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

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Due: Wednesday, March 23, in class

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

- the shift in apparent position of the asteroid,
as seen from two different locations on Earth
- the distance between those two locations
- 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?