# 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?