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Central-forces perturbations of conical orbits
Maurizio M. DEliseo
Osservatorio S.Elmo - Via A.Caccavello 22, 80129 Napoli
Italy
The relativistic precession of a planetary orbit is treated from
the point of view of the analyticgeometry of a conic on the
plane.
I. THE UNPERTURBED ORBIT
We need the vectorial expression of the conic describedby an
unperturbed planet (considered as a test-body)around a star located
at the origin. If we let r = r r be the position vector, where r =
r r , we have therst-order system
dr dt
= v , (1)
dv dt
= r 2
r , (2)
where = GM is the standard gravitational parameterof the star.
Operating on Eq. (2) by r , we get
r dv dt
=ddt
(r v ) =
r 2r r = 0 , (3)
and integrating,
r v = h = const . (4)Since
(r v ) r = 0 , (5)the motion takes place in the plane h r = 0.
We willmatch this plane with the xy-plane. By considering
theinnitesimal triangle formed by the vectors r and dr , wesee that
|r v | is twice the area swept out by the radiusvector per unit
time. In polar coordinates, twice the areaof the triangle swept out
in time dt and per unit time arerespectively
r ( r d), r 2 , (6)
so we have
h = h h = r 2 h . (7)
To get the orbit, we need one more integral of motion:the
Laplace integral. We consider the time derivative of the cross
product
ddt
(v h ) =dv dt h =
hr 2
r h = = dr dt
, (8)
which may be integrated to give the Laplace integral
e = v h r , (9)where e is a constant vector. The factor is
needed toget a natural geometric interpretation for e , namely
that
being the eccentricity vector, directed toward the perihe-lion,
and whose length is the usual scalar eccentricity e.The
eccentricity vector is situated, as one can see, in theplane of
motion. So, we have constantly a closed vectortriangle formed by
the variable vectors v h , r andthe constant vector e . The orbit
is obtained dottingEq. (9) with r
e r = ( v h ) r r , (10)but
r (v h ) = ( r v ) h = h h = h2 ,
and so we get
h h
= r r + e r . (11)h 2
= r + er cos( ). (12)
where we have assumed as reference direction for anglesthe
positive x-axis, and denoted with the xed angleformed by e .
Equation (12) is the polar expression of themagnitude of the radius
vector from which, comparingwith the polar equation of a conic, we
deduce that h 2 =a(1
e2 ), being a the semimajor axis of the orbit. We
will assume that e < 0, that is the conic is an
ellipse.Notice, in particular, that r is comprised between
a(1 e) r a(1 + e), (13)and that the minimum occurs at the point
of orbit when = .
II. THE PERTURBED ORBIT
We study now the effects of a perturbing force f whenthe test
body is initially moving on a conic described byEq. (11). The
method of variation of the arbitrary con-stants assumes that the
motion happens always on anorbit of the same type, but whose
constants h, e slowlyvary in time in a form depending from the
nature of theperturbing force. The problem is thus shifted from
thesolution of the unperturbed equation of motion to thesolution of
the differential equations related to these un-known functions. The
values they assume at each instantt identify the osculating orbit,
that is the orbit the planetwould follow if the perturbing force in
this same instantwas put equal to zero. From the point of view of
the dif-ferential equation of motion, the values of r (t) and r
(t)
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at a particular instant t can be seen as the initial valuesof
the unperturbed conic the body would follow, and sor and r are the
same at any time for the two paths, thereal and the elliptical.
Only the second derivative willreveal the presence of the
disturbing force. We introduceso a derivative operator /dt with the
property that
r
dt = 0
v
dt =f
, (14)and, consequently, we have also
r dt
= 0 ,r dt
= 0 . (15)
Let rst submit h to the action of this operator. Wehave, by Eq.
(11)
hdt
=dt
(r v ) = r v dt
= r f (16)After we have, by Eq. (9)
edt
=1
v dt h + v
hdt
=1
f h + v hdt
.
If the the perturbing force is central, that is of the typef = f
( r ) r , we have
hdt
= r f = f (r r ) = 0 , (17)edt
=fh
( r h ) = . (18)
A consequence of Eq.() is that
2
dt= 2
dt
= 0 , (19)
2
dt=
dt
[a(1 e2 )] = (1 e
2 )adt 2ae
edt
= 0 ,(20)
and by separating the variables
a
adt
=2e
1 e2edt
= h (21)
where h is a constant. Now, a and e are two independentconstants
in the unperturbed motion so, for continuityreasons, we can put h =
0.
We will suppose f ( r ) to be of the form
f ( r ) = k
r 5(22)
III. THE VARIABLE PHASE
To nd higher order corrections to the relativistic pre-cession
formula, we must nd more and more approx-imate solutions to the Eq.
( ?? ), expressing bound pe-riodical motions. Equation ( ?? )
indicates two possiblealternative on the method one can follow to
achieve thisresult. We can act by searching the frequency or
thephase variations. The study of the phase is more sim-ple,
because it leads directly to the perihelion formulawithout the need
of any nal transformation.
Electronic address: [email protected] M. M. DEliseo, The
rst-order orbital equation, Am. J.
Phys. 75 , 352 (2007),2 A. dInverno, Introducing Einsteins
Relativity , Oxford Uni-
verity Press, Oxford, 1992.
3 A. H. Nayfeh, Perturbation Methods , Interscience, NewYork,
1973.
4 J.Kevorkian, J. D. Cole, Perturbation Methods in Applied
Mathematics Springer-Verlag, New York, 1981.
