.. _astropy-coordinates-velocities:

Working with Velocities in Astropy Coordinates
**********************************************

Using Velocities with ``SkyCoord``
==================================

The best way to start getting a coordinate object with velocities is to use the
|SkyCoord| interface.

Examples
--------

..
  EXAMPLE START
  Using SkyCoord to Get Coordinate Objects with Velocities

A |SkyCoord| to represent a star with a measured radial velocity but unknown
proper motion and distance could be created as::

    >>> from astropy.coordinates import SkyCoord
    >>> import astropy.units as u
    >>> sc = SkyCoord(1*u.deg, 2*u.deg, radial_velocity=20*u.km/u.s)
    >>> sc  # doctest: +FLOAT_CMP
    <SkyCoord (ICRS): (ra, dec) in deg
        (1., 2.)
     (radial_velocity) in km / s
        (20.,)>
    >>> sc.radial_velocity  # doctest: +FLOAT_CMP
    <Quantity 20.0 km / s>

|SkyCoord| objects created in this manner follow all of the same transformation
rules and will correctly update their velocities when transformed to other
frames. For example, to determine proper motions in Galactic coordinates for
a star with proper motions measured in ICRS::

    >>> sc = SkyCoord(1*u.deg, 2*u.deg, pm_ra_cosdec=.2*u.mas/u.yr, pm_dec=.1*u.mas/u.yr)
    >>> sc.galactic  # doctest: +FLOAT_CMP
    <SkyCoord (Galactic): (l, b) in deg
      ( 99.63785528, -58.70969293)
    (pm_l_cosb, pm_b) in mas / yr
      ( 0.22240398,  0.02316181)>

..
  EXAMPLE END

For more details on valid operations and limitations of velocity support in
`astropy.coordinates` (particularly the :ref:`current accuracy limitations
<astropy-coordinate-finite-difference-velocities>`), see the more detailed
discussions below of velocity support in the lower-level frame objects. All
these same rules apply for |SkyCoord| objects, as they are built directly on top
of the frame classes' velocity functionality detailed here.

.. _astropy-coordinate-custom-frame-with-velocities:

Creating Frame Objects with Velocity Data
=========================================

The coordinate frame classes support storing and transforming velocity data
(alongside the positional coordinate data). Similar to the positional data that
use the ``Representation`` classes to abstract away the particular
representation and allow re-representing from (e.g., Cartesian to Spherical),
the velocity data makes use of ``Differential`` classes to do the
same. (For more information about the differential classes, see
:ref:`astropy-coordinates-differentials`.) Also like the positional data, the
names of the differential (velocity) components depend on the particular
coordinate frame.

Most frames expect velocity data in the form of two proper motion components
and/or a radial velocity because the default differential for most frames is the
`~astropy.coordinates.SphericalCosLatDifferential` class. When supported, the
proper motion components all begin with ``pm_`` and, by default, the
longitudinal component is expected to already include the ``cos(latitude)``
term. For example, the proper motion components for the ``ICRS`` frame are
(``pm_ra_cosdec``, ``pm_dec``).

Examples
--------

..
  EXAMPLE START
  Creating Frame Objects with Proper Motions

To create frame objects with velocity data in the form of proper motion
components::

    >>> from astropy.coordinates import ICRS
    >>> ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...      pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr)  # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec) in deg
        (8.67, 53.09)
     (pm_ra_cosdec, pm_dec) in mas / yr
        (4.8, -15.16)>
    >>> ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...      pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr,
    ...      radial_velocity=23.42*u.km/u.s)  # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec) in deg
        (8.67, 53.09)
     (pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
        (4.8, -15.16, 23.42)>

For proper motion components in the ``Galactic`` frame, the names track the
longitude and latitude names::

    >>> from astropy.coordinates import Galactic
    >>> Galactic(l=11.23*u.degree, b=58.13*u.degree,
    ...          pm_l_cosb=21.34*u.mas/u.yr, pm_b=-55.89*u.mas/u.yr)  # doctest: +FLOAT_CMP
    <Galactic Coordinate: (l, b) in deg
        (11.23, 58.13)
     (pm_l_cosb, pm_b) in mas / yr
        (21.34, -55.89)>

Like the positional data, velocity data must be passed in as
`~astropy.units.Quantity` objects.

..
  EXAMPLE END

..
  EXAMPLE START
  Changing the Differential Class when Creating Frame Objects

The expected differential class can be changed to control the argument names
that the frame expects. By default the proper motion components are expected to
contain the ``cos(latitude)``, but this can be changed by specifying the
`~astropy.coordinates.SphericalDifferential` class (instead of the default
`~astropy.coordinates.SphericalCosLatDifferential`)::

    >>> from astropy.coordinates import SphericalDifferential
    >>> Galactic(l=11.23*u.degree, b=58.13*u.degree,
    ...          pm_l=21.34*u.mas/u.yr, pm_b=-55.89*u.mas/u.yr,
    ...          differential_type=SphericalDifferential)  # doctest: +FLOAT_CMP
    <Galactic Coordinate: (l, b) in deg
        (11.23, 58.13)
     (pm_l, pm_b) in mas / yr
        (21.34, -55.89)>

This works in parallel to specifying the expected representation class, as long
as the differential class is compatible with the representation. For example, to
specify all coordinate and velocity components in Cartesian::

    >>> from astropy.coordinates import (CartesianRepresentation,
    ...                                  CartesianDifferential)
    >>> Galactic(u=103*u.pc, v=-11*u.pc, w=93.*u.pc,
    ...          U=31*u.km/u.s, V=-10*u.km/u.s, W=75*u.km/u.s,
    ...          representation_type=CartesianRepresentation,
    ...          differential_type=CartesianDifferential)  # doctest: +FLOAT_CMP
    <Galactic Coordinate: (u, v, w) in pc
        (103., -11., 93.)
     (U, V, W) in km / s
        (31., -10., 75.)>

Note that the ``Galactic`` frame has special, standard names for Cartesian
position and velocity components. For other frames, these are just ``x,y,z`` and
``v_x,v_y,v_z``::

    >>> ICRS(x=103*u.pc, y=-11*u.pc, z=93.*u.pc,
    ...      v_x=31*u.km/u.s, v_y=-10*u.km/u.s, v_z=75*u.km/u.s,
    ...      representation_type=CartesianRepresentation,
    ...      differential_type=CartesianDifferential)  # doctest: +FLOAT_CMP
    <ICRS Coordinate: (x, y, z) in pc
        (103., -11., 93.)
     (v_x, v_y, v_z) in km / s
        (31., -10., 75.)>

..
  EXAMPLE END

..
  EXAMPLE START
  Shorthands for Convenient Access to Velocity Data in Frame Objects

For any frame with velocity data with any representation, there are also
shorthands that provide easier access to the underlying velocity data in
commonly needed formats. With any frame object with 3D velocity data, the 3D
Cartesian velocity can be accessed with::

    >>> icrs = ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...             distance=171*u.pc,
    ...             pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr,
    ...             radial_velocity=23.42*u.km/u.s)
    >>> icrs.velocity # doctest: +FLOAT_CMP
    <CartesianDifferential (d_x, d_y, d_z) in km / s
        ( 23.03160789,  7.44794505,  11.34587732)>

There are also shorthands for retrieving a single `~astropy.units.Quantity`
object that contains the two-dimensional proper motion data, and for retrieving
the radial (line-of-sight) velocity::

    >>> icrs.proper_motion # doctest: +FLOAT_CMP
    <Quantity [  4.8 ,-15.16] mas / yr>
    >>> icrs.radial_velocity # doctest: +FLOAT_CMP
    <Quantity 23.42 km / s>

..
  EXAMPLE END

Adding Velocities to Existing Frame Objects
===========================================

Another use case similar to the above comes up when you have an existing frame
object (or |SkyCoord|) and want an object with the same position but with
velocities added. The most conceptually direct way to do this is to
use the differential objects along with
`~astropy.coordinates.BaseCoordinateFrame.realize_frame`.

Examples
--------

..
  EXAMPLE START
  Adding Velocities to Existing Frame Objects

The following snippet accomplishes a well-defined case where the desired
velocities are known in the Cartesian representation. To add the velocities to
the existing frame using
`~astropy.coordinates.BaseCoordinateFrame.realize_frame`::

    >>> icrs = ICRS(1*u.deg, 2*u.deg, distance=3*u.kpc)
    >>> icrs # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec, distance) in (deg, deg, kpc)
        (1., 2., 3.)>
    >>> vel_to_add = CartesianDifferential(4*u.km/u.s, 5*u.km/u.s, 6*u.km/u.s)
    >>> newdata = icrs.data.to_cartesian().with_differentials(vel_to_add)
    >>> icrs.realize_frame(newdata) # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec, distance) in (deg, deg, kpc)
        (1., 2., 3.)
     (pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
        (0.34662023, 0.41161335, 4.29356031)>

A similar mechanism can also be used to add velocities even if full 3D coordinates
are not available (e.g., for a radial velocity observation of an object where
the distance is unknown). However, it requires a slightly different way of
specifying the differentials because of the lack of explicit unit information::

    >>> from astropy.coordinates import RadialDifferential
    >>> icrs_no_distance = ICRS(1*u.deg, 2*u.deg)
    >>> icrs_no_distance
    <ICRS Coordinate: (ra, dec) in deg
        (1., 2.)>
    >>> rv_to_add = RadialDifferential(500*u.km/u.s)
    >>> data_with_rv = icrs_no_distance.data.with_differentials({'s':rv_to_add})
    >>> icrs_no_distance.realize_frame(data_with_rv) # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec) in deg
        (1., 2.)
     (radial_velocity) in km / s
        (500.,)>

Which we can see yields an object identical to what you get when you specify a
radial velocity when you create the object::

    >>> ICRS(1*u.deg, 2*u.deg, radial_velocity=500*u.km/u.s) # doctest: +FLOAT_CMP
    <ICRS Coordinate: (ra, dec) in deg
        (1., 2.)
     (radial_velocity) in km / s
        (500.,)>

..
  EXAMPLE END

.. _astropy-coordinate-transform-with-velocities:

Transforming Frames with Velocities
===================================

Transforming coordinate frame instances that contain velocity data to a
different frame (which may involve both position and velocity transformations)
is done exactly the same way as transforming position-only frame instances.

Example
-------

..
  EXAMPLE START
  Transforming Coordinate Frames with Velocities

To transform a coordinate frame that contains velocity data::

    >>> from astropy.coordinates import Galactic
    >>> icrs = ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...             pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr)  # doctest: +FLOAT_CMP
    >>> icrs.transform_to(Galactic()) # doctest: +FLOAT_CMP
    <Galactic Coordinate: (l, b) in deg
        (120.38084191, -9.69872044)
     (pm_l_cosb, pm_b) in mas / yr
        (3.78957965, -15.44359693)>

..
  EXAMPLE END

However, the details of how the velocity components are transformed depends on
the particular set of transforms required to get from the starting frame to the
desired frame (i.e., the path taken through the frame transform graph). If all
frames in the chain of transformations are transformed to each other via
`~astropy.coordinates.BaseAffineTransform` subclasses (i.e., are matrix
transformations or affine transformations), then the transformations can be
applied explicitly to the velocity data. If this is not the case, the velocity
transformation is computed numerically by finite-differencing the positional
transformation. See the subsections below for more details about these two
methods.

Affine Transformations
----------------------

Frame transformations that involve a rotation and/or an origin shift and/or
a velocity offset are implemented as affine transformations using the
`~astropy.coordinates.BaseAffineTransform` subclasses:
`~astropy.coordinates.StaticMatrixTransform`,
`~astropy.coordinates.DynamicMatrixTransform`, and
`~astropy.coordinates.AffineTransform`.

Matrix-only transformations (e.g., rotations such as
`~astropy.coordinates.ICRS` to `~astropy.coordinates.Galactic`) can be performed
on proper-motion-only data or full-space, 3D velocities.

Examples
^^^^^^^^

..
  EXAMPLE START
  Affine Frame Transformations

To perform a matrix-only transformation::

    >>> icrs = ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...             pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr,
    ...             radial_velocity=23.42*u.km/u.s)
    >>> icrs.transform_to(Galactic())  # doctest: +FLOAT_CMP
    <Galactic Coordinate: (l, b) in deg
        (120.38084191, -9.69872044)
     (pm_l_cosb, pm_b, radial_velocity) in (mas / yr, mas / yr, km / s)
        (3.78957965, -15.44359693, 23.42)>

The same rotation matrix is applied to both the position vector and the velocity
vector. Any transformation that involves a velocity offset requires all 3D
velocity components (which typically require specifying a distance as well),
for example, `~astropy.coordinates.ICRS` to `~astropy.coordinates.LSR`::

    >>> from astropy.coordinates import LSR
    >>> icrs = ICRS(ra=8.67*u.degree, dec=53.09*u.degree,
    ...             distance=117*u.pc,
    ...             pm_ra_cosdec=4.8*u.mas/u.yr, pm_dec=-15.16*u.mas/u.yr,
    ...             radial_velocity=23.42*u.km/u.s)
    >>> icrs.transform_to(LSR())  # doctest: +FLOAT_CMP
    <LSR Coordinate (v_bary=(11.1, 12.24, 7.25) km / s): (ra, dec, distance) in (deg, deg, pc)
        (8.67, 53.09, 117.)
     (pm_ra_cosdec, pm_dec, radial_velocity) in (mas / yr, mas / yr, km / s)
        (-24.51315607, -2.67935501, 27.07339176)>

..
  EXAMPLE END

.. _astropy-coordinate-finite-difference-velocities:

Finite Difference Transformations
---------------------------------

Some frame transformations cannot be expressed as affine transformations.
For example, transformations from the `~astropy.coordinates.AltAz` frame can
include an atmospheric dispersion correction, which is inherently nonlinear.
Additionally, some frames are more conveniently implemented as functions, even
if they can be cast as affine transformations. For these frames, a finite
difference approach to transforming velocities is available. Note that this
approach is implemented such that user-defined frames can use it in
the same manner (i.e., by defining a transformation of the
`~astropy.coordinates.FunctionTransformWithFiniteDifference` type).

This finite difference approach actually combines two separate (but important)
elements of the transformation:

  * Transformation of the *direction* of the velocity vector that already exists
    in the starting frame. That is, a frame transformation sometimes involves
    reorienting the coordinate frame (e.g., rotation), and the velocity vector
    in the new frame must account for this. The finite difference approach
    models this by moving the position of the starting frame along the velocity
    vector, and computing this offset in the target frame.
  * The "induced" velocity due to motion of the frame *itself*. For example,
    shifting from a frame centered at the solar system barycenter to one
    centered on the Earth includes a velocity component due entirely to the
    Earth's motion around the barycenter. This is accounted for by computing
    the location of the starting frame in the target frame at slightly different
    times, and computing the difference between those. Note that this step
    depends on assuming that a particular frame attribute represents a "time"
    of relevance for the induced velocity. By convention this is typically the
    ``obstime`` frame attribute, although it is an option that can be set when
    defining a finite difference transformation function.

Example
^^^^^^^

..
  EXAMPLE START
  Transforming Velocity Data Between Frames Using a Finite Difference Scheme

It is important to recognize that the finite difference transformations
have inherent limits set by the finite difference algorithm and machine
precision. To illustrate this problem, consider the AltAz to GCRS  (i.e.,
geocentric) transformation. Let us try to compute the radial velocity in the
GCRS frame for something observed from the Earth at a distance of 100 AU with a
radial velocity of 10 km/s:

.. plot::
    :context: reset
    :include-source:

    import numpy as np
    from matplotlib import pyplot as plt

    from astropy import units as u
    from astropy.time import Time
    from astropy.coordinates import EarthLocation, AltAz, GCRS

    time = Time('J2010') + np.linspace(-1,1,1000)*u.min
    location = EarthLocation(lon=0*u.deg, lat=45*u.deg)
    aa = AltAz(alt=[45]*1000*u.deg, az=90*u.deg, distance=100*u.au,
               radial_velocity=[10]*1000*u.km/u.s,
               location=location, obstime=time)
    gcrs = aa.transform_to(GCRS(obstime=time))

    fig, ax = plt.subplots()
    ax.scatter(time.datetime, gcrs.radial_velocity.to(u.km / u.s))
    ax.set_ylabel("RV [km/s]")

This seems plausible: the radial velocity should indeed be very close to 10 km/s
because the frame does not involve a velocity shift.

Now let us consider 100 *kiloparsecs* as the distance. In this case we expect
the same: the radial velocity should be essentially the same in both frames:

.. plot::
    :context:
    :include-source:

    time = Time('J2010') + np.linspace(-1,1,1000)*u.min
    location = EarthLocation(lon=0*u.deg, lat=45*u.deg)
    aa = AltAz(alt=[45]*1000*u.deg, az=90*u.deg, distance=100*u.kpc,
               radial_velocity=[10]*1000*u.km/u.s,
               location=location, obstime=time)
    gcrs = aa.transform_to(GCRS(obstime=time))

    ax.scatter(time.datetime, gcrs.radial_velocity.to(u.km / u.s))
    ax.set_ylabel("RV [km/s]")

But this result is nonsense, with values from -1000 to 1000 km/s instead of the
~10 km/s we expected. The root of the problem here is that the machine
precision is not sufficient to compute differences on the order of kilometers
over distances on the order of kiloparsecs. Hence, the straightforward finite
difference method will not work for this use case with the default values.

.. testsetup::

    >>> import numpy as np
    >>> from astropy.coordinates import EarthLocation, AltAz, GCRS
    >>> from astropy.time import Time
    >>> time = Time('J2010') + np.linspace(-1,1,1000) * u.min
    >>> location = EarthLocation(lon=0*u.deg, lat=45*u.deg)
    >>> aa = AltAz(alt=[45]*1000*u.deg, az=90*u.deg, distance=100*u.kpc,
    ...            radial_velocity=[10]*1000*u.km/u.s,
    ...            location=location, obstime=time)

It is possible to override the timestep over which the finite difference occurs.
For example::

    >>> from astropy.coordinates import frame_transform_graph, AltAz, CIRS
    >>> trans = frame_transform_graph.get_transform(AltAz, CIRS).transforms[0]
    >>> trans.finite_difference_dt = 1 * u.year
    >>> gcrs = aa.transform_to(GCRS(obstime=time))  # doctest: +REMOTE_DATA
    >>> trans.finite_difference_dt = 1 * u.second  # return to default

In the above example, there is exactly one transformation step from
`~astropy.coordinates.AltAz` to `~astropy.coordinates.GCRS`.  In general, there
may be more than one step between two frames, or the single step may perform
other transformations internally.  One can use the context manager
:func:`~astropy.coordinates.TransformGraph.impose_finite_difference_dt` for the
transformation graph to override ``finite_difference_dt`` for *all*
finite-difference transformations on the graph::

    >>> from astropy.coordinates import frame_transform_graph
    >>> with frame_transform_graph.impose_finite_difference_dt(1 * u.year):
    ...     gcrs = aa.transform_to(GCRS(obstime=time))  # doctest: +REMOTE_DATA

But beware that this will *not* help in cases like the above, where the relevant
timescales for the velocities are seconds. (The velocity of the Earth relative
to a particular direction changes dramatically over the course of one year.)

..
  EXAMPLE END

Future versions of Astropy will improve on this algorithm to make the results
more numerically stable and practical for use in these (not unusual) use cases.

.. _astropy-coordinates-rv-corrs:

Radial Velocity Corrections
===========================

Separately from the above, Astropy supports computing barycentric or
heliocentric radial velocity corrections. While in the future this may
be a high-level convenience function using the framework described above, the
current implementation is independent to ensure sufficient accuracy (see
:ref:`astropy-coordinates-rv-corrs` and the
`~astropy.coordinates.SkyCoord.radial_velocity_correction` API docs for
details).

Example
-------

..
  EXAMPLE START
  Computing Barycentric or Heliocentric Radial Velocity Corrections

This example demonstrates how to compute this correction if observing some
object at a known RA and Dec from the Keck observatory at a particular time. If
a precision of around 3 m/s is sufficient, the computed correction can then be
added to any observed radial velocity to determine the final heliocentric
radial velocity::

    >>> from astropy.time import Time
    >>> from astropy.coordinates import SkyCoord, EarthLocation
    >>> # keck = EarthLocation.of_site('Keck')  # the easiest way... but requires internet
    >>> keck = EarthLocation.from_geodetic(lat=19.8283*u.deg, lon=-155.4783*u.deg, height=4160*u.m)
    >>> sc = SkyCoord(ra=4.88375*u.deg, dec=35.0436389*u.deg)
    >>> barycorr = sc.radial_velocity_correction(obstime=Time('2016-6-4'), location=keck)  # doctest: +REMOTE_DATA
    >>> barycorr.to(u.km/u.s)  # doctest: +REMOTE_DATA +FLOAT_CMP
    <Quantity 20.077135 km / s>
    >>> heliocorr = sc.radial_velocity_correction('heliocentric', obstime=Time('2016-6-4'), location=keck)  # doctest: +REMOTE_DATA
    >>> heliocorr.to(u.km/u.s)  # doctest: +REMOTE_DATA +FLOAT_CMP
    <Quantity 20.070039 km / s>

Note that there are a few different ways to specify the options for the
correction (e.g., the location, observation time, etc.). See the
`~astropy.coordinates.SkyCoord.radial_velocity_correction` docs for more
information.

..
  EXAMPLE END

Precision of `~astropy.coordinates.SkyCoord.radial_velocity_correction`
------------------------------------------------------------------------

The correction computed by `~astropy.coordinates.SkyCoord.radial_velocity_correction`
uses the optical approximation :math:`v = zc` (see :ref:`astropy-units-doppler-equivalencies`
for details). The correction can be added to any observed radial velocity
to provide a correction that is accurate to a level of approximately 3 m/s.
If you need more precise corrections, there are a number of subtleties of
which you must be aware.

The first is that you should always use a barycentric correction, as the
barycenter is a fixed point where gravity is constant. Since the heliocenter
does not satisfy these conditions, corrections to the heliocenter are only
suitable for low precision work. As a result, and to increase speed, the
heliocentric correction in
`~astropy.coordinates.SkyCoord.radial_velocity_correction` does not include
effects such as the gravitational redshift due to the potential at the Earth's
surface. For these reasons, the barycentric correction in
`~astropy.coordinates.SkyCoord.radial_velocity_correction` should always
be used for high precision work.

Other considerations necessary for radial velocity corrections at the cm/s
level are outlined in `Wright & Eastman (2014) <https://ui.adsabs.harvard.edu/abs/2014PASP..126..838W>`_.
Most important is that the barycentric correction is, strictly speaking,
*multiplicative*, so that you should apply it as:

.. math::

    v_t = v_m + v_b + \frac{v_b v_m}{c},

Where :math:`v_t` is the true radial velocity, :math:`v_m` is the measured
radial velocity and :math:`v_b` is the barycentric correction returned by
`~astropy.coordinates.SkyCoord.radial_velocity_correction`. Failure to apply
the barycentric correction in this way leads to errors of order 3 m/s.

The barycentric correction in `~astropy.coordinates.SkyCoord.radial_velocity_correction` is consistent
with the IDL implementation of
the Wright & Eastmann (2014) paper to a level of 10 mm/s for a source at
infinite distance. We do not include the Shapiro delay nor the light
travel time correction from equation 28 of that paper. The neglected terms
are not important unless you require accuracies of better than 1 cm/s.
If you do require that precision, see `Wright & Eastmann (2014) <https://ui.adsabs.harvard.edu/abs/2014PASP..126..838W>`_.
