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# Using the SpectralCoord Class

Warning

The |SpectralCoord| class at this time does not fully support cases where the observer and target are moving relativistically relative to each other, so care should be taken in those cases. The behavior of instances of multiple spectral values combined with multiple target frames as an N * M dimensional |SpectralCoord| has also not yet been finalised. It is possible that there will be API changes in future versions of Astropy based on user feedback.

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 sub-class 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 telescope-centric (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 sub-class 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 possible 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 the |SpectralCoord|

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 x-axis 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 re-applied 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 well-defined 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('2019-04-24T02:32:10'))
    >>> alma
    <ITRS Coordinate (obstime=2019-04-24T02:32:10.000, location=(0., 0., 0.) km): (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=2019-04-24T02:32:10.000, location=(0., 0., 0.) km): (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)
                 (radial_velocity) in km / s
                    (23.9,)>
        observer to target (computed from above):
          radial_velocity=41.03594953774002 km / s
          redshift=0.00013689056329480032)
      [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 commonly-used 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=2019-04-24T02:32:10.000, 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
                      (4.33251262e-09, 8.96175625e-08, -1.49258412e-08)>
        target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
                    (65.497625, 19.53511111, 144.321)
                 (radial_velocity) in km / s
                    (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)
                 (radial_velocity) in km / s
                    (23.9,)>
        observer to target (computed from above):
          radial_velocity=23.9 km / s
          redshift=7.97249967898761e-05)
      [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 ~10-20 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)
                 (radial_velocity) in km / s
                    (23.9,)>
        observer to target (computed from above):
          radial_velocity=12.50698856018455 km / s
          redshift=4.171969349386906e-05)
      [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)
                 (radial_velocity) in km / s
                    (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 x-axis 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)
                 (radial_velocity) in km / s
                    (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=1-0 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)
                 (radial_velocity) in km / s
                    (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)
      [-11.90160348, -27.50828539, -30.10939905] 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
--- | ---
`'gcrs'` | Geocentric frame (defined as stationary relative to the GCRS origin)
`'icrs'` | Barycentric frame (defined as stationary relative to the ICRS origin)
`'hcrs'` | Heliocentric frame (defined as stationary relative to the HCRS origin)
`'lsrk` | 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].
`'lsrd'` | 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].
`'lsr'` | 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 1976-recommended 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
                      (-1.42108547e-14, -300., 2.84217094e-14)>
        target: <ICRS Coordinate: (ra, dec, distance) in (deg, deg, pc)
                    (65.497625, 19.53511111, 144.321)
                 (radial_velocity) in km / s
                    (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]