How to do astrometry

How to do astrometry with SIP

Along with brightness measurments of celestial objects, position measurements provide the other most basic measurement possible, and one that is of vital importance to astronomy. Position measurments let astronomers track and understand the motions of all objects in the solar system. Beyond the solar system, measuring the motions of nearby stars has enabled astronomers to map out the locations, including distance, of the stars in our neigborhood, and by extension, the distances to more distant stars, then galaxies, and finally the size of the observable universe. On a more "personal" note, accurate measurements of the positions and therefore motions of asteroids and comets in our solar system will one day inform us of a potential disaster involving the collision of one of those objects with Earth. If we know well in advance of the collision, it can possibly be averted. Your instructor may have specific instructions on what you should do to perform astrometric measurements in your images using SIP. This page gives a basic outline of the process, pointing out tools and websites that can be of specific use.

In addition to discussion of how you can carry out basic astrometry tasks, this page discusses an Example Image Set for Astrometry.

Accurate astrometric measurements of the position of an object, an asteroid for example, have as a goal the determination of the sky coordinates of that object. Typical sky coordinates are the equatorial coordinates "Right Ascension" (RA) and "Declination" (Dec), analogous to longitude and latitude on the Earth. One must also specify the specific "epoch," or year, of the equatorial coordinate system being used, since the coordinate grid is defined by the Earth's orientation, and the Earth slowly "precesses" or wobbles (just as a spinning top gradually wobbles as it spins). One standard epoch is "2000.0" (the beginning of the year 2000).

RA and Dec coordinates

If you already know what RA and Dec are, you might want to skip this section.

RA is expressed in a time-like format. For example, RA = 13 hours 23 minutes 13.456 seconds, or simply, 13 23 13.456. There are a 24 full hours of RA in a complete trip around the sky along the celestial equator (analogous to the Earth's equator). The origin of this time-like expression for RA is in the fact that it takes about 24 hours of time for the sky to appear to rotate around the Earth.

Dec is expressed in a similar format, but: "degrees" takes the place of "hours," with zero degrees being the Dec of an object on the celestial equator, and a positive number of degrees being used for objects north of the celestial equator (up to +90 degrees at the north celestial pole); "arcminutes" takes the place of "minutes," with 60 arcminutes being equivalent to one degree; "arcseconds" takes the place of "seconds," with 60 arcseonds being equal to 1 arcminute. For example, Dec = +34 degrees 45 arcminutes 23.456 arcseconds, or +34 45 23.456.

How to determine the RA and Dec of an object in your image

To determine the RA and Dec coordinates of some object of interest, you first need to set up a mathematical translation, called a "transformation" between the x,y pixel coordinates in your image and RA,Dec coordinates. In all you need to: (1) identify a set of stars in your image which have known RA,Dec coordinates (in some stellar catalog), (2) determine the x,y pixel coordinates of these stars in your image, (3) use these data to calculate what transformation is necessary to determine RA,Dec given x,y, and (4) use the transformation to calculate the RA,Dec of the object of interest. This page tells you how to accomplish steps 1 and 2. Steps 3 and 4 can be accomplished using the astrometry calculator, a javascript program I supply as part of the SIP project (look under "Astrometry" on the left side of the SIP homepage). Since the calculator uses 4 to 10 stars, you only need to identify and use at most 10 somewhat widely spaced stars in your image.

Step 1: Identify a set of stars in your image which have known RA,Dec

Your instructor may have a specific way for you to accomplish this task. The way described here makes use of Lowell Observatory's asteroid services.

The ASTPLOT service at the Lowell Observatory website will provide you with a plot of the stars in the region covered by your image. The stars plotted are those in the US Naval Observatory's Astrometric Standards catalog (USNO-A2.0). The form on the ASTPLOT page requires a set of responses used to construct your plot. Specifying the exact "Observatory," "Date," and "Time" are not important unless you are trying to also plot the exact position of a known asteroid. The only important entries are the "RA" (approximate RA of the center of your image), "Decl" (approximate Dec of the center of your image), "Limiting V Mag" (faintest magnitude to plot), "FOV RA (arscec)" (the field of view of your image along the RA, or east-west direction), and the "FOV Decl (arcsec)" (the field of view of your image along the Dec, or north-south direction). A limiting magnitude which produces some dozen or so stars is adequate. The larger the magnitude number, the fainter the stars plotted, and the more numerous those stars will be. (You might want to review the concept of stellar magnitudes.) Your instructor should be able to tell you the field of view of your image, and it's approximate center RA and Dec. The plot your produce appears as a new webpage, which you can print out, and compare with your image. You should be able to see a correspondence between stars in the plot and stars in your image. The north sky direction is up on the plot; the east sky direcion is to the left.

The REFNET service at the Lowell Observatory website will provide you with a listing of the stars in the region covered by your image. The list includes the RA and Dec of each star. The form on the REFNET page has similar entries as on the ASTPLOT page; again the RA Width, Dec Height, RA Center, and Dec Center are important entries. Also important is the "Red mag high limit" and "Blue mag high limit" (the limiting magnitude(s), which should be set to values like you used in the ASTPLOT form). The produced listing will be in order of increasing RA (i.e., from right to left across an image if north is up in the image). Besides RA and Dec, each star will have a number stating how far (in arcseconds) the star is from the center RA,Dec (called the "Radius" in the listing), and a number specifying the direction of the star from the center (called the "Angle" in the listing). The Radius and Angle listings can greatly help you in identifying which plotted stars in your ASTPLOT correspond to which stars listed in your REFNET listing. The ASTPLOT plot has a scale (in arcseconds) along the edges of the plot. The Angle is defined such that 0 degrees means directly north of center (up, in the plot), 90 degrees is directly east (left) of center, etc. For example, a star with Radius = 300, Angle = 225 is 300 arcseonds from the center of the ASTPLOT plot, in the direction away from the center toward the lower right --- use a protractor to determine the exact direction.

Once you have identified 4 to 10 stars of known RA,Dec that lie in your image, you may proceed to step 2.

Step 2: Determine the x,y pixel coordinates of these stars in your image

Use SIP to open your image. Set appropriate display parameters so you can clearly see the stars in the image (e.g., use the "Automatic Contrast Adjustment" selection under the View menu item, but note that fields containing only stars are generally best viewed with a higher Display Max than is produced by the "Automatic Constrast Adjustment" --- use "Change Image Display Parameters..." to set that value). Select the "Determine Centroid or Instrumental Magnitude..." selection under the Analyze menu item. Adjusting the location of the green (object) box so it surrounds one of your stars will produce a value for the "centroid" of that star. The centroid is the "center of light" analogous to the "center of mass" of an object. It is a measurement of the x,y pixel coordinates of the center of the star's image, including the decimal fraction of a pixel. These x,y values are to be used in step 3.

Note that you can play with the size of the green box, as well as the size and location of the red (background) box obtaining different values of the star's centroid coordinates. The "mean" (average) of the values in the background box and the "root-mean-square" (rms) value for those background pixels are used in computing the centroid x,y for the object in the green box: only pixels in the green box with values greater than the background mean plus 5 times the background rms are used in the centroid calculation. Essentially, the background box's mean value is a "sea level" value for the determination of the location of the "mountain peak" of intensity in the green box, and the background box's rms value is a typical height of the random "foothills" in the background box (called the "noise") --- it is used by the centroid calculation to ignore all but the true mountain peak created by the star. It is important to set the red box near the green box to get background values of relevance to the star. You can also try experimenting with the Background Annulus approach to determining the background values (the background mean and rms are determined in a square red "ring" centered on the green box). In the end, you should obtain values for the centroid that are not very sensitive to changes in the size or location of the green (object) or red (background) box or annulus. Then you have good values for the centorid x,y coordinates.

Repeat the determination of the centroid pixel coordinates for each of the stars you will use in the astrometric transformation calculation.

Finally, detemine the x,y centroid pixel coordinates for the object whose RA,Dec you want to determine. This object could be an asteroid or comet, for example.

Steps 3 and 4: Determine the x,y-to-RA,Dec transformation, and compute the RA,Dec of the object

Now that you have the x,y pixel coordinates of the centroid of your object, and a list of the RA,Dec and x,y coordinates for 4 to 10 stars in your image, you are ready to determine the RA,Dec of your object. The astrometry calculator, a javascript program I supply as part of the SIP project (look under "Astrometry" on the left side of the SIP homepage), will determine the x,y-to-Ra,Dec transformation for your image, and determine the RA,Dec coordinates of your object.

The method used to determine the x,y-to-RA,Dec coordinate transformation is somewhat complicated in practice (i.e., in the program). In principle, it's not difficult to understand. Given the x,y and RA,Dec coordinates for your 4 to 10 stars, the program determines a mapping of one system of coordinates to the other --- a conversion. This conversion is almost a simply rescaling (like converting from km to miles), but there are typically slight distortions of the sky when imaging it in the telescope/camera system. Most important is the fact that the sky (essentially the inside of a large spherical ball) is imaged down onto a flat image. The program takes account of these distortions when setting up the transformation.

The actual transformation equations are
u = x + ax + by + c
v = y + dx + ey + f
where u,v are angular sky coordinates (related to RA,Dec) measured relative to the center of the image and x,y are pixel coordinates in the image, and a through f are the transformation constants determined by the program (these constants are the so-called "plate constants," a name which derives from when this procedure was done with photographic plates). For more information on the astrometric reduction procedure see, for example, "How to Reduce Plate Measurements," by Brian G. Marsden, in Sky & Telescope magazine (September 1982, page 284), and "Measuring Positions on a Photograph," by Jordon D. Marche, also in Sky & Telescope (July 1990, page 71).

Example Image Set for Astrometry

Images http://www1.phys.vt.edu/~jhs/SIP/images/pax/pax_1.fit through pax_4.fit are images of the field around the asteroid 679 Pax taken with an SBIG ST-7 CCD camera on the 0.4m f/4 reflector at the Martin Observatory, Virginia Tech. (For experts: 2x2 binning was used, along with a Bessell V photometric filter.) In each image North is up, East is to the left (approximately). The field of view is approximately 15 arcminutes (East-West) and 10 arcminutes (North-South). The center of each image is approximately at RA = 5 25 55 Dec = +20 04 05. The images were taken by students Michael Cooley, Eric Lang, and Chris Logie at Virginia Tech. The date and time of the observation (in the FITS header) are the beginning of the exposure in Universal Time (UT); each image has an exposure time of 25 seconds. These images are supplied so users can try out the astrometry techniques described on this page. These images are dark and flat field corrected (but not bias corrected --- which will not matter to the performance of the astrometry). You may need to open both pax_1.fit and pax_4.fit in two separate SIP windows in order to see which object in the images is moving (i.e., which object is the asteroid 679 Pax).

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