Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.

Some basic astrometry

There are times when it is important to identify precisely the location of a star or other target on the sky; when descriptions like "about one arminute north and a little east" aren't good enough. At these times, one must know enough about astrometry -- the science of measuring positions of stars -- to provide the required location.

The goal of today's lecture is to show you how to calculate positions both differentially:

and explicitly in Right Ascension and Declination

Equinox: due to precession of Earth's axis

Two key astrometric terms which are sometimes confused are equinox and epoch. What do they mean?

Equinox is concerned with the very slow but inevitable precession of the Earth's rotational axis. Over a period of about 26,000 years, the rotational axis of the Earth traces a circular path around the sky.

It's like a top spinning on a table: the top's spin axis precesses very rapidly due to the torque caused by gravity on its center of mass. The Earth's precession is also caused by a gravitational torque, due to the combined pulls of the Sun and Moon. So, although the Earth's rotational axis currently points close to Polaris (thus making it the "North Star"), it will eventually point elsewhere. Halfway through the 26,000-year cycle, it will point about 23.5 + 23.5 = 47 degrees away from Polaris, in the general vicinity of Vega.

Astronomers want to keep a coordinate system which is locked to a reference frame attached to the Earth. Therefore, they want "Dec = +90" always to mean "in the direction of the Earth's rotational axis." As the Earth precesses, the location of "Dec = +90" must move relative to the stars. In fact, the location of every (RA, Dec) coordinate will move very slightly. A star located at "RA = 0, Dec = +90" today won't be there next year (though the change is very small):

  date         RA          Dec
  1950      23:58:43     89:43:18
  2000      00:00:00     90:00:00
  2050      00:01:17     89:43:18

So, when an astronomer provides a position, he needs to specify the coordinate system in which the measurements are made. Astronomers generally define systems at 50-year intervals:

You will sometimes see these abbreviated to "B1950" and "J2000". Why "B" and "J"? Go look it up!

Epoch: due to proper motion of individual stars

There is another complication to consider when comparing positions today to those measured years ago. Proper motion is the shift of an individual star relative to its neighbors, due to its own intrinsic motion through space. A typical value for the speed of a star in its orbit around the Milky Way galaxy is 200 km/sec. Each individual star, however, has its own random velocity on top of this average motion. The magnitude of this individual component varies from around 10 km/s for young stars in the disk to around 40 km/s for older stars in the disk. Stars in the Milky Way's halo population may have very high speeds, over 100 km/s relative to the stars in the disk.

Of course, these stars are all very far away from us. How large will their apparent angular motion appear? Consider the following case: a star at a distance of 50 parsecs, which has tangential velocity 20 km/s.

  1. How large an angle does it move over a full century? Express your value in arcseconds.
  2. What is its proper motion? Express your value in arcsec/century.

Clearly, the position of a star with large proper motion depends on the date at which is was measured. The epoch of an observation is simply the date on which the data were acquired. It is usually expressed in fractional years: an image taken at 03:43 UT on Apr 28, 2002, has epoch 2002.321.

Example: position of asteroid 1107 Lictoria

In order to explain the steps one must take to perform simple astrometric measurements, I will employ a concrete example: an image of asteroid 1107 Lictoria, taken at 21:48:30 EDT on Apr 11, 2002. The chart below shows a section of the image about 20 arcmin wide by about 14 arcmin high; North is down, and East to the right.

We will use stars "A" and "B" to determine the plate scale and orientation precisely (yes, in real life, one would use more than a single pair of stars), and then apply those values to find the position of asteroid Lictoria, marked with an "L". The steps will be

  1. look up the coordinates of stars "A" and "B"
  2. measure the separations of stars "A" and "B", and "B" and asteroid
  3. determine plate scale (arcsec/pix)
  4. determine rotation angle (row/col relative to North/East)
  5. measure distance of asteroid from star "B", in pixels
  6. convert distance to arcseconds North and East
  7. calculate (RA, Dec) of the asteroid, using star "B" as a reference

Step 1: (RA, Dec) coordinates of stars A and B

Using Aladin, this isn't hard at all. We simply acquire an image of the field (near RA = 07:17:21.45, Dec = +24 29 16, equinox 2000), ask for an overlay of stellar positions from the USNO-A2.0 catalog, and note the values for our stars. We can convert the USNO-A2.0 values from decimal degrees to sexigesimal notation if we wish.

 star       RA    (J2000)  Dec              RA    (J2000)     Dec
  A      109.413731   +24.452589          07:17:39.3       +24:27:09
  B      109.309950   +24.474787          07:17:14.4       +24:28:29

Now, the USNO-A2.0 catalog is based upon photographs taken in the first Palomar Sky Survey. You can see from the detailed entry on each star that the epoch of the measurements is quite a while ago: 1954.972. That means that we must be careful: it's possible that one or both of these stars may have appreciable proper motion, in which case the positions in the catalog will NOT be the same as their current positions. But we'll ignore that point for now.

Step 2: pixel coordinates of stars A, B and asteroid

Again, this isn't very hard. We can simply display the images with XVista's tv command, move the cursor to each star, and then press the 'a' key.

                                        distance from star B
star           row         col             drow       dcol
 A            72.78       372.75          -30.77    121.42
 B           103.55       251.33
 L           120.05       283.88           16.50     32.55

Step 3: calculating the plate scale (arcsec/pixel)

The basic idea is to measure the distance between stars A and B in units of arcsec (via the catalog positions), and in units of pixels (via the image coordinates). Then we find the ratio of the two distances. It's simpler to break these distances down into components in each step.

So, first, the distance between A and B in arcseconds.

    RA:  (A - B)  =  109.413731 - 109.309950  =   0.103781   degrees   (??)
   Dec:  (A - B)  =   24.452589 -  24.474787  =  -0.022198   degrees
But wait: this isn't quite correct. The separation in the RA direction must be corrected for the change in scale of RA with Declination. What's that? Use an analogy with coordinates on the surface of the Earth. Image a man flying a plane around the world, along a line of constant latitude. At the equator (latitude = 0), he must cover a circumference of about 40,000 km. But if he flies along the line delimiting latitude = 40 degrees North, his path is much shorter, only about 30,700 km. If he goes very close to the North Pole, he can fly in a tiny circle of latitude = 89.9 degrees and go "all the way around the Earth" in a mere 70 km.

In order to calculate distances properly, we must project the spherical sky onto a flat plane at the position of the image. As long as the distance between two stars is small -- just a fraction of a degree -- we can do so pretty easily:

     separation in RA  =  delta_RA * cos(avg. Dec)

                       =  0.103781 * cos(24.46 degrees)

                       =  0.094467  degrees     

                       =  340.1 arcsec

     separation in Dec = -0.022198  degrees     (no correction needed)

                       = -79.9  arcsec

The total distance between the stars is

         sqrt [ (340.1*340.1)  +  (-79.9*-79.9) ]   =  349.4  arcsec

Next, we must find the separation between the stars in pixels. This is easy -- there are no corrections to make: sqrt [ (-30.77*-30.77) + (121.42*121.42) ] = 125.3 pixels

Therefore, the plate scale of the image is

                        349.4 arcsec
     plate scale  =  -----------------  =  2.79  arcsec/pixel
                        124.3 pixels

In real life, one would use more than a single pair of stars to determine this conversion factor.

Step 4: determine image rotation angle

Astronomers try to arrange their cameras so that the rows and columns run along the cardinal directions, but never get it exactly right. One must determine the rotation of the camera's rows and cols relative to North and East in order to transform pixel offsets into proper astrometric values.

A simple way to find the rotation angle is to calculate the angle between the same pair of stars in each coordinate system, as shown. Let's do it for our pair of stars. The column labelled "angle" is simply the inverse tangent of the two component distances. It has not (necessarily) been placed into the proper quadrant.

                   delta             delta              angle (*)
  RA/Dec:       340.1 arcsec       -79.9 arcsec        76.78 degrees
  col/row:      121.42 pixels      -30.77 pixels       75.78 degrees

Notice that I've figured out that "column" is the direction corresponding to RA, and "row" the column corresponding to Declination; just look at the image, and it's obvious.

The rotation angle between the two systems is exactly 1.00 degrees. If we wish to convert a vector from (col,row) to (RA, Dec), we must

  1. multiply the vector by the plate scale (pixels -> arcsec)
  2. rotate the vector by +1.0 degrees (col,row -> RA,Dec)

  1. Draw a picture which shows two coordinate axes: one for (row, col), and the other for (RA, Dec). Make the (row, col) axis straight up/down/left/right on your paper, and the (RA, Dec) axes at an angle. Exaggerate the angle between them for clarity, but show properly the sense of the angle of rotation.

Step 5: measure distance of asteroid from star "B", in pixels

We already did that, back in step 2. See the table there.

   vector V  from B to L  =  (  +16.50 rows,  +32.55 cols )

                          =     36.49 pixels  @  63.1 degrees 

We measure the angle 63.1 degrees as "degrees from positive row direction, moving towards the positive col direction". Or, in other words, angle away from straight down on the image, measured towards the right.

Step 6: convert distance to arcseconds North and East

Following the procedure listed above in step 4,

  1. scale the vector to convert to arcseconds
            V  =    36.49 pixel  *   2.79 arcsec/pixel  =   101 arcseconds

  2. rotate the vector by +1.0 degrees
            V  =    63.1 degrees  +  1.0 degrees        =    64.1 degrees

The result is a vector expressing the separation between star B and the asteroid:

       L - B  =  101 arcseconds  @  64.1 degrees  East of North

              =  ( 44.1 arcsec N,  90.8 arcsec E )

It is always a good idea at this point to go back to your picture of the field and make sure that you have kept the rotation angles correct. Looking back at the finding chart, we see that, yes, the asteroid is both North and East of star B. Good. It's even a greater distance East than North from star B, which matches our offset. Double good.

Step 7: calculate (RA, Dec) of the asteroid, using star "B" as a reference

Finally, we need to convert this offset into an (RA, Dec) position. Once again, we need to correct for the shrinking of the RA coordinate with Declination. = ( 44.1 arcsec N, 90.8 arcsec E )

       RA (asteroid)  =  RA (star B)       +  (90.8 arcsec) * cos(24.46)

                      =  109.309950  deg   +   82.7 arcsec

                      =  109.309950  deg   +    0.02296 deg

                      =  109.3329  deg

      Dec (asteroid)  =  Dec (star B)      +   44.1 arcsec

                      =   24.474787 deg    +    0.01225 deg

                      =   24.4870  deg

It's even possible to convert this position to sexigesimal notation, to make it look more official:

        asteroid at   RA = 07:17:19.9   Dec = +24:29:13

Creative Commons License Copyright © Michael Richmond. This work is licensed under a Creative Commons License.