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THE ROTATIONAL RESONANCES
OF MERCURY AND VENUS
Copyright © 1976 by Lynn E. Rose.

[* This article is one of 22 essays contained in an Anthology presented to Dr. Immanuel Velikovsky on December 5, 1975, in honor of Dr. Velikovsky and the 25th anniversary of Worlds in Collision; it is our hope to publish the Anthology in its entirety. - The Ed.]

In discussing the rotation of a planet, we might have occasion to use either the sidereal "day" or the solar "day" of that planet. (Words like "day" and "year" will be placed in quotation marks whenever they refer to planets other than Earth.) The sidereal "day" is the interval between the time when one face of the planet is turned toward a given point among the fixed stars and the next time when that same face is turned toward that same point. The mean solar "day" of a planet is the average interval between the time when one face of the planet is turned toward the Sun and the next time when that same face is turned toward the Sun (after having turned all the way around with respect to the Sun).

MERCURY

The Sun-Locked Mercury Model.

For many years, astronomers believed that Mercury completed one rotation on its axis during one revolution around the Sun -- in other words, that Mercury always kept the same face turned toward the Sun, just as the Moon always keeps the same face turned toward Earth. By the mid-1960's, however, improved radar techniques had indicated that Mercury's period of sidereal rotation was about 59 3 days.(1) Following a suggestion made by Colombo,(2) it was widely expected that the exact figure would be 58.646 days, which is two-thirds of Mercury's 87.969-day "year" or period of revolution around the Sun; in that case, the mean solar "day" on Mercury would last 175.938 of our days and would stand in a ratio of 2:1 to the "year" of Mercury. Thus, for Mercury, the conventional model requires that:

1 mean solar "day" = 3 sidereal "days" = 2 "years".

Mercury is a planet whose rotation is very slow (many times slower, in fact, than that of any other planet except Venus); whose orbital eccentricity is relatively high (e = 0.2056, which is more than twice that of any other planet except Pluto); and whose proximity to the Sun is, of course, greater than that of any other planet. These factors are usually seen as sufficient to explain the commensurability or 66 resonance": it is presumed that the Sun was able to get a gravitational "lock" on an axial asymmetry in the distribution of Mercury's mass, in such a way that the axial asymmetry tends to be aligned with the Sun at each perihelion passage, when Mercury passes closest to the Sun and the Sun's gravitational pull on Mercury is at its greatest. Three points should be emphasized about this conventional explanation:

(1) For none of the planets is the interval from one perihelion passage to the next quite the same as the sidereal "year". The anomalistic "Year" is defined as the interval between one perihelion passage and the next, as opposed to the sidereal "year" that is the interval between one orientation with respect to the stars and the next such orientation. Too often, those discussing this resonance refer to the sidereal "year",(3) but what should be used is the anomalistic "year", since the mechanism that is usually envisioned depends upon the Sun's greater pull at perihelion. In this paper, any years" that are possibly commensurable with mean solar "days" are to be understood as anomalistic. For now, this distinction may not be of major importance, since the anomalistic "year" of Mercury exceeds the sidereal "year" by only about one part in one million and current measurements of the rate of rotation are not that fine. Eventually, however, it might be possible to determine empirically whether the rotation of Mercury is locked to the anomalistic "year" (as the conventional theory expects) or to the sidereal "year".

(2) The axial asymmetry that is usually postulated might be thought of as an arrow passing through Mercury. The arrow will be pointed toward the Sun at one perihelion passage, away from the Sun at the next perihelion passage, toward the Sun at the next perihelion passage, and so on. The arrow will never be aligned with the Sun except at perihelion and at two other points that occur shortly before perihelion and shortly after perihelion(4) The reason that the arrow would be aligned with the Sun at those two points is that the orbital angular velocity temporarily exceeds the rotational angular velocity. "To an observer on one surface of the planet the Sun, a huge fiery disc of apparent diameter 2-3 times as large as we see on the Earth would rise in the east and set in the west every 176 terrestrial days. At the time of each perihelion passage the Sun would appear to pause in the sky and actually reverse the direction of its motion... 11(5) This prolonged alinement with the Sun during the time near perihelion, together with the other factors that were mentioned earlier, might help to make the lock that the Sun would have on Mercury's rotation a very strong and effective one.

(3) The rotational and orbital commensurability of Mercury is usually explained as if Mercury and the Sun were alone in the solar system; no reference is made to the influence of any other body. But if Mercury could develop a rotational and orbital resonance influenced only by the Sun, it seems at least possible that other planets could also develop rotational and orbital resonances governed by the Sun.

The Venus-Locked Mercury Model.

It is rather mysterious that astronomers so quickly assumed that the rotation of Mercury has to be locked to the Sun. A rotation rate of 58.646 days would indeed imply that the rotation of Mercury is Sun-locked, but what was overlooked is that a rotation rate of 58.370 days which was also quite consistent with the radar estimate of 593 days -- would imply that the rotation of Mercury is locked to Venus, in the sense that Mercury would turn the same face toward Venus every time Mercury passed between Venus and the Sun. (The fact that astronomers speak of Venus turning the same face toward Earth every time Venus passes between Earth and the Sun renders it even more peculiar that they did not consider the possibility of a similar lock between Mercury and Venus; later on, in discussing the Venus situation, we will see how the fact that astronomers speak of a Sun-locked Mercury model also renders it somewhat peculiar that they have not even considered the possibility of a Sun-locked Venus model.)

The mean synodic period of Venus and Mercury may be defined as the average interval between one inferior conjunction of Mercury, as seen from Venus, and the next; it is equal to 144.566 days. If the mean solar "day" of Mercury were exactly commensurable with the mean synodic period of Venus and Mercury, then five mean solar "days" of Mercury would be equal to six mean synodic periods of Venus and Mercury, and one mean solar "day" of Mercury would be equal to 173.479 days, or a little less than the value of 175.938 days that we found in the case where the rotation of Mercury is locked to the anomalistic "year" of Mercury. The corresponding sidereal "day" that would have caused the same face of Mercury to be turned toward the planet Venus every time Mercury passes between the Sun and Venus Would be equal to 58.370 days. As was mentioned earlier, this figure, like the figure of 58.646 days, was well within the radar estimates of 59 3 days. Thus the radar measurements were compatible with either the Venus-locked model or the Sun-locked model for Mercury. Yet the astronomers seem not even to have noticed the figure of 58.370 days.(6) Eventually, photographic work by Smith and Reese (7) seemed to support the Sun-locked model and -- although no one noticed it -- to disconfirm the Venus-locked model for Mercury. Throughout the mid-1960's, however, there was no justification whatsoever for neglecting the Venus-locked model.

Since the astronomers did (as we shall see) recognize that five mean solar "days" of Venus are about equal to one mean synodic period of Earth and Venus, why did they not notice that five mean solar "days" of Mercury were about equal to six mean synodic periods of Venus and Mercury? It is notorious that modem astronomy is cluttered with relics of geocentrism. Could this oversight be another instance of geocentric thinking? Why is it so readily conceivable that the rotation of one of the planets might be dominated by Earth, but inconceivable -- or at least unconceived -- that the rotation of one of the planets might be dominated by some planet other than Earth?

VENUS

One suggestion of this paper is that if there was or is a Sun-locked rotational and orbital resonance of Mercury, that state of affairs might in some respects have been mirrored in a Sun-locked rotational and orbital resonance of Venus. This Venus resonance might have been in effect twenty-seven or twenty-eight centuries ago, just after Venus achieved its present orbit.(8) Indeed, such a resonance might still be in effect, although it seems more likely that at some time after Earth achieved its present orbit, the rate of rotation of Venus was changed very slightly under the remote influence of Earth. The orbital eccentricity of Venus is lower than that of any other planet, and the differential pull exerted by the Sun at the time of the perihelion passages of Venus might have been outweighed and might have yielded control of Venus' rotation to Earth. At every inferior conjunction of Venus, Earth would tend to adjust the rotation of Venus to the synodic period of Venus and Earth, by tugging on the axial asymmetry so as to keep it pointed toward Earth at successive inferior conjunctions. Thus the rotation of Venus might have been changed from a Sun-locked resonance to an Earth-locked resonance.

The earliest radar measurements of the sidereal rotation of Venus were rather tentative and imprecise, but the rate seemed to be about 100-300 days in a retrograde direction; this was soon narrowed to a range of about 230-250 days.(9) It is peculiar that astronomers almost immediately seized upon the possibility of a 1:5 ratio between the mean solar "day" of Venus and the mean synodic period of Venus and Earth, but seem never even to have considered the possibility of a 13:12 ratio between the sidereal "day" of Venus and the "year" of Venus. Astronomers neglected the possibility of a Sun-locked Venus in favor of an Earth-locked Venus, just as they neglected the possibility of a Venus-locked Mercury in favor of a Sun-locked Mercury. (Of course, the very fact that they did consider a Sun-locked Mercury renders it even more peculiar that they did not consider a Sun-locked Venus.) Later on,(10) more precise measurements indicated that the rate of the sidereal rotation of Venus was 243.1 .1 days, which, as we shall see, favors the Earth-locked model and casts some doubt upon the Sun-locked model. But it must be stressed that the astronomers had disregarded the Sun-locked model even before the more precise measurements became available. And even the "best" measurements available today seem not to be fully decisive. Further investigation will be needed before we can confidently eliminate either the Earth-locked model or the Sun-locked model.

The Sun-Locked Venus Model.

Let us first consider the Sun-locked model, which features a commensurability between the "day" of Venus and the "year" of Venus. This state of affairs might have existed twenty-seven centuries ago, and then been disrupted by Earth's arrival on its present orbit; or it might be the case that this Sun-locked model exists even today: we still need better evidence. Whether this model does or does not exist today, however, let us examine what it might have been like twenty-seven centuries ago. Let us assume that the "year" of Venus has changed not at all during that time and that the "day" has changed very little if any. If the mean solar "day" of Venus was resonant with the "year", then the length of the mean solar "day" would have been equal to 116.844 days, and the length of the sidereal "day" would have been equal to 243.425 days. (This figure seems slightly high, in relation to the "best" radar measurements of 243.1 .1 days.)

The sidereal "day" of Venus would have stood in a 13:12 ratio to the "year" of Venus. The rotation of Venus is -- and was presumably twenty-seven centuries ago also -- retrograde; in that case, the mean solar "day" of Venus would have stood in a 13:25 ratio to the "year" of Venus. The face that Venus turned toward the Sun at perihelion would also have been turned toward the Sun at twenty-four other points on the orbit, but, after 25 mean solar "days" (= 12 sidereal "days") had passed, that same face would have been turned again toward the Sun at perihelion. This is not as effective an arrangement as we saw with the Sun-locked model for Mercury, where the axial asymmetry is aligned with the Sun only at perihelion (and at two other points that occur shortly before perihelion and shortly after perihelion), and where the orbital eccentricity of the planet is so much greater than that of most other planets, thus exaggerating the gravitational differential at perihelion. Nevertheless, let us assume that the Sun could have held the rotation of Venus to a resonance with the "year" of Venus. These relationships would then have obtained:

25 mean solar "days" = 12 sidereal "days" = 13 "years".

It might be noted that the difference between the sidereal and the anomalistic "years" of Venus is only about one-quarter of a second of arc, or about two parts in ten million, and may therefore be regarded as negligible.

It might be noted also that, assuming that the astronomers had noticed the Sun-locked model for Venus, there would still have remained some grounds for hesitation about that model: (1) the low orbital eccentricity, as mentioned, offers very little gravitational differential at perihelion; (2) the more recent radar estimates do seem to favor the Earth-locked model over the Sun-locked model; and (3) the integers involved in the Sun-locked model - 25 solar "days", 12 sidereal "days", 13 "years" - do seem rather high to be significant. How high such integers might be and remain significant will be discussed later.

The Earth-Locked Venus Model.

The mean synodic period of Earth and Venus may be defined as the average interval between one inferior conjunction of the inner planet, as seen from the outer planet, and the next; it is equal to 583.916 days. If the mean solar "day" of Venus were exactly commensurable with the mean synodic period of Earth and Venus, then five mean solar "days" of Venus would be equal to one mean synodic period of Earth and Venus, and one mean solar "day" of Venus would be equal to 116.783 days, or a little less than the mean solar "day" of 116.844 days that would have been resonant with the 4 6 year" of Venus. The corresponding sidereal "day" that would cause the same face of Venus to be turned toward Earth at every inferior conjunction of Venus would be 243.161 days.

The value of 243.16 days was first suggested by Goldstein in 1962,(11) and since then has remained more or less consistent with the radar studies. If the rate of sidereal rotation does indeed lie within the estimated range of 243.1 .1 days, then there seems to be strong evidence for the reality of the Earth-locked resonance of Venus, and we would be entitled to rule out any Sun-locked resonance of Venus at the present time. It should be noted, however, that radar estimation of planetary rotations is a statistical and indirect procedure, not a matter of direct measurement; it should also be noted that there have been some radar results that seem to conflict both with the Sunlocked model and with the Earth-locked model.(12) Thus the situation remains quite fluid - and we should certainly not rule out the Sun-locked model just yet, especially since the astronomers seem not even to have noticed it, let alone investigated it. In general, we still await some assurance from the radar astronomers that the 243.425-day rotation has been definitely ruled out, and that there is therefore no possibility of a Sun-locked rotation of Venus at present. (Even if that is the case, of course, there might have been such a Sun-locked arrangement twenty-seven centuries ago.)

It must be emphasized that for some years the radar measurements both of Venus and of Mercury remained imprecise enough that they were consistent both with the solar and with the planetary models. Eventually, the photographic studies by Smith and Reese(13) established a figure of 58.663 0.021 days for the rotation of Mercury. It was only after these photographic studies of Mercury (which do seem to rule out the Venus-locked Mercury model) and after improved radar measurements of Venus (which do seem to call into question the Sun-locked Venus model), that the astronomers' neglect of these two models might have been justified. But the important thing is that they neglected these models, and accepted the alternative models, years before they had any evidence that might support their positions. (As far as I have been able to determine, no one has ever mentioned either a Venus-locked Mercury model or a Sun-locked Venus model.)

No Commensurabilities? Resonance Without Commensurability?

We have seen that a mean solar "day" of Venus that was in a 13:25 resonance with the "year" of Venus and was changed to a 1:5 resonance with the mean synodic period of Earth and Venus would thereby have been reduced from 116.844 days to 116.783 days: a reduction by one part in two thousand. Instead of using the mean solar "day", we could use the sidereal "day"; the latter was in a 13:12 resonance with the sidereal "year" and would have been reduced from 243.425 days to 243.161 days: a reduction by one part in one thousand. But where is the resonance, where is the exact commensurability that corresponds to the exact 1:5 ratio of the mean solar "day" to the mean synodic period? The fact is that, if we use the sidereal "day" in speaking of the way Venus turns the same face toward Earth at each inferior conjunction, this "resonance" does not involve an exact commensurability. And yet, if we use the mean solar "day" (which is really what is involved when Venus turns the same face out toward Earth and away from the Sun), there is an exact 1:5 commensurability. Similarly, in the Venus-locked model that we considered for Mercury, if we use the sidereal "day", the "resonance" does not involve any commensurability; and yet, if we use the mean solar "day", there is an exact 6:5 commensurability between the mean solar "day" of Mercury and the mean synodic period of Venus and Mercury.

The very essence of resonance is commensurability. Hence it is rather awkward to speak of a resonance that involves no commensurability, especially when a different definition of "day" is available that would involve an exact commensurability. In studying the Mercury-Venus lock, or the Venus-Earth lock, it would therefore have been preferable to use the ancient definition of rotation based upon a return to the same orientation with respect to the central body, rather than the modem definition of rotation based upon a return to the same orientation with respect to the stars (without regard to the orbital movements of the body in the meantime). Not all that is modem is thereby superior.

Too Many Commensurabilities For a Velikovskian Sequence of Events?

There is a further lesson to be learned here. it is often said that the fact that there are a great many resonances in the solar system is proof of a long slow evolution featuring a gradual development of delicate near-Commensurabilities, and that recently colliding planets would never have ended up in such arrangements. What we found in the case of Venus, however, is that one commensurability is so close to another possible commensurability that we still cannot with much confidence determine empirically which, if either, of the Commensurabilities actually obtains. And what we found in the case of Mercury is that, in the mid-1960's and prior to the photographic studies that finally seemed to decide the matter, one commensurability was again so close to another one that the available evidence was insufficient for discriminating between the two possible commensurabilities.

The fact is that almost any conceivable state of affairs is relatively close to some state of commensurability. It becomes a question of how many of the lower integers - the first ten? the first fifty? we are willing to accept as components in the ratios. The 2:1 ratio of the Sun-locked rotation of Mercury would be acceptable. Presumably the 5:6 ratio of the Venus-locked rotation of Mercury would have been acceptable. And the nearly 8:13 ratio of Venus' 11 year" and Earth's "year" is usually accepted, too. But what about the possible 13:25 ratio between the mean solar "day" of Venus and the "year" of Venus? Is that acceptable, or are the integers becoming too high to be significant? After all, if you allow large enough integers, you can always find a near-commensurability to be the case. (Thus, it has been pointed out that there is an exact 11:28 ratio between the year of Earth and the mean "year" of the asteroid Ivar; (13) and that there is a very nearly exact 13:54 ratio between the "year" of Mercury and the year of Earth.(14))

The existence of numerous near-commensurabilities is hardly proof that the solar system has not undergone any drastic rearrangements within historical times. Throw any two planets onto new orbits with new periods, and the chances are good that those new periods will not be far from some state of commensurability. (Thus it would have been easy for Velikovskian sequences of events to feature numerous close commensurabilities.(15) Even if -- as seems to be the case -- the number of "small-integer" near-commensurabilities is greater today than might have been expected by chance, perhaps all that that means is that the present state of the solar system is not the most likely outcome of a period of planetary turmoil, but is something of a long shot. Perhaps our apparent safety at present is an improbable outcome of a long series of unstable situations, with intersecting orbits and Worlds in Collision. It may be just a matter of luck and chance that we have finally arrived at an arrangement of the solar system in which no further collisions seem very likely in the immediate future. (It has been argued by Bass(16) that after a near-collision the planets involved might end up by chance in a state not far from commensurability, and then "in a few centuries" be drawn into a genuine state of resonance and onto "truly orbitally stable" paths.)

Consequences For Velikovskian Theory.

According to Velikovsky's theory, Venus and Earth both underwent radical perturbations, after their last close encounter that occurred some thirty-four centuries ago.(17) Any Earth-Venus resonance -- existing, say, thirty-three centuries ago -- would hardly have survived these radical perturbations of the orbits of both planets by Mars (although fortuitous but numerically similar ratios involving different orbits may have occurred in later periods). It is also important that, according to Velikovsky's sequence of events, Venus would have been established at its present location in the solar system, with very nearly its present "year" and presumably with very nearly its present "day", prior to the time that Earth was perturbed onto its present orbit and ipso facto prior to the time that the mean synodic period of Venus acquired its present value.

Earth might have introduced a slight change in the rotation of Venus after the year 687 before this era, when Earth reached its present orbit, but presumably the rotation of Venus was already rather close to its present value. That present value is apparently locked to the mean synodic period of Earth and Venus rather than to the anomalistic period of Venus, but suppose that earlier it was locked to the anomalistic period of Venus. That earlier lock or resonance would have been similar in certain respects to the conventionally-recognized orbital-rotational resonance of Mercury; the Venus situation might not have been without precedent. Presumably the orbital-rotational resonance of Venus came into being after the last near-collision with Mars -- its origin would have been immediate and fortuitous but only approximate -- and then gradually became locked in to the Sun because of the gravitational influence of the Sun at perihelion. All this would presumably have occurred without any special help from Earth, which was not yet even on its present orbit and thus could not yet have begun playing its present role involving the 1:5 resonance between the mean solar "day" of Venus and the mean synodic period of Earth and Venus.

Notice that the existence of a resonance between the mean solar "day" of Venus and the mean synodic period of Earth and Venus does not show that Earth and Venus ever passed near each other. The resonance can be fitted into Velikovsky's sequence of events, as I have been attempting to do, but in Velikovsky's sequence of events the resonance could not have developed until more than seven centuries after the last Earth-Venus contact, and is not the result of near contact between Earth and Venus. The resonance must have developed at long range (about twenty-five million miles separate Earth and Venus at inferior conjunction). Supporters of Velikovsky are ill-advised when they cite this resonance as proof that Earth and Venus underwent near-collisions with each other thirty-four centuries ago.

Thus the resonance between the mean solar "day" of Venus and the mean synodic period of Earth and Venus cannot be used as evidence that Earth and Venus underwent a near-collision, nor can it be used as evidence that Earth and Venus did not undergo a near collision. The resonance may show that planets can influence each other over vast distances in ways that have not always been appreciated and that are even now not fully understood, but that should be the extent of what is inferred from the resonance.

NOTES

1. See Pettengill and Dyce (1965), page 1240, and Dyce, Pettengill, and Shapiro (1967), pages 351-352.
2. Colombo (1965), page 575.
3. See, for example, Kopal (1973), page 68.
4. See Soter and Ulrichs (1967), page 1315.
S. Kopal (1973), page 68.
6. See McGovern, Gross, and Rasool (1965), page 375. These authors studied old drawings of Mercury and inferred a rotation period of 5 8.4 0.4 days. It is striking that they did not even realize how close they were to the 58.370 days of the Venus locked model for Mercury.
7. Smith and Reese (1968), page 1276.
8. See Velikovsky (1950), pages 202-203, 252, and 259.
9. For background, see Dyce, Pettengill, and Shapiro (1967), page 351.
10. See, for example, Kopal (1973), page 50.
11. See Shapiro (1967), pages 423 and 425.
12. A value of 242.98 0.04 days was found by Carpenter (1970), pages 61 and 66, and a value of 243 0.1 days was found by Jurgens (1970), pages 435-436 and 441442. Both of these values seem too low for either model. The value of 245.1 0.7 days, however, which was found by Dyce, Pettengill, and Shapiro (1967), page 355, seems too high for either model. After having arrived at this value, Dyce, Pettengill, and Shapiro decide that it would be "prudent to increase the error associated with the period to 2 days". They do not indicate just how they derived the "2 days", but the result, of course, is that their altered value of 245.1 2 days now includes the 243.161 days of the Earth-locked model for Venus.
13. See Ip and Mehra (1973), page 146.
14. See Smith and Reese (1968), page 1276.
15. For illustrations of this, see Rose and Vaughan (1 974), pages 30-31.
16. Bass (1974), page 11.
17. See Velikovsky (1950), pages 238-297.

REFERENCES

  • Bass, Robert W.: "Did Worlds Collide?" Pensee , Summer, 1974, pages 8-20.
  • Carpenter, R. L.: "A Radar Determination of the Rotation of Venus", Astronomical Journal (75), February, 1970, pages 61-66.
  • Colombo, G.: "Rotational Period of the Planet Mercury", Nature (208), November 6, 1965, page575.
  • Dyce, R. B., Pettengill, G. H., and Shapiro, I. I.: "Radar Determination of the Rotations of Venus and Mercury", Astronomical Journal (73), April, 1967, pages 351-359.
  • Ip, W. H and Mehra, H.: "Resonances and Librations of Some Apollo and Amor Asteroids with the Earth", Astronomical Journal (78), February, 1973, page-S 142-147.
  • Jurgens, Raymond F.: "Some Preliminary Results of the 70-cm Radar Studies of Venus", Radio Science (5), February, 1970, pages 435442.
  • Kopal, Zdenek: The Solar System. Oxford: University Press, 1973.
  • McGovern, W. E., Gross, S. H., and Rasool, S. I.: "Rotation Period of the Planet Mercury", Nature (208), October 23, 1965, page 375.
  • Pettengill, G. H., and Dyce, R. B.: "A Radar Determination of the Rotation of the Planet Mercury", Nature (206), June 19, 1965, page 1240.
  • Rose, Lynn E., and Vaughan, Raymond C.: "Velikovsky and the Sequence of Planetary Orbits", Pensee, Summer, 1974, pages 27-34.
  • Shapiro, Irwin I.: "Resonance Rotation of Venus", Science (158), July 28, 1967, pages 423-425.
  • Smith, B. A., and Reese, E. I.: "Mercury's Rotation: Photographic Confirmation", Science (162), December 13, 1968, pages 1275-1277.
  • Soter, Steven, and Ulrichs, Juris: "Rotation and Heating of the Planet Mercury", Nature (214), June 24, 1967, pages 1315-1316.
  • Velikovsky, Immanuel: Worlds in Collision, New York: Macmillan, 1950.

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