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Kepler’s Orbits and Special Relativity
in
Introductory Classical Mechanics
Tyler J. Lemmon∗ and Antonio R. Mondragon†
(Dated: April 21, 2016)
Kepler’s orbits with corrections due to Special Relativity are explored using the Lagrangian
formalism. A very simple model includes only relativistic kinetic energy by defining a Lagrangian
that is consistent with both the relativistic momentum of Special Relativity and Newtonian gravity.
The corresponding equations of motion are solved in a Keplerian limit, resulting in an approximate
relativistic orbit equation that has the same form as that derived from General Relativity in the
same limit and clearly describes three characteristics of relativistic Keplerian orbits: precession
of perihelion; reduced radius of circular orbit; and increased eccentricity. The prediction for the
rate of precession of perihelion is in agreement with established calculations using only Special
Relativity. All three characteristics are qualitatively correct, though suppressed when compared to
more accurate general-relativistic calculations. This model is improved upon by including relativistic
gravitational potential energy. The resulting approximate relativistic orbit equation has the same
form and symmetry as that derived using the very simple model, and more accurately describes
characteristics of relativistic orbits. For example, the prediction for the rate of precession of perihelion
of Mercury is one-third that derived from General Relativity. These Lagrangian formulations of the
special-relativistic Kepler problem are equivalent to the familiar vector calculus formulations. In this
Keplerian limit, these models are supposed to be physical based on the likeness of the equations
of motion to those derived using General Relativity. The resulting approximate relativistic orbit
equations are useful for a qualitative understanding of general-relativistic corrections to Keplerian
orbits. The derivation of this orbit equation is approachable by undergraduate physics majors and
nonspecialists whom have not had a course dedicated to relativity.
PACS numbers: 45.20.Jj, 03.30.+p, 45.50.Pk, 04.25.Nx
I. INTRODUCTION
The relativistic contribution to the rate of precession
of perihelion of Mercury is calculated accurately using
General Relativity [1–6]. However, the problem is
commonly discussed in undergraduate and graduate
classical mechanics textbooks, without introduction of
an entirely new, metric theory of gravity. One approach
[7–10] is to define a Lagrangian that is consistent with
both the momentum-velocity relation of Special Relativity
and Newtonian gravity. The resulting equations of motion
are solved perturbatively, and an approximate rate of
precession of perihelion of Mercury is extracted. This
approach is satisfying in that a familiar element of Special
Relativity—relativistic momentum—produces a small
modification to a familiar problem—Kepler’s orbits—and
results in a characteristic of general-relativistic orbits—
precession of perihelion. On the other hand, one must
be content with an approximate rate of precession that
is one-sixth the correct value. Another approach [11–15]
∗ Colorado College† Colorado College; [email protected]
is that of a history lesson and mathematical exercise. A
modification to Newtonian gravity is postulated, resulting
in an equation of motion that is the same as that derived
from General Relativity. The equation of motion is
solved perturbatively, and the correct rate of precession
of perihelion of Mercury is extracted. This method
is satisfying in that the modification to Newtonian
gravity results in the observed value for the relativistic
contribution to perihelic precession. On the other hand,
one must be content with a mathematical exercise, rather
than an understanding of the metric theory of gravity from
which the modification of Newtonian gravity is derived.
Both approaches provide an opportunity for students of
introductory classical mechanics to learn that relativity is
responsible for a small contribution to perihelic precession
and to calculate that contribution.
A review of the approach using only Special Relativity
and an alternative solution of the equations of motion
in a Keplerian limit are presented—resulting in an
approximate relativistic orbit equation. This orbit
equation has the same form as that derived using
General Relativity and clearly describes three relativistic
corrections to Keplerian orbits: precession of perihelion,
reduced radius of circular orbit, and increased eccentricity.
arX
iv:1
012.
5438
v6 [
astr
o-ph
.EP]
20
Apr
201
6
2
The approximate rate of perihelic precession is in
agreement with established calculations using only Special
Relativity. The method of solution makes use of a simple
change of variables and the correspondence principle,
rather than standard perturbative techniques, and is
approachable by undergraduate physics majors.
Two models are considered. A very simple model
(Secs. II and III) consists of relativistic kinetic energy
and unmodified Newtonian gravity. An improved
model (Sec. IV) includes both relativistic kinetic energy
and relativistic modification to Newtonian gravity. A
special-relativistic force is approximated in a Keplerian
limit, resulting in a conservative approximate relativistic
force, from which a relativistic potential energy is
derived—thereby enabling the usage of the Lagrangian
formalism. For both models, the Lagrangian formalism
is demonstrated to be equivalent to the vector calculus
formalism. The keplerian limit and validity of the
approximate relativistic orbit equations is discussed
thoroughly in Sec. VI. These models are supposed to be
physical based on the likeness of the equations of motion
to those derived using General Relativity (Sec. VI B). The
physical relevance of these models is emphasized in Sec. V.
A method of construction of similar models using more
general modifications to Newtonian gravity is discussed,
including a more broadly-defined Keplerian limit that is
useful in the derivation of orbit equations for such models
(App. A and Sec. VI C).
II. RELATIVISTIC KINETIC ENERGY
Conceptually, the simplest relativistic modification to
Kepler’s orbits is to define a Lagrangian with a kinetic
energy term that is consistent with both the momentum-
velocity relation of Special Relativity and the Newtonian
gravitational potential energy [7–9, 16–21];
L = −mc2γ−1 +GMm
r, (1)
where γ−1 ≡√
1− v2/c2, and v2 = r2 + r2θ2. (G is
Newton’s universal gravitational constant, M is the mass
of the sun, and c is the speed of light in vacuum.) The
equations of motion follow from Lagrange’s equations
d
dt
∂L
∂qi− ∂L
∂qi= 0, (2)
where qi ≡ dqi/dt for each of {qi} = {θ, r}. The results
are
d
dt(γr2θ) = 0, (3)
and
γr + γr +GM
r2− γrθ2 = 0. (4)
Using Eq. (3), a relativistic analogue to the Newtonian
equation for conservation of angular momentum per unit
mass is defined
` ≡ γr2θ = constant. (5)
This is used to eliminate the explicit occurrence of θ in
the equation of motion Eq. (4)
γrθ2 =`2
γr3. (6)
Time is eliminated by successive applications of the chain
rule, together with conserved angular momentum [22, 23];
r = − `γ
d
dθ
1
r, (7)
and, therefore,
γr = −γr − `2
γr2
d2
dθ2
1
r. (8)
Substituting Eqs. (6) and (8) into the equation of motion
Eq. (4) results in
`2d2
dθ2
1
r− γGM +
`2
r= 0. (9)
Anticipate a solution of Eq. (9) that is near Keplerian and
introduce the radius of a circular orbit for a nonrelativistic
particle with the same angular momentum, rc ≡ `2/GM .
The result is
d2
dθ2
rc
r+rc
r= 1 + λ, (10)
where λ ≡ γ − 1 is a velocity-dependent correction to
Newtonian orbits due to Special Relativity. The conic
sections of Newtonian mechanics [24, 25] are recovered by
setting λ = 0 (c→∞)
d2
dθ2
rc
r+rc
r= 1, (11)
resulting in the well-known orbit equation
rc
r= 1 + e cos θ, (12)
where e is the eccentricity. Kepler’s orbits are described
by 0 < e < 1.
3
III. KEPLERIAN LIMIT AND ORBIT
EQUATION
The planets of our solar system are described by
near-circular orbits (e � 1) and require only small
relativistic corrections (v/c� 1). Mercury has the largest
eccentricity (e ≈ 0.2), and the next largest is that of
Mars (e ≈ 0.09). Therefore, λ [defined after Eq. (10)] is
taken to be a small relativistic correction to near-circular
orbits of Newtonian mechanics—Keplerian orbits. This
correction is approximated using the first-order series
γ ≈ 1 + 12 (v/c)2, and neglecting the radial component of
velocity v ≈ rθ
λ ≈ 12 (rθ/c)2. (13)
See Sec. VI for a thorough discussion of this approximation.
Using angular momentum Eq. (5) to eliminate θ results
in λ ≈ 12 (`/rc)2(1 + λ)−2, or
λ ≈ 12 (`/rc)2. (14)
The equation of motion Eq. (10) is now expressed
approximately as
d2
dθ2
rc
r+rc
r≈ 1 + 1
2ε(rc
r
)2. (15)
where ε ≡ (GM/`c)2. The conic sections of Newtonian
mechanics, Eqs. (11) and (12), are now recovered by
setting ε = 0 (c → ∞). The solution of Eq. (15) for
ε 6= 0 approximately describes Keplerian orbits with small
corrections due to Special Relativity.
If ε is taken to be a small relativistic correction to
Keplerian orbits, it is convenient to make the change
of variable 1/s ≡ rc/r − 1 � 1. The last term on
the right-hand-side of Eq. (15) is then approximated
as (rc/r)2 ≈ 1 + 2/s, resulting in a linear differential
equation for 1/s(θ)
d2
dθ2
2
εs+
2(1− ε)εs
≈ 1. (16)
The additional change of variable α ≡ θ√
1− ε results in
the familiar form
d2
dα2
sc
s+sc
s≈ 1, (17)
where sc ≡ 2(1− ε)/ε. The solution is similar to that of
Eq. (11)
sc
s≈ 1 +A cosα, (18)
where A is an arbitrary constant of integration. In terms
of the original coordinates
rc
r≈ 1 + e cos κθ, (19)
where
rc ≡ rc1− ε
1− 12ε
(20)
e ≡12εA
1− 12ε
(21)
κ ≡ (1− ε) 12 . (22)
According to the correspondence principle, Kepler’s orbits
[Eq. (12) with 0 < e < 1] must be recovered in the limit
ε → 0 (c → ∞), so that 12εA ≡ e is the eccentricity of
Newtonian mechanics. To first order in ε, Eqs. (20)–(22)
are
rc ≈ rc(1− 12ε) (23)
e ≈ e(1 + 12ε) (24)
κ ≈ 1− 12ε, (25)
so that relativistic orbits in this limit are described
concisely by
rc(1− 12ε)
r≈ 1 + e(1 + 1
2ε) cos (1− 12ε)θ. (26)
When compared to Kepler’s orbits [Eq. (12) with
0 < e < 1], this orbit equation clearly displays three
characteristics of near-Keplerian orbits: precession of
perihelion; reduced radius of circular orbit; and increased
eccentricity. This approximate orbit equation has the
same form as that derived from General Relativity in this
limit [26]
rc(1− 3ε)
r≈ 1 + e(1 + 3ε) cos (1− 3ε)θ. (27)
The equations of motion, Eqs. (3) and (4), are
identical to those derived using the simple force
equation (Newton’s 3rd Law) p = −GMmr/r2, where
p = pr r + pθθ, and {pr, pθ} = {γmr, γmrθ}, verifying
that the unfamiliar relativistic kinetic energy term in
the Lagrangian, T ≡ −mc2γ−1 in Eq. (1), is consistent
with the familiar definition of relativistic momentum
p = γmv. For example, with p = pr r + pθθ, and using
{dr/dt, dθ/dt} = {θθ,−rθ}, the conserved relativistic
angular momentum Eq. (3) is verified
pθ/m =1
r
d
dt(γr2θ) = 0. (28)
IV. RELATIVISTIC GRAVITATIONAL
POTENTIAL ENERGY
The effect of using the special-relativistic γ factor to
define both relativistic kinetic energy and relativistic
gravitational potential energy is explored [27–33]. A
4
relativistic gravitational potential energy is derived from
a conservative approximate gravitational force. In this
Keplerian limit, a Lagrangian formulation of this problem
is demonstrated to be equivalent to the vector calculus
formulation.
A. Vector Calculus Formalism
The central-mass problem including both relativistic
kinetic energy and relativistic gravitational force is
described by the simple force equation
p = −γGMm
r2r, (29)
where p = γmv. The equations of motion are derived
as described in the final paragraph in Sec. III and are
identical to Eqs. (3) and (4), except that the gravitational
force is multiplied by the relativistic γ factor
` ≡ γr2θ = constant, (30)
and
γr + γr + γGM
r2− γrθ2. (31)
An approximate relativistic orbit equation is derived as
described in Secs. II and III
d2
dθ2
rc
r+rc
r= 1 + λ, (32)
where λ ≡ γ2 − 1 ≈ (rθ/c)2. Notice that λ = 2λ, where
λ ≡ γ − 1 ≈ 12 (rθ/c)2 is defined after Eq. (10) and in
Eq. (13). Therefore, the orbit equation is found using the
simple replacement ε→ 2ε in Eq. (26)
rc(1− ε)r
≈ 1 + e(1 + ε) cos (1− ε)θ. (33)
This approximate relativistic orbit equation has the
same form as that derived in Secs. II and III using only
relativistic kinetic energy Eq. (26) and is more accurate—
when compared to that derived from General Relativity
in this same limit Eq. (27).
B. Lagrangian Formalism
An equivalent Lagrangian formulation of this problem
is accomplished by defining a conservative approximate
relativistic gravitational force, from which a corresponding
potential energy is derived. A relativistic gravitational
force is defined as the Newtonian force due to gravity
with the replacement m→ γm
Fg ≡ −γGMm
r2= −GMm
r2(1 + λ), (34)
where λ ≡ γ − 1 is a small correction to Newtonian
gravity due to Special Relativity. Also, relativistic
angular momentum (per unit mass) is defined to be
that of Newtonian mechanics with the replacement
m→ γm, ` ≡ γr2θ. Anticipating near-circular orbits, λ is
approximated using the first-order series γ ≈ 1 + 12 (v/c)2,
neglecting the radial component of velocity v ≈ rθ, and
using angular momentum to eliminate θ, as described in
the first paragraph in Sec. III
λ ≈ 12ε(rc
r
)2
. (35)
The result is a conservative approximate relativistic
gravitational force Eq. (34) that describes a small
correction to Kepler’s orbits due to Special Relativity
Fg ≈ −GMm
r2
[1 + 1
2ε(rc
r
)2]. (36)
This force is integrated to derive an approximate
relativistic gravitational potential energy
U(r) = −GMm
r
[1 + 1
6ε(rc
r
)2]. (37)
The Lagrangian
L = −mc2γ−1 − U(r) (38)
results in the equations of motion
d
dt(γr2θ) = 0, (39)
and
γr + γr +GM
r2
[1 + 1
2ε(rc
r
)2]− γrθ2. (40)
The first equation Eq. (39) verifies the definition of
relativistic angular momentum, and the second equation
Eq. (40) includes a small relativistic correction to Kepler’s
orbits due to Special Relativity. Compare these equations
of motion to those derived using only relativistic kinetic
energy in Sec. II, Eqs. (3) and (4). A relativistic orbit
equation is derived as described in Secs. II and III
d2
dθ2
rc
r+rc
r= 1 + λ, (41)
where λ ≡ γ[1 + 12ε(rc/r)
2] − 1 is a small correction to
Keplerian orbits due to Special Relativity. Using the
first-order series γ ≈ 1 + 12 (v/c)2, neglecting the radial
component of velocity v ≈ rθ, and keeping terms first
order in ε results in λ ≈ 2λ, where λ ≡ γ−1 ≈ 12 (rθ/c)2 is
defined after Eq. (10) and in Eq. (13). Therefore, the orbit
equation is found using the simple replacement ε→ 2ε in
5
Eq. (26), thereby reproducing the orbit equation Eq. (33)
derived using the vector calculus formalism in Sec. IV A
rc(1− ε)r
≈ 1 + e(1 + ε) cos (1− ε)θ. (42)
Characteristics of near-Keplerian orbits are most easily
understood by comparing the approximate relativistic
orbit equations to that derived from the nonrelativistic
Kepler problem Eq. (12). For this purpose, it is convenient
to express the orbit equation as
rc
r≈ 1 + e cos κθ, (43)
where, using the simple replacement ε→ 2ε in Eqs. (20)–
(25),
rc ≡ rc1− 2ε
1− ε ≈ rc(1− ε) (44)
e ≡ e 1
1− ε ≈ e(1 + ε) (45)
κ ≡ (1− 2ε)12 ≈ 1− ε. (46)
V. CHARACTERISTICS OF
NEAR-KEPLERIAN ORBITS
The approximate relativistic orbit equation Eq. (42)
predicts a shift in perihelion through an angle
∆θ ≡ 2π(κ−1 − 1) ≈ 2πε (47)
per revolution. This prediction is twice that derived
using the standard approach [7–9, 34] to incorporating
Special Relativity into the Kepler problem, and is
compared to observations assuming that relativistic and
Keplerian angular momenta are approximately equal. For
a Keplerian orbit [24, 25] `2 = GMa(1 − e2), where
G = 6.670× 10−11 m3/kg·s2, M = 1.989× 1030 kg is the
mass of the Sun, and a and e are the semimajor axis
and eccentricity of the orbit, respectively. Therefore, the
relativistic correction defined after Eq. (15),
ε ≈ GM
c2a(1− e2), (48)
is largest for planets closest to the Sun and for
planets with very eccentric orbits. For Mercury
[35, 36] a = 5.79× 1010 m and e = 0.2056, so that
ε ≈ 2.66 × 10−8. (The speed of light is taken to
be c2 = 8.987554× 1016 m2/s2.) According to Eq. (47),
Mercury precesses through an angle
∆θ ≈ 2πGM
c2a(1− e2)= 1.67× 10−7 rad (49)
per revolution. This angle is very small and is usually
expressed cumulatively in arc seconds per century. The
rsinθ/r c
r cos θ/rc
KeplerEinstein
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5 1 1.5
FIG. 1. A relativistic orbit in a Keplerian limit (solid)
Eq. (42) is compared to a Keplerian orbit (dotted) Eq. (12)
with the same angular momentum. Precession of perihelion
is one characteristic of relativistic orbits and is illustrated
here for 0 ≤ θ ≤ 6π. Precession of perihelion is also predicted
by General Relativity Eq. (27) in greater magnitude. This
characteristic of relativistic orbits is exaggerated by both the
choice of eccentricity (e = 0.25) and relativistic correction
parameter (ε = 0.1) for purposes of illustration. Precession is
present for smaller (non-zero) reasonably chosen values of e
and ε as well. (The same value of e is chosen for both orbits.)
orbital period of Mercury is 0.24085 terrestrial years, so
that
∆Θ ≡ 100 yr
0.24085 yr× 360× 60× 60
2π×∆θ (50)
≈ 14.3 arcsec/century. (51)
Precession, as predicted by Special Relativity is illustrated
in Fig. 1.
The general-relativistic (GR) treatment of this problem
results in a prediction of 43.0 arcsec/century [26, 34–55],
and agrees with the observed precession of perihelia
of the inner planets [36–45, 56–59]. Historically, this
contribution to the precession of perihelion of Mercury’s
orbit precisely accounted for the observed discrepancy,
serving as the first triumph of the general theory of
relativity [1–4]. The present approach, using only Special
Relativity, accounts for approximately one-third of the
observed discrepancy Eq. (51).
The approximate relativistic orbit equation, Eq. (42)
with e = 0, predicts a reduced radius of circular orbit—
when compared to that of Newtonian mechanics, Eq. (12)
with e = 0. This characteristic is not discussed in the
6
rc
Rc
(rc/`)
2Veff
r/rc
Newtonian Mechanics
General Relativity
−0.75
−0.5
−0.25
0
0.25
0 0.5 1 1.5 2 2.5 3
FIG. 2. The effective potential commonly defined in the
Newtonian limit to General Relativity (dashed) Eq. (52) is
compared to that derived from Newtonian mechanics (solid)
with the same angular momentum. The vertical (dotted)
lines identify the radii of circular orbits, Rc and rc, as
calculated using General Relativity and Newtonian mechanics,
respectively. General Relativity predicts a smaller radius of
circular orbit, when compared to that predicted by Newtonian
mechanics. This reduction in radius of a circular orbit Eq. (53),
Rc − rc ≈ −3εrc, is three times that predicted by the present
treatment using only Special Relativity Eq. (44), rc−rc ≈ −εrc.A curve representing an effective potential including small
corrections predicted by Special Relativity is expected to be
nearly identical to that for Newtonian mechanics (solid), with a
slightly smaller radius of circular orbit, rc. The value ε = 0.06
is chosen for purposes of illustration. Reduction in radius
of circular orbit is present for smaller (non-zero) reasonably
chosen values of ε as well.
standard approach to incorporating Special Relativity
into the Kepler problem, but is consistent with the GR
description. An effective potential naturally arises in the
GR treatment of the central-mass problem [26, 35, 39, 40,
42, 45, 52, 53, 60, 61],
Veff ≡ −GM
r+
`2
2r2− GM`2
c2r3, (52)
that reduces to the Newtonian effective potential in
the limit c → ∞. In the Keplerian limit, the GR
angular momentum per unit mass ` is also taken to be
approximately equal to that for a Keplerian orbit [26, 35–
42, 45, 53, 61, 62]. Minimizing Veff with respect to r
results in the radius of a stable circular orbit,
Rc =1
2rc +
1
2rc
√1− 12ε ≈ rc(1− 3ε), (53)
rsinθ/r c
r cos θ/rc
KeplerEinstein
−1.5
−1
−0.5
0
0.5
1
1.5
−2 −1.5 −1 −0.5 0 0.5 1
FIG. 3. A relativistic orbit in a Keplerian limit (solid)
Eq. (42) is compared to a Keplerian orbit (dotted) Eq. (12)
with the same angular momentum. Precession of perihelion
has been removed from the relativistic orbit equation, Eq. (42)
with (1 − ε)θ → θ, to emphasize two other characteristics
of relativistic orbits—reduced orbital radii and increased
eccentricity. These two characteristics are also predicted
by General Relativity Eq. (27) in greater magnitude. These
characteristics of relativistic orbits are exaggerated by both the
choice of eccentricity (e = 0.45) and the relativistic correction
parameter (ε = 0.15) for purposes of illustration. Reduced
orbital radii and increased eccentricity are present for smaller
(non-zero) reasonably chosen values of e and ε as well. (The
same value of e is chosen for both orbits.)
so that the radius of circular orbit is predicted to be
reduced, Rc − rc ≈ −3εrc. (There is also an unstable
circular orbit, as illustrated in Fig. 2.) This reduction
in radius of a circular orbit is three times that predicted
by the present treatment using only Special Relativity
Eq. (44), for which rc − rc ≈ −εrc. Reduced size of an
orbit, as predicted by Special Relativity, is illustrated in
Fig. 3.
Many discussions of the GR effective potential Eq. (52)
emphasize relativistic capture. The 1/r3 term in Eq. (52)
contributes negatively to the effective potential, resulting
in a finite—rather than infinite—centrifugal barrier and
affecting orbits very near the central mass (large-velocity
orbits), as illustrated in Fig. 2. This purely GR effect is
not expected to be described by the approximate orbit
equation Eq. (42), which is derived using only Special
Relativity and implicitly assumes orbits very far from the
central mass (small-velocity orbits).
An additional characteristic of relativistic orbits is that
7
of increased eccentricity. The relativistic orbit equation
Eq. (42) predicts increased eccentricity, when compared
to a Keplerian orbit Eq. (12) with the same angular
momentum Eq. (45), e − e ≈ εe. This characteristic
of relativistic orbits is not discussed in the standard
approach to incorporating Special Relativity into the
Kepler problem, but is consistent with the GR description.
The GR orbit equation in this Keplerian limit Eq. (27)
predicts an increase in eccentricity e− e ≈ 3εe, which is
three times that predicted by the present treatment using
only Special Relativity. Increased eccentricity of an orbit,
as predicted by Special Relativity, is illustrated in Fig. 3.
VI. DISCUSSION
The debate concerning the usage of relativistic inertial
mass and relativistic gravitational mass [63–70] is
irrelevant in the present context due to the fundamental
incompatibility of Special Relativity and gravitation
[71]. Accordingly, language describing the replacement
m → γm is intentionally omitted. Instructors may
feel more comfortable using the equivalent replacements
v → γv to introduce relativistic momentum, and G→ γG
to introduce relativistic gravitational force. Rather,
Lagrangians are constructed using familiar elements of
Special Relativity for the purpose of simulating general-
relativistic orbital effects using only Special Relativity
in a well-defined Keplerian limit. One purpose of
this presentation is to provide students with interesting
and tractable problems that arise from small special-
relativistic modifications to a familiar problem—Kepler’s
orbits, the solutions of which provide a qualitative
understanding of corrections to Kepler’s orbits due to
General Relativity. In this Keplerian limit, these models
are supposed to be physical based on the likeness of
the equations of motion to those derived using General
Relativity [26]. Another purpose is to present methods
by which similar models may be constructed and solved.
A. Domain of Validity
The following discussion concerning the derivation,
validity, and scope of the approximate special-relativistic
orbit equations follows the presentation in Secs. II and III,
in which only relativistic kinetic energy is included. The
arguments and conclusions also apply to the presentation
in Sec. IV, in which both relativistic kinetic energy and
relativistic gravitational potential energy are included.
The approximate relativistic orbit equation Eq. (26)
provides small corrections to Kepler’s orbits Eq. (12) due
to Special Relativity. A systematic verification may be
carried out by substituting Eq. (26) into Eq. (15), and
only keeping terms of orders e, ε, and eε. The domain of
validity is expressed by subjecting the solution Eq. (26)
to the condition
rc
r− 1� 1 (54)
for the smallest value of r. Evaluating the orbit equation
Eq. (42) at perihelion rp results in
rc
rp=
1 + e(1 + 12ε)
1− 12ε
. (55)
Substituting this into Eq. (54) results in the domain of
validity
e(1 + 12ε) + ε� 1. (56)
Therefore, the relativistic eccentricity e = e(1 + 12ε)� 1,
and Eq. (26) is limited to describing relativistic corrections
to near-circular (Keplerian) orbits. Also, the relativistic
correction ε� 1, and thus the orbit equation Eq. (26) is
valid only for small relativistic corrections.
The correction to Keplerian orbits due to Special
Relativity λ ≡ γ − 1 [defined after Eq. (10) and in
Eq. (13)] is approximated using the first-order series
γ ≈ 1 + 12 (v/c)2, and neglecting the radial component of
the velocity v ≈ rθ. Neglecting the radial component of
velocity in the relativistic correction λ is consistent with
the assumption of approximately Keplerian (near-circular)
orbits, and is supported by the condition rc/r − 1 � 1
preceding Eq. (16). It is emphasized that the radial
component of velocity is neglected only in the relativistic
correction λ; it is not neglected in the derivation of the
relativistic equation of motion Eq. (10). That there is
no explicit appearance of r in the relativistic equation
of motion, other than in the definition of γ, is due to a
fortunate cancellation after Eq. (8).
The presentation in Sec. IV includes relativistic
gravitational potential energy using the replacement
m→ γm in the Newtonian gravitational force. Although
the use of relativistic gravitational mass in the special-
relativistic Kepler problem is discouraged by some
authors [68], there are several useful results in the
Keplerian limit. This relativistic gravitational force is
approximated, resulting in a conservative force, from
which an approximate relativistic potential energy is
derived—thereby enabling the use of the Lagrangian
formalism. The resulting approximate relativistic orbit
equation Eq. (42) is more accurate, when compared
to that derived from General Relativity in the same
limit Eq. (27), than that derived using only relativistic
kinetic energy Eq. (26). Specifically, this more accurate
orbit equation demonstrates that—in this Keplerian
8
limit—the only consequence of neglecting relativistic
gravitational potential energy is that corrections due to
Special Relativity are decreased by a factor of two.
B. Structure of the Models
These models are appealing because they produce
equations of motion that are similar to those derived
using General Relativity. Compare the special-relativistic
(SR) equation of motion, Eq. (9) in Sec. II [using
γGM = GM +GM(γ − 1)], to the general-relativistic
(GR) equation of motion, Eq. (10) in Ref. 26,
(SR) `2d2
dθ2
1
r−GM +
`2
r−GM(γ − 1) = 0 (57)
(GR) ¯2 d2
dϕ2
1
r−GM +
¯2
r−GM
( 3¯2
c2r2
)= 0. (58)
The terms in parentheses describe corrections to
Newtonian orbits due to Special Relativity Eq. (57)
and General Relativity Eq. (58); these terms are zero
in the Newtonian limit c → ∞. Note that both of
these equations are exact in the sense that they are
derived directly from Lagrangians without making any
approximations. The SR equation of motion Eq. (57)
has a correction to Newtonian orbits GM(γ − 1). Using
the first-order expansion γ ≈ 1 + 12 (v/c)2, neglecting the
radial component of velocity v ≈ rθ, and using angular
momentum to eliminate θ results in
GM(γ − 1) ≈ GM( `2
2c2r2
). (59)
Aside from a constant, this term is identical to the
term that naturally arises in the GR derivation Eq. (58),
and is directly responsible for the 12ε corrections that
appear in the approximate relativistic orbit equation
Eq. (26). The factor of γ that is responsible for
this term is a direct result of the γ that appears
in the definition of angular momentum (per unit
mass) Eq. (5) ` ≡ γr2θ. Fundamentally, this angular
momentum has a γ factor due to the definition
of relativistic kinetic energy T ≡ −mc2γ−1. The GR
angular momentum is found to be simply ¯ ≡ r2ϕ
[Eq. (4) in Ref. 26] because that problem is solved using
proper time τ rather than coordinate time t, so that
ϕ ≡ dϕ/dτ . A transformation to coordinate time results
in ¯≡ γr2ϕ, where γ ≡ dt/dτ = [1 + 2V (r)/c2]−1/2, and
V (r) ≡ −GM/r is the Newtonian gravitational potential.
The vector calculus formalism in Sec. IV A includes a γ
factor in the definition of the relativistic gravitational
force. The γ factor from the relativistic angular
momentum propagates through the derivation of the
orbit equation as described in the preceding paragraph,
resulting in a correction to Newtonian orbits GM(γ2− 1).
Using the first-order expansion γ2 ≈ 1+(v/c)2, neglecting
the radial component of velocity v ≈ rθ, and using angular
momentum to eliminate θ results in
GM(γ2 − 1) ≈ GM( `2
c2r2
). (60)
This is exactly twice the correction term that results
from using only relativistic kinetic energy Eq. (59), and is
directly responsible for the ε corrections that appear in the
approximate relativistic orbit equation, Eqs. (33) and (42).
Using the Lagrangian formalism in Sec. IV B, there is an
additional requirement that the relativistic correction
to the Newtonian gravitational force does not depend
explicitly on θ or θ, so that the simple γ factor appears
in the definition of angular momentum.
More generally, any function f(γ) may be used in the
definition of relativistic gravitational force, provided a
first-order expansion of the form γf(γ) − 1 ≈ α(v/c)2
exists, where α > 0 is a constant. For example, choosing
a simple power-law dependence f(γ) = γn, so that
Fg ≡ γnGMm/r2, results in an approximate relativistic
orbit equation identical to Eq. (26) with the replacement
ε → (n + 1)ε. Notice the physical condition n ≥ 0 that
is necessary to insure the proper direction of precession,
reduced orbital radii, and increased eccentricity.
Using these types of models, the Keplerian limit is
defined precisely by only approximating the correction
term that represents a modification of the Keplerian
equation of motion. The approximation of this correction
term consists of: a series expansion to first order in (v/c)2;
neglecting the higher-order r2 term; and making a change
of variable subject to the condition rc/r − 1 ≡ 1/s� 1,
allowing the linearization (rc/r)n ≈ 1 + n/s.
With appropriate conditions and approximations,
modifications to Newtonian gravity that depend, more
generally, on velocity and radial coordinate may be used.
This is most easily described by example. A toy model is
presented in App. A and discussed in the following section
that describes methods—including a more broadly-defined
Keplerian limit—that are useful for solving problems with
more general modifications to Newtonian gravity.
C. A Toy Model
There are many attempts to motivate a physically
appropriate modification to Newtonian gravity in the
literature [27–33]. The possibility of simply replacing
m→ γm in the Newtonian gravitational potential energy
and using the Lagrangian formalism is explored in App. A
and discussed in this section. This toy model is useful
for describing methods for solving problems with more
9
general modifications to Newtonian gravity, for which
a more broadly-defined Keplerian limit is needed. The
Lagrangian for this model is
L = −mc2γ−1 + γGMm
r. (61)
Lagrange’s equations result in an abstruse set of
differential equations, Eqs. (A2) and (A3), that—using
an appropriate approximation—reduce to the equations
of motion derived in Sec. IV, Eqs. (30) and (31), thereby
reproducing the orbit equation, Eqs. (33) and (42)
rc(1− ε)r
≈ 1 + e(1 + ε) cos (1− ε)θ. (62)
Another solution to the differential equations,
Eqs. (A2) and (A3), is derived using a more broadly-
defined Keplerian limit. For near-circular approximately
Newtonian orbits, r and γ are very slowly varying
functions, so that the higher-order terms γr and
(γ/c)2V (r)γr2/r are taken to be negligible. The result
is the Newtonian equation of motion with a relativistic
correction term Eq. (A10)
˜2 d2
dθ2
1
r−GM +
˜2
r−GMλ = 0, (63)
where
λ ≡ γ2[1− (γ/c)2V (r)
]− 1, (64)
˜≡ γr2θ[1− (γ/c)2V (r)
]is a constant of motion, and
V (r) ≡ −GM/r is the Newtonian gravitational potential.
Compare this equation of motion to Eqs. (57) and (58).
This equation of motion has a small relativistic correction
to Keplerian orbits Eq. (A12)
GMλ ≈ GM[ ˜2
c2r2− V (r)
c2
]. (65)
Compare this to the correction terms used in the two
simple models, Eqs. (59) and (60). The corresponding
orbit equation in the Keplerian limit Eq. (A16),
rc(1− 2ε)
r≈ 1 + e(1 + ε) cos (1− 3
2 ε)θ, (66)
does not have the symmetry of those derived using the
two simple models, Eqs. (26) and (42). The methods and
approximations used to derive this orbit equation may
be applied to other, more physical, models that include
relativistic corrections to Kepler’s orbits.
VII. CONCLUSION
The Lagrangian formalism is useful for describing small
relativistic corrections to Kepler’s orbits. The simplest
model includes only relativistic kinetic energy using the
replacement m→ γm in the Newtonian linear momentum,
p = γmv. A solution to the corresponding equations of
motion in a Keplerian limit results in an approximate
relativistic orbit equation Eq. (26) that has the same
form as that derived from General Relativity in this limit
Eq. (27) and is easily compared to that describing Kepler’s
orbits Eq. (12). This form is that of elliptical orbits
of Newtonian mechanics with corrections to radius and
eccentricity, and exhibiting precession. Specifically, the
approximate relativistic orbit equation clearly describes
three characteristics of relativistic orbits: precession of
perihelion; reduced radius of circular orbit; and increased
eccentricity. The predicted rate of precession of perihelion
of Mercury is in agreement with that of established
calculations using only Special Relativity. Each of these
characteristics of relativistic Keplerian orbits is exactly
one-sixth of the corresponding correction described by
General Relativity in this limit—providing a qualitative
description of corrections to Keplerian orbits due to
General Relativity.
This simple model is improved upon by including
relativistic gravitational force using the replacement
m → γm in Newtonian gravity, Fg = γGMm/r2. An
approximation consistent with the Keplerian limit results
in a conservative force, from which a relativistic potential
energy is derived that is useful in a Lagrangian formulation
of the special-relativistic Kepler problem. A solution of
the corresponding equations of motion in a Keplerian
limit results in an approximate relativistic orbit equation
Eq. (42) that has the same form as that derived using
the simplest model Eq. (26), thereby describing the same
three characteristics of relativistic orbits. Each of these
characteristics of relativistic Keplerian orbits is exactly
one-third of the corresponding correction described by
General Relativity in this limit Eq. (27).
The Lagrangian formalism applied to the special-
relativistic Kepler problem is instructive, providing
several challenges appropriate for an introductory classical
mechanics course, including: solve Newton’s force
equation using vector calculus to verify the unfamiliar
relativistic kinetic energy term in the Lagrangian—as
outlined in the last paragraph of Sec. III and in Sec. IV A;
derive a potential energy function from a conservative
force; apply Lagrange’s equations to derive the conserved
relativistic angular momentum and equation of motion;
and transform and solve a differential equation to derive
an approximate relativistic orbit equation in terms
of planar coordinates. This approach also provides
an opportunity to use less familiar problem solving
strategies, including: variable transformations to cast the
differential equation into familiar form; approximation
10
methods that simplify the differential equation; and
usage of the correspondence principle to identify a
constant of integration. Most importantly, students
are rewarded with a clear understanding that a small
relativistic modification to a familiar problem results in
an approximate relativistic orbit equation that clearly
demonstrates that relativity is responsible for a small
contribution to perihelic precession, and the satisfaction
of calculating that contribution.
These models are appealing because they are easy to
motivate and, most importantly, they produce equations
of motion that are similar to those derived using General
Relativity, as discussed in Sec. VI B. A larger class
of models may be solved using similar methods and
approximations. This is demonstrated using a toy model
in App. A and in Sec. VI C, wherein a more broadly-
defined Keplerian limit is described. Exact solutions of
the special-relativistic Kepler problem require a thorough
understanding of special relativistic mechanics [72, 73] and
are, therefore, inaccessible to many undergraduate physics
majors. The present approach and method of solution is
understandable to nonspecialists, including undergraduate
physics majors whom have not had a course dedicated to
relativity.
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Appendix A: A Toy Model
The possibility of including relativistic gravitationalpotential by simply replacing m→ γm in the Newtoniangravitational potential energy and using the Lagrangianformalism is explored. This toy model is useful fordescribing methods for solving problems with moregeneral modifications to Newtonian gravity, for whicha more broadly-defined Keplerian limit is needed. TheLagrangian for this model is
L = −mc2γ−1 + γGMm
r. (A1)
Lagrange’s equations result in an abstruse set ofdifferential equations
d
dt
{γr2θ
[1− (γ/c)2V (r)
]}= 0, (A2)
and
γr[1− (γ/c)2V (r)
]+ γr
[1− 3(γ/c)2V (r)
]+ γ
GM
r2− γrθ2
[1− (γ/c)2V (r)
]+ (γ/c)2V (r)γ
r2
r= 0,
(A3)
where V (r) ≡ −GM/r is the Newtonian gravitationalpotential. Mercury’s orbit is approximately circularwith radius of the same order of magnitude as itssemimajor axis r ∼ a ≈ 5.79 × 1010 m, and its velocityis very small when compared to the speed of light, sothat (γ/c)2V (r) ∼ V (a)/c2 ∼ −10−8. Ignoring termsproportional to (γ/c)2V (r) results in the equations ofmotion derived in Sec. IV, Eqs. (30) and (31), therebyreproducing the orbit equation, Eqs. (33) and (42)
rc(1− ε)r
≈ 1 + e(1 + ε) cos (1− ε)θ. (A4)
Another solution to the equations of motion,Eqs. (A2) and (A3), is derived using a more broadly-defined Keplerian limit. For near-circular approximately
Newtonian orbits, r and γ are very slowly varyingfunctions, so that the higher-order terms γr and(γ/c)2V (r)γr2/r are taken to be negligible. [ForMercury’s orbit (γ/c)2V (a)γ/a ∼ −10−18 m−1.]
˜≡ γr2θ[1− (γ/c)2V (r)
]= constant, (A5)
and
γr[1− (γ/c)2V (r)
]+ γ
GM
r2
− γrθ2[1− (γ/c)2V (r)
]≈ 0.
(A6)
Conservation of angular momentum Eq. (A5) is used toeliminate the explicit occurrence of θ in the equation ofmotion Eq. (A6)
γrθ2[1− (γ/c)2V (r)
]=
˜2
γr3 [1− (γ/c)2V (r)]. (A7)
Time is eliminated by successive applications of the chainrule, together with the conserved angular momentum;
r = −˜
γ[1− (γ/c)2V (r)]
d
dθ
1
r, (A8)
and, therefore, (again taking γr to be negligible)
γr[1−(γ/c)2V (r)] ≈ −˜2
γ[1− (γ/c)2V (r)]r2
d2
dθ2
1
r. (A9)
Substituting Eqs. (A7) and (A9) into the equation ofmotion Eq. (A6) results in
˜2 d2
dθ2
1
r− γ2[1− (γ/c)2V (r)]GM +
˜2
r= 0. (A10)
Anticipate a solution of Eq. (A10) that is near Keplerianand introduce the radius of a circular orbit for anonrelativistic particle with the same angular momentum,rc ≡ ˜2/GM . The result is
d2
dθ2
rc
r+rc
r= 1 + λ, (A11)
where λ ≡ γ2[1 − (γ/c)2V (r)] − 1 is a correction toNewtonian orbits due to Special Relativity. [Tildenotation is used to emphasize that quantities dependon the exact conserved quantity ˜ defined in Eq. (A5),rather than the angular momentum ` defined in Eq. (30).]
The orbit equation is derived following the methoddescribed in Sec. III. The correction term λ isapproximated by expanding to first-order in 1/c2,neglecting the radial component of velocity, and usingangular momentum to eliminate θ
λ ≈ (˜/rc)2 − V (r)/c2. (A12)
The equation of motion Eq. (A11) is now expressedapproximately as
d2
dθ2
rc
r+rc
r≈ 1 + ε
rc
r+ ε( rc
r
)2, (A13)
13
where ε ≡ (GM/˜c)2. An orbit equation is derived, asdescribed in Sec. III. The equation of motion Eq. (A13)is linearized using rc/r = 1 + 1/s, and (rc/r)
2 ≈ 1 + 2/s.The additional change of variable α ≡ θ
√1− 3ε results
in the familiar differential equation
d2
dα2
sc
s+sc
s≈ 1, (A14)
where sc ≡ (1− 3ε)/(2ε). The solution is similar to thatof Eq. (11)
sc
s≈ 1 +A cosα, (A15)
where A is an arbitrary constant of integration. In termsof the original coordinates, and defining e ≡ 2εA, an orbitequation in the Keplerian limit is described concisely by
rc(1− 2ε)
r≈ 1 + e(1 + ε) cos (1− 3
2 ε)θ. (A16)
This orbit equation does not have the symmetry of thosederived using the two simple models, Eqs. (26) and (42).The methods and approximations used to derive this orbitequation may be applied to other, more physical, modelsthat include relativistic corrections to Kepler’s orbits. Seethe discussions in Sec. VI B and Sec. VI C.