an easy 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 conjuction with support for Magnitudes and other Logarithmic Units. For 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/”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. Two example conversions:

>>> import astropy.units as u
>>> H0_70 = 70 * u.km/u.s / u.Mpc
>>> distance = 100 * (u.Mpc/u.littleh)
>>> distance.to(u.Mpc, u.with_H0(H0_70))  
<Quantity 70.0 Mpc>
>>> luminosity = 1 * u.Lsun * u.littleh**-2
>>> luminosity.to(u.Lsun, u.with_H0(H0_70))  
<Quantity 0.49 solLum>

Note the unit name littleh - while this unit is usually expressed in the literature as just h, here it is littleh to not cause 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 67.74 Mpc>

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

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

Writing new equivalencies

An equivalence list is just a list of tuples, where each tuple has 4 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 simply declares that the two units should be taken as equivalent.

For example, 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 units:

>>> 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 slightly more complicated 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. For 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 one has an involved calculation where one is regularly switching back between equivalent units. For these cases, one can set equivalencies that will by default be used, in a way similar to which one can enable other units.

For instance, to enable radian to be treated as a dimensionless unit, simply do:

>>> 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 one could add, e.g., 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) >