Two examples of astrometry

Michael Richmond
Jan 16, 2009

Today, our topic is astrometry, the science of measuring positions of celestial objects. You will have two different little projects to do today, to help illustrate two different facets of this field.

1. In order to determine the parallax of an asteroid, you will need to make simple relative measurements of the position of an asteroid relative to a few nearby stars. Because the field of view is small, you can use simple plane geometry and get an answer which is good enough.
2. In order to calculate the (RA, Dec) positions of a large number of stars in a large field, you'll need to deal with the conversion from a spherical coordinate system to a plane system and back again. Fortunately, there are plenty of tools you can use for this job.

Get connected

1. Log into computers in the CIS lab.
2. Start the "Terminal" program
3. Type the following:

               ssh -Y -l student spiff.rit.edu
   

   When asked for a password, supply the value I will tell you.
You should now be connected to a computer in my office, which has some images of a cluster of stars.
4. Type the following commands:

               propinit
   
               cd grant         (or your own first name here)
   
               ls
   

You should now be in the directory with the images you'll need. The two images you'll use for the parallax experiment are

           rit.fit              an image taken from the RIT Observatory
           usna.fit             an image taken from the US Naval Academy

You can display one of the image by using the tv command, like so:

           tv rit.fit zoom=2

A new window should pop up like this:

The zoom=2 argument zooms into the image, making it twice as big as it really is. You can use zoom=0.5 to make an image appear smaller, or leave out the zoom keyword entirely to see the image at its normal size.

If you move your cursor into the image, you should see the coordinates (bottom of the window) change as you move the cursor. If the coordinates DON'T change, then click on the window's bar at the top. If they STILL don't change, left-click once while pointing anywhere in the image.

There are three keys you can use to measure stellar properties.

• a press this when you are pointing at a star and you'll see a line of information appear in your terminal window, like this:

   ( 499.69  563.62) flux   9108.6 npix  49.5 mag 15.101  sky   32.0  [ 4.0  8 15]
     

  The number after "mag" will give you the instrumental magnitude of the star in this image.
• r press this when you are pointing at a star and you should see a radial profile.
• z press this, and you will create a new, smaller window which is zoomed-in at the location of the cursor.

To destroy an image window, you can right-click while the cursor is inside it, or use the Mac red-color-button at the top left of the window.

Measure the parallax of an asteroid and compute its distance

One way to measure the distance to objects is via parallax, which involves measuring the apparent shift in position of an object when it is viewed from two different locations. You are probably most familiar with heliocentric parallax, in which measurements of a star are made 6 months apart and the orbit of the Earth provides the baseline.

In this case, however, two images were taken simultaneously. I took one from the RIT Observatory, and a colleague that US Naval Academy in Annapolis, Maryland, took the other. You can use the locations of our two observatories to determine the baseline for the measurements.

                             latitude        longitude
    ---------------------------------------------------
    RIT Observatory          43:04:33         77:39:53

    US Naval Academy         38:58:41         76:29:31

In order to determine the angle by which the asteroid appeared to shift, you need to measure the position of the asteroid relative to some other star in each image. You'll also need to determine the plate scale of each image -- in other words, the relationship between pixels and arcseconds. You can use this chart as a guide:

In this chart, stars B and D are 704 arcseconds apart.

1. how many pixels apart are stars B and D in the RIT image?
2. how many pixels are in one arcsecond in the RIT image?
3. how many pixels apart are stars B and D in the USNA image?
4. how many pixels are in one arcsecond in the USNA image?

The next step is to measure the location of the asteroid relative to some reference point in each image. One approach is to set up a coordinate system using the line connecting stars B and D. If you measure the vectors B-D and B-asteroid, you should be able to compute the x-component and y-component of the vector B-asteroid. Do that in both images, and you can then find the shift in the position of the asteroid.

1. How far is the asteroid from star B in the RIT image? Express your answer in arcseconds.
2. Break this distance in an X-component and Y-component, so you can write the distance between the asteroid and star B as a vector.
3. How far is the asteroid from star B in the USNA image? Express your answer in arcseconds.
4. Break this distance in an X-component and Y-component, so you can write the distance between the asteroid and star B as a vector.
5. What is the shift in position of the asteroid between the two images? Write the magnitude of the shift in arcseconds.

Once you have the parallax angle θ, you can compute the distance to the asteroid.

Note that there are some complications that I've ignored in this exercise, which mean that your value is only a lower limit to the distance. Can you explain what these complications might be?

Computing the positions of many stars in a large field

If we take a picture showing a large area of the sky, then we run into a big problem: the positions of stars will be distorted, because our telescope projects the image of a curved, three-dimensional sky onto a flat, two-dimensional surface.

This distortion is the same reason that maps of large portions of the Earth's surface can look strange. This map make Greenland look as large as Africa, but it actually has less than one-tenth the area!

This Mercator projection of the Earth courtesy of Stefan Kuhn and Wikimedia Commons

So, one ingredient in a successful astrometrical recipe is a good method for accounting for the distortions involved in projecting the celestial sphere onto a plane.

Another ingredient is a good set of reference stars with known positions. In the optical regime, the best astrometric catalogs at the present time are

• Hipparcos, the new Reduction of the Raw data (for stars brighter than mag V = 8 ).
• The Tycho-2 catalog (for stars 8 < V < 11)
• The USNO B1.0 catalog includes stars to about V = 21 .

When you find yourself with an image of some portion of the sky, and you want to compute the positions of stars in the image, you can follow these steps:

1. measure the positions of many stars in the image itself, in pixel coordinates (x, y)
2. starting with one of the astrometric catalog, extract a list of stars with known positions in that area, with positions in (RA, Dec)
3. project the (RA, Dec) positions of the catalog stars onto a plane, into what are called the "standard coordinates" or projected coordinates (xi, eta)
4. match the catalog stars to stars detected in the image
5. using the matched pairs, compute a coordinate transformation which transforms pixel coords (x, y) into projected coordinates (xi, eta)
6. apply the transformation to ALL the detected stars, taking them all into the (xi, eta) system
7. de-project the plane coordinate back into the spherical coordinate system (RA, Dec)

Today, I'd like simply to illustrate the procedure, so I'll do most of the work for you. We'll start with an image of this pretty galaxy: m51.fit.

I've measured the positions and instrumental magnitudes of the brightest stars in this image. You can find them in the file m51.pht. The columns are

0. an index number
1. row coordinate = y-coord (pixels)
2. col coordinate = x-coord (pixels)
3. local sky value
4. uncertainty in local sky value
5. instrumental magnitude
6. uncertainty in instrumental magnitude
7. flag indicating how "clean" the measurement is (0 = good)

I've also gathered the (RA, Dec) positions of many stars in this region from the USNO B1.0 catalog. You can find these positions in the file m51cat.dat. In this file, the important columns are

  0 - an index number 
  9 - the Right Ascension (decimal degrees)
 10 - the Declination (decimal degrees)
 11 - the R-band magnitude 

So, given all this preliminary work, your job is pretty easy.

project the catalog (RA, Dec) positions onto a plane (xi, eta)

The image of the galaxy is centered at

              RA  =  202.4676    degrees
              Dec =  +47.1932    degrees
      

We will use a program which is part of the "match" group of programs. This one projects (RA, Dec) onto a plane, centered at some specific position. Here's how to run it:

         project_coords  m51cat.dat 9 10 202.4676 47.1932  outfile=m51cat.proj
      

Compare the two files m51cat.dat and m51cat.proj. What is the same? What has changed? The position of each object has been converted from degrees into radians, and the center of the coordinate system is the (RA, Dec) value you typed in the command.

match the image (x, y) positions to catalog (xi, eta) positions

Okay, now we have two lists of objects, both of which have positions in a plane: the objects measured from the image, m51.pht, and the projected coordinates of stars in the catalog, m51cat.proj. We now run a program which will try to match items in the first list with items in the second list, and then use those matches to derive a transformation which converts (x, y) to (xi, eta), like this:

        xi   =  a  +  bx  +  cy
        eta  =  d  +  ex  +  fy
   

To run this program, type (all on one line)

        match m51.pht 1 2 5 m51_proj.dat 9 10 11 id1=0 id2=0 
            matchrad=0.000024 nobj=30 linear trirad=0.001 recalc 
      

If all goes well, you'll see a set of numbers printed out, something like this:

      TRANS: a=0.004369603     b=-0.000008157    c=-0.000002399    
             d=-0.003236631    e=-0.000002400    f=0.000008158     
             sig=4.5759e-12 Nr=85 Nm=85 sx=1.5888e-06 sy=1.8051e-06 
      

The numbers listed for a, b, c, d, e, f are the coefficients of the linear transformation which will take (x, y) coordinates and turn them into (xi, eta). We are almost done!

Disclaimer: this step is often the trickiest one in the whole process. There are many, many parameters which control the matching procedure. Read the match package manual for details, or talk to me, if you ever run into trouble.

convert the image (x, y) position to projected (xi, eta) positions, and then de-project those positions onto the sky (RA, Dec)

The final step in the procedure is to apply these a, b, c, d, e, f coefficients to the (x, y) positions of all the stars we detected in the image, and THEN to de-project the results back onto a spherical surface and put them into the (RA, Dec) coordinate system. That's a big job, but fortunately, you can do it all with just one more command (again, type this on one line)

         apply_match m51.pht 1 2 202.4676 47.1932 linear
                a   b   c   d   e    f
                outfile=m51.radec
      

Now, in this command, you need to replace the a, b, etc., with the numerical values you found in the previous step.

If all goes well, you'll find that the m51.radec file contains a list of all the detected stars with the instrumental magnitudes -- just like before -- but now, the columns for (x, y) positions have been replaced with (RA, Dec) values.

In the list of objects detected in the image, there was one faint star at (677.84, 132.47). What are the (RA, Dec) coordinates of this star?

For more information

• The "match" suite of programs for matching two lists of coordinates and projecting/de-projecting onto the sky