Homework: using parallax to determine the distance to asteroid 1998WT

Due: Wednesday, March 23, in class

How can we measure the distance to an asteroid? We need to know three things:

  1. the shift in apparent position of the asteroid, as seen from two different locations on Earth
  2. the distance between those two locations
  3. a little trigonometry

On March 4, 2005, at UT 02:45:00, astronomers at Gettysburg Observatory, in Gettysburg, PA, and at Yerkes Observatory, in Williams Bay, WI, measured the position of asteroid 1998WT. They found:

                         RA (J2000)              Dec (J2000)
   Gettysburg          07:26:33.65             +00:46:13.7
   Yerkes              07:26:34.69             +00:46:10.0
   difference         -00:00:01.04              00:00:03.7

We can convert these differences in position into units of arcseconds. First, the difference in Right Ascension; this calculation is a little complicated, both because we have to convert from seconds of time into arcseconds, and because we need to include a factor for the Declination. The average Declination of the positions is about Dec = +00:46:12 = 0.77 degrees.

diff RA  =  (diff in seconds of time)*(15 arcsec/second of time)*cos(Dec)

         =  (-1.04 sec of time) * (15 arcsec/sec of time) * cos(0.77 degrees)

         =  -15.60 arcseconds

The difference in Declination is simple:

diff Dec =  3.7 arcseconds

The total shift in apparent position of the asteroid is therefore

                                      2                     2
  total shift  =  sqrt (  (diff_in_RA)      +  (diff_in_Dec)   )

                                         2                 2
               =  sqrt (  (-15.60 arcsec)   +  (3.7 arcsec)    )

               =  sqrt (  257 arcsec)

               =  16.03 arcsec

What is the distance between the two locations on Earth?