Nonconservative Extension of Keplerian Integrals and a New Class

Home , Hyperbolic spiral, Inverse curve

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E-mail: Con- . turba- inter- rpit1 coe 08Cmie sn NA L MNRAS using Compiled 2018 October 11 Preprint sys- nts it- i- d u oterpoon implications. profound their to due attention important received perturbations conservative erln ogrhlswe rnfrigtecrua restr ( case circular elliptic the the transforming to when problem holds three-body Ja longer the example, no For tegral straightforward. not is generalization explo integrals transformations. first ilarity of existence for conditions necessary ehdfrfidn h ieso ftemnfl nwihorbi syst the which that in integrals isolating manifold of the number the of i.e. dimension evolve, the finding for method connec transformations define to order them. in problems circular a rbe noasmlfidoe euigtedgeso fre orig of degrees the the transforms reducing and one, ( simplified motion a of into problem constants nal transforma the of similarity one nonisol a and embeds introduced isolating study of Their concepts integrals. the reviewed and laws tion ( Contopoulos l,te o nyetne ote’ hoe,btas t i its also but theorem, Noether’s extended nonconservativ only variationa Hamilton’s not included they on ple, Relying and derivation. the theorem in forces Noether’s on advanced asfrobt fsaleccentricities. small of orbits for laws rode l 2007 al. et Arnold 1964 h odtosfreitneo nerl fmto ne no under motion of integrals of existence for conditions The osrainlw r estv oprubtosadtheir and perturbations to sensitive are laws Conservation eetdfo h iiaiisbtenteelpi n th and elliptic the between similarities the from benefited ) éo n Heiles and Hénon ( 1967 §3.2). , a bet n prxmt conservation approximate find to able was ) ( n fti e rbe r discussed, are problem new this of ons 1964 epolm fe hrceiigthe characterizing After problem. he Carpintero utpouso,adsm instances some and propulsion, rust h oino atce etre by perturbed particles of motion the rac oasmlruniparametric simpler a to urbance uha adod n logarithmic, and cardioids as such s ,prblc n yeblc Kep- hyperbolic. and parabolic, c, so h rtitgasi special a in integrals first the of ms oes h ytmi perturbed is system The models. l rmmnu.Nehrstheorem Noether’s momentum. ar dd hnst h conservation the to Thanks ided. ae nteWirtaselliptic Weierstrass the on based epndi h aueo conserva- of nature the in deepened ) e ihohrkonintegrable known other with ted oain nKpe’ problem Kepler’s in rotations d gniladnra directions. normal and ngential jkcadVujanovic and Djukic ( 2008 i 1993 Xia zbhl n Giacaglia and Szebehely rpsdanumerical a proposed ) mc acceleration – amics A ls of class T .Nevertheless, ). E tl l v3.0 file style X madmits. em ntepast the in tn sim- iting inthat tion princi- l oiin- cobi ( nverse 1975 edom ating icted ting n- ts i- e e ) 2 J. Roa form and the Noether-Bessel-Hagen and Killing equations. Later cursively in the literature because they render constant radial ac- studies by Djukic and Sutela (1984) sought integrating factors that celerations, relevant for the design of spacecraft trajectories pro- yield conservation laws upon integration. Examples of application pelled by continuous-thrust systems. The pioneering work by Tsien of Noether’s theorem to constrained nonconservative systems can (1953) provided the explicit solution to the problem in terms of el- be found in the work of Bahar and Kwatny (1987). Honein et al. liptic integrals, as predicted by Whittaker (1917, p. 81). By means (1991) arrived to a compact formulation using what was later of a special change of variables, Izzo and Biscani (2015) arrived to called the neutral action method. Remarkable applications exploit- an elegant solution in terms of the Weierstrass elliptic functions. ing Noether’s symmetries span from cosmology (Capozziello et al. These functions were also exploited by MacMillan (1908) when he 2009; Basilakos et al. 2011) to string theory (Beisert et al. 2008), solved the dynamics of a particle attracted by a central force de- 5 field theory (Halpern et al. 1977), and fluid models (Narayan et al. creasing with r− . Urrutxua et al. (2015a) solved the Tsien problem 1987). In the book by Olver (2000, Chaps. 4 and 5), an exhaus- using the Dromo formulation, which models orbital motion with a tive review of the connection between symmetries and conservation regular set of elements (Peláez et al. 2007; Urrutxua et al. 2015b; laws is provided within the framework of Lie algebras. We refer to Roa et al. 2015). Advances on Dromo can be found in the works Arnold et al. (2007, Chap. 3) for a formal derivation of Noether’s by Baù et al. (2015) and Roa and Peláez (2015). The case of a con- theorem, and a discussion on the connection between conservation stant radial force was approached by Akella and Broucke (2002) laws and dynamical symmetries. from an energy-driven perspective. They studied in detail the roots Integrals of motion are often useful for finding analytic or of the polynomial appearing in the denominator of the equation to semi-analytic solutions to a given problem. The acclaimed solu- integrate, and connected their nature with the form of the solution. tion to the satellite main problem by Brouwer (1959) is a clear General considerations on the integrability of the problem can be example of the decisive role of conserved quantities in deriving found in the work of San-Juan et al. (2012). solutions in closed form. By perturbing the Delaunay elements, Another relevant example of an integrable system in celestial Brouwer and Hori (1961) solved the dynamics of a satellite sub- mechanics is the Stark problem, governed by a constant acceler- ject to atmospheric drag and the oblateness of the primary. They ation fixed in the inertial frame. Lantoine and Russell (2011) pro- proved the usefulness of canonical transformations even in the con- vided the complete solution to the motion relying extensively on text of nonconservative problems. Whittaker (1917, pp. 81–82) ap- elliptic integrals and Jacobi elliptic functions. A compact form of proached the problem of a central force depending on powers of the solution involving the Weierstrass elliptic functions was later the radial distance, rn, and found that there are only fourteen val- presented by Biscani and Izzo (2014), who also exploited this for- ues of n for which the problem can be integrated in closed form malism for building a secular theory for the post-Newtonian model using elementary functions or elliptic integrals. Later, he discussed (Biscani and Carloni 2012). The Stark problem provides a simpli- the solvability conditions for equations involving square roots of fied model of radiation pressure. In the more general case, the dy- polynomials (Whittaker and Watson 1927, p. 512). Broucke (1980) namics subject to this perturbation cannot be solved in closed form. advanced on Whittaker’s results and found six potentials that are An intuitive simplification that makes the problem integrable con- a generalization of the integrable central forces discussed by the sists in assuming that the force due to the solar radiation pressure latter. These potentials include the referred fourteen values of n follows the direction of the Sun vector. The dynamics are equiv- as particular cases. Numerical techniques for shaping the potential alent to those governed by a Keplerian potential with a modified given the orbit solution were published by Carpintero and Aguilar gravitational parameter. (1998). Classical studies on the integrability of systems governed The present paper introduces a new class of integrable system, by central forces are based strongly on Newton’s theorem of re- governed by a biparametric nonconservative perturbation. This ac- volving orbits.1 The problem of the orbital precession caused by celeration unifies various force models, including special cases of central forces was recently recovered by Adkins and McDonnell solar radiation pressure, low-thrust propulsion, and some particular (2007), who considered potentials involving both powers and log- configurations in general relativity. The problem is formulated in arithms of the radial distance, and the special case of the Yukawa Sec. 2, where the biparametric acceleration is defined and then re- potential (Yukawa 1935). Chashchina and Silagadze (2008) relied duced to a uniparametric forcing thanks to a similarity transforma- on Hamilton’s vector to simplify the analytic solutions found by tion. The conservation laws for the energy and angular momentum Adkins and McDonnell (2007). More elaborated potentials have are generalized to the nonconservative case by exploiting known been explored for modeling the perihelion precession (Schmidt symmetries of Kepler’s problem. Before solving the dynamics ex- 2008). The dynamics of a particle in Schwarzschild space-time plicitly, we will prove that there are four cases that can be solved can also be regarded as orbital motion perturbed by an effec- in closed form using elementary or elliptic functions. Sections 3–6 tive potential depending on inverse powers of the radial distance present the properties of each family of orbits and the correspond- (Chandrasekhar 1983, p. 102). ing trajectories are derived analytically. Section 7 is a summary of Potentials depending linearly on the radial distance appear re- the solutions, which are unified in Sec. 8 introducing the Weier- strass elliptic functions. Finally, Sec. 9 discusses the connection with known solutions to similar problems, and with Schwarzschild 1 Section IX, Book I, of Newton’s Principia is devoted to the motion of geodesics. bodies in moveable orbits (De Motu Corporum in Orbibus mobilibus, deq; motu Apsidum, in the original latin version). In particular, Thm. XIV states that “The difference of the forces, by which two bodies may be made to move equally, one in a quiescent, the other in the same orbit revolving, is in 2 DYNAMICS a triplicate ratio of their common altitudes inversely”. Newton proved this The motion of a body orbiting a central mass under the action of an theorem relying on elegant geometric constructions. The motivation behind arbitrary perturbation a is governed by this result was the development of a theory for explaining the precession p of the orbit of the Moon. A detailed discussion about this theorem can be d2r µ = r + a , (2.1) found in the book by Chandrasekhar (1995, pp. 184–201) dt2 − r3 p

MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 3 where µ denotes the gravitational parameter of the attracting body, r is the radiusvector of the particle, and r = r . In the case ap = 0, Eq. (2.1) reduces to Kepler’s problem, which|| || can be solved in closed form. It is well known that under the action of a radial perturbation of the form ξµ a = r, (2.2) p r3 in which ξ is an arbitrary constant, the system (2.1) remains integrable. Perturbations of this type model various physical phe- nomena, like the solar radiation pressure acting on a surface per- pendicular to the Sun vector, or simple control laws for spacecraft with solar electric propulsion. Indeed, the perturbed problem can be written Figure 1. Geometry of the problem. 2 d r µ(1 ξ) µ∗ = − r = r, (2.3) dt2 − r3 − r3 coordinates (r,θ) to formulate the problem. The radial velocity is which is equivalent to Kepler’s problem with a modified gravita- defined as tional parameter, µ∗ = µ(1 ξ). − (v r) The gravitational acceleration written in the intrinsic frame r˙ = · = v cos ψ, (2.13) I = t, n, b , with r { } whereas the projection of the velocity normal to the radiusvector t = v/v, b = h/h, n = b t (2.4) × takes the form defined in terms of the velocity and angular momentum vectors, v rθ˙ = v sin ψ. (2.14) and h respectively, reads µ µ Consequently, these geometric relations unveil the dynamical equa- a = r = (cos ψ t sin ψ n). (2.5) g − r3 − r2 − tions: dr The flight-direction angle ψ is given by = v cos ψ dt (r v) cos ψ = · . (2.6) dθ v rv = sin ψ (2.15) dt r 3 Consequently, perturbations of the form ap = ξµ (r/r ) result in v = r˙2 + (rθ˙)2. ξµ ap = (cos ψ t sin ψ n). (2.7) q 2 The dynamics are referred to an inertial frame I = iI, jI, kI , with r − { } kI h and using the xI-axis as the origin of angles. Figure 1 depicts Since the component normal to the velocity does not perform any k work, Bacon (1959) neglected its contribution and found that the configuration of the problem. We restrict the study to prograde motions (θ>˙ 0) without losing generality. µ a = cos ψ t (2.8) p 2r2 furnishes another integrable problem: the trajectory of the particle 2.1 Similarity transformation is a logarithmic spiral. Consider a linear transformation S of the form Motivated by this line of thought, we can treat the normal component of the acceleration in Eq. (2.7) separately by introduc- S :(t, r,θ, r˙, θ˙) (τ,ρ,θ,ρ′,θ′) (2.16) ing a second parameter, η: 7→ defined explicitly by the positive constants α, β, and δ = α/β: µ a = (ξ cos ψ t + η sin ψ n). (2.9) p r2 t r r˙ τ = , ρ = , ρ′ = , θ′ = β θ.˙ (2.17) The dimensionless parameters ξ and η are assumed constant. The β α δ forcing parameter ξ controls the power exerted by the perturbation, The constants α, β, and δ have units of length, time, and velocity, respectively. The symbol ′ denotes derivatives with respect to τ, dEk ξµ = ap v = v cos ψ, (2.10) ˙ dt · r2 whereas is reserved for derivatives with respect to t. The scaling factor α can be seen as the ratio of a homothetic transformation that with E the Keplerian energy of the system. The second parameter k simply dilates or contracts the orbit. Similarly, β represents a time η scales the terms that do not perform any work. Introducing the dilation or contraction. The velocity of the particle transforms into accelerations (2.5) and (2.9) into Eq. (2.1) yields v d2r µ v˜ = . (2.18) = δ 2 2 (1 ξ)(cos ψ t γ sin ψ n), (2.11) dt − r − − In addition, β and δ are defined in terms of α by virtue of having replaced η by the new parameter 1 + η α3 α µγ(1 ξ) γ = . (2.12) β = and δ = = − . (2.19) 1 ξ s µγ(1 ξ) β r α − − Since the motion is planar we shall introduce a system of polar We assume γ> 0 and ξ< 1 for consistency.

MNRAS 000, 1–16 (2016) 4 J. Roa

Equation (2.11) then becomes This equation and the condition F q′ f , 0 furnish the gener- i − i alized Noether-Bessel-Hagen (NBH) equations (Trautman 1967; d2ρ 1 1 = cos ψ t sin ψ n , (2.20) Djukic and Vujanovic 1975; Vujanovic et al. 1986). The NBH dτ2 − ρ2 γ − ! equations involve the full derivative of the gauge function and the which is equivalent to the normalized two-body problem generators with respect to τ, meaning that Eq. (2.29) depends on d2ρ ρ the partial derivatives of Ψ, Fi, and f with respect to time, the coor- = + a˜ p (2.21) dinates, and the velocities. By expanding the convective terms the dτ2 − ρ3 NBH equations decompose in the system of Killing equations: perturbed by the purely tangential acceleration L L ∂ f ∂ ∂Fi ∂ f ∂Ψ γ 1 + qi′ = , a˜ = cos ψ t. (2.22) ∂q′ ∂q′ ∂q′ − ∂q′ ∂q′ p −2 j i i  j j  j γρ X     This result shows that S establishes a similarity transformation ∂  ∂L  ∂ f ( f L Ψ) + F + L q′ + Q (F q′ f ) between the two-body problem (2.1) perturbed by the accelera- ∂τ − ∂q i ∂q i i i − i i ( i i tion (2.9), and the simpler problem in Eq. (2.21) and perturbed X L by (2.22). That is, S 1 transforms the solution to Eq. (2.20), ρ(τ), ∂ ∂Fi ∂ f ∂Fi ∂ f ∂Ψ − + q′ + q′ q′q′ q′ = 0. ∂q ∂τ − i ∂τ ∂q j − i j ∂q − ∂q i into the solution to Eq. (2.11), r(t). i′  j j j ! i ) Using q = (ρ,θ) and q = (ρ ,θ ) as the generalized coor-  X  ′ ′ ′   (2.30) dinates and velocities, respectively, the dynamics of the problem   The system (2.30) decomposes in three equations that can be solved abide by the Euler-Lagrange equations for the generators Fρ, Fθ, and f given a certain gauge. If the trans- d ∂L ∂L = Q . (2.23) formation deﬁned in Eq. (2.26) satisﬁes the NBH equations, then dτ ∂q − ∂q i the system admits the integral of motion (2.28). i′ ! i The Lagrangian of the transformed system takes the form

1 2 2 2 1 2.2.1 Generalized equation of the energy L = (ρ′ + ρ θ′ ) + (2.24) 2 ρ The Lagrangian in Eq. (2.24) is time-independent. Thus, the action and the generalized forces Qi read is not aﬀected by arbitrary time transformations. In the Keplerian 2 case a simple time translation reveals the conservation of the en- (γ 1) ρ′ (γ 1) ρ′θ′ Q = − and Q = − . (2.25) ergy. Motivated by this fact, we explore the generators ρ γ ρv˜ θ γ v˜2 ! ! f = 1, Fρ = 0, and Fθ = 0. (2.31)

2.2 Integrals of motion and dynamical symmetries Solving Killing equations (2.30) with the above generators leads to the gauge function Let us introduce an infinitesimal transformation R: γ 1 Ψ= − . (2.32) τ τ∗ = τ + ε f (τ; qi, q′) → i (2.26) γr q q∗ = q + εF (τ; q , q′), i → i i i i i Provided that the NBH equations hold, the system admits the inte- defined in terms of a small parameter ε 1 and the generators F gral of motion ≪ i and f . For transformations that leave the action unchanged up to an v˜2 1 κ˜ = Λ 1 , (2.33) exact differential, 2 − γρ − ≡ 2 dq∗ dq L i L i written in terms of the constantκ ˜1 = 2Λ. This term can be solved τ∗; qi∗, dτ∗ τ; qi, dτ = ε dΨ(τ; qi, qi′), (2.27) − dτ − dτ from the initial conditions ∗ ! ! Ψ τ , 2 2 with ( ; qi qi′) a given gauge, Noether’s theorem states that κ˜ = v˜ . (2.34) 1 0 − γρ ∂L ∂L 0 Fi + f L q′ Ψ(τ; qi, q′) =Λ (2.28) When γ = 1 the perturbation (2.22) vanishes and Eq. (2.33) re- ∂q ∂q i i i i′ − i′ − X ( ! " ! #) duces to the normalized equation of the Keplerian energy. In fact, is a first integral of the problem. Here Λ is a certain constant of mo- in this caseκ ˜1 becomes twice the Keplerian energy of the system, tion. We refer the reader to the work of Efroimsky and Goldreich κ˜1 = 2E˜ k. Moreover, the gauge vanishes and Eq. (2.28) furnishes (2004) for a refreshing look into the role of gauge functions in ce- the Hamiltonian of Kepler’s problem. The integral of motion (2.33) lestial mechanics. is a generalization of the equation of the energy. Since the perturbation in Eq. (2.22) is not conservative, we In the Keplerian case (γ = 1 and ξ = 0) the sign of the energy shall focus on the extension of Noether’s theorem to nonconserva- determines the type of solution. Negative values ofκ ˜1 yield elliptic tive systems by Djukic and Vujanovic (1975). It must be F q′ f , orbits, positive values correspond to hyperbolas, and the orbits are i − i 0 for the conservation law to hold (Vujanovic et al. 1986). For the parabolic for vanishingκ ˜1. We shall extend this classification to case of nonconservative systems the generators Fi, f , and the gauge the general case γ , 1: the solutions will be classified as elliptic Ψ need to satisfy the following relation: (˜κ1 < 0), parabolic (˜κ1 = 0), and hyperbolic (˜κ1 > 0) orbits. ∂L ∂L F + (F′ q′ f ′) + Q (F q′ f ) ∂q i ∂q i i i i i 2.2.2 Generalized equation of the angular momentum i i i′ − − X ( ! ! ) ∂L In the unperturbed problem θ is an ignorable coordinate. Indeed, a + f ′L + f = Ψ′. (2.29) ∂τ simple translation in θ (a rotation) with f = Fρ =Ψ= 0 and Fθ = 1

MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 5

2 yields the conservation of the angular momentum. In order to ex- Since δ > 0 no matter the values of ξ or γ, the sign of κ1 is not tend this first integral to the perturbed case, we consider the same affected by the transformation S . If the solution to the original generator Fθ = 1. However, solving for the gauge and the remain- problem (2.1) is elliptic, the solution to the reduced problem (2.21) ing generators in Killing equations yields the nontrivial functions will be elliptic too, and vice-versa. The transformation reduces to a series of scaling factors affecting each variable independently. The integral of the angular momentum transforms into ρ′ γ 1 F = (1 v˜ − ), ρ 2 γ 1 ˙ θ′ − r v − θ κ2 γ κ˜2 = = = κ2 = αδ κ˜2. (2.44) γ 1 αδγ αδγ ⇒ 1 v − 2 γ 3 f = − + (1 γ)ρ θ′v˜ − , γ θ′ − The constantκ ˜2 remains positive, although the scaling factor αδ 1 2 γ 1 γ 1 2 can modify its value significantly. Ψ= v˜ ρ (3 γ)ρv˜ − + 2 v˜ − [2γ (3 γ)ρ′ ρ] 2ρθ′ − − − − − The transformation is defined in terms of three parameters: α, n h γ 3 4 4 i 4 2 ξ and γ. For ξ = ξγ, with + v˜ − [ρ(ρ θ′ ρ′ ) 2(1 γ)ρ′ ] . − − − α o (2.35) ξγ = 1 , (2.45) − γ They satisfy F q , 0. Noether’s theorem holds and Eq. (2.28) i i′ S reduces to the identity map. Choosing α = 1 the special values furnishes the integral− of motion of ξ that yield trivial transformations for γ = 1, 2, 3 and 4 are, re- 2 γ 1 ρ v˜ − θ′ =Λ κ˜2. (2.36) spectively, ξ = 0, 1/2, 2/3 and 3/4. The similarity transformation ≡ γ ff This first integral is none other than a generalized form of the can be understood from a di erent approach: solving the simplified conservation of the angular momentum. Indeed, making γ = 1 problem (2.20) is equivalent to solving the full problem (2.11), but = Eq. (2.36) reduces to setting ξ ξγ. Combining Eqs. (2.15) renders 2 ρ θ′ = κ˜ , (2.37) 2 dθ tan ψ = . (2.46) whereκ ˜2 coincides with the angular momentum of the particle. In dρ ρ addition, the generators F and f , and the gauge vanish when γ ρ The right-hand side of this equation can be written as a function of equals unity. ρ alone thanks to By recovering Eq. (2.15) the integral of motion (2.36) can be written: nκ˜2 tan ψ = . (2.47) ρv˜γ sin ψ = κ˜ (2.38) (˜vγρ)2 κ˜2 2 − 2 in terms of the coordinates intrinsic to the trajectory. The fact that The parameterq n is n = +1 for orbits in a raising regime (˙r > 0), sin ψ 1 forces and n = 1 for a lowering regime (˙r < 0). The velocity is solved ≤ − γ 2 2γ 2 from Eq. (2.33). Integrating Eq. (2.46) furnishes the solution θ(ρ), κ˜2 v˜ ρ = κ˜2 v˜ ρ . (2.39) ≤ ⇒ ≤ which can then be inverted to define the trajectory ρ(θ). The second step is possible because all variables are positive. Com- bining this expression with Eq. (2.33) yields γ 2.4 Solvability 2 κ˜2 κ˜ + ρ2. (2.40) 2 ≤ 1 γρ The trajectory of the particle is obtained upon integration and in- ! Depending on the values of γ,Eq.(2.40) may define upper or lower version of Eq. (2.46). This equation can be written limits to the values that the radius ρ can reach. In general, this con- dθ nκ˜ = 2 dition can be resorted to provide the polynomial constraint , (2.48) dρ Psol(ρ)

Pnat(ρ) 0, (2.41) ≥ where Ppsol(ρ) is a polynomial in ρ, in particular where P (ρ) is a polynomial of degree γ in ρ whose roots dic- nat P (ρ) = ρ2P (ρ). (2.49) tate the nature of the solutions. This inequality will be useful for sol nat defining the different families of orbits. The roots of Psol(ρ) determine the form of the solution and coincide with those of Pnat(ρ) (obviating the trivial ones). The integration of Eq. (2.48) depends on the factorization of Psol(ρ). This polynomial 2.3 Properties of the similarity transformation expression can be expanded thanks to the binomial theorem: S The main property of the similarity transformation is that it does γ k γ 2 4 k γ k 2 not change the type of the solution, i.e. the sign ofκ ˜ is not altered. ρ = ρ κ − ρ κ 1 Psol( ) k − ˜1 ˜2 (2.50) 1 k γ − Applying the inverse similarity transformation S to Eq. (2.33) k=0 − X   yields   with γ , 0 an integer. For γ 4 the polynomial is of degree four: = = ≤ v2 2α v2α 2α α 2µ when γ 1 or γ 2 there are two trivial roots and Eq. (2.48) κ˜ = = = v2 (1 ξ) can be integrated using elementary functions; when γ = 3 or 1 δ2 − γr µγ(1 ξ) − γr µγ(1 ξ) − r − − − " # γ = 4 it yields elliptic integrals. Negative values of γ or positive (2.42) values greater than four lead to a polynomial Psol(ρ) with degree and results in five or above. The solution can no longer be reduced to elemen- 2µ 2µ tary functions nor elliptic integrals (Whittaker and Watson 1927, κ = δ2 κ˜ = v2 (1 ξ) = v2 ∗ . (2.43) 1 1 − r − − r p. 512), for it is given by hyperelliptic integrals. This special class

MNRAS 000, 1–16 (2016) 6 J. Roa of Abelian integrals can only be inverted in very speciﬁc situations 4 CASE γ = 2: GENERALIZED LOGARITHMIC (see Byrd and Friedman 1954, pp. 252–271). Thus, we shall focus SPIRALS on the solutions to the cases γ = 1, 2, 3 and 4. The case γ = 2 yields the family of generalized logarithmic spirals The following sections 3–6 present the corresponding families found by Roa et al. (2016a) in the context of interplanetary mission of orbits. For γ = 1 the solutions to the reduced problem are Keple- design. An extended version and the solution to the spiral Lambert rian orbits. For the case γ = 2 the solutions are called generalized problem can be found in following sequels (Roa and Peláez 2016; logarithmic spirals, because forκ ˜ = 0 the particle describes a log- 1 Roa et al. 2016b). Negative values ofκ ˜ deﬁne the so called elliptic arithmic spiral (Roa et al. 2016a). The cases γ = 3 and γ = 4 yield 1 spirals, positive values yield hyperbolic spirals, and the limit case generalized cardioids and generalized sinusoidal spirals, respec- κ˜ = 0 corresponds to parabolic spirals, which turn out to be pure tively (forκ ˜ = 0 the orbits are cardioids and sinusoidal spirals). 1 1 logarithmic spirals.

4.1 Elliptic motion

3 CASE γ = 1: CONIC SECTIONS For the caseκ ˜1 < 0 the inequality in Eq. (2.40) translates into ρ < ρmax, with For γ = 1 the integrals of motion (2.33) and (2.36) reduce to the 1 κ˜ normalized equations of the energy and angular momentum, re- ρ = − 2 . (4.1) max ( κ˜ ) spectively. The condition on the radius given by Eq. (2.40) becomes − 1 Here ρmax behaves as the apoapsis of the elliptic spiral. In addition, it forcesκ ˜2 1. Spirals of this type initially in raising regime 2 2 ≤ Pnat(ρ) κ˜1ρ + 2ρ κ˜2 0. (3.1) will reach ρ , then transition to lowering regime and fall toward ≡ − ≥ max the origin. If they are initially in lowering regime the radius will For the caseκ ˜ < 0 (elliptic solution) this translates into ρ 1 ∈ decrease monotonically. The velocity at apoapsis is the minimum [ρmin,ρmax], where possible velocity, and reads 1 2 κ˜ κ˜ κ˜ ρmin = 1 1 + κ˜1κ˜2 2 1 2 ( κ˜ ) − v˜m = = − . (4.2) 1 ρ 1 κ˜ − q (3.2) s max r 2 1 2 − ρmax = 1 + 1 + κ˜1κ˜2 . ( κ˜1) Introducing the spiral anomaly ν: − q ℓ These limits are none other than the periapsis and apoapsis radii, ν(θ) = (θ θm) (4.3) κ˜2 − provided thatκ ˜1 andκ ˜2 relate to the semimajor axis and eccentricity by means of: and with ℓ = 1 κ˜2, Eq. (2.46) can be integrated and inverted to − 2 provide the equation of the trajectory: 1 2 q = a˜ and 1 + κ˜1κ˜2 = e˜. (3.3) ( κ˜1) ρ(θ) 1 + κ˜2 − q = . (4.4) ρmax 1 + κ˜2 cosh ν Forκ ˜1 = 0 (parabolic case) the semimajor axis becomes infinite, 2 The angle θm defines the orientation of the apoapsis, i.e. ρ(θm) = and Eq. (3.1) has only one root corresponding to ρ = κ˜2/2. Note that it must beκ ˜2 > 1/κ˜ . Similarly, for hyperbolic orbits (˜κ > 0) ρmax. It can be solved form the initial conditions: 2 − 1 1 it is ρ ρmin as ρmax becomes negative. 2 ≥ nκ˜2 1 ℓ Thus, the solution is simply a conic section: θm = θ0 + arccosh 1 + . (4.5) ℓ − κ˜2 κ˜1ρ0 " !# ˜ 2 κ˜2 The trajectory is symmetric with respect to θ , provided that ρ(θ + h 2 m m ρ(θ) = = . (3.4) ∆ = ∆ 1 + e˜ cos(θ θm) 2 θ) ρ(θm θ),and it is plotted in Fig. 2a (see Roa 2016, Chap. 9, − 1 + 1 + κ˜1κ˜ cos(θ θm) − 2 − for details on the symmetry properties). q The angle θm = Ω + ω defines the direction of the line of apses in frame I, meaning that θ θm is the true anomaly. Ifκ ˜1 < 0, then 4.2 Parabolic motion: the logarithmic spiral ρ(θ ) = ρ , and ρ(θ + π−) = ρ . The velocityv ˜ follows from the m min m max κ integral of the energy: For parabolic spirals Eq. (2.40) reduces to ˜2 1, meaning that there are no limit radii. Spirals in raising regime≤ escape to infinity, and spirals in lowering regime fall to the origin. The limit v˜ = κ˜1 + 2/ρ. (3.5) limρ v˜ = 0 shows that the particle reaches infinity along a spiral →∞ It isp minimum at apoapsis and maximum at periapsis. branch. 1 Applying the similarity transformation S − to the previous The particle follows a pure logarithmic spiral, solution leads to the extended integral (θ θ0)cot ψ ρ(θ) = ρ0 e − , (4.6) 2 2 κ1 κ˜1 v µ v µ∗ = = δ2 = (1 ξ) = . (3.6) keeping in mind that cot ψ nℓ/κ˜2. See Fig. 2b for an example. In 2 2 2 − r − 2 − r the reduced problem the velocity matches the local circular velocity,v ˜ = 1/ρ, and the parameter ξ modifies the velocity in the full The factor µ∗ = µ(1 ξ) behaves as a modified gravitational pa- − problem: rameter. This kind of solutions arise from, for example, the effect p of the solar radiation pressure directed along the Sun-line on a par- 2µ(1 ξ) 2µ∗ ticle following a heliocentric orbit (McInnes 2004, p. 121). v = δv˜ = − = . (4.7) r r r r MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 7

(a) Elliptic (b) Parabolic (c) Hyperbolic Type I (d) Hyperbolic Type II (e) Hyperbolic transition Figure 2. Examples of generalized logarithmic spirals (γ = 2). A zoomed view of the periapsis of Type II hyperbolic spirals is included.

2 2 Note that when ξ = 1/2 (or µ∗ = µ/2) this is the true circular The spiral anomaly ν(θ) is deﬁned in Eq. (4.3), with ℓ = κ˜2 1. velocity. The line of apses follows the direction of −

2 nκ˜2 1 ℓ θm = θ0 arccos 1 . (4.13) − ℓ κ˜2 κ˜1ρ0 − 4.3 Hyperbolic motion " !# Equation (4.12 ) depends on the spiral anomaly by means of cos ν. κ > Under the assumption ˜1 0 the polynomial constraint in Thus, the trajectory is symmetric with respect to θ : the apse line Eq. (2.40) transforms into ρ ρ , with m ≥ min is the axis of symmetry. The symmetry of the trajectory proves that κ˜2 1 there are two values of ν that cancel the denominator: there are two ρmin = − . (4.8) κ˜1 asymptotes, deﬁned explicitly by

This equation yields two different cases: whenκ ˜2 1 the peri- κ˜2 1 ≤ θas = θm arccos . (4.14) apsis radius ρ becomes negative, which means that the spiral ± ℓ − κ˜ min 2 ! κ > reaches the origin when in lowering regime. Conversely, for ˜2 1 Both asymptotes are symmetric with respect to the line of apses. there is an actual periapsis; a spiral in lowering regime will reach The shape of the hyperbolic spirals of Type II can be analyzed in ρ min, then transition to raising regime and escape. Hyperbolic spi- Fig. 2d. The figure includes a zoomed view of the periapsis region. rals withκ ˜2 < 1 are of Type I, whereasκ ˜2 > 1 defines hyperbolic spirals of Type II. Hyperbolic spirals reach infinity with a finite, nonzero velocity 4.3.3 Transition between Type I and Type II hyperbolic spirals limρ v˜ = v˜ √κ˜1. That is, the generalized constant of the →∞ ∞ ≡ 2 Hyperbolic spirals of Type I have been defined forκ ˜2 < 1, whereas energyκ ˜1 is equivalent to the characteristic energyv ˜ = C3. ∞ κ˜2 > 1 yields hyperbolic spirals of Type II. In the limit caseκ ˜2 = 1 the equations of motion simplify noticeably; the resulting spiral is

4.3.1 Hyperbolic spirals of Type I 2/κ˜1 ρ(θ) = . (4.15) ν(ν + 2n) The trajectory described by a hyperbolic spiral withκ ˜2 < 1 (Type I) takes the form The angular variable ν(θ) is defined with respect to the orientation of the asymptote, i.e. ℓ2/κ˜ ρ(θ) = 1 . (4.9) ν ν ν ν(θ) = θas θ. (4.16) 2 sinh 2 sinh 2 + ℓ cosh 2 − The asymptote is fixed by In this case the spiral anomaly ν(θ) reads nℓ 2 ν(θ) = (θ θ), (4.10) θas = θ0 n 1 1 + . (4.17) κ˜ as − −  − s κ˜1ρ0  2     where the direction of the asymptote is solved from Qualitatively, this type of solution is equivalent to a Type II spiral with ρmin 0. The trajectory is similar to Type I spirals, or to one nκ˜ κ˜ (ℓ cos ψ + 1 κ˜ sin ψ ) → θ = θ + 2 ln 2 | 0| − 2 0 . (4.11) half of a Type II spiral (see Fig. 2e). as 0 ℓ (˜κ sin ψ )(1 + ℓ) " 2 − 0 # An example of a Type I hyperbolic spiral connecting the origin with infinity can be found in Fig. 2c. The dashed line represents the 5 CASE γ = 3: GENERALIZED CARDIOIDS asymptote. The condition in Eq. (2.40), yields the polynomial inequality: 4˜κ 8 P (ρ) κ˜3 ρ3 + 2˜κ2 ρ2 + 1 κ˜2 ρ + 0. (5.1) 4.3.2 Hyperbolic spirals of Type II nat ≡ 1 1 3 − 2 27 ≥ ! Hyperbolic spirals of Type II are defined by The discriminant ∆ of the polynomial Pnat(ρ) predicts the nature of the roots. It is ρ(θ) 1 + κ˜2 = . (4.12) 3 4 2 + ∆= 4˜κ κ˜2(3˜κ1 κ˜2). (5.2) ρmin 1 κ˜2 cos ν − 1 − MNRAS 000, 1–16 (2016) 8 J. Roa

The intermediate value theorem shows that there is at least one real 5.2 Parabolic motion: the cardioid root. For the elliptic case (˜κ1 < 0) it is ∆ < 0, meaning that the other When the constant of the generalized energyκ ˜1 vanishes the con- two roots are complex conjugates. In the hyperbolic case (˜κ1 > 0) 2 dition in Eq. (2.40) translates into ρ<ρmax, where the maximum the sign of the discriminant depends on the values ofκ ˜2:ifκ ˜2 > 3˜κ1 2 radius ρmax takes the form it is ∆ > 0, and forκ ˜2 < 3˜κ1 the discriminant is negative. This 8 behavior yields two types of hyperbolic solutions. = ρmax 2 . (5.14) 27˜κ2 5.1 Elliptic motion Parabolic generalized cardioids, unlike logarithmic spirals or Kep- lerian parabolas, are bounded (they never escape the gravitational The nature of elliptic motion is determined by the polynomial con- attraction of the central body). straint in Eq. (5.1). The only real root is given by The line of apses is deﬁned by: 2 3 2 Λ(Λ+ 2˜κ1) + 3˜κ1κ˜2 π 27 ρ = (5.3) 2 1 3 θm = θ0 + n + arcsin 1 κ˜2ρ0 . (5.15) 3( κ˜1) Λ 2 − 4 − " !# with The equation of the trajectory reveals that the orbit is in fact a pure 1/3 cardioid:2 Λ= ( κ˜ ) 3( κ˜ )˜κ2 3˜κ (˜κ2 3˜κ ) + 3˜κ . (5.4) − 1 − 1 2 − 1 2 − 1 1 ρ(θ) 1 q = [1 + cos(θ θm)]. (5.16) Equation (5.1) reduces to ρ ρ 0 and we shall write ρmax 2 − − 1 ≤ This curve is symmetric with respect to θ . Figure 3b depicts the ρmax ρ1. (5.5) m ≡ geometry of the solution. Thus, elliptic generalized cardioids never escape to inﬁnity because they are bounded by ρmax. When in raising regime they reach the apoapsis radius ρmax, then transition to lowering regime and fall 5.3 Hyperbolic motion toward the origin. The velocity at apoapsis, The inequality in Eq. (2.40) determines the structure of the solu- v˜m = κ˜1 + 2/(3ρmax), (5.6) tions and the sign of the discriminant (5.2) governs the nature of its roots. There are two types of hyperbolic generalized cardioids: is thep minimum velocity in the cardioid. forκ ˜ < √3˜κ the cardioids are of Type I, and forκ ˜ > √3˜κ the Equation (2.46) can be integrated from the initial radius r to 2 1 2 1 0 cardioids are of Type II. the apoapsis of the cardioid, and the result provides the orientation of the line of apses:

nκ˜2 5.3.1 Hyperbolic cardioids of Type I θm = θ0 + [2K(k) F(φ0, k)], (5.7) 3/2 √AB( κ˜1) − For hyperbolic cardioids of Type I there is only one real root. The − where K(k) and F(φ0, k) are the complete and incomplete elliptic other two are complex conjugates. The real root is integrals of the first kind, respectively. Introducing the auxiliary Λκ˜2(2 +Λκ˜ ) + 3˜κ2 = 1 1 2 term ρ3 3 , (5.17) − 3˜κ1Λ λ = ρ ρ (5.8) ij i − j having introduced the auxiliary parameter their argument and modulus read 1/3 κ2 3˜2 2 2 2 Λ= 3 κ˜1 + 3(3˜κ1 κ˜2) . (5.18) Bλ10 Aρ0 ρmax (A B) κ˜9/2 − φ0 = arccos − , k = − − . (5.9)  1  Bλ + Aρ 4AB  p q  " 10 0 # r Provided that Λ > 0, Eq. (5.17) shows that ρ < 0. Therefore, there   3 The previous definitions involve the auxiliary parameters: are no limits to the values that ρ can take. As a consequence, the cardioid never transitions between regimes. If it is initially ψ < A = (ρ b )2 + a2, B = b2 + a2 (5.10) 0 max − 1 1 1 1 π/2 it will always escape to infinity, and fall toward the origin for q q and ψ0 > π/2. Λ(Λ 4˜κ2) + 3˜κ3κ˜2 (Λ2 3˜κ3κ˜2)2 The equation of the trajectory for hyperbolic generalized car- b = − 1 1 2 , a2 = − 1 2 . (5.11) dioids of Type I is 1 3 1 6 2 6˜κ1Λ 12˜κ1Λ ρ(θ) (A + B) sn2(ν, k) + 2B[cn(ν, k) 1] Recall the definition of Λ in Eq. (5.4). = − . (5.19) ρ A (A + B)2 sn2(ν, k) 4AB The equation of the trajectory is obtained by inverting the 3 − function θ(ρ), and results in: It is defined in terms of

1 2 2 2 2 ρ(θ) A 1 cn(ν, k) − A = b + a , B = (ρ3 b1) + a , (5.20) = 1 + − . (5.12) 1 1 − 1 ρmax B 1 + cn(ν, k) q q ( " #) It is defined in terms of the Jacobi elliptic function cn(ν, k). The 2 The cardioid is a particular case of the limaçons, curves first studied by anomaly ν reads the amateur mathematician Étienne Pascal in the 17th century. The general 3/2 form of the limaçon in polar coordinates is r(θ) = a + b cos θ. Depending ( κ˜1) ν(θ) = − √AB (θ θm). (5.13) on the values of the coefficients the curve might reach the origin and form κ˜2 − 1 loops. It is worth noticing that the inverse of a limaçon, r(θ) = (a+b cos θ)− , Equation (5.12) is symmetric with respect to the apse line, ρ(θm + results in a conic section. Limaçons with a = b are considered part of the ∆θ) = ρ(θ ∆θ), as shown in Fig. 3a. family of sinusoidal spirals. m − MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 9

(a) Elliptic (b) Parabolic (c) Hyperbolic Type I (d) Hyperbolic Type II (e) Hyperbolic transition Figure 3. Examples of generalized cardioids (γ = 3). which require spirals are interior, whereas for ρ ρ they are exterior spirals. ≥ 1 2 2 2 3 2 2 The geometry of the forbidden region can be analyzed in Fig. 3d. Λκ˜ (Λκ˜1 4) + 3˜κ (Λ κ˜ 3˜κ ) b = 1 − 2 , a2 = 1 − 2 . (5.21) The particle cannot enter the barred annulus, whose limits coincide 1 3Λ 1 Λ2 6 6˜κ1 6 κ˜1 with the periapsis and apoapsis of the exterior and interior orbits, There are no axes of symmetry. Therefore, the anomaly is referred respectively. directly to the initial conditions: The axis of symmetry of interior spirals is given by

3/2 κ˜ 2nκ˜2[K(k) F(φ0, k)] 1 θm = θ0 + − (5.29) ν(θ) = √AB (θ θ0) + F(φ0, k). (5.22) 3/2 κ˜2 − κ˜1 √ρ1λ23 The moduli of both the Jacobi elliptic functions and the elliptic in terms of the arguments: integral, and the argument of the latter, are

ρ0λ23 ρ2λ13 (A + B)2 ρ2 λ ρ φ0 = arcsin , k = . (5.30) 3 A 03 B 0 ρ2λ03 ρ1λ23 k = − , φ0 = arccos − . (5.23) s s s 4AB Aλ03 + Bρ0 " # The trajectory simplifies to: The cardioid approaches infinity along an asymptotic branch, with v˜ v˜ as ρ . The orientation of the asymptote follows from ρ(θ) 1 = → ∞ → ∞ 2 . (5.31) the limit ρ3 1 + (λ32/ρ2) dc (ν, k) nκ˜ θ = lim θ(ρ) = θ + 2 F(φ , k) F(φ , k) . (5.24) Here we made use of Glaisher’s notation for the Jacobi elliptic as 0 3/2 0 ρ √ ∞ − 3 →∞ κ˜1 AB functions, so dc(ν, k) = dn(ν, k)/ cn(ν, k). The spiral anomaly ν Here, the value of φ , takes the form ∞ 3/2 A B κ˜1 √ρ1λ23 φ = arccos − , (5.25) ν(θ) = (θ θm). (5.33) ∞ A + B 2˜κ2 − is defined as φ = limρ φ. An example of a hyperbolic cardioid ∞ →∞ The trajectory is symmetric with respect to the line of apses defined of Type I with its corresponding asymptote is presented in Fig. 3c. by θm. For the case of exterior spirals the largest root ρ1 behaves as 5.3.2 Hyperbolic cardioids of Type II the periapsis. A cardioid initially in lowering regime will reach ρ1, then it will transition to raising regime and escape to infinity. Equa- For hyperbolic cardioids of Type II the polynomial in Eq. (2.40) tion (2.46) is integrated from the initial radius to the periapsis to admits three distinct real roots, ρ1,ρ2,ρ3 , given by provide the orientation of the line of apses: { } π 2˜κ2 1 √3˜κ1 2 2nκ˜2 F(φ0, k) ρ + = cos (2k + 1) arccos (5.26) = k 1 θm θ0 3/2 , (5.34) 3 3 − 3 κ˜2 − 3˜κ1 − κ˜ √ρ1λ23 3˜κ1 " !# 1 q with with k = 0, 1, 2. The roots are then sorted so that ρ1 > ρ2 > ρ3. Since λ23λ01 ρ2λ13 8 φ0 = arcsin , k = (5.35) ρ ρ ρ = < , λ13λ02 ρ1λ23 1 2 3 3 0 (5.27) r s − 27˜κ1 the argument and modulus of the elliptic integral. then ρ3 < 0 and also ρ1 > ρ2 > 0 for physical coherence. The polynomial constraint reads

(ρ ρ )(ρ ρ )(ρ ρ ) 0. (5.28) 3 − 1 − 2 − 3 ≥ Glaisher’s notation establishes that if p,q,r are any of the four letters s,c,d,n, then: and holds for both ρ > ρ1 and ρ < ρ2. The integral of mo- pr(ν, k) 1 tion (2.33) shows that both situations are physically admissible, pq(ν, k) = = . (5.32) 2 2 becausev ˜ (ρ1) > 0 andv ˜ (ρ2) > 0. There are two families of so- qr(ν, k) qp(ν, k) lutions that lie outside the annulus ρ < [ρ ,ρ ]. When ρ ρ the Under this notation repeated letters yield unity. 2 1 0 ≤ 2 MNRAS 000, 1–16 (2016) 10 J. Roa

The trajectory of exterior spirals is obtained upon inversion of Two example trajectories with n = +1 are plotted in Fig. 3e. The the equation for the polar angle, dashed line corresponds to m = +1 and the solid line to m = 1. The trajectories terminate/emanate from a circular orbit of radius− ρ λ sn2(ν, k) ρ λ ρ(θ) = 2 13 − 1 23 . (5.36) ρ . λ sn2(ν, k) λ lim 13 − 23 The anomaly is redeﬁned as 3/2 6 CASE γ = 4: GENERALIZED SINUSOIDAL SPIRALS κ˜1 √ρ1λ23 ν(θ) = (θ θm). (5.37) 2˜κ2 − Setting γ = 4inEq.(2.40) gives rise to the polynomial inequality This variable is referred to the line of apses, given in Eq. (5.34). The 2 2 4(˜κ1ρ + κ˜1 + κ˜2)ρ + 1 4(˜κ1ρ + κ˜1 κ˜2)ρ + 1 0, (6.1) form of the solution shows that hyperbolic generalized cardioids of − ≥ whichh governs the subfamiliesih of the solutionsi to the problem. The Type II are symmetric with respect to θm. Due to the symmetry of Eq. (5.36) the trajectory exhibits two four roots of the polynomial are symmetric asymptotes, deﬁned by κ˜ κ˜ √κ˜ (˜κ 2˜κ ) =+ 2 1 2 2 1 ρ1,2 − ± 2 − 2˜κ2F(φ , k) 2˜κ1 θ = θ ∞ . (5.38) (6.2) as m ± 3/2 κ˜ + κ˜ √κ˜ (˜κ + 2˜κ ) κ˜1 √ρ1λ23 1 2 2 2 1 ρ , = ∓ 3 4 − 2˜κ2 The argument φ reads 1 ∞ and the discriminant of Pnat(ρ) is λ23 φ = arcsin . (5.39) 6 κ˜2 ∞ λ13 ∆= 2 2 r 20 (˜κ2 4˜κ1). (6.3) κ˜1 − 5.3.3 Transition between Type I and Type II hyperbolic cardioids The sign of the discriminant determines the nature of the four roots.

The limit caseκ ˜2 = √3˜κ1 defines the transition between hyperbolic cardioids of Types I and II. The discriminant ∆ vanishes: the roots 6.1 Elliptic motion are all real and one is a multiple root, ρlim ρ1 = ρ2. The region of ≡ Whenκ ˜1 < 0 there are two subfamilies of elliptic sinusoidal spirals: forbidden motion degenerates into a circumference of radius ρlim. of Type I, withκ ˜2 > 2˜κ1 (∆ > 0), and of Type II, withκ ˜2 < 2˜κ1 The roots take the form: − − (∆ < 0). Both types are separated by the limit caseκ ˜2 = 2˜κ1, 8 1 which makes ∆= 0. − ρ3 = < 0 and ρlim ρ1 = ρ2 = . (5.40) − 3˜κ1 ≡ 3˜κ1 The condition (ρ ρ )(ρ ρ )2 0 holds naturally for ρ > 0. − 3 − lim ≥ 6.1.1 Elliptic sinusoidal spirals of Type I When the cardioid reaches ρlim the velocity becomesv ˜lim = √3˜κ1 = For the caseκ ˜ > 2˜κ the discriminant is positive and the four 1/ √ρlim. It coincides with the local circular velocity. Moreover, 2 − 1 from the integral of motion (2.36) one has roots are real, with ρ1 > ρ2 > ρ3 > ρ4. Since ρ3,4 < 0 Eq. (6.1) reduces to 3 κ˜2 = ρlimv˜lim sin ψlim = sin ψlim = 1 (5.41) ⇒ (ρ ρ1)(ρ ρ2) 0, (6.4) meaning that the orbit becomes circular as the particle approaches − − ≥ meaning that it must be either ρ>ρ or ρ<ρ . The integral of ρ . When ψ = π/2 the perturbing acceleration in Eq. (2.22) 1 2 lim lim motion (2.33) reveals that only the latter case is physically possible, vanishes. As a result, the orbit degenerates into a circular Keple- 2 becausev ˜ (ρ1) < 0. Thus, ρ2 is the apoapsis of the spiral: rian orbit. A cardioid with ρ0 < ρlim and in raising regime will κ˜ κ˜ √κ˜ (˜κ 2˜κ ) reach ρlim and degenerate into a circular orbit with radius ρlim. This = 2 1 2 2 1 ρmax ρ2 − − 2 − (6.5) phenomenon also appears in cardioids with ρ0 >ρlim and in lower- ≡ 2˜κ1 ing regime. and ρ ρ . Equation (2.48) is then integrated from ρ to ρ to ≤ max 0 max The trajectory reduces to define the orientation of the apoapsis, ρ(θ) cosh ν 1 = 2nκ˜2[F(φ0, k) K(k)] − , (5.42) θm = θ0 − . (6.6) ρlim cosh ν + 5/4 2 − κ˜1 √λ13λ24 which is written in terms of the anomaly The arguments of the elliptic integrals are θ θ ρ ν(θ) = − 0 + 2mn arctanh 3 0 . (5.43) λ24λ03 λ23λ14 φ0 = arcsin , k = . (6.7) √3 8ρlim + ρ0 λ λ λ λ r ! r 23 04 r 13 24 The integer m = sign(1 ρ0/ρlim) determines whether the particle Recall that λ = ρ ρ . − ij i − j is initially below (m = +1) or above (m = 1) the limit radius Elliptic sinusoidal spirals of Type I are defined by = = − ρlim. The limit limθ ρ ρlim 1/(3˜κ1) shows that the radius ρ λ cd2(ν, k) ρ λ →∞ = 4 23 3 24 converges to ρlim. This limit only applies to the cases m = n = +1 ρ(θ) 2 − . (6.8) λ23 cd (ν, k) λ24 and m = n = 1. When the particle is initially below ρlim and in − lowering regime,− m =+1, n = 1 , it falls toward the origin. In the The spiral anomaly reads { − } opposite case, m = 1, n = +1 , the cardioid approaches infinity nκ˜2 { − } 1 along an asymptotic branch with ν(θ) = (θ θm) λ13λ24. (6.9) 2˜κ2 − ρ The trajectory is symmetricp with respect to θ , which corresponds θ = θ 2 √3 mn arctanh 3 0 + arctanh(3) . (5.44) m as 0 − 8ρ + ρ to the line of apses. " r lim 0 ! # MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 11

6.1.2 Elliptic sinusoidal spirals of Type II 6.2 Parabolic motion: sinusoidal spiral (off-center circle) Whenκ ˜ < 2˜κ two roots are real and the other two are com- Makingκ ˜ = 0, the condition in Eq. (6.1) simplifies to 2 − 1 1 plex conjugates. In this case the real roots are ρ1,2 and the inequal- 1 ity (6.1) reduces again to Eq. (6.4), meaning that ρ ρ ρ . ρ ρmax = , (6.20) 2 max ≤ 4˜κ However, the form of the solution is different from≤ the trajectory≡ 2 described by Eq. (6.8), being meaning that the spiral is bounded by a maximum radius ρmax. It is equivalent to the apoapsis of the spiral. Its orientation is given by ρ1 B ρ2A (ρ2A + ρ1 B)cn(ν, k) ρ(θ) = − − . (6.10) π ρ0 B A (A + B)cn(ν, k) θm = θ0 + n arcsin . (6.21) − − 2 − ρmax The spiral anomaly is " !# The trajectory reduces to a sinusoidal spiral,4 and its definition 2 nκ˜1 can be directly related to θm: ν(θ) = √AB (θ θm). (6.11) κ˜2 − ρ(θ) = cos(θ θm). (6.22) It is referred to the orientation of the apse line, θm. This variable is ρmax − given by The spiral defined in Eq. (6.22) is symmetric with respect to the

nκ˜2 line of apses. The resulting orbit is a circle centered at (ρmax/2,θm) θm = θ0 + [2K(k) F(φ0, k)] (6.12) 2 √ − (Fig. 4b). Circles are indeed a special case of sinusoidal spirals. κ˜1 AB considering the arguments: 6.3 Hyperbolic motion 2 2 Aλ02 + Bλ10 (A + B) λ12 Given the discriminant in Eq. (6.3), for the caseκ ˜ > 0 the values of φ = arccos , k = − . (6.13) 1 0 Aλ Bλ s 4AB the constantκ ˜ define two different types of hyperbolic sinusoidal " 02 − 10 # 2 spirals: spirals of Type I (˜κ < 2˜κ ), and spirals of Type II (˜κ > The coefficients A and B are defined in terms of 2 1 2 2˜κ1). κ˜ (˜κ + 2˜κ ) κ˜ + κ˜ a2 = 2 2 1 and b = 1 2 , (6.14) 1 − 4˜κ4 1 − 2˜κ2 1 1 6.3.1 Hyperbolic sinusoidal spirals of Type I namely Ifκ ˜2 < 2˜κ1, then ρ1,2 are complex conjugates and ρ3,4 are both real 2 2 2 2 but negative. Therefore, the condition in Eq. (6.1) holds naturally A = (ρ1 b1) + a , B = (ρ2 b1) + a . (6.15) − 1 − 1 for any radius and there are no limitations to the values of ρ. The q q The fact that the Jacobi function cn(ν, k) is symmetric proves that particle can either fall to the origin or escape to infinity along an elliptic sinusoidal spirals of Type II are symmetric. asymptotic branch. The equation of the trajectory is given by: ρ B ρ A + (ρ B + ρ A) cn(ν, k) ρ(θ) = 3 − 4 3 4 , (6.23) B A + (A + B) cn(ν, k) 6.1.3 Transition between spirals of Types I and II − where the spiral anomaly can be referred directly to the initial con- In this is particular case of elliptic motion, 2˜κ1 = κ˜2, the roots of ditions: − polynomial Pnat(ρ) are 2 nκ˜1 ν(θ) = √AB (θ θ0) + F(φ0, k). (6.24) 3 + 2 √2 3 2 √2 1 κ˜2 − ρ1 = , ρ2 ρmax = − , ρ3,4 = . (6.16) κ˜2 ≡ κ˜2 − κ˜2 This definition involves an elliptic integral of the first kind with These results simplify the definition of the line of apses to argument and parameter:

2 2 √ + 2 Aλ04 + Bλ30 (A + B) λ 2(1 ρ0κ˜2) (3 ρ0κ˜2) 8 = = − 34 θm = θ0 + √2 n ln − − − . (6.17) φ0 arccos , k . (6.25) 1 + ρ κ˜ Aλ04 Bλ30 s 4AB  p0 2  " − #   The coefficients A and B require the terms: Introducing the spiral anomaly ν(θ),  κ˜ + κ˜ κ˜ (˜κ + 2˜κ ) = 1 2 2 = 2 2 1 √2 b1 2 , a1 4 , (6.26) ν(θ) = (θ θ ), (6.18) − 2˜κ1 − 4˜κ1 2 − m being the equation of the trajectory takes the form A = (ρ b )2 + a2, B = (ρ b )2 + a2. (6.27) ρ(θ) 5 4 √2 cosh ν + cosh(2ν) 3 − 1 1 4 − 1 1 = − . (6.19) q q ρmax (3 2 √2)[3 cosh(2ν)] − − 4 It was the Scottish mathematician Colin Maclaurin the first to study si- The trajectory is symmetric with respect to the line of apses θm nusoidal spirals. In his “Tractatus de Curvarum Constructione & Mensura”, thanks to the symmetry of the hyperbolic cosine. published in Philosophical Transactions in 1717, he constructed this family Figure 4a shows the three types of elliptic spirals. It is impor- of curves relying on the epicycloid. Their general form is rn = cos(nθ) and tant to note that in all three cases the condition in Eq. (6.1) trans- different values of n render different types of curves; n = 2 correspond to − forms into Eq. (6.4), equivalent to ρ<ρmax. As a result, there are no hyperbolas, n = 1 to straight lines, n = 1/2 to parabolas, n = 1/3 to − − − differences in their nature although the equations for the trajectory Tschirnhausen cubics, n = 1/3 to Cayley’s sextics, n = 1/2 to cardioids, are different. n = 1 to circles, and n = 2 to lemniscates.

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(a) Elliptic (b) Parabolic (c) Hyperbolic Type I (d) Hyperbolic Type II (e) Hyperbolic transition Figure 4. Examples of generalized sinusoidal spirals (γ = 4).

The direction of the asymptote is defined by and it is symmetric with respect to θm. The geometry of the solution is similar to that of hyperbolic cardioids of Type II, mainly nκ˜2 θas = θ0 + F(φ , k) F(φ0, k) . (6.28) because of the existence of the forbidden region plotted in Fig. 4d. κ˜2 √AB ∞ − 1 The asymptotes follow the direction of This definition involves the argument 2˜κ2F(φ , k) λ24 A B θ = θ ∞ , with φ = arcsin . (6.37) as m 2 φ = arccos − . (6.29) ± κ˜ √λ13λ24 ∞ λ14 ∞ A + B 1 r The velocity of the particle when reaching infinity isv ˜ = √κ˜1. Hyperbolic sinusoidal spirals of Type I are similar to the hy∞ perbolic 6.3.3 Transition between Type I and Type II spirals solutions of Type I with γ = 2 and γ = 3. Figure 4d depicts and Whenκ ˜ = 2˜κ the radii ρ and ρ coincide, ρ = ρ ρ , and example trajectory and the asymptote defined by θ . 2 1 1 2 1 2 lim as become equal to the limit radius ≡ 1 6.3.2 Hyperbolic sinusoidal spirals of Type II ρlim = . (6.38) 2˜κ1 In this case the four roots are real and distinct, with ρ3,4 < 0. The equation of the trajectory is a particular case of Eq. (6.36), 2 The two positive roots ρ1,2 are physically valid, i.e.v ˜ (ρ1) > 0 obtained withκ ˜2 2˜κ1: andv ˜2(ρ ) > 0. This yields two situations in which the condition → 2 1 (ρ ρ )(ρ ρ ) 0 is satisfied: ρ>ρ (exterior spirals) and ρ<ρ ρ(θ) 8 − − 1 − 2 ≥ 1 2 = 1 . (6.39) (interior spirals). ρlim − 4 + m(sinh ν 3 cosh ν) " − # Interior spirals take the form The spiral anomaly can be referred to the initial conditions, ρ λ ρ λ sn2(ν, k) ρ(θ) = 2 13 − 1 23 . (6.30) 2 nm 2(1 + 2˜κ1ρ0) + 2 + 8˜κ1ρ0(3 + κ˜1ρ0) λ13 λ23 sn (ν, k) − ν(θ) = (θ θ0) + ln , √2 − m(1 2˜κ1ρ0) The spiral anomaly is  p−   (6.40) 2   κ˜1   ν(θ) = λ13λ24 (θ θm). (6.31) 2˜κ − avoiding additional parameters. Here m = sign(1 ρ0/ρlim) deter- 2 − The orientationp of the line of apses is solved from mines whether the spiral is below or over the limit radius ρlim. The asymptote follows from 2nκ˜2F(φ0, k) θm = θ0 + , (6.32) 2 θas = θ0 n √2 log(1 √2/2). (6.41) κ˜1 √λ13λ24 − − with Like in the case of hyperbolic cardioids, for m = n = 1 and − m = n = +1 spirals of this type approach the circular orbit of ra- λ13λ20 λ23λ14 φ0 = arcsin , k = . (6.33) dius ρlim asymptotically, i.e. limθ ρ = ρlim. When approaching λ λ λ λ →∞ r 23 10 r 13 24 a circular orbit the perturbing acceleration vanishes and the spiral Interior hyperbolic spirals are bounded and their shape is similar to converges to a Keplerian orbit. See Fig. 4e for examples of hy- that of a limaçon. perbolic sinusoidal spirals with m = +1, n = +1 (dashed) and The line of apses of an exterior spiral is defined by m = 1, n =+1 (solid). { } { − } 2nκ˜ F(φ , k) θ = θ 2 0 . (6.34) m 0 2 − κ˜1 √λ13λ24 The modulus and the argument of the elliptic integral are 7 SUMMARY The solutions presented in the previous sections are summarized in λ24λ01 λ23λ14 φ0 = arcsin , k = . (6.35) Table 1, organized in terms of the values of γ. Each family is then λ λ λ λ r 14 02 r 13 24 divided in elliptic, parabolic, and hyperbolic orbits. The table in- The trajectory becomes cludes references to the corresponding equations of the trajectories ρ λ + ρ λ sn2(ν, k) for convenience. The orbits are said to be bounded if the particle ρ(θ) = 1 24 2 41 (6.36) 2 can never reach infinity, because r < r . λ24 + λ41 sn (ν, k) lim

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Table 1. Summary of the families of solutions.

Family Type γ κ˜1 κ˜2 Bounded Trajectory

Elliptic 1 < 0 > √ 1/κ˜1 Y Eq. (3.4) − Conic sections Parabolic 1 = 0 N Eq. (3.4) − Hyperbolic 1 > 0 N Eq. (3.4) − Elliptic 2 < 0 < 1 Y Eq. (4.4) Generalized Parabolic 2 = 0 1 N Eq. (4.6) ≤ ∗ logarithmic Hyperbolic T-I 2 > 0 < 1 N Eq. (4.9) spirals Hyperbolic T-II 2 > 0 > 1 N Eq. (4.12) Hyperbolic trans. 2 > 0 = 1 N Eq. (4.15)

Elliptic 3 < 0 < 1 Y Eq. (5.12) Parabolic 3 = 0 1 Y Eq. (5.16) ≤ † Generalized Hyperbolic T-I 3 > 0 < √3˜κ1 N Eq. (5.19) cardioids Hyperbolic T-II (int) 3 > 0 > √3˜κ1 Y Eq. (5.31) Hyperbolic T-II (ext) 3 > 0 > √3˜κ1 N Eq. (5.36) Hyperbolic trans. 3 > 0 = √3˜κ1 Y/N Eq. (5.42)

Elliptic T-I 4 < 0 > 2˜κ1 Y Eq. (6.8) − Elliptic T-II 4 < 0 < 2˜κ1 Y Eq. (6.10) − Generalized Elliptic trans. 4 < 0 = 2˜κ1 Y Eq. (6.19) − sinusoidal Parabolic 4 = 0 Y Eq. (6.22) − ‡ spirals Hyperbolic T-I 4 > 0 < 2˜κ1 N Eq. (6.23) Hyperbolic T-II (int) 4 > 0 > 2˜κ1 Y Eq. (6.30) Hyperbolic T-II (ext) 4 > 0 > 2˜κ1 N Eq. (6.36) Hyperbolic trans. 4 > 0 = 2˜κ1 Y/N Eq. (6.39)

∗ Logarithmic spiral. † Cardioid. ‡ Sinusoidal spiral (oﬀ-center circle).

8 UNIFIED SOLUTION IN WEIERSTRASSIAN Table 2. Coefficients ai of the polynomial f (s), and factor k. FORMALISM γ a a a a a k The orbits can be unified introducing the Weierstrass elliptic func- 0 1 2 3 4 2 tions. Indeed, Eq. (2.48) furnishes the integral expression 1κ ˜1 1/2 κ˜2/6 0 0 nκ˜2 2 − 2 r 2κ ˜1 κ1/2 (1 κ˜2)/6 0 0 nκ˜2 k ds 3 2 − 2 √ θ(r) θ0 = (8.1) 3 27˜κ1 27˜κ1/2 6˜κ1 9˜κ2/2 2 0 27 nκ˜2 4 3 2 − 2 − r0 f (s) 4 16˜κ 8˜κ 4˜κ 8˜κ /3 2˜κ1 1 4nκ˜2 Z 1 1 1 − 2 with p 4 3 2 f (s) Psol(s) = a0 s + 4a1 s + 6a2 s + 4a3 s + a4 (8.2) Symmetric spirals reach a minimum or maximum radius ρm, ≡ which is a root of f (ρ). Thus, Eq. (8.4) can be simplified if referred and a0,1 , 0. Introducing the auxiliary parameters to ρm instead of ρ0:

ϑ = f ′(ρ0)/4 and ϕ = f ′′(ρ0)/24 (8.3) f ′(ρm)/4 ρ(θ) ρm = . (8.7) Eq. (8.1) can be inverted to provide the equation of the trajectory − ℘(zm) f ′′(zm)/24 − (Whittaker and Watson 1927, p. 454), This is the unified solution for all symmetric solutions, with zm = 1 (θ θm)/k. Practical comments on the implementation of the Weier- ρ θ =ρ + − 2 ( ) 0 2 (iv) 2[℘(z) ϕ] f (ρ0) f (ρ0)/48 strass elliptic functions can be found in Biscani and Izzo (2014), − − for example. Although ℘(z) = ℘( z), the derivative ℘′(z) is an − odd function in z, ℘′( z) = ℘′(z). Therefore, the integer n needs [℘(z) ϕ]2ϑ ℘′(z) f (ρ0) + f (ρ0) f ′′′(ρ0)/24 . (8.4) × − − to be adjusted according− to the− regime of the spiral when solving The solutionn is written in termsp of the Weierstrass elliptic ofunction Eq. (8.4): n = 1 for raising regime, and n = 1 for lowering regime. − ℘(z) ℘(z; g , g ) (8.5) ≡ 2 3 and its derivative ℘′(z), where z = (θ θ )/k is the argument and − 0 the invariant lattices g2 and g3 read 9 PHYSICAL DISCUSSION OF THE SOLUTIONS 2 Each family of solutions involves a fundamental curve: the case g2 = a0a4 4a1a3 + 3a2 − (8.6) γ = 1 relates to conic sections, γ = 2 to logarithmic spirals, γ = 3 g = a a a + 2a a a a3 a a2 a2a . 3 0 2 4 1 2 3 − 2 − 0 3 − 1 4 to cardioids, and γ = 4 to sinusoidal spirals. This section is devoted The coefficients ai and k are obtained by identifying Eq. (8.1) with to analyzing the geometrical and dynamical connections between Eq. (2.48) for different values of γ. They can be found in Table 2. the solutions and other integrable systems.

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9.1 Connection with Schwarzschild geodesics The Schwarzschild metric is a solution to the Einstein field equations of the form 2M (dr)2 (ds)2 = 1 (dt)2 r2(dφ)2 r2 sin2 φ (dθ)2, (9.1) − r − 1 2M/r − − ! − written in natural units so that the speed of light and the gravitational constant equal unity, and the Schwarzschild radius reduces to 2M. In this equation M is the mass of the central body, φ = π/2 is the colatitude, and θ is the longitude. The time-like geodesics are governed by the differential equation (a) γ = 3 (b) γ = 4 dr 2 1 2M = (E2 1) r4 + r3 r2 + 2Mr, (9.2) Figure 5. Schwarzschild geodesic (solid line) compared to generalized car- dθ L2 − L2 − ! dioids and generalized sinusoidal spirals (dashed line). where L is the angular momentum and E is a constant of motion related to the energy and defined by outer radii equal to the limit radii of Schwarzschild geodesics. Like

2M dts in the previous example the interior solutions coincide, while the E = 1 (9.3) exterior solutions separate in time. In order for two completely dif- − r dtp ! ferent forces to yield a similar trajectory it suffices that the differ- in terms of the proper time of the particle tp and the Schwarzschild ential equation dr/dθ takes the same form. Even if the trajectory is time ts. Its solution is given by similar the integrals of motion might not be comparable, because r nL ds the angular velocities may be different. As a result, the velocity θ(r) θ0 = (9.4) along the orbit and the time of flight between two points will be dif- − r0 f (s) Z ferent. Equation (2.46) shows that the radial motion is governed by and involves thep integer n = 1, which gives the raising/lowering the evolution of the flight-direction angle. For γ = 3 and γ = 4 the ± regime of the solution. Here f (s) is a quartic function defined acceleration in Eq. (2.22) makes the radius to evolve with the polar like in Eq. (8.2). The previous equation is formally equivalent angle just like in the Schwarzschild metric. Consequently, the orbits to Eq. (8.1). As a result, the Schwarzschild geodesics abide by may be comparable. However, since the integrals of motion (2.33) Eqs. (8.4) or (8.7), which are written in terms of the Weierstrass and (2.36) do not hold along the geodesics, the velocities will not elliptic functions and identifying the coefficients necessarily match. 2 2 a0 = E 1, a1 = M/2, a2 = L /6, − − (9.5) 2 a3 = ML /2, a4 = 0, k = nL. 9.2 Newton's theorem of revolving orbits Compact forms of the Schwarzschild geodesics using the Weier- Newton found that if the angular velocity of a particle following a strass elliptic functions can be found in the literature (see for ex- Keplerian orbit is multiplied by a constant factor k, it is then possi- ample Hagihara 1930, §4). We refer to classical books like the one ble to describe the dynamics by superposing a central force depend- by Chandrasekhar (1983, Chap. 3) for an analysis of the structure ing on an inverse cubic power of the radius. The additional perturb- of the solutions in Schwarzschild metric. ing terms depend only on the angular momentum of the original The analytic solution to Schwarzschild geodesics involves el- orbit and the value of k. liptic functions, like generalized cardioids (γ = 3) and general- Consider a Keplerian orbit defined as ized sinusoidal spirals (γ = 4). If the values ofκ ˜1 andκ ˜2 are ad- h˜ 2 justed in order the roots of P to coincide with the roots of f (r) in ρ(θ) = 1 . (9.8) sol 1 + e˜ cos ν Schwarzschild metric, the solutions will be comparable. For exam- 1 Here ν = θ θ denotes the true anomaly. The radial motion ple, when the polynomial f (r) has three real roots and a repeated 1 − m one the geodesics spiral toward or away from the limit radius of hyperbolic spirals of Type II —Eq. (4.12)— resembles a Kep- lerian hyperbola. The difference between both orbits comes from 2 2 1 L + 4M the angular motion, because they revolve with different angular rlim = . (9.6) 2 2 2M − 2ML(2L + √L 12M ) velocities. Indeed, recovering Eq. (4.12), identifyingκ ˜2 = e˜ and − ρ (1 + κ˜ ) = h˜2, and calling the spiral anomaly ν = k(θ θ ), it If we make this radius coincide with the limit radius of a hyperbolic min 2 1 2 − m is cardioid withκ ˜2 = √3˜κ1 [Eq. (5.40)] it follows h˜ 2 1 1 ρ(θ) = 1 . (9.9) κ˜1 = andκ ˜2 = . (9.7) 1 + e˜ cos ν2 3rlim √rlim When equated to Eq. (9.8), it follows a relation between the spiral Figure 5a compares hyperbolic generalized cardioids and (ν2) and the true (ν1) anomalies: Schwarzschild geodesics with the same limit radius. The interior solutions coincide almost exactly, whereas exterior solutions sep- ν2 = kν1. (9.10) arate slightly. Similarly, when f (r) admits three real and distinct The factor k reads roots the geodesics are comparable to interior and exterior hyper- κ˜2 κ˜2 bolic solutions of Type II. For example, Fig. 5b shows interior and k = = . (9.11) ℓ exterior hyperbolic sinusoidal spirals of Type II with the inner and κ˜2 1 2 − q MNRAS 000, 1–16 (2016) Nonconservative extension of Keplerian integrals 15

Replacing ν1 by ν2/k in Eq. (9.8) and introducing the result in the 10 CONCLUSIONS equations of motion in polar coordinates gives rise to the radial ac- The dynamical symmetries in Kepler’s problem hold under a spe- celeration that renders a hyperbolic generalized logarithmic spiral cial nonconservative perturbation: a disturbance that modifies the of Type II, namely tangential and normal components of the gravitational acceleration 1 h˜2 h˜ 2 in the intrinsic frame renders two integrals of motion, which are a˜ = + 1 (1 k2) = a˜ + 1 (1 k2). (9.12) r,2 − ρ2 ρ3 − r,1 ρ3 − generalized forms of the equations of the energy and angular momentum. The existence of a similarity transformation that reduces This is in fact the same result predicted by Newton’s theorem of the original problem to a system perturbed by a tangential unipara- revolving orbits. metric forcing simplifies the dynamics significantly, for the integra- The radial accelerationa ˜ yields a hyperbolic generalized r,2 bility of the system is evaluated in terms of one single parameter. logarithmic spiral of Type II. This is a central force which pre- The algebraic properties of the equations of motion dictate what serves the angular momentum, but not the integral of motion in values of the free parameter make the problem integrable in closed Eq. (2.36). Thus, a particle accelerated bya ˜ describes the same r,2 form. trajectory as a particle accelerated by the perturbation in Eq. (2.22), The extended integrals of motion include the Keplerian ones but with different velocities. As a consequence, the times between as particular cases. The new conservation laws can be seen as gen- two given points are also different. The acceleration derives from eralizations of the original integrals. The new families of solutions the specific potential are defined by fundamental curves in the zero-energy case, and

V(ρ) = Vk(ρ) + ∆V(ρ), (9.13) there are geometric transformations that relate different orbits. The orbits can be unified by introducing the Weierstrass elliptic func- where Vk(ρ) denotes the Keplerian potential, and ∆V(ρ) is the per- tions. This approach simplifies the modeling of the system. turbing potential: The solutions derived in this paper are closely related to dif- ˜2 ferent physical problems. The fact that the magnitude of the accel- h1 ∆V(ρ) = (1 k2). (9.14) 2 2ρ2 − eration decreases with 1/r makes it comparable with the perturbation due to the solar radiation pressure. Moreover, the inverse similarity transformation converts Keplerian orbits into the conic sec- 9.3 Geometrical and physical relations tions obtained when the solar radiation pressure is directed along The inverse of a generic conic section r(θ) = a + b cos θ, using one the radial direction. The structure of the solutions, governed by the of its foci as the center of inversion, defines a limaçon. In particular, roots of a polynomial, is similar in nature to the Schwarzschild the inverse of a parabola results in a cardioid. Let us recover the geodesics. This is because under the considered perturbation and equation of the trajectory of a generalized parabolic cardioid (a true in Schwarzschild metric the evolution of the radial distance takes cardioid) from Eq. (5.16), the same form. The perturbation can also be seen as a control law for a continuous-thrust propulsion system. Some of the solutions ρmax ρ(θ) = [1 + cos(θ θm)]. (9.15) are comparable with the orbits deriving from potentials depending 2 − on different powers of the radial distance. Although the trajectory Taking the cusp as the inversion center defines the inverse curve: may take the same form, the velocity will be different, in general.

1 2 1 = [1 + cos(θ θm)]− . (9.16) ρ ρmax − Identifying the terms in this equation with the elements of a ACKNOWLEDGEMENTS parabola it follows that the inverse of a generalized parabolic car- This work is part of the research project entitled “Dynamical dioid with apoapsis ρ is a Keplerian parabola with periapsis max Analysis, Advanced Orbit Propagation and Simulation of Com- 1/ρ . The axis of symmetry remains invariant under inversion; max plex Space Systems” (ESP2013-41634-P) supported by the Spanish the lines of apses coincide. Ministry of Economy and Competitiveness. The author has been The subfamily of elliptic generalized logarithmic spirals is a funded by a “La Caixa” Doctoral Fellowship for he is deeply grate- generalized form of Cotes’s spirals, more specifically of Poinsot’s ful to Obra Social “La Caixa”. The present paper would not have spirals: been possible without the selfless help, support, and advice from 1 Prof. Jesús Peláez. An anonymous reviewer made valuable contri- = a + b cosh ν. (9.17) ρ butions to the final version of the manuscript. Cotes’s spirals are known to be the solution to the motion immerse in a potential V(r) = µ/(2r2)(Danby 1992, p. 69). − The radial motion of interior Type II hyperbolic and Type I el- REFERENCES liptic sinusoidal spirals has the same form, and is also equal to that of Type II hyperbolic generalized cardioids, except for the sorting Adkins, GS., McDonnell, J. (2007) Orbital precession due to central-force of the terms. perturbations. Phys Rev D 75(8):082,001 It is worth noticing that the dynamics along hyperbolic sinu- Akella, MR., Broucke, RA. (2002) Anatomy of the constant radial thrust problem. J Guid Control Dynam 25(3):563–570 soidal spirals withκ ˜2 = 2˜κ1 are qualitatively similar to the motion 5 Arnold, VI., Kozlov, VV., Neishtadt, AI. (2007) Mathematical aspects of under a central force decreasing with r− . Indeed, the orbits shown classical and celestial mechanics, 3rd edn. Springer in Fig. 4e behave as the limit case γ = 1 discussed by MacMillan Bacon, RH. (1959) Logarithmic spiral: An ideal trajectory for the in- (1908, Fig. 4). On the other hand, parabolic sinusoidal spirals (off- terplanetary vehicle with engines of low sustained thrust. Am J Phys center circles) are also the solution to the motion under a central 27(3):164–165 5 force proportional to r− . Bahar, LY., Kwatny, HG. (1987) Extension of Noether’s theorem to

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