This lesson is being piloted (Beta version)

Coordinate Transformations

Overview

Teaching: 75 min
Exercises: 18 min
Questions
• How do we transform celestial coordinates from one frame to another and save results in files?

Objectives
• 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.

2. Coordinates and Units

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

In this lesson, we’ll pick up where we left off and write a query to select stars from a particular region of the sky.

Outline

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.

• Store the results in a FITS file.

Working with Units

The measurements we will work with 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.

dir(u)

['A',
'AA',
'AB',
'ABflux',
'ABmag',
'AU',
'Angstrom',
'B',
'Ba',
'Barye',
'Bi',
[Output truncated]


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

angle = 10 * u.degree
type(angle)

astropy.units.quantity.Quantity


The result is a Quantity object. Jupyter knows how to display Quantities like this:

angle

<Quantity 10. deg>


$10 \; \mathrm{^{\circ}}$

Quantities provide a method called to that converts to other units. For example, we can compute the number of arcminutes in angle:

angle_arcmin = angle.to(u.arcmin)
angle_arcmin

<Quantity 600. arcmin>


$600 \; \mathrm{^{\prime}}$

If you add quantities, Astropy converts them to compatible units, if possible:

angle + 30 * u.arcmin

<Quantity 10.5 deg>


$10.5 \; \mathrm{^{\circ}}$

If the units are not compatible, you get an error. For example:

angle + 5 * u.second


causes a UnitConversionError.

Exercise (3 minutes)

Create a quantity that represents 5 arcminutes and assign it to a variable called radius.

Then convert it to degrees.

Solution

radius = 5 * u.arcmin



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 objects in a circular region centered at (88.8, 7.4) with a search radius of 5 arcmin (0.08333 deg).

query_cone = """SELECT
TOP 10
source_id
WHERE 1=CONTAINS(
POINT(ra, dec),
CIRCLE(88.8, 7.4, 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_cone)
job

Created TAP+ (v1.2.1) - Connection:
Host: gea.esac.esa.int
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 0x7f277785fa30>

results = job.get_results()
results

<Table length=10>
source_id
int64
-------------------
3322773965056065536
3322773758899157120
3322774068134271104
3322773930696320512
3322774377374425728
3322773724537891456
3322773724537891328
[Output truncated]


Exercise (5 minutes)

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 = """SELECT
COUNT(source_id)
WHERE 1=CONTAINS(
POINT(ra, dec),
CIRCLE(88.8, 7.4, 0.08333333))
"""

job = Gaia.launch_job(query)
results = job.get_results()
results


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 these coordinates with Astropy.

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.

ICRS is the “International Celestial Reference System”, adopted in 1997 by the International Astronomical Union.

from astropy.coordinates import SkyCoord

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

coord_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')
coord_galactic

<SkyCoord (Galactic): (l, b) in deg
(199.79693102, -8.95591653)>


Notice that in the Galactic frame, the coordinates are called l and b, not ra and dec.

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()
gd1_frame

<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)
coord_gd1

<SkyCoord (GD1Koposov10): (phi1, phi2) in deg
(-94.97222038, 34.5813813)>


Notice that the coordinates are called phi1 and phi2. These are the coordinates shown in the figure from the paper, above.

Exercise (10 minutes)

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

origin_gd1 = SkyCoord(0*u.degree, 0*u.degree, frame=gd1_frame)

# OR

origin_gd1 = SkyCoord(phi1=0*u.degree,
phi2=0*u.degree,
frame=gd1_frame)

# Note: because ICRS is built into Astropy,
# we can identify it by string name
origin_gd1.transform_to('icrs')

# More formally, we could instantiate it
from astropy.coordinates import ICRS
icrs_frame = ICRS()
origin_gd1.transform_to(icrs_frame)


Notice that the origin of the GD-1 frame maps to ra=200, exactly, in ICRS. That’s by design.

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 * u.degree
phi1_max = -45 * u.degree
phi2_min = -8 * u.degree
phi2_max = 4 * u.degree


To create a rectangle, we’ll use the following function, which takes the lower and upper bounds as parameters.

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


The return value is a tuple containing a list of coordinates in phi1 followed by a list of coordinates in phi2.

phi1_rect, phi2_rect = make_rectangle(
phi1_min, phi1_max, phi2_min, phi2_max)


phi1_rect and phi2_rect contains 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. First we’ll put them into a SkyCoord object.

corners = SkyCoord(phi1=phi1_rect, phi2=phi2_rect, frame=gd1_frame)
corners

<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.

corners_icrs = corners.transform_to('icrs')
corners_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 (non-rectangular) polygon.

Defining 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)
"""


SkyCoord provides to_string, which produces a list of strings.

t = corners_icrs.to_string()
t

['146.275 19.2619',
'135.422 25.8774',
'141.603 34.3048',
'152.817 27.1361',
'146.275 19.2619']


We can use the Python string function join to join t into a single string (with spaces between the pairs):

s = ' '.join(t)
s

'146.275 19.2619 135.422 25.8774 141.603 34.3048 152.817 27.1361 146.275 19.2619'


That’s almost what we need, but we have to replace the spaces with commas.

s.replace(' ', ', ')

'146.275, 19.2619, 135.422, 25.8774, 141.603, 34.3048, 152.817, 27.1361, 146.275, 19.2619'


The following function combines these steps.

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


Here’s how we use it.

point_list = skycoord_to_string(corners_icrs)
point_list

'146.275, 19.2619, 135.422, 25.8774, 141.603, 34.3048, 152.817, 27.1361, 146.275, 19.2619'


Assembling the query

Now we’re ready to assemble the query. We need columns again (as we saw in the previous lesson).

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


And here’s the query base we used in the previous lesson:

query3_base = """SELECT
TOP 10
{columns}
WHERE parallax < 1
AND bp_rp BETWEEN -0.75 AND 2
"""


Now we’ll add a WHERE clause to select stars in the polygon we defined.

query4_base = """SELECT
TOP 10
{columns}
WHERE parallax < 1
AND bp_rp BETWEEN -0.75 AND 2
AND 1 = CONTAINS(POINT(ra, dec),
POLYGON({point_list}))
"""


The query base contains format specifiers for columns and point_list.

We’ll use format to fill in these values.

query4 = query4_base.format(columns=columns,
point_list=point_list)
print(query4)

SELECT
TOP 10
source_id, ra, dec, pmra, pmdec
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.

job = Gaia.launch_job_async(query4)
print(job)

INFO: Query finished. [astroquery.utils.tap.core]
<Table length=10>
name    dtype    unit                              description
--------- ------- -------- ------------------------------------------------------------------
source_id   int64          Unique source identifier (unique within a particular Data Release)
ra float64      deg                                                    Right ascension
dec float64      deg                                                        Declination
pmra float64 mas / yr                         Proper motion in right ascension direction
pmdec float64 mas / yr                             Proper motion in declination direction
Jobid: 1615815873808O
Phase: COMPLETED
[Output truncated]


Here are the results.

results = job.get_results()
results

<Table length=10>
source_id              ra         ...        pmdec
deg         ...       mas / yr
int64             float64       ...       float64
------------------ ------------------ ... -------------------
637987125186749568 142.48301935991023 ...   2.941813096629439
638285195917112960 142.25452941346344 ... -12.165984395577347
638073505568978688 142.64528557468074 ...  -7.950659620550862
638086386175786752 142.57739430926034 ...  -2.584105480335548
638049655615392384 142.58913564478618 ...  -4.941079187010136
638267565075964032 141.81762228999614 ...  1.8838892877285924
[Output truncated]


Finally, we can remove TOP 10 run the query again.

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

query5_base = """SELECT
{columns}
WHERE parallax < 1
AND bp_rp BETWEEN -0.75 AND 2
AND 1 = CONTAINS(POINT(ra, dec),
POLYGON({point_list}))
"""

query5 = query5_base.format(columns=columns,
point_list=point_list)
print(query5)

SELECT
source_id, ra, dec, pmra, pmdec
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))

job = Gaia.launch_job_async(query5)
print(job)

INFO: Query finished. [astroquery.utils.tap.core]
<Table length=140339>
name    dtype    unit                              description
--------- ------- -------- ------------------------------------------------------------------
source_id   int64          Unique source identifier (unique within a particular Data Release)
ra float64      deg                                                    Right ascension
dec float64      deg                                                        Declination
pmra float64 mas / yr                         Proper motion in right ascension direction
pmdec float64 mas / yr                             Proper motion in declination direction
Jobid: 1615815886707O
Phase: COMPLETED
[Output truncated]

results = job.get_results()
len(results)

140339


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.

We can use getsize to confirm that the file exists and check the size:

from os.path import getsize

MB = 1024 * 1024
getsize(filename) / MB

5.36407470703125


Summary

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.

Key Points

• 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.