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# Create a new coordinate class (for the Sagittarius stream)

This document describes in detail how to subclass and define a custom spherical coordinate frame, as discussed in Defining a New Frame and the docstring for `BaseCoordinateFrame`. In this example, we will define a coordinate system defined by the plane of orbit of the Sagittarius Dwarf Galaxy (hereafter Sgr; as defined in Majewski et al. 2003). The Sgr coordinate system is often referred to in terms of two angular coordinates, \(\Lambda,B\).

To do this, we need to define a subclass of `BaseCoordinateFrame` that knows the names and units of the coordinate system angles in each of the supported representations. In this case we support `SphericalRepresentation` with “Lambda” and “Beta”. Then we have to define the transformation from this coordinate system to some other built-in system. Here we will use Galactic coordinates, represented by the `Galactic` class.

See also

The gala package

  Defines a number of Astropy coordinate frames for stellar stream coordinate systems.

Majewski et al. 2003

  “A Two Micron All Sky Survey View of the Sagittarius Dwarf Galaxy. I. Morphology of the Sagittarius Core and Tidal Arms”, https://arxiv.org/abs/astro-ph/0304198

Law & Majewski 2010

  “The Sagittarius Dwarf Galaxy: A Model for Evolution in a Triaxial Milky Way Halo”, https://arxiv.org/abs/1003.1132

David Law’s Sgr info page

  https://www.stsci.edu/~dlaw/Sgr/

    >>> import matplotlib.pyplot as plt
    >>> import numpy as np
    >>> import astropy.coordinates as coord
    >>> from astropy import units as u
    >>> from astropy.coordinates import frame_transform_graph
    >>> from astropy.coordinates.matrix_utilities import matrix_transpose, rotation_matrix

The first step is to create a new class, which we’ll call `Sagittarius` and make it a subclass of `BaseCoordinateFrame`:

    >>> class Sagittarius(coord.BaseCoordinateFrame):
    ...     """A Heliocentric spherical coordinate system defined by the orbit
    ...     of the Sagittarius dwarf galaxy, as described in
    ...         https://ui.adsabs.harvard.edu/abs/2003ApJ...599.1082M
    ...     and further explained in
    ...         https://www.stsci.edu/~dlaw/Sgr/.
    ...
    ...     Parameters
    ...     ----------
    ...     representation : `~astropy.coordinates.BaseRepresentation` or None
    ...         A representation object or None to have no data (or use the other keywords)
    ...     Lambda : `~astropy.coordinates.Angle`, optional, must be keyword
    ...         The longitude-like angle corresponding to Sagittarius' orbit.
    ...     Beta : `~astropy.coordinates.Angle`, optional, must be keyword
    ...         The latitude-like angle corresponding to Sagittarius' orbit.
    ...     distance : `~astropy.units.Quantity`, optional, must be keyword
    ...         The Distance for this object along the line-of-sight.
    ...     pm_Lambda_cosBeta : `~astropy.units.Quantity`, optional, must be keyword
    ...         The proper motion along the stream in ``Lambda`` (including the
    ...         ``cos(Beta)`` factor) for this object (``pm_Beta`` must also be given).
    ...     pm_Beta : `~astropy.units.Quantity`, optional, must be keyword
    ...         The proper motion in Declination for this object (``pm_ra_cosdec`` must
    ...         also be given).
    ...     radial_velocity : `~astropy.units.Quantity`, optional, keyword-only
    ...         The radial velocity of this object.
    ...     """
    ...     default_representation = coord.SphericalRepresentation
    ...     default_differential = coord.SphericalCosLatDifferential
    ...     frame_specific_representation_info = {
    ...         coord.SphericalRepresentation: [
    ...             coord.RepresentationMapping("lon", "Lambda"),
    ...             coord.RepresentationMapping("lat", "Beta"),
    ...             coord.RepresentationMapping("distance", "distance"),
    ...         ]
    ...     }

Breaking this down line-by-line, we define the class as a subclass of `BaseCoordinateFrame`. Then we include a descriptive docstring. The final lines are class-level attributes that specify the default representation for the data, default differential for the velocity information, and mappings from the attribute names used by representation objects to the names that are to be used by the `Sagittarius` frame. In this case we override the names in the spherical representations but do not do anything with other representations like cartesian or cylindrical.

Next we have to define the transformation from this coordinate system to some other built-in coordinate system; we will use Galactic coordinates. We can do this by defining functions that return transformation matrices, or by simply defining a function that accepts a coordinate and returns a new coordinate in the new system. Because the transformation to the Sagittarius coordinate stem is just a spherical rotation from Galactic coordinates, we will define a function that returns this matrix. We will start by constructing the transformation matrix using pre-determined Euler angles and the `rotation_matrix` helper function:

    >>> SGR_PHI = (180 + 3.75) * u.degree  # Euler angles (from Law & Majewski 2010)
    >>> SGR_THETA = (90 - 13.46) * u.degree
    >>> SGR_PSI = (180 + 14.111534) * u.degree

Generate the rotation matrix using the x-convention (see Goldstein):

    >>> SGR_MATRIX = (
    ...     np.diag([1.0, 1.0, -1.0])
    ...     @ rotation_matrix(SGR_PSI, "z")
    ...     @ rotation_matrix(SGR_THETA, "x")
    ...     @ rotation_matrix(SGR_PHI, "z")
    ... )

Since we already constructed the transformation (rotation) matrix above, and the inverse of a rotation matrix is just its transpose, the required transformation functions are very simple:

    >>> @frame_transform_graph.transform(
    ...     coord.StaticMatrixTransform, coord.Galactic, Sagittarius
    ... )
    ... def galactic_to_sgr():
    ...     """Compute the Galactic spherical to heliocentric Sgr transformation matrix."""
    ...     return SGR_MATRIX

The decorator `@frame_transform_graph.transform(coord.StaticMatrixTransform, coord.Galactic, Sagittarius)` registers this function on the `frame_transform_graph` as a coordinate transformation. Inside the function, we return the previously defined rotation matrix.

We then register the inverse transformation by using the transpose of the rotation matrix (which is faster to compute than the inverse):

    >>> @frame_transform_graph.transform(
    ...     coord.StaticMatrixTransform, Sagittarius, coord.Galactic
    ... )
    ... def sgr_to_galactic():
    ...     """Compute the heliocentric Sgr to spherical Galactic transformation matrix."""
    ...     return matrix_transpose(SGR_MATRIX)

Now that we have registered these transformations between `Sagittarius` and `Galactic`, we can transform between any coordinate system and `Sagittarius` (as long as the other system has a path to transform to `Galactic`). For example, to transform from ICRS coordinates to `Sagittarius`, we would do:

    >>> icrs = coord.SkyCoord(280.161732 * u.degree, 11.91934 * u.degree, frame="icrs")
    >>> sgr = icrs.transform_to(Sagittarius)
    >>> print(sgr)
    <SkyCoord (Sagittarius): (Lambda, Beta) in deg
        (346.81830652, -39.28360407)>

Or, to transform from the `Sagittarius` frame to ICRS coordinates (in this case, a line along the `Sagittarius` x-y plane):

    >>> sgr = coord.SkyCoord(
    ...     Lambda=np.linspace(0, 2 * np.pi, 128) * u.radian,
    ...     Beta=np.zeros(128) * u.radian,
    ...     frame="sagittarius",
    ... )
    >>> icrs = sgr.transform_to(coord.ICRS)
    >>> print(icrs)
    <SkyCoord (ICRS): (ra, dec) in deg
        [(284.03876751, -29.00408353), (287.24685769, -29.44848352),
         (290.48068369, -29.81535572), (293.7357366 , -30.1029631 ),
         ...]>

As an example, we will now plot the points in both coordinate systems:

    >>> fig, axes = plt.subplots(2, 1, figsize=(8, 10), subplot_kw={"projection": "aitoff"})
    >>> axes[0].set_title("Sagittarius")
    >>> axes[0].plot(
    ...     sgr.Lambda.wrap_at(180 * u.deg).radian,
    ...     sgr.Beta.radian,
    ...     linestyle="none",
    ...     marker=".",
    ... )
    >>> axes[0].grid(visible=True)
    >>> axes[1].set_title("ICRS")
    >>> axes[1].plot(
    ...     icrs.ra.wrap_at(180 * u.deg).radian, icrs.dec.radian, linestyle="none", marker="."
    ... )
    >>> axes[1].grid(visible=True)

This particular transformation is just a spherical rotation, which is a special case of an Affine transformation with no vector offset. The transformation of velocity components is therefore natively supported as well:

    >>> sgr = coord.SkyCoord(
    ...     Lambda=np.linspace(0, 2 * np.pi, 128) * u.radian,
    ...     Beta=np.zeros(128) * u.radian,
    ...     pm_Lambda_cosBeta=np.random.uniform(-5, 5, 128) * (u.mas / u.yr),
    ...     pm_Beta=np.zeros(128) * (u.mas / u.yr),
    ...     frame="sagittarius",
    ... )
    >>> icrs = sgr.transform_to(coord.ICRS)
    >>> print(icrs)
    <SkyCoord (ICRS): (ra, dec) in deg
        [(284.03876751, -29.00408353), (287.24685769, -29.44848352),
         ...,
         ...]>
    >>> fig, axes = plt.subplots(3, 1, figsize=(8, 10), sharex=True)
    >>> axes[0].set_title("Sagittarius")
    >>> axes[0].plot(
    ...     sgr.Lambda.degree, sgr.pm_Lambda_cosBeta.value, linestyle="none", marker="."
    ... )
    >>> axes[0].set_xlabel(r"$\Lambda$ [deg]")
    >>> axes[0].set_ylabel(
    ...     rf"$\mu_\Lambda \, \cos B$ [{sgr.pm_Lambda_cosBeta.unit.to_string('latex_inline')}]"
    ... )
    >>> axes[0].grid(visible=True)
    >>> axes[1].set_title("ICRS")
    >>> axes[1].plot(icrs.ra.degree, icrs.pm_ra_cosdec.value, linestyle="none", marker=".")
    >>> axes[1].set_ylabel(
    ...     rf"$\mu_\alpha \, \cos\delta$ [{icrs.pm_ra_cosdec.unit.to_string('latex_inline')}]"
    ... )
    >>> axes[1].grid(visible=True)
    >>> axes[2].set_title("ICRS")
    >>> axes[2].plot(icrs.ra.degree, icrs.pm_dec.value, linestyle="none", marker=".")
    >>> axes[2].set_xlabel("RA [deg]")
    >>> axes[2].set_ylabel(rf"$\mu_\delta$ [{icrs.pm_dec.unit.to_string('latex_inline')}]")
    >>> axes[2].grid(visible=True)
    >>> plt.draw()

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