Here are three image of asteroid 8567 (96HW1) obtained at New Mexico Tech on 23/24 July 2008 by Erika DeBenedictis. They were 90 second exposures. Within a degree, north is up and east is to the left, the normal way of showing celestial images. We determined that the asteroid had an apparent motion of 46.65 arcsec per hour northward and 64.82 arcsec per hour eastward against the background of stars. Notice how the asteroid is moving with respect to the two stars it is passing between.

One can determine the coordinates of field stars by first downloading a fits image from the Digitized Sky Survey website (click here ) and displaying it with the display tool DS9. Using 10 field stars surrounding the position of the asteroid I determined the right ascension and declination of the asteroid using the IRAF tasks ccmap and cctrans.

Here is an image obtained by SSP observers in Ojai, California, at just about the same time as the third image taken in New Mexico. Note that the asteroid is much closer to the line between the two stars in this image compared to its position in the third NMT image. This is owing to the parallax of the asteroid as observed at the two sites. Note that it is observed further east from the Ojai site. I have transformed the Ojai image to exactly north up, east to the left using the DSS map of the field.

Here are the coordinates from the three NMT images and the one Ojai image:

Image     mean UT          RA            DEC            rms (RA)     rms (DEC)         
                                                             (arc seconds)
NMT#1    07:52:04.17    21:27:05.52   +15:52:43.03      +/- 0.29      +/- 0.39
NMT#2    08:06:34.53    21:27:06.62   +15:52:54.34          0.33          0.38
NMT#3    08:17:27.80    21:27:07.42   +15:53:02.77          0.31          0.36

Ojai      see below     21:27:07.77   +15:53:02.25          0.52          0.44
Using Google Earth, we determined that the Ojai site (Besant Hill School) has latitude N 34 deg 26' 03.96" and longitude W 119 deg 11' 22.58". The NMT observatory has latitude N 34 deg 04' 21.7" and longitude W 106 deg 54' 50.1". The baseline between the observatories is the straight line through an edge of the Earth connecting the two sites. It amounts to about 1130 km. Given the longitudes of the two sites, a star transits about 49 minutes later at the Ojai site compared to the NMT site.

How do we determine the distance to the asteroid given this set of imagery and these RA's and DEC's? The most straightforward way is as follows. First, we need two simultaneous images of the asteroid from the two sites. From a geometrical standpoint it would be simplest if our simultaneous observations were taken when the asteroid was 24.5 minutes of time west of the celestial meridian in NM and 24.5 minutes of time east of the meridian in Ojai. Then, the baseline and the asteroid would make a very long skinny isosceles triangle. But we do not quite have that situation.

The observers at the two sites used cell phones and short wave radio time signals to start their exposures at the same time. Never mind what was the UT given in the file header of the Ojai image. There seems to have been a systematic error in that time, at least with respect to computer time at the NMT observatory. It is just a given that the two observations were taken at about the same time (maybe started within 2 or 3 seconds).

Whatever the RA of the asteroid was, as viewed from the center of the Earth at the moment the near-simultaneous images were obtained, the observed RA's of the asteroids were different by 0.35 seconds of time. Given the declination of the asteroid, that amounts to 5.05 arcsecs, with an uncertainty of about +/- 0.61 (the square root of the sum of squares of 0.52 and 0.31).

From W. M. Smart, Textbook on Spherical Astronomy , p. 209, we have a formula to correct the observed right ascension to what would be observed from the center of the Earth:

Delta RA (radians) = RA (topocentric) - RA (at Earth center) = - (rho/R) * sin (H) * cos(phi') * sec(DEC),

where rho = radius of Earth (6378.1 km), R = distance to asteroid in km, H = hour angle, phi' = latitude of observer, and DEC = declination of asteroid.

We are concentrating here on the corrections of the RA's. Because the two sites had almost the same latitude, we have essentially no baseline in the north-south direction to do a parallax experiment.

Imagine that the asteroid is in the eastern half of the sky, so H is negative. The observed RA of the asteroid is further east than it would be measured from the center of the Earth. So we need a negative correction for the observed RA to produce the RA for the center of the Earth.

There exists some correction X such that the right ascension observed in Ojai, corrected to the center of the Earth, will match the right ascension observed in NM, corrected to center of the Earth by Y. And both X and Y are functions of the size of the Earth, the distance to the asteroid, and various trig functions multiplied together (all of which we can determine). Thus, as long as we have simultaneous observations of the asteroid or can reduce the coordinates to same moment in time, we can actually solve for the distance to the asteroid. In short, if we consider the seconds only part of the observed RA's of the asteroid,

7.77 - X = 7.42 - Y.

Or X-Y = 0.35 seconds of time = 5.05 +/- 0.61 arcsec at the declination of the asteroid. Thus,

(5.05 +/- 0.61)/206265 = (R_Earth/r*1.496E8)*function of (phi',DEC, and H for Ojai) - (R_Earth/r*1.496E8)*function of (phi',DEC, and H for NMT) ,

where r is the distance to the asteroid in AU's and 1.496E8 is the number of km in an AU.

From NMT the hour angle of the asteroid was -1.899 deg. If the Ojai image was taken at the same time as the third NMT image, the hour angle of the asteroid at Ojai was -14.176 deg.

We can thus solve explicitly for the distance of the asteroid in AU's:

r = [206265/(5.05 +/- 0.61)]*(6378.1/1.496E08)*(0.210010 - 0.028541).

We obtain a distance to the asteroid of 0.316 +/- 0.038 AU. According the ephemeris of asteroid 8567 from the JPL Horizons website (click here ), the true distance of the asteroid was 0.290 AU on 24 July 2008 at 0817 UT. Thus, we have obtained the correct answer to within one standard deviation.

Given our baseline of 1130 km, our parallax experiment was a reasonable success because we were observing a near Earth asteroid. If we had observed a main belt asteroid somewhere between Mars and Jupiter, it might have been 3 AU distant instead of 0.3 AU. Then the observed parallax would have been 0.5 arcsec instead of 5 arcsec. The only way we could successfully measure the parallax of a main belt asteroid would be to have a longer baseline. A way to do that from a single site would be to observe an asteroid early in the evening at 4 hours east hour angle, then observe it 8 hours later at 4 hours west hour angle. Such a parallax experiment would require knowing the east-west motion of the asteroid very accurately, as that would have to be subtracted off.

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