Astrometry

Astronomical Laboratory 29:137, Fall 2013
by Philip Kaaret

Reading

Equipment


Introduction

In this lab, we will do astrometry on images obtained using the Rigel robotic telescope located at the Iowa Robotic Observatory which is located at the Winer Observatory in southern Arizona. The telescope is a 37cm f/14 classical Cassegrain system on an equatorial fork mount. The imaging camera is a Finger Lakes Instruments (FLI) Proline CCD camera with a Kodak 16803 front-illuminated sensor cooled to -35 C. The sensor has 4096 x 4096 (16M) 9-micron pixels, a gain of 1.4 e/ADU, and a noise of 10 e rms. The field of view is 25’ x 25’, with a plate scale 0.73" per pixel using 2x2 pixel binning. The camera has a seven position filter wheel.  The telescope is scheduled and operated remotely from the University of Iowa in Iowa City using Talon telescope control software.

We will be looking at a pair of images of the open star cluster M39 obtained in the B-band.  One image has the cluster centered in the telescope field of view and the other image is offset by 5 arcminutes in declination.  We will also use frames for bias, dark current, and flat field.  All of the images have 2x2 pixel binning.  Download all of the images now and complain (loudly) to your instructor if they are not accessible or damaged.    These are the same images from the Aperture Photometry lab.  Used the reduced versions of the images that you calculated for that lab.  We will also be using a reference image that is a B-band image of M39 from the Second Palomar Observatory Sky Survey (POSS II) downloaded from the STScI Digitized Sky Survey.


World Coordinate System

Load the POSS2 image of M39 into ds9.  As you move the pointer around on the image, you see the astronomical coordinates of the pointer position listed in the two boxes next to the letters WCS or DSS.  The astronomical coordinates are calculated from the pixel position using the so-called 'World Coordinate System' (WCS) defined in the FITS header of the image.  In ds9, do File/Display Header and then scroll down until you see the lines describing the FITS header that start with 'WCSAXES'.

WCSAXES gives the number of WCS axes, usually 2 when dealing with 2-dimensional images.  WCSNAME gives the name of the WCS coordinate system.  RADESYS gives the particular coordinate reference system used.  We will use using the International Celestial Reference System (ICRS).

CTYPE1, CTYPE2 specify the projection used for the coordinate system.  The projection is how pixels in the image are mapped onto the sky.  Recall that the celestial sphere is, indeed, a sphere.  Conversely, the surface of a CCD is usually a plane.  For CCD images with relatively small fields of view, the CCD surface can be thought of as a small, flat square stuck onto the celestial sphere at one particular point, usually the center of the image.  This is called the tangent projection because the flat square is tangent to the sphere at the point of attachment.  In the DSS/POSS2 images, these keywords have the values CTYPE1='RA---TAN' and CTYPE2='DEC--TAN', meaning that the WCS uses a tangent projection and that the first coordinate is Right Ascension and the second coordinate is Declination.  The CUNIT1 and CUNIT2 keywords specify the units of the WCS coordinates, which are usually degrees.

The four keywords CRPIX1, CRVAL1, CRPIX2, CRVAL2 specify the tangent point where the flat CCD surface is stuck onto the celestial sphere.  We need to know both the coordinates of the tangent point both on the CCD, given by the two CRPIXs, and the tangent on the celestial sphere, given by the CRVALs.

There are two things left to specify to complete the transformation from CCD coordinates to astronomical coordinates.  One is how the CCD pixel size translates into astronomical coordinates, or equivalently the angular size of the image pixels, s.  The other is how the CCD image is rotated relative to the axes of the celestial sphere.  Both of these are described in the 'rotation matrix', which is slightly more general than the usual mathematical rotation matrix.  A rotation matrix is used to perform a rotation in Euclidean space.   The typical two dimensional rotation matrix is

R = \begin{bmatrix} \cos \theta &
          -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}

which rotates a vector with 2 elements (x, y) counter-clockwise through an angle θ.  Usually a rotation matrix is defined with its determinant = 1, which means that the magnitude of the vector is not changed when multiplied by the matrix.  If we multiply the rotation matrix by an overall factor, then the length of the vector will change.  We can use this to translate the pixel size into astronomical coordinates.  Our equation for the transformation from CCD coordinates into astronomical coordinates is then

      a = s R u

where

      a is a vector of astronomical coordinates relative to the tangent point, a = (RA-CRVAL1, DEC-CRVAL2),

      u is  vector of pixel coordinates relative to the tangent point, u  = (x-CRPIX1, y-CRPIX2),

      s is the angular size of the pixels, and

      R specifies the rotation of the CCD image relative to celestial North.

Note that defined in this way, the two diagonal values of the matrix will be equal and the diagonal values will be equal but with opposite sign.  If the CCD pixel axes are reflected relative to the celestial axis, it may be necessary to change the sign of one or both of the components of u.  This can be done by changing the signs of the appropriate components in the rotation matrix.  If the two diagonal components of the rotation matrix have different magnitudes, this means that the angular size of the pixels is not quite the same in RA and DEC.  Today, we will assume that the CCD pixels are square and that they transform to square pixels on the sky.

Using the values of the rotation matrix in the FITS header keywords of the POSS2 image, find the angular size of the pixels in terms of arcseconds/pixel and the approximate rotation angle (accurate to 1 degree).


Finding Coordinates

When finding the WCS for an astronomical image, one usually knows the approximate pointing of the telescope as a starting point.  One compares the CCD image to an existing image or catalog of stars in that field and then tries to find correspondences.  Typically, one uses on the order of 8 to 30 matches to calculate the best WCS transformation.

Today, we'll simplify that procedure and calculate a WCS using a minimal number of stars.  We'll use the following parameters in WCS transformation:

      x, y of the tangent point in CCD coordinates
      RA, DEC of the tangent point in astronomical coordinates
      s = angular size of pixels
      θ = angle of rotation of CCD coordinates relative to astronomical coordinates

We can choose the RA and DEC of the tangent point as we please, so this leaves four parameters that we need to determine from the image.  The position of a star gives us two values, its x and y coordinates, so we need two stars to fully determine the WCS.  Additional stars could be used to test the accuracy of the WCS or 'improve' the WCS by allowing additional parameters in the transformation, such as different pixel sizes in RA and DEC.

To set the tangent point, we will use a bright star.  The brightest star, located near the center of the field of view, is HD 205210 and is saturated, so not suitable for astrometric nor photometric calibration.  Instead, we will use BD +47 3462, located in the green circle in the image below, as the calibration star.  Use the SIMBAD Astronomical Database to determine the coordinates of this star.  We want our WCS to use ICRS coordinates, so use the ICRS coordinates from Simbad.  Use your centroiding code from the photometry lab to determine the CCD coordinates of the star.  Note that you need to be careful to make sure that your python code returns coordinates that are the same as the image coordinates reported by ds9.  Worry about how ds9, python, and your code numbers pixels (e.g. is the first pixel 0 or 1?).


M39


Now pick a second star that is visible on both the POSS2 image and the Rigel image.  To get the best WCS, the star should be quite far from the first star and not be saturated.  It is probably best to load both two images into ds9 and view them simultaneously using Frame/Tile Frame.  Make sure that you see the whole of each image using Zoom/Zoom to Fit Frame.  Find approximate coordinates for the star using ds9 on the POSS2 image and then identify the star using the Query by coordinates option in Simbad.  Find the CCD coordinates of your star using your python centroiding code.

Now find the angular distance between the two stars.  Note that you need to use spherical trigonometry.  Write down the equation for the angular distance in terms the RA and DEC of the two stars and record your calculations.  You may want to write a python function to calculate angular distances as you will be doing this for several stars below.  Now find the distance in pixels between the CCD coordinates of the two stars.  Do you need to use spherical trigonometry?  Think about the fact that we are using a tangent projection for the coordinate system.  Take the ratio of the two values, this is your value for s = angular size of pixels.

Now find the angle between a line draw from BD +47 3462 to your second star in astronomical coordinates and a line draw due north from BD +47 3462.  Also find the angle between a  line draw from BD +47 3462 to your second star in CCD coordinates and a line draw towards positive x from BD +47 3462.  This is the rotation angle of the image. 

Write a python program to read in the reduced image in FITS format (or calculate the reduced image from the object and bias images), add keywords to the header for the WCS, and then write out a new FITS file with the new header (as is traditional with FITS file, keep all the old keywords as long as they don't conflict with your new ones).  Your code should specify the six input parameters for the transformation as listed above (write the parameters into the code so that Jake can run the code without any inputs).  Your code will need to use the rotation angle to calculate a matrix and then multiply by s to find the rotation matrix for your WCS.  Note that the definitions of the angles are a bit tricky, so you might have to do a bit of trial by error in generating the correct rotation matrix for your WCS.  If you prefer, you can write your code to start with the two sets of coordinates (sky, image) for the two stars and do the calculations for s and θ.  You might find this page useful in learning about how to deal with FITS headers in python: http://pythonhosted.org/pyfits/users_guide/users_tutorial.html.  This code might be useful to get started.

When you have you new FITS file with WCS keywords in its header, load the image into ds9 and check the coordinates of a few stars versus the POSS2 image and/or the star positions in Simbad.  Record the positions, both catalog and from your image, of at least three stars from different corners of the image, find the angular distance between your coordinates and the catalog coordinates, and estimate the accuracy of your WCS solution.