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ENERGY AND ECCENTRICITY

Velikovsky and the Sequence of Planetary Orbits

 Lynn E. Rose and Raymond C. Vaughan

Dr.  Rose is Professor of Philosophy at the State University of New York (Buffalo) where, among other courses, he teaches the history and philosophy of science.  Mr. Vaughan is a senior technician at Carborundum Corporation, Niagara, New York.

The orbits we proposed in 1972 (1) were intended to refute the claim that Velikovsky's theory is astronomically impossible.  They demonstrate that Keplerian orbits can be proposed that not only cross each other—so that collisions or near-collisions will tend to occur—but also conserve total angular momentum and do not increase total orbital energy.  Further investigation of these orbits was carried out by Ransom and Hoffee (2).  In this article, we propose several alternative sequences of Keplerian orbits, accompanied by discussion of our methods, assumptions, and sources.  All these orbits belong to the relatively calm intervals that were separated by the catastrophes; the interactions during the near-collisions have not been investigated in detail.  We conclude with a discussion of two problems that are not yet satisfactorily resolved: eccentricity-damping and energy-disposal.

It should be noted that the development of this article has relied greatly on an unpublished paper by Vaughan entitled "Orbits and Their Measurements" (3).  The map of orbits presented there has been useful in several ways: as a slide-rule-like device for calculating orbital parameter values, as a graphical demonstration of the parameter interrelationships, and as a worksheet on which real and hypothetical orbits can be represented.

Our units of measurement are geobasic units (4): the unit of mass is Earth's mass; the unit of length is the astronomical unit, or present mean distance of Earth from the Sun; and the unit of time is Earth's present sidereal year.  The traditional symbols are used for the planets: (Venus),  Å (Earth), (Mars), and ʮ (Jupiter).

Two examples showing how limits may be put on specific orbits will be presented before we discuss the sequences of orbits.

The Pre-Exodus Orbit of Venus

Limits on Venus' orbit prior to its first encounter with Earth might be derived from the following assumptions: (A) Venus originated from Jupiter. (B) Venus later was involved in a near-collision with Earth. (C) The orbit of Venus was not changed substantially from the time of its last proximity to Jupiter to the time of its first encounter with Earth. (D) The orbit of Jupiter has not changed substantially since its last encounter with Venus. (E) The orbit of Earth was somewhat smaller than it is now, with its aphelion being perhaps eight-tenths of an astronomical unit.

It follows from assumptions A, B, and C that Venus' orbit extended at least as far from the Sun as the perihelion (rmin) of Jupiter and at least as close to the Sun as the aphelion (rmax) of Earth.  Incorporating assumption D, it follows that (rmax)_ ³ (rmin)_ = 4.95 and, incorporating assumption E, it follows that (rmin)_ <(rmax)Å = 0.8. Using the map format presented by Vaughan (3), one can see that the orbit of Venus must lie above the rmax = 4.95 contour and below the rmin = 0.8 contour.  These two contours are plotted on the map in Figure 1; the orbit must lie within the shaded area in the upper right-hand corner of the map.  The lowest possible eccentricity for such an orbit is 0.722, requiring a semimajor axis of 2.875 a.u. The smallest possible semimajor axis is approximately 2.5 a.u., requiring an eccentricity between 0.95 and 1.0.

The Post-Beth-horon Orbits of Earth

A second example will show how dynamical considerations might be used to set limits on the various orbits of Earth since its final encounter with Venus at the time of the battle of Beth-horon.  Earth's present orbit is included in this period of time, as well as Earth's orbits before and during the Earth-Mars encounters; all would lie within these limits.  The limits could be derived from the following assumptions: (A) The orbit of Earth was not changed substantially from the time of its final encounter with Venus to the time of its first encounter with Mars in the eighth century. (B) The orbit of Venus has not been changed substantially since its final encounter with Mars in the ninth or eighth century. (C) The present orbit of Venus has a semimajor axis a = 0.72 and an eccentricity e = 0.007. (D) The masses of Venus, Earth, and Mars have remained approximately the same since Venus' final encounter with Earth. (E) The total orbital energy of the three planets has either remained the same or decreased since Venus' final encounter with Mars (the present total is: H = -43.61). (F) The present total orbital angular momentum (I = 11.54) of the three planets has remained the same since Venus' final encounter with Mars. (G) The orbits of all three planets lay more or less in the present ecliptic plane; none of the three was moving in a retrograde orbit.

This derivation of limits applies to the period since the final Venus-Mars encounter in the ninth or eighth century; all post-Beth-horon orbits of Earth are included in this period of time.  Using assumptions B, C, and D, one can calculate that Venus has had an orbital energy H = -22.24 and an angular momentum I = 4.36 throughout this period of time.  By subtracting Venus' values from the total values given in assumptions E and F, it follows that the sum of Earth's and Mars' orbital energies was greater than or equal to -21.37, while the sum of Earth's and Mars' orbital angular momenta was equal to 7.18.

The orbital energy of a planet must be less than zero, in the sense that a greater energy would cause the planet to escape from the Sun's gravitational field.  Zero is thus the extreme limit for the greatest possible orbital energy of Mars.  By subtracting this Mars limit of zero from the minimum Mars-Earth total of -21.37, it follows that the minimum value for Earth's orbital energy was -21.37. Since the mass of Earth (i.e., the Earth-Moon system) is 1.01, the minimum value for Earth's orbital energy per unit mass must have been -21.2. Thus, the location of Earth's orbit on the map of orbits must be above the orbital energy per unit mass contour H/m = -21.2. This contour is shown in Figure 2.

The smallest possible orbit for Mars would be an orbit whose angular momentum per unit mass is approximately 0.56, since a smaller orbit would pass within the Roche limit of the Sun, in which case Mars would tend to break up as the tidal force of the Sun exceeded the planet's own gravity and structural cohesion. (For Mars, the radius of the Sun's Roche limit is 0.0081 astronomical units, so that the practical lower limit of orbits on the map of orbits would be the contour rmin= 0.0081, which coincides more or less with the contour I/m = 0.56.)  The mass of Mars is 0.107; the smallest possible value for Mars' angular momentum would thus be 0.56 x 0.107 = 0.06.  By subtracting this extreme minimum value for Mars from the Mars-Earth total of 7.18, it follows that the maximum value for Earth's orbital angular momentum was 7.12.  Division by Earth's mass gives 7.05 as the maximum value for Earth's orbital angular momentum per unit mass.  Thus, the location of Earth's orbit on the map of orbits must be below the angular momentum per unit mass contour I/m = 7.05.  This contour and the contour derived in the preceding paragraph are plotted on Figure 2; these two contours are the extreme limits (according to our seven assumptions) on Earth's orbit since its last encounter with Venus.

In these two examples, changes in our assumptions would have varying effects on our orbital conclusions.  As an example of a quantitative change, it might be mentioned that a more realistic narrowing of the extreme limits used for Mars' orbit in the second example would result in a relatively slight narrowing of the limits derived for Earth, since Mars has only one-ninth the mass of Earth.  One qualitative change affecting assumption E in the second example will be discussed in greater detail later; we have assumed that near-collisions could be either completely elastic (causing no change in orbital energy) or partially inelastic (causing a loss of orbital energy).  The latter possibility may turn out to be severely limited.

Most of the other past orbits cannot be put within limits as close as in these two examples, so that there remains plenty of room for conjecture.  As of now, there is just not enough information available, either from Velikovsky's theory or from independent ancient sources, for any further strict narrowing of this kind.

 THE TABLES

We have tentatively worked out a more detailed sequence of events—indeed, three alternative sequences of events (Tables 1, 2, and 3)—that are to varying degrees consistent with Velikovsky's views as stated in Worlds in Collision.  By making relatively minor adjustments in the various parameters, we have been able to incorporate into the models some Velikovskian features such as 50-"year" and 4-"year" cycles of Venus and 15-"year" cycles of Mars. (Words such as "year" will appear in quotation marks whenever they are not necessarily the same as our present units.)

In all three of the Tables, Stage I occurs before the first Venus-Earth contact at the time of the Exodus.  Stage 2 is the interval of 50-52 "years" between the Exodus and the battle of Beth-horon, at which time Earth and Venus again came near each other.  Stage 3 is the post-Beth-horon period, extending from the fifteenth through the eighth centuries.  Stage 4 is after the first Earth-Mars contact and before the last Earth-Mars contact.  Stage 5 is since -687.  The numbering of the Stages reflects the successive orbits of Earth; the lettered subdivisions of Stages reflect those changes that did not affect Earth.

How Long Was 360 "Days"?

Velikovsky maintains that the post-Beth-horon "year" contained 360 "days" (5), but he does not claim to know the exact length of the post-Beth-horon "day."  Since the post-Beth-horon "year" had fewer "days" than the present year, it might seem probable that the post-Beth-horon "year" was slightly shorter than the present year and that the orbit was consequently smaller than the present orbit of Earth.  This is not as easy as it sounds.

If the orbit of Earth at present is greater than during the fifteenth through eighth centuries (and if the present eccentricity of 0.017 is not greater than before), then Earth must have gained angular momentum at the expense of Mars.  Thus we have some forbidding limits: the angular momentum of Mars on its former orbit must have been greater than that possessed by Mars today, and yet the perihelion of that orbit must have been quite near the present distance of Venus from the Sun.  These conditions cannot be satisfied by any stable orbit, as can be shown either on the map of orbits or by algebraic calculation.  On the assumptions that the perihelion of this elliptical orbit was .7 a.u. and that the angular momentum of this orbit was greater than Mars' present angular momentum of .826, we have

rmin= a(1×e) = .7

and

I = 2πm (a (1-e2 ))1/2

   = (2π) x (.107) x (a(1-e2 ) )1/2 > .826,

from which it can be deduced that e > 1.16.  But it is not possible for an elliptical orbit to have e >- 1.0. And if the perihelion were any lower than .7, that would only serve to raise still higher the required eccentricity.  Tables 1, 2, and 3 provide three different solutions to this angular momentum obstacle.

Table 1

One way out of this difficulty is to make the "year" of 360 "days" longer than the present year of 365 1/4 days, as in Table 1, and to suppose that Mars gained angular momentum from its cumulative contacts with Earth.  Although we did not spell them out in our original proposal of orbits in Pensée (1), these were the considerations that lay behind our decision to have the present semimajor axis of Earth's orbit shorter than it was during the fifteenth through eighth centuries.

Table 1 is, for the most part, consistent with views expressed by Velikovsky.  Thus Table 1 incorporates a 7.144-"year" mean synodic period of Venus during the 50"year" interval between the Exodus and the battle of Beth-horon: since 7 x 7.144 = 50.01, provision has been made for the possibility that Venus might collide with Earth again at the close of the seventh of these synodic periods.  Note also the 4-"year" synodic period of Venus in Stage 3a (related to the olympiad?), the 1.6-"year" mean synodic periods in Stages 3b and 5 (1.6 x 5 = 8.0), and the 1.333-"year" mean synodic period in Stage 4 (1.333 x 3 = 4.0).

Table 1 also incorporates five different 15-"year" cycles for Mars: in Stages 3a, 3b, 3c, and 4, where 15 "years" is very nearly equal to an integral number (41, 14, 8, and 8, respectively) of sidereal periods of Mars, and also to an integral number (26, 1, 7, and 7, respectively) of synodic periods of Mars; and in the present stage (Stage 5) of the solar system, where favorable oppositions of Mars occur about every 15 years.  Velikovsky suggests that this latter phenomenon might be regarded as a "vestige" of the approaches of Mars every 15 "years" during the eighth and seventh centuries (6).

It may seem that we have been over-zealous in our efforts to find opportunities to incorporate cycles of 15 "years" into the model.  But we do not claim that all of the 15-"year" cycles in our model actually occurred, nor do we claim that those cycles that did occur were of exactly 15 "years."  Our only intention has been to illustrate the relative ease with which several different sets of orbits could have involved Martian cycles of approximately 15 "years."  The other cycles that we have introduced are subject to similar qualifications.

Stage 1 and Stages 3a, 3b, and 3c also incorporate the ratio of 210:400 that is reported in the Shitah Mekubbezet (7): the sidereal period of .593 is 210/400 of the later sidereal period of 1.1295. (This source is discussed in greater detail in a separate article in this issue, in relation to the vital statistics of persons in the Bible.)  Velikovsky mentions this Midrash from the Shitah Mekubbezet, but he is reluctant to accept the exact ratio of 210:400 (8).  Velikovsky's caution is justified, but it should be stressed that such a ratio is not at all incompatible with Velikovsky's general sequence of events.

TABLE  1

                Semi-      Eccen-    Mass    Sidereal   Mean       Peri-     Aphelion   Minimum  Maximum    Energy   Angular
                major      tricity                    Period     Synodic   helion                      Velocity    Velocity                      Momentum
                Axis                                                       Period*

 Stage I
   Earth      .706      .070       1.012       .593                            .656         .755            6.972         8.022          -28.307          5.330
   Mars      .555      .075         .107       .413           2.299        .513         .596            7.825         9.094            -3.807            .499
   Venus  3.000      .794         .870     5.196           1.129        .618        5.382           1.229        10.705           -5.724          5.756

-37.839        11.585

Stage 2
    Earth     .985        .260        1.012       .978                          .729        1.241           4.852           8.261         -20.285         6.095
    Mars      .555       .075          .107        .413           .732        .513         .596            7.825           9.094           -3.807           .499
    Venus  1.089       .450        .8501      .137         7.144        .599        1.579           3.708           9.775         -15.405          4.978

-39.497        11.572

Stage 3a
    Earth    1.085      .090        1.012      1.1295                         .987        1.182          5.513           6.603        -18.422         6.597
    Mars     .555        .075        .107        .413             .577        .513          .596          7.825           9.094          -3.807           .499
    Venus   .935        .472        .835        .904           4.000        .494        1.376          3.892          10.851       -17.634         4.472

-39.863       11.56

Stage 3b
    Earth     1.085      .090       1.012      1.1295                         .987         1.182          5.513           6.603        -18.422        6.597
    Mars     1.136      .570        .107       1.210        15.038        .488         1.783          3.086          11.267         -1.860          .589
    Venus   .785        .305        .825        .695           1.600        .545         1.024          5.176          9.719          -20.753        4.373

-41.036       11.558

Stage 3c
    Earth     1.085      .090       1.012      1.1295                        .987          1.182          5.513           6.603        -18.422        6.597
    Mars     1.649      .725        .107        2.118        2.143        .454          2.845          1.954          12.254         -1.281          .595
    Venus   .723        .007.        815          .615        1.196        .718            .728          7.338           7.438        -22.241        4.355

-41.944       11.546

Stage 4
    Earth    1.050      .200        1.012      1.077                          .840         1.261           5.005          7.508         -19.020        6.387
    Mars     1.597      .340        .107       2.019         2.143      1.054         2.140           3.489          7.084           -1.322          .799
    Venus   .723        .007        .815         .6151         .333        .718           .728           7.338          7.438         -22.241        4.355

-42.584       11.541

Stage 5
Earth       1.000       .017        1.012      1.000                         .983          1.017          6.179          6.389         -19.981        6.359
Mars       1.524       .093          .107      1.881      2.135       1.381          1.666           4.635          5.590           -1.386          .826
Venus       .723       .007          .815        .6151      .599        .718             .728           7.338          7.438         -22.241        4.355

-43.608      11.640

 * The mean synodic periods are expressed in terms of the "year" that obtained at
    that time, all other values are expressed in geobasic units.

 

TABLE 2

                Semi-      Eccen-    Mass    Sidereal   Mean       Peri-     Aphelion   Minimum  Maximum    Energy   Angular
                major      tricity                    Period     Synodic   helion                      Velocity    Velocity                      Momentum
                Axis                                                       Period*

Stage 1
    Earth       .706      .070        1.012        .593                           .656          .755         6.972          8.022        -28.307        5.330
    Mars       .574      .080          .107        .4352       .748           .528           .620        7.655          8.986          -3.680          .508
    Venus   3.000      .794          .870      5.196       1.129           .618         5.382        1.229        10.705          -5.724        5.756

-37.712       11.594

Stage 2
    Earth       .985      .260         1.012       .978                           .7291         .241         4.852          8.261        -20.285        6.095
    Mars       .574      .080           .107       .435          .801           .528          .620         7.655          8.986          -3.680          .508
    Venus   1.089      .450           .850     1.137       7.144           .599         1.579         3.708          9.775        -15.405        4.978

-39.370     11.580

Stage 3a
    Earth       .990       .090         1.012        .985                         .901        1.079         5.769          6.910         -20.176       6.303
    Mars       .574       .080           .107        .435        .789          .528          .620         7.655          8.986           -3.680         .508
    Venus   1.003       .424           .835      1.005    51.618          .577        1.429         3.987          9.869         -16.428       4.758

-40.285     11.569

Stage 3b
    Earth      .990       .090         1.012       .985                           .901        1.079         5.769          6.910          -20.176      6.303
    Mars     1.218      .510           .107     1.344         3.750         .597        1.839         3.243          9.995            -1.734        .638
    Venus     .854      .275           .825       .789         4.003         .619        1.088         5.129          9.019          -19.080      4.604

-40.991      11.546

Stage 4a
    Earth    1.077      .070          1.012     1.118                         1.002        1.152          5.645          6.494         -18.552       6.584
    Mars      .590      .723            .107       .453           .682         .163        1.016          3.280        20.402           -3.580         .357
    Venus    .854      .275            .825       .789          2.396         .619       1.088          5.129          9.019         -19.080       4.604

-41.213      11.545

Stage 4b
    Earth    1.077      .070          1.012      1.118                         1.002        1.152          5.645          6.494        -18.552       6.584
    Mars    1.128      .535            .107      1.197         15.006       .524        1.731          3.257        10.750          -1.873         .603
    Venus    .723      .007          15             .615           1.224       .718          .728          7.338          7.438        -22.241       4.355

-42.666      11.542

Stage 5
    Earth    1.000      .017         1.012       1.000                            .983        1.017          6.179          6.389        -19.981        6.359
    Mars    1.524      .093           .107       1.881           2.135        1.381      1.666         4.6355           .590          -1.386          .826
    Venus    .723       .007          .815       .6151             .599         .718         .728           7.338         7.438        -22.241        4.355

-43.608      11.540

*  The mean synodicc periods are expressed in terms of the "year" that obtained at
    that time, all other values are expressed in geobasic units.

Another solution to the angular momentum limitations on the orbit of Mars involves abandoning the tacit but crucial assumption that the last Mars-Venus contact preceded the first Earth-Mars contact.  If we abandon that assumption, and have the first Earth-Mars contact precede the last Mars-Venus

contact, it then becomes possible for Mars to obtain angular momentum at the expense of Venus, lose it to Earth, gain more from Venus, regain most of what it had previously surrendered to Earth, and then proceed eventually to its present orbit.  This sequence of events is illustrated quantitatively in Table 2.

Table 2, like Table 1, contains a number of Velikovskian features.  There is a 7.144-"year" mean synodic period of Venus in Stage 2, and there are mean synodic periods of 51.618 "years," 4.003 "years," and 1.599 years in Stages 3a, 3b, and 5, respectively.  Table 2 also incorporates five different 15-"year" cycles for Mars: in Stages 3a, 3b, 4a, and 4b, where 15 "years" is very nearly equal to an integral number (34, 11, 37, and 14, respectively) of sidereal periods of Mars, and also to an integral number (19, 4, 22, and 1, respectively) of synodic periods of Mars; and in Stage 5, where favorable oppositions occur every 15 years or so.

Table 3

There is still a third-and extremely speculative-way to resolve this question.  If we allow Venus to move spontaneously from right to left along its I/m contour on the map of orbits—conserving angular momentum but losing both eccentricity and orbital energy—we would make it possible for the energy loss over the past four or five thousand years to occur gradually, rather than only at those times when near-collisions were in progress.  It should be noted, however, that we would still not be in a position to identify the form or the mechanism or the destination of this lost energy.

With this effect operative on a sufficiently large scale, Earth's orbit during the post-Beth-horon period could be smaller than the present orbit of Earth, without requiring the first Earth-Mars collision to precede the last Mars-Venus collision.  For this effect might allow Venus to emerge from the last Mars-Venus contact with an aphelion substantially farther from the Sun than Venus' present aphelion, and might allow Venus gradually to circularize its orbit, that is, to reduce its semimajor axis slightly and to reduce its eccentricity greatly.  It then becomes possible for Mars, with a very high eccentricity and a rather long period, to lose angular momentum in its collisions with Earth and for Earth to gain a very slight amount of angular momentum, just enough for the transition from a "year" of 360 "days" to a year of 3651/4 days to represent a genuine lengthening.

Thus we have prepared Table 3, which incorporates certain Velikovskian features such as a 7.144-"year" mean synodic period of Venus, with the seventh such period ending about 50 "years" after the beginning of the first such period; a 52.171-"year" mean synodic period of Venus; a 4-"year" synodic period of Venus (that may be associated with the olympiad); and a number of 15-"year" cycles of Mars (the situations in Stages 3a, 3b, 3c, 3d, 3e, and 4, where 15 "years" is very nearly equal to an integral number (34, 34, 14, 11, 1, and 8, respectively) of sidereal periods of Mars, and also to an integral number ( 19, 19, 1, 4, 14, and 7, respectively) of synodic periods of Mars; and the 15-year cycle of favorable oppositions in Stages 5a and 5b).

"Days...... Months," and Eclipses

According to Velikovsky, the post-Beth-horon "year" (Stage 3) contained 360 "days" and 12 "months."  If, as Table 1 indicates, the post-Beth-horon "year" was equal to 1.1295 of our present years, some important consequences follow: The length of the post-Beth-horon "day" would have been 27.5 modern hours.  The length of the post-Beth-horon "month"—that is, the synodic period of the Moon—would have been 34.38 modern days; thus the sidereal period of the Moon would have been 31.74 modern days (which is 1.1615 times its present value), and the semi-major axis of the Moon's orbit would have been (1.1615)2/3 = 1.105 times its present value.  The semimajor axis of Earth's orbit would have been (1.1295)2/3 = 1.085 times its present value.  While both the Sun and the Moon would have been at greater distances from Earth than at present, the Moon's distance was greater by 1.105 times and the Sun's distance was greater by only 1.085 times.  This means that it would have been slightly more difficult (but by no means impossible, given appropriate eccentricities) for the Moon to cover the Sun completely.

Both in Table 2 and in Table 3 the post-Beth-horon "day" would have been almost exactly 24 present hours. The "month" would have been almost exactly 30 present days. The sidereal period of
 

TABLE  3

                Semi-      Eccen-    Mass    Sidereal   Mean       Peri-     Aphelion   Minimum  Maximum    Energy   Angular
                major      tricity                    Period     Synodic   helion                      Velocity    Velocity                      Momentum
                Axis                                                       Period*

Stage 1
   Earth      .706       .070       1.012         .593                             .656       .755             6.972          8.022        -28.307         5.330
   Mars      .574       .080         .107         .435           2.748         .528       .620             7.655          8.986          -3.680           .508
   Venus  3.000       .794       .8705         .196           1.129         .618     5.382             1.229        10.705          -5.724         5.756

-37.712        11.594

Stage 1b
   Earth      .706       .070       1.012          .593                            .656       .755             6.972          8.022        -28.307         5.330
   Mars      .574       .080         .107          .435         2.748          .528       .620             7.655          8.986          -3.680           .508
   Venus  2.100       .687         .870        3.043         1.242          .657     3.543             1.867        10.067          -8.178         5.756

-40.165        11.594

Stage 2
   Earth     .962       .225        1.012         .943                             .745     1.178            5.096           8.055       -20.776           6.077
   Mars     .574       .080          .107         .435           .855           .528       .620            7.655           8.986         -3.680             .508
   Venus 1.063       .417          .850       1.097         7.144           .620     1.507            3.908           9.499       -15.778           5.006

-40.234       11.590

Stage 3a
   Earth      .990      .090         1.012        .985                              .901     1.079           5.769           6.910       -20.176          6.303
   Mars      .574      .080           .107        .435           .789            .528       .620           7.655           8.986         -3.680            .508
   Venus  1.003      .424           .838      1.005       52.171            .577      1.429          3.988           9.869       -16.499          4.778

-40.356       11.589

Stage 3b
   Earth      .990      .090         1.012        .985                              .901     1.079           5.769          6.910        -20.176          6.303
   Mars      .574      .080           .107        .435            .789           .528       .620           7.655          8.986          -3.680            .508
   Venus    .969      .389           .838        .954        30.164           .592     1.346           4.234          9.623        -17.081          4.778

-40.937       11.589

Stage 3c
   Earth      .990       .090       1.012        .985                                .901      1.079          5.769          6.910        -20.176         6.303
   Mars    1.037       .428         .107      1.056         15.034            .593      1.481          3.905          9.750          -2.037           .619
   Venus    .877       .315         .832        .822           5.023            .601      1.154          4.841          9.294        -18.727         4.650

-40.940       11.572

Stage 3d
   Earth      .990       .090        1.012        .985                                .901     1.079          5.769          6.910        -20.176         6.303
   Mars      .218       .490          .107      1.344          3.750             .621     1.814          3.331          9.732          -1.734           .647
   Venus    .853       .280          .825        .788          4.000             .614     1.092          5.101          9.069        -19.082         4.597

-40.993       11.547

Stage 3e
   Earth      .990        .090        1.012        .985                               .901     1.079          5.769          6.910        -20.176        6.303
   Mars    6.023        .843          .107    14.782          1.071            .946    11.101           .747          8.772            -.351          .888
   Venus    .784        .278          .815        .694          2.383            .566      1.002         5.332          9.444        -20.520        4.355

-41.047      11.546

Stage 4
   Earth      .039        .166        1.012      1.069                                 .866     1.211          5.213          7.289       -19.232         6.393
   Mars    1.580        .340          .107      1.986          2.143            1.043     2.117          3.508          7.123         -1.337           .795
   Venus    .780        .270          .815        .689          1.861              .570       .990          5.396          9.380       -20.625         4.355

-41.194      11.542

Stage 5a
   Earth   1.000        .017        1.012      1.000                                  .983     1.017         6.179          6.389       -19.981         6.359
   Mars   1.524        .093          .107      1.881          2.135             1.381     1.666         4.635          5.590         -1.386           .826
   Venus   .776        .261          .815        .684          2.160               .574       .978         5.463          9.313       -20.731         4.355

-42.098      11.540

Stage 5b
   Earth   1.000        .017        1.012      1.000                                   .983    1.017          6.179          6.389      -19.981         6.359
   Mars   1.524        .093          .107      1.881          2.135              1.381    1.666          4.635          5.590        -1.386           .826
   Venus   .723        .007          .815        .615          1.599                .718      .728          7.338          7.438      -22.241         4.355

-43.608      11.540

*The mean synodic periods are expressed in terms of the "year" that obtained at
   that time, all other values are expressed in geobasic units.

the Moon would have been 27.69 days; hence, the semi-major axis of the Moon's orbit would have been 1.009 times its present value.  Both in Table 2 and in Table 3, the Sun would have been slightly closer to Earth than at present, while the Moon would have been slightly farther from Earth than at present.  The relative distances of the Sun and Moon would again suggest that total solar eclipses were less frequent in the post-Beth-horon period than now.

ECCENTRICITY AND ENERGY

In constructing each of the Tables, we have followed this guideline: If the transition from one stage to another is marked by a near-collision of planets A and B, then the presumed point of that near-collision, expressed in terms of its distance from the Sun, should be greater than the perihelions of A and B in both the earlier and later stages, and less than the aphelions of A and B in both the earlier and later stages.  The presumed "point" of such a near-collision is of course not a precise point, because the planets are not only some distance apart during their near-collision, but also may both be moving in the same general direction for some time, so that the "point" where the planets "are colliding" may actually move over some distance.  The guideline also needs to be qualified insofar as it is quite possible for a larger planet to "capture" a smaller planet at one distance from the Sun and "release" it at another distance from the Sun, so that the earlier and the later orbits of the smaller planet need not overlap at all.

But this guideline cannot be followed in the transition from Stage 4 to the present Stage 5. The present orbits of Earth and Mars do not intersect, and it is obvious that the two planets did not leave the sector of their near-collision on the same Keplerian orbits that they have followed to this day.  As Juergens describes this problem, "the final encounter must necessarily leave at least one participant traveling on a highly eccentric orbit-one that must return the body again and again to at least one point of possible collision with its late antagonist" (9).  Yet neither Mars nor Earth is presently on an orbit eccentric enough to carry it to a point on the orbit of the other.

There are two major problems here.  First, there is the problem of identifying a process capable of reducing orbital eccentricity following the last encounter.  Second, there is apparently an energy-disposal problem that accompanies this final eccentricity-reduction and may also accompany the entire sequence proposed by Velikovsky.  The following discussion involving Venus, although it puts off the question of the final eccentricity-reduction involving Earth and Mars, will illustrate the nature of these two problems.

Reduction of Eccentricity

The planets continuously undergo deformation by the tidal force of the Sun.  Tidal friction occurs insofar as the movement of a tidal deformation (either its rotation around the planet due to the planetary rotation, or variation in its magnitude caused by the orbital eccentricity) is opposed by internal friction within the planet; the effect is to convert rotational and/or orbital energy to thermal energy.  Such a process was suggested by Sherrerd (10) to explain the last stages of the reduction of Venus' orbital eccentricity.  Sherrerd's proposal is attractive, but is apparently not sufficient for our more general purposes here.  Even if we assume an unlimited ability for conversion of orbital energy to heat, it is difficult to explain the destination of this heat.  How much heat can Venus have absorbed and retained, and how much can have been lost as thermal radiation?  Since we know that the present surface temperature of Venus is approximately 750°K., we can make some educated guesses.  It would appear that Venus cannot have absorbed and retained much tidal-friction heat (especially since a high initial temperature is postulated for the tidal-friction process), and it would appear that the surface temperature of Venus has not been great enough to radiate a significant amount of energy in terms of our orbital requirements, even if we assume Venus to be a perfect black-body (which is certainly not true now, since its clouds and atmosphere are opaque to many parts of the electromagnetic spectrum).

According to Stefan's law, the rate of radiation from a black-body is proportional to the area and to the fourth power of the temperature.  A black-body the size of Venus would have a surface area of 4.83 x 1018 cm2.  At a surface temperature of 1000°K., the rate of radiation would be 2.75 x 1026 ergs/ second or 6.5 x 10-6 geobasic energy units/year; at 2000°K., the rate would be 4.4 x 1027 ergs/second or 1.0 x 10-4 geobasic energy units/year (4).  A Venus surface temperature as high as 2000°K. within the historical past is purely hypothetical; such an extreme temperature seems to require an unacceptably high rate of cooling, and is more than is needed for the incandescence claimed by Velikovsky (the approximate relationship between observed incandescence and temperature is: dark red heat, 925 to 1025°K.; bright red heat, 1125 to 1225° K.; yellowish red heat, 1325 to 1425°K.; white heat, 1725°K. and up (11)).  Nevertheless, as an extreme example, how much energy could be radiated during 2000 years at 2000°K.?  The amount would be 2.8 x 1038 ergs or 0.21 geobasic energy units.

The present orbital energy of Venus is -22.24 geobasic energy units (see Tables).  Venus' orbital energy thus would have been -22.24 + 0.21 = -22.03 geobasic energy units prior to this hypothetical energy dissipation.  Expressed as a percentage, Venus' loss of orbital energy would have been 0.95%, or just under one percent.

An orbital change due to planetary tidal friction would presumably involve a loss of orbital energy accompanied by either a loss of, or no change in, angular momentum.  The decrease of eccentricity per unit decrease of energy is greatest when the angular momentum remains unchanged; this would be the most favorable case from our viewpoint.  Such an orbital change can be visualized on the map of orbits as movement (of the point representing the orbit) toward the left along an 1/m contour.  Changes of eccentricity along the contour can be readily correlated with changes of energy, although direct measurement on the map is impractical for determining relative changes along the contour near the lefthand edge of the map (eccentricities between zero and one-tenth).  In this zone, where the I/m contour is practically horizontal, the percent change of energy is very nearly equal to 100 (e22 - e12 ) as the eccentricity changes from e1 to e2. Our example falls in this zone; thus, since the percent change of energy was -0.95 = 100(0.000049-e12), the former eccentricity would have been e1 = (0.009549)1/2 = 0.098.  A greater loss of eccentricity due to tidal friction would not seem possible; even this value, being based on some seemingly unrealistic assumptions, appears too high.  Earth and Mars, of course, have much lower surface temperatures than Venus, so that tidal friction seems quite useless in explaining the transition from their final encounter to their present orbits.

The General Problem of Energy

There is a still more general problem of energy disposal.  Consider the present orbits of Venus, Earth, and Mars in comparison with their orbits prior to the initial encounter between Earth and Venus some thirty-five centuries ago.  No definite orbits are known for the pre-Exodus period; however, certain limits follow from Velikovsky's hypotheses, especially for Venus, as was shown in Figure 1.  The Table 1 orbits for the pre-Exodus period can be used as an example: the total orbital energy for Stage 1 of Table 1 is -37.84 geobasic energy units.  The total for the present orbits (shown as Stage 5) is -43.61 geobasic energy units.  This implies that 5.77 geobasic energy units (i.e., approximately 1040 ergs) have somehow been disposed of within the past thirty-five centuries.

This amount of energy presents a formidable problem.  An interesting book by Lane (12) gives a scale of energy magnitudes that is useful for reference.  A letter to Nature by Urey (13) is concerned with a similar situation, but on a smaller scale: the disposal of 1031 ergs and its terrestrial effects.  We are talking about the disposal of one billion (i.e., 109) times as much energy.  While this energy would not all have resulted from a single encounter, but from several encounters occurring over a span of nearly 1000 years, it nevertheless seems unlikely that this amount of energy could be absorbed and stored by Venus, Earth, and Mars, or that it could be dissipated into space by the three planets during the time available.

The Energy Problem: Alternatives

Dissipation into space would normally occur by means of electromagnetic radiation.  The limitations according to Stefan's law have already been discussed; however, that discussion did not consider the possibility that the radiation was emitted from an excited plasma surrounding the particular planet, rather than from the planet's surface.  If this were the case, a much higher temperature would be available, resulting in a higher rate of energy emission, but this emission would seemingly consist of a greater proportion of short-wavelength (ultra-violet, x-ray) radiation—more than enough, perhaps, to produce the mutations described by Velikovsky.

Likewise, it seems questionable whether substantial amounts of energy were absorbed and stored by Venus, Earth, and Mars.  The total mass of the three planets is approximately 1.2 x 1028 grams; if 5 x 1039 ergs were absorbed, the average energy storage requirement would be 4 x 1011 ergs, or 10,000 calories, per gram of planetary material.  Current theories do not leave any room for such a possibility, but it should perhaps be mentioned that our knowledge of the planetary interiors and their histories is composed from a great deal of circumstantial and indirect evidence.  Suppose, hypothetically, that 1039 or 1040 ergs were suddenly delivered to Earth's core: how long would it take for the effects to become manifest at the surface?  How hot are the interiors of Earth and the other planets, and what latent energies reside there (i.e., what is the energy difference between the present state of Earth and a uniformly cold Earth)?  Cook considers the deep interior of all celestial bodies to be high-density plasma, or pressure-induced metallic states, which are characterized by a deep "energy well" (14).  It remains to be seen whether this sort of energy capacity is available to satisfy a significant part of our requirements.

Can different pre-Exodus orbits be selected that would eliminate the energy problem?  We have been using a pre-Exodus orbit near (a = 3.0, e = 0.8) for Venus; a smaller, less eccentric orbit would be ideal.  But our reasons for fixing Venus' orbit within the limits shown in Figure 1 have already been described; unless these reasons are invalidated, we can only vary the orbits of Earth and Mars.  There do exist two sets of pairs of orbits for Earth and Mars that (in combination with the Venus orbit) exactly conserve both total orbital energy and total orbital angular momentum at their present levels, but none of these orbital pairs seems satisfactory, since none of the pairs will tend to produce a Velikovskian sequence of encounters, while, at the same time, one or the other planet is required to be too close to the Sun.  The best possible (i.e., least objectionable) pairs occur when both orbital eccentricities are zero: the semimajor axes of Earth and Mars are, respectively, 0.87 and 0.14 a.u. for the best pair in one set and 0.53 and 5.1 a.u. for the best pair in the other set (15).  Thus, there is no apparent solution to this energy problem by choosing different pre-Exodus orbits.  The energy problem could be resolved, with a variety of possible orbits for Earth and Mars, by assuming that total angular momentum was not conserved, but this seems to raise even more difficult questions than the energy-disposal problem.

The participation of at least one other body besides Venus, Earth, and Mars in the encounters that have occurred since Venus' final departure from the vicinity of Jupiter could provide an easy solution to this energy-disposal problem (16).  Although such a proposal might be described as deus ex machine, the possibility should not be completely overlooked.  To provide the desired effect, the additional body would have gained orbital energy from Venus, Earth, or Mars.  For example, it is possible that a small planet or planetoid was on a relatively small (a < 3.0) orbit around the Sun, and that various near-collisions either have propelled it right out of the solar system or else have placed it on a highly eccentric orbit with such a large semimajor axis that it is noticeable now only at its perihelion passages, which may be separated by many centuries. (This type of occurrence might explain why deities major enough to be "planetary" no longer have planets to which we may assign them.)

The involvement of an additional body of this sort would permit the Velikovskian sequence of events to occur with a strict balancing both of energy and of angular momentum; if we permit this "other body" to interact with Mars after -687, then the problem of eccentricity damping will evaporate as well.  It should be emphasized, however, that Velikovsky has never endorsed such an approach, and has preferred instead to explain the final stages of his sequence of events in terms of electromagnetic processes.

It is also possible that a once-separate body, on a relatively small orbit, collided directly and thereby merged with the original Venus, or that a former part or parts of Venus (or Mars) was split off in an encounter and is now one or more separate bodies following relatively large orbits around the Sun.  In the latter case, it is possible that the original mass of Venus was substantially greater than its present mass and that the debris of Venus' stormy career carried off substantial amounts of orbital energy.

There are a number of seemingly unlikely ideas that might lead to a solution of the energy problem.  Much attention has been given recently to the possible existence of black holes in the universe; it would be worth investigating the effects of a black hole passing through the solar system-or even passing directly through Jupiter.  It is interesting to note that the energy needed for the original escape of Venus from Jupiter is roughly comparable to the energy that Venus must thereafter lose to reach its present orbit; no energy need ultimately be gained or lost if it could somehow be "borrowed."  Perhaps there have been fields of force acting in the solar system whose effects, including potential energy, we do not fully appreciate.

In looking for possible answers to the energy problem, we are speculating freely, but with a definite purpose in mind.  We believe that it will ultimately be possible to assemble enough numerical historical data to settle the question of orbital changes; so far, the question has seldom been asked, much less answered.  If there turns out to be consistent evidence for currently unexplained orbital changes within the historical past, science will need to seek hypotheses that fit the historical record, rather than demanding that history conform to the expectations of present-day science.

Earth and Mars

The energy-disposal alternatives in the preceding discussion also apply to the final reduction of the orbital eccentricity of Earth or Mars or both.  There are no orbits for Earth and Mars that cross each other and that also have values of orbital energy and angular momentum that add up to the present Earth-Mars totals.  There do exist two sets of pairs of orbits for Earth and Mars that exactly conserve both total orbital energy and total orbital angular momentum at their present levels, but none of these paired orbits will cross each other.  Conversely, there exist various paired orbits that cross each other while conserving the total orbital angular momentum at its present level, but, for every pair, the total orbital energy is at least 0.3 geobasic energy units or 4 x 1038 ergs higher than the present level.  While this excess energy is less than the 1040 ergs in the preceding problem, it nevertheless remains difficult to explain, especially since the energy disposal must have occurred within a relatively short time following the last encounter.

In summary, the two major energy-disposal problems can be put in the form of questions: Has there been an overall loss of orbital energy since the first Earth-Venus encounter?  Has there been an overall loss of orbital energy since the final Earth-Mars encounter?  Both these questions imply others: If so, by what process was the orbital energy lost, and where has it gone?  If not, how could the various orbits have resulted from or led to a Velikovskian sequence of encounters?  The resolution of these questions is crucial to Velikovsky's theory.

CONCLUSIONS

We have suggested several alternative models for Velikovsky's sequence of planetary orbits.  At this point, we are inclined to think that any well-founded establishment of one model rather than another will await the discovery, or the recognition, of further historical information.  Ancient reports are usually of isolated events, but the determination of orbits requires a series of observations.  As far as we know, the only dated sequence of observations from before -687 is on the so-called "Venus Tablets of Ammizaduga."  We have given preliminary accounts of our work on those tablets (17), (18); our investigations are continuing, but we do not yet have any final results to report.  Additional historical data of the sort found on these tablets will be needed for the development of more definitive models.  But not all of the information that we need would have to be precise or sophisticated; even casual reports could conceivably be quite important.  Thus a report that something happened at night rather than during the day could be decisive in the choice between one model and another.

It should be pointed out that even if we could choose one model rather than another, the models that we have been considering are in a sense only outlines of the possible orbital sequences: the precise description of a planetary orbit would involve six different elements.  Our approach has been to treat all planetary orbits as lying in the ecliptic plane and to focus our consideration on two elements, the semimajor axis and the eccentricity, and on various other parameters that are functions of those two elements.  There is much that can be accomplished using this restricted approach, but a fully-detailed model would of course have to treat all six orbital elements.  Such a fully-detailed model does not seem imminent—mainly because of a lack of clues about precise longitudes.

We close with a cautionary note regarding energy disposal, eccentricity damping, and electromagnetic processes in astronomy.  These may indeed be consequences or ramifications of Velikovsky's theory, but at this point it is largely a matter of attitude whether one sees these as vulnerable points of the theory or as strong opportunities for discovery.  An analogy could be drawn to the Copernican theory, which entailed that either there was a measurable stellar parallax or the stars were fantastically farther away than anyone had guessed.  Since there was at that time no measurable parallax, some saw this "problem" as an insuperable barrier to the Copernican theory and rejected the theory as absurd.  Others, like Giordano Bruno, saw clearly that the Copernican theory entailed an enormous universe, and Bruno concluded that the stars were so far away that they must be suns.  The parallax "problem," which some saw as a reason for rejecting the Copernican theory, Bruno saw as the key to his own discoveries that the stars are suns and that those suns are probably centers of planetary systems of their own.

Let us not suppose that our relative ignorance, at present, of the processes that damp eccentricities or dissipate energy or produce electromagnetic effects in the solar system shows that Velikovsky is wrong.  Instead, let us study the consequences of Velikovsky's theory and engage in serious and systematic search for the information and the understanding that will enable us to evaluate the role of those processes toward which Velikovsky points.  We do not stand before a wall: we stand before a door.

NOTES AND REFERENCES

(1)       Lynn Rose and Raymond Vaughan, "The Orbits of Mars, Earth and Venus," Pensée 2 (May, 1972): 43.

(2)       C. J. Ransom and L. H. Hoffee, "The Orbits of Venus," Pensée 3 (Winter, 1973): 22.

(3)       Raymond C. Vaughan, "Orbits and Their Measurements." Unpublished.  ABSTRACT: A diagram or "map" is developed that shows the values (±1%) of various orbital parameters for a given orbit, or establishes the orbit when parameter values are given.  The map shows graphically the interdependence of parameters and thereby illustrates the limitations on orbital change.  First, a brief description is given of orbital parameters, Kepler's laws, and ellipses.  Elliptical orbits are then classified and represented as points on a semi-log grid, using coordinates a and e, in order to form the basic map.  Other kinematic parameters are introduced that are functions of a and e, and a network of parameter contours is superimposed on the map, allowing parameter values to be read from the map by direct measurement. Conservation of energy and of angular momentum are discussed insofar as they define the limitations on orbital change.

(4)       See (3) regarding geobasic units and cgs-mks-geobasic conversion factors.

(5)       Immanuel Velikovsky, Worlds in Collision (New York: Macmillan, 1950), pp. 330-342.

(6)       Velikovsky, Worlds in Collision, p. 363.

(7)       [Bezalel ben Abraham Ashkenazi?], Shitah Mekubbezet.  Nedarim. (Berlin: Kornegg, 1860).

(8)       Velikovsky, Worlds in Collision, p. 124.

(9)       Ralph Juergens, "Reconciling Celestial Mechanics and Velikovskian Catastrophism," Pensée 2 (Fall, 1972): 6.

(10)     Chris Sherrerd, "Venus' Circular Orbit," Pensée 2 (May, 1972): 43.

(11)     "Color Scale of Temperature," in Handbook of Chemistry and Physics, 43rd ed. (Cleveland, Ohio: The Chemical Rubber Publishing Co., 1961), p. 2321.

(12)     Frank W. Lane, The Elements Rage (Philadelphia: Chilton, 1965).

(13)     Harold C. Urey, "Cometary Collisions and Geological Periods," Nature 242 (March 2, 1973): 32.

(14)     Melvin A. Cook, untitled remarks, Number V in "Special Supplement: On Celestial Mechanics," Pensée 3 (Winter, 197 3): 55.

(15)     We assume that neither Earth nor Mars was moving in a retrograde orbit.

(16)     One candidate to play the role of this other body would be Jupiter: although it seems unlikely, it is possible that ordinary gravitational perturbations by Jupiter would be sufficient to accomplish this.

(17)     Lynn E. Rose, "Babylonian Observations of Venus," Pensée 3 (Winter, 1973): 18.

(18)     Lynn E. Rose and Raymond C. Vaughan, "Analysis of the Babylonian Observations of Venus." Unpublished.  Presented at the symposium, "Velikovsky and the Recent History of the Solar System," McMaster University, Hamilton, Ontario, June 19, 1974.

PENSEE Journal VIII

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