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Kepler's Orbits and Special Relativity in Introductory Classical Mechanics

Kepler's Orbits and Special Relativity in Introductory Classical Mechanics

Kepler’s Orbits and in Introductory Classical

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 by defining a Lagrangian that is consistent with both the relativistic of Special Relativity and Newtonian . 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 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 . 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 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 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 is that of a history lesson and mathematical exercise. A modification to Newtonian gravity is postulated, resulting The relativistic contribution to the rate of precession in an equation of motion that is the same as that derived of perihelion of Mercury is calculated accurately using from General Relativity. The equation of motion is General Relativity [1–6]. However, the problem is solved perturbatively, and the correct rate of precession commonly discussed in undergraduate and graduate of perihelion of Mercury is extracted. This method textbooks, without introduction of is satisfying in that the modification to Newtonian an entirely new, metric theory of gravity. One approach gravity results in the observed value for the relativistic [7–10] is to define a Lagrangian that is consistent with contribution to perihelic precession. On the other hand, both the momentum- relation of Special Relativity one must be content with a mathematical exercise, rather and Newtonian gravity. The resulting equations of motion than an understanding of the metric theory of gravity from are solved perturbatively, and an approximate rate of which the modification of Newtonian gravity is derived. precession of perihelion of Mercury is extracted. This Both approaches provide an opportunity for students of arXiv:1012.5438v6 [astro-ph.EP] 20 Apr 2016 approach is satisfying in that a familiar element of Special introductory classical mechanics to learn that relativity is Relativity—relativistic momentum—produces a small responsible for a small contribution to perihelic precession modification to a familiar problem—Kepler’s orbits—and and to calculate that contribution. results in a characteristic of general-relativistic orbits— precession of perihelion. On the other hand, one must A review of the approach using only Special Relativity be content with an approximate rate of precession that and an alternative solution of the equations of motion is one-sixth the correct value. Another approach [11–15] 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 ∗ Colorado College corrections to Keplerian orbits: precession of perihelion, † Colorado College; [email protected] reduced radius of circular orbit, and increased eccentricity. 2

The approximate rate of perihelic precession is in where q˙i dqi/dt for each of qi = θ, r . The results ≡ { } { } agreement with established calculations using only Special are Relativity. The method of solution makes use of a simple d change of variables and the correspondence principle, (γr2θ˙) = 0, (3) dt rather than standard perturbative techniques, and is and approachable by undergraduate physics majors. GM γr¨ +γ ˙r˙ + γrθ˙2 = 0. (4) Two models are considered. A very simple model r2 − (Secs.II andIII) consists of relativistic Using Eq. (3), a relativistic analogue to the Newtonian and unmodified Newtonian gravity. An improved equation for conservation of per unit model (Sec.IV) includes both relativistic kinetic energy is defined and relativistic modification to Newtonian gravity. A special-relativistic is approximated in a Keplerian ` γr2θ˙ = constant. (5) limit, resulting in a conservative approximate relativistic ≡ force, from which a relativistic potential energy is This is used to eliminate the explicit occurrence of θ˙ in derived—thereby enabling the usage of the Lagrangian the equation of motion Eq. (4) formalism. For both models, the Lagrangian formalism `2 is demonstrated to be equivalent to the γrθ˙2 = . (6) formalism. The keplerian limit and validity of the γr3 approximate relativistic orbit equations is discussed Time is eliminated by successive applications of the chain thoroughly in Sec.VI. These models are supposed to be rule, together with conserved angular momentum [22, 23]; physical based on the likeness of the equations of motion ` d 1 to those derived using General Relativity (Sec.VIB). The r˙ = , (7) physical relevance of these models is emphasized in Sec.V. −γ dθ r A method of construction of similar models using more and, therefore, general modifications to Newtonian gravity is discussed, including a more broadly-defined Keplerian limit that is `2 d2 1 γr¨ = γ˙r˙ . (8) useful in the derivation of orbit equations for such models − − γr2 dθ2 r (App.A and Sec.VIC). Substituting Eqs. (6) and (8) into the equation of motion Eq. (4) results in

d2 1 `2 `2 γGM + = 0. (9) dθ2 r − r II. RELATIVISTIC KINETIC ENERGY Anticipate a solution of Eq. (9) that is near Keplerian and introduce the radius of a circular orbit for a nonrelativistic 2 Conceptually, the simplest relativistic modification to particle with the same angular momentum, rc ` /GM. ≡ Kepler’s orbits is to define a Lagrangian with a kinetic The result is energy term that is consistent with both the momentum- 2 d rc rc velocity relation of Special Relativity and the Newtonian + = 1 + λ, (10) dθ2 r r energy [7–9, 16–21]; where λ γ 1 is a velocity-dependent correction to ≡ − Newtonian orbits due to Special Relativity. The conic GMm L = mc2γ−1 + , (1) sections of Newtonian mechanics [24, 25] are recovered by − r setting λ = 0 (c ) → ∞ 2 p d rc rc where γ−1 1 v2/c2, and v2 = r˙2 + r2θ˙2.(G is + = 1, (11) ≡ − dθ2 r r ’s universal , M is the mass of the sun, and c is the in vacuum.) The resulting in the well-known orbit equation equations of motion follow from Lagrange’s equations r c = 1 + e cos θ, (12) r d ∂L ∂L where e is the eccentricity. Kepler’s orbits are described = 0, (2) dt ∂q˙i − ∂qi by 0 < e < 1. 3

III. KEPLERIAN LIMIT AND ORBIT where EQUATION 1  r¯c rc − (20) ≡ 1 1  The planets of our solar system are described by − 2 1 A near-circular orbits (e 1) and require only small e¯ 2 (21)  1 relativistic corrections (v/c 1). Mercury has the largest ≡ 1 2   − 1 eccentricity (e 0.2), and the next largest is that of κ¯ (1 ) 2 . (22) ≈ Mars (e 0.09). Therefore, λ [defined after Eq. (10)] is ≡ − ≈ taken to be a small relativistic correction to near-circular According to the correspondence principle, Kepler’s orbits orbits of Newtonian mechanics—Keplerian orbits. This [Eq. (12) with 0 < e < 1] must be recovered in the limit correction is approximated using the first-order series  0 (c ), so that 1 A e is the eccentricity of → → ∞ 2 ≡ γ 1 + 1 (v/c)2, and neglecting the radial component of Newtonian mechanics. To first order in , Eqs. (20)–(22) ≈ 2 velocity v rθ˙ are ≈ 1 2 1 λ (rθ/c˙ ) . (13) r¯c rc(1 ) (23) ≈ 2 ≈ − 2 e¯ e(1 + 1 ) (24) See Sec.VI for a thorough discussion of this approximation. ≈ 2 ˙ κ¯ 1 1 , (25) Using angular momentum Eq. (5) to eliminate θ results ≈ − 2 in λ 1 (`/rc)2(1 + λ)−2, or ≈ 2 so that relativistic orbits in this limit are described λ 1 (`/rc)2. (14) concisely by ≈ 2 1 rc(1 ) The equation of motion Eq. (10) is now expressed − 2 1 + e(1 + 1 ) cos (1 1 )θ. (26) approximately as r ≈ 2 − 2 d2 r r r 2 When compared to Kepler’s orbits [Eq. (12) with c + c 1 + 1  c . (15) dθ2 r r ≈ 2 r 0 < e < 1], this orbit equation clearly displays three characteristics of near-Keplerian orbits: precession of where  (GM/`c)2. The conic sections of Newtonian ≡ perihelion; reduced radius of circular orbit; and increased mechanics, Eqs. (11) and (12), are now recovered by eccentricity. This approximate orbit equation has the setting  = 0 (c ). The solution of Eq. (15) for → ∞ same form as that derived from General Relativity in this  = 0 approximately describes Keplerian orbits with small 6 limit [26] corrections due to Special Relativity. If  is taken to be a small relativistic correction to r (1 3) c − 1 + e(1 + 3) cos (1 3)θ. (27) Keplerian orbits, it is convenient to make the change r ≈ − of variable 1/s r /r 1 1. The last term on ≡ c −  The equations of motion, Eqs. (3) and (4), are the right-hand-side of Eq. (15) is then approximated identical to those derived using the simple force as (r /r)2 1 + 2/s, resulting in a linear differential c ≈ equation (Newton’s 3rd Law) p˙ = GMmˆr/r2, where equation for 1/s(θ) − p = prˆr + pθθˆ, and pr, pθ = γmr,˙ γmrθ˙ , verifying { } { } d2 2 2(1 ) that the unfamiliar relativistic kinetic energy term in + 1. (16) 2 −1 2 − the Lagrangian, T˜ mc γ in Eq. (1), is consistent dθ s s ≈ ≡ − with the familiar definition of relativistic momentum The additional change of variable α θ√1  results in ˆ ≡ − p = γmv. For example, with p˙ =p ˙rˆr +p ˙θθ, and using the familiar form dˆr/dt, dθˆ/dt = θˆθ,˙ ˆrθ˙ , the conserved relativistic { } { − } d2 s s angular momentum Eq. (3) is verified c + c 1, (17) dα2 s s ≈ 1 d 2 ˙ where s 2(1 )/. The solution is similar to that of p˙θ/m = (γr θ) = 0. (28) c ≡ − r dt Eq. (11) s c 1 + A cos α, (18) IV. RELATIVISTIC GRAVITATIONAL s ≈ POTENTIAL ENERGY where A is an arbitrary constant of integration. In terms of the original coordinates The effect of using the special-relativistic γ factor to r¯ define both relativistic kinetic energy and relativistic c 1 +e ¯cosκθ, ¯ (19) r ≈ gravitational potential energy is explored [27–33]. A 4 relativistic gravitational potential energy is derived from where λ γ 1 is a small correction to Newtonian ≡ − a conservative approximate gravitational force. In this gravity due to Special Relativity. Also, relativistic Keplerian limit, a Lagrangian formulation of this problem angular momentum (per unit mass) is defined to be is demonstrated to be equivalent to the vector calculus that of Newtonian mechanics with the replacement formulation. m γm, ` γr2θ˙. Anticipating near-circular orbits, λ is → ≡ approximated using the first-order series γ 1 + 1 (v/c)2, ≈ 2 neglecting the radial component of velocity v rθ˙, and A. Vector Calculus Formalism ≈ using angular momentum to eliminate θ˙, as described in the first paragraph in Sec.III The central-mass problem including both relativistic 2 kinetic energy and relativistic gravitational force is rc  λ 1  . (35) described by the simple force equation ≈ 2 r GMm The result is a conservative approximate relativistic p˙ = γ ˆr, (29) − r2 gravitational force Eq. (34) that describes a small correction to Kepler’s orbits due to Special Relativity where p = γmv. The equations of motion are derived as described in the final paragraph in Sec.III and are GMm  r 2 F˜ 1 + 1  c . (36) identical to Eqs. (3) and (4), except that the gravitational g ≈ − r2 2 r force is multiplied by the relativistic γ factor This force is integrated to derive an approximate ` γr2θ˙ = constant, (30) ≡ relativistic gravitational potential energy and  2 GMm 1 rc  GM 2 U˜(r) = 1 +  . (37) γr¨ +γ ˙r˙ + γ γrθ˙ . (31) 6 r2 − − r r An approximate relativistic orbit equation is derived as The Lagrangian described in Secs.II andIII 2 −1 ˜ 2 L = mc γ U(r) (38) d rc rc ¯ − − 2 + = 1 + λ, (32) dθ r r results in the equations of motion where λ¯ γ2 1 (rθ/c˙ )2. Notice that λ¯ = 2λ, where ≡ − ≈ d 2 λ γ 1 1 (rθ/c˙ )2 is defined after Eq. (10) and in (γr θ˙) = 0, (39) ≡ − ≈ 2 dt Eq. (13). Therefore, the orbit equation is found using the and simple replacement  2 in Eq. (26) → GM  r 2 γr¨ +γ ˙r˙ + 1 + 1  c γrθ˙2. (40) r (1 ) 2 2 c − 1 + e(1 + ) cos (1 )θ. (33) r r − r ≈ − The first equation Eq. (39) verifies the definition of This approximate relativistic orbit equation has the relativistic angular momentum, and the second equation same form as that derived in Secs.II andIII using only Eq. (40) includes a small relativistic correction to Kepler’s relativistic kinetic energy Eq. (26) and is more accurate— orbits due to Special Relativity. Compare these equations when compared to that derived from General Relativity of motion to those derived using only relativistic kinetic in this same limit Eq. (27). energy in Sec.II, Eqs. (3) and (4). A relativistic orbit equation is derived as described in Secs.II andIII B. Lagrangian Formalism d2 r r c + c = 1 + λ,¯ (41) dθ2 r r An equivalent Lagrangian formulation of this problem is accomplished by defining a conservative approximate where λ¯ γ[1 + 1 (r /r)2] 1 is a small correction to ≡ 2 c − relativistic gravitational force, from which a corresponding Keplerian orbits due to Special Relativity. Using the potential energy is derived. A relativistic gravitational first-order series γ 1 + 1 (v/c)2, neglecting the radial ≈ 2 force is defined as the Newtonian force due to gravity component of velocity v rθ˙, and keeping terms first ≈ with the replacement m γm order in  results in λ¯ 2λ, where λ γ 1 1 (rθ/c˙ )2 is → ≈ ≡ − ≈ 2 defined after Eq. (10) and in Eq. (13). Therefore, the orbit ˜ GMm GMm Fg γ 2 = 2 (1 + λ), (34) equation is found using the simple replacement  2 in ≡ − r − r → 5

Eq. (26), thereby reproducing the orbit equation Eq. (33) 1.5 derived using the vector calculus formalism in Sec.IVA Kepler Einstein r (1 ) 1 c − 1 + e(1 + ) cos (1 )θ. (42) r ≈ − Characteristics of near-Keplerian orbits are most easily 0.5 understood by comparing the approximate relativistic c orbit equations to that derived from the nonrelativistic Kepler problem Eq. (12). For this purpose, it is convenient θ/r 0 sin

to express the orbit equation as r r¯ 0.5 c 1 +e ¯cosκθ, ¯ (43) − r ≈ where, using the simple replacement  2 in Eqs. (20)– 1 → − (25), 1 2 1.5 r¯c rc − rc(1 ) (44) − ≡ 1  ≈ − 1.5 1 0.5 0 0.5 1 1.5 − − − − 1 r cos θ/rc e¯ e e(1 + ) (45) ≡ 1  ≈ − 1 FIG. 1. A relativistic orbit in a Keplerian limit (solid) κ¯ (1 2) 2 1 . (46) ≡ − ≈ − Eq. (42) is compared to a Keplerian orbit (dotted) Eq. (12) with the same angular momentum. Precession of perihelion V. CHARACTERISTICS OF is one characteristic of relativistic orbits and is illustrated NEAR-KEPLERIAN ORBITS 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 The approximate relativistic orbit equation Eq. (42) choice of eccentricity (e = 0.25) and relativistic correction predicts a shift in perihelion through an angle parameter ( = 0.1) for purposes of illustration. Precession is present for smaller (non-zero) reasonably chosen values of e ∆θ 2π(¯κ−1 1) 2π (47) ≡ − ≈ and  as well. (The same value of e is chosen for both orbits.) per revolution. This prediction is twice that derived using the standard approach [7–9, 34] to incorporating orbital period of Mercury is 0.24085 terrestrial years, so Special Relativity into the Kepler problem, and is that compared to observations assuming that relativistic and Keplerian angular momenta are approximately equal. For 100 yr 360 60 60 ∆Θ × × ∆θ (50) a Keplerian orbit [24, 25] `2 = GMa(1 e2), where ≡ 0.24085 yr × 2π × − − G = 6.670 10 11 m3/kg s2, M = 1.989 1030 kg is the 14.3 arcsec/century. (51) × · × mass of the Sun, and a and e are the semimajor axis ≈ and eccentricity of the orbit, respectively. Therefore, the Precession, as predicted by Special Relativity is illustrated relativistic correction defined after Eq. (15), in Fig.1. The general-relativistic (GR) treatment of this problem GM  , (48) results in a prediction of 43.0 arcsec/century [26, 34–55], ≈ c2a(1 e2) − and agrees with the observed precession of perihelia is largest for planets closest to the Sun and for of the inner planets [36–45, 56–59]. Historically, this planets with very eccentric orbits. For Mercury contribution to the precession of perihelion of Mercury’s [35, 36] a = 5.79 1010 m and e = 0.2056, so that orbit precisely accounted for the observed discrepancy, ×  2.66 10−8. (The speed of light is taken to serving as the first triumph of the general theory of ≈ × be c2 = 8.987554 1016 m2/s2.) According to Eq. (47), relativity [1–4]. The present approach, using only Special × Mercury precesses through an angle Relativity, accounts for approximately one-third of the observed discrepancy Eq. (51). 2πGM ∆θ = 1.67 10−7 rad (49) The approximate relativistic orbit equation, Eq. (42) ≈ c2a(1 e2) × − with e = 0, predicts a reduced radius of circular orbit— per revolution. This angle is very small and is usually when compared to that of Newtonian mechanics, Eq. (12) expressed cumulatively in arc seconds per century. The with e = 0. This characteristic is not discussed in the 6

0.25 1.5 Newtonian Mechanics General Relativity 1 0 0.5 eff c V 2 ) 0.25 θ/r 0 Kepler /` − Einstein c sin r r (

r 0.5 0.5 c − − Rc 1 − 0.75 − 0 0.5 1 1.5 2 2.5 3 1.5 − 2 1.5 1 0.5 0 0.5 1 r/rc − − − − r cos θ/rc

FIG. 2. The effective potential commonly defined in the FIG. 3. A relativistic orbit in a Keplerian limit (solid) Newtonian limit to General Relativity (dashed) Eq. (52) is Eq. (42) is compared to a Keplerian orbit (dotted) Eq. (12) compared to that derived from Newtonian mechanics (solid) with the same angular momentum. Precession of perihelion with the same angular momentum. The vertical (dotted) has been removed from the relativistic orbit equation, Eq. (42) lines identify the radii of circular orbits, R and r , as c c with (1 − )θ → θ, to emphasize two other characteristics calculated using General Relativity and Newtonian mechanics, of relativistic orbits—reduced orbital radii and increased respectively. General Relativity predicts a smaller radius of eccentricity. These two characteristics are also predicted circular orbit, when compared to that predicted by Newtonian by General Relativity Eq. (27) in greater magnitude. These mechanics. This reduction in radius of a circular orbit Eq. (53), characteristics of relativistic orbits are exaggerated by both the R − r ≈ −3r , is three times that predicted by the present c c c choice of eccentricity (e = 0.45) and the relativistic correction treatment using only Special Relativity Eq. (44), r¯ −r ≈ −r . c c c parameter ( = 0.15) for purposes of illustration. Reduced A curve representing an effective potential including small orbital radii and increased eccentricity are present for smaller corrections predicted by Special Relativity is expected to be (non-zero) reasonably chosen values of e and  as well. (The nearly identical to that for Newtonian mechanics (solid), with a same value of e is chosen for both orbits.) slightly smaller radius of circular orbit, r¯c. The value  = 0.06 is chosen for purposes of illustration. Reduction in radius of circular orbit is present for smaller (non-zero) reasonably chosen values of  as well. so that the radius of circular orbit is predicted to be reduced, R r 3r . (There is also an unstable c − c ≈ − c circular orbit, as illustrated in Fig.2.) This reduction standard approach to incorporating Special Relativity in radius of a circular orbit is three times that predicted into the Kepler problem, but is consistent with the GR by the present treatment using only Special Relativity description. An effective potential naturally arises in the Eq. (44), for which r¯ r r . Reduced size of an c − c ≈ − c GR treatment of the central-mass problem [26, 35, 39, 40, orbit, as predicted by Special Relativity, is illustrated in 42, 45, 52, 53, 60, 61], Fig.3. GM `2 GM`2 Many discussions of the GR effective potential Eq. (52) 3 Veff + , (52) emphasize relativistic capture. The 1/r term in Eq. (52) ≡ − r 2r2 − c2r3 contributes negatively to the effective potential, resulting that reduces to the Newtonian effective potential in in a finite—rather than infinite—centrifugal barrier and the limit c . In the Keplerian limit, the GR affecting orbits very near the central mass (large-velocity → ∞ angular momentum per unit mass ` is also taken to be orbits), as illustrated in Fig.2. This purely GR effect is approximately equal to that for a Keplerian orbit [26, 35– not expected to be described by the approximate orbit 42, 45, 53, 61, 62]. Minimizing Veff with respect to r equation Eq. (42), which is derived using only Special results in the radius of a stable circular orbit, Relativity and implicitly assumes orbits very far from the central mass (small-velocity orbits). 1 1 R = r + r √1 12 r (1 3), (53) c 2 c 2 c − ≈ c − An additional characteristic of relativistic orbits is that 7 of increased eccentricity. The relativistic orbit equation carried out by substituting Eq. (26) into Eq. (15), and Eq. (42) predicts increased eccentricity, when compared only keeping terms of orders e, , and e. The domain of to a Keplerian orbit Eq. (12) with the same angular validity is expressed by subjecting the solution Eq. (26) momentum Eq. (45), e¯ e e. This characteristic to the condition − ≈ of relativistic orbits is not discussed in the standard r c 1 1 (54) approach to incorporating Special Relativity into the r −  Kepler problem, but is consistent with the GR description. The GR orbit equation in this Keplerian limit Eq. (27) for the smallest value of r. Evaluating the orbit equation predicts an increase in eccentricity e¯ e 3e, which is Eq. (42) at perihelion rp results in − ≈ three times that predicted by the present treatment using 1 rc 1 + e(1 + 2 ) only Special Relativity. Increased eccentricity of an orbit, = 1 . (55) rp 1  as predicted by Special Relativity, is illustrated in Fig.3. − 2 Substituting this into Eq. (54) results in the domain of validity VI. DISCUSSION 1 e(1 + 2 ) +  1. (56) The debate concerning the usage of relativistic inertial  mass and relativistic gravitational mass [63–70] is Therefore, the relativistic eccentricity e¯ = e(1 + 1 ) 1, 2  irrelevant in the present context due to the fundamental and Eq. (26) is limited to describing relativistic corrections incompatibility of Special Relativity and gravitation to near-circular (Keplerian) orbits. Also, the relativistic [71]. Accordingly, language describing the replacement correction  1, and thus the orbit equation Eq. (26) is  m γm is intentionally omitted. Instructors may valid only for small relativistic corrections. → feel more comfortable using the equivalent replacements The correction to Keplerian orbits due to Special v γv to introduce relativistic momentum, and G γG Relativity λ γ 1 [defined after Eq. (10) and in → → ≡ − to introduce relativistic gravitational force. Rather, Eq. (13)] is approximated using the first-order series Lagrangians are constructed using familiar elements of γ 1 + 1 (v/c)2, and neglecting the radial component of ≈ 2 Special Relativity for the purpose of simulating general- the velocity v rθ˙. Neglecting the radial component of ≈ relativistic orbital effects using only Special Relativity velocity in the relativistic correction λ is consistent with in a well-defined Keplerian limit. One purpose of the assumption of approximately Keplerian (near-circular) this presentation is to provide students with interesting orbits, and is supported by the condition r /r 1 1 c −  and tractable problems that arise from small special- preceding Eq. (16). It is emphasized that the radial relativistic modifications to a familiar problem—Kepler’s component of velocity is neglected only in the relativistic orbits, the solutions of which provide a qualitative correction λ; it is not neglected in the derivation of the understanding of corrections to Kepler’s orbits due to relativistic equation of motion Eq. (10). That there is General Relativity. In this Keplerian limit, these models no explicit appearance of r˙ in the relativistic equation are supposed to be physical based on the likeness of of motion, other than in the definition of γ, is due to a the equations of motion to those derived using General fortunate cancellation after Eq. (8). Relativity [26]. Another purpose is to present methods The presentation in Sec.IV includes relativistic by which similar models may be constructed and solved. gravitational potential energy using the replacement m γm in the Newtonian gravitational force. Although → the use of relativistic gravitational mass in the special- A. Domain of Validity relativistic Kepler problem is discouraged by some authors [68], there are several useful results in the The following discussion concerning the derivation, Keplerian limit. This relativistic gravitational force is validity, and scope of the approximate special-relativistic approximated, resulting in a conservative force, from orbit equations follows the presentation in Secs.II andIII, which an approximate relativistic potential energy is in which only relativistic kinetic energy is included. The derived—thereby enabling the use of the Lagrangian arguments and conclusions also apply to the presentation formalism. The resulting approximate relativistic orbit in Sec.IV, in which both relativistic kinetic energy and equation Eq. (42) is more accurate, when compared relativistic gravitational potential energy are included. to that derived from General Relativity in the same The approximate relativistic orbit equation Eq. (26) limit Eq. (27), than that derived using only relativistic provides small corrections to Kepler’s orbits Eq. (12) due kinetic energy Eq. (26). Specifically, this more accurate to Special Relativity. A systematic verification may be orbit equation demonstrates that—in this Keplerian 8 limit—the only consequence of neglecting relativistic resulting in a correction to Newtonian orbits GM(γ2 1). − gravitational potential energy is that corrections due to Using the first-order expansion γ2 1+(v/c)2, neglecting ≈ Special Relativity are decreased by a factor of two. the radial component of velocity v rθ˙, and using angular ≈ momentum to eliminate θ˙ results in  `2  B. Structure of the Models GM(γ2 1) GM . (60) − ≈ c2r2 These models are appealing because they produce This is exactly twice the correction term that results equations of motion that are similar to those derived from using only relativistic kinetic energy Eq. (59), and is using General Relativity. Compare the special-relativistic directly responsible for the  corrections that appear in the (SR) equation of motion, Eq. (9) in Sec.II [using approximate relativistic orbit equation, Eqs. (33) and (42). γGM = GM + GM(γ 1)], to the general-relativistic − Using the Lagrangian formalism in Sec.IVB, there is an (GR) equation of motion, Eq. (10) in Ref. 26, additional requirement that the relativistic correction to the Newtonian gravitational force does not depend d2 1 `2 (SR) `2 GM + GM(γ 1) = 0 (57) explicitly on θ or θ˙, so that the simple γ factor appears dθ2 r − r − − in the definition of angular momentum. d2 1 `¯2  3`¯2  (GR) `¯2 GM + GM = 0. (58) More generally, any function f(γ) may be used in the dϕ2 r − r − c2r2 definition of relativistic gravitational force, provided a The terms in parentheses describe corrections to first-order expansion of the form γf(γ) 1 α(v/c)2 − ≈ Newtonian orbits due to Special Relativity Eq. (57) exists, where α > 0 is a constant. For example, choosing and General Relativity Eq. (58); these terms are zero a simple -law dependence f(γ) = γn, so that in the Newtonian limit c . Note that both of F˜ γnGMm/r2, results in an approximate relativistic → ∞ g ≡ these equations are exact in the sense that they are orbit equation identical to Eq. (26) with the replacement derived directly from Lagrangians without making any  (n + 1). Notice the physical condition n 0 that → ≥ approximations. The SR equation of motion Eq. (57) is necessary to insure the proper direction of precession, has a correction to Newtonian orbits GM(γ 1). Using reduced orbital radii, and increased eccentricity. − the first-order expansion γ 1 + 1 (v/c)2, neglecting the Using these types of models, the Keplerian limit is ≈ 2 radial component of velocity v rθ˙, and using angular defined precisely by only approximating the correction ≈ momentum to eliminate θ˙ results in term that represents a modification of the Keplerian equation of motion. The approximation of this correction  `2  GM(γ 1) GM . (59) term consists of: a series expansion to first order in (v/c)2; − ≈ 2c2r2 neglecting the higher-order r˙2 term; and making a change Aside from a constant, this term is identical to the of variable subject to the condition rc/r 1 1/s 1, n − ≡  term that naturally arises in the GR derivation Eq. (58), allowing the linearization (rc/r) 1 + n/s. 1 ≈ and is directly responsible for the 2  corrections that With appropriate conditions and approximations, appear in the approximate relativistic orbit equation modifications to Newtonian gravity that depend, more Eq. (26). The factor of γ that is responsible for generally, on velocity and radial coordinate may be used. this term is a direct result of the γ that appears This is most easily described by example. A toy model is in the definition of angular momentum (per unit presented in App.A and discussed in the following section mass) Eq. (5) ` γr2θ˙. Fundamentally, this angular that describes methods—including a more broadly-defined ≡ momentum has a γ factor due to the definition Keplerian limit—that are useful for solving problems with of relativistic kinetic energy T˜ mc2γ−1. The GR more general modifications to Newtonian gravity. ≡ − angular momentum is found to be simply `¯ r2ϕ˙ ≡ [Eq. (4) in Ref. 26] because that problem is solved using τ rather than t, so that C. A Toy Model ϕ˙ dϕ/dτ. A transformation to coordinate time results ≡ in `¯ γr¯ 2ϕ˙, where γ¯ dt/dτ = [1 + 2V (r)/c2]−1/2, and There are many attempts to motivate a physically ≡ ≡ V (r) GM/r is the Newtonian gravitational potential. appropriate modification to Newtonian gravity in the ≡ − The vector calculus formalism in Sec.IVA includes a γ literature [27–33]. The possibility of simply replacing factor in the definition of the relativistic gravitational m γm in the Newtonian gravitational potential energy → force. The γ factor from the relativistic angular and using the Lagrangian formalism is explored in App.A momentum propagates through the derivation of the and discussed in this section. This toy model is useful orbit equation as described in the preceding paragraph, for describing methods for solving problems with more 9 general modifications to Newtonian gravity, for which model includes only relativistic kinetic energy using the a more broadly-defined Keplerian limit is needed. The replacement m γm in the Newtonian linear momentum, → Lagrangian for this model is p = γmv. A solution to the corresponding equations of motion in a Keplerian limit results in an approximate GMm L = mc2γ−1 + γ . (61) relativistic orbit equation Eq. (26) that has the same − r form as that derived from General Relativity in this limit Lagrange’s equations result in an abstruse set of Eq. (27) and is easily compared to that describing Kepler’s differential equations, Eqs. (A2) and (A3), that—using orbits Eq. (12). This form is that of elliptical orbits an appropriate approximation—reduce to the equations of Newtonian mechanics with corrections to radius and of motion derived in Sec.IV, Eqs. (30) and (31), thereby eccentricity, and exhibiting precession. Specifically, the reproducing the orbit equation, Eqs. (33) and (42) approximate relativistic orbit equation clearly describes r (1 ) three characteristics of relativistic orbits: precession of c − 1 + e(1 + ) cos (1 )θ. (62) r ≈ − perihelion; reduced radius of circular orbit; and increased eccentricity. The predicted rate of precession of perihelion Another solution to the differential equations, of Mercury is in agreement with that of established Eqs. (A2) and (A3), is derived using a more broadly- calculations using only Special Relativity. Each of these defined Keplerian limit. For near-circular approximately characteristics of relativistic Keplerian orbits is exactly Newtonian orbits, r˙ and γ˙ are very slowly varying one-sixth of the corresponding correction described by functions, so that the higher-order terms γ˙r˙ and General Relativity in this limit—providing a qualitative 2 2 (γ/c) V (r)γr˙ /r are taken to be negligible. The result description of corrections to Keplerian orbits due to is the Newtonian equation of motion with a relativistic General Relativity. correction term Eq. (A10) This simple model is improved upon by including d2 1 `˜2 `˜2 GM + GMλ˜ = 0, (63) relativistic gravitational force using the replacement dθ2 r − r − m γm in Newtonian gravity, F˜ = γGMm/r2. An → g where approximation consistent with the Keplerian limit results in a conservative force, from which a relativistic potential 2  2  λ˜ γ 1 (γ/c) V (r) 1, (64) energy is derived that is useful in a Lagrangian formulation ≡ − − of the special-relativistic Kepler problem. A solution of `˜ γr2θ˙ 1 (γ/c)2V (r) is a constant of motion, and ≡ − the corresponding equations of motion in a Keplerian V (r) GM/r is the Newtonian gravitational potential. ≡ − limit results in an approximate relativistic orbit equation Compare this equation of motion to Eqs. (57) and (58). Eq. (42) that has the same form as that derived using This equation of motion has a small relativistic correction the simplest model Eq. (26), thereby describing the same to Keplerian orbits Eq. (A12) three characteristics of relativistic orbits. Each of these  `˜2 V (r) characteristics of relativistic Keplerian orbits is exactly GMλ˜ GM . (65) ≈ c2r2 − c2 one-third of the corresponding correction described by General Relativity in this limit Eq. (27). Compare this to the correction terms used in the two simple models, Eqs. (59) and (60). The corresponding The Lagrangian formalism applied to the special- orbit equation in the Keplerian limit Eq. (A16), relativistic Kepler problem is instructive, providing several challenges appropriate for an introductory classical r˜c(1 2˜) mechanics course, including: solve Newton’s force − 1 + e(1 + ˜) cos (1 3 ˜)θ, (66) r ≈ − 2 equation using vector calculus to verify the unfamiliar does not have the symmetry of those derived using the relativistic kinetic energy term in the Lagrangian—as two simple models, Eqs. (26) and (42). The methods and outlined in the last paragraph of Sec.III and in Sec.IVA; approximations used to derive this orbit equation may derive a potential energy function from a conservative be applied to other, more physical, models that include force; apply Lagrange’s equations to derive the conserved relativistic corrections to Kepler’s orbits. relativistic angular momentum and equation of motion; and transform and solve a differential equation to derive an approximate relativistic orbit equation in terms VII. CONCLUSION of planar coordinates. This approach also provides an opportunity to use less familiar problem solving The Lagrangian formalism is useful for describing small strategies, including: variable transformations to cast the relativistic corrections to Kepler’s orbits. The simplest differential equation into familiar form; approximation 10 methods that simplify the differential equation; and Relativity, as discussed in Sec.VIB. A larger class usage of the correspondence principle to identify a of models may be solved using similar methods and constant of integration. Most importantly, students approximations. This is demonstrated using a toy model are rewarded with a clear understanding that a small in App.A and in Sec.VIC, wherein a more broadly- relativistic modification to a familiar problem results in defined Keplerian limit is described. Exact solutions of an approximate relativistic orbit equation that clearly the special-relativistic Kepler problem require a thorough demonstrates that relativity is responsible for a small understanding of special relativistic mechanics [72, 73] and contribution to perihelic precession, and the satisfaction are, therefore, inaccessible to many undergraduate physics of calculating that contribution. majors. The present approach and method of solution is These models are appealing because they are easy to understandable to nonspecialists, including undergraduate motivate and, most importantly, they produce equations physics majors whom have not had a course dedicated to of motion that are similar to those derived using General relativity.

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[68] G. Oas, “On the abuse and use of relativistic mass,” Newtonian orbits, r˙ and γ˙ are very slowly varying (2005), arXiv: physics/0504110v2 [physics.ed-ph]. functions, so that the higher-order terms γ˙r˙ and G. Oas, “On the Use of Relativistic Mass in Various (γ/c)2V (r)γr˙2/r are taken to be negligible. [For Published Works,” (2005), arXiv: physics/0504111v1 Mercury’s orbit (γ/c)2V (a)γ/a 10−18 m−1.] [physics.ed-ph]. ∼ − [69] C. G. Adler, “Does mass really depend on velocity, dad?,” `˜ γr2θ˙ 1 (γ/c)2V (r) = constant, (A5) Am. J. Phys. 55 (8), 739–743 (1987). ≡ − [70] S. A. Vasiliev, “On the Notion of the of and in the Special Relativity Theory,” App. Phys. Res. 4 (2), GM γr¨1 (γ/c)2V (r) + γ 136–143 (2012). − r2 [71] Reference 37, ch. 7. (A6) 2  2  [72] W. Rindler, Special Relativity (Oliver and Boyd, London, γrθ˙ 1 (γ/c) V (r) 0. − − ≈ 1960), pp. 79–103. [73] J. L. Synge, Relativity: The Special Theory (North- Conservation of angular momentum Eq. (A5) is used to ˙ Holland Publishing Company, Amsterdam, 1958), pp. 394– eliminate the explicit occurrence of θ in the equation of 399. motion Eq. (A6) `˜2 γrθ˙2 1 (γ/c)2V (r) = . (A7) Appendix A: A Toy Model − γr3 [1 (γ/c)2V (r)] − Time is eliminated by successive applications of the chain The possibility of including relativistic gravitational rule, together with the conserved angular momentum; potential by simply replacing m γm in the Newtonian → `˜ d 1 gravitational potential energy and using the Lagrangian r˙ = , (A8) −γ[1 (γ/c)2V (r)] dθ r formalism is explored. This toy model is useful for − describing methods for solving problems with more and, therefore, (again takingγ ˙r˙ to be negligible) general modifications to Newtonian gravity, for which `˜2 d2 1 a more broadly-defined Keplerian limit is needed. The γr¨[1 (γ/c)2V (r)] . (A9) Lagrangian for this model is − ≈ −γ[1 (γ/c)2V (r)]r2 dθ2 r − GMm Substituting Eqs. (A7) and (A9) into the equation of L = mc2γ−1 + γ . (A1) − r motion Eq. (A6) results in Lagrange’s equations result in an abstruse set of d2 1 `˜2 differential equations `˜2 γ2[1 (γ/c)2V (r)]GM + = 0. (A10) dθ2 r − − r d n o γr2θ˙ 1 (γ/c)2V (r) = 0, (A2) Anticipate a solution of Eq. (A10) that is near Keplerian dt − and introduce the radius of a circular orbit for a and nonrelativistic particle with the same angular momentum,  2   2  γr¨ 1 (γ/c) V (r) +γ ˙r˙ 1 3(γ/c) V (r) r˜ `˜2/GM. The result is − − c ≡ 2 GM 2  2  d r˜c r˜c + γ γrθ˙ 1 (γ/c) V (r) (A3) + = 1 + λ,˜ (A11) r2 − − dθ2 r r r˙2 + (γ/c)2V (r)γ = 0, where λ˜ γ2[1 (γ/c)2V (r)] 1 is a correction to r ≡ − − Newtonian orbits due to Special Relativity. [Tilde where V (r) GM/r is the Newtonian gravitational ≡ − notation is used to emphasize that quantities depend potential. Mercury’s orbit is approximately circular on the exact conserved quantity `˜ defined in Eq. (A5), with radius of the same order of magnitude as its rather than the angular momentum ` defined in Eq. (30).] semimajor axis r a 5.79 1010 m, and its velocity ∼ ≈ × The orbit equation is derived following the method is very small when compared to the speed of light, so described in Sec.III. The correction term λ˜ is 2 2 −8 that (γ/c) V (r) V (a)/c 10 . Ignoring terms approximated by expanding to first-order in 1/c2, ∼ 2 ∼ − proportional to (γ/c) V (r) results in the equations of neglecting the radial component of velocity, and using motion derived in Sec.IV, Eqs. (30) and (31), thereby angular momentum to eliminate θ˙ reproducing the orbit equation, Eqs. (33) and (42) λ˜ (`/rc˜ )2 V (r)/c2. (A12) r (1 ) ≈ − c − 1 + e(1 + ) cos (1 )θ. (A4) r ≈ − The equation of motion Eq. (A11) is now expressed Another solution to the equations of motion, approximately as Eqs. (A2) and (A3), is derived using a more broadly- d2 r˜ r˜ r˜ r˜ 2 c + c 1 + ˜ c + ˜ c , (A13) defined Keplerian limit. For near-circular approximately dθ2 r r ≈ r r 13 where ˜ (GM/`c˜ )2. An orbit equation is derived, as where A is an arbitrary constant of integration. In terms ≡ described in Sec.III. The equation of motion Eq. (A13) of the original coordinates, and defining e 2A, an orbit ≡ is linearized using r /r = 1 + 1/s, and (r /r)2 1 + 2/s. equation in the Keplerian limit is described concisely by c c ≈ The additional change of variable α θ√1 3˜ results ≡ − r˜ (1 2˜) in the familiar differential equation c − 1 + e(1 + ˜) cos (1 3 ˜)θ. (A16) r ≈ − 2 d2 s s c + c 1, (A14) This orbit equation does not have the symmetry of those dα2 s s ≈ derived using the two simple models, Eqs. (26) and (42). where s (1 3˜)/(2˜). The solution is similar to that The methods and approximations used to derive this orbit c ≡ − of Eq. (11) equation may be applied to other, more physical, models that include relativistic corrections to Kepler’s orbits. See s c 1 + A cos α, (A15) the discussions in Sec.VIB and Sec.VIC. s ≈