Vol.3, No.4, 268-274 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.34034
New components of the mercury’s perihelion
precession
Joseph J. Smulsky
Institute of Earth’s Cryosphere of Siberian Branch of Russian Academy of Sciences, Tyumen, Russia;
Received 28 January 2011; revised 27 February 2011; accepted 27 March 2011.
ABSTRACT
The velocity of perihelion rotation of Mercury’s
orbit relatively motionless space is computed. It
is prove that it coincides with that calculated by
the Newtonian interaction of the planets and of
the compound model of the Sun’s rotation.
Keywords: Mercury; Perihelion; Observation;
Calculation; Sun’s Compound Model
1. INTRODUCTION
There are many lasting puzzles in the Solar system the
understanding of which is crucial, and that for decades
and some for centuries have not received a final expla-
nation. One such phenomenon is the precession of the
perihelion of Mercury’s orbit. The encyclopedia Wikipe-
dia [1] posted an article “Tests of general relativity” of
evidence confirming the General Relativity (GR) by the
observations. In particular, Table 1 gives the confirma-
tion of General Relativity in the perihelion of Mercury.
The data in Table 1 are well known in theoretical phys-
ics [2-5]. However, due to lack of awareness of theoreti-
cal physicists, several lines of this table are incorrect.
Therefore, we consider the rotation of the perihelion of
Mercury in detail.
2. OBSERVED MOTION OF THE
VERNAL EQUINOX
The rotation of the Mercury perihelion can be deter-
mined as a result of the analysis of changes of several
parameters of planets orbits. For this purpose we shall
consider, what changes of the planets orbits occur and
from which points the readout of angles is carried out.
On Figure 1 in heliocentric equatorial system of coor-
dinates xyz the orbit plane of planet (Mercury) draws an
arc of circle DAB on celestial sphere, and the projection
of the orbit’s perihelion is marked by point B. The mo-
tionless planes of equator 00
A
A

and ecliptics 00 EE

are
fixed on the certain epoch ТS, for example, 1950.0 or
2000.0. Other planes of equator AA', of ecliptics EE' and
of Mercury DAB in epoch Т are moving in space.
The angles between planes are submitted on Figure 2,
there corresponds Figure 2(a) of our work [6] in which
the results are given for Mars, but they are fair for any
planet, including Mercury. As the planes of Earth equa-
tor AA' and of Earth orbit EE' on Figure 2 move in space,
therefore the vernal equinox point
 retreats on an arc 
2

from motionless equator plane 00
A
A with velocity
5025 6412223
ct
pT



, (1)
where pc - velocity in arcsec/century, Tt - time in tropi-
cal centuries from epoch of 1900.0. It needs to note, that
in geocentric system the Sun passes point
 at spring,
and in heliocentric system the Earth passes point
 at
autumn.
The Eq.1 is derived by S. Newcomb [7] as a result of
approximation of observation data on an interval several
hundreds years. It gives velocity of retreating of point 
from a motionless equator plane 00
A
A, which is equal
to −5026.75 arcsec/century for 1950.0 and −5027.86
arcsec/century for 2000.0. As the point  moves clock-
wise therefore the velocity is written down with this sign
Table 1. Sources of the precession of perihelion for Mercury
according to the encyclopedia Wikipedia [1].
Amount
(arcsec/Julian century)Cause
5028.83 ± 0.04 [2] Coordinate (due to the precession of
the equinoxes)
530 [3] Gravitational tugs of the other planets
0.0254 Oblateness of the Sun
(quadrupole moment)
42.98 ± 0.04 [4] General relativity
5603.24 Total
5599.7 [5] Observe
−3.54 (−0.0632%) Discrepancy

J. J. Smulsky / Natural Science 3 (2011) 268-274 269
Figure 1. The basic planes on celestial sphere C: 00
A
A

- a
motionless plane of the Earth’s equator for epoch ТS; 00 EE

- a
motionless plane of the Earth’s orbit for epoch ТS (a plane of
motionless ecliptic); AA' - a mobile plane of the Earth’s equa-
tor in epoch Т; EE' - a mobile plane of the Earth’s orbit in ep-
och Т (the inclination for presentation is increased); 
 - vernal
equinox point of epoch ТS;  - a point on a line of crossing of
mobile equator in epoch Т with a mobile ecliptic (vernal equi-
nox point in epoch Т); DAB - a plane of the Mercury’s orbit in
epoch Т.
“-”. Note that in modern treatment of observation data
by J. L. Simon et al. [8], the velocity of removal of the
point
 is - 5028.82 arcsec per century.
Thus, the number 5028.83 arcsec per century in the
first row in the Table 1 represents the motion of the ver-
nal equinox
 relative to the motionless space.
3. THE RELATIVE VELOCITY OF THE
PERIHELION ON THE
OBSERVATIONS
In Astronomy the moton of perihelion point B is de-
fined by a longitude a, which as result of approximation
of observation data S. Newcomb [7] represents as a
polynomial of the third power of time:
23
π334 1305 536626 73
0467500043
aj
jj
T
TT
 

 


, (2)
where Tj - time counted in Julian centuries for 36 525
days from fundamental epoch 1900.0.
Value a represents the sum of two different arcs (see
Figure 2)
πa
A
AB

, (3)
Figure 2. The parts of the Mercury’s perihelion rotation on the
celestial sphere. The designations of planes it is given on Fig-
ure 1; 
G - an arc of the big circle, which is perpendicular to
plane of the Mercury’s orbit GDAB; B - the heliocentric pro-
jection of the Mercury’s perihelion to celestial sphere; А - as-
cending node of the Mercury’s orbit on mobile ecliptic; D -
ascending node of the Mercury’s orbit on motionless equator
of epoch ТS; the parameters of the Mercury’s orbit in inertial
equatorial reference frame: 
 =

0D;

р.= DB; i = 

0DG; iE -
inclination of the Earth mobile orbit (mobile ecliptic); and in
mobile ecliptic system Ωa =  A; ωa = AB; πa =  A + AB = Ωa +
ωa; iеа = iе =   AG; index “e” - the angles relatively to mobile
ecliptic; index “a” - by results of approximation of observation
data.
where arc  A = a refers as longitude of ascending node
of Mercury’s orbit.
From the Eq.2 the velocity of perihelion rotation on a
way of arcs
 A + AB is equal 5602.9 arcsec/century for
1950.0 and 5601.9 arcsec/century for 2000.0. For ele-
ments of J. L. Simon et al. [8] it is equal to 5603.0
arcsec par century to 2000.0. From Figure 2 it is seen,
that at definition of perihelion point B by value a, in
velocity of perihelion will enter: 1) velocity of move-
ment of point
 on mobile ecliptic EE'; 2) velocity of
displacement of point A of mobile ecliptic EE' due to its
rotation around point N and 3) velocity of displacement
of ascending node A of Mercury orbit GDAB on mobile
ecliptic EE', caused by rotation of plane GDAB.
Thus, the value 5599.7 arcsec per century in the 6-th
row of Table 1 gives the velocity of the perihelion rota-
tion from the moving point  of the vernal equinox with
the inclusion of velocities of change of the ecliptic and
of the Mercury orbit. It is slightly differs from the values
5602 ÷ 5603 arcsec per century determined by the ele-
ments of S. Newcomb [7] and J. L. Simon et al. [8]. This
velocity is not absolute but relative. In addition, as
shown above, it includes the rates of change of the eclip-
tic and the orbit of Mercury.

J. J. Smulsky / Natural Science 3 (2011) 268-274 270
4. ABSOLUTE VELOCITY OF THE
PERIHELION ON OBSERVATIONS
To these velocities did not affect the velocity of
perihelion movement, it should be counted from a mo-
tionless point. As such point we have taken the point G,
which is on crossing of the circle GDAB with perpen-
dicular circle to it

0G. In work [6] we have derived the
formula (29) for arc GB which depends on parameters of
mobile orbit planes of the Earth (EE') and of the Mer-
cury (GDAB) and have the following form:



0
2
0.5
2
π
arcsinsinsinsin
arccoscos1sinsin
a
paa
a
Ea a
aaa
GB
ii
i












(4)
The designations are given in the caption signature to
Figure 2. For angles a

 and iEa in work [6] are also
given formulas, which depend on the ecliptic angles of
orbits: Ωa, iеа etc. As a result of approximation of obser-
vation data S. Newcomb [7] has presented ecliptic an-
gles as polynomials of the third degree on time which
example is the Eq.2. J. L. Simon et al. [8] have modified
Newcomb’s results for epoch of 2000.0 and have given
as polynomials of 6-th degree.
The Eq.4 gives velocity of perihelion rotation of
Mercury orbit relatively motionless space which equals
582.05 arcsec/century for 1950.0 and 583.15 for 2000.0
[9,10]. It is velocity of perihelion rotation according to
observation. Due to orbits elements by J. L. Simon et al.
[8] it equals 582.53 arcsec/century for 2000.0. It is ab-
solute velocity of the perihelion on observations data.
Thus, in the Table 1 the velocity of the perihelion
rota- tion on observation data does not been determined.
It is equal to 582 - 583 arcsec per century relatively to
the motionless space.
5. THE ROTATION OF PERIHELION FOR
MERCURY UNDER ACTION OF THE
PLANETS AND THE SUN
5.1. Gravitational Tugs of the Other Planets
The interaction of the Solar system bodies under the
Newton law of gravitation results in change of their or-
bits (see Figure 3), including the rotation of perihelion.
We have developed a refined approach for integrating
the equations of celestial and space dynamics, on which
is created the program Galactica. In many of our works,
for example [6,11], the periods and the amplitudes of the
changes of the planets orbits elements are received at
dif- ferent time spans, including up to 100 million years.
In these calculations the bodies are considered as mate-
rial points, which interact under Newton’s law of gravity.
On Figure 3 in span of 7 thousand years the points 1
show the evolution of the elements of the Mercury orbit,
ob- tained with the Galactica. For comparison, the ap-
proxima- tions of observation data, made in 1895 by S.
Newcomb [7] (line 2) and a hundred years later by J. L.
Simon et al. [8] (line 3) are given. The evolution of the
Mercury perihelion angle relative to the motionless point
G in Figure 2 is seen on points 1 on Figure 3 for ele-
ment

р0. Its velocity is 529.86 arcsec per century. This
differs from the observation data at 53 arcsec per century,
but not at 43 arcsec per century, as previously thought.
For finding out of the reason of difference of calcu-
lated on Newton interaction and according to observa-
tion of the velocity of Mercury perihelion rotation we
have carried out various researches. First, we have es-
tablished, that such difference of velocity of perihelion
rotation is presented only for the Mercury, which is the
closest planet to the Sun. Second, the calculated on
Newton interaction other parameters of Mercury orbit
and velocity of their change practically coincided with
the data of observation [9,10].
5.2. Influence of Finite Propagation Speed
of Gravity
We investigated influence of gravity propagation ve-
locity on results of interaction of two bodies. The Gen-
eral Theory of Relativity was created to take into ac-
count final velocity of gravity. A. Einstein has based it
on the equations and results received by Paul Gerber.
Paul Gerber has thought up such mechanism of gravity
propagation with velocity of light that it is explained
rotation of perihelion in 43 arcsec/century [12]. However,
as we have shown in paper [10], this mechanism is
proved by nothing and is erroneous. Besides as it is
above shown, the difference of calculations on Newton
interaction and observation is equal to not 43 arcsec
/century but 53 arcsec/century.
In nature only one mechanism of interaction propaga-
tion with speed of light is known: it is mechanism of
propagation of electromagnetic interaction. From ex-
perimental laws of electromagnetism we have derived
equation [13-15] for force of interaction of two particles
with charges q1 and q2:



 
2
32
22
1
,k
r
 

r
Frv
r 
, (5)
where k = ke = q1q2/ε, 1 c

v

, r and v - distance and
velocity of one particle relatively another; ε - dielectric
permeability of media between particles, and c1 - speed
of light in media.
Apparently from (5), at not instant interaction the force

J. J. Smulsky / Natural Science 3 (2011) 268-274 271

р0


Figure 3. The secular changes in orbital elements of Mercury in the interval 7 thousand years and oscillations of
semi-major axis (a) and orbital period (P): Δa and ΔP are deviations from the means for 7 thousand years; 1 is result of
the numerical solution of the program Galactica; 2 is secular variations on Newcomb S., 3 is secular variations on Simon
J.L. et al.; e is eccentricity; i is the angle of inclination of the orbital plane to the plane of the motionless Earth’s equator
at the epoch 2000.0, 
 is the angular position of the ascending node of the orbit from the motionless equatorial plane;

р0 = GB (see Figure 2). The angles are in radians, the time T in centuries from 1949 December 30.0, Δa in meters and
ΔP in years.
depends not only on distance r between particles, but
also from their relative velocity v. If for gravity to accept
the same mechanism of interaction propagation the for-
mula (5) will define the gravity force at k = kG =
−G·m1·m2, where m1 and m2 - masses of interacting bod-
ies, and G - gravitational constant. With force


,
F
rv
we have calculated a trajectory of movement of one body
relatively another at all possible changes of an eccentric-
ity and velocity of body in a perihelion [14-18]. In case
of an elliptic orbit the perihelion rotates and the more
strongly, than there is the more velocity of body in a
perihelion. In such orbit the length of the semi-major
axis and the period change in comparison with the orbit
received at interaction of two bodies under the Newton
law of gravity. The changes of the semi-major axis and
the period have the same order as the change of the
perihelion angle.
The calculation of rotation of Mercury perihelion at
force


,
F
rv has given velocity 0.23 arcsec/century,
i.e. almost in 200 times smaller than velocity 43

J. J. Smulsky / Natural Science 3 (2011) 268-274 272
arcsec/century explained by Paul Gerber [12] and ac-
cepted in GR. The conclusion from here follows that
surplus of the perihelion rotation in 52 ÷ 53
arcsec/century may not be explained by the mechanism
of gravity interaction propagation with speed of light.
5.3. The Precession of the Perihelion under
Action of the Sun
The explanation of surplus of perihelion rotation of
the Sun oblateness now is complicated with complexities
of model of interaction and absence of knowledge of
distribution of the Sun density on radius and along an
axis of the Sun. Therefore the executed calculations of
influence of the Sun oblateness, most likely, are doubtful.
If inside the Mercury orbit there was a planet of the
certain mass, it might make necessary rotation of the
Mercury perihelion and at the same time not to render
appreciable influence on other planets. Such planet is not
present. But the Sun rotates about its axis, and the mov-
ing masses of its substance may influence Mercury the
same as the planet offered above. During two centuries
these ideas were put forward in various forms. However,
the satisfactory solutions have not been received. In
2007, the problem of axisymmetric gravitational n-body
interaction has been solved exactly (see [14,15,19]). Be-
fore this solution, there was only one exact solution of
the problem of bodies interaction, namely for two bodies,
it was received by I. Newton 300 years ago. In works
[15] and [19] the exact solution for a symmetrically lo-
cated on a plane bodies has obtained for all possible
cases. The bodies, as in the case of the two bodies, can
move in a circle, ellipse, parabola, hyperbola and
straight lines. This solution is allowed to create a com-
pound model of the Earth’s rotation (see [11,20]), in
which a part of the Earth’s mass is distributed between
the bodies symmetrically located in the equator plane of
Earth. The orbital plane of one of these bodies simu-
lates the evolution of the Earth’s equatorial plane under
the action of other Solar system bodies.
With assistance of the compound model of the Sun’s
rotation (see [9,10,21]) it is considered the inverse prob-
lem: to what changes in the planets motion would such
model of the Sun’s rotation? The papers [9,10] consid-
ered various variants for the action of the compound of
the Sun’s rotation with jointly action of other Solar sys-
tem bodies. Appearently, that at the certain mass of pe-
riph- eral bodies of model the same velocity of rotation
of perihelion may be received as observed one, i.e. 583
arcsec/century. In this case, the velocity of change of
other parameters of the Mercury’s orbit essentially does
not change. The velocity of the Venus perihelion does
not change essentially, and parameters of planets are
more distant from the Sun and also change still to a
lesser degree. Let's note that the compound model of the
Sun rotation takes into account of the Sun oblateness and
rotation of its masses.
To make sure that the additional rotation of the peri-
helion is due only to the compound model, it was studied
the action only one compound model of the Sun’s rota-
tion (without planets) on Mercury [21]. The differential
equations of motion of all bodies have been integrated
numerically and have been studied the evolution of
Mercury for three thousand years. In this case the rota-
tion of its orbit is received of 53 arcsec per century, i.e.
precisely the surplus, which is available at the joint in-
fluence of compound model of the Sun at the planets.
As a result of the carried out researches we shall write
down the basic characteristics of rotation of Mercury
perihelion in such form (see Table 2) that they could be
compared to the data on Wiki site. Apparently from
Table 2. Velocity of rotation of the Mercury perihelion on observation for 2000.0 on orbit elements of S. Newcomb [7] (Ncb) and J.
L. Simon et al. [8] (Sim) and on Newton interaction. To compare in round brackets - accordingly Wikipedia [1].
Amount
(arcsec/century) Explanation
On observation data
−5027.86 – Ncb
−5028.82 – Sim
(5028.83 – Wiki)
Velocity of movement of vernal equinox point  relatively motionless space (according form. (1)).
5601.9 – Ncb
5603.0 – Sim
(5599.7 – Wiki)
Velocity of perihelion rotation relatively the mobile vernal equinox point  with including velocities changes of ecliptic
and of Mercury orbit (according form. (2)).
583.15 – Ncb
582.53 – Sim Velocity of perihelion rotation relatively motionless space (according form. (4)).
By results of interaction under the Newton law of gravity. Velocity of rotation of a perihelion relatively motionless space.
530 (530 – Wiki) Planets and the Sun interact as material points.
582 Planets interact as material points, and the oblateness and rotation of the Sun is taken into account as compound model.

J. J. Smulsky / Natural Science 3 (2011) 268-274 273
above-stated, the problem of perihelion rotation is de-
fined by many circumstances. Here we have not men-
tioned a problem of reliability of observation data ap-
proximation. We have tried to state other problems
clearly and with necessary explanatory that everyone
might pass on this way and be convinced of our conclu-
sions.
We have briefly outlined a number of stages of the re-
search phenomenon of rotation of the Mercury perihe-
lion, which were performed by computer algorithms. We
used the numerical integration of the differential equa-
tions systems, a variety of calculations with geometric
transformations, mathematical treatment of time series
and other computer calculations. Due to them, it was
found that the components of the perihelion rotation of
the Mercury’s orbit can be explained by the correct ac-
count of Newton’s gravitational force in the interaction
of the celestial bodies.
6. CONCLUSIONS
1) The velocity of perihelion rotation relatively mo-
tionless space accordingly observation data is equal to
583 arcsec per century.
2) The velocity of perihelion rotation relatively mo-
tionless space as a result of interaction of the planets
under the Newton law of gravity is 530 arcsec per cen-
tury.
3) The Newtonian interaction of planets and of the
compound model of the Sun’s rotation gives the obser-
ved Mercury’s perihelion precession.
7. ACKNOWLEDGEMENTS
I am grateful to David Weber for being interested in the problem and
his work to notify society of our received results. Many calculations in
the above-mentioned studies were performed on supercomputers of
Siberian Supercomputer Center of Russian Academy of Science.
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