Gerver's scheme involves observing a meteor as it penetrates the earth's atmosphere. A meteor, which is debris from a comet or a chunk of material from the asteroid belts, heats up rapidly as it falls through the atmosphere, becoming so hot that its glow is visible from the ground. Nearly all meteors burn up before they leave the upper atmosphere.

Gerver's method is to determine a meteor's speed with respect to the earth by dividing the duration of the glow into the length of the meteor trail. If the meteor is orbiting the sun, the upper limit to its speed with respect to the sun is related to the earth's speed around the sun. By measuring the meteor's speed through the atmosphere you can calculate the earth's speed and then the radius of the earth's orbit of the sun.

Gerver's method is put into practice during a time of meteor showers. To apply the method you should arrange for several observers to be separated from one another by tens of kilometers. Have them record the time and duration of any meteors they sight and also mark the path of the meteors on a star map. The duration of a meteor burn should be timed by a chant such as "One one thousand, two one thousand" and so on to count off the seconds. Later examine the collected data for any common sighting. If you find one, you can employ the relative positions of the observers and their measurements of the meteor to calculate the height of the meteor's end point, which is where it was last seen.

At this stage Gerver introduces a check on the results. From the computed end point and the observers' perspectives of it he calculates the compass headings between the observers. If the calculated headings approximate the true ones, he knows he is on the right track.

You see meteors against the celestial sphere, an enormous imaginary structure centered on the earth. Since the stars are

so distant, they seem to lie on the inside surface of the sphere. The point there at which a meteor shower appears to originate is called the radiant. The shower takes its name from the nearest constellation. In the course of a shower you may see meteors streaming in all directions from the radiant. The appearance is an illusion. The trails are in fact approximately parallel. Their apparent divergence arises from your different perspective of the various paths.

One advantage of having several distant observers is that you can extrapolate the observed trails backward until they cross at approximately the radiant. Gerver extrapolates the trails by holding a straightedge against the night sky, aligning it first along the trail recorded by one of the observers and then along the trail recorded by another observer. He extrapolates each trail to the point where they all cross. (On a flat star map the extrapolations are curved lines because of the distortions imposed in mapping the curved surface of the celestial sphere onto a flat surface.) After locating the radiant you can calculate the height of the meteor when it was first seen and the length of its trail.

The meteor's speed in relation to the earth is the ratio of the calculated length of its trail to the duration of its burn. The speed is only approximate because the duration of the burn is estimated. You could greatly improve the precision of the experiment by timing the burn with an instrument. Better yet, you could videotape a meteor shower and then replay it to time a burn. Such refinements, however, would spoil the fun. Gerver's aim is to calculate the distance to the sun without instruments.

The next step of computation requires a knowledge of the orbital characteristics of meteors. Many meteors, certainly those that are associated with repeated showers, move in an elliptical orbit around the sun before the earth's gravity captures them. When one of them passes through the earth's orbit, its speed with respect to the sun can be no more than the square root of 2 times the earth's speed. A meteor with a greater speed orbits the sun only once and then flies off to an effectively infinite distance from the sun.

Since Gerver limits his observations to meteors associated with repeated showers, he assumes that any meteor figuring in his calculations will be moving at a speed (with respect to the sun) no greater than the upper limit. Armed with this limit and the measured speed of the meteor with respect to the earth, he sets out to calculate the speed of the earth.

The velocity of an object is a vector: a line with an orientation that represents the direction of the object and a length that

represents the object's speed. The velocities of the meteor with respect to the earth and to the sun and the earth's velocity in relation to the sun can be represented in what is termed a vector diagram. The length of the earth's velocity vector is arbitrarily drawn. The length of the meteor's velocity with respect to the sun is the square root of 2 times as much, under the assumption that the speed of the meteor approximates the upper limit.

The third vector, which extends from the head of the earth's velocity vector to the head of the vector representing the velocity of the meteor with respect to the sun, represents the meteor's velocity in relation to the earth. Its size is the speed you calculate from observations of the meteor trail. Note that the vectors form a triangle for which one side is known and the other sides represent the unknown speed of the earth. To find the earth's speed from the triangle, you must first determine trigonometrically the angle between the earth's velocity and the meteor's observed velocity.

The meteor's velocity in relation to the earth points toward you from the radiant. Mark the radiant on a star map. Determining the direction of the earth's velocity by means of the map is somewhat trickier. The map is marked in right ascension along one axis and in declination along the other axis. The declination is measured in degrees. During the night stars on the celestial sphere move through your view parallel to a line of declination. Right ascension is often measured in units of hours because the celestial sphere rotates through a certain angle in one hour. Since a full rotation of 360 degrees takes 24 hours, each hour of rotation is equivalent to 15 degrees.

Celestial objects can be positioned on the map or in the sky by means of declination and right ascension. Imagine an extension of the earth's Equator to the inside surface of the sphere. Also imagine a spherical grid, similar to the latitude and longitude lines on a map, laid on that surface. The grid represents declination (analogous to latitude) and right ascension (analogous to longitude).

In the course of a year the sun seems to move with respect to the celestial sphere along a curved path called the ecliptic; the movement is measured in degrees. (The motion is, of course, only apparent because the earth in fact orbits the sun in what is called the ecliptic plane.) For the day of your meteor observations figure out where the sun is on the ecliptic. Then move 90 degrees to the east along the ecliptic. It is from the corresponding position in the sky that the earth's velocity vector then points.

You now have determined two velocity vectors on the star map, one for the meteor's velocity with respect to the earth and another for the earth's velocity in relation to the sun. Measure the distance between the two marks in units of degrees according to the scale on either of the axes. For example, you can mark the distance on the edge of a sheet of paper and then align the edge along the declination axis to convert the distance into degrees. The result is only an estimate of the angle between the two vectors because of the distortion of the flat map.

The supplement of the angle you measure is the angle in the vector diagram between the sides representing the earth's

speed and the measured meteor speed. Substitute that angle and the lengths of the triangle's sides into the trigonometric cosine rule, which relates the sides to the cosine of one of the angles, to solve for the only remaining unknown, the earth's speed. The calculation yields a lower limit to the earth's speed because it includes the assumption that the meteor's speed with respect to the sun is at its upper limit. The meteor could be slower, in which case the earth's speed would be greater than you had calculated.

Since the earth's orbit is nearly circular, the circumference of the orbit is approximately 21rr, where r is the distance between the sun and the earth. The product of the earth's speed and one year (expressed in suitable units) equals the circumference. From this relation you can calculate a lower limit to the distance to the sun.

On three occasions over three years Gerver collected data during meteor showers. He was assisted by his brother Michael and Stephen Alessandrini, Marcus Wright, Jonathan Singer, L. Thomas Ramsey, Herb Doughty, Peter Gaposhkin, Ned Phipps, Agnis Kaugars and Cynthia Burnham. He needed such a throng of observers because the chance that widely separated observers would sight the same meteor was fairly small. The group encountered problems such as overcast skies and traffic jams. The first time Gerver attempted the experiment he and an assistant 10 miles away spotted a meteor at the same time and in the same part of the sky. When Gerver made his computations, however, the distance to the meteor turned out to be negative. Apparently he and the assistant saw different meteors.

Gerver and his team achieved their first successful observation at 10:56 P.M. Pacific Standard Time last August 11. Gerver, Ramsey and Singer were stationed on Route 17 in the Santa Cruz Mountains midway between Los Gatos and Santa Cruz, Calif., and Michael Gerver was 31 kilometers distant in Palo Alto. Call the first position A and the second one B. A observers spotted a meteor that was as bright as Jupiter and had an unusually long trail. Although most of the meteors visible that night were part of the Perseid shower, Gerver believes the bright one seen by the observers was part of the Delta Aquarid shower, whose path is perpendicular to that of the Perseid group.

The observers recorded the altitude and azimuth of the meteor's end point. The altitude is the angle between the end point and the point on the horizon directly below it. The azimuth is the horizontal angle measured clockwise (as it would be seen from above) from north to just below the end point. From A the altitude of the end point was 58 degrees and the azimuth was 225 degrees. From B the altitude was 43 degrees, the azimuth 200 degrees.

Let C be the point on the ground directly below the end point and h be the height of the end point. Construct a right

triangle with one leg equal to h and another leg equal to the distance between A and C. The altitude is the angle between the hypotenuse and the leg along the ground. By means of the tangent of the angle determine the ground-level leg in terms of h. It is .62 h in length.

Next take an overhead view of A and C. Construct a right triangle in which the hypotenuse is the A-to-C separation of .62 h, one leg running past and west and the other leg north and south. Since the azimuth to C as it is seen from A is 225 degrees, the angle between the hypotenuse and the north-south leg must be 45 degrees. The cosine of 45 degrees equals the ratio of the north-south leg to the hypotenuse. Solve the equation for the north-south leg in terms of h: the leg is equal to .44 h. Similarly solve for the east-west leg in terms of h with the aid of the sine of 45 degrees. The east-west leg is also .44 h. Hence A is .44 h north and .44 h east of C.

Repeat the procedure for the observations from B. The ground separation between B and C is 1.1 h. Since C has an azimuth of 200 degrees, the angle in the ground-level triangle between the hypotenuse and the north-south leg is 20 degrees. After solving the appropriate trigonometric relations you will find that B is 1.05 h north and .38 h east of C.

Next draw a ground-level triangle that includes A, B and C. Subtract the north-south distance between A and C from the north-south distance between B and C to find that B is .61 h north of A. Subtract the east-west distance between B and C from the east-west distance between A and C to find that B is .06 h west of A. The north-south and east-west distances between A and B form the legs of a right triangle that has a hypotenuse equal to the actual distance between the points. By means of the Pythagorean theorem calculate the hypotenuse, which turns out to be .61 h. Since this symbolic result must equal the actual separation of 31 kilometers, h must be 50 kilometers. You now have the height at which the meteor was last seen.

As a check on the calculations, reconsider the ground-level triangle in which the separation of A and B is the hypotenuse. With h equal to 50 kilometers, the north-south leg of the triangle must be 31 kilometers and the east-west leg three kilometers. Since the tangent of the angle between the hypotenuse and the north-south leg equals the ratio of three kilometers to 31 kilometers, the corresponding angle must be six degrees. Hence the compass heading from A to B must be six degrees less than 360 degrees, or 354. The true compass heading was 345 degrees. Since the discrepancy is only nine degrees, the calculations appear to be trustworthy.

From A the beginning of the meteor trail was at an altitude of 52 degrees and an azimuth of 160 degrees. If Gerver were certain that both sets of observers spotted the beginning of the trail simultaneously, he could find that point the same way he did the end point. Without that certainty he must find the starting point by means of the radiant. Several nights after the observations he extrapolated the observed paths of the meteor back over the sky by means of his straightedge. The trails intersected just east of the star Tau Aquarii at an altitude of 20 degrees and an azimuth of 120 degrees. This approximate radiant is a few degrees from the star Delta Aquarii, which Gerver believes is the true radiant for the meteor.

To calculate the length of the trail and the height of the beginning point as Gerver did construct an overhead view of the trail's projection onto the ground. Let D be the point on the ground directly below the trail's onset point. The orientation of the trail is set by the azimuth of the radiant, which is 120 degrees. Find the angle CAD within the triangle ACD by subtracting the azimuth of D from the azimuth of C. Find the angle ADC oh the basis that the acute angle between AD and the north-south line is 20 degrees.

From a previous calculation you know that the length of AC is .62 h, which is 31 kilometers. By means of the trigonometric law of sines, which relates the sines of two of the angles to the sides of the triangle opposite those angles, find the length of CD (44 kilometers). Next draw a right triangle in which the hypotenuse represents the length of the meteor trail, one leg represents the distance between C and D and the other leg represents the vertical extent of the meteor's trail. The angle between the trail and the horizontal is the altitude of the meteor's radiant (20 degrees). Calculate the trail's h length with a cosine function; the answer works out to 47 kilometers. With a tangent function calculate the vertical extent of the trail (16 kilometers). You have now determined the length 9 and orientation of the meteor's trail.

The meteor was visible for about 2.5 seconds. In order to travel 47 kilometers in that time, its speed with respect to the earth must be about 19 kilometers per second. On the night of the observations the earth was moving toward the constellation Aries. Gerver found from a star map that the angle between the earth's velocity in relation to the sun and the meteor's velocity with respect to the earth was about 1 12 degrees. From a triangle of those two vectors and the calculated speed of 19 kilometers per second you can calculate that the earth's speed with respect to the sun must be at least 13 kilometers per second. Divide the product of the speed and the number of seconds in a year by 27. The result (65 million kilometers) is a lower limit for the distance to the sun.

Gerver puts forward several possible explanations for the fact that the result is only about half the true distance to the sun. The meteor may not have been traveling at the upper limit of its speed in relation to the sun before it entered the atmosphere. Moreover, the slowing effect of the atmosphere is not considered in the calculation. The most serious source of error is the estimate of the duration of the burn. Gerver believes it may be off by as much as 20 percent.

Gerver also observed Perseid meteors, but he had no distant assistants. Perseid meteors follow the orbit of comet Swift-Tuttle, which has a period of 120 years. When they pass through the earth's orbit, their speed in relation to the sun should be near the upper limit of a sun-orbiting object. By extrapolation Gerver estimated their radiant to be a few degrees east of Epsilon Cassiopeiae, which is about 10 degrees from the accepted radiant near Eta Persei. One of the meteors seemed to move from Eta Herculis to Beta Herculis in about .25 to .5 second. At the time the altitude of Eta Herculis was 69 degrees and the azimuth was 274 degrees; the altitude of Beta Herculis was 58 degrees and the azimuth was 240 degrees. The radiant estimated by Gerver was at an altitude of 25 degrees and an azimuth of 22 degrees.

Assuming that the initial height of the meteor was 66 kilometers, Gerver calculated that the final height was 57 kilometers and the total length of the trail was 23 kilometers. Estimating the burn time to be .35 second, he found that the meteor's speed in relation to the earth was 65 kilometers per second. The estimated radiant was about 50 degrees from the direction of travel of the earth. From a vector diagram he calculated a lower limit to the earth's velocity in relation to the sun as being 36 kilometers per second. That result puts the distance to the sun at 180 million kilometers.

Gerver employed star maps, but he points out that you can make altitude measurements with a primitive astrolabe. As a trial he sighted stars with a pad of paper held vertically with one bottom corner in front of his right eye and the bottom of the pad exactly horizontal. He rolled his eye upward until h he saw a star aligned with a point on the far vertical edge of the pad. He marked the apparent position of the star on that edge of the paper. With a ruler he then measured the width of the pad and the distance from the bottom edge to the mark. From the ratio of the lengths he calculated the angle between the horizontal and the direction to the star. He found that for any given star he was able to ascertain the angle to within five degrees of the true altitude of the star.

Gerver offers several guides on how best to repeat his experiment. The observers should be familiar with the constellations so that they can accurately locate a meteor trail in the night sky and mark it on a star map. At least two people should be at each observing site. One person should record the duration of the burn while the other memorizes the trail before marking it on the map. Having more than two observing sites improves the accuracy of the calculations.

The weakest stage in Gerver's procedure is estimating the duration of the burn, particularly when it is less than a second. Sometimes, after a meteor had disappeared, Gerver attempted to sweep a finger of his outstretched arm across the sky at about the speed of the meteor. During the sweep he chanted to measure the time. The duration of the meteor burn is the product of the duration of the sweep and the ratio of the angle through which the meteor passed to the angle through which his finger moved.

Bibliography

BASIC PHYSICS OF THE SOLAR SYSTEM. V. M. Blanco and S. W. McCuskey. Addison-Wesley Publishing Co., 1961.

THE AMATEUR SCIENTIST. Jearl Walker in Scientific American, Vol. 240, No. 5, pages 172-182; May, 1979.

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