Of Time and Space and Other Things

by Isaac Asimov

  But our average reader may reason to himself as fol lows: "I see only one face of the Moon and I say it is not rotating. An observer on the Sun sees all parts of the Moon and he says it is rotating. Clearly, I am more im portant than the Sun observer since, firstly, I exist and be doesn't, and, secondly, even if he existed, I am me and he isn't. Therefore, I insist on having it my way. The Moon does not rotate!"

  There has to be a way out of this confusion, so let's think things through a little more systematically. And to do so, let's start with the rotation of the Earth itself, since that is a point nearer to all our hearts.

  One thing we can admit to begin with: To an observer on the Earth,,the Earth is not rotating. If you stay in one place from now till doomsday, you will see but one portion of the Earth's surface and no other. As far as you are concerned, the planet is standing still. Indeed, through most of civilized human history, even the wisest of men insisted that "reality" (whatever that may be) exactly matched the appearance and that the Earth "really" did not rotate. As late as 1633, Galileo found himself in a spot of trouble for maintaining otherwise.

  But suppose we had an observer on a star situated (for simplicity's sake) in the plane of the EartWs equator; or, to put it another way, on the celestial equator (see Chapter 3). Let us further suppose that the observer was equipped with a device that made it possible for Mm to study the Earth's surface in detail. To him, it would seem that the Earth rotated, for little by little he would see every part of its surface pass before his eyes. By taking note of some particular small feature (for example, you and I standing on some point on the equator) and timing its return, he could even determine the exact period of the Earth's rotation-that is, as far as he is concerned.

  We can duplicate his feat, for when the observer on the star sees us exactly in the center of that part of Earth's surface visible to himself, we in turn see. the observer's star directly overhead. And just as he would time the periodic return of ourselves to that centrally located position, so we could time the return of his star to the overhead point.

  The period determined wiU be the same in either case.

  (Let's measure this time in minutes, by the way. A minute can be defined as 60 seconds, where I second is equal to 1/31,556,925.9747 of the tropical year.)

  The period of Earth's rotation with respect to the star is just about 1436 minutes. It doesn't matter which star we use, for the apparent motion of the stars with respect to one another, ii viewed from the Earth, is so vanishingly small that the constellations can be considered as moving all in one piece.

  The period of 1436 minutes is called Earth's "sidereal day." The word "sidereal" comes from a Latin word for "star," and the phrase therefore means, roughly speaking, "the star-based day."

  Suppose, though, that we were considering an observer on the Sun. If he were watching the Earth, he, too, would observe it rotating, but the period of rotation would not seem the same to him as to the observer on the star. Our solar observer would be much closer to the Earth; close enough, in fact, for Earth's motion about the Sun to intro duce a new factor. In the course of a single rotation of the Earth (judging by the star's observer), the Earth would have moved an appreciable distance through space, and the solar observer would find himself viewing the planet from a different angle. The Earth would have to turn for four more minutes before it adjusted itself to the new angle of view.

  We could interpret these results from the point of view of an observer on the Earth. To duplicate the measure ments of the solar observer, we on Earth would have to measure the period of time from one passage of the Sun overhead to the next (from noon to noon, in other words).

  Because of the revolution of the Earth about the Sun, the Sun seems to move from west to east against the back ground of the stars. After the passage of one sidereal day, a particular star would have returned to the overhead posi tion, but the Sun would have drifted eastward to a point where four more minutes would be required to make it pass overhead. The solar day is therefore 1440 minutes long, 4 ' ates longer than the sidereal day.

  Next, suppose we have an observer on the Moon. He is even closer to the @h and the apparent motion of the Earth against the stars is some thirteen times greater for him than for an observer on the Sun. Therefore, the dis crepancy between what he sees and what the star observer sees is about thirteen times greater than is the Sun/star discrepancy.

  If we consi er this same si tion from t. ie Earth, we would be measuring the time between successive passages of the Moon exactly overhead. The Moon drifts eastward against the starry background at thirteen times the rate the Sun does. After one sidereal day is completed, we have to wait a total of 54 additional minutes for the Moon to pass overhead again. The Earth's "lunar day" is therefore 1490 minutes long.

  We could also figure out the periods of Earth's rotation with respect to an ob 'server on Venus, on Jupiter, on Halley's Comet, on an artificial satellite, and so on, but I shall have mercy and refrain. We can instead summarize the little we do have as follows: sidereal day 1436 minutes solar day 1440 minutes lunar day 1490 minutes By now it may seem reasonable to ask: But which is the day? The real day?

  The answer to that question is that the question is not a reasonable one at all, but quite unreasonable; and that there is no real day, no real period of rotation. There are only different apparent periods, the lengths of which de pend upon the position of the observer. To use a prettier sounding phrase, the length of the period of the Earth's rotation depends on the frame of reference, and all frames of reference are equally valid.

  But if all frames of reference are equally valid, are we left nowhere?

  Not at all! Frames of reference may be equally valid, but they are usually not equally useful. In one respect, a particular frame of reference may be most useful; in an other respect, another frame of reference may be most useful. We are free to pick and choose, using now one, now another, exactly as suits our dear little hearts.

  For instance, I said that the solar day is 1440 minutes long but actually thafs a He. Because the Earth's axis is tipped to the plane of its orbit and because the Earth is sometimes closer to the Sun and sometimes farther (so that it moves now faster, now slower in its orbit), the solar day is sometimes a little longer than 1440 minutes and sometimes a little shorter. If you mark off "noons" that are exactly 1440 minutes apart all through the year, there will be times during the year when the Sun will pass over head fully 16 minutes ahead of schedule, and other times when it will pass overhead fully 16 minutes behind schedule. Fortunately, the Fffors cancel out and by the end of the year all is even agmn.

  For that reason it is not the solar day itself that is 1440 minutes Ion amp; but the average length of all the solar days of the year; this average is the "mean solar day." And at noon of all but four days a year, it is not the real Sun that crosses the overhead point but a fictitious body called the "mean Sun." The mean Sun is located where the real Sun would be if the real Sun moved perfectly evenly.

  The lunar day is even more uneven than the solar day, but the sidereal day is a really steady affair. A particular star passes overhead every 1436 minutes virtually on the dot.

  If we7re going to measure time, then, it seems obvious that the sidereal day is the most useful, since it is the most constant. Where the sidereal day is used as the basis for checking the clocks of the world by the passage of a star across the hairline of a telescope eyepiece, then the Earth itself, as it rotates with respect to the stars, is serving as the reference clocl The second can then be defined as 1/1436.09 of a sidereal day. (Actually, the length of the year is even more constant than that of the sidereal day, which is why the second is now officially defined as a frac tion of the tropical year.)

  The solar day, uneven as it is, carries one important advantage. It is based on the position of the Sun, and the position of the Sun determines whether a particular por tion of the Earth is in light or in shadow. In short, the solar day is equal to one period of light (daytime) plus one period of
darkness (night). The average man through out history has managed to remain unmoved by the posi tion of the stars, and couldn't have cared less when one of them moved overhead; but he certainly couldn't help noticing, and even being deeply concerned, by the fact that it might be day or night at a particular moment; surmise or sunset; noon or twilight.

  It is the solar day, therefore, which is by far the most useful and important day of all. It was the original basis of time measurement and it is divided into exactly 24 hours, each of which is divided into 60 minutes (and 24 times 60 is 1440, the number of minutes in a solar day).

  On this basis, the sidereal day is 23 hours 56 minutes long and the lunar day is 24 hours 50 minutes long.

  So useful is the solar day, in fact, that mankind has become accustomed to thinking of it as the day, and of thinking that the Earth "really" rotates in exactly 24 hours, where actually this is only its apparent rotation with re spect to the Sun, no more "real" or "unreal" than its ap parent rotation with respect to any other body. It is no more "real" or "unreal," in fact, than the apparent rota tion of the Earth with respect to an observer on the Earth - that is, to the apparent lack of rotation altogether.

  The lunar day has its uses, too. If we adjusted our watches to lose 2 minutes 5 seconds every hour, it would then be running on a lunar day basis. In that case, we would find that high tide (or low tide) came exactly twice a day and at the same times every day-indeed, at twelve hour intervals (with minor variations).

  And extremely useful is the frame of reference of the Earth itself; to wit, the assumption that the Earth is not rotating at all. In judging a billiard shot, in throwing a baseball, in planning a trip cross-country, we never take into account any rotation of the Earth. We always assume the Earth is standing still.

  Now we can pass on to the Moon. For the viewer from the Earth, as I said earlier, it does not rotate at all so that its "terrestrial day" is of infinite length. Nevertheless, we can maintain that the Moon rotates if we shift our frame of reference (-usually without warning or explana' tion so that the reader has trouble following). As a matter of fact, we can shift our plane of reference to either the Sun or the stars so that not only can the-Moon be con sidered to rotate but to do so in either of two periods.

  With respect to the stars, the period of the Moon's rota tion is 27 days, 7 hours, 43 minutes, 11.5 seconds, or 27.3217 days (where the day referred to is the 24-hour mean solar day). This is the Moon's sidereal day. It is also the period (with respect to the stars) of its revolu tion about the Earth, so it is almost invariably called the "sidereal month."

  In one sidereal month, the Moon moves about 1/1.1 of the length of its orbit about the Sun, and to an observer on the Sun the change in angle of viewpoint is consider able. The Moon must rotate for over two more days to make up for it. The period of rotation of the Moon with respect to the Sun is the same as its perio,d of revolution about the Earth with respect to the Sun, and this may be called the Moon's solar day or, better still, the solar month. (As a matter of fact, as I shall shortly point out, it is called neither.) The solar month is 29 days, 12 hours, 44 minutes, 2.8 seconds long, or 29.5306 days long.

  Of these two months, the solar month is far more useful to mankind because the phases of the Moon depend on the relative positions of Moon and -Sun. It is therefore 29-5306 days, or one solar month, from new Moon to new Moon, or from full Moon to full Moon. In ancient times, when the phases of the Moon were used to mark off the seasons, the solar month became the most iinpor tant unit of time.

  Indeed, great pains were taken to detect the exact day on which successive new Moons appeared in order that the calendar be accurately kept (see Chapter 1). It was the place of the priestly caste to take care of this, and the very word "calendar," for instance, comes from the Latin word meaning "to proclaim," because the beginning of each month was proclaimed with much ceremony. An assembly of priestly officials, such as those that, in ancient times, might have proclaimed the beginning of each month, is called a "synod." Consequently, what I have been calling the solar month (the logical name) is, actually, called the synodic month."

  The farther a planet is from the Sun and the faster it turns with respect to the stars, the smaller the discrepancy between its sidereal day and solar day. For the planets beyond Earth, the discrepancy can be ignored.

  For the two planets closer to the Sun than the Earth the discrepancy is very great. Both Mercury and Venus turn one face eternally to the Sun and have no solar day. They turn with respect to the stars, however, and have a sidereal day which @ out to be as long as the period of their revolution about the Sun (again with respect to the stars).

  If the various true satellites of the Solar System (see Chapter 7) keep one face to their primaries at all times, as is very likely true, their sidereal day would be equal to their period of revolution about their primary.

  If this is so I can prepare a table (not quite like any I have ever seen) listing the sidereal period of rotation for each of the 32 major bodies of the Solar System: the Sun, the Earth, the eight other planets (even Pluto, which has a rotation figure, albeit an uncertain one), the Moon, and the 21 other true satellites. For the sake of direct corn parison I'll give the period in minutes and list them in the order of length. After each satellite I shall put the name of the primary in parentheses and give a number to represent the position of that satellite, counting outward from the primary.

  Sidereal Day

  Body (minutes)

  Venus 324,000

  Mercury 129,000

  Iapetus (Satum-8) 104,000

  Moon (Earth-1) 39,300

  Sun 35,060

  Hyperion (Saturn-7) 30,600

  Callisto (Jupiter-5) 24,000

  Titan (Satum-6) 23,000

  Oberon (Uranus-5) 19,400

  Titania (Uranus-4) 12,550

  Ganymede (Jupiter-4) 10,300

  Pluto 8650

  Triton (Neptune-1)

  Rhea (Saturn-5) 6500

  Umbriel (Uranus-3) 5950

  Europa (Jupiter-3) 5100

  Dione (Satum-4) 3950

  Ariel (Uranus-2) 3630

  Tethys (Saturn-3) 2720 lo (Jupiter-2) 2550

  Miranda (Uranus-1) 2030

  Enceladus (Saturn-2) 1975

  Deimos (Mars-2) 1815

  Mars 1477

  Earth 1436

  Mimas (Satum-1) 1350

  Neptune 948

  Amaltheia (Jupiter-1) 720

  Uranus 645

  Saturn 614

  Jupiter 590

  Phobos 460

  These figures-represent the time it takes for stars to make a complete circuit of the skies from the frame of reference of an observer on the surface of the body in question. If you divide each figure by 720, you get the number of minutes it would take a star (in the region of the body's celestial equator) to travel the width of the Sun or Moon as seen from the Earth.

  On Earth itself, this takes about 2 minutes and no more, believe it or not. On Phobos (Mars's inner satellite), it takes only a little over half a minute. The stars will be whirling by at four times their customary rate, while a bloated Mars hangs motionless in the sky. What a sight that would be to see.

  On the Moon, on the other hand, it would take 55 minutes for a star to cover the apparent width of the Sun.

  Heavenly bodies could be studied over continuous sustained intervals nearly thirty times as long as is possible on the Earth. I have never seen this mentioned as an advantage for a Moon-based telescope, but, combined with the absence of clouds or other atmospheric, interference, it makes a lunar observatory something for which astron omers ought to be willing to undergo rocket trips.

  On Venus, it would take 450 minutes or 7'h hours for a star to travel the apparent width of the Sun as we see it. What a fix astronomers could get on the heavens there - if only there were no clouds.

  7. Just Mooning Around

  Almost every book on astronomy I have ever seen, large or small, contains a little table of the Solar System. F
or each planet, there's given its diameter, its distance from the sun, its time of rotation, its albedo, its density, the number of its moons, and so on.

  Since I am morbidly fascinated by numbers, I jump on such tables with the perennial hope of finding new items of information. Occasionally, I am rewarded with such things as surface temperature or orbital velocity, but I never really get enough.

  So every once in a while' when the ingenuity-circuits in my brain are purring along with reasonable smoothness, I deduce new types of data for myself out of the material on band, and while away some idle hours. (At least I did this in the long-gone days when I had idle hours.)

  I can still do it, however, provided I put the results into formal essay-form; so come join me and we will just moon around together in this fashion, and see what turns up.

  Let's begin this way, for instance…

  According to Newton, every object in the universe attracts every other object in the universe with a force (i) that is proportional to the product of the masses (ml and M2) of the two objects divided by the square of the distance (d) between them, center to center. We multiply by the gravitational constant (g) to convert the propor tionality to an equality, and we have-. f = 9MIM2 (Equation 1) d2

  This means, for instance, that there is an attraction be tween the Earth and the Sun, and also between the Earth and the Moon, and between the Earth and each of the various planets and, for that matter, between the Earth and any meteorite or piece of cosmic dust in the heavens.

  Fortunately, the Sun is so overwhelmingly massive corn 'Pared with everything else in the Solar System that in calculating the orbit of the Earth, or of any other planet, an excellent first approximation is attained if only the planet and the Sun are considered, as though they were alone in the Universe. The effect of other bodies can be calculated later for relatively minor refinements.

  In the same way, the orbit of a satellite can be worked out first by supposing that it is alone in the Universe with its primary.