Coordinates and units

This is the second in a series of notebooks related to astronomy data.

As a running example, we are replicating parts of the analysis in a recent paper, “Off the beaten path: Gaia reveals GD-1 stars outside of the main stream” by Adrian M. Price-Whelan and Ana Bonaca.

In the first notebook, we wrote ADQL queries and used them to select and download data from the Gaia server.

In this notebook, we’ll pick up where we left off and write a query to select stars from the region of the sky where we expect GD-1 to be.


We’ll start with an example that does a “cone search”; that is, it selects stars that appear in a circular region of the sky.

Then, to select stars in the vicinity of GD-1, we’ll:

  • Use Quantity objects to represent measurements with units.

  • Use Astropy to convert coordinates from one frame to another.

  • Use the ADQL keywords POLYGON, CONTAINS, and POINT to select stars that fall within a polygonal region.

  • Submit a query and download the results.

  • Store the results in a FITS file.

After completing this lesson, you should be able to

  • Use Python string formatting to compose more complex ADQL queries.

  • Work with coordinates and other quantities that have units.

  • Download the results of a query and store them in a file.

Selecting a region

One of the most common ways to restrict a query is to select stars in a particular region of the sky.

For example, here’s a query from the Gaia archive documentation that selects “all the objects … in a circular region centered at (266.41683, -29.00781) with a search radius of 5 arcmin (0.08333 deg).”

query = """
TOP 10 source_id
FROM gaiadr2.gaia_source
  POINT(ra, dec),
  CIRCLE(266.41683, -29.00781, 0.08333333))

This query uses three keywords that are specific to ADQL (not SQL):

  • POINT: a location in ICRS coordinates, specified in degrees of right ascension and declination.

  • CIRCLE: a circle where the first two values are the coordinates of the center and the third is the radius in degrees.

  • CONTAINS: a function that returns 1 if a POINT is contained in a shape and 0 otherwise.

Here is the documentation of CONTAINS.

A query like this is called a cone search because it selects stars in a cone.

Here’s how we run it.

from astroquery.gaia import Gaia

job = Gaia.launch_job(query)
Created TAP+ (v1.2.1) - Connection:
	Use HTTPS: True
	Port: 443
	SSL Port: 443
Created TAP+ (v1.2.1) - Connection:
	Use HTTPS: True
	Port: 443
	SSL Port: 443
<astroquery.utils.tap.model.job.Job at 0x7f59bd93e490>
result = job.get_results()
Table length=10


When you are debugging queries like this, you can use TOP to limit the size of the results, but then you still don’t know how big the results will be.

An alternative is to use COUNT, which asks for the number of rows that would be selected, but it does not return them.

In the previous query, replace TOP 10 source_id with COUNT(source_id) and run the query again. How many stars has Gaia identified in the cone we searched?

# Solution

query = """
FROM gaiadr2.gaia_source
  POINT(ra, dec),
  CIRCLE(266.41683, -29.00781, 0.08333333))

Getting GD-1 Data

From the Price-Whelan and Bonaca paper, we will try to reproduce Figure 1, which includes this representation of stars likely to belong to GD-1:

The axes of this figure are defined so the x-axis is aligned with the stars in GD-1, and the y-axis is perpendicular.

  • Along the x-axis (\(\phi_1\)) the figure extends from -100 to 20 degrees.

  • Along the y-axis (\(\phi_2\)) the figure extends from about -8 to 4 degrees.

Ideally, we would select all stars from this rectangle, but there are more than 10 million of them, so

  • That would be difficult to work with,

  • As anonymous Gaia users, we are limited to 3 million rows in a single query, and

  • While we are developing and testing code, it will be faster to work with a smaller dataset.

So we’ll start by selecting stars in a smaller rectangle near the center of GD-1, from -55 to -45 degrees \(\phi_1\) and -8 to 4 degrees \(\phi_2\).

But first we let’s see how to represent quantities with units like degrees.

Working with coordinates

Coordinates are physical quantities, which means that they have two parts, a value and a unit.

For example, the coordinate \(30^{\circ}\) has value 30 and its units are degrees.

Until recently, most scientific computation was done with values only; units were left out of the program altogether, often with catastrophic results.

Astropy provides tools for including units explicitly in computations, which makes it possible to detect errors before they cause disasters.

To use Astropy units, we import them like this:

import astropy.units as u

u is an object that contains most common units and all SI units.

You can use dir to list them, but you should also read the documentation.


To create a quantity, we multiply a value by a unit.

quantity = 30 *

The result is a Quantity object.

Jupyter knows how to display Quantities like this:

\[30 \; \mathrm{{}^{\circ}}\]

Transforming coordinates

Astropy provides a SkyCoord object that represents sky coordinates relative to a specified frame.

The following example creates a SkyCoord object that represents the approximate coordinates of Betelgeuse (alf Ori) in the ICRS frame.

from astropy.coordinates import SkyCoord

ra = 88.8 *
dec = 7.4 *
coord_icrs = SkyCoord(ra=ra, dec=dec, frame='icrs')

<SkyCoord (ICRS): (ra, dec) in deg
    (88.8, 7.4)>

SkyCoord provides a function that transforms to other frames. For example, we can transform coords_icrs to Galactic coordinates like this:

coord_galactic = coord_icrs.transform_to('galactic')
<SkyCoord (Galactic): (l, b) in deg
    (199.79693102, -8.95591653)>

To transform to and from GD-1 coordinates, we’ll use a frame defined by Gala, which is an Astropy-affiliated library that provides tools for galactic dynamics.

Gala provides GD1Koposov10, which is “a Heliocentric spherical coordinate system defined by the orbit of the GD-1 stream

from gala.coordinates import GD1Koposov10

gd1_frame = GD1Koposov10()
<GD1Koposov10 Frame>

We can use it to find the coordinates of Betelgeuse in the GD-1 frame, like this:

coord_gd1 = coord_icrs.transform_to(gd1_frame)
<SkyCoord (GD1Koposov10): (phi1, phi2) in deg
    (-94.97222038, 34.5813813)>


Let’s find the location of GD-1 in ICRS coordinates.

  1. Create a SkyCoord object at 0°, 0° in the GD-1 frame.

  2. Transform it to the ICRS frame.

Hint: Because ICRS is built into Astropy, you can specify it by name, icrs (as we did with galactic).

# Solution

coord_gd1 = SkyCoord(0*, 0*, frame=gd1_frame)

# Note: because ICRS is built into Astropy, 
# we can identify it by name

# More formally, we could instantiate it
from astropy.coordinates import ICRS
icrs_frame = ICRS()
<SkyCoord (ICRS): (ra, dec) in deg
    (200., 59.4504341)>

Selecting a rectangle

Now we’ll use these coordinate transformations to define a rectangle in the GD-1 frame and transform it to ICRS.

The following variables define the boundaries of the rectangle in \(\phi_1\) and \(\phi_2\).

phi1_min = -55 *
phi1_max = -45 *
phi2_min = -8 *
phi2_max = 4 *

To represent a rectangle, we’ll use two lists of coordinates and multiply by their units.

def make_rectangle(x1, x2, y1, y2):
    """Return the corners of a rectangle."""
    xs = [x1, x1, x2, x2, x1]
    ys = [y1, y2, y2, y1, y1]
    return xs, ys
phi1_rect, phi2_rect = make_rectangle(
    phi1_min, phi1_max, phi2_min, phi2_max)

phi1_rect and phi2_rect represent the coordinates of the corners of a rectangle in the GD-1 frame.

In order to use them in a Gaia query, we have to convert them to ICRS.

import gala.coordinates as gc

corners = SkyCoord(phi1=phi1_rect, phi2=phi2_rect, frame=gd1_frame)
<SkyCoord (GD1Koposov10): (phi1, phi2) in deg
    [(-55., -8.), (-55.,  4.), (-45.,  4.), (-45., -8.), (-55., -8.)]>

Now we can use transform_to to convert to ICRS coordinates.

import astropy.coordinates as coord

corners_icrs = corners.transform_to('icrs')
<SkyCoord (ICRS): (ra, dec) in deg
    [(146.27533314, 19.26190982), (135.42163944, 25.87738723),
     (141.60264825, 34.3048303 ), (152.81671045, 27.13611254),
     (146.27533314, 19.26190982)]>

Notice that a rectangle in one coordinate system is not necessarily a rectangle in another. In this example, the result is a polygon.

Selecting a polygon

In order to use this polygon as part of an ADQL query, we have to convert it to a string with a comma-separated list of coordinates, as in this example:

POLYGON(143.65, 20.98, 
        134.46, 26.39, 
        140.58, 34.85, 
        150.16, 29.01)

The following function does the job:

def skycoord_to_string(skycoord):
    """Convert SkyCoord to string."""
    t = skycoord.to_string()
    s = ' '.join(t)
    return s.replace(' ', ', ')

SkyCoord provides to_string, which returns a list of strings.

We use join to make a single string with spaces between the coordinates.

Then we use replace to add commas between the coordinates.

point_list = skycoord_to_string(corners_icrs)
'146.275, 19.2619, 135.422, 25.8774, 141.603, 34.3048, 152.817, 27.1361, 146.275, 19.2619'

Before we can assemble the query, we need columns again (as we saw in the previous notebook).

columns = 'source_id, ra, dec, pmra, pmdec, parallax, parallax_error, radial_velocity'

Here’s the base for the query, with format specifiers for columns and point_list.

query_base = """SELECT {columns}
FROM gaiadr2.gaia_source
WHERE parallax < 1
  AND bp_rp BETWEEN -0.75 AND 2 
  AND 1 = CONTAINS(POINT(ra, dec), 

And here’s the result:

query = query_base.format(columns=columns, 
SELECT source_id, ra, dec, pmra, pmdec, parallax, parallax_error, radial_velocity
FROM gaiadr2.gaia_source
WHERE parallax < 1
  AND bp_rp BETWEEN -0.75 AND 2 
  AND 1 = CONTAINS(POINT(ra, dec), 
                   POLYGON(146.275, 19.2619, 135.422, 25.8774, 141.603, 34.3048, 152.817, 27.1361, 146.275, 19.2619))

As always, we should take a minute to proof-read the query before we launch it.

The result will be bigger than our previous queries, so it will take a little longer.

job = Gaia.launch_job_async(query)
INFO: Query finished. [astroquery.utils.tap.core]
<Table length=140339>
      name       dtype    unit                              description                             n_bad 
--------------- ------- -------- ------------------------------------------------------------------ ------
      source_id   int64          Unique source identifier (unique within a particular Data Release)      0
             ra float64      deg                                                    Right ascension      0
            dec float64      deg                                                        Declination      0
           pmra float64 mas / yr                         Proper motion in right ascension direction      0
          pmdec float64 mas / yr                             Proper motion in declination direction      0
       parallax float64      mas                                                           Parallax      0
 parallax_error float64      mas                                         Standard error of parallax      0
radial_velocity float64   km / s                                                    Radial velocity 139373
Jobid: 1609277169233O
Owner: None
Output file: async_20201229162609.vot
Results: None

Here are the results.

results = job.get_results()

There are more than 100,000 stars in this polygon, but that’s a manageable size to work with.

Saving results

This is the set of stars we’ll work with in the next step. But since we have a substantial dataset now, this is a good time to save it.

Storing the data in a file means we can shut down this notebook and pick up where we left off without running the previous query again.

Astropy Table objects provide write, which writes the table to disk.

filename = 'gd1_results.fits'
results.write(filename, overwrite=True)

Because the filename ends with fits, the table is written in the FITS format, which preserves the metadata associated with the table.

If the file already exists, the overwrite argument causes it to be overwritten.

To see how big the file is, we can use ls with the -lh option, which prints information about the file including its size in human-readable form.

!ls -lh gd1_results.fits
-rw-rw-r-- 1 downey downey 8.6M Dec 29 11:47 gd1_results.fits

The file is about 8.6 MB. If you are using Windows, ls might not work; in that case, try:

!dir gd1_results.fits


In this notebook, we composed more complex queries to select stars within a polygonal region of the sky. Then we downloaded the results and saved them in a FITS file.

In the next notebook, we’ll reload the data from this file and replicate the next step in the analysis, using proper motion to identify stars likely to be in GD-1.

Best practices

  • For measurements with units, use Quantity objects that represent units explicitly and check for errors.

  • Use the format function to compose queries; it is often faster and less error-prone.

  • Develop queries incrementally: start with something simple, test it, and add a little bit at a time.

  • Once you have a query working, save the data in a local file. If you shut down the notebook and come back to it later, you can reload the file; you don’t have to run the query again.