How to Search for Small Comets
Can the naked eye spot a small comet before it disrupts in the atmosphere? Can a pair of binoculars? Can a small telescope? Dr. Louis A. Frank explains how you can--or why you can't--in this brief technical note.
First of all, consider the naked, rested eye. The apparent brightness of
the small comets at the distance previously detected with the Spacewatch
Camera at a range of 137,000 km (85,600 miles) is typically V=19
magnitude. At a target range of 1000 km (625 miles) the apparent
brightness is about V=8.3 because you have gained V=10.7. For the naked
eye, the threshold brightness has been estimated by Opik and Millman in
terms of the percentage of objects detected as a function of brightness.
Opik claims 8% and 50% of the objects are detected at V=5 and V=4,
respectively. Millman estimates less than 0.5% and 25% at V=5 and V=2,
respectively. Clearly, under the best viewing conditions, the threshold
of the naked human eye is about V=5. Thus the small comets would be
dimmer than the threshold of the naked eye by V=3.3, or a factor of about
20 in light intensity. The naked eye simply cannot detect the small
comets at a range of 1000 km (625 miles).
Can the small comets be detected with binoculars? We have previously estimated the detection rate with telescopes owned by amateur astronomers and have shown that the detection rate is about 1 or 2 per night of clear-sky viewing, if the telescope is pointed in the proper position just after dusk and just prior to dawn. But let us take a look at the binocular situation for a range to the small comet of 1000 km (625 miles). We will use binoculars with a magnification of 7X and a clear entrance pupil of 50 mm in diameter. For the human eye, the entrance pupil varies but a diameter of 5 mm is typical. Thus the binoculars increase the light-gathering capability of the human eye by a factor of 100, or V=5. As seen through the binoculars then, the brightness of a small comet with V=8.3 is V= (8.3 - 5) = 3.3 and is detectable with the eye.
Consequently, the frequency of detection of the small comets with the binoculars is of interest. The viewing geometry is quite constrained because the objects will not be seen in the darkness of Earth's shadow and certainly not through the sunlit atmosphere. In fact, the atmospheric brightness is not less than V=3.1, and similar to the brightness of the small comet, until about 72 minutes after sunset and during the period 72 minutes before sunrise. We estimate that there are approximately 30-minute windows for detection of these small comets at solar phase angles in the range of 18 degrees to 40 degrees which are contiguous to and on the dark side of the above two windows. Importantly, the binoculars must be pointed at the intersection of Earth's shadow and the 1000-km (625-mile) range. The apparent motions of the small comets will be in the range of about 1 degree/s and will remain in the field-of-view of binoculars with a field-of-view of 10 degrees for about 10 seconds.
Now we must calculate the frequency of the small comet detections with your binoculars. The viewing is through the Earth's shadow so that part of the detection area must be subtracted because, obviously, if the sun doesn't shine on the comets they are essentially invisible.(See the viewing geometry in the accompanying drawing.) With an apex at your binoculars and the two lines extending to 1000 km (625 miles), a rough estimate of the area in sunlight is approximately 2/3 of the total area of the triangle. If your field-of-view is 10 degrees then the detection area in sunlight is about 6 x 104 km2. For the present purposes of simple estimation we use the average interplanetary rate of 1.2 x 10-9 small comets/km2-s intercepting the above detection area and neglect the effects of Earth's gravitational focusing. Thus the detection rate is about (1.2 x 10-9) x (6 x 104) = 7 x 10-5 events/s, or 1 event every 5 hours. Since your viewing time is about 30 minutes each for the pre-dawn and post-dusk samples, one would expect to see one event every 5 days of viewing with these binoculars. The short period for viewing is due to the rotation of the Earth and the rapid geometric degradation of the viewing geometry.
For telescopes with larger apertures the viewing geometry remains the same as shown in the accompanying figure. The viewing geometry is generally parallel to the ecliptic plane. Basically, you can estimate your detection rate by comparing your telescope's characteristics with those for the binoculars as given above. For example, if you have a collecting aperture of 90-mm diameter, then the range of your telescope is approximately proportional to the diameter, or about 1800 km (1125 miles), since the object brightness decreases as r-2. At this distance, for an object with an apparent speed of 10 km/s, the apparent angular speed is about 0.3 degree/second. The event rate is proportional to the fan collecting area, i.e., proportional to the square of the above range times the viewing angle of your telescope. If the field-of-view of your telescope is the same as that of the above binoculars (probably smaller) then your event rate will be (1800/1000)2 x (10/10) = 3.2 greater than with the binoculars. This is about 1 event every 1.5 hours for the average interplanetary rate. The occurrence frequency is greater than this average for the months of June through the first week of November due to seasonal variations by factors in the range of 2 to 3. The rates are lesser for the other months with January at the minimal rates.
For a summary of the capability of the human eye for the detection of faint meteors, see D.W.R. McKinley, Meteor Science and Engineering, McGraw-Hill Book Company, New York, pp. 106-107, 1961.
For the variation of sky brightness throughout twilight, see C.W. Allen, Astrophysical Quantities, 3rd Edition, Athlone Press, London, p. 134, 1973.
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