ABSTRACT

INTRODUCTION

SUMMARY

NOMINAL

TIDAL EFFECTS

PRECESSION

CALCULATIONS FOR GR

THEORETICAL

TIDAL EFFECTS

BINARY
STAR
PRECESSIONS

PROPAGATION

SPEED OF GRAVITY

HYPERBOLIC

CREATION MODEL

DISCUSSION

CONCLUSIONS

REFERENCES

APPENDIX

GENERAL RELATIVITY or
NEWTONIAN TIDAL EFFECTS?

ABSTRACT

In this
paper, theoretical tidal effects are derived, (similar to the earth-moon
system), and then orbital precession comparisons are made with GR for
twenty-two celestial bodies. TOP OF
PAGE

INTRODUCTION

Six prior papers
investigated the orbital precessions of the planet Mercury, the moon, the
major satellites of Jupiter, Saturn, Uranus, and four binary stars.
Special attention was given to the possibility that Newtonian tidal
effects may account for excess precession, rather than resorting to
general relativity for explanations. In this paper, theoretical tidal
effects are derived, (similar to the earth-moon system), and then orbital
precession comparisons are made with general relativity for twenty-two
celestial bodies. TOP OF
PAGE

SUMMARY

1) Quite surprisingly,
the derived nominal tidal effects model duplicates general relativity
precessions for all celestial bodies in the solar system. However, for two
binary stars, (DI Herculis and AS Camelopardalis), GR predicts double the
measured orbital precessions, while the NTE model duplicates the measured
values. The results of this investigation indicate that NTE provides
better correlation with measured orbital precessions than does GR.

2) For the binary
pulsar, (PSR1913+16), GR duplicates the measured orbital precession of 422
degrees per century, with a solar mass combination of 1.4/1.4 and assuming
both bodies are neutron stars. With neutron stars the extreme density
excludes NTE, which calculates to be only 189 degrees per century.
However, a solar mass combination of 4.17/2.5 also duplicates the orbit of
PSR1913+16, with NTE producing the measured 422 degrees per century. The
4.17 solar mass might be a neutron star, but the hidden component (2.5
solar mass) could be capable of producing the NTE precession. For this
latter mass combination NTE excludes GR, which now calculates to be 727
degrees per century.

3) The orbital period
decay of this pulsar has been attributed to energy being dissipated in the
form of gravitational waves, which is a GR effect. However, this decay
could also be attributed to a Newtonian propagation speed of gravity which
is several million times the speed of light. With a speed of gravity
several million times the speed of light, special relativity and general
relativity appear to be violated, since nothing can presumedly travel
faster than the speed of light.

4) From the results of
this investigation, I believe that the reality of GR is subject to
question. Newtonian tidal effects with a speed of gravity several million
times the speed of light offers a reasonable alternative. Light bending
around the sun has also been considered, and may be explained as a
Newtonian refraction passing through the sun's corona, combined with the
sun's gravitational pull on light photons as they pass by the sun. The
implication is that the universe is ordinary three-dimensional space and
time; that is, Euclidean, rather than conforming to GR.

5) A Euclidean
universe that is just a few thousand years old would not have had time to
collapse upon itself. Of course, with a very young universe we could not
see light from stars that take billions of years to get here unless the
transit speed of light were much higher. One possibility is that the
average transit speed of light increment (above 186,000 miles per second)
is proportional to its light source distance from the solar system divided
by the age of the universe. This decaying super-light speed always arrives
in the solar system at 186,000 miles per second regardless of its source
distance. This possibility is referred to as the Hyperbolic Creation Model
(HCM).

6) It is interesting
to consider Supernova 1987 with its distance of 170,000 light-years, which
was observed exploding in 1987. When did it actually explode? With light
speed constant at 186,000 miles per second one would say 170,000 years
earlier than 1987. However, using the HCM with a universe age of 6000
years, the explosion would have taken place 5783 years before 1987, just
204 years after creation. Further, the actual duration time of the
explosion would have lasted only seven per cent of the time observed on
earth.

7) What about a galaxy
that is 12 billion light-years distance from the earth? Are we really
seeing it as it was 12 billion years ago? Once again, using the HCM with a
universe age of 6000 years, we would be seeing this galaxy as it was just
24 hours after creation, 6000 years ago. Further, its actual spin rate
would have been a million times faster than the relatively slow spin rate
that we now observe from earth. If it could have been seen on site it
would have looked like a whirling fireworks display. TOP
OF PAGE

NOMINAL TIDAL EFFECTS

The sun-tide produced
by a planet may be understood if one imagines the sun being divided into
two halves, connected by a powerful spring trying to hold the two halves
together. From Newton's law of gravity, the half closest to the planet is
pulled more strongly toward the planet than is the farthest half. This
force differential causes the two halves to separate, limited by the
powerful spring, but elongating the sun with a tidal bulge that has the
closest half oriented in the direction of the planet. This orientation is
maintained in spite of the fact that the sun rotates once a month on its
own axis. If the planet's orbit were a perfect circle around the sun then
this force differential would be constant, with no change in the radial
distance due to tidal effects.

However, with an
elliptical orbit this force differential cycles with the planet's period.
At aphelion, when the planet is farthest from the sun, the force
differential between halves is at its lowest value, resulting in a
positive incremental change (+KRF) in the orbital radius (R). Similarly,
at perihelion, when the planet is closest to the sun, the force
differential between halves is at its highest value, resulting in a
negative incremental change (-KRF) in the orbital radius (R).

The NTE sun-tide
amplitude is proportional to a planet's orbital eccentricity, as given in
equation (1). The constant (14,808) was derived from Mercury, which has an
eccentricity of 0.20563, with a resulting sun-tide amplitude of 3045 feet
(KRF). As will be shown later, this sun-tide amplitude (3045 feet)
produces an incremental change in Mercury's orbital precession of 43 arc
seconds per century, the same as that calculated from general relativity.
TOP OF
PAGE

NOMINAL TIDAL
EFFECTS EQUATIONS

1)
KRF = 14,808 x (W / Wsun) x Ecc

2) dR = KRF
x cos (360 x (Day / DayOrb))

3) Rcalc =
R + dR

KRF = tidal
amplitude (feet)
W / Wsun = solar mass of major component
Ecc =
orbital eccentricity
dR = radius increment (feet)
Day = days from
apoapse ( greatest separation)
DayOrb = orbital period (days)
R =
elliptical radial distance (feet)
Rcalc = actual radial distance
(feet)

The cyclical change in
the orbital radius distance is in phase (resonant) with a planet's orbital
period, as may be noted in the equations. It might have taken several
years for this resonant tidal amplitude to be established.

How can one say that
this tidal effect is happening on the sun? Because the earth-moon tidal
interaction is similar. We always see the same side of the moon because
its orbital period (27.3217 days) is the same as its rotation rate. The
tidal elongation of the moon is maintained and aligned with the earth,
because the near side has a stronger force acting on it than does the far
side, keeping it pointed toward the earth.

The moon causes an
oceanic bulge on the earth which is aligned with the moon. As the earth
rotates, two tides a day result. A less well-known earth-tide also is
aligned with the moon, which has an amplitude of about two to three feet
due to the elasticity of the earth. Sun-tides would be expected to be much
larger than earth-tides, because of the sun's lower density (1.4 compared
to earth's 5.52), and the sun being primarily gaseous.

Every atom of the
sun-Mercury system would gravitationally interact with every other atom,
according to Newton's law. In addition, complex elastic forces are
involved. Therefore, any tidal effect analysis can only approximate the
actual tidal amplitude.

However, if one
assumes that the orbital precession increment for Mercury (43 arc seconds
per century) is due entirely to sun-tide effects, (rather than GR), then a
tidal amplitude may be determined. This calculation involves use of a
numerical integration simulation of Mercury and the sun, (plamd-2.bas),
which is described in the Appendix of this paper.

The calculation
results in a sun-tide amplitude of about 0.6 miles (KRF = 3045 feet) due
to Mercury's gravitational pull on the sun. This is less than one
millionth of the sun's diameter (862,000 miles). Dividing 3045 by
Mercury's eccentricity (0.20563) results in a sun-tide amplitude of 14,808
times a planet's eccentricity. (see NTE equations). TOP OF
PAGE

PRECESSION
CALCULATIONS FOR GR

Calculated precessions
for general relativity were accomplished using the equation below which
was adapted from Tauber[5].
This equation results in the same predictions as Taylor[7,8]
obtains for the binary pulsar PSR1913+16.

GENERAL RELATIVITY
PRECESSION EQUATION

R = 12 x { [ (PIE
x Rmean) / (186,000 x T) ]\2 } / { 1 - e\2 }

PREC = R x
(360 / T) x 86,400 x 36,525 x 3600

R = precession ratio
for one orbit
Rmean = mean orbital radius (miles)
186,000 speed of
light (miles / second)
T = orbital period (seconds per orbit)
e =
orbital eccentricity
PREC = orbital precession (arc seconds per
century)

For comparison
purposes the nominal eccentricity of Mercury (0.20563) was varied,
computing corresponding orbital precessions for both GR and NTE. The
comparison indicated a remarkable similarity between GR and NTE, even as
the eccentricity approached zero, (41.5 arc seconds per century for both
GR and NTE). Of course, at zero eccentricity the orbit is circular with no
definable perihelion longitude. With an eccentricity of 0.6, the orbital
precession for NTE is 60 arc seconds per century as compared with 63 for
GR. TOP OF
PAGE

THEORETICAL
TIDAL EFFECTS

NTE does not include
the mass contributions of the minor orbiting component, which is defined
in this section for the five closest planets. First the spring rate for
the sun is calculated to be 4.1133D+15 lb. per ft., based on the 3045 ft.
amplitude for Mercury's sun-tide. The procedure for this calculation is
outlined in the Appendix (tidal-2.bas). Using this derived spring rate for
the sun the theoretical sun-tide amplitudes for Venus, Earth, Mars and
Jupiter were calculated (tidal-2.bas). Next, the orbital precessions for
TTE and NTE were calculated (plamd-2.bas). These results are compared with
GR in the accompanying TABLE A. TOP OF
PAGE

TABLE A
Planetary Nominal and
Theoretical TE Precessions

Mercury

43.15

43.09

43.09

3045

3045

Venus

8.66

8.78

17.09

100

197

Earth

3.85

3.89

3.54

247

225

Mars

1.35

1.37

0.04

1383

39

Jupiter

0.06

0.06

0.12

718

1484

[1]

[2]

[3]

[4]

[5]

[1] GR precession
(arc seconds per century)
[2] Nominal NTE precession
(arc seconds per century)
[3] Theoretical tidal
precession (arc seconds per century)
[4] Nominal NTE sun-tide
amplitude (feet)
[5] Theoretical sun-tide amplitude (feet)

The comparison
indicates that NTE precessions duplicate the GR precessions for these five
planets. The theoretical tidal precession for the earth also agrees with
GR precession (3.89 and 3.54 arc seconds per century). For Venus, the TTE
precession is 17.09 compared to 8.78 for NTE. The orbit of Venus is nearly
circular, (eccentricity = 0.00678), which makes its perihelion longitude
difficult to determine from observations. For example, Mercury's
eccentricity is thirty times greater than for Venus. Added to this is the
complexity of extracting the excess precession from observed precessions,
which include the gravitational interactions of the other planets.

The theoretical
sun-tide amplitude for Mars is just 39 feet, as compared to the NTE value
of 1383 feet. This results in an orbital precession of 0.04 for TTE as
compared to 1.37 for NTE. Jupiter has 0.06 for NTE and 0.12 for TTE. All
of these differences are expected to be within the accuracy of observed
measurements.

NTE calculations were
also made for the moon, an asteroid Toutatis, Jupiter's four Galilean
moons, Saturn's Titan, and the five major moons of Uranus. These
calculations duplicate GR precessions. One might think that NTE is just
another way of describing GR, until it is realized that GR fails the test
for two binary stars, DI Herculis and AS Camelopardalis[4].
NTE duplicates the measured orbital precessions of these two binary stars,
as will be shown in the next section. TOP OF
PAGE

BINARY STAR
PRECESSIONS

The measured orbital
precessions for the binary stars DI Herculis and AS Camelopardalis are
respectively 1.05 and 15.0 degrees per century. These observations cover a
time span of about 18 years by Dr. Edward Guinan[4].
Using NTE the precessions are respectively 1.19 and 15.4, close to the
measured values. GR predicts precessions of 2.35 and 26.8 degrees per
century, about double the measured precessions. It should be noted that
the article[4]
quotes GR's contribution for AS Camelopardalis as 8.5 rather than 26.8
degrees per century calculated in this paper using the GR equation above.
I believe that the latter 26.8 degrees per century is the correct value.
The eccentricities for the two binaries are respectively 0.489 and 0.150,
as personally reported by Guinan. TABLE B presents comparative data for
three binary stars and a nominal and optional mass combination for
PSR1913+16. TOP OF
PAGE

TABLE B
Binary Star
Orbital Comparisons (*RW Tauri eccentricity
assumed)

Binary
Stars

DI
Herculis

2.35 deg

1.19 deg

5.2/4.5

0.489

37,654

0.200768

10.55

AS Camelo

26.8 deg

15.4 deg

3.3/2.5

0.150

7330

0.050379

1.715

RW Tauri

9.37 deg

7.42 deg

2.55/0.55

*0.4

15,104

0.056012

2.75

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Binary Pulsar
PSR1913+16

Nominal

408 deg

189 deg

1.4/1.4

0.600

12,438

0.012983

0.322917

Option

727 deg

422 deg

4.17/2.5

0.600

37,050

0.017336

0.322917

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[1] GR orbital
precession (per century)
[2] NTE orbital precession
(per century)
[3] solar masses
[4] orbital
eccentricity
[5] NTE tidal amplitude (feet)
[6] mean orbital radius
(AU)
[7]
orbital period (day)

The same article
states, "MIT physicist Sean Carroll, who has worked with Guinan on DI
Her, has tried without success to modify general relativity so it is
consistent with DI Her and the many situations in which general relativity
works."[4].
Another physicist, John Moffat of the University of Toronto, is reported
to have come up with a modified version of GR[4].
However, it does not appear to be practical to change GR as defined by
Einstein.

Joseph Taylor and
Russell Hulse, having studied the binary pulsar PSR1913+16, have
determined that the pulsar's apsidal motion (423 degrees per century)
"...fits in beautifully with general relativity."[4]
They have interpreted the pulsar's orbit in terms of GR, excluding tidal
effects by assuming that it is made up of two neutron stars, having
extremely high densities, so great that Newtonian tidal effects are not
possible, which would "muddle" the apsidal motion.

For this reason Taylor
considers the binary pulsar to be a much stronger test of general
relativity than DI Her; saying, "My suspicion is that for reasons none
of us have thought of, something else is causing this discrepancy. It
would be very hard for me to conclude on the basis of DI Her alone that
the place you should be looking for fault is in the fundamental gravity
itself."[4]
Of course, for Taylor, GR is an integral part of "fundamental
gravity".

It may be noted from
TABLE B that for PSR1913+16 there is a nominal interpretation (conforming
to GR), and an optional interpretation (conforming to NTE). For nominal
solar masses of 1.4/1.4 the 422 degrees per century orbital precession is
matched with GR, (408 is approximately equal to 422). Using GR the 408
number increases to 422 using an eccentricity of 0.6171308 reported by
Kleppner[10],
rather than 0.6 used in this paper. NTE calculates a precession of 189
degrees per century. For solar masses of 4.17/2.5 the 422 degrees per
century is matched with NTE, while GR now calculates 727 degrees per
century.

The two pulsar
component orbits are identical in size for either mass combination.
However, the mean orbital radius for the 4.17/2.5 combination is 1.335
times the 1.4/1.4 combination, which when squared in the GR equation
results in the increased GR precession calculation from 408 to 727 degrees
per century. The nominal tidal amplitude is increased from 12,438 feet to
37,050 feet, resulting in an orbital precession increase from 189 to 422
degrees per century.

With mean specific
densities of 11.34 or higher for either mass combination, physical contact
at periastron is avoided. Lead has a specific density of 11.34. Periastron
is the closest approach of the two orbiting bodies to each other. Higher
densities provide greater clearance at periastron. The pulsar signal
(16.94 beeps per second) may be the spin rate of the inner core of the
pulsating component, rather than demanding that the entire component be a
neutron star spinning at 16.94 cycles per second.

However, if the
pulsating component with the solar mass of 4.17 is a very high density
neutron star (diameter 13.6 miles), then the 2.5 solar mass hidden
component could have a calculated diameter of 1.3 million miles. The mean
specific density for the hidden component would be just 1.073, being
primarily gas. At periastron, with the pulsar traveling at 726 miles per
second relative to the hidden component, its spin rate of 16.94
revolutions per second would be maintained as it makes gentle contact with
the hidden companion's surface.

According to the
Cambridge Atlas of Astronomy[12],
a photograph obtained by J. A. Tyson at Kitt Peak has been suggested by
some researchers as possibly being the companion of PSR1913+16. For either
mass combination the hidden component would not block the pulsating
signal, because the orbital plane is tilted 45 degrees as viewed from the
earth[8].
The decrease in the pulsar's orbital period is reported to be 0.076
seconds per year[12].
However, the decrease is probably closer to 0.4 seconds per year, based on
a linear slope of data presented by Weisberg[8].

This decrease has been
explained as due to energy being carried away by gravitational waves, a
by-product of Einstein's general theory of relativity[8].
However, either decrease in period may also be calculated due to a speed
of gravity which is respectively 17 and 3 million times the speed of
light. This calculation utilizes the numerical integration simulation in
the Appendix (plamd-2.bas). The simulation is analogous to connecting the
two orbiting bodies with a hydraulic strut (linear damping), which tends
to circularize the orbit with a decreasing period, mean radius,
eccentricity, and a loss of orbital energy. Energy would be radiated in
the form of heat. TOP OF
PAGE

PROPAGATION
SPEED OF GRAVITY

Sir Arthur Eddington[6]
in keeping with general relativity states, "...gravitation is
propagated with the speed of light, and there is no discordance with
observation". His argument is valid for circular orbits but makes no
allowance for a changing radius with an eccentric orbit. For a speed of
gravity equal to the speed of light there can be a very significant lag in
the R distance in Newton's equation. Using the earth-sun as an example,
with a speed of gravity equal to the speed of light, the decrease in the
earth's orbital period would be 15 seconds per year, with a corresponding
decrease in its mean distance from the sun of 30 miles per year. These
large changes are not supported by observations.

Again using the
earth-sun simulation, with a speed of gravity three million times the
speed of light, the earth's distance from the sun would decrease about 0.6
inches per year, which is about 1.6 centimeters per year. It is
interesting to note that laser measurements from reflectors on the moon,
(placed there by astronauts), result in an increase in the moon's distance
from the earth of about four centimeters per year. Perhaps the earth's
orbit is shrinking, pulling away from the moon, rather than the moon's
orbit expanding. The increasing separation may be due to a speed of
gravity several million times the speed of light. The earth's primary
attractor is the sun and the moon's primary attractor is the much closer
earth. TOP OF
PAGE

HYPERBOLIC CREATION
MODEL

The HCM cosmogony
mentioned in the Summary is defined below. Although the rate of decay of
the assumed transit super-speed light is not defined, it always arrives in
the solar system at 186,000 miles per second, which is the same as one
light-year per year.

HYPERBOLIC CREATION
MODEL EQUATIONS

1)
dC = D / T

2) C = 1 +
dC

3) Tapp = T
- (D / C)

dC = average light
speed increment (light-years per year)
D = light source distance from
the sun (light-years)
T = age of universe since creation (years)
C
= average light speed (light-years per year)
1 = nominal light speed
(one light-year per year)
Tapp = apparent age of light source (years)

Interesting Newtonian
time dilation effects due to this decaying super-speed light were noted in
the Summary with regard to Supernova 1987 and a galaxy 12 billion
light-years distance from earth. The time dilation equations are presented
below.

TIME DILATION USING
HCM EQUATIONS

Tapp1 = Tapp at T1 = T - DELT (HCM eq: 1,2,3)

Tapp2 =
Tapp at T2 = T + DELT (HCM eq: 1,2,3)

4)
Tdil = (T2 - T1) / (Tapp2 -
Tapp1)

DELT = time
increment (years) about T / 1000
Tdil = ratio of earth-time divided
by source-time

HCM time dilation
effects may also be involved with PSR1913+16. With the pulsar's distance
of 15,000 light-years, the time interval between observed pulses on earth
may be greater than the actual interval at the pulsar. This would result
in computing an orbit for PSR1913+16 which is not the same as the actual
orbit[14].
For example, using the HCM with a universe age of 6000 years and the
4.17/2.5 mass combination, the beep rate would actually be 34.67 beeps per
second, rather than the measured 16.94. On site orbital precession would
be 866 rather than 423 degrees per century. NTE would match the 866 value,
but GR would now compute 1488 rather than 727 degrees per century, still
the same discrepancy ratio. TOP OF
PAGE

DISCUSSION

The results of this
investigation indicate that NTE provides a better overall correlation with
measured orbital precession than does GR. More measured data is desirable.
Io (one of Jupiter's satellites) with its excess precession of about 270
arc seconds per century would appear to be a likely candidate. However,
its gravitational interaction precession resulting from the other three
major satellites is about 200 degrees per century[2].
This is 2700 times the NTE or GR precession increment, making this
extraction unreliable. RW Tauri (a well-known binary star) might be
another candidate for measurement with its predicted precession of 7 to 9
degrees per century, which is about ten times that of DI Herculis. TOP
OF PAGE

CONCLUSIONS

General relativity can
hardly be relied upon to predict the characteristics of the entire
universe, if it can only predict orbital precession increments in the
solar system, and then fails the test for binary stars. The proofs for
general relativity have always been subject to controversy, searching for
proof in the noise level of data. It appears that we should go back to
square one with Newton; that is, Euclidean space and time for the entire
universe. Nominal Tidal Effects (NTE) combined with the Hyperbolic
Creation Model (HCM) described in this paper offer a reasonable
alternative to a universe cosmogony which is explained using general
relativity (GR). The HCM cosmogony is heliocentric (sun-centered), as also
is the well-known background Red Shift. The cause of the Red Shift
frequencies could be a decaying transit super-speed of light. Finally, the
possibility that the age of the universe is just a few thousand years has
been demonstrated. TOP OF
PAGE

REFERENCES

[1] Sanctuary, G. E., "Orbital Precession", 1999, Feb.,
unpublished.

[2] Sanctuary, G. E., "Orbital Precession of Jupiter's
Four Galilean Satellites", 1999, May, unpublished.

[3] Sanctuary, G. E., "Orbital Precession of Five
Uranian Satellites", 1999, June, unpublished.

[4] Naeye, R., "Was Einstein Wrong? The mystery of DI
Herculis",
1995, Astronomy, Nov. p.54-59.

[5] Tauber, G. E., "Albert Einstein's Theory of General
Relativity",
1979, Crown Publishers, Inc., p.127, p.128.
TOP OF
PAGE

[6] Eddington, S. A., "Space, Time & Gravitation",
1920, 1987 reprint edition, Cambridge University Press, p.94-95,
pp.123-125.

[7] Taylor, J. H., et al, "Measurements of General
Relativistic Effects on the Binary Pulsar PSR1913+16, 1979, Nature,
pp.277,437.

[8] Weisberg, J. M., et al, "Gravitational Waves from
an Orbiting Pulsar", 1981, Scientific American, Oct. p.74.

[9] Piran, Tsvi, "Binary Neutron Stars", 1995,
Scientific American,
May p.54,58.

[10] Kleppner, D., "The Gem of General Relativity",
Physics Today,
April 1993, Vol. 46 Issue 4, p.9 TOP OF
PAGE

[11] Stetson, H. T., "Sunspots in Action", 1947, The
Ronald Press Company, New York, pp.132-145.

[12] Audouze, j. & Israel, G., "Cambridge Atlas of
Astronomy",
1994, Cambridge University Press, 1994, p.289.

[13] Sanctuary, G. E., "General Relativity ~~ or
Newtonian Tidal Effects?", 1999, unpublished, Aug., p.8.

[14] Sanctuary, G. E., "Newtonian Time Dilation and
Pulsar PSR1913+16", 1999, unpublished, Aug., pp.1-4

[15] Sanctuary, G. E., "Hyperbolic Creation Model, (an
alternate understanding of the Red Shift frequencies)", 1997,
unpublished,
Nov. pp.1-6. TOP OF
PAGE

APPENDIX

QUICKBASIC PROGRAMS

TIDAL-3.bas

Calculates earth tidal
amplitude (ft.) resulting from the moon's gravitational pull, using a
specified elastic spring rate of the solid earth (lb./ft.), diameter of
the earth (ft.), mass of the earth (lb.), mass of the moon (lb.), mean
separation distance (AU), and the moon's orbital eccentricity around the
earth. Assumes that the earth is divided into two equal halves, separated
by D/2, and held together by a powerful spring. The force differential
between halves (lb.) is calculated first at apogee radius (greatest
earth-moon separation, ft.), and then at perigee (closest earth-moon
separation, ft.). The difference between these two differential forces is
then divided by two times the elastic spring rate to get the earth tidal
amplitude (ft.). Varying the specified spring rate to satisfy the desired
earth tidal amplitude (0.3 ft.) was accomplished to calculate an earth
spring rate of -2.024D+17 (lb./ft.). Decimal point is 17 places to the
right.

TIDAL-2.bas

Calculates sun-tide
amplitude resulting from a planet's gravitational pull on the sun, in the
same way as TIDAL-3.bas was done for the earth (above). The spring rate
for the sun is -4.1133D+15 (lb./ft.) for all planets. A sun-tide amplitude
of 3045 ft. results from Mercury's gravitational pull on the sun, (using
this spring rate), and causes an incremental change in Mercury's orbital
precession of 43 arc seconds per century, calculated using the PLAMD-2.bas
program. The theoretical sun-tide amplitudes for five planets are
presented in TABLE A. Based on these tidal amplitudes, the orbital
precessions were then calculated using PLAMD-2.bas program, which are also
presented in TABLE A.

PLAMD-2.bas

This program is a
two-body numerical integration simulation using double precision for
calculating longitudinal orbital precession of the periapse, (the closest
approach of the two orbiting bodies). Specified constants are the major
component mass (WE, lb.), the minor component mass (WM, lb.), the orbital
eccentricity, the orbital period (DAYORBM), and the tidal amplitude (KRF,
ft.). The initial conditions for the simulation are established at
apoapse, balancing the two components to orbit about their common center
of mass, using Kepler's area law (equal areas swept out in equal times).
An incremental adjustment is made in the initial tangential velocity to
ascertain that the orbital period and eccentricity are closely matched to
nine decimal places during the integration process, with the tidal
amplitude (KRF) set equal to zero. A time increment about 0.0001 (or less)
times the orbital period is recommended. A baseline run of 100 orbits is
made (KRF = 0), with periapse longitudes calculated for each orbit cycle.
This determination is accomplished by reducing the time increment to
1000th of its nominal value as periapse is approached and crossed for the
reading. Then the time increment is returned to its nominal value.
Averages for 20 orbits are printed out for day, periapse longitude, and
eccentricity. Five blocks are printed for a total of 100 orbits. After
completing the baseline run, the baseline orbital precession is determined
by taking the difference in the perihelion longitude from block one and
five, then dividing by the day difference from block one and five. A
second run of 100 orbits is made after changing (KRF = 0) to the desired
tidal effect amplitude (KRF). The orbital precession for the second run
may then be determined in the same way as was for the baseline run.
Subtracting the baseline precession from the second run precession gives
the actual precession. The orbital precession is very linear with KRF, so
that any other precession (for the same initial conditions) may be
determined by the KRF ratio times the calculated precession. TOP
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THE HYPERBOLIC
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