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Characterizing Planetary Orbits and the Trajectories of Light F.T. Hioe* and David Kuebel Department of Physics, St. John Fisher College, Rochester, NY 14618 and Department of Physics & Astronomy, University of Rochester, Rochester, NY 14627 December 16, 2018 Abstract Exact analytic expressions for planetary orbits and light trajectories in the Schwarzschild geometry are presented. A new parameter space is used to characterize all possible planetary orbits. Dierent regions in this parameter space can be associated with dierent characteristics of the orbits. The boundaries for these regions are clearly dened. Observational data can be directly associated with points in the regions. A possible extension of these considerations with an additional parameter for the case of Kerr geometry is brie y discussed. PACS numbers: 04.20.Jb, 02.90.+p 1 Introduction Nearly a century after Einstein's theory of general relativity was found to cor- rectly predict the precession of the planet Mercury around the sun and the de ection of light by the sun's gravitational eld, the problem of understand- ing orbital trajectories around very massive objects still retains interest as it relates to current astrophysical topics [1] such as the study of gravitational waves. Among the numerous works on this subject, we should mention the classic publications of Whittaker [2], Hagihara [3] and Chandrasekhar [4] on the Schwarzschild geometry, and the more recent work of Levin and Perez-Giz [5] on the Schwarzschild and Kerr geometries. In the work of Chandrasekhar and that of Hagihara, the orbits are classied into various types according to the roots of a certain cubic equation, while in the work of Levin and Perez-Giz, the orbits are classied topologically by a triplet of numbers that indicate the numbers of zooms, whirls and vertices. In the work of Levin and Perez-Giz, the orbits were obtained by numerically integrating the integrable equations. These authors used the planet's energy and angular momentum as the principal physical parameters, and made extensive use of an 1arXiv:1001.0031v1 [gr-qc] 30 Dec 2009eective potential for describing the Schwarzschild orbits, as most studies on the topics of general relativity do. In this paper, we rst present, in Section 2, three explicit analytic expressions for the orbits in the Schwarzschild geometry: one is for periodic orbits [6] (in- cluding a special case that we call asymptotic), and two are terminating orbits. The explicit analytic expressions that we derive not only describe the precise features of the orbits (periodic, precessing, non-periodic, terminating, etc.) but also clearly indicate two physical parameters which can be used to characterize these orbits. These two dimensionless parameters are specic combinations of the following physical quantities: the total energy and angular momentum of the planet, the masses of the massive object and the planet, and, of course, the universal gravitation constant Gand the speed of light c. These two physical dimensionless parameters were rst used by one of us in ref.6. We shall refer to these two quantities as the energy eccentricity parameter eand the gravitational eld parameter srespectively (or simply as the energy parameter and the eld parameter). They will be dened in Section 2. We will use neither the common convention of setting G=c= 1, nor the energy and angular momentum of the planet by themselves, as the physical parameters for characterizing the orbits. With the energy parameter (0 e1) plotted on the horizontal axis and the eld parameter (0 s1 ) plotted on the vertical axis, the parameter space for all possible orbits will be shown to be divisible into three sectors, which we call Regions I, II and II', that have clearly dened boundaries. Region I permits periodic and terminating orbits. Regions II and II' terminating orbits only. In Section 3, we describe Region I for the orbits in greater detail. We rst divide Region I by lines each of which represents orbits described by elliptic functions of the same modulus k. We then give a more physical division of Region I which consists of nearly horizontal lines each of which represents orbits that have the same precession angle , and of bent vertical lines each of which represents orbits that have the same "true" eccentricity ". The terminating orbits will be characterized by two parameters one of which is the angle at which the planet enters the center of the blackhole, and the other being the initial distance of the planet from the star or blackhole. In Sections 4 and 5, we describe Regions II and II' in which all orbits are terminating, and we again divide Region II by curves of constant modulus keach of which describes orbits with the same modulus. Regions II and II' are separated by the Schwarzschild horizon. Thus the grid of our "map" can be used to describe all possible orbits in the Schwarzschild geometry in their entirety. The observational data related to a planet's orbit about some giant star or blackhole can be directly identied with a point having certain coordinates ( e;s) on our map, which can then be used for estimating the physical characteristics associated with the star or blackhole and that of the planet itself, assuming that the star or blackhole is not spinning very fast. For the Kerr geometry, another dimensionless quantity, which is clearly the ratio of the spin angular momentum per unit mass of the blackhole to the orbital angular momentum per unit mass of the planet, should enter into the consideration. In Section 6, we brie y discuss a possible extension of our results to the case involving a slowly spinning blackhole, at least to the rst order 2perturbation, by rescaling the physical parameters involved. In Section 7, we study the de ection of light by the gravitational eld of a very massive object. A single dimensionless parameter will be used to charac- terize the region. We show that here too, we should divide the region into three sectors, which we again call Regions I, II and II', and we present three analytic expressions for the trajectories of light applicable in these dierent regions. Re- gion I has trajectories of light that get de ected, and Regions II and II' have trajectories of light that are absorbed by and terminate at the blackhole. In Section 8, we give a summary of our results. Proofs of many interesting analytic relations among the parameters appearing in these studies are given in several appendices. Since our results presented in this paper cover gravitational elds of all ranges, from the weak eld produced by the sun of our solar sys- tem, for example, to the very strong eld produced by a blackhole, we want to avoid referring to the massive object that produces the gravitational eld as a blackhole, and prefer to refer to it as the star or blackhole, and we shall refer to the object of a much smaller mass that orbits around it as the planet or the particle. We have supplemented our many analytic results with numerous tables that present various physical quantities such as the minimum and maximum distances of the planet from the star and the angles of precession of the orbits that are calculated from our analytic expressions, as well as numerous gures that show various kinds of orbits of the planet and various kinds of de ection of light. 2 Analytic Expressions for the Orbits We consider the Schwarzschild geometry, i.e. the static spherically symmetric gravitational eld in the empty space surrounding some massive spherical object such as a star or a blackhole of mass M. The Schwarzschild metric for the empty spacetime outside a spherical body in the spherical coordinates r;; is [1] dl2=c2 1 r dt2 1 r 1 dr2 r2d2 r2sin2d2(1) where =2GM c2(2) is known as the Schwarzschild radius, Gis the universal gravitation constant, andcis the speed of light. If [ x] = (t;r;; ), then the worldline x(), where is the proper time along the path, of a particle moving in the equatorial plane ==2, satises the equations [1] 1 r t=; (3) c2 1 r t2 1 r 1r2 r2 2 =c2; (4) 3r2 =h; (5) where the derivative representsd=d . The constant his identied as the angular momentum per unit mass of the planet, and the constant is identied to be =E m0c2; whereEis the total energy of the planet in its orbit and m0is the rest mass of the planet at r=1. Substituting eqs.(3) and (5) into (4) gives the 'combined' energy equation [1] r2 +h2 r2 1 r c2 r=c2(2 1): (6) Substituting dr=d = (dr=d )(d=d ) = (h=r2)(dr=d ) into the combined energy equation gives the dierential equation for the orbit of the planet du d2 =u3 u2+Bu+C (7) whereu= 1=r,B= 2GM=h2,C=c2(2 1)=h2. Following Whittaker [2], it is convenient to change variable from uto a dimensionless quantity Udened by U=1 4 r 1 3 =1 4 u 1 3 ; (8) oru= 4U= + 1=(3) so that eq.(7) becomes dU d2 = 4U3 g2U g3 (9) where g2=1 12 s2 g3=1 216 1 12s2+1 4(1 e2)s4; (10) and where e= 1 +h2c2(2 1) (GM)21=2 (11) and s=GM hc: (12) 4The two dimensionless parameters eands, which are dened by the two above equations and which we call the energy and eld parameters respectively, will be the principal parameters we shall use for characterizing the orbit of a planet. It will be noted that the constant c2(2 1) which is <0 for a bound orbit, can be identied with 2 E0=min the Newtonian limit, where E0is the sum of the kinetic and potential energies and is <0 for a bound orbit, and m is the mass of the planet (which approaches m0), and that e' 1 +2E0h2 m(GM)21=2 (13) is the "eccentricity" of the orbit. Also, in the small slimit, the orbit equation can be shown to be given by 1 r'GM h2[1 ecos(1 )]; (14) where'3(GM)2=(hc)2. Thusrassumes the same value when increases to+ 2=(1 ). Comparing this with the increase of fromto+ 2, the ellipse will rotate about the focus by an amount which is the angle of precession '2 1 2'2=6(GM)2 h2c2: (15) This is the well known approximate expression for the precession angle for the case of very small s. The limiting case for = 0 is the well known orbit equation in Newtonian mechanics. We should note that while the limit s= 0 (and thus= 0) cannot be strictly correct in principle so long as M6= 0, this limit can be used for many practical cases with great accuracy as evidenced by the predictions of Newtonian mechanics. A special case of these Newtonian orbits is the circular orbit of radius r=h2=GM fore= 0. We now derive the exact analytic solutions of eq.(9) and classify the three possible solutions from a purely mathematical viewpoint, and we shall consider their physical interpretations in the next section. We rst dene the discrimi- nant of the cubic equation 4U3 g2U g3= 0 (16) by = 27g2 3 g3 2: (17) The three roots of the cubic equation (16) are all real for the case 0. We call the three roots e1;e2;e3and arrange them so that e1> e2> e3; the special cases when two of the roots are equal will be considered also. For the case >0, the cubic equation (16) has one real root and two roots that are complex conjugates. The analytic solutions of eq.(9) that we shall present will give the distance rof the planet from the star or blackhole in terms of the Jacobian elliptic functions that have the polar angle in their argument and 5that are associated with a modulus kthat will be dened. Instead of writing r, we use the dimensionless distance qmeasured in units of the Schwarzschild radiusand dened by qr 1 u: (18) The dimensionless distance qis related to Uof eq.(8) by 1 q=1 3+ 4U: (19) We now give the three analytic solutions of eq.(9) in the following. Solution (A1) For 0,e1>e2U >e 3. Writing the right-hand side of eq.(9) as 4( e1 U)(e2 U)(U e3), eq.(9) can be integrated with expressed in terms of the inverse Jacobian snfunction [7]. After a little algebra and some rearrangement, the equation for the orbit is found to be 1 q=1 3+ 4e3+ 4(e2 e3)sn2( ;k ) =1 3+ 4e3+ 4(e2 e3)1 cn(2 ;k ) 1 +dn(2 ;k ); (20) where the point at = 0 has been chosen to give U=e3. The constant appearing in the argument, and the modulus k, of the Jacobian elliptic functions are given in terms of the three roots of the cubic equation (16) by = (e1 e3)1=2; (21) k2=e2 e3 e1 e3: (22) wheree1;e2;e3are given by e1= 2g2 121=2 cos 3 ; e2= 2g2 121=2 cos 3+4 3 ; e3= 2g2 121=2 cos 3+2 3 ; (23) and where cos=g327 g3 21=2 : (24) 6Equation (20) was rst given in ref.6 using a slightly dierent approach that was initiated by Whittaker [2]. In addition, eq.(20) was shown to reduce to eq.(14) for the case of very small swhich in turn gave the known approximate precession angle given by eq.(15). The modulus kof the elliptic functions has a range 0k21. Since the elliptic functions sn,cnanddnare all periodic functions of for 0k2<1, we shall call this solution for the orbit the periodic solution. For the special case of k2= 1, sincesn( ;1) = tanh( ), cn( ;1) =dn( ;1) = sech( ), the solution is no longer periodic, and we shall refer to it as the asymptotic periodic solution. The period of cn(2 ;k ) is 4K(k), and the period of dn(2 ;k ) and of sn2( ;k ) is 2K(k), whereK(k) is the complete elliptic integral of the rst kind [7]. For k= 0,sn(x;0) = sinx; cn (x;0) = cosx; dn (x;0) = 1. As k2 increases from 0 to 1, K(k) increases from =2 to1. The distance rof the planet from the center of the star or blackhole assumes the same value when its polar angle increases from to+ 4K=(2 ) =+ 2K= . Comparing this with the increase of fromto+ 2in one revolution, the perihelion (or the aphelion) will rotate by an amount =2K(k) 2; (25) which is the exact expression for the precession angle. For k2close to the value 1, the planet can make many revolutions around the star or blackhole before assuming a distance equal to its initial distance. Thus if nis the largest integer for which 2 K(k)= is equal to or greater than 2 n, the angle of preces- sion should be more appropriately dened as 2 K(k)= 2n. For the sake of consistency, however, we shall stick to the denition given by eq.(25). For the case of very small sand to the order of s2, it was shown in ref.6 that '[1 (3 e)s2]=2,k2'4es2,K(k)'(1 +es2)=2, and substituting these into eq.(25) gives the well known approximate result given by eq.(15). For these periodic orbits, we note that the maximum distance rmax(the aphelion) of the planet from the star or blackhole and the minimum distance rmin (the perihelion) of the planet from the star or blackhole, or their corresponding dimensionless forms qmax(=rmax=) andqmin(=rmin=);are obtained from eq.(20) when = 0 and when =K(k) respectively, and they are given by 1 qmax=1 3+ 4e3; (26) and 1 qmin=1 3+ 4e2; (27) wheree2ande3are determined from eqs.(23), (24) and (10) in terms of e ands. Although we call the orbits given by this solution for 0 k2<1 periodic, they are not necessarily closed orbits. It is seen from the precession discussed 7above that for =f, unlessfis a rational number, the orbit will not close and it is not strictly a closed periodic orbit. However, for all practical purposes, any irrational number when truncated becomes a rational number, and thus the orbit will be closed. The distinction of closed and non-closed orbits depending on whether fis rational or irrational is of course of profound theoretical interest [5]. For a general periodic orbit that precesses, the general or true eccentricity " of the orbit is dened by "rmax rmin rmax+rmin=qmax qmin qmax+qmin; (28) whereqmaxandqminare given by eqs.(26) and (27). We shall show in the following section that the true eccentricity "is in general not equal to the energy eccentricity parameter edened by eq.(11), but that "!ein the limit of s!0, i.e. in the Newtonian limit. For the special case of "= 1 however, we shall show that it coincides with the special case of e= 1 for all applicable values of s, and that it signies an unbounded orbit. We now proceed to present the second solution. Solution (A2) For 0,U >e 1>e2>e3. We write the right-hand side of eq.(9) as 4( U e1)(U e2)(U e3) and eq.(9) can be integrated with expressed in term of the inverse Jacobian sn function [7]. The equation for the orbit is found to be 1 q=1 3+ 4e1 e2sn2( ;k ) cn2( ;k ); (29) where ,k,e1,e2ande3are given by eqs.(21)-(24) as in the rst solution. This solution gives a terminating orbit. The point at = 0 has been chosen to be given by 1 q1=1 3+ 4e1: (30) The planet, starting from the polar angle = 0 at a distance q1from the blackhole, plunges into the center of the blackhole when its polar angle 1is given bycn( 1;k) = 0, i.e. when 1=K(k) ; where andkare given by eqs.(21) and (22). The region of ( e;s) where orbits given by solutions A1 and A2 are applicable will be called Region I, and it will be described in greater detail in Section 3. Thus each point ( e;s) of parameter space in Region I represents two distinct orbits, one periodic and one terminating. At the same coordinate point, the characteristic quantities that describe the two distinct orbits are related. For 8example, by noting e1+e2+e3= 0 and from eqs.(26) and (27), q1can be expressed as 1 q1= 1 1 qmin+1 qmax ; (31) whereqminandqmaxare the minimum and maximum distances for the pe- riodic orbit at the same coordinate points ( e;s). It will be noted that q1is less thanqmin, i.e. for the terminating orbit the planet is assumed initially to be closer to the blackhole than the qminfor the associated periodic orbit, except at k2= 1 where q1=qminand the planet has a circular instead of a terminating orbit that will be explained later. We note that the terminating orbit equation (29) presented has no singularity at the Schwarzschild horizon q= 1, because, as is well known, q= 1 is a coordinate singularity and not a physical singularity. The orbit obtained from continuing beyond the value 1=K(k)= may become interesting if the concept of "whitehole" turns out to be of physical relevance. For now, the orbits in Region I are characterized mathematically by 0. We now present the third solution. Solution (B) For >0. Dene A=1 2 g3+r 27!1=3 ; B=1 2 g3 r 27!1=3 ; (32) whereg3and are dened by eqs.(10) and (17). The real root of the cubic equation (16) is given by a=A+B (33) and the two complex conjugate roots bandbare (A+B)=2(A B)p 3i=2. We further dene = [3(A2+AB+B2)]1=4(34) and k2=1 2 3(A+B) 4p 3(A2+AB+B2)=1 2 3a 4 2: (35) Writing the right-hand side of eq.(9), with Ua, as 4(U a)(U b)(U b), eq.(9) can be integrated with expressed in terms of the inverse Jacobian cn function [7]. We nd the equation for the orbit to be 91 q=1 3+ 4 2+a ( 2 a)cn(2 ;k ) 1 +cn(2 ;k ): (36) This is a terminating orbit. The initial distance q2of the planet at = 0 has been chosen to be given by 1 q2=1 3+ 4a: (37) It plunges into the center of the blackhole when its polar angle =2is given by 2=K(k) ; where andkare given by eqs.(34) and (35). Again, we note that the orbit equation (36) has no singularity at q= 1. The region of ( e;s) where orbits given by eq.(36) are applicable will be divided into two sectors called Regions II and II', the boundary between which will be dened later. They have terminating orbits only. For now, the orbits in Regions II and II' are characterized mathematically by >0. As for the initial points of the orbits discussed above, by comparing eq.(19) with the orbit equations (20), (29) and (36), and with eqs.(26), (30) and (37), we already noted that our choice of = 0 in our orbit equations is such that it givesU=e3,e1, andarespectively that in turn give q=qmax,q1, andq2 as the initial distances of the planet from the star or blackhole. We then note from eq.(9) that dU=d = 0 and hence dr=d = 0 for the planet at these initial points of the trajectories, i.e. the trajectory or more precisely the tangent to the trajectory at = 0 is perpendicular to the line joining the planet to the star or blackhole. All this will be seen in the gures presented later, and all our references to the initial position of the planet from here onward assume that the trajectory (as increases from 0) of the planet at its initial position is perpendicular to the line joining the planet to the star or blackhole. 3 Region I Consider the orbits expressed by eqs.(20) and (29) given by solutions A1 and A2 and characterized mathematically by 0. We call the region covered by the associated range of values for ( e;s) Region I. To gain a preliminary perspective, consider the earth (as the planet) and the sun (as the star) in our solar system. Substituting the mass of the sun M= MS= 1:991030kgand the angular momentum of the earth per unit mass of the earthh= 4:481015m2=s, we nds= 0:98310 4. The energy eccentricity parametere, which is equal to the true eccentricity "of the earth's orbit for such a very small svalue, is known to be about 0 :017. The approximate relation k2'4es2gives the squared modulus of the elliptic functions that describe the earth's orbit to be k2= 0:65710 9. We see that for the planetary system that 10is familiar to us, the values of sandk2are very small indeed. We may also note that the Schwarzschild radius = 2GMS=c2'3kmwould be well inside the sun which has a radius of 6 :4103km. The earth's dimensionless distance is q'5107from the sun's center. For this value of s, withqmin'qmax'5107, the orbit given by eq.(29) from solution A2 would require the initial position q1 of a planet to be '1 according to eq.(31), i.e. the planet would have to be at a distance equal to the Schwarzschild radius from the center of the sun for it to have a terminating orbit which plunges to the center of the sun. Therefore the terminating orbit given by eq.(29) is inapplicable for our solar system. The periodic orbits, on the other hand, are perfectly valid. However, for cases when the massive object is a gigantic mass concentrated in a small radius such as a blackhole, all the possibilities presented here may arise. As the eld parameter sincreases from 0, the modulus kof the elliptic functions that describe the planet's orbits also increases. From eqs.(21)-(24), it is seen that several steps are needed to relate k2toeands. In Appendix A, we show that a direct relationship between k2andeandscan be established, and it is given by 1 18s2+ 54(1 e2)s4 (1 12s2)3=2=(2 k2)(1 +k2)(1 2k2) 2(1 k2+k4)3=2(38) = cos: (39) The cosof eq.(39) is the same cos that appears in eq.(24), and, in partic- ular, it is equal to 1 ;0; 1 fork2= 0;1=2;1 respectively. The curve represented by k2= 1, after setting cos = 1 in eq.(39), can be readily shown to give a quadratic equation 27(1 e2)2s4 2(1 9e2)s2 e2= 0 that gives s2 1=1 9e2+p (1 9e2)2+ 27e2(1 e2)2 27(1 e2)2; (40) for 0e<1, ands2 1= 1=16 fore= 1. Equation (40) representing k2= 1 gives the upper boundary (for the values of s) of Region I (the uppermost heavy solid line in Fig.1); it extends from s1=p 2=27 = 0:272166 for e= 0 to s1= 1=4 = 0:250000 for e= 1, i.e. a line that is nearly parallel to the e-axis. Thus Region I is a region bounded by 0 e1, and by 0ss1wheres1 is given by eq.(40), in which the squared modulus of the elliptic functions that describe the orbits cover the entire range 0 k21. We now use eq.(38) to give a plot of lines of constant k2= 0:001;0:01;0:1;0:3;:::;1 as shown in Fig.1. These lines conveniently divide Region I into regions of in- creasing eld strengths as k2increases from 0 to 1. On a point representing a particulark2and a particular evalue,scan be determined from eq.(38) and the orbit is then given by eq.(20) using eqs.(98), (10) and (21). The values of s on these constant k2lines for the values of e= 0:1;0:2;:::;1:0 are given in Table 1 which thus give the coordinates ( e;s) of the points on the lines representing dierent values of k2. These coordinate points ( e;s) from Table 1 are used to 11give the following tables: Tables 2 and 3 give the values of qmaxandqminfor the orbits obtained from eqs.(26) and (27). Note that the dimensionless distance q is in units of the Schwarzschild radius which depends on the mass Mof the star or blackhole corresponding to that particular coordinate point, and thus one should not compare qat two dierent coordinate points just by their abso- lute values alone. Table 4 presents the values of the precession angle in units of, i.e. =, obtained from eq.(25). Table 5 presents the values of the true eccentricity "obtained from eq.(28). Tables 2-5 are to be used in conjunction with Table 1 for identifying the locations ( e;s) of the corresponding quantities that are presented. The physical quantities presented in Tables 2-5 together with the coordinates ( e;s) given in Table 1 now give all possible periodic orbits in the Schwarzschild geometry in its entirety. That is, the coordinates ( e;s) of a planet orbiting a non-spinning blackhole can be identied if the observation data onrmin;rmax;"and can be collected. Region I shown in Fig.1 is where orbits given by eqs.(20) and (29) apply. In Sections 4 and 5, we shall discuss Regions II and II' which are shown above Region I in Fig.2 where orbits given by eq.(36) apply. As an example of application of Tables 1-5, from the second row and second column of Tables 1-5 and using only two signicant gures, for orbits with e= 0:10;s= 0:11;k2= 0:010;we nd from Tables 2-5 that qmax= 50;qmin= 34;= = 0:079 or = 14;and"= 0:19, i.e. orbits with those seemingly small values of sandk2give a precession angle of 14per revolution that is already very large compared to those encountered in our solar system for which the precession angle is only 3 :8" per century for the earth's orbit (for which s'0:98310 4;k2'0:65710 9;"'e'0:017), and the value of the true eccentricity "of these orbits is already quite dierent from their energy parameter e. We thus appreciate that the range of values for s given by 0ss1for Region I, where s1ranges from 0 :276166 for e= 0 to 0:25 fore= 1, is not as small as it seems (noting also that 0 k21), and that the classical Newtonian orbits are restricted to a very narrow strip of the region indeed for which s'0 andk2'0;and for which "'efor 0e1. Although the lines of constant k2in Region I conveniently associate the orbits with the orbit equations for the periodic and terminating orbits given by eqs.(20) and (29) and with the physical parameters given in Tables 2-5, the precession angle and the true eccentricity "are more physically meaningful parameters that can be associated with the description of the orbit. The expressions for and"in terms of kandsare given by eq.(99) in Appendix A and eq.(100) in Appendix B. For a given value of and ofe, we can use eqs.(99) and (38) to solve for s(andk) (using a numerical program such as FSOLVE in MAPLE) and thus locate its coordinate ( e;s); and similarly for a given value of "and ofe, we can use eqs.(100) and (38) to solve for s(andk). The relationship of eands with"is simpler for k2= 1 and will be discussed later (see eqs.(49) and (50)). In Fig.3, we present lines of constant =(that are nearly horizontal) and lines of constant "(that are bent vertical) in Region I, and the corresponding tables for their coordinates are presented in Tables 6 and 7. We note that because given by eq.(25) depends on given by eq.(21) as well as on K(k), the line of constant does not coincide with the line of constant k2except for 12k2= 1. We note also that the line of constant "does not coincide with the (vertical) line of constant eexcept for"=e= 1. We show in Appendix B that it is only for a very thin strip of region, where sis between zero and some very small positive value, that "'ewhich applies in the Newtonian limit. We also show in Appendix B that "=ewhene= 1 exactly. The distinction between edened by eq.(11) or eq.(13) with "dened by eq.(28) in the Newtonian or non-Newtonian theory has never been clearly recognized previously. With Fig.3 which has curves of constant = and constant "in place, Region I is now partitioned into cells with the coordinate points specied by (=;" ). We have a clear idea what the orbits of a planet would be like at points within each cell in terms of their precession angle and true eccentric- ity, and the coordinates of these orbits ( e;s) then give the energy and eld parameters corresponding to these orbits. In Fig.4, we present examples of pe- riodic and unbounded orbits, plotted in polar coordinates ( q;), corresponding to various precession angles of =6;=3;=2;;3=2;2;1(vertically from top to bottom) for values of e= 0;0:5;1 (horizontally from left to right), where the star or blackhole is located at the origin. We rst note that the orbits for which e <1 are periodic and closed because fis a rational number in =ffor each one of them. The precession angle can be seen from the heavy solid line that marks the trajectory (as increases) from the initial point at = 0 to the rst point at which the distance from the origin is equal to the distance at = 0. The true eccentricity of the orbits is "given by eq.(28). For example, for the orbit of Fig.4 (a1) for ==6,e= 0,"is far from zero which can be seen from theqminandqmaxin the gure, and it can be more accurately calculated to be equal to 0 :22629. For each of the unbounded orbits characterized by e= 1, the incoming trajectory coming from innity at = 0 makes an angle with the outgoing trajectory going to innity given by = 2K(k)= = 2+from eq.(25), as can be seen in some of the gures presented. The case ==1 corresponding to the special case of k2= 1 will be discussed later in this section for which the planet starting from qmaxends up circling the blackhole with a radius that approaches qmin(see Fig.6d). Generally, if we are given a coordinate point in Fig.3, for example, a point one= 0:5 just above the = = 1=3 line slightly to the left of the "= 0:6 curve (where "= 0:581431:::ands= 0:194229:::), then we nd = 60:4706::: degrees or == 0:33594:::, and part of the orbit is shown in Fig.5. Whether the orbit will close on itself depends on whether = is or is not a rational number in principle, even though, as we mentioned before, a truncated number in practice is always a rational number and the orbit will be a closed one. We only show part of the orbit in Fig.5 as the subsequent path is clear from the angle of precession and true eccentricity of the orbit and we are not concerned with how many "leaves" the orbit is going to create. Figure 3 (or one with even more curves of constant =and constant ") is a very useful map that can be used fruitfully with any observation data that are obtained for any planet. Besides the special case k2= 1, the case of k2= 1=2 is also somewhat special in that it allows many relationships to be expressed simply and explicitly. We present some of these simple relations for k2= 1=2 in Appendix C. It is to be 13noted from Fig.1 that the line of constant k2= 1=2 is very close to the boundary given byk2= 1. The line of constant k2= 1=2 for Region II, on the other hand, is closer to dividing the region approximately into two halves, as shown in Fig.2. Thek2= 1=2 curve for Region II will be discussed in Section 4. The terminating orbits in Region I given by eq.(29) can be characterized by the planet's initial position q1given by eq.(31), and by the angle 1at which the planet enters the center of the blackhole. It is interesting to note that even for these terminating orbits, the precession angle still has an "extended" meaning and use that we shall describe. It is clear from eq.(29) that the orbit terminates, i.e. qbecomes zero when 1=K(k), but if the orbit is continued (by continuing to increase ),qwould assume its initial value at = 0 when 0= 2K(k), producing a "precession angle" of =0 2= 2K(k)= 2 which is equal to the precession angle for the corresponding periodic orbit at the same coordinate point ( e;s). Since0= 21, the polar angle 1at which the path of the terminating orbit enters the center of the blackhole is related simply to the precession angle of the periodic orbit by 1= =2 +, or 1 =1 2 + 1: As1=can be easily calculated from = for the periodic orbits given in Table 4, we do not tabulate it separately. The values of q1are presented in Table 8, and we note the small range 1 q12:25 for the entire Region I. Examples of these terminating orbits are presented in Fig.6. The dotted line represents the continuation of the orbit when is continued beyond 1. Before we discuss Regions II and II', we want to describe three special cases: the case of k2= 0 which, as we shall see, is not of any interest but must be included for completeness; the case of k2= 1 which gives the upper boundary of Region I (and lower boundary of Region II); and the case of e= 1 which gives the right boundary of Region I (and of Regions II and II') (see Figs.1 and 2). (i) The Special Case of k2= 0 The line of k2= 0 coincides with the s= 0 axis in Fig.1. To show this, we note thatk2= 0 implies = 0 from eq.(95). Substituting = 0 into eq.(24) givess= 0 when we use the expressions in eq.(10) for g2andg3. The we nd g2= 1=12 andg3= 1=216, and from eq.(98), we nd e1=1 6; e2=e3= 1 12: Equation (20) then gives 1 =q= 0 orq=1, i.e. it is the limiting case of zero gravitational eld. As we pointed out earlier, the classical Newtonian case is given by only a very narrow strip represented by k2'0 ands'0 for which qis large but nite. (ii) The Special Case of k2= 1 14It follows from eqs.(38) and (39) that on the line of k2= 1, cos= 1. Thus from eqs.(24) and (17), we have = 0 (41) which can be identied as the "boundary" between Solutions A and B in Section 2. The range of svalues for = 0 is 0 :25s0:272166 for 1e0 (see the discussion below eq.(40)), and for that range of svalues,s1=2p 3 = 0:288675 ors21=12 and therefore g2>0 (see eq.(10)). From eq.(41), the relation between g2andg3can be more precisely expressed as 3pg3= rg2 3 after noting that g3is negative and g2is positive for the values of salong the linek2= 1. Also from eq.(98), we note that e1=e2=rg2 12; e3= rg2 3: (42) The periodic orbits given by eq.(20) become 1 q=1 3+ 2rg2 31 5 sech(2 ) 1 + sech(2 ); (43) where =3g2 41=4 (44) and where the values of g2(andg3) are those given by the values of eands on the line k2= 1 that are obtained from eq.(40). The orbit is not a periodic orbit; it is what we call an asymptotic periodic orbit. The planet starts from an initial position qmaxat= 0 given by 1 qmax=1 3+ 4e3=1 3 4rg2 3(45) and ends up at =1circling the star or blackhole with a radius that asymptotically approaches qmingiven by 1 qmin=1 3+ 4e2=1 3+ 2rg2 3: (46) Equations (43)-(46) are explicit and simple equations that give the orbit equation,qmax, andqminfork2= 1. In particular, it is seen from Table 3, for example, that qminranges from 2 for e= 0 to 9=4 = 2:25 fore= 1, i.e.qminis 15still no less than twice the Schwarzschild radius for the strongest gravitational eld that permits the periodic orbits. However, it is a very small number indeed compared to, say, qmin'5107for the earth's orbit around the sun. On this upper boundary k2= 1 of Region I, the terminating orbit given by eq.(29) from Solution A2 becomes a circular orbit with a radius qc=q1, where q1is the initial distance of the planet from the star or blackhole given by eq.(30). From eqs.(30) and (31) and noting that e1=e2fork2= 1, we nd that qc=q1=qmin (47) given by eq.(46) (see Tables 3 and 8 for k2= 1). We shall refer to the orbits given by eqs.(43) and (47) as the asymptotic periodic and the asymptotic termi- nating orbits respectively of Region I. Thus the special cases given by eqs.(43) and (47) for k2= 1 of the periodic and terminating orbits given by eqs.(20) and (29) for solutions A1 and A2 respectively clearly exhibit completely dier- ent behaviors from their counterparts for 0 k2<1. Examples of asymptotic periodic orbits are shown in Fig.4g. Asymptotic terminating orbits are simply circles of radius equal to q1, as shown in Fig.6(d). Using eqs.(28), (42), (45) and (46), for k2= 1 the true eccentricity "can be shown to be expressible in terms of g2by "=9p g2=3 1 3p g2=3; (48) which can be solved to give sin terms of ", and then ein terms of "using eq.(38). We nd that the coordinates ( e;s) of a given 0 :6"1 on the line k2= 1 are given by e=p (1 +")( 3 + 5") (3 "); (49) and s=p (3 ")(1 +") 2(3 +"): (50) It is interesting that eqs.(49) and (50) can be used in place of eq.(40) as parametric equations for determining the coordinates ( e;s) of the line k2= 1 as"takes the values from 0 :6 to 1. In particular, eqs.(49) and (50) allow us to see that the "=const: curves are not vertical (except for "=e= 1), and they intersect the upper boundary s1of Region I for 0 :6"1 (see Fig.3). The"= 0:6 curve, the boundary curve s1, and thes-axis are concurrent at e= 0,s=p 2=27. We can conclude that periodic orbits with e= 0 have true eccentricity in the range 0 "0:6. As another example it can be shown using eqs.(49) and (50) that periodic orbits with e<3p 5=11 = 0:609836 have "<0:8. (iii) The Special Case of e= 1 16The maximum or boundary value for ewhich ise= 1 is also a special case of interest. From its denition given by eq.(11), since 21,ecannot be greater than 1. In Appendix B, we show that e= 1 always gives an unbounded orbit for the periodic and the asymptotic periodic orbits of Region I and the terminating orbits of Region II, but not the asymptotic terminating orbit of Region I which has a radius given by eq.(47) independent of eand thus is not an unbounded orbit. Thus e= 1 is the boundary for efor Region I (as well as for Region II). Many explicitly simple relationships among s;k;q min;q1;etc. have been found on the boundary line e= 1, and they are given and proved in Appendix B. In particular, we have, on e= 1 in Region I, that s2=k2 4(1 +k2)2; (51) =1 4(1 +k2)1=2 ; (52) qmin=1 +k2 k2; (53) and q1= 1 +k2: (54) Examples of unbounded orbits for e= 1 are shown in Fig.4 (a3-g3). We shall now describe Regions II and II' for the orbit equation (36) given by solution B for the case >0. 4 Region II Consider the orbits expressed by eq.(36) given by solution B and characterized mathematically by >0. The associated values for ( e;s) in this case satisfy s>s 1, wheres1is the upper boundary of Region I given by eq.(40). This region of parameter space dened by s>s 1can be naturally divided into two sectors which we call Region II and II' with Region II bordering Region I (see Fig.2). The boundary between Regions II and II' is determined by the Schwarzschild radius in a manner to be described later in this section. We rst want to prove that the lower boundary (for s) of Region II, charac- terized by = 0 as it is for the upper boundary of Region I, also gives k2= 1, wherek2is calculated from eq.(35) for Solution B (In Section 3, we showed that fork2calculated from eq.(22) for Solution A, k2= 1 implies = 0). Substituting = 0 into eq.(32) gives A=B=1 23pg3= 1 2rg2 3: After noting that A(=B) is a negative value for the range of svalues for = 0, substituting the above into eq.(35) gives k2= 1. We also nd from 17eq.(34) that 2= 33pg3=2 =p3g2=2 which agrees with the given by eq.(44), and we nd from eq.(33) that a=3pg3= rg2 3: (55) Substituting these into eq.(36) gives the same orbit equation (43) for the terminating orbit in Region II on its lower boundary as that for the asymptotic periodic orbit in Region I on its upper boundary. Thus on the boundary k2= 1 the equation for the orbits in Region II does not represent a terminating orbit but is the same as the asymptotic periodic orbit for Region I given by eq.(43) (see Fig.4, g1-g3). Also, from eqs.(42) and (55), we see that the smallest root in eq.(23) in Solution (A) is identied with the real root given by eq.(33) of Solution (B), i.e. e3=a. Thus from eqs.(26) and (37), q2=qmaxwhenk2= 1, i.e. the initial distance q2of the terminating orbit in Region II can be identied as the continuation of qmaxof the periodic orbit from Region I. On the boundary of Regions I and II, the two other real roots e1=e2given by eq.(42) of the cubic equation (16) agree with b=bgiven below eq.(33). The line k2= 1 dened by eq.(40) is the boundary between Regions I and II; it is the upper boundary for Region I and is the lower boundary for Region II (see Fig.2). The above discussion also illustrates the transition that takes place: from a periodic orbit to an asymptotic periodic orbit to a terminating orbit, as one crosses the boundary from Region I to II. We now consider the upper boundary of Region II. We dene this boundary to be that obtained by requiring the planet's initial position to be just at the Schwarzschild horizon, i.e. that obtained by setting q= 1 initially at = 0. Settingq= 1 in eq.(36) for = 0 which is 1 =q= 1=3 + 4a, we require a= 1=6, whereais the real root of the cubic equation (16). We then use the equation 41 63 1 6 g2 g3= 0; (56) and substitute the expressions for g2andg3given in eq.(10) into eq.(56) and solve fors. We nd s2 2=1 1 e2(57) which we shall use as the equation for the upper boundary of Region II, for 0e1. Thus Region II is a region bounded between e= 0 ande= 1, and betweens1given by eq.(40) (the lower heavy solid line in Fig.2) and s2given by eq.(57) (the upper heavy solid line in Fig.2), i.e. s1<ss2. The region dened bys2<s1 and bounded between e= 0 ande= 1 will be called Region II', for which the planet's initial position ranges from just inside the Schwarzschild horizon up to the center of the blackhole. Since the same terminating orbit equation (36) applies in Regions II and II', the division into two regions may seem unnecessary. However, the Schwarzschild radius is of physical signicance, and it is useful to know the location of the curve s2in the (e;s) plot which 18indicates that the initial position of the planet is at the Schwarzschild horizon. Separating out Region II' also makes it possible to realize and appreciate that a very large region of the characterizing parameter s2<s1 is of relevance only to a very small physical region 0 q <1 for the case where the initial position of the planet is inside the Schwarzschild horizon. For Region II, as sincreases its value above those on its lower boundary s=s1on whichk2= 1, the value of k2calculated from eq.(35) decreases from 1. The curves of constant k2fork2= 0:9;0:8;:::can be easily obtained from eq.(35) where AandBare expressed in terms of sande(again using MAPLE FSOLVE) and they are presented in Fig.2. However, the value of k2has a minimum value that is not 0 in Region II. First, we show in Appendix C that thek2= 1=2 curve is given by s2=1 6(1 e2) 1 +r 1 + 2e2 3! : (58) The signicance of this k2= 1=2 curve is that on it, as e!1,s!1 , just like the curve for the upper boundary of Region II represented by eq.(57). For k2>0:5, the constant k2curves intersect the e= 1 line at some nite value of s, whereas for k2<0:5, the constant k2curves intersect the upper boundary curve given by eq.(57) at points for which the values of eare less than 1. We then nd that the minimum value of k2in Region II is equal to 1 =2 1=(2p 5) = 0:276393 which is obtained by setting e= 0;s= 1 in eqs.(32) and (35), and this value of k2appears at one coordinate point only at e= 0 ands= 1. There is no orbit whosek2is less than 0 :276393 in Region II (see Fig.2), and k2is thus restricted to the range 0 :276393k21. In Table 9, we present the coordinates ( e;s) of these curves of constant k2 between 0:276393 and 1. In Table 10, we present the values of q2given by eq.(37), the initial distance of the planet from the blackhole. Note that unlike q1for the terminating orbits in Region I whose range is nite and small, q2can be innite (for e= 1 andk2>0:5). Like the terminating orbits of Region I, the terminating orbits of Region II can be characterized by q2and the angle 2at which the planet enters the center of the blackhole. If we dene the "precession angle" for the terminating orbits as in eq.(25), with kand dened by eqs.(35) and (34), then 2=K(k)= = =2 +, or 2 =1 2 + 1: (59) In Table 11, we present the values of 2. Tables 10 and 11 are to be used in conjunction with Table 9 that give the coordinates of the constant k2curves. Ex- amples of these terminating orbits obtained from eq.(36) are shown in Fig.7(b)- (d). Again the dotted line shows the continuation of the orbit beyond 2. A planet coming from very far away, i.e. an unbounded orbit with e= 1, with an initial trajectory perpendicular to the line joining it to the blackhole, can terminate at the blackhole; the condition for this to happen is s>0:25. Figures 8 (a)-(c) show three unbounded orbits ( e= 1) assincreases from just below 19to just above the critical eld parameter s= 0:25. Figure 8(a) also shows an example of a precession angle in which the planet makes more than three revolu- tions around a blackhole before assuming a distance equal to its initial distance (which is innity) from the blackhole. As noted after eq.(25), the actual pre- cession angle in this case should be more appropriately given by 2 K(k)= 6 which can be obtained from the presented value of == 4:6378 (where is dened by eq.(25)) and gives 0 :6378. Thus for Fig.8(a), 0 :6378gives the angle between the initial incoming trajectory from very far away at = 0 and the nal outgoing trajectory going to innity, i.e. 0 :6378is the polar angle of the direction of the outgoing trajectory going to innity with respect to the x-axis (but we have not extended the outgoing trajectory far enough to show the accuracy of this angle). We now present some useful simple expressions for the following special cases. (i) Special Case on the Upper Boundary of Region II Given By Eq.(57) We show in Appendix D that on the upper boundary of Region II given by eq.(57), the values of k2and given by eqs.(35) and (34) become k2=1 2 1 4p 1=3 g2; (60) =1 41 3 g21=4 ; (61) where thesvalues forg2are given by eq.(57). (ii) Special Case on the Right Boundary e= 1 of Region II Just as for Region I, there are simple and interesting relations among k2,s and on the right boundary e= 1 of Region II, and they are shown in Appendix D. In particular, we have, on e= 1 in Region II, that for s>1=4, k2=1 2+1 8s; (62) or that, for 1k2>1=2, s=1 8(k2 1=2); (63) and that =rs 2: (64) Appendix D also presents a special case given by s2= 1=12 that is notable. 5 Region II' While we may call the entire sector s > s 1given by eq.(40) above Region I in Fig.2 just one region that allows only terminating orbits given by eq.(36), 20it is useful to divide it into Regions II and II' using the curve s=s2given by eq.(57). Region II' is the region of parameter space in ( e;s) for which s>s 2and 0e <1. The heavy solid curve labeled s2in Fig.2 delineates the boundary of Region II' which separates it from Region II. Despite the apparent large size of Region II', the terminating orbits here have little variety in the sense that the range of initial distances q20that are given by 1 =q20= 1=3 + 4a(see eq.(37)), is limited (0 q20<1) and the range of the angle 20=K(k)= at which the planet enters the blackhole is also limited. It can be shown that the range of 20is 020<0:789. An example of a terminating orbit obtained from eq.(36) in Region II' is shown as the solid line in Fig.7(e); the dotted line shows the continuation of the orbit beyond 20. It may be of some mathematical interest to note that as s!1 in Region II', the modulus of the Jacobian elliptic functions used to describe the orbits does not go to zero; insteadk2!(2 p 3)=4 = 0:0669873, and thus k2in Region II' is restricted to the range 0 :0669873k2<0:5. Tables 1-8, the orbit equations (20), (29) and (36), and the description of the orbits and the three regions where these orbit equations apply, complete our characterization of all possible planetary orbits in the Schwarzschild geometry. We now brie y discuss how all this may be used for the Kerr geometry when the spinning blackhole has a spin angular momentum per unit mass of the blackhole that is relatively small compared to the orbital angular momentum per unit mass of the planet. 6 Kerr Geometry The spinning blackhole is assumed to have a spin angular momentum Jgiven by [1] J=Mac; (65) whereaccan be identied as the spin angular momentum per unit mass of the blackhole and is the quantity to be compared with h, the orbital angu- lar momentum per unit mass of the planet. The Kerr geometry becomes the Schwarzschild geometry in the limit ac=h!0. The worldline of a particle moving in the equatorial plane ==2 satises the equations [1] t=1 D r2+a2+a2 r ah cr ; (66) =1 Dhac r+ 1 r hi ; (67) whereDr2 r+a2. For the equatorial trajectories of the planet in the Kerr geometry, the combined energy equation is r2 +h2 a2c2(2 1) r2 (h ac)2 r3 c2 r=c2(2 1): (68) 21Provided that a2=2<1 andac=h<< 1, to the rst order in ac=h , it is not dicult to see, by comparing eqs.(66)-(68) with eqs.(3)-(6), that we can re-scale to0=(1 2ac=h ),stos0=s(1 ac=h ), andto0=[1 b(ac=h )], wherebis some approximation constant, such that the results we have presented for the orbits in the Schwarzschild geometry are approximately applicable for the orbits in the Kerr geometry in terms of the scaled parameters. That is, the orbits in the equatorial plane and their characterization for the Schwarzschild and Kerr geometries are qualitatively very similar to the rst order in ac=h except that the basic parameters s,andhave to be slightly rescaled. Levin and Perez-Giz [5] obtained their orbits in the Kerr geometry from numerically integrating eqs.(66)-(68) and it would be interesting to study and examine when and how the planet's orbits in the Kerr geometry that they obtained can be related with our results with the scaled parameters, and when and how they begin to dier signicantly from those in the Schwarzschild geometry that we presented in this paper. 7 Trajectory of Light We now consider the de ection of light by a gravitational eld. We cannot use the proper time as a parameter. So we use some ane parameter along the geodesic [1]. Considering motion in the equatorial plane, the geodesic equations give eqs.(3) and (5), and we replace the r-equation (4) by the rst integral of the null geodesic equation, and we have [1] 1 r t=; (69) c2 1 r t2 1 r 1r2 r2 2 = 0; (70) r2 =h; (71) where the derivative representsd=d . Substituting eqs.(69) and (71) into (70) gives the 'combined' energy equation r2 +h2 r2 1 r =c22: (72) Substituting dr=d = (dr=d )(d=d ) = (h=r2)(dr=d ) andu= 1=rinto the combined energy equation gives the dierential equation for the trajectories of light in the presence of a gravitational eld du d2 =u3 u2+c22 h2: (73) The constants andhhave a physical signicance through their ratio =h as follows: Let Rdenote the distance of the light beam to the center of a star or 22blackhole when the trajectory of the light beam is such that du=d = 0.Rcan either be associated with the distance of closest approach of the light beam to the blackhole or with the initial distance to the blackhole of the light beam. The latter case is associated with light trajectories that terminate at the blackhole. WithRso dened and letting u11=R, we can set c22=h2to be equal to u2 1 u3 1[8]. It is again convenient to consider the problem in terms of the dimensionless inverse distance Udened by U= r=u=1 q: (74) Udened here is slightly dierent from the Udened by eqs.(8) and (19) previously. In terms of Uof eq.(74), eq.(73) becomes dU d2 =U3 U2+c222 h2: (75) SincedU=d = 0 atr=R, one rootUwhich we call U1 Ru1 (76) of the cubic equation U3 U2+c222=h2= 0 is known, and the term c222=h2on the right hand side of eq.(75) can be replaced by U3 1+U2 1, and the other two roots of the cubic equation U3 U2 U3 1+U2 1= 0 can be found from solving a quadratic equation. We denote the three roots of the cubic equation by e1;e2;e3. Thus writing eq.(75) as dU d2 =U3 U2 U3 1+U2 1 (77) the trajectory of light represented by an equation for Uas a function of the polar angle obtained from integrating eq.(77) can be characterized by a single parameterU1which essentially species either the distance of the closest ap- proach or the initial distance of the light beam to the blackhole (These distances are scaled by the Schwarzschild radius of the blackhole). As in our discussion of the planets, our references to the initial position of the light beam assume that the trajectory of the light beam at that initial position is perpendicular to the line joining that position to the star or blackhole. The range of U1is clearly between 0 and1, whereU1= 0 means that the light beam is innitely far away from the blackhole, U1= 1 means that the light beam is at the Schwarzschild radius at its closest approach or its initial position, and U1=1means that the light beam is at the center of the blackhole. As we show in the following, the region 0U11 can be appropriately divided into three sectors which we again call Regions I, II and II'. The similarity between the characterization of these three regions with that for the planetary orbits discussed in the pre- vious sections will become apparent. Not surprisingly perhaps, only a single parameter which we choose to be U1, is needed for the characterization of the 23trajectories of a light beam in contrast to the two parameters (which we choose to beeands), which we needed for the characterization of the orbits of a planet. The relationship between U1andR, from eqs.(76) and (2), is R=2 U1GM c2 : Region I: 0U12=3;or1>R3GM=c2 HereRdenotes the distance of closest approach of a light beam that comes from a great distance. We let e1=1 2[1 U1+ (1 + 2U1 3U2 1)1=2]; e2=U1; e3=1 2[1 U1 (1 + 2U1 3U2 1)1=2]; (78) withe1>e2>e3;and we consider the region e1>e2>Ue3, and write eq.(77) as dU d2 = (e1 U)(e2 U)(U e3): (79) Equation (79) can be integrated [7] with expressed in terms of an inverse snfunction. After a little algebra and re-arrangement, we nd the trajectory's equation in terms of the Jacobian elliptic functions of modulus kto be 1 q=(e1 e3)e2 (e2 e3)e1sn2( ;k ) (e1 e3) (e2 e3)sn2( ;k ); (80) where =(e1 e3)1=2 2; k2=e2 e3 e1 e3: (81) The angle of de ection can be obtained as follows. If we set q=1and also set==2 + =2 as the incoming angle in eq.(80) (see Fig.9a for the special case of =2 = 45) where denotes the total angle of de ection of light by the mass M, we get the following equation for determining exactly: sn2 ( 2+ 2);k =(e1 e3)e2 (e2 e3)e1; wheree1;e2;e3; ;k; are given by eqs.(78) and (81). It can also be expressed as 24= +2 sn 1( ;k); (82) where =(e1 e3)e2 (e2 e3)e11=2 Equations (80)-(82) were rst given by one of us in ref.6. Examples of these trajectories obtained from eq.(80) are presented in polar coordinates ( q;) in Fig.9, where the blackhole is located at the origin. By setting the angles of de- ection presented in Fig.9 to be =2;;3=2;2, the corresponding values ofU1can be determined from eqs.(82) and (78) using the MAPLE FSOLVE program, and they are found to correspond to the distances of closest approach R= 4:6596GM=c2, 3:5206GM=c2, 3:2085GM=c2, 3:0902GM=c2respectively. The case of R= 3:5206GM=c2is interesting as it corresponds to the light ray being turned around by 180. That the upper boundary of Region I character- ized byU1= 2=3 orR= 3GM=c2is a very special case can be seen mathemati- cally because it results in e1=e2= 2=3;e3= 1=3;and hencek2= 1, = 1=2 andU= 2=3 =const: from eqs.(81) and (80). Physically, it results in the light circling the blackhole with a radius R= 3GM=c2even though the trajectory has been shown to be an unstable one [1]. This known result can also be simply obtained from the equation of motion d2U=d2= (3=2)U2 UforU=const: and thusU= 2=3. If one compares the size of the unstable circular photon orbit with the allowed limiting radii of the planetary asymptotic periodic orbits (2qmin2:25 or 4GM=c2rmin4:5GM=c2), one can see that the radius of the asymptotic circular path of a planet around a blackhole is still a little larger than that for a photon, but not by much. The lower boundary of Region I characterized by U1= 0 orR=1gives e1= 1;e2=e3= 0,k2= 0 and = 1=2, and thus gives U= 0 orr=1 which is a limiting case as the light ray that is innitely far away at its closest approach to the blackhole is completely unde ected. As in the case of the Region I particle orbits discussed in Section 3, the squared modulus k2of the elliptic functions that describe the trajectories of light here also covers the entire range 0 k21; it varies from 0 at the lower boundary to 1 at the upper boundary. For smallU1, the trajectory of light given by eq.(80) has been shown [6] to reduce to 1 r'cos R+GM c2R2(1 + cos+ sin2); (83) and the total de ection of light to reduce to the well known result '4GM c2R: (84) 25It can be shown from our exact result given by eq.(82) that this approximate expression (84) still gives an accuracy of two signicant gures for U1= 0:1 or R= 20GM=c2. AsU1approaches 2 =3, or asRapproaches 3 GM=c2, we may let U1= 2=3 , where(2=3)(1 3GM=c2R) is a small positive number. From eqs.(78) and (81), we can express the quantities 2 = , andkappearing in eq.(82) in power series in and nd, to the rst order in , 2= '4(1 =2 +:::), '1 =2+:::, andk'1 +:::. Substituting these into eq.(82) immediately gives an expression for which is correct to the rst order in . If an attempt is made to nd an expansion of sn 1( ;k) neark= 1, since sn 1( ;1) = tanh 1 = ln[(1 + )=(1 )]1=2, the expansion would involve terms in ln (which is a large number for small ) and ordering the expansion terms in the right way can be tricky. Dierent forms of such expansions have been given and studied by various authors [9]. As we showed above and in Fig.9, our exact expressions given by eqs.(80) and (82) can be used simply and directly for all cases in Region I. AsU1increases beyond 2 =3 or as the distance of closest approach Rof the light beam to the blackhole becomes smaller than 3 GM=c2, the light is not just de ected but is absorbed by and terminates at the blackhole. It is useful to divide the region 2 =3<U 11 or 3GM=c2>R0 into two regions that we call Region II and II' that are separated by the Schwarzschild horizon, as we discuss below. Region II is for Rfrom 3GM=c2up to the Schwarzschild horizon, and Region II' is for Rfrom the Schwarzschild horizon up to the center of the blackhole. Region II: 2 =3<U 11;or 3GM=c2>R2GM=c2 HereRdenotes the initial distance to the blackhole of the light beam which has initial trajectory (as increases from 0) perpendicular to the line joining it to the blackhole. AsU1increases beyond 2 =3,U1becomes greater than [1 U1+ (1 + 2U1 3U2 1)1=2]=2, and the order of the three roots must be changed to maintain the inequalitye1>e2>e3. We write e1=U1; e2=1 2[1 U1+ (1 + 2U1 3U2 1)1=2]; e3=1 2[1 U1 (1 + 2U1 3U2 1)1=2]: (85) We consider the region U >e 1>e2>e3, and write eq.(77) as dU d2 = (U e1)(U e2)(U e3): (86) Equation (86) can be integrated [7] with expressed in terms of an inverse snfunction. After some rearrangement, we nd 261 q=e1 e2sn2( ;k ) cn2( ;k ); (87) where andk2are calculated using the same expressions given by eq.(81) but withe1;e2;e3given by eq.(85). The expressions for e1;e2;e3given by eqs.(78) and (85) coincide at k2= 1 for whiche1=e2= 2=3;e3= 1=3, and both equations (80) and (87) give U= 2=3 orr= 3GM=c2independent of . Eq.(87) gives a trajectory of light which terminates at the blackhole when =2=K(k)= . As in our discussion of the terminating orbits for the planet, the terminating light ray trajectories can be characterized by the angle 2with which the light beam enters the center of the blackhole. AsU1increases from 2 =3 to 1,k2covers the entire range 1 k20; it decreases from 1 to 0. When U1= 1, i.e. when the light beam grazes the Schwarzschild horizon, e1= 1;e2=e3= 0;k2= 0; = 1=2, and we have the trajectory of light given by 1 q=1 cos2(=2); (88) which gives, for = 0;U= 1 orr=, and for=;U =1orr= 0, i.e. the light is absorbed at the center of the blackhole. Examples of the trajectories of light obtained from eqs.(87) and (88) for U1= 5=6 = 0:83333 (R= 2:4GM=c2) and 1 (R= 2GM=c2) in Region II are shown as the solid lines in Fig.10 (a) and (b). The path that emerges from the center of the blackhole whenis continued beyond 2(shown as a dotted line in Fig.10) again may be interesting if the concept of whitehole is of any physical relevance. When the distance Rto the blackhole at = 0 is inside the Schwarzschild horizon, the terminating path takes on a somewhat dierent form as we show below. Region II': 1 <U 11;or 2GM=c2>R0 HereRhas the same meaning as that in Region II. As U1increases beyond 1, i.e. when Ris less than the Schwarzschild radius, e1in eq.(85) remains real whilee2ande3become complex. We now write the three roots of the cubic equationU3 U2 U3 1+U2 1= 0 asa,bandbgiven by a=U1; b=1 2[1 U1+i(3U2 1 2U1 1)1=2]; b=1 2[1 U1 i(3U2 1 2U1 1)1=2]: (89) We consider the region U >a , and write eq.(77) as dU d2 = (U a)(U b)(U b): (90) 27This equation can be integrated [7] with expressed in terms of an inverse cnfunction. After a little algebra, we nd 1 q= 2+a ( 2 a)cn( ;k ) 1 +cn( ;k ); (91) where = [U1(3U1 2)]1=4(92) and k2=1 2 3U1 1 4p U1(3U1 2)=1 2 3a 1 4 2: (93) Equation (91) gives the trajectory of light when Ris inside the Schwarzschild horizon and it terminates at the blackhole when =20= 2K(k)= , wherek and are given by eqs.(93) and (92). On the boundary with Region II where U1= 1, andk2= 0, = 1 from eqs.(93) and (92), eq.(91) becomes eq.(88) and thus there is no discontinuity in the orbit as it makes a transition from Region II to Region II' across U1= 1. We note that as in the case of Region II' for the planetary orbits, Region II' for light trajectories covers a semi-innite range of the parameter character- izing it (1< U 11 ) but is of relevance only to a very small physical region 2GM=c2>R0 for the initial position of a light beam inside the Schwarzschild horizon. The terminating orbits of light rays in Region II' are also of very little variety as20is restricted to a limited range of 0 20. An example of a terminating trajectory obtained from eq.(91) is shown as the solid line in Fig.10(c) for U1= 10 (R= 0:2GM=c2); the dotted line again represents a tra- jectory of light coming out from the center of the blackhole as is continued beyond20. It may be of some mathematical interest to note that as U1!1 or R!0, the squared modulus of the Jacobian elliptic functions used to describe the trajectories k2approaches a value (2 p 3)=4 = 0:0669873 that is the same as that given in Section 5 for the case of Region II' for the planetary orbits. Thus the squared modulus of the elliptic functions that describe the terminating light trajectories in Region II' is restricted to a very small range 0 <k20:0669873 even as Region II' consists of a very large interval 1 <U 11. 8 Summary We have presented exact analytic expressions given by eqs.(20), (29) and (36) for the planetary orbits in the Schwarzschild geometry. The equations relate the distancerof the planet from the star or blackhole to the polar angle and are described explicitly by Jacobian elliptic functions of modulus k. Equation (20) gives periodic orbits that describe a planet precessing around a star or blackhole, while eqs.(29) and (36) give terminating orbits that describe a planet plunging into the center of a blackhole. One of the most important aspects of our analysis 28is the construction of a map with coordinates ( e;s) that we use to view all possible orbits in their entirety, where the two dimensionless parameters eand sare dened by eqs.(11) and (12) which we call the energy and eld parameters respectively. For 0 e1, we show that there are three regions which we call Regions I (0 ss1), II (s1< ss2) and II' (s2< s1 ) where these orbits are applicable, and where s1ands2that depend on eare given by eqs.(40) and (57) respectively (Fig.2). Region I has periodic and terminating orbits given by eqs.(20) and (29). Regions II and II' have terminating orbits only given by eq.(36). We have divided Region I into grids that consist of lines of constant precession angle 0 1 given by eq.(25) and lines of constant true eccentricity 0 "1 dened by eq.(28) (Fig.3); the lines of constant are obtained from solving eqs.(38) and (99), and those of constant "from solving eqs.(38) and (100). These grids make the identication of all possible periodic orbits, including the unbounded and asymptotic ones, convenient and precise. Numerous numerical results for orbits in Region I are presented in Tables 1-6, and examples of precessing orbits, including the unbounded ones, are shown in Figs.4 and 5. Among the interesting results, for example, Table 6 for = 2; e= 1; s= 0:248804 and Fig.4 f3 show that a planet coming from innity at zero polar angle, makes a complete loop about the blackhole, and returns to innity with a polar angle that approaches 2 . Figure 3, or a more rened version of it that can be constructed using the expressions for and"that we presented, can be fruitfully used with the experimental observation data. As for the terminating orbits, our orbit equations assume the planet's initial trajectory to be perpendicular to the line joining it to the star or blackhole. The terminating orbits of Region I require the planet to be at a distance q1<2 from the star or blackhole, and those of Region II' requires q201, i.e. they require the planet to be initially very close to the blackhole or within the Schwarzschild horizon. The terminating orbits of Region II, on the other hand, are more interesting as the planet can be initially at a distance 1 < q21 . We have shown how the periodic orbits of Region I become the asymptotic periodic orbits ass!s1, and then become terminating orbits as sbecomes greater than s1. It is interesting to note that a planet initially at innity with e= 1 and in a trajectory perpendicular to the line joining it to a blackhole can be absorbed by the blackhole if sis greater than 0 :25. We have also presented exact analytic expressions given by eqs.(80), (87) and (91) for the trajectories of light in the presence of a star or blackhole depending on the value of one parameter U1that has a range which can be divided into three regions: Regions I (0 U12=3), II (2=3< U 11) and II' (1< U 11 ), whereU1is dened by eq.(76). In Region I, the de ection of light can range from small angles to going continuously around the star or blackhole in a circle. In Regions II and II', light is absorbed into the center of the blackhole. Among the interesting results, a de ection of light by 180 requires a distance of closest approach Rto the blackhole equal to 3 :5206GM=c2 (Fig.9b), and for R< 3GM=c2, light will be absorbed by the blackhole. We have thus presented a complete map that can help to identify character- istics of stars and blackholes (that are not spinning too fast) from the observed 29characteristics of objects or light beams that are aected by them. 9 References *Electronic address: fhioe@sjfc.edu [1] M.P. Hobson, G. Efstathiou and A.N. Lasenby: General Relativity, Cam- bridge University Press, 2006, Chapters 9 and 10. [2] E.T. Whittaker: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th Edition, Dover, New York, 1944, Chapter XV. [3] Y. Hagihara, Japanese Journal of Astronomy and Geophysics VIII, 67 (1931). [4] S. Chandrasekhar: The Mathematical Theory of Black Holes, Oxford University Press, 1983, Chapter 3. [5] J. Levin and G. Perez-Giz, Phys. Rev. D 77, 103005 (2008). [6] F.T. Hioe, Phys. Lett. A 373, 1506 (2009). [7] P.F. Byrd and M.D. Friedman: Handbook of Elliptic Integrals for En- gineers and Scientists, 2nd Edition, Springer-Verlag, New York, 1971. See in particular p.72, 74, 81, 86 for the formulas used in this paper. [8] See e.g. J. L. Martin: General Relativity: A Guide to its Consequences for Gravity and Cosmology, Ellis Horwood Ltd., Chichester,1988, Chapter 4. [9] S.V. Iyer and A.O. Petters, Gen. Relat. and Grav. 39, 1563 (2007), C.R. Keeton and A.O. Petters, Phys. Rev. D 72, 104006 (2005). 10 Appendix A Relation Among s; eandk2 In this Appendix, we derive the relation among s;eandk2given by eqs.(38) and (39). Substituting eq.(23) into eq.(22) and after a little algebra, we nd tan 3=p 3k2 2 k2; (94) and hence we nd sin 3=p 3k2 2p 1 k2+k4; (95) cos 3=2 k2 2p 1 k2+k4: (96) We then nd cos 3+4 3 = 1 + 2k2 2p 1 k2+k4; cos 3+2 3 = 1 k2 2p 1 k2+k4: (97) 30Equations (96)-(98) and (23) allow e1;e2;e3to be expressed in terms of g2 andk2as e1=rg2 122 k2 p 1 k2+k4; e2=rg2 12 1 + 2k2 p 1 k2+k4; e3=rg2 12 1 k2 p 1 k2+k4: (98) Substituting eq.(96) into the relation cos = 3 cos(=3) + 4 cos3(=3), and substituting the result into eq.(24) gives the relation among s;eandk2given by eqs.(38) and (39). For k2= 1, we get cos = 1 from eqs.(38) and (39), and from eq.(98) we get e1=e2=p g2=12 = (1=2)e3. From eqs.(25), (21), (98) and (10), we nd an expression for the precession angle given by eq.(25) in terms of kandsto be = 4K(k)1 k2+k4 1 12s21=4 2: (99) 11 Appendix B The Energy Parameter eand the True Eccentricity "in Region I In this Appendix, we show the relation between the energy parameter eand the true eccentricity ". The energy parameter ein all three regions I, II and II' is dened by eq.(11). The general or true eccentricity "is dened by eq.(28). From eq.(28), (26) and (27) and using the expressions (98) of e1;e2;e3in terms ofk2given in Appendix A, we nd that "can be expressed as "=6(e2 e3) 1 + 6(e2+e3) or "=3k2p 1 12s2 2p 1 k2+k4 (2 k2)p 1 12s2: (100) For smalls2andk2, we nd that "'3k2=2 1 (1 6s2)'k2 4s2'e (101) becausek2'4es2for very small k2ands2[6]. We thus conrm the iden- tication of edened in eq.(11) with the eccentricity of the orbit in Newtonian mechanics. As we pointed out in the text, "is generally not equal to e. How- ever, as we show below, "=eexactly when e= 1 and in this case the orbits are unbounded. 31For the possibility of unbounded orbits in Region I, we set the initial q=1 at= 0 in eq.(20), and get 1 3+ 4e3= 0; (102) or e3= 1 12: (103) Using eqs.(23), (10) and (97) that give e3in terms of sandk2, and after a little algebra, we nd the simple equation that relates stok2one= 1 to be given by eq.(51). This simple equation (51) between sandk2fore= 1 can be used for any 0k21. For example, we nd that for k2= 1,s= 1=4 = 0:25, and for k2= 1=2,s= 1=(3p 2) = 0:235702. For e= 1, and for small k2, we have s2'k2=4 which is a special case of k2'4es2that is valid more generally for 0e1. In addition, we nd g2= 1=12 s2= 1=12 k2=[4(1 +k2)2] and therefore g2=1 k2+k4 12(1 +k2)2; (104) and from the expression for e1ande2given by eq.(98), we nd e1=2 k2 12(1 +k2); e2= 1 + 2k2 12(1 +k2): (105) Thus from the expressions (27), (30), and (21) for qmin,q1, and , we nd that whene= 1, they have the simple expressions given by eqs.(53), (54) and (52). Also, substituting eq.(48) into eq.(100) shows that "= 1 whene= 1, i.e." andecoincide at e= 1. 12 Appendix C Some Simple Relations for the Special Case of k2= 1=2 It is known that in elliptic functions, the squared modulus k2= 1=2 is a special value for which many simple relations arise. We rst consider the case of k2= 1=2 in Region I. We note that substituting k2= 1=2 into eq.(22) gives a relation e1+e3= 2e2, and substituting the expressions of e1;e2;e3from eq.(23) into this relation gives =3 ==6, or==2. Thus cos = 0 which results in g3= 0 (106) 32from eq.(24), which in turn gives a simple relationship s2=1 6(1 e2) 1 r 1 + 2e2 3! : (107) For example, we have s=q (3 p 3)=18 = 0:265408 for e= 0, and s= 1=(3p 2) = 0:235702 for e= 1 (using L'Hospital rule). These two values represent the two terminal coordinates of the constant k2= 1=2 line in Region I (see Fig.1). The other solution of eq.(106) is eq.(58) which is applicable for Region II as we shall show later in this Appendix. We also nd from eq.(23) that e1= e3=pg2 2; e2= 0; (108) and =4pg2; (109) and the orbit equation (20) becomes 1 q=1 3 2pg2cn2( ;1=p 2); where thesvalue forg2in this case is given by eq.(107) for Region I. The precession angle can be found from eq.(25) and from K(1=p 2) = 1:85407. It is given by =1:18034 4pg2 2: From eqs.(26)-(28) and (108), we nd "=6(e2 e3) 1 + 6(e2+e3)=3pg2 1 3pg2; (110) which can be inverted and solved for sin terms of ", giving s2=3 + 6" |