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Open letter to science editors


KRONOS Vol X, No. 2



In our first installment of this paper,(1) M. S. Bodner and I described the possible effects of a massive body incursion into the solar system. These effects were asserted to be both electromagnetic and gravitational in nature. The gravitational force between two objects A and B is given by:

G mA mB
F = 末末末

This is the classical Newtonian gravitational force acting along the line of centers, and treats the bodies as if their masses were concentrated at their center points.

The tidal force exerted by object A on object B acts on points about the center of object B. The tidal force is therefore a differential gravitational force resulting from the fact that massive objects in space are not point masses, but, rather, have extensibility (finite volume). If the approach velocity of the massive object is much less than the speed of light ("c" equal to 300,000 kilometers per second), the tidal effect of such a passage on the planets would most likely be minimal.

For example, if the object is the same distance from the Earth as the Sun, and has solar mass, then its tidal pull would be approximately equal to that of the Sun. This particular tidal effect is only about one-half of the tidal pull exerted by the Moon upon the Earth, and therefore not very significant.

However, if the object were travelling at a velocity near that of light, its mass might be very great as a result of special relativistic effects. For example, if the object had a rest mass equal to that of the Sun, and velocity of 98 percent of the speed of light, its observed mass would be 5 times that of the Sun. At .99c its mass would jump to 10 times that of the Sun. If this particular object was one astronomical unit from the Earth (the mean distance between the Earth and Sun) it would exert a tidal force 5 times larger than the force exerted by the Moon upon the Earth. This may be a significant enough effect to cause disturbances in the Earth's crust and oceans.

The speed of the encounter may even enhance any devastating effects by producing an impulsive-type force interaction governed by the well-known mechanics equation:

F D T = M D V

Such a high speed event might be akin to hitting the Solar System with a galactic-sized baseball bat, and may damage the very fabric of space time in the vicinity of the planets.

Putting aside these speculations for now, we shall concentrate in greater detail upon the more concrete effects of classical gravitation in this follow up work. The gravitational effects of an ultramassive object intrusion into the Solar System can be accurately described within the context of the so-called N-body problem in physics.

To explore these effects in greater detail I have developed a computer model that simulates the motions of the objects involved (Sun, massive object, and planets). Pluto has been excluded from the simulations since its effect is relatively minor; and by excluding it, computational time is comfortably decreased. The model enables the user to experiment with different scenarios by injecting objects of varying mass, velocity, and starting position into the vicinity of the Solar System.


The simulation utilizes a mathematical technique known as Euler numerical integration(2) (a modified version of the Runge-Kutta method) to calculate the acceleration upon each of the objects. The acceleration is produced by the summation of the gravitational pulls on an object at each instant of time and can be expressed vectorially as (3)

[*!* Image]

where A (P,D) is the acceleration on the Pth object in the Dth direction (the Cartesian coordinate system is used in the simulation with dimensions X, Y, and Z, and with the origin at the position of the Sun). G(K) is the gravitational pull exerted by the Kth object and is equal to that object's mass times Newton's Universal Gravitational Constant, G. R(K, D) is the position of the Kth object in the Dth direction, and [*!* Image] is the cubed distance between objects K and P.

To compute the acceleration terms we must, therefore, calculate the position and velocity of each object at each instant of time. Our computational strategy then consists of:

1. Assign initial positions, velocities, and masses to each object. Table I provides a list of these parameters.(4)

2. Compute the acceleration on each object resulting from the forces exerted by every other object.

3. Compute the new position and velocity for each object.

4. Go to the next instant of time and repeat this process starting with step 2

The Taylor series expansion method is used to compute the positions, velocities, and accelerations according to the following set of relations:

[*!* Image]

Making use of these equations within the Euler method, our position measurement error is reduced to order t4.

The objects in the simulation are treated mathematically and physically as point masses that would experience elastic collisions similar to the interactions of billiard balls in a game of pool. This is certainly not a perfect model of the planets since it does not take into account planetary volume or rotation effects, however, it is accurate from a macroscopic viewpoint (our frame of reference is one of an observer far from the Solar System).

In the simulation, the planets begin their orbits in a coplanar fashion (initial positions along the positive X-axis according to the radii values listed in Table I ) with their elliptical paths occurring as a natural consequence of the equations of motion.

Table I. Program Simulation Object Data

Object Radius (AU) Mean Orbital
Velocity (AU/YR)  
Acceleration (AU/YR2)
Sun 0.00 0.00 39.470
Mercury 0.3874 10.1042 6.549 x 10-6
Venus 0.7228 7.3821 0.949 x 10-4
Earth 1.000 6.2832 1.183 x 10-4
Mars 1.5233 5.0889 1.2630 x 10-5
Jupiter 5.2025 2.7559 3.915 x 10-2
Saturn 9.5407 2.0350 1.1650 x 10-2
Uranus 19.190 1.4353 1.584 x 10-3
Neptune 30.086 1.1472 2.154 x10-3
Massive Object Values entered by program user.


The simulation was originally written in the BASIC programming language for use on a personal microcomputer. This version was too slow and inefficient and was rewritten in the C programming language(5) for execution on a 32 bit system. (See Appendix A for information regarding where the source code may be obtained.)

The simulation user interactively enters the mass of the intruding object in Sun masses, the initial X, Y, and Z positions of the object from the origin in astronomical units (AU), and the initial X, Y, and Z components of velocity in units of the speed of light, c.

A relativistic mass correction is then applied to the object using the Lorentz transformation:


m =


SQRT(1 - v2 / c2)

where m0 is the initial mass of the object, and v its initial speed. Processing then begins with the position of each object being updated every 0.001 sidereal year (approximately every nine hours). The objects are plotted onto the computer graphics screen in pseudo three dimensions. The paths of the objects are then visible from time zero to the present time (time at which the simulation is halted).


The elapsed time for each execution of the simulation is approximately one sidereal year. Figure I shows the normal configuration of the planets without intervention of a massive intruding object. The Y-axis is "into the paper" providing the illusion of depth. In the simulation, the planets start out along the positive X-axis and orbit in a counterclockwise fashion. The Sun is positioned at the cross point of the three axes (origin of the coordinate system) and is suppressed in this particular view by those lines. The planets not shown are included in all calculations; however, only the inner planets were plotted in order to focus attention on the alterations in their orbits.

Figure 2 shows an object of 1.5 Solar mass traversing from left to right, parallel to the X-axis at avelocity of 0.001c . The major effect here is that the Sun was displaced from its position as it started to track with the object. In turn, the planets spiralled upward to keep "in step" with the Sun. Although this is a relatively slow moving object (approximately 63 AU/yr), it is far removed from the Solar System at the time this view was recorded.

A note on the plotting method is in order here. The optimal method of observing the simulation is by viewing it as it executes on a color graphics terminal where the relative motions of the objects are easily displayed. Since we could not reproduce color photographs or illustrations in a journal article, the next best thing we could do was to generate plots on a computer dot matrix printer. The orbital traces shown in the figures are produced by a series of dots (one dot for each object per instant of integration time). For the planets, the dots are so close together they appear to produce a continuous line segment. In a qualitative sense, the closer the dots, the slower the planet is travelling. In these views, therefore, all of the planets are travelling at approximately the same (slow) velocity (again refer to Table I), however the massive object is observed to be travelling relatively faster since its dots are spread apart slightly.

[*!* Image] Figure 1. Normal configuration of inner planets without massive object intervention. [Labels:KEY TO FIGURES 1 = path of the Sun. 2 = path of Mercury. 3 = path of Venus. 4 = path of Earth. 5 = path of Mars. 6 = path of massive object. NOTE: path directions are indicated with arrows.

[*!* Image] Figure 2. Massive object starting parameters: mass = 1.5 Solar mass, position (in AU) = -2X, 0Y, 0.7Z,velocity (fraction of c) = 0.001X, 0Y, 0Z.

[*!* Image] Figure 3. Massive object starting parameters: mass = 1 Solar mass, position (AU) = 0X, -2.5Y, 0.6Z,velocity (c) = 0X, 0.001Y, 0Z.

[*!* Image] Figure 4. Massive object starting parameters: mass = 0.5 Solar mass, position (AU) = 2X, 0Y, -0.5Z,velocity (c) = -0.001X, 0Y, 0.0007Z.

[*!* Image] Figure 5. Massive object starting parameters: mass = 0.3 Solar mass, position (AU) = 0.85X, -3Y, 0Z,velocity (c) = 0X, 0.0015Y, 0Z.

Figure 3 illustrates the effects of a 1 Solar mass object travelling in the positive Y direction at 0.001c. Although this is a less massive object, the effects are observed to be just as dramatic as in Figure 2.

Figure 4 incorporates an even smaller object (0.5 Solar mass) at approximately the same velocity moving between the orbits of Mars and Earth. Mars has been severely perturbed and appears to be leaving the Solar System.

In Figure 5, a 0.3 Solar mass object is injected between the orbits of Venus and Earth. Here, not only was the Sun displaced from the origin of the coordinate system, but there was at least one point of near interaction between Earth and Venus. Also notice how enlarged the Earth orbit became.


The results of this study show that an object of relatively large mass can cause an appreciable alteration of the orbits of (at least) the inner planets of the Solar System. Even an object of one-third Solar mass can seriously disturb the stability of the Solar System as shown in Figure 5.

The most obvious (and perhaps least drastic) effect upon the Earth would be a change in the length of the year since its orbit has been altered by the trajectory of the massive object. This would most likely produce serious consequences for life on the planet. If the intruding object has a net charge (as would be the case for a charged rotating black hole) then an additional acceleratory term of the form:

[*!* Image]

might need to be included in the simulation. This would introduce an electromagnetic-gravitational coupling(6) among the interacting objects that would increase the effects if the charge were substantial.

Although we have not pursued the effects of near light speed objects in this study these would almost certainly produce appreciable torque, and tidal effects.

It is conceivable that the model described here may have been the agent behind the interactions among the inner planets described by Velikovsky in Worlds in Collision. It may not have taken too much force for a massive object to free a proto-Venus from the Jovian System and send it spiraling toward Mars, Earth, and Mercury, especially if that object had sufficient mass, or was travelling at a high enough velocity. This object would not need to collide with Jupiter to cause this to happen. A fly-by (near encounter) would prove to be sufficient.

There are innumerable combinations of parameters for the intruding object that could have caused the near-collisions between Earth, Venus, and Mars, and all it would have taken is one such occurrence! However, attempting to recreate the events described by Velikovsky will require a more precise understanding of the physical parameters of the Solar System at that time, as well as the use of a larger and faster computer.

In the months and years ahead this technology will become more accessible and easy to use, thereby substantially increasing our chances of arriving at a solution to the problem.


1. Bodner, M. S., and Brandt, M. E., "The Gravitational-Electromagnetic Effects of Ultramassive Objects on the Solar System", KRONOS VI:2 (Winter 1981), pp. 3-17.
2. Boyce, W. E., and DiPrima, R. C., Elementary Differential Equations and Boundary Value Problems (N. Y., 2nd Ed., 1969), pp.330-357.
3. Bennett, W. R., Jr., Scientific and Engineering Problem-Solving with the Computer (N.Y., 1976), pp.199-255.
4. Weast, R., editor, The Handbook of Chemistry and Physics, 56th Edition, CRC Press, 1976.
5. Kernighan, B . W., and Ritchie, D . M., The C Programming Language (N. Y., 1978) .
6. Brandt, M. E., and Bodner, M. S., "Electromagnetic-Gravitational Coupling Phenomena in the Saturn Ring System", KRONOS VI:3 (Spring 1981), pp. 63-78.


The simulation is written in the C programming language, and was executed using the Masscomp MC-500 computer under the UNIX operating system. The MC-500 is a 32-bit microcomputer with color graphics, and one megabyte of random access memory. Execution speed is approximately 45 times faster than the BASIC version written for the Apple II microcomputer.

Contact the author at:

University of Texas Medical Branch
Dept. of Psychiatry and Behavioral Sci.
205 Administration Annex D43
Galveston, TX 77550

for a documented source listing of the program (C and BASIC versions) as well as additional sample outputs.

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