International Journal of Astronomy and Astrophysics, 2012, 2, 249-255
http://dx.doi.org/10.4236/ijaa.2012.24032 Published Online December 2012 (http://www.SciRP.org/journal/ijaa)
Angular Precession of Elliptic Orbits. Mercury
Javier Bootello
Engineer, Málaga, Spain
Received August 31, 2012; revised October 2, 2012; accepted October 13, 2012
ABSTRACT
The relativistic precession of Mercury −43.1 seconds of arc per century, is the result of a secular addition of 5.02 × 10−7
rad. at the end of every orbit around the Sun. The question that arises in this paper, is to analyse the angular precession
at each single point of the elliptic orbit and determine its magnitude and oscillation around the mean value, comparing
key theoretical proposals. Underline also that, this astronomical determination has not been yet achieved, so it is con-
sidered that Messenger spacecraft, now orbiting the planet or the future mission BepiColombo, should provide an op-
portunity to perform it. That event will clarify some significant issues, now that we are close to reach the centenary of
the formulation and first success of General Relativity.
Keywords: Relativistic Precession; Mercury
1. The Theoretical G.R. Angular Precession
The trajectory of a target around a massive object (M), is
defined starting from the Schwarzschild solution, in a
geometry and a space-time with spherical symmetry. The
G.R. equation of motion with 1ur is [1-3]:
2
2
222
d3
d 
uGMGM
uu
hc 
We can write the relativistic orbit as a slight perturba-
tion of the newtonian ellipse as:
 
21cos
GM
ue
h
 
h = angular momentum per unit of mass; e = eccentricity;

= true anomaly; p = semi-latus;


 is a very
small function that produces the G.R. orbit differences,
from the newtonian-kepler ellipse: an orbit precession.
On that basis, a first approximation and particular so-
lution of this differential equation, neglecting second
order terms, and assuming a geodesic orbit, is presented
in the classic relativity textbook “Gravitation” by W.
Misner [4]:

 
01cos sin
1cos1 2π
1cos 1cos1
pp
reKe
e
pp
e
eK


 









 

(1)

2
3sin sin GM eKe
cp

  
were δ 
0/2π = constant angular precession = K.
As result of it, angular instantaneous precession in
each point of the trajectory -δ(
 )-, is constant referred to
 , so that the gradual addition along the orbit, orbital
precession -Δ(
 )-, has a linear accumulation till its final
value Δ(2π) (Figure 1).
Final one orbit precession is: Δ(
 ) = K × 

7
2
6π
2π2πrad orbit5.0210rad orbit
43.1 secarccentury
GM
Kcp

 

Figure 1. Angular (δ) and orbital (Δ) precession.
C
opyright © 2012 SciRes. IJAA

J. BOOTELLO
250
This particular solution with a constant angular pre-
cession was, in my opinion, the first result obtained by
Einstein in 1915. [5]:
“··· That contribution from the radius vector and de-
scribed angle between the perihelion and the aphelion is
obtained from the elliptical integral:

2
11
23
22
dd
2
x2
f
xx
Axx x
BB



 




(2)
where α1 and α2 (··· reciprocal values of the maximal
and minimal distance from the Sun ···) are the corre-
sponding first roots of the equation:
23
22
20 Axx x
BB

. (3)
Coefficients of Equation (3), were also confirmed by
Schwarzschild and other authors:
2
23
222 2
d22 2
d
uE GMGM
uu u
hchc


 


where E = Energy per unit.
The coefficients, must be also consistent with the
complete orbit precession of Mercury:
23
22 3
29
22.95
1113.610
21.0210.
GM c
Bh
AB






2
10;
;
Equation (2) represented by function f(x), has the fol-
lowing graphic expression (Figures 2 and 3).
We can remark that f(x), has virtually the same values
Figure 2. f(x): General graphic. Mercury (blue).
Figure 3. Graphic focused on mercury. α1, α2.

J. BOOTELLO 251
both in the aphelion as in the perihelion and also through
the rest of the orbit. It means that this solution, involves a
constant angular precession -δ(  )- along the whole orbit
and also a linear accumulation of the orbital precession
-Δ(
 )-, with a K proportion relative to the true anomaly 
(Figure 1).
2. G.R. Angular Instantaneous Precession.
Periodic Oscillations
General Relativity accepts also small periodic oscilla-
tions that should be insignificant contributions and their
only effect is to change slightly the position of the peri-
helion and the interpretation of rmin and e [6].
Usual formulation of G.R. fluctuations about the av-
erage constant precession, based also in a particular solu-
tion of the Schwarzschild’s methodological approach is
[1-3,7]:


2
2
2
2
311
1cos2si
26
31sin
2


 
 






GM ee
cp
GMe je
cp
n






 

We will analyse the range of the periodic oscillations
produced by function

j

related with the mean value
that involves the last term sine

(Figure 4). It must
be underlined that the cumulative effect produced by


K
e

has also a periodic origin and implication; it
really represents


sin

eK



that makes



 a
function consistent in Equation (1) and, as a result of it,
the perturbation’s effect -Δ(
 )-, is shaped definitely as an
angle, a real precession.
Function

j

involves very small variations. Its
amplitude is about 3/100 of the mean constant value.
Professor M. Berry [6], presents another



 func-
tion with larger amplitude of oscillations:
Figure 4. Angular precession oscillations: j(  ).


2
22
2
13
32coscos
3sin


 






GM e
ee
e cp
e
2







2
2323 sin
B
GM ej e
cp






Standing out from Figure 5, there are significant os-
cillations, but with the same final orbital relativistic pre-
cession. The range is equivalent to the magnitude of the
theoretical constant precession. The eccentricity of the
orbit has clear effects on the angular precession, increas-
ing the amplitude as the eccentricity decreases.
3. G.R. Perturbing Gravitational
Potential/Force
Trying to analyse the oscillations of the angular preces-
sion, we can also study the effects of a perturbing poten-
tial or force. This procedure should allow even more ac-
curate results than those obtained solving the second or-
der differential equation of motion.
The effective G.R. potential is displayed in Equation
(4), where the last term, is the perturbation potential
added to the classic newtonian one [1,8-10].
22
223
2
eff
GM hGMh
Vrrcr
  (4)
 
2
23 G.R. perturbingpot.
GMh
Vr cr  ;
and,



2
24
3G.R.perturbing force
Vr GMh
Fp rrcr

 

Figure 5. Angular precession oscillations: jB(  ).

J. BOOTELLO
252
We will now analyse some approaches and methods
that explore the orbital precession produced by any po-
tential or perturbation force.
1) In 1982, B. Davies [11] presented a solution to the
orbital precession, based on the Laplace-Runge-Lenz
vector, located in the same plane as the orbit and pointing
in the direction of the perihelion. Vector’s angular veloc-
ity, measures the precession if there is any external dis-
turbance.
The magnitude of the total force would be equal to the
usual newtonian, added with a function

--
g
r as a
perturbing factor.
 
21
GMm
F
rg
r
 r




The solution to the orbital precession is then:

π
0
2cosdrad.gr
e
  
If we apply this method to G.R. perturbing force and also
considering an elliptic orbit:

2
22
3h
gr cr ; ;
2
hGMp1cos
p
re

;
and then, in agreement with the orbit’s symmetry:

 
π
2
2
0
2π2π
2
2
00
31
21cos cosd
31
1coscos ddrad.
D
GM e
e cp
GM e
e cp


 
 


Then, the instantaneous angular precession referred to 
radians is (Figure 6):
 
2
11coscosrad. rad.
DKe
e
  
and referred to time:




rad. sec.
DD
t  
The integral of this angular precession gives exactly
the relativistic final orbital precession, equivalent to 43.1
seconds arc/century, however with significant intermedi-
ate oscillations. Orbital precession is (Figure 6):
 
 
0
2
drad.
34 111
sinsin 2sin 3
212
D
e
Ke
e










 





Davies also remarks that the factor cos

, brings a
positive sign to the precession in the part of the orbit
when the planet is closer to the focus than the average
distance (p); the rest is negative. Therefore, that state-
ment supports that one half of the relativistic precession,
is in the opposite direction to the advance of the planet in
its orbit.
2) In 2005, M.G. Stewart [9] started also his approach
from the Laplace-Runge-Lenz vector, but providing the
following alternative formulation:


22
cosrad. sec.
GR
rFprh
GMe

 

Figure 6. Theoretic approaches to angular (δ) and orbital (Δ) precession.

J. BOOTELLO 253
with GR

, angular velocity of L-R-L vector.
If we apply this method to G.R. perturbing force:

 
2π
00
2π2π
2
2
0
d1d
31
1cos cosdd
 




GR GR
D
t
GM e
e cp



0




This result is just the same to the previous one.
3) In 2007, professor G. Adkins [10], studies the pre-
cession solving the equation of motion, with a right
method, adding a perturbing potential in the fol-
lowing equation:

Vu

2
22 2
drad.
2
22
u
Vu
EGM
uu
mh hmh



By the change of variables

1uez p , he obtains
the next formulation of the orbital precession:

1
22
1
d
2d
d
1
Vz
pzz
z GMme z


 

and for a power-law potential, this alternative formula-
tion:


 1
1
n
n
Vr r 







1
1
12
1
21 d1
1
nn
nn
nzz ez
GMmp ez





 


If we apply this method to G.R. perturbing potential and
changing cosz

:
 
2π2π
2
2
00
31
1coscos ddrad.
D
GM e
e cp
  

Again it is exactly the same result, however starting
from a different hypothesis.
4) We will check these results with an accurate test,
based on a new approach. This is the Langrange Plane-
tary Equations applied to a slight perturbation with an
energy and forces conservative framework. The preces-
sion is referred to the argument of the periapsis whose
derivative in this elliptic orbit, is equivalent to the true
anomaly.
2
2
dd 1
dd
eR
tt na e
 
 e
where R is a perturbing function.
If we consider a plane elliptic orbit with a central po-
tential, we have the following relation:

2
Vr
Rap
ee
r



;
2
1ep
na h

And then, 2
dd
dd
hp
thae e r

R





2
2
d
d
Vr
p
e he



V(r) is the relativistic perturbation potential, so


22
32
31cos cos
Vr hGM e
epc


,
then:
 
2
11coscosrad. rad.
DKe
e
  
Solution that is just the same as before and also with
identical result to that obtained through the Gauss Plane-
tary Equations, based on the perturbing force:

2
dd 1cos
dd
eFp r
ttnae




2
224
d13
cos
d
hpGMh
he rcr



and then:
 
2
11coscosrad. rad.
DKe
e
  
5) We will finally verify this proposals with the Lan-
dau & Lifshitz formulation [12], which defines the pre-
cession produced by a perturbing potential-Energy. This
formula is valid as a theorem, suitable for any small per-
turbation whatever could be its physical origin and re-
turning the exact value. Integration is performed over an
unperturbed orbit [13]:

π
2
0
2d mrU
MM





 


where M = mh = angular momentum, δU = perturbing
potential-Energy = 3 m V(r) (for a three dimension tar-
get).
Then, the angular instantaneus precession is:
 
2
1
3
LrV r
hh




.
If we apply this method to G.R. perturbing potential:
 
2
31co
L
GM he
hp cs
 

 






derivates referred to h are:
2
11
2;
ppe e
hhhhe



 [10];
and then:

J. BOOTELLO
254
 
2
1
1coscosrad. rad.
L
e
Ke e
 


 






1
1cosrad.rad.
LKe
 




Orbital precession is (Figure 6):
 
0
1
drad.sinrad
LKe

 

 


.
The solution is different, however very similar to the
previous ones, and also with identical value of the final
orbital relativistic precession. The maximum values are
somewhat lower at 0 and 2π but higher at π. The angular
precession is null at  = 1.77 rad. and  = 4.50 rad.
while previously was null at π/2 y 3π/2.
The results obtained, shows that the oscillation is such
that between π/2 and 3π/2, the orbital precession turns
back, opposite to Mercury’s own progress in its orbit. At
these points (maximum and minimum) there is an
equivalent lead/lag of 1.9 sec.arc./century related with
the magnitude of the orbital precession at the final/ initial
point of the orbit (Figure 6).
Another issue is the clear influence that has the eccen-
tricity in the magnitude of oscillations. The lower is the
eccentricity, the greater the fluctuation of the angular
precession because they are inversely proportional.
In case of Mars (e = 0.093), there would be a lead/lag
of 1.3 sec.arc./century equivalent itself to the magnitude
of the relativistic precession at the final/ initial point of
the orbit. The Earth (e = 0.017) should have a lead/lag of
37.1 sec.arc./century, nearly ten times the relativistic
precession and Venus (e = 0.0068) should have 203.4
sec.arc./century, 24 times the final precession (Figure 7).
If this theoretical formulation is correct, these results
should have significant observational data records, in the
Registered orbital precession of these planets.
4. Mercury’s Orbit as an Open Free-Fall
Path
The currently precession of Mercury, is far larger to the
one with only a relativistic origin. This is due to the ef-
fect produced by the rest of the planets, causing also
precessions that must be added.
The largest precession is produced by Venus (277
sec.arc./cent.) followed by Jupiter (154 sec.arc./cent.),
the Earth-Moon system (91 sec.arc./cent.) and the rest of
the planets for a total of 532 sec.arc./cent. Relativistic
Precession is 43 sec.arc./cent., therefore we can conclude
that the real precession detected in astronomical observa-
tions is equivalent to 575 sec.arc./cent.
There are other perturbations as solar oblatness or
Lense-Thirring secular precessions, but with magnitudes
some orders lower.
To study the oscillations of the angular precession re-
lated to the final magnitude in each orbit, it would be
necessary to have for at least one year, data from the posi-
tion of Mercury with the best possible accuracy. These
data should be reduced with the other perturbations of the
planets, as well as considering the effect of the equi-
oxes’s precession. In this way, we could examine Mer- n
Figure 7. Eccentricity, angular (δ) and orbital (Δ) precession.

J. BOOTELLO 255
cury’s orbit as an open free-fall path, isolated from other
planets gravitational interference. It is certainly a difficult
and complex duty but clearly available with the current
development of our technology and also not expensive.
Messenger spacecraft, now orbiting the planet, should
provide an excellent opportunity to perform it, giving
precise radiometric data on the day to day real position of
Mercury. A detailed study and related tests on relativistic
and gravitational effects that could be achieved with a
Mercury orbiter mission, is summarized in [14]. Another
alternative is to wait till the Bepi Colombo be launched
in 2015, an European mission to Mercury where, testing
relativistic gravity is recognized as a crucial scientific
objective.
To assess the influence of each planet in the orbit of
Mercury, is not enough to replace it by the approxima-
tion due to a uniform ring of matter. We need to perform
a software calculation based on elliptical and inclined
orbits, positioning each planet in every moment.
5. Conclusions and Open Comments
1) A first solution is a constant angular precession and
a lineal accumulation along the orbit.
2) Angular precession may oscillate about a mean
value. The magnitude depends on the alternative theo-
retical method we use. There are significant differences
and coincidences between them. In all of them, angular
precession has a non-zero effect in the perihelion neither
the aphelion, nodes where radial velocity is null.
3) The orbital precession produced by the perturbing
potential, involves oscillations with a negative advance
and turns back, opposite to Mercury’s own progress in its
orbit. Any elliptic orbit with eccentricity e < 0.22, would
have the same behaviour with a lead/lag related to the
final/initial precession. However, the final one orbit pre-
cession does not change in any case and is always ex-
actly the expected relativistic one.
4) Eccentricity should have great influence in the
magnitude of oscillations of the angular precession.
5) The astronomical determination of the angular and
orbital precession at each single point of the orbit, has
not been yet achieved, so it is considered that Messenger
spacecraft, now orbiting the planet or the future mission
BepiColombo, should provide an opportunity to perform
it.
6) Close to reach the centenary of the formulation and
first success of General Relativity, there are still some
open issues: Is it right to accept a constant precession?
How large is the magnitude of oscillations if there are
any? Has the orbital precession any turn back? Which of
these theoretic proposals fits on the real trajectory of
Mercury?
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