Cent. Eur. J. Phys. • 6(3) • 2008 • 754-758
DOI: 10.2478/s11534-008-0081-6
Central European Journal of Physics
Relativistic perihelion precession of orbits of Venus
and the Earth
Short Communication
Abhijit Biswas, Krishnan R. S. Mani ∗
Department of Physics, Godopy Center for Scientific Research, Calcutta 700 008, India
Received 29 February 2008; accepted 2 April 2008
Abstract:
Among all the theories proposed to explain the “anomalous” perihelion precession of Mercury’s orbit first
announced in 1859 by Le Verrier, the general theory of relativity proposed by Einstein in November 1915
alone could calculate Mercury’s “anomalous” precession with the precision demanded by observational
accuracy. Since Mercury’s precession was a directly derived result of the full general theory, it was viewed
by Einstein as the most critical test of general relativity from amongst the three tests he proposed. With the
advent of the space age, the level of observational accuracy has improved further and it is now possible
to detect this precession for other planetary orbits of the solar system – viz., Venus and the Earth. This
conclusively proved that the phenomenon of “anomalous” perihelion precession of planetary orbits is a
relativistic effect. Our previous papers presented the mathematical model and the computed value of
the relativistic perihelion precession of Mercury’s orbit using an alternate relativistic gravitational model,
which is a remodeled form of Einstein’s relativity theories, and which retained only experimentally proven
principles. In addition this model has the benefit of data from almost a century of relativity experimentation,
including those that have become possible with the advent of the space age. Using this model, we present
in this paper the computed values of the relativistic precession of Venus and the Earth, which compare
well with the predictions of general relativity and are also in agreement with the observed values within the
range of uncertainty.
PACS (2008): 04.25.D-, 04.50.Kd, 03.30.+p, 45.50.Pk
Keywords:
perihelion precession • relativistic precession • Venus • Earth • numerical relativity • relativity theory
© Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
In our solar system, the potential manifestations of relativistic effect are so tiny that, in 1916, Einstein could think
of only three tests in which his General relativity theory
(GRT) differs from Newton’s gravitational theory: starlight
deflection, perihelion precession of Mercury, and gravita-
tional red-shift. With the advent of the space age in the
1960s, the level of observational accuracy and available
computational capability improved further, making possible the following additional tests: Shapiro time delay, and
relativistic perihelion precession of Venus and the Earth.
∗
E-mail: [email protected]
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2.
Historical background
Relativity experimentation began with renewed vigor in
the sixties and, with the utilization of new space-age technologies, the accuracy of observational data and the derived values of planetary masses and of other astronomical
constants improved considerably. As a result it became evident that the phenomena of relativistic precession exists
for perihelion of the orbits of planets like Venus and the
Earth, even though these are further away from the Sun
and their orbits are not as highly elliptical as the orbit of
Mercury. However, the centennial rate of relativistic precession for Venus and the Earth as compared to Mercury is
lower by almost an order of magnitude, and the centennial
rate as derived in the seventies and the mid-eighties contained large uncertainties. However, GRT estimates for
the relativistic precession for both Venus and the Earth
compared well with their respective observational values
and lay within the inherent levels of uncertainty. This
proved conclusively that the phenomena of “anomalous”
perihelion precession of planetary orbits is a relativistic effect and hence cannot be explained by any of the
“ad-hoc” hypotheses [1] that were offered before Einstein
presented his GRT.
3.
Gravitational model
It may be mentioned here that the results of numerical
simulations presented in this paper are based on an alternate relativistic gravitational model titled the Remodeled
Relativity Theory (RRT), as given in our earlier papers
[2–4].
It may be noteworthy to mention here that RRT, as explained in detail in our earlier paper [4] retained only the
experimentally proven principles from Einstein’s relativity
theories, and that RRT has been enriched by the benefits
of almost a century of relativity experimentation including
those made possible with the advent of the space age.
RRT adopted and expressed the conservation law of momentum vector direction as its generalized law for spinning
and rotational motions, from which vectorially all the expressions for inertial forces and torques (viz., centrifugal
and Coriolis forces, as well as gyroscopic precessional
torques for spinning tops and gyroscopes) could be derived. In fact, as mentioned below, this law involving the
inertial forces and torques has been successfully used for
the precision computation of planetary and lunar orbits
[3, 4].
The conservation laws of energy and momentum are the
most fundamental principles of the RRT. Based on almost
a century worth of data from relativity experiments, two
fundamental principles were adopted for RRT [4]: one,
that energy level is the underlying cause for relativistic
effects and two, that mass is expressed by the relativistic
mass-energy equation as enunciated by Einstein.
Utilizing the space age ephemeris generation experience
and following the methodology of nature to conserve energy and momentum, we found the reason to replace the
concept of “relativity of all frames” with that of “nature’s
preferred frame”, as explained in our earlier paper [4]. This
is strongly supported by the fact that the least-squaresadjusted (LSA) astronomical constants (e.g., the planetary
masses, etc.) of nature, which are an outcome of global
fits done during the generation of a particular ephemeris,
are consequences of not only the gravitational model, but
also of the coordinate frame. In other words, the constants
of nature are linked to the coordinate frame. This conclusively tells us that today one has to accept the existence of
only one set of the constants of nature as a concomitant
of only one appropriate “preferred frame”, and the relevant orbit or orbits linked to them. This “preferred frame”
according to the RRT has been termed as the “nature’s
preferred frame” [4].
Based on the few well-proven basic principles cited above,
a comprehensive remodeling effort led to the RRT that has
been used to consistently and successfully simulate numerically the results of all the “well-established” tests of
the GRT at their current accuracy levels [2, 3], and for
the precise calculation of relativistic effects observed in
the case of the Global Positioning System (GPS) applications, the accurate macroscopic clock experiments and
other tests of the special relativity theory [4].
The mathematical model used to compute the results presented here is exactly the same as that utilized for computing the centennial rate of the relativistic perihelion
precession of Mercury’s orbit [3]. However, for the convenience of the readers a few basic equations are being
presented here with self-consistent notations.
Equation (1) given below is the most precise form of the
relativistic mass equation according to the “matter model”
of RRT [3, 4], which is being applied here for the case of a
typical heliocentric planetary orbit. This equation is the
expression for the rest mass m0 of a planet (inclusive of
the mass of its satellites and atmosphere, if any) at an
infinite distance (where the gravitational influence of any
other body also is insignificant), in terms of its relativistic
mass mr , as it is pulled and moved to its new location at
a radial coordinate r, under the gravitational potential of
the sun.
p
m0 = mr 1 − βr2 ,
(1)
where:
mr = relativistic mass of the moving or orbiting planet
at a radial coordinate r in its natural coordinate system
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Relativistic perihelion precession of orbits of Venus and the Earth
(c.s.), i.e., the SCSF (sun centered space fixed) frame for
a planetary orbit,
βr = velocity ratio of the orbiting planet at a radial coordinate r, given by
βr = [v/cr ] ,
The equation of the gravitational acceleration of the orbiting planet due to all bodies in the solar system, other
than the central body, is given by
r̈J =
X
J6=c
where:
v = velocity of the orbiting planet at a radial coordinate
r, in the SCSF frame, and
cr = magnitude of light velocity at a distance r from the
center of the SCSF frame.
Simulation of a photon’s orbit in the same SCSF frame,
using an almost similar “photon model” as that mentioned
in previous papers [2, 4], reveals that the following relationship exists for the velocities of light, at the current
accuracy level:
c = cr · [1 − F ] ,
(3)
where:
c = magnitude of light velocity at infinite distance (where
the gravitational influence of any other body is also insignificant) from the center of the same SCSF frame, and
F = gravitational red-shift factor (strictly speaking, it
should be called a “blue-shift” when the photon is moving
closer to the gravitating mass) at a radial coordinate r, in
the SCSF frame, given by
F=
GMs
,
cr 2 r
(4)
where:
Ms = mass of the sun, and
G = Gravitational constant.
As mentioned in previous papers [2, 4], the principle of
equality of the relativistic transformation factors for energy level and time (and length), was first adopted and
later proven as a fundamental principle of RRT. Accordingly, the relativistic transformation factor for time for a
planet moving under the gravitational influence of the sun,
in the SCSF frame, may be written as:
dt = dτ [1 − F ] ,
#
(rJ − r) mJ rJ
G
·
− 2 ·
.
rJ rJ
|rJ − r|2 |rJ − r|
"
(2)
(5)
where:
dt = coordinate time, that is, the proper time at a point
situated at an infinite distance from the center of sun,
where the gravitational influence of any other body is also
insignificant, and
dτ = proper time at a radial coordinate r, in the SCSF
frame.
mJ
(6)
where:
mJ = mass of the J-th body in the solar system,
rJ = Position vector of the J-th body in the SCSF frame,
and
r = Position vector of the orbiting planet.
The resultant gravitational acceleration vector due to all
bodies including the central body, is given by
r̈ = r̈J −
G(Ms + m) r
· .
r2
r
(7)
where:
r̈ = Resultant gravitational acceleration of the orbiting
planet caused by all celestial bodies in the solar system
including the sun,
r = Position vector of the center-of-mass of the orbiting
planet, and
m = mass of the orbiting planet.
As usual, the double dots above a symbol signify the double differential with respect to time, while the single dot
above a symbol signifies the single differential. This convention has been used throughout this paper.
The resultant gravitational acceleration is numerically integrated to obtain the epochwise velocity and position
vectors of the orbiting planet.
The radial component of the planet’s epochwise magnitudes of velocity, designated by vr , is obtained from vector
dot-product as follows:
vr = r̂ · ṙ,
(8)
where:
r̂ = the unit vector corresponding to the position vector of
the orbiting planet.
The tangential component of the planet’s epochwise magnitudes of velocity, designated by vφ , is obtained as follows:
p
vφ = v 2 − vr2 ,
(9)
The epochwise magnitudes of the angle between the vector
directions of vφ and v, designated by λv , is derived from
the magnitude of two velocity vectors, as follows:
λv = Arctan
vr
vφ
,
(10)
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Abhijit Biswas, Krishnan R. S. Mani
The term “relativistic perihelion precession” can be obtained as explained in our earlier paper [3].
It may be mentioned here that no work using a similar
approach could be found in the relevant literature, for the
calculation of the relativistic precession of Venus and the
Earth.
additional feature of our program is the existence of a
counter-checking method that numerically integrates the
ODE’s corresponding to the acceleration vector calculated
from the so-called “inertial forces”, which are exactly equal
and opposite to the corresponding “real forces”. This enables the counter-checking of the velocity and position
vectors obtained from the integration of “real forces”. This
also enables us to ensure that the conservation laws of
energy, linear and angular momentum are obeyed during
all stages of epochwise calculations.
4.
5.
In this paper, we present the result of numerical simulation of the “relativistic perihelion precession” of two more
planetary orbits of the solar system – namely, Venus and
the Earth.
Numerical simulation
Computations were performed on the heliocentric
equatorial-of-date co-ordinate system. JPL’s (Jet Propulsion Laboratory, Caltech) ephemeris DE405 provided the
data for the heliocentric positions of all planets other
than the orbiting planet. Ordinary differential equations
(ODE’s) were generated for the equations of motion based
on the relevant mathematical model of RRT [3]. All the
ODE’s were solved simultaneously using the variable step
differential equation solver, namely the Gear’s method [5].
This method is comparable to the best available in the
field for astro-dynamical calculations in controlling the
integration error within the specified value and has been
tested against various JPL integration algorithms. The
appropriate codes provided by JPL were used for extraction and interpolation of data from DE405. All the Fortran codes have been developed by the authors using the
above-mentioned mathematical model as presented in the
earlier paper [3] with the exception of a few codes provided by JPL for using ephemeris data, while making use
of a few of the algorithms for rotation matrices given in
JPL’s Technical Report [6] and in Newhall et al [7]. The
results of the computation are presented and discussed in
the following section.
In fact, our program can generate the ephemeris data –
that is, the epochwise position and velocity vector data
of any orbiting planet of the solar system whilst reading the corresponding position and velocity vector data of
other planets, and only the orbital starting epoch’s position and velocity vector data of the orbiting planet, from
JPL’s DE405. This is done by numerically integrating the
ODE’s corresponding to the resultant gravitational acceleration vector. This part is similar to the JPL’s ephemeris
generation process in the sense that the acceleration vector is calculated from the so-called “real forces”. But, it
differs from JPL in the sense that JPL uses the general
relativistic equation of motion, whereas we use the equations of motion that are based on the RRT model, and
are able to achieve comparable levels of accuracy. An
Discussion of results
Using the model developed by the authors, the results presented in Table 1 have been computed for the centennial
rate of the “relativistic perihelion precession” for the orbits
of Venus and the Earth. Computations for a time frame of
over two centuries were carried out. Observational results
for the “relativistic perihelion precession” have been presented from other sources for which the references have
been cited.
For Venus, we have presented very recent observational
results (that are yet to be published) derived from recent
Magelan doppler data near Venus1 .
A close scrutiny of Table 1 will show that our computed
values for Venus and the Earth not only agree with the respective values predicted by GRT but also agree with the
recent observational results within their respective range
of uncertainty.
6.
Conclusion
It can thus be seen from the presentation of data above
and discussion thereon, that RRT leads to computed values that compare well with the prediction of GRT and the
recent observational results, for the relativistic perihelion
precession of Venus and the Earth. Thus, the results presented in this paper can be said to confirm again the consistency of the RRT.
Acknowledgements
We thank Dr. E.V. Pitjeva, the head of Laboratory of
Ephemeris Astronomy of the Institute of Applied Astronomy, Russian Academy of Sciences, for helpful clarifica1
E.V. Pitjeva, private communication (2007)
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Relativistic perihelion precession of orbits of Venus and the Earth
Table 1.
Computed values of the relativistic perihelion precession in arcseconds per century compared with GRT value and observational results.
Orbiting
Celestial
body
Observational Uncertainty
results
Reference
GRT value
Relativistic perihelion precession,
computed using
the RRT model
Venus
8.6247
0.0005
Pitjeva (2007)a
8.6247
8.62473
Earth
3.8387
0.0004
[8], Pitjeva (2005)b
3.8387
3.83868
a
b
E.V. Pitjeva, private communication (2007)
E.V. Pitjeva, private communication (2005)
tions, and for help in obtaining data on observational results and their uncertainties. We thank also Dr. Lorenzo
Iorio for giving us encouraging help.
References
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Summer Institute on Particle Physics, Ed. L.
Dixon, 1998, SLAC, Stanford, California (Stanford Linear Accelerator Center, Stanford 1998),
15
http://www.slac.stanford.edu/gen/meeting/ssi/-
1998/manu list.html, arXiv:gr-qc/9811036
[2] A. Biswas, K.R.S. Mani, Cent. Eur. J. Phys. 2, 687
(2004)
[3] A. Biswas, K.R.S. Mani, Cent. Eur. J. Phys. 3, 69 (2005)
[4] A. Biswas, K.R.S. Mani, Phys. Essays (in press)
[5] C.W. Gear, Comm. of the ACM, 14, 176 (1971)
[6] T.D. Moyer, In: J. P. L. Tech. Rept., 32-1527, (Jet
Propulsion Laboratory, Pasadena, California, USA,
1971)
[7] X.X. Newhall, E.M. Standish, J.G. Williams, Astron.
Astrophys. 125, 150 (1983)
[8] L. Iorio, arXiv:0710.2610v1
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