It is possible to determine the approximate height, speed, distance and orbit of an artificial satellite without seeing it-if the satellite broadcasts radio signals. The method, which requires a modified radio receiver, was described in this department last January. The same article told how to make an astrolabe out of three boards to measure the height of satellites visually. Now it turns out that one does not need even this simple equipment.

Within 12 days of the date the first satellite had been launched Jane Shelby, a high-school student in Teaneck, N.J., observed it without instruments, announced its period of orbit with an error of less than three seconds, and accurately computed its velocity, height and orbital inclination. For this accomplishment she was selected as a finalist in the Westinghouse Science Talent Search last year, placed third and received a $5,000 scholarship.

Her method of observing the orbit of a satellite requires the use of a watch with a second hand, a few friends and the ability to consult a table of trigonometric functions. "In 1956," she writes, "I joined one of the Moonwatch teams then being organized all over the world to track the artificial satellites to be launched as part of the International Geophysical Year. My group planned to observe from the roof of the RCA building in New York. But midway in our preparations the U.S.S.R. put Sputnik I on orbit and caught us by surprise. After the first two or three alerts, it was necessary for the group to shift its base of operations to the borough of Queens until more permanent arrangements for the RCA-building site could be made. This was too far away for me to travel in the early-morning hours, so I decided to organize an informal group in my home town of Teaneck.

"The team of eight members was recruited from the Bergen County Astronomical Society. We had no equipment and only two experienced observers. Accordingly the technique of observing had to be simple. One team member was designated as timekeeper and the others as observers. The timekeeper's watch was set beforehand by telephone time signals and checked afterward to determine whether the recorded times should be corrected.

"Each observer was assigned a section of the sky in which to keep naked-eye watch for the satellite. As soon as the object was sighted, all observers followed its motion across the sky. This technique had the advantage of conforming to human nature: most observers, especially novices, watch the satellite as soon as it is spotted anyway. Each individual called out 'fix' at the instant the satellite formed a recognizable configuration with the stars it was passing. The eye is surprisingly accurate in judging such shapes. The timekeeper recorded the time to the nearest second and the name of the observer who made the fix. The observers plotted the position of each fix on a star chart. The coordinates of the plotted positions were read from the charts after the satellite had passed out of sight.

"Four good fixes of the third-stage rocket of Sputnik I were recorded on the morning of October 15, 1957. The following morning we made several good fixes on the rocket and also one on the satellite itself. The data were forwarded immediately to the Smithsonian Astrophysical Observatory as a possible check on the orbit calculated from reports submitted by the official Moon watch teams.

"My objective was to learn as much about the orbit as possible from the observations made by the Teaneck team. The observations had been plotted as hours, minutes and seconds of right ascension, and degrees and minutes of declination. This is the system used to designate the positions of stars. The celestial sphere is divided by imaginary coordinates resembling the lines denoting latitude and longitude on the earth.

Celestial latitudes are measured in degrees, minutes and seconds of the angle made by the earth's axis and an imaginary line connecting the object with the center of the earth. Longitude on the celestial sphere is measured in hours, minutes and seconds from the meridian which passes near the right-hand star in the 'W' of the constellation Cassiopeia. This is designated 0^h 00^m (0 hours and 00 minutes) right ascension. Observations made in this way were converted into those of the so-called 'horizon' system. In this system the point beneath the observer is called the nadir; and the one directly overhead, the zenith. The altitude of a stellar object is specified in the system as the height of the object above the horizon in degrees. Its azimuth with respect to the observer is its compass direction measured in degrees away from true north. Because of the earth's rotation the position of zero longitude on earth with respect to celestial zero longitude changes with the clock. Hence in computing the positions and motions of celestial bodies, time must be taken into account.

"The hour angle which was on the meridian at the time of our observations was calculated from the fact that the sun entered the constellation Libra at 2:27 a.m. Eastern Standard Time on September 23. This meant that on that day right ascension 0^h 00^m was on the meridian at 00:00 hours (midnight) E.S.T. Teaneck is less than one degree from the 75th meridian-so close that no correction for local time is necessary. Star time, as measured by the 0^h 0^m celestial meridian, differs from sun time by a little less than four minutes per day. So four minutes for each day since September 23 were subtracted from our local time. To this I added the time after midnight. The result showed that at 5:55 a.m., the time at which we began our observations on October 15, the hour angle on the meridian at Teaneck was 7^h 55^m

"When it is plotted on a globe, the position of a satellite forms one corner of a triangle on the celestial sphere; the other two corners are formed by the observer's zenith and the celestial north pole.[see illustration on this page]. The angular distance between the zenith ['A' in the illustration] and the North Pole ['C' in the illustration] is the complement of the observer's latitude ['B']. Similarly the line a is the complement of the satellite's declination, and the angle at C is the difference between the hour angle on the meridian and the right ascension of the satellite. The complement of the altitude of the satellite is C, and the angle at A is the satellite's azimuth if the object is in the eastern half of the sky, or 360 degrees minus the angle of azimuth if it is in the western half of the sky.

"The angle at C, and a and b are known. The unknowns can be found by simple arithmetic employing the values of the sines and cosines of the three known quantities, as taken from a table of trigonometric functions. The cosine of a is multiplied by that of b. The sine of a is next multiplied by the sine of b, and the product is multiplied by the cosine of the angle at C. The sum of the two products is equal to the cosine of C. The angle at A is determined in the same way. Its sine is equal to the sine of the angle at C multiplied by the sine of a, the product then being divided by the sine of C.

"The positions thus found are then plotted on an ordinary globe of the earth. If the land markings are ignored and only the coordinates used, this is an excellent way to visualize positions on the celestial sphere. A thread stretched through the points as accurately as possible on a great circle makes interpolation easy. By this means I found the points on the globe where the satellite crossed an east-west line through the zenith, and estimated the times at which the crossings occurred on each morning. The interval was 23 hours, 53 minutes and 51 seconds, which gave a figure of 95 minutes, 35 seconds for the orbital period of the satellite, a result which is probably accurate to within two or three seconds.

"By extrapolating the apparent path of the satellite to the two points where it crosses the horizon, I found the angle at which the satellite crossed an east-west line through the point of observation. The orbit of the satellite, together with the meridian passing through Teaneck and the equator, forms the sides of a spherical triangle, as shown in the accompanying illustration [Figure 3]. Here g represents the orbit, f the meridian and e the equator. Because the angle at E as well as f (the latitude of Teaneck) and the right angle at G are known, the inclination of the orbit to the equator can be found. The cosine of the angle at F is equal to the cosine of f multiplied by the sine of the angle at E.

"This calculation gave a value of 68 degrees for the inclination of the orbit. The method is not too accurate unless the satellite passes nearly overhead and the observations are made close to the horizon.

"Having established the period of the satellite, I calculated its average height and velocity by means of the elementary formulas of mechanics. The gravitational pull acting on the satellite must just equal the force necessary to keep a body at that altitude. The two forces may therefore be equated:

(GMm)/R² =( mV²)/ R

"In this equation G is the universal constant of gravitation (6.66 X 10^-8). M is the mass of the earth (5.975 X 10²⁷ grams). The mass of the satellite (m) appears on each side of the equation and hence cancels out. V is the velocity of the satellite, and R is the radius of the orbit. Because the velocity of the satellite is also equal to 2R/T, in which T is the period of the satellite in seconds, it is possible to write an equation for the height (h) of the satellite in which T is the only variable quantity:

"The average velocity of the satellite in miles per second is given by the equation: V = (2/T) X (h + 3,959). Performing the indicated arithmetic gave values of 325 miles as the average height of Sputnik I, and 4.68 miles per second (17,200 miles per hour) as its average velocity.

f the amateur does not wish to recruit observers, he can record satellite transits with an ordinary camera. The shutter is opened at the beginning of the transit and closed at the end. The time is noted at the beginning and end of. the exposure. The stars appear as short trails, as shown in the photograph in Figure 1. At a film speed of A.S.A. 650 (Royal X Pan) the aperture should be set at f/4.5.

For amateurs who prefer to study satellites indoors, David Berger, who is 14 and lives in Croton-on-Hudson, N. Y., submits an "orbit simulator."

"This simulator," he writes, "consists of a rubber sheet stretched over a circular hole in a piece of plywood, the rubber being depressed in the center by a short stick as shown in the accompanying illustration [below]. The depression forms a three-dimensional curve and represents the gravitational field in the vicinity of a body in space. Assuming that the edge of the depressed sheet is r level, a freely rolling body such as a steel ball, when placed near the edge of the sheet, will roll toward the center of the sheet. In effect the ball is 'attracted' to the center by a 'force' which depends on the depth to which the center is depressed. If the freely rolling ball is brought near the edge and given a circular push, it will orbit around the depression as a 'planet.' The push simulates the kick given to a satellite by the final stage rocket.

"The friction of rolling simulates the drag of the atmosphere on a satellite. The ball accordingly spirals closer to the center of the depression. As the average radius of the orbit decreases, the curve becomes tighter and the ball speeds up, just as a satellite does when it spirals closer to the earth and is acted upon by the increased pull of the earth's gravitational field. One can also demonstrate the eccentric orbit of a comet and an orbit in which a body escapes.

"The effect of two bodies acting on a satellite in space can be simulated by inserting two sticks spaced a few inches apart under the supporting bridge. With this arrangement and variations of it a variety of orbits can be simulated: an earth-to-moon orbit in the form of a figure eight in which the satellite returns to earth, an S-shaped earth-to-moon orbit which terminates on the moon, and an earth-to-moon elliptical orbit which returns to earth."

Roger Hayward, who illustrates this department, admits to being something of a table-top-satellite enthusiast. "David Berger," he writes, "deserves a real pat on the back for calling the attention of high-school-physics teachers to this nice demonstration of celestial mechanics. In the interest of strict accuracy, however, the curve can be improved. The displacement of a point on a circular membrane of rubber which is loaded with a force at the center is equal to (2F/T) In (a/r). F and T have to do with the forces applied and the character of the membrane, In is the natural logarithm, a is the radius of the membrane and r is the radius of the point in question. This formula indicates that the force acting toward the center on a ball rolling on the curve would vary inversely with the radius. Such a force would give rise to elliptical orbits. But the 'attracting' body would be at the center of the ellipse instead of at one focal point, and thus would present a most uncelestial appearance.

"The required surface has a profile of the form x = 1/y, the slope of which varies inversely as the square of the distance. The difference between this profile and that of the rubber membrane is shown in the accompanying illustration [Figure 5 ]. I turned such a surface in a disk of plaster with a lathe. The actual shape required is a petal curve which is removed from the x = l/y curve by a distance equal to the radius of the ball. Such a curve is difficult to compute but easy to make if the face of the cutter has the same radius as the ball. In making my surface I turned a series of rings in the plaster with the tool depicted here [bottom]. The rings were spaced 1/10 inch apart. The compound w rest of the lathe was used to set the depth of each cut, and the ridges between cuts were subsequently smoothed off by hand. Plaster cuts nicely but corrodes steel, so the lathe should be covered for protection.

"It is not always easy to prepare a plaster-cast free of bubbles. This difficulty can be overcome by proper mixing. Fill a container of adequate size with water and sprinkle the plaster into the center of the bowl by hand. This enables you to feel and remove any lumps. Plaster should be added until the pile rises about an inch above the water in the center. The dry heap acts as an escape duct for trapped air. Allow about five minutes for the air to escape. Then the mixture may be stirred gently. Under no circumstances should the plaster be stirred before water has penetrated to the center of the pile, because the dry portion will be broken into small volumes from which air cannot escape.

"I cast the disk from which my surface was turned in a mold formed by wrapping a strip of gummed paper around a thick metal disk so that half the strip extended above the edge of the metal. The resulting cast was attached by sealing wax to a faceplate for machining. Never pat or trowel wet plaster or it will harden unevenly and increase the difficulty of accurate machining. Permit it to dry thoroughly. Wet plaster loads sandpaper quickly.

"Perhaps some amateur might like to investigate an orbit simulator based on the attraction of a magnet for a small sphere of soft iron. The magnetic force varies as the square of the distance, as in the case of gravitational attraction. Such an arrangement might consist of a carefully leveled plate of glass with a magnet below its center. If the magnetic field is generated by a solenoid, perhaps the current could be pulsed to keep the 'satellite' going. If the pulse is relatively light, it should not greatly distort the shape of the orbit."

The procedure of aligning the telescopes of a binocular is called "collimation," and consists in repositioning the lenses and prisms of the instrument according to the results of an optical test. Both the test and adjustment are simple in principle. The mechanical work may prove difficult, however, because many binoculars are sealed against moisture by a preparation which tends to freeze the threaded parts. This may discourage some from attempting to adjust their own instruments. But one can at least make the test for collimation and, if desired, have the work performed by an optical repair shop.

A simple and reasonably precise test is described by Allen Naber of Pittsford, N. Y. "Basically," he writes, "this test involves pointing the objectives of the binocular directly toward the sun when it is at a low altitude and projecting the beams from the eyepieces on a white screen in the shade. When the instrument is focused on infinity, images of the sun will appear as two overlapping disks of light on the screen. If the instrument is supported so that the two telescopes are level with each other, the images will appear at the same height on the screen and the space between their centers will match the space between the centers of the eyepieces-if the instrument is in good adjustment.

"Although a wall may be used for a quick check, accuracy can be increased by providing a screen, perhaps two by three feet in size, which has been ruled with horizontal reference lines an inch or so apart. A pair of vertical rulings spaced three inches apart near one side are also useful as references. The binocular may be supported by a conventional tilt-top tripod or even a block of wood clamped to the window sill. Any method of support is satisfactory which permits the glass to be kept pointed toward the sun and level laterally. The screen may be tipped forward or backward as much as 10 degrees without affecting the accuracy, but it must be kept as close to a right angle with the sun as possible.

"In normal use one's eyes never diverge, either laterally or vertically, although they do converge from zero in looking at a fairly distant object to as much as six or eight degrees in close reading. This should be kept in mind when collimating a binocular. Errors should be in the direction of convergence, not divergence. In other words, the distance between the images on the screen should be equal to or slightly greater than that between the eyepieces, never less. Incidentally, the measurement may be made between the edges of the images and the corresponding edges of the eyepiece lenses instead of at the respective centers. It is a good idea to mask one of the objectives while observing the images to make sure that the beams do not cross. Masking the right-hand objective, for example, should cause the right-hand image to disappear. In the case of the average binocular placed 10 feet from the screen, a convergence or divergence of three degrees results in a difference of about an eighth of an inch in the distances between the eyepieces and the images.

"The objective lenses of most binoculars are clamped by retaining rings in a short length of internally threaded brass tubing bored off-center. These 'eccentric cells,' in turn, telescope into and are secured by eccentric rings, also short brass tubes with an off-center bore. One or more thin slots usually appear in the edge of the retaining rings that faces outward. These slots take the spanner wrench used for loosening or tightening the retaining rings. Such wrenches may either be made or procured from an optical supply house. When the retaining rings are loosened, the eccentric cell and ring may be rotated with respect to each other. This adjusts the radial position of the lens in any desired direction. The image on the screen will move in the same direction as the lens. The collimation procedure consists in finding the position of the lenses that yields the desired image-separation as described.

"In the case of many instruments, particularly the inexpensive imported variety, the retaining rings are made of soft aluminum and the metal may tear away at the slots when pressure is applied with the wrench. When this happens the ring must be cut with a sharp chisel such as a hand-engraving tool, removed, and replaced with a new ring. Usually replacements are not available, so you have to make them yourself. In some designs the rings are locked by small set-screws. They are loosened with a watchmaker's screwdriver. Frequently the heads twist off. The old screw is then drilled out and, after the hole has been retapped, replaced by a larger screw.

Suppliers and Organizations

The Society for Amateur Scientists
5600 Post Road, #114-341
East Greenwich, RI 02818
Phone: 1-401-823-7800

Internet: http://www.sas.org/