# Working with velocities in Astropy coordinates¶

## Using velocities with `SkyCoord`

¶

The easiest way of getting a coordinate object with velocities is to use the
`SkyCoord`

interface. For example, 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
<SkyCoord (ICRS): (ra, dec) in deg
( 1., 2.)
(radial_velocity) in km / s
( 20.,)>
>>> sc.radial_velocity
<Quantity 20.0 km / s>
```

`SkyCoord`

objects created in this manner follow all 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
<SkyCoord (Galactic): (l, b) in deg
( 99.63785528, -58.70969293)
(pm_l_cosb, pm_b) in mas / yr
( 0.22240398, 0.02316181)>
```

For more details on valid operations and limitations of velocity support in
`astropy.coordinates`

(particularly the current accuracy limitations ), 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.

## Creating frame objects with velocity data¶

The coordinate frame classes support storing and transforming velocity data
(along side 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
Differentials and derivatives of Representations). 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
`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`

):

```
>>> 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)
<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)
<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)
<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
`Quantity`

objects.

The expected differential class can be changed to control the argument names
that the frame expects. As mentioned above, by default the proper motions
components are expected to contain the `cos(latitude)`

, but this can be
changed by specifying the `SphericalDifferential`

class
(instead of the default `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)
<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)
<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)
<ICRS Coordinate: (x, y, z) in pc
(103., -11., 93.)
(v_x, v_y, v_z) in km / s
(31., -10., 75.)>
```

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
<CartesianDifferential (d_x, d_y, d_z) in km / s
( 23.03160789, 7.44794505, 11.34587732)>
```

There are also shorthands for retrieving a single `Quantity`

object that contains the two-dimensional proper motion data, and for retrieving
the radial (line-of-sight) velocity:

```
>>> icrs.proper_motion
<Quantity [ 4.8 ,-15.16] mas / yr>
>>> icrs.radial_velocity
<Quantity 23.42 km / s>
```

## 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 transforming position-only frame instances:

```
>>> 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)
>>> icrs.transform_to(Galactic)
<Galactic Coordinate: (l, b) in deg
(120.38084191, -9.69872044)
(pm_l_cosb, pm_b) in mas / yr
(3.78957965, -15.44359693)>
```

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
`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
`BaseAffineTransform`

subclasses:
`StaticMatrixTransform`

,
`DynamicMatrixTransform`

, and
`AffineTransform`

.

Matrix-only transformations (e.g., rotations such as
`ICRS`

to `Galactic`

) can be performed
on proper-motion-only data or full-space, 3D velocities:

```
>>> 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)
<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 requires specifying a distance as well),
for example, `ICRS`

to `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)
<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)>
```

### Finite Difference Transformations¶

Some frame transformations cannot be expressed as affine transformations.
For example, transformations from the `AltAz`

frame can
include an atmospheric dispersion correction, which is inherently non-linear.
Additionally, some frames are simply more easily 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 exact
same manner as (by defining a transformation of the
`FunctionTransformWithFiniteDifference`

type).

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

- Transformation of the
directionof the velocity vector that already exists in the starting frame. That is, a frame transformation sometimes involves re-orienting 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`obtime`

frame attribute, although it is an option that can be set when defining a finite difference transformation function.

However, 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. Lets 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:

```
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))
plt.plot_date(time.plot_date, gcrs.radial_velocity.to(u.km/u.s))
plt.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’s consider 100 *kiloparsecs* as the distance. In this case we expect
the same: the radial velocity should be essentially the same in both frames:

```
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))
plt.plot_date(time.plot_date, gcrs.radial_velocity.to(u.km/u.s))
plt.ylabel('RV [km/s]')
```

but this result is clearly nonsense, with values from -1000 to 1000 km/s. The root of the problem here is that the machine precision is not sufficient to computedifferences of order km over distances of order kiloparsecs. Hence, the straightforward finite difference method will not work for this use case with the default values.

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))
>>> trans.finite_difference_dt = 1*u.second # return to default
```

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).

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.

## Radial Velocity Corrections¶

Separately from the above, Astropy supports computing barycentric or
heliocentric radial velocity corrections. While in the future this may simply
be a high-level convenience function using the framework described above, the
current implementation is independent to ensure sufficient accuracy (see
Radial Velocity Corrections and the
`radial_velocity_correction`

API docs for
details).

An example of this is below. It 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)
>>> barycorr.to(u.km/u.s)
<Quantity 20.077135 km / s>
>>> heliocorr = sc.radial_velocity_correction('heliocentric', obstime=Time('2016-6-4'), location=keck)
>>> heliocorr.to(u.km/u.s)
<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
`radial_velocity_correction`

docs for more
information.

### Precision of `radial_velocity_correction`

¶

The correction computed by `radial_velocity_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 you must be aware of.

The first is that one should always use a barycentric correction, as the barycenter is a fixed
point where gravity is constant. Since the heliocentre does not satisfy these conditions, corrections
to the heliocentre are only suitable for low precision work. As a result, and
to increase speed, the heliocentric correction in
`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 `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 & Eastmann (2014). Most important
is that the barycentric correction is, strictly speaking, *multiplicative*, so that one should apply it
as

where \(v_t\) is the true radial velocity, \(v_m\) is the measured radial velocity and \(v_b\)
is the barycentric correction returned by `radial_velocity_correction`

.
Failure to apply the barycentric correction in this way leads to errors of order 3 m/s.

The barycentric correction in `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 any effect related to the finite distance or proper motion of the source.
The Shapiro delay is unlikely to be important unless you seek mm/s precision, but the effects of the
source’s parallax and proper motion can be important at the cm/s level. These effects are likely to be
added to future versions of Astropy along with velocity support for `SkyCoord`

, but in the meantime
see Wright & Eastmann (2014).