Proper Motion

This is the third 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 lesson, we wrote ADQL queries and used them to select and download data from the Gaia server.

In the second lesson, we wrote a query to select stars from the region of the sky where we expect GD-1 to be, and saved the results in a FITS file.

Now we’ll read that data back and implement the next step in the analysis, identifying stars with the proper motion we expect for GD-1.


Here are the steps in this lesson:

  1. We’ll read back the results from the previous lesson, which we saved in a FITS file.

  2. Then we’ll transform the coordinates and proper motion data from ICRS back to the coordinate frame of GD-1.

  3. We’ll put those results into a Pandas DataFrame, which we’ll use to select stars near the centerline of GD-1.

  4. Plotting the proper motion of those stars, we’ll identify a region of proper motion for stars that are likely to be in GD-1.

  5. Finally, we’ll select and plot the stars whose proper motion is in that region.

After completing this lesson, you should be able to

  • Select rows and columns from an Astropy Table.

  • Use Matplotlib to make a scatter plot.

  • Use Gala to transform coordinates.

  • Make a Pandas DataFrame and use a Boolean Series to select rows.

  • Save a DataFrame in an HDF5 file.

Reload the data

In the previous lesson, we ran a query on the Gaia server and downloaded data for roughly 100,000 stars. We saved the data in a FITS file so that now, picking up where we left off, we can read the data from a local file rather than running the query again.

If you ran the previous lesson successfully, you should already have a file called gd1_results.fits that contains the data we downloaded.

If not, you can run the following cell, which downloads the data from our repository.

import os
from wget import download

filename = 'gd1_results.fits'
path = ''

if not os.path.exists(filename):

Now here’s how we can read the data from the file back into an Astropy Table:

from astropy.table import Table

results =

The result is an Astropy Table.

We can use info to refresh our memory of the contents.
<Table length=140340>
      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
       parallax float64      mas                                                           Parallax
 parallax_error float64      mas                                         Standard error of parallax
radial_velocity float64   km / s                                                    Radial velocity

Selecting rows and columns

In this section we’ll see operations for selecting columns and rows from an Astropy Table. You can find more information about these operations in the Astropy documentation.

We can get the names of the columns like this:


And select an individual column like this:

<Column name='ra' dtype='float64' unit='deg' description='Right ascension' length=140340>

The result is a Column object that contains the data, and also the data type, units, and name of the column.


The rows in the Table are numbered from 0 to n-1, where n is the number of rows. We can select the first row like this:

Row index=0
degdegmas / yrmas / yrmasmaskm / s

As you might have guessed, the result is a Row object.


Notice that the bracket operator selects both columns and rows. You might wonder how it knows which to select.

If the expression in brackets is a string, it selects a column; if the expression is an integer, it selects a row.

If you apply the bracket operator twice, you can select a column and then an element from the column.


Or you can select a row and then an element from the row.


You get the same result either way.

Scatter plot

To see what the results look like, we’ll use a scatter plot. The library we’ll use is Matplotlib, which is the most widely-used plotting library for Python.

The Matplotlib interface is based on MATLAB (hence the name), so if you know MATLAB, some of it will be familiar.

We’ll import like this.

import matplotlib.pyplot as plt

Pyplot part of the Matplotlib library. It is conventional to import it using the shortened name plt.

Pyplot provides two functions that can make scatterplots, plt.scatter and plt.plot.

  • scatter is more versatile; for example, you can make every point in a scatter plot a different color.

  • plot is more limited, but for simple cases, it can be substantially faster.

Jake Vanderplas explains these differences in The Python Data Science Handbook

Since we are plotting more than 100,000 points and they are all the same size and color, we’ll use plot.

Here’s a scatter plot with right ascension on the x-axis and declination on the y-axis, both ICRS coordinates in degrees.

x = results['ra']
y = results['dec']
plt.plot(x, y, 'ko')

plt.xlabel('ra (degree ICRS)')
plt.ylabel('dec (degree ICRS)');

The arguments to plt.plot are x, y, and a string that specifies the style. In this case, the letters ko indicate that we want a black, round marker (k is for black because b is for blue).

The functions xlabel and ylabel put labels on the axes.

This scatter plot has a problem. It is “overplotted”, which means that there are so many overlapping points, we can’t distinguish between high and low density areas.

To fix this, we can provide optional arguments to control the size and transparency of the points.


In the call to plt.plot, use the keyword argument markersize to make the markers smaller.

Then add the keyword argument alpha to make the markers partly transparent.

Adjust these arguments until you think the figure shows the data most clearly.

Note: Once you have made these changes, you might notice that the figure shows stripes with lower density of stars. These stripes are caused by the way Gaia scans the sky, which you can read about here. The dataset we are using, Gaia Data Release 2, covers 22 months of observations; during this time, some parts of the sky were scanned more than others.

# Solution

# x = results['ra']
# y = results['dec']
# plt.plot(x, y, 'ko', markersize=0.1, alpha=0.1)

# plt.xlabel('ra (degree ICRS)')
# plt.ylabel('dec (degree ICRS)');

Transform back

Remember that we selected data from a rectangle of coordinates in the GD1Koposov10 frame, then transformed them to ICRS when we constructed the query. The coordinates in results are in ICRS.

To plot them, we will transform them back to the GD1Koposov10 frame; that way, the axes of the figure are aligned with the orbit of GD-1, which is useful for two reasons:

  • We can identify stars that are likely to be in GD-1 by selecting stars near the centerline of the stream, where \(\phi_2\) is close to 0.

  • We expect stars in GD-1 to have similar proper motion along the \(\phi_1\) axis.

To do the transformation, we’ll put the results into a SkyCoord object.

from astropy.coordinates import SkyCoord
import astropy.units as u

skycoord = SkyCoord(

Most of the arguments we send to SkyCoord come directly from results.

We provide distance and radial_velocity to prepare the data for reflex correction, which we explain below.

The result is an Astropy SkyCoord object, which we can transform to the GD-1 frame.

from gala.coordinates import GD1Koposov10

gd1_frame = GD1Koposov10()
transformed = skycoord.transform_to(gd1_frame)

The result is another SkyCoord object, now in the GD1Koposov10 frame.

The next step is to correct the proper motion measurements from Gaia for reflex due to the motion of our solar system around the Galactic center.

When we created skycoord, we provided distance and radial_velocity as arguments, but we did not use the measurements provided by Gaia. Instead, we use fixed values for these parameters.

That might seem like a strange thing to do, but here’s the motivation:

  • Because the stars in GD-1 are so far away, the distance estimates we get from Gaia, which are based on parallax, are not very precise. So we replace them with our current best estimate of the mean distance to GD-1, about 8 kpc. See Koposov, Rix, and Hogg, 2010.

  • For the other stars in the table, this distance estimate will be inaccurate, so reflex correction will not be correct. But that should have only a small effect on our ability to identify stars with the proper motion we expect for GD-1.

  • The measurement of radial velocity has no effect on the correction for proper motion; the value we provide is arbitrary, but we have to provide a value to avoid errors in the reflex correction calculation.

We are grateful to Adrian Price-Whelen for his help explaining this step in the analysis.

With this preparation, we can use reflex_correct from Gala (documentation here) to correct for solar reflex motion.

from gala.coordinates import reflex_correct

gd1_coord = reflex_correct(transformed)

The result is a SkyCoord object that contains

  • phi1 and phi2, which represent the transformed coordinates in the GD1Koposov10 frame.

  • pm_phi1_cosphi2 and pm_phi2, which represent the transformed and corrected proper motions.

We can select the coordinates like this:

phi1 = gd1_coord.phi1
phi2 = gd1_coord.phi2

And plot them like this:

plt.plot(phi1, phi2, 'ko', markersize=0.1, alpha=0.2)

plt.xlabel('ra (degree GD1)')
plt.ylabel('dec (degree GD1)');

Remember that we started with a rectangle in GD-1 coordinates. When transformed to ICRS, it’s a non-rectangular polygon. Now that we have transformed back to GD-1 coordinates, it’s a rectangle again.

Pandas DataFrame

At this point we have two objects containing different subsets of the data. results is the Astropy Table we downloaded from Gaia.


And gd1_coord is a SkyCoord object that contains the transformed coordinates and proper motions.


On one hand, this division of labor makes sense because each object provides different capabilities. But working with multiple object types can be awkward.

It will be more convenient to choose one object and get all of the data into it. We’ll use a Pandas DataFrame, for two reasons:

  1. It provides capabilities that are pretty much a superset of the other data structures, so it’s the all-in-one solution.

  2. Pandas is a general-purpose tool that is useful in many domains, especially data science. If you are going to develop expertise in one tool, Pandas is a good choice.

However, compared to an Astropy Table, Pandas has one big drawback: it does not keep the metadata associated with the table, including the units for the columns.

It’s easy to convert a Table to a Pandas DataFrame.

import pandas as pd

results_df = results.to_pandas()
(140340, 8)

DataFrame provides shape, which shows the number of rows and columns.

It also provides head, which displays the first few rows. It is useful for spot-checking large results as you go along.

source_id ra dec pmra pmdec parallax parallax_error radial_velocity
0 637987125186749568 142.483019 21.757716 -2.516838 2.941813 -0.257345 0.823721 1.000000e+20
1 638285195917112960 142.254529 22.476168 2.662702 -12.165984 0.422728 0.297472 1.000000e+20
2 638073505568978688 142.645286 22.166932 18.306747 -7.950660 0.103640 0.544584 1.000000e+20
3 638086386175786752 142.577394 22.227920 0.987786 -2.584105 -0.857327 1.059607 1.000000e+20
4 638049655615392384 142.589136 22.110783 0.244439 -4.941079 0.099625 0.486224 1.000000e+20

Python detail: shape is an attribute, so we display its value without calling it as a function; head is a function, so we need the parentheses.

Now we can extract the columns we want from gd1_coord and add them as columns in the DataFrame. phi1 and phi2 contain the transformed coordinates.

results_df['phi1'] = gd1_coord.phi1
results_df['phi2'] = gd1_coord.phi2
(140340, 10)

pm_phi1_cosphi2 and pm_phi2 contain the components of proper motion in the transformed frame.

results_df['pm_phi1'] = gd1_coord.pm_phi1_cosphi2
results_df['pm_phi2'] = gd1_coord.pm_phi2
(140340, 12)

Detail: If you notice that SkyCoord has an attribute called proper_motion, you might wonder why we are not using it.

We could have: proper_motion contains the same data as pm_phi1_cosphi2 and pm_phi2, but in a different format.

Exploring data

One benefit of using Pandas is that it provides functions for exploring the data and checking for problems.

One of the most useful of these functions is describe, which computes summary statistics for each column.

source_id ra dec pmra pmdec parallax parallax_error radial_velocity phi1 phi2 pm_phi1 pm_phi2
count 1.403400e+05 140340.000000 140340.000000 140340.000000 140340.000000 140340.000000 140340.000000 1.403400e+05 140340.000000 140340.000000 140340.000000 140340.000000
mean 6.792378e+17 143.822971 26.780161 -2.484410 -6.100784 0.179474 0.518068 9.931167e+19 -50.091337 -1.803264 -0.868980 1.409215
std 3.792015e+16 3.697824 3.052639 5.913923 7.202013 0.759622 0.505558 8.267982e+18 2.892321 3.444439 6.657700 6.518573
min 6.214900e+17 135.425699 19.286617 -106.755260 -138.065163 -15.287602 0.020824 -1.792684e+02 -54.999989 -8.029159 -115.275637 -161.150142
25% 6.443515e+17 140.967807 24.592348 -5.038746 -8.341641 -0.035983 0.141108 1.000000e+20 -52.603097 -4.750410 -2.948851 -1.107074
50% 6.888056e+17 143.734183 26.746169 -1.834971 -4.689570 0.362705 0.336103 1.000000e+20 -50.147567 -1.671497 0.585038 1.987196
75% 6.976578e+17 146.607180 28.990490 0.452995 -1.937833 0.657636 0.751171 1.000000e+20 -47.593466 1.160632 3.001761 4.628859
max 7.974418e+17 152.777393 34.285481 104.319923 20.981070 0.999957 4.171221 1.000000e+20 -45.000086 4.014794 39.802471 79.275199


Review the summary statistics in this table.

  • Do the values makes senses based on what you know about the context?

  • Do you see any values that seem problematic, or evidence of other data issues?

# Solution

# A few issues that are likely to come up:

# 1. Why are some of the parallax values negative?
#    Some parallax measurements are inaccurate, especially
#    stars that are far away.

# 2. Why are some of the radial velocities 1e20?
#    It seems like this value is used to indicate invalid data.
#    Notice that the 25th percentile is 1e20, which indicates
#    that at least 75% of these values are invalid.

Plot proper motion

Now we are ready to replicate one of the panels in Figure 1 of the Price-Whelan and Bonaca paper, the one that shows the components of proper motion as a scatter plot:

In this figure, the shaded area is a high-density region of stars with the proper motion we expect for stars in GD-1.

  • Due to the nature of tidal streams, we expect the proper motion for most stars to be along the axis of the stream; that is, we expect motion in the direction of phi2 to be near 0.

  • In the direction of phi1, we don’t have a prior expectation for proper motion, except that it should form a cluster at a non-zero value.

To locate this cluster, we will select stars near the centerline_df of GD-1.

Selecting the centerline

As we can see in the following figure, many stars in GD-1 are less than 1 degree from the line phi2=0.

So stars near this line have the highest probability of being in GD-1.

To select them, we will use a “Boolean mask”. We’ll start by selecting the phi2 column from the DataFrame:

phi2 = results_df['phi2']

The result is a Series, which is the structure Pandas uses to represent columns.

We can use a comparison operator, >, to compare the values in a Series to a constant.

phi2_min = -1.0 * u.deg
phi2_max = 1.0 * u.deg

mask = (phi2 > phi2_min)

The result is a Series of Boolean values, that is, True and False.

0    False
1    False
2    False
3    False
4    False
Name: phi2, dtype: bool

The & operator computes “logical AND”, which means the result is true where elements from both Boolean Series are true.

mask = (phi2 > phi2_min) & (phi2 < phi2_max)

Python note: We need the parentheses around the conditions; otherwise the order of operations is incorrect.

The sum of a Boolean Series is the number of True values, so we can use sum to see how many stars are in the selected region.


A Boolean Series is sometimes called a “mask” because we can use it to mask out some of the rows in a DataFrame and select the rest, like this:

centerline_df = results_df[mask]

centerline_df is a DataFrame that contains only the rows from results_df that correspond to True values in mask; that is, in contains the stars near the centerline of GD-1.

Plotting proper motion

Here’s a scatter plot of proper motion for the selected stars.

pm1 = centerline_df['pm_phi1']
pm2 = centerline_df['pm_phi2']

plt.plot(pm1, pm2, 'ko', markersize=0.1, alpha=0.1)
plt.xlabel('Proper motion phi1 (GD1 frame)')
plt.ylabel('Proper motion phi2 (GD1 frame)');

Looking at these results, we see a large cluster around (0, 0), and a smaller cluster near (0, -10).

We can use xlim and ylim to set the limits on the axes and zoom in on the region near (0, 0).

pm1 = centerline_df['pm_phi1']
pm2 = centerline_df['pm_phi2']

plt.plot(pm1, pm2, 'ko', markersize=0.3, alpha=0.3)
plt.xlabel('Proper motion phi1 (GD1 frame)')
plt.ylabel('Proper motion phi2 (GD1 frame)')

plt.xlim(-12, 8)
plt.ylim(-10, 10);

Now we can see the smaller cluster more clearly.

You might notice that our figure is less dense than the one in the paper. That’s because we started with a set of stars from a relatively small region. The figure in the paper is based on a region about 10 times bigger.

In the next lesson we’ll go back and select stars from a larger region. But first we’ll use the proper motion data to identify stars likely to be in GD-1.

Filtering based on proper motion

The next step is to select stars in the “overdense” region of proper motion, which are candidates to be in GD-1.

In the original paper, Price-Whelan and Bonaca used a polygon to cover this region, as shown in this figure.

We’ll use a simple rectangle for now, but in a later lesson we’ll see how to select a polygonal region as well.

Here are bounds on proper motion we chose by eye:

pm1_min = -8.9
pm1_max = -6.9
pm2_min = -2.2
pm2_max =  1.0

To draw these bounds, we’ll make two lists containing the coordinates of the corners of the rectangle.

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
pm1_rect, pm2_rect = make_rectangle(
    pm1_min, pm1_max, pm2_min, pm2_max)

Here’s what the plot looks like with the bounds we chose.

plt.plot(pm1, pm2, 'ko', markersize=0.3, alpha=0.3)
plt.plot(pm1_rect, pm2_rect, '-')
plt.xlabel('Proper motion phi1 (GD1 frame)')
plt.ylabel('Proper motion phi2 (GD1 frame)')

plt.xlim(-12, 8)
plt.ylim(-10, 10);

To select rows that fall within these bounds, we’ll use the following function, which uses Pandas operators to make a mask that selects rows where series falls between low and high.

def between(series, low, high):
    """Make a Boolean Series.
    series: Pandas Series
    low: lower bound
    high: upper bound
    returns: Boolean Series
    return (series > low) & (series < high)

The following mask select stars with proper motion in the region we chose.

pm1 = results_df['pm_phi1']
pm2 = results_df['pm_phi2']

pm_mask = (between(pm1, pm1_min, pm1_max) & 
           between(pm2, pm2_min, pm2_max))

Again, the sum of a Boolean series is the number of True values.


Now we can use this mask to select rows from results_df.

selected_df = results_df[pm_mask]

These are the stars we think are likely to be in GD-1. Let’s see what they look like, plotting their coordinates (not their proper motion).

phi1 = selected_df['phi1']
phi2 = selected_df['phi2']

plt.plot(phi1, phi2, 'ko', markersize=0.5, alpha=0.5)

plt.xlabel(r'$\phi_1$ (degree GD1)')
plt.ylabel(r'$\phi_2$ (degree GD1)');

Now that’s starting to look like a tidal stream!

Saving the DataFrame

At this point we have run a successful query and cleaned up the results; this is a good time to save the data.

To save a Pandas DataFrame, one option is to convert it to an Astropy Table, like this:

selected_table = Table.from_pandas(selected_df)

Then we could write the Table to a FITS file, as we did in the previous lesson.

But Pandas provides functions to write DataFrames in other formats; to see what they are find the functions here that begin with to_.

One of the best options is HDF5, which is Version 5 of Hierarchical Data Format.

HDF5 is a binary format, so files are small and fast to read and write (like FITS, but unlike XML).

An HDF5 file is similar to an SQL database in the sense that it can contain more than one table, although in HDF5 vocabulary, a table is called a Dataset. (Multi-extension FITS files can also contain more than one table.)

And HDF5 stores the metadata associated with the table, including column names, row labels, and data types (like FITS).

Finally, HDF5 is a cross-language standard, so if you write an HDF5 file with Pandas, you can read it back with many other software tools (more than FITS).

We can write a Pandas DataFrame to an HDF5 file like this:

filename = 'gd1_dataframe.hdf5'

results_df.to_hdf(filename, 'results_df')

Because an HDF5 file can contain more than one Dataset, we have to provide a name, or “key”, that identifies the Dataset in the file.

We could use any string as the key, but in this example I use the variable name results_df.


We’re going to need centerline_df and selected_df later as well. Write a line or two of code to add it as a second Dataset in the HDF5 file.

# Solution

centerline_df.to_hdf(filename, 'centerline_df')
selected_df.to_hdf(filename, 'selected_df')

Detail: Reading and writing HDF5 tables requires a library called PyTables that is not always installed with Pandas. You can install it with pip like this:

pip install tables

If you install it using Conda, the name of the package is pytables.

conda install pytables

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

!ls -lh gd1_dataframe.hdf5
-rw-rw-r-- 1 downey downey 20M Dec 29 11:48 gd1_dataframe.hdf5

If you are using Windows, ls might not work; in that case, try:

!dir gd1_dataframe.hdf5

We can read the file back like this:

read_back_df = pd.read_hdf(filename, 'results_df')
(140340, 12)

Pandas can write a variety of other formats, which you can read about here.


In this lesson, we re-loaded the Gaia data we saved from a previous query.

We transformed the coordinates and proper motion from ICRS to a frame aligned with the orbit of GD-1, and stored the results in a Pandas DataFrame.

Then we replicated the selection process from the Price-Whelan and Bonaca paper:

  • We selected stars near the centerline of GD-1 and made a scatter plot of their proper motion.

  • We identified a region of proper motion that contains stars likely to be in GD-1.

  • We used a Boolean Series as a mask to select stars whose proper motion is in that region.

So far, we have used data from a relatively small region of the sky. In the next lesson, we’ll write a query that selects stars based on proper motion, which will allow us to explore a larger region.

Best practices

  • When you make a scatter plot, adjust the size of the markers and their transparency so the figure is not overplotted; otherwise it can misrepresent the data badly.

  • For simple scatter plots in Matplotlib, plot is faster than scatter.

  • An Astropy Table and a Pandas DataFrame are similar in many ways and they provide many of the same functions. They have pros and cons, but for many projects, either one would be a reasonable choice.