Using the SpectralCoord Class¶
Warning
The SpectralCoord
class is new in Astropy v4.1 and should be considered
experimental at this time. Note that we do not fully support cases
where the observer and target are moving relativistically relative to each
other, so care should be taken in those cases. It is possible that there
will be API changes in future versions of Astropy based on user feedback. If
you have specific ideas for how it might be improved, please let us know on
the astropydev mailing list or at http://feedback.astropy.org.
The SpectralCoord
class provides an interface for representing and
transforming spectral coordinates such as frequencies, wavelengths, and photon
energies, as well as equivalent Doppler velocities. While the plain Quantity
class can also represent these kinds of physical quantities, and allow
conversion via dedicated equivalencies (such as u.spectral or the u.doppler_* equivalencies), SpectralCoord
(which is
a subclass of Quantity
) aims to make this more straightforward, and can also
be made aware of the observer and target reference frames, allowing for example
transformation from telescopecentric (or topocentric) frames to e.g.
Barycentric or Local Standard of Rest (LSRK and LSRD) velocity frames.
Creating SpectralCoord Objects¶
Since the SpectralCoord
class is a subclass of Quantity
, the simplest way
to initialize it is to provide a value (or values) and a unit, or an existing
Quantity
:
>>> from astropy import units as u
>>> from astropy.coordinates import SpectralCoord
>>> sc1 = SpectralCoord(34.2, unit='GHz')
>>> sc1
<SpectralCoord 34.2 GHz>
>>> sc2 = SpectralCoord([654.2, 654.4, 654.6] * u.nm)
>>> sc2
<SpectralCoord [654.2, 654.4, 654.6] nm>
At this point, we are not making any assumptions about the observer frame, or
the target that is being observed. As we will see in subsequent sections, more
information can be provided when initializing SpectralCoord
objects, but first
we take a look at simple unit conversions with these objects.
Unit conversion¶
By default, unit conversions between spectral units will work without having to specify the u.spectral equivalency:
>>> sc2.to(u.micron)
<SpectralCoord [0.6542, 0.6544, 0.6546] micron>
>>> sc2.to(u.eV)
<SpectralCoord [1.89520328, 1.89462406, 1.89404519] eV>
>>> sc2.to(u.THz)
<SpectralCoord [458.25811373, 458.11805929, 457.97809044] THz>
As is the case with Quantity
and the Doppler equivalencies, it is also posible to convert these
absolute spectral coordinates into velocities, assuming a particular rest
frequency or wavelength (such as that of a spectral line). For example, to
convert the above values into velocities relative to the Halpha line at 656.65
nm, assuming the optical Doppler convention, you can do:
>>> sc3 = sc2.to(u.km / u.s,
... doppler_convention='optical',
... doppler_rest=656.65 * u.nm)
>>> sc3
<SpectralCoord
(doppler_rest=656.65 nm
doppler_convention=optical)
[1118.5433977 , 1027.23373258, 935.92406746] km / s>
The rest value for the Doppler conversion as well as the convention to use are
stored in the resulting sc3
SpectralCoord
object. You can then convert
back to frequency without having to specify them again:
>>> sc3.to(u.THz)
<SpectralCoord
(doppler_rest=656.65 nm
doppler_convention=optical)
[458.25811373, 458.11805929, 457.97809044] THz>
or you can explicitly specify a different convention or rest value to use:
>>> sc3.to(u.km / u.s, doppler_convention='relativistic')
<SpectralCoord
(doppler_rest=656.65 nm
doppler_convention=relativistic)
[1120.63005892, 1028.99362163, 937.38499411] km / s>
It is also possible to set doppler_convention
and doppler_rest
from the
start, even when creating a SpectralCoord
in frequency, energy, or
wavelength:
>>> sc4 = SpectralCoord(343 * u.GHz,
... doppler_convention='radio',
... doppler_rest=342.91 * u.GHz)
>>> sc4.to(u.km / u.s)
<SpectralCoord
(doppler_rest=342.91 GHz
doppler_convention=radio)
78.68338987 km / s>
Reference frame transformations¶
If you work with any kind of spectral data, you will often need to determine
and/or apply velocity corrections due to different frames of reference, or apply
or remove the effects of redshift. There are two main ways to do this using the
SpectralCoord
class:
You can specify or change the velocity offset or redshift between the observer and the target without having to specify the absolute observer and target, but rather specify a velocity difference. For example, that you know that there is a velocity difference of 15km/s along the line of sight, or that you are observing a galaxy at z=3.2. This can be useful for quick analysis but will not determine any frame transformations (e.g. from topocentric to barycentric) for you.
You can specify the absolute position of the observer and the target, as well as the date of observation, which means that
SpectralCoord
can then compute different frame transformations. If information about the observer and target are available, this is the recommended approach, although it requires you to specify more information when setting up theSpectralCoord
In the next two sections we will look at each of these in turn.
Specifying radial velocity or redshift manually¶
As an example, we will consider an example of a SpectralCoord
which represents
frequencies which form the xaxis of a (small) spectrum. We happen to know that
the target that was observed appears to be at a redshift of z=0.5, and we will
assume that any frequency shifts due to the Earth’s motion are unimportant. In
the reference frame of the telescope, the spectrometer provides 10 values
between 500 and 900nm:
>>> import numpy as np
>>> wavs = SpectralCoord(np.linspace(500, 900, 9) * u.nm, redshift=0.5)
>>> wavs
<SpectralCoord
(observer to target:
radial_velocity=115304.79153846153 km / s
redshift=0.5)
[500., 550., 600., 650., 700., 750., 800., 850., 900.] nm>
We have set redshift=0.5 here so that we can keep track of what frame of reference
our spectral values are in. The radial_velocity
property gives the recession
velocity equivalent to that redshift, and it is indeed large enough that we don’t need
to worry about the rotation of the Earth on itself around the Sun (which would be
at most a ~30km/s contribution).
Note
In the context of SpectralCoord
, we use the full relativistic relation
between redshift and velocity, i.e. \(1 + z = \sqrt{(1 + v/c)/(1  v/c)}\)
We now want to shift the wavelengths so that they would be in the rest frame of
the galaxy. We can do this using the
to_rest()
method:
>>> wavs_rest = wavs.to_rest()
>>> wavs_rest
<SpectralCoord
(observer to target:
radial_velocity=0.0 km / s
redshift=0.0)
[333.33333333, 366.66666667, 400. , 433.33333333, 466.66666667,
500. , 533.33333333, 566.66666667, 600. ] nm>
The wavelengths have decreased by 1/3, which is what we expect for z=0.5. Note
that the redshift
and radial_velocity
properties are now zero, since we
are in the reference frame of the target. We can also use the
with_radial_velocity_shift()
method to more
generically apply redshift and velocity corrections. The simplest way to use
this method is to give a single value that will be applied to the target  if
this value does not have units, it is interpreted as a redshift:
>>> wavs_orig = wavs_rest.with_radial_velocity_shift(0.5)
>>> wavs_orig
<SpectralCoord
(observer to target:
radial_velocity=115304.79153846153 km / s
redshift=0.5)
[500., 550., 600., 650., 700., 750., 800., 850., 900.] nm>
This returns an object equivalent to the one we started with, since we’ve
reapplied a redshift of 0.5. We could also provide a velocity as a Quantity
:
>>> wavs_rest.with_radial_velocity_shift(100000 * u.km / u.s)
<SpectralCoord
(observer to target:
radial_velocity=100000.0 km / s
redshift=0.41458078170200463)
[471.52692723, 518.67961996, 565.83231268, 612.9850054 , 660.13769813,
707.29039085, 754.44308357, 801.5957763 , 848.74846902] nm>
which shifts the values to a frame of reference at a redshift of approximately 0.33 (that is, if the spectrum did contain a contribution from an object at z=0.33, these would be the rest wavelengths for that object.
Specifying an observer and a target explicitly¶
To use the more advanced functionality in SpectralCoord
, including the ability
to easily transform between different welldefined velocity frames, you will
need to give it information about the location (and optionally velocity) of
the observer and target. This is done by passing either coordinate frame objects
or SkyCoord
objects. To take a concrete example, let’s assume that we are now
observe the source T Tau using the ALMA telescope. To create an observer object
corresponding to this, we can make use of the EarthLocation
class:
>>> from astropy.coordinates import EarthLocation
>>> location = EarthLocation.of_site('ALMA')
>>> location
<EarthLocation (2225015.30883296, 5440016.41799762, 2481631.27428014) m>
The three values in meters are geocentric coordinates, i.e. the 3D coordinates
relative to the center of the Earth. See EarthLocation
for more details about
the different ways of creating these kinds of objects.
Once you have done this, you will need to convert location
to a coordinate
object using the get_itrs()
method,
which takes the observation time (which is important to know for any kind of
velocity frame transformation):
>>> from astropy.time import Time
>>> alma = location.get_itrs(obstime=Time('20190424T02:32:10'))
>>> alma
<ITRS Coordinate (obstime=20190424T02:32:10.000): (x, y, z) in m
(2225015.30883296, 5440016.41799762, 2481631.27428014)>
ITRS here stands for International Terrestrial Reference System which is a 3D coordinate frame centered on the Earth’s center and rotating with the Earth, so the observatory will be stationary in this frame of reference.
For the target, the simplest way is to use the SkyCoord
class:
>>> from astropy.coordinates import SkyCoord
>>> ttau = SkyCoord('04h21m59.43s +19d32m06.4', frame='icrs',
... radial_velocity=23.9 * u.km / u.s,
... distance=144.321 * u.pc)
In this case we specified a radial velocity and a distance for the target (using the T Tauri SIMBAD entry, but it is also possible to not specify these, which means the target is assumed to be stationary in the frame in which it is observed, and are assumed to be at large distance from the Sun (such that any parallax effects would be unimportant if relevant). The radial velocity is assumed to be in the frame used to define the target location, so it is relative to the ICRS origin (the Solar System barycenter) in the above case.
We now define a set of frequencies corresponding to the channels in which fluxes have been measured (for the purposes of the example here we will assume we have only 11 frequencies):
>>> sc_ttau = SpectralCoord(np.linspace(200, 300, 11) * u.GHz,
... observer=alma, target=ttau)
>>> sc_ttau
<SpectralCoord
(observer: <ITRS Coordinate (obstime=20190424T02:32:10.000): (x, y, z) in m
(2225015.30883296, 5440016.41799762, 2481631.27428014)
(v_x, v_y, v_z) in km / s
(0., 0., 0.)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=41.03594947739035 km / s
redshift=0.00013689056309340586)
[200., 210., 220., 230., 240., 250., 260., 270., 280., 290., 300.] GHz>
We can already see above that SpectralCoord
has computed the difference in
velocity between the observatory and T Tau, which includes the motion of the
observatory around the Earth, the motion of the Earth around the Solar System
barycenter, and the radial velocity of T Tau relative to the Solar System
barycenter. We can get this value directly with:
>>> sc_ttau.radial_velocity
<Quantity 41.03594948 km / s>
If you work with any kind of spectral data, you will often need to determine and/or apply velocity corrections due to different frames of reference. For example if you have observations of the same object on the sky taken at different dates, it is common to transform these to a common velocity frame of reference, so that your spectral coordinates are those that would have applied if the observer had been stationary relative to e.g. the Solar System Barycenter. You may also want to transform your spectral coordinates so that they would be in a frame at rest relative to the local standard of rest (LSR), the center of the Milky Way, the Local Group, or even the Cosmic Microwave Background (CMB) dipole.
We can transform our frequencies for the observations of T Tau to different
velocity frames using the
with_observer_stationary_relative_to()
method. This method can take the name of an existing coordinate/velocity frame,
a BaseCoordinateFrame
instance, or any arbitrary
3D position and velocity coordinate object defined either as a
BaseCoordinateFrame
or a SkyCoord
object. Most
commonlyused frames are accessible using strings. For example to transform to a
velocity frame stationary with respect to the center of the Earth (so removing
the effect of the Earth’s rotation), we can use the 'gcrs'
which stands for
Geocentric Celestial Reference System (GCRS):
>>> sc_ttau.with_observer_stationary_relative_to('gcrs')
<SpectralCoord
(observer: <GCRS Coordinate (obstime=20190424T02:32:10.000, obsgeoloc=(0., 0., 0.) m, obsgeovel=(0., 0., 0.) m / s): (x, y, z) in m
(5878853.86171412, 192921.84773269, 2470794.19765021)
(v_x, v_y, v_z) in km / s
(4.33251262e09, 8.96175625e08, 1.49258412e08)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=40.674086368345165 km / s
redshift=0.00013568335316072044)
[200.00024141, 210.00025348, 220.00026555, 230.00027762, 240.00028969,
250.00030176, 260.00031383, 270.0003259 , 280.00033797, 290.00035004,
300.00036211] GHz>
As you can see, the frequencies have changed slightly, which is because we have
removed the Doppler shift caused by the Earth’s rotation (this can also be seen
in the radial_velocity
property, which has changed by ~0.35 km/s. To use a
velocity reference frame relative to the Solar System barycenter, which is the
origin of the International Celestial Reference System (ICRS) system, we can use:
>>> sc_ttau.with_observer_stationary_relative_to('icrs')
<SpectralCoord
(observer: <ICRS Coordinate: (x, y, z) in m
(1.25867767e+11, 7.48979688e+10, 3.24757657e+10)
(v_x, v_y, v_z) in km / s
(0., 0., 0.)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=23.9 km / s
redshift=7.97249967898761e05)
[200.0114322 , 210.01200381, 220.01257542, 230.01314703, 240.01371864,
250.01429025, 260.01486186, 270.01543347, 280.01600508, 290.01657669,
300.0171483 ] GHz>
Note that in this case the total radial velocity between the observer and the target matches what we specified when we set up the target, since it was defined relative to the ICRS origin (the Solar System barycenter). The observer location is still as before, but the observer velocity is now ~1020 km/s in x, y, and z, which is because the observer is now stationary relative to the barycenter so has a significant velocity relative to the surface of the Earth.
We can also transform the frequencies to the Kinematic Local Standard of Rest (LSRK) frame of reference, which is a reference frame commonly used in some branches of astronomy (such as radio astronomy):
>>> sc_ttau.with_observer_stationary_relative_to('lsrk')
<SpectralCoord
(observer: <LSRK Coordinate: (x, y, z) in m
(1.25867767e+11, 7.48979688e+10, 3.24757657e+10)
(v_x, v_y, v_z) in km / s
(0., 0., 0.)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=12.50698856018455 km / s
redshift=4.171969349386906e05)
[200.01903338, 210.01998505, 220.02093672, 230.02188839, 240.02284006,
250.02379172, 260.02474339, 270.02569506, 280.02664673, 290.0275984 ,
300.02855007] GHz>
See Common velocity frames for a list of common velocity frames
available as strings on the SpectralCoord
class.
Since we can give any arbitrary SkyCoord
to the
with_observer_stationary_relative_to()
method, we can also specify the target itself, to find the frequencies in the
rest frame of the target:
>>> sc_ttau_targetframe = sc_ttau.with_observer_stationary_relative_to(sc_ttau.target)
>>> sc_ttau_targetframe
<SpectralCoord
(observer: <ICRS Coordinate: (x, y, z) in m
(1.25867767e+11, 7.48979688e+10, 3.24757657e+10)
(v_x, v_y, v_z) in km / s
(9.34149908, 20.49579745, 7.99178839)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=0.0 km / s
redshift=0.0)
[200.02737811, 210.02874702, 220.03011592, 230.03148483, 240.03285374,
250.03422264, 260.03559155, 270.03696045, 280.03832936, 290.03969826,
300.04106717] GHz>
The radial_velocity
, which is the velocity offset between observer and
target, is now zero.
SpectralCoord
is intended to be versatile and be useful for representing any spectral
values  not just the xaxis of a spectrum, but also for example the
frequencies of spectral features. For example, if we now consider that we found a
spectral feature that appears to have components at the following frequencies
in the frame of reference of the telescope:
>>> sc_feat = SpectralCoord([115.26, 115.266, 115.267] * u.GHz,
... observer=alma, target=ttau)
We can convert these to the rest frame of the target using:
>>> sc_feat_rest = sc_feat.with_observer_stationary_relative_to(sc_feat.target)
>>> sc_feat_rest
<SpectralCoord
(observer: <ICRS Coordinate: (x, y, z) in m
(1.25867767e+11, 7.48979688e+10, 3.24757657e+10)
(v_x, v_y, v_z) in km / s
(9.34149908, 20.49579745, 7.99178839)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=0.0 km / s
redshift=0.0)
[115.27577801, 115.28177883, 115.28277896] GHz>
The frequencies are very close to the rest frequency of the 12CO J=10 molecular line transition, which is 115.2712018 GHz. However, they are not exactly the same, so if the features we see are indeed from 12CO, then they are Doppler shifted compared to what we consider the rest frame of T Tau. We can convert these frequencies to velocities assuming the Doppler shift equation (in this case with the radio convention):
>>> sc_feat_rest.to(u.km / u.s, doppler_convention='radio', doppler_rest=115.27120180 * u.GHz)
<SpectralCoord
(observer: <ICRS Coordinate: (x, y, z) in m
(1.25867767e+11, 7.48979688e+10, 3.24757657e+10)
(v_x, v_y, v_z) in km / s
(9.34149908, 20.49579745, 7.99178839)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=0.0 km / s
redshift=0.0
doppler_rest=115.2712018 GHz
doppler_convention=radio)
[11.90160353, 27.50828545, 30.1093991 ] km / s>
Note that these resulting velocities are different from the radial_velocity
property (which is still zero here)  the latter is the difference in velocity
between observer and target, while the former are how much the spectral values
are Doppler shifted by relative to the rest frequency or wavelength.
So if the features are indeed from 12CO, they have velocities of approximately 11.9, 27.5 and 30.1 km/s relative to the T tau rest frame.
Common velocity frames¶
Any valid astropy coordinate frame can be passed to the
with_observer_stationary_relative_to()
method, including string aliases such as icrs
. Below we list some of the
frames commonly used to define spectral coordinates in:
The velocity frames available as constants on the SpectralCoord
class are:
Frame name 
Description 


Geocentric frame (defined as stationary relative to the GCRS origin) 

Barycentric frame (defined as stationary relative to the ICRS origin) 

Heliocentric frame (defined as stationary relative to the HCRS origin) 

Kinematic Local Standard of Rest (LSRK), defined as having a velocity of 20 km/s towards 18h +30d (B1900) relative to the Solar System Barycenter 1. 

Dynamical Local Standard of Rest (LSRD), defined as having a velocity of U=9 km/s, V=12 km/s, and W=7 km/s in Galactic coordinates (equivalent to 16.552945 km/s towards l=53.13 and b=25.02) 2. 

A more recent definition of the Local Standard of rest, with U=11.1 km/s, V=12.24 km/s, and W=7.25 km/s in Galactic coordinates 3. 
Defining custom velocity frames¶
As mentioned in the earlier examples on this page, it is possible to pass any
arbitrary BaseCoordinateFrame
or SkyCoord
object
to the with_observer_stationary_relative_to()
method,
and the observer will be updated to be stationary relative to those coordinates.
As an example, we can define an object that can be used to define a velocity
frame that moves with the local group of galaxies. There is not a unique definition
of this, but for the purposes of this example we use the IAU 1976recommended
value which states that the Solar System barycenter is moving at 300 km/s towards
l=90 and b=0 in the velocity frame of the local group of galaxies 4. Given
this value, we can define the velocity frame using:
>>> from astropy.coordinates import Galactic
>>> localgroup_frame = Galactic(u=0 * u.km, v=0 * u.km, w=0 * u.km,
... U=0 * u.km / u.s, V=300 * u.km / u.s, W=0 * u.km / u.s,
... representation_type='cartesian',
... differential_type='cartesian')
Note that here we specify the velocity as 300, because what we need here is the
velocity of the local group relative to the Solar System barycenter. With this
object, we can then transform a SpectralCoord
so that the observer is stationary
in that frame of reference:
>>> sc_ttau.with_observer_stationary_relative_to(localgroup_frame)
<SpectralCoord
(observer: <Galactic Coordinate: (u, v, w) in m
(8.8038652e+10, 5.31344273e+10, 1.09238291e+11)
(U, V, W) in km / s
(1.42108547e14, 300., 2.84217094e14)>
target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
(65.497625, 19.53511111, 144.321)
(pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
(1.37949782e15, 1.46375638e15, 23.9)>
observer to target (computed from above):
radial_velocity=42.33062895275233 km / s
redshift=0.00014120974955456056)
[199.99913628, 209.9990931 , 219.99904991, 229.99900673, 239.99896354,
249.99892036, 259.99887717, 269.99883398, 279.9987908 , 289.99874761,
299.99870443] GHz>
References¶
 1
Meeks, M. L. 1976, Methods of experimental physics. Vol._12. Astrophysics. Part C: Radio observations, Section 6.1 by Gordon, M. A. [ADS].
 2
Delhaye, J. 1965, Galactic Structure. Edited by Adriaan Blaauw and Maarten Schmidt. Published by the University of Chicago Press, p61 [ADS].
 3
Schönrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829 [ADS].
 4
Transactions of the IAU Vol. XVI B Proceedings of the 16th General Assembly, Reports of Meetings of Commissions: Comptes Rendus Des Séances Des Commissions, Commission 28. [DOI]