1001.0031.txt

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 dened. 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 classied
into various types according to the roots of a certain cubic equation, while in the
work of Levin and Perez-Giz, the orbits are classied 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 specic 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 dened 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 dened 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 identied 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
r1
dr2r2d2r2sin2d2(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, satises the equations [1]

1
r

t=; (3)
c2
1
r

t2

1
r1
r2
r2

2
=c2; (4)
3r2

=h; (5)
where the derivative 
representsd=d . The constant his identied as the
angular momentum per unit mass of the planet, and the constant is identied
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(21): (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
=u3u2+Bu+C (7)
whereu= 1=r,B= 2GM=h2,C=c2(21)=h2. Following Whittaker [2],
it is convenient to change variable from uto a dimensionless quantity Udened
by
U=1
4
r1
3
=1
4
u1
3
; (8)
oru= 4U= + 1=(3) so that eq.(7) becomes
dU
d2
= 4U3g2Ug3 (9)
where
g2=1
12s2
g3=1
2161
12s2+1
4(1e2)s4; (10)
and where
e=
1 +h2c2(21)
(GM)21=2
(11)
and
s=GM
hc: (12)
4The two dimensionless parameters eands, which are dened 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(21) which is <0 for a bound
orbit, can be identied 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[1ecos(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
12'2=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 dene the discrimi-
nant 
 of the cubic equation
4U3g2Ug3= 0 (16)
by

 = 27g2
3g3
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 dened. Instead of writing
r, we use the dimensionless distance qmeasured in units of the Schwarzschild
radiusand dened 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( e1U)(e2U)(Ue3), 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(e2e3)sn2(
;k )
=1
3+ 4e3+ 4(e2e3)1cn(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

= (e1e3)1=2; (21)
k2=e2e3
e1e3: (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 dened as 2 K(k)=
2n. For the sake of
consistency, however, we shall stick to the denition given by eq.(25).
For the case of very small sand to the order of s2, it was shown in ref.6 that

'[1(3e)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 dened by
"rmaxrmin
rmax+rmin=qmaxqmin
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 edened 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 signies 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( Ue1)(Ue2)(Ue3) 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+ 4e1e2sn2(
;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= 11
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.
Dene
A=1
2 
g3+r


27!1=3
;
B=1
2 
g3r


27!1=3
; (32)
whereg3and 
 are dened 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(AB)p
3i=2.
We further dene

= [3(A2+AB+B2)]1=4(34)
and
k2=1
23(A+B)
4p
3(A2+AB+B2)=1
23a
4
2: (35)
Writing the right-hand side of eq.(9), with Ua, as 4(Ua)(Ub)(Ub),
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(
2a)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 dened 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= 0:983104. 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= 0:657109. 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
118s2+ 54(1e2)s4
(112s2)3=2=(2k2)(1 +k2)(12k2)
2(1k2+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)2s42(19e2)s2e2= 0
that gives
s2
1=19e2+p
(19e2)2+ 27e2(1e2)2
27(1e2)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 identied 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 signicant 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'0:983104;k2'0:657109;"'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
edened by eq.(11) or eq.(13) with "dened 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 specied 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 innity at = 0 makes an angle with
the outgoing trajectory going to innity 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 
 =02= 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 identied 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
315 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
34rg2
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
13p
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 denition 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 dened 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 identied 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 identied
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
dened 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 dene 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
g2g3= 0; (56)
and substitute the expressions for g2andg3given in eq.(10) into eq.(56) and
solve fors. We nd
s2
2=1
1e2(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 dened
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 signicance,
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(1e2) 
1 +r
1 + 2e2
3!
: (58)
The signicance 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 =21=(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 innite (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 dene the "precession
angle" 
for the terminating orbits as in eq.(25), with kand
dened 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 innity) 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 dened 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 innity, i.e. 0 :6378is the polar angle
of the direction of the outgoing trajectory going to innity 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
21
4p
1=3g2; (60)

=1
41
3g21=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(k21=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!(2p
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 identied 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 satises
the equations [1]


t=1
D
r2+a2+a2
r
ah
cr
; (66)


=1
Dhac
r+
1
r
hi
; (67)
whereDr2r+a2. For the equatorial trajectories of the planet in the
Kerr geometry, the combined energy equation is

r2
+h2a2c2(21)
r2(hac)2
r3c2
r=c2(21): (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=(12ac=h ),stos0=s(1ac=h ), andto0=[1b(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 signicantly 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
r1
r2
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
=u3u2+c22
h2: (73)
The constants andhhave a physical signicance 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 dened and letting u11=R, we can set c22=h2to be equal to
u2
1u3
1[8].
It is again convenient to consider the problem in terms of the dimensionless
inverse distance Udened by
U=
r=u=1
q: (74)
Udened here is slightly dierent from the Udened by eqs.(8) and (19)
previously. In terms of Uof eq.(74), eq.(73) becomes
dU
d2
=U3U2+c222
h2: (75)
SincedU=d = 0 atr=R, one rootUwhich we call
U1
Ru1 (76)
of the cubic equation U3U2+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 U3U2U3
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
=U3U2U3
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 species 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 innitely 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[1U1+ (1 + 2U13U2
1)1=2];
e2=U1;
e3=1
2[1U1(1 + 2U13U2
1)1=2]; (78)
withe1>e2>e3;and we consider the region e1>e2>Ue3, and write
eq.(77) as
dU
d2
= (e1U)(e2U)(Ue3): (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=(e1e3)e2(e2e3)e1sn2(
;k )
(e1e3)(e2e3)sn2(
;k ); (80)
where

=(e1e3)1=2
2;
k2=e2e3
e1e3: (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
=(e1e3)e2
(e2e3)e1;
wheree1;e2;e3;
;k; are given by eqs.(78) and (81). It can also be expressed
as
24
=+2

sn1( ;k); (82)
where
 =(e1e3)e2
(e2e3)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)U2UforU=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 innitely 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 signicant 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)(13GM=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 sn1( ;k) neark= 1, since sn1( ;1) =
tanh1 = 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[1U1+ (1 + 2U13U2
1)1=2];
e3=1
2[1U1(1 + 2U13U2
1)1=2]: (85)
We consider the region U >e 1>e2>e3, and write eq.(77) as
dU
d2
= (Ue1)(Ue2)(Ue3): (86)
Equation (86) can be integrated [7] with expressed in terms of an inverse
snfunction. After some rearrangement, we nd
261
q=e1e2sn2(
;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
equationU3U2U3
1+U2
1= 0 asa,bandbgiven by
a=U1;
b=1
2[1U1+i(3U2
12U11)1=2];
b=1
2[1U1i(3U2
12U11)1=2]: (89)
We consider the region U >a , and write eq.(77) as
dU
d2
= (Ua)(Ub)(Ub): (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(
2a)cn(
;k )
1 +cn(
;k ); (91)
where

= [U1(3U12)]1=4(92)
and
k2=1
23U11
4p
U1(3U12)=1
23a1
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-innite 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 dened 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 dened 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 identication 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 innity at
zero polar angle, makes a complete loop about the blackhole, and returns to
innity with a polar angle that approaches 2 . Figure 3, or a more rened
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 innity 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 dened 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
2k2; (94)
and hence we nd
sin
3=p
3k2
2p
1k2+k4; (95)
cos
3=2k2
2p
1k2+k4: (96)
We then nd
cos
3+4
3
=1 + 2k2
2p
1k2+k4;
cos
3+2
3
=1k2
2p
1k2+k4: (97)
30Equations (96)-(98) and (23) allow e1;e2;e3to be expressed in terms of g2
andk2as
e1=rg2
122k2
p
1k2+k4;
e2=rg2
121 + 2k2
p
1k2+k4;
e3=rg2
121k2
p
1k2+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)1k2+k4
112s21=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 dened by eq.(11). The general or true eccentricity "is dened 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(e2e3)
1 + 6(e2+e3)
or
"=3k2p
112s2
2p
1k2+k4(2k2)p
112s2: (100)
For smalls2andk2, we nd that
"'3k2=2
1(16s2)'k2
4s2'e (101)
becausek2'4es2for very small k2ands2[6]. We thus conrm the iden-
tication of edened 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=12s2= 1=12k2=[4(1 +k2)2] and therefore
g2=1k2+k4
12(1 +k2)2; (104)
and from the expression for e1ande2given by eq.(98), we nd
e1=2k2
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(1e2) 
1r
1 + 2e2
3!
: (107)
For example, we have s=q
(3p
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
32pg2cn2(
;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
4pg22:
From eqs.(26)-(28) and (108), we nd
"=6(e2e3)
1 + 6(e2+e3)=3pg2
13pg2; (110)
which can be inverted and solved for sin terms of ", giving
s2=3 + 6""2
36(1 +")2: (111)
Substituting eq.(111) into eq.(106) gives ein terms of "as
e2= 112(1 +")2(1")(1 + 3")
(3 + 6""2)2: (112)
Since fore= 0,s=q
(3p
3)=18 whenk2= 1=2 as we showed above, sub-
stituting this svalue into eq.(106) gives "= (2=p
3 + 2p
31)1= 0:516588.
33Fore= 1,s= 1=(3p
2) whenk2= 1=2, and substituting this svalue into
eq.(106) gives "= 1 as it should. Thus eqs.(111) and (112) are the parametric
equations for the line of constant k2= 1=2 which can be used instead of eq.(107)
as"takes the values between 0 :516588 and 1.
Also, substituting e2= 0 from eq.(108) into eq.(27) gives
qmin= 3 (113)
independent of efork2= 1=2, as shown in Table 3.
We now consider k2= 1=2 in Region II. From eqs.(35) and (33), we have
a=A+B= 0: (114)
From eq.(32), and from A=B, andA3=B3, we arrive again at eq.(106)
with the same g3given by eq.(10). This explains why we stated after eq.(107)
that the other solution of eq.(106) given by eq.(58) gives the relation between s
andefor Region II. The two equations (108) and (58) that give simple relations
betweensandein two dierent regions and that arise as two dierent solutions
of the same equation (106) show a rather remarkable symmetry exhibited by
the special case k2= 1=2.
It also follows from eqs.(32) and (34) that
A=1
2g2
31=2
=1
12p
12s21; (115)

=
3A2
1=4=1
21
3(12s21)1=4
; (116)
and that the orbit equation (36) becomes
1
q=1
3+ 4
21cn(2
;1=p
2)
1 +cn(2
;1=p
2); (117)
where thesvalue for the above equations is given by eq.(58) for 0 e1.
Sincea= 0, the initial distance q2of the planet from the blackhole is q2= 3
from eq.(37), independent of e, as shown in Table 10.
13 Appendix D The Boundaries of Region II
On the upper boundary s2
2= 1=(1e2) of Region II, the planet starts from
the Schwarzschild horizon given by q= 1, which implies that for = 0, 1=q=
1=3 + 4afrom eq.(36), or
1 =1
3+ 4a: (118)
Hence
34a=A+B=1
6: (119)
Since from eqs.(32) and (17),
AB=1
4
g2
3

271=3
=g2
12; (120)
we can conclude from eqs.(35) and (34) that k2and
are given on the
boundarys2
2= 1=(1e2) of Region II by eqs.(60) and (61).
In particular, for e= 0,s= 1, we nd 
= (5=16)1=4= 0:747674,k2=
(11=p
5)=2 = 0:276393, and this is the minimum value of k2in Region II.
We now consider the right boundary e= 1 of Region II. Consider the un-
bounded orbit of a planet coming from innity that requires, from eq.(36), that
1
3+ 4a= 0; (121)
or
a=1
12: (122)
From
A+B=a=1
12; (123)
and
AB=g2
12; (124)
and from eqs.(34) and (35), we nd eqs.(62) and (64) that give k2and
in
terms ofson the boundary e= 1 of Region II.
Finally, there is a special case of g2= 0 ors2= 1=12 (s= 1=2p
3 =
0:288675135) in Region II that is somewhat interesting. If g2of eq.(10) is
equal to zero, then 
 = 27 g2
3from eq.(17), and from eqs.(32)-(35), we nd
A= (2g3)1=3=2,B= 0,a=A,
= (3A2)1=4, and noting that A < 0 for
s2= 1=12 and 0e1, we have
k2=1
2p
3A
4jAj=1
2+p
3
4= 0:933012702
which is independent of e, i.e. the constant k2= (2 +p
3)=4 curve in Region
II just above the boundary curve of Regions I and II in Fig.2 is a horizontal
line. Thus the terminating orbits represented by eq.(36) for any point along this
horizontal line are represented by elliptic functions of the same squared modulus
given above. It should be remembered, however, that the orbits and the initial
positions of the planet (the initial trajectory of which is perpendicular to the
35line joining it to the blackhole) depend also on 
andathat are dependent on
the value of e. Thus the orbit for e= 1, for example, that gives a=1=12
(and
= 1=(2
31=4) = 0:379917843) is a terminating orbit for a planet that is
initially innitely far away from a blackhole.
36Table 1: Values of sfor various values of eand k2in Region I
s e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 0.0554787 0.0454170 0.0346617 0.0286392 0.0248892 0.0222952 0.0203686 0.0188667 0.0176539 0.0166480 0.0157963
k2/u003d0. 01 0.114809 0.109191 0.0969713 0.0851063 0.0757980 0.0686999 0.0631762 0.0587575 0.0551328 0.0520954 0.0495050
k2/u003d0. 1 0.210213 0.208385 0.203281 0.195887 0.187374 0.178683 0.170376 0.162703 0.155729 0.149428 0.143740
k2/u003d0. 2 0.238703 0.237612 0.234488 0.229734 0.223875 0.217424 0.210787 0.204236 0.197932 0.191956 0.186339
k2/u003d0. 3 0.252575 0.251809 0.249595 0.246160 0.241814 0.236873 0.231612 0.226239 0.220900 0.215689 0.210663
k2/u003d0. 4 0.260533 0.259944 0.258236 0.255562 0.252131 0.248163 0.243857 0.239372 0.234829 0.230311 0.225877
k2/u003d0. 5 0.265408 0.264926 0.263523 0.261314 0.258458 0.255122 0.251460 0.247600 0.243642 0.239658 0.235702
k2/u003d0. 6 0.268462 0.268045 0.266831 0.264913 0.262422 0.259494 0.256258 0.252821 0.249269 0.245666 0.242061
k2/u003d0. 7 0.270350 0.269973 0.268875 0.267137 0.264873 0.262203 0.259239 0.256076 0.252791 0.249443 0.246076
k2/u003d0. 8 0.271452 0.271099 0.270069 0.268436 0.266305 0.263787 0.260985 0.257986 0.254863 0.251671 0.248452
k2/u003d0. 9 0.272006 0.271665 0.270668 0.269088 0.267025 0.264583 0.261863 0.258948 0.255908 0.252796 0.249653
k2/u003d1. 0 0.272166 0.271828 0.270840 0.269276 0.267232 0.264812 0.262116 0.259225 0.256209 0.253120 0.250000
Table 2: Values of qmaxfor various values of eand k2in Region I
qmax e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 174.74 273.42 522.17 871.99 1345.9 2012.0 3012.8 4681.4 8018.4 18025 /u221e
k2/u003d0. 01 43.673 49.688 68.266 99.742 145.78 212.38 313.52 482.96 822.60 1842.4 /u221e
k2/u003d0. 1 14.306 14.851 16.550 19.608 24.429 31.802 43.389 63.171 103.21 223.98 /u221e
k2/u003d0. 2 11.377 11.697 12.693 14.481 17.300 21.627 28.458 40.168 63.938 135.75 /u221e
k2/u003d0. 3 10.282 10.534 11.314 12.713 14.914 18.291 23.625 32.774 51.357 107.53 /u221e
k2/u003d0. 4 9.7280 9.9479 10.630 11.849 13.766 16.704 21.342 29.298 45.460 94.318 /u221e
k2/u003d0. 5 9.4118 9.6145 10.243 11.365 13.128 15.828 20.089 27.397 42.239 87.109 /u221e
k2/u003d0. 6 9.2220 9.4148 10.012 11.078 12.751 15.314 19.356 26.285 40.359 82.903 /u221e
k2/u003d0. 7 9.1078 9.2946 9.8733 10.906 12.527 15.008 18.920 25.627 39.246 80.413 /u221e
k2/u003d0. 8 9.0421 9.2256 9.7938 10.808 12.399 14.833 18.672 25.252 38.612 78.997 /u221e
k2/u003d0. 9 9.0094 9.1912 9.7542 10.759 12.335 14.747 18.549 25.066 38.299 78.296 /u221e
k2/u003d1. 0 9.0000 9.1814 9.7429 10.745 12.317 14.722 18.514 25.013 38.209 78.095 /u221e
Table 3: Values of qminfor various values of eand k2in Region I
qmin e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 149.15 215.27 343.84 466.77 574.79 669.11 751.92 825.10 890.20 948.48 1001.0
k2/u003d0. 01 31.135 33.980 41.469 50.945 60.399 69.110 76.948 83.963 90.248 95.899 101.00
k2/u003d0. 1 7.0549 7.1489 7.4154 7.8125 8.2864 8.7891 9.2876 9.7635 10.209 10.621 11.000
k2/u003d0. 2 4.7480 4.7753 4.8530 4.9703 5.1133 5.2691 5.4278 5.5830 5.7311 5.8703 6.0000
k2/u003d0. 3 3.8359 3.8465 3.8765 3.9220 3.9777 4.0389 4.1018 4.1639 4.2237 4.2803 4.3333
k2/u003d0. 4 3.3289 3.3325 3.3428 3.3583 3.3773 3.3983 3.4199 3.4413 3.4619 3.4815 3.5000
k2/u003d0. 5 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000
k2/u003d0. 6 2.7667 2.7645 2.7585 2.7494 2.7382 2.7260 2.7134 2.7009 2.6889 2.6774 2.6667
k2/u003d0. 7 2.5911 2.5877 2.5778 2.5629 2.5446 2.5247 2.5042 2.4840 2.4645 2.4460 2.4286
k2/u003d0. 8 2.4535 2.4491 2.4366 2.4178 2.3949 2.3698 2.3442 2.3189 2.2946 2.2716 2.2500
k2/u003d0. 9 2.3422 2.3371 2.3228 2.3012 2.2751 2.2465 2.2174 2.1887 2.1613 2.1354 2.1111
k2/u003d1. 0 2.2500 2.2445 2.2288 2.2052 2.1767 2.1458 2.1142 2.0833 2.0538 2.0259 2.0000
37Table 4: Values of /u0394/u03c6//u03c0for various values of eand k2in Region I
/u0394/u03c6//u03c0 e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 0.018906 0.012572 0.0072747 0.0049522 0.0037347 0.0029941 0.0024975 0.0021419 0.0018748 0.0016669 0.0015004
k2/u003d0. 01 0.088019 0.078760 0.060819 0.046033 0.036087 0.029413 0.024739 0.021316 0.018713 0.016671 0.015029
k2/u003d0. 1 0.42211 0.41041 0.37969 0.33964 0.29902 0.26259 0.23177 0.20625 0.18522 0.16780 0.15323
k2/u003d0. 2 0.69762 0.68449 0.64888 0.59968 0.54604 0.49423 0.44737 0.40639 0.37108 0.34079 0.31478
k2/u003d0. 3 0.95650 0.94206 0.90245 0.84652 0.78377 0.72123 0.66290 0.61042 0.56410 0.52353 0.48808
k2/u003d0. 4 1.2200 1.2043 1.1608 1.0987 1.0281 0.95649 0.88857 0.82648 0.77085 0.72150 0.67786
k2/u003d0. 5 1.5029 1.4858 1.4383 1.3701 1.2917 1.2116 1.1348 1.0638 0.99972 0.94234 0.89123
k2/u003d0. 6 1.8226 1.8038 1.7519 1.6770 1.5905 1.5015 1.4157 1.3360 1.2635 1.1982 1.1398
k2/u003d0. 7 2.2072 2.1866 2.1293 2.0465 1.9507 1.8517 1.7559 1.6666 1.5851 1.5114 1.4453
k2/u003d0. 8 2.7169 2.6938 2.6295 2.5365 2.4286 2.3170 2.2087 2.1075 2.0150 1.9312 1.8558
k2/u003d0. 9 3.5401 3.5129 3.4374 3.3281 3.2011 3.0696 2.9419 2.8224 2.7130 2.6139 2.5247
k2/u003d1. 0 /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e
Table 5: Values of /u03b5for various values of eand k2in Region I
/u03b5 e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 0.079005 0.11900 0.20592 0.30268 0.40148 0.50088 0.60054 0.70032 0.80015 0.90002 1
k2/u003d0. 01 0.16760 0.18774 0.24420 0.32383 0.41413 0.50896 0.60587 0.70380 0.80227 0.90105 1
k2/u003d0. 1 0.33947 0.35009 0.38115 0.43017 0.49343 0.56695 0.64738 0.73227 0.81998 0.90946 1
k2/u003d0. 2 0.41109 0.42021 0.44683 0.48894 0.54373 0.60819 0.67964 0.75594 0.83548 0.91710 1
k2/u003d0. 3 0.45659 0.46503 0.48963 0.52845 0.57889 0.63825 0.70412 0.77454 0.84802 0.92344 1
k2/u003d0. 4 0.49009 0.49813 0.52152 0.55832 0.60598 0.66190 0.72378 0.78978 0.85847 0.92880 1
k2/u003d0. 5 0.51659 0.52436 0.54692 0.58231 0.62797 0.68133 0.74014 0.80261 0.86737 0.93341 1
k2/u003d0. 6 0.53845 0.54603 0.56798 0.60232 0.64644 0.69778 0.75410 0.81364 0.87508 0.93743 1
k2/u003d0. 7 0.55703 0.56445 0.58594 0.61944 0.66233 0.71200 0.76623 0.82327 0.88183 0.94096 1
k2/u003d0. 8 0.57314 0.58044 0.60155 0.63437 0.67623 0.72449 0.77692 0.83179 0.88781 0.94410 1
k2/u003d0. 9 0.58733 0.59454 0.61534 0.64759 0.68857 0.73560 0.78645 0.83939 0.89316 0.94690 1
k2/u003d1. 0 0.60000 0.60713 0.62766 0.65942 0.69963 0.74558 0.79502 0.84623 0.89798 0.94943 1
Table 6: Values of sfor constant values of /u0394/u03c6in Region I
s e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
/u0394/u03c6/u003d/u03c0/18 0.0929975 0.0929922 0.0929763 0.0929498 0.0929129 0.0928655 0.0928078 0.0927400 0.0926622 0.0925745 0.0924773
/u0394/u03c6/u003d/u03c0/6 0.150971 0.150946 0.150871 0.150747 0.150574 0.150353 0.150087 0.149776 0.149424 0.149031 0.148601
/u0394/u03c6/u003d/u03c0/3 0.195246 0.195185 0.195000 0.194694 0.194273 0.193741 0.193104 0.192372 0.191551 0.190651 0.189680
/u0394/u03c6/u003d/u03c0/2 0.220477 0.220377 0.220080 0.219591 0.218920 0.218080 0.217085 0.215951 0.214696 0.213336 0.211888
/u0394/u03c6/u003d/u03c0 0.254214 0.254018 0.253437 0.252492 0.251216 0.249650 0.247838 0.245823 0.243650 0.241354 0.238971
/u0394/u03c6/u003d3/u03c0/20.265371 0.265111 0.264346 0.263113 0.261468 0.259478 0.257210 0.254729 0.252091 0.249347 0.246537
/u0394/u03c6/u003d2/u03c0 0.269502 0.269206 0.268334 0.266938 0.265091 0.262877 0.260379 0.257671 0.254819 0.251875 0.248804
/u0394/u03c6/u003d/u221e 0.272166 0.271828 0.270840 0.269276 0.267232 0.264812 0.262116 0.259225 0.256209 0.253120 0.250000
38Table 7: Values of sfor constant values of /u03b5in Region I
s e/u003d0. 00 e/u003d0. 02 e/u003d0. 04 e/u003d0. 06 e/u003d0. 08 e/u003d0. 10 e/u003d0. 12 e/u003d0. 14 e/u003d0. 16 e/u003d0. 18 e/u003d0. 20
/u03b5/u003d0. 2 0.135153 0.134536 0.132663 0.129457 0.124776 0.118388 0.109916 0.0987083 0.0835085 0.0611485 0
s e/u003d0. 00 e/u003d0. 04 e/u003d0. 08 e/u003d0. 12 e/u003d0. 16 e/u003d0. 20 e/u003d0. 24 e/u003d0. 28 e/u003d0. 32 e/u003d0. 36 e/u003d0. 40
/u03b5/u003d0. 4 0.234806 0.234069 0.231803 0.227832 0.221831 0.213264 0.201255 0.184309 0.159570 0.120022 0
s e/u003d0. 00 e/u003d0. 06 e/u003d0. 12 e/u003d0. 18 e/u003d0. 24 e/u003d0. 30 e/u003d0. 36 e/u003d0. 42 e/u003d0. 48 e/u003d0. 54 e/u003d0. 60
/u03b5/u003d0. 6 0.272166 0.272037 0.271572 0.270524 0.268439 0.264552 0.257565 0.245130 0.222468 0.177299 0
s e/u003d0. 61 e/u003d0. 62 e/u003d0. 64 e/u003d0. 66 e/u003d0. 68 e/u003d0. 70 e/u003d0. 72 e/u003d0. 74 e/u003d0. 76 e/u003d0. 78 e/u003d0. 80
/u03b5/u003d0. 8 0.261834 0.261450 0.259987 0.257309 0.252931 0.246121 0.235710 0.219699 0.194219 0.149735 0
Table 8: Values of q1for various values of eand k2in Region I (terminating orbits)
q1 e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d0. 001 1.0126 1.0084 1.0048 1.0033 1.0025 1.0020 1.0017 1.0014 1.0012 1.0011 1.0010
k2/u003d0. 01 1.0582 1.0521 1.0403 1.0306 1.0240 1.0196 1.0165 1.0142 1.0124 1.0111 1.0100
k2/u003d0. 1 1.2685 1.2614 1.2427 1.2180 1.1928 1.1699 1.1504 1.1341 1.1206 1.1094 1.1000
k2/u003d0. 2 1.4255 1.4182 1.3983 1.3703 1.3394 1.3089 1.2810 1.2563 1.2348 1.2161 1.2000
k2/u003d0. 3 1.5575 1.5502 1.5299 1.5007 1.4672 1.4332 1.4008 1.3711 1.3445 1.3209 1.3000
k2/u003d0. 4 1.6756 1.6683 1.6481 1.6186 1.5841 1.5483 1.5135 1.4809 1.4511 1.4242 1.4000
k2/u003d0. 5 1.7844 1.7773 1.7574 1.7281 1.6935 1.6570 1.6210 1.5869 1.5552 1.5263 1.5000
k2/u003d0. 6 1.8864 1.8795 1.8601 1.8315 1.7973 1.7610 1.7248 1.6900 1.6575 1.6275 1.6000
k2/u003d0. 7 1.9831 1.9764 1.9578 1.9301 1.8968 1.8612 1.8254 1.7908 1.7582 1.7279 1.7000
k2/u003d0. 8 2.0754 2.0691 2.0513 2.0248 1.9929 1.9584 1.9236 1.8898 1.8577 1.8278 1.8000
k2/u003d0. 9 2.1642 2.1583 2.1415 2.1164 2.0860 2.0532 2.0198 1.9872 1.9562 1.9271 1.9000
k2/u003d1. 0 2.2500 2.2445 2.2288 2.2052 2.1767 2.1458 2.1142 2.0833 2.0538 2.0259 2.0000
Table 9: Values of sfor various values of eand k2in Region II
s e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d1. 0 0.272166 0.271828 0.270840 0.269276 0.267232 0.264812 0.262116 0.259225 0.256209 0.253120 0.250000
k2/u003d0. 9 0.297739 0.297917 0.298442 0.299293 0.300443 0.301865 0.303537 0.305444 0.307575 0.309927 0.312500
k2/u003d0. 8 0.329945 0.330718 0.333024 0.336843 0.342169 0.349050 0.357613 0.368098 0.380913 0.396727 0.416667
k2/u003d0. 7 0.371926 0.373406 0.377868 0.385401 0.396232 0.410826 0.430060 0.455531 0.490285 0.540844 0.625000
k2/u003d0. 6 0.429234 0.431592 0.438767 0.451100 0.469328 0.494871 0.530424 0.581374 0.659978 0.802847 1.25000
k2/u003d0. 5 0.512730 0.516250 0.527046 0.545908 0.574478 0.615920 0.676462 0.769603 0.930895 1.30267 ——
k2/u003d0. 4 0.646974 0.652192 0.668307 0.696858 0.741019 0.806949 0.907116 1.07001 1.37756 2.21530 ——
k2/u003d0. 3 0.901890 0.910054 0.935412 0.980848 1.05231 —— —— —— —— —— ——
39Table 10: Values of q2for various values of eand k2in Region II
q2 e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d1. 0 9.0000 9.1814 9.7429 10.745 12.317 14.722 18.514 25.013 38.209 78.095 /u221e
k2/u003d0. 9 7.6760 7.7965 8.1685 8.8291 9.8596 11.426 13.833 18.073 26.550 52.109 /u221e
k2/u003d0. 8 6.4074 6.4806 6.7060 7.1041 7.7200 8.6471 10.084 12.507 17.353 31.823 /u221e
k2/u003d0. 7 5.1997 5.2381 5.3557 5.5620 5.8780 6.3467 7.0597 8.2346 10.522 17.148 /u221e
k2/u003d0. 6 4.0606 4.0750 4.1191 4.1959 4.3120 4.4809 4.7310 5.1276 5.8592 7.8120 /u221e
k2/u003d0. 5 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 ——
k2/u003d0. 4 2.0324 2.0259 2.0059 1.9717 1.9217 1.8523 1.7574 1.6241 1.4235 1.0692 ——
k2/u003d0. 3 1.1805 1.1733 1.1516 1.1148 1.0618 —— —— —— —— —— ——
Table 11: Values of /u03c62//u03c0for various values of eand k2in Region II
/u03c62//u03c0 e/u003d0. 0 e/u003d0. 1 e/u003d0. 2 e/u003d0. 3 e/u003d0. 4 e/u003d0. 5 e/u003d0. 6 e/u003d0. 7 e/u003d0. 8 e/u003d0. 9 e/u003d1. 0
k2/u003d1. 0/u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e /u221e
k2/u003d0. 9 2.6599 2.6469 2.6099 2.5551 2.4890 2.4176 2.3448 2.2733 2.2043 2.1385 2.0761
k2/u003d0. 8 2.1586 2.1480 2.1175 2.0711 2.0132 1.9479 1.8781 1.8055 1.7309 1.6541 1.5742
k2/u003d0. 7 1.8169 1.8079 1.7819 1.7412 1.6889 1.6274 1.5584 1.4822 1.3977 1.3010 1.1817
k2/u003d0. 6 1.5360 1.5283 1.5060 1.4703 1.4231 1.3654 1.2977 1.2186 1.1237 1.0002 0.78496
k2/u003d0. 5 1.2822 1.2758 1.2569 1.2262 1.1843 1.1315 1.0671 0.98815 0.88720 0.74053 ——
k2/u003d0. 4 1.0373 1.0321 1.0166 0.99105 0.95540 0.90923 0.85112 0.77757 0.68006 0.53257 ——
k2/u003d0. 3 0.78606 0.78213 0.77030 0.75047 0.72230 —— —— —— —— —— ——
400.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.000.050.100.150.200.25
es
0.0010.010.11.0
Fig. 1. Region I plots of k2/u003d0. 001, 0. 01, 0. 1, 0. 3, 0. 5, 0. 7, 1. 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.2
es
10.5s(2)
0.70.60.4
0.3
Region IRegion IIRegion II'
s(1)
Fig. 2. Region II plots of k2/u003d1. 0, 0. 9, 0. 8, 0. 7, 0. 6, 0. 5, 0. 4, 0. 3
410.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.000.020.040.060.080.100.120.140.160.180.200.220.240.260.28
es
0.2 0.4 0.6 0.81/181/61/31/21
Fig. 3./u0394/u03c6//u03c0/u003d1/18, 1/6, 1/3, 1/2, 1, 3/2, 2 a nd/u03b5/u003d0. 2, 0. 4, 0. 6, 0. 8 ( Region I)
42Fig. 4. Region I: Periodic(a1-f1,a2-f2), unbounded(a3-g3), asymptotic period ic orbits(g1-g3) for ∆߮ equal to 
(a) /ߨ6 , (b) /ߨ3, (c) /ߨ2, (d) ,ߨ( e) 3/ߨ2,  (f) 2,ߨ( g) ∞ and for e equal to (1) 0, (2) 0.5, (3) 1. 
,
43Fig. 5. Precessional orbit for ∆߮ ൌ ቀଵ
ଷ൅0.002614…ቁ߳,ߨ ൌ 0.58143… , ݁ ൌ 0.5, ݏ ൌ 0.194229…
44Fig. 6. Terminating orbits in region I                                    
(a) ݇ଶൌ 0.01, ݁ ൌ 0, ݏ ൌ 0.114809,ఝభ
గൌ 1.0440, ݍ ଵൌ 1.0582  
(b) ݇ଶൌ 0.1, ݁ ൌ 0.5, ݏ ൌ 0.178683,ఝభ
గൌ 1.1313, ݍ ଵൌ 1.1699  
(c) ݇ଶൌ 0.9, ݁ ൌ 0.9, ݏ ൌ 0.252796,ఝభ
గൌ 2.3070, ݍ ଵൌ 1.9271   
(d) ݇ଶൌ1 , ݁ൌ1 , ݏൌ0 . 2 5 , ߮ ଵൌ ∞ , ݍ ଵൌ2  
45Fig. 7. Terminating orbits in Region II (a-d) and Region II’ (e)          
(a) ݇ଶൌ1 , ݁ൌ1 , ݏൌ0 . 2 5 ,ఝమ
గൌ ∞ , ݍ ଶൌ∞  
(b) ݇ଶൌ0 . 6 , ݁ൌ1 , ݏൌ1 . 2 5 ,ఝమ
గൌ  0.78496, ݍ ଶൌ ∞  
(c) ݇ଶൌ 0.5, ݁ ൌ 0.5, ݏ ൌ 0.615920,ఝమ
గൌ 1.1315, ݍ ଶൌ3 
(d) ݇ଶൌ 0.3, ݁ ൌ 0, ݏ ൌ 0.901890,ఝమ
గൌ 0.78606, ݍ ଶൌ 1.1805  
(e) ݇ଶ ൌ  0.095385, ݁ ൌ 0, ݏ ൌ 10,ఝమ
గൌ 0.13684, ݍ ଶ = 0.03267 
46Fig. 8. Unbounded orbits for  
(a) ݇ଶൌ 0.99, ݁ ൌ 1, ݏ ൌ 0.2499968435,∆ఝ
గൌ 4.6378      
(b) ݇ଶൌ1 , ݁ൌ1 , ݏൌ0 . 2 5 ,∆ఝ
గൌ ∞       
(c) ݇ଶൌ 0.9998000799, ݁ ൌ 1, ݏ ൌ 0.2501 ,ఝమ
గൌ 5.0816            
 
                                 
47Fig. 9.  Trajectories of light for  
(a) ܷଵൌ 0.42922, ∆߮ ൌ  /ߨ2  
(b) ܷଵ ൌ 0.56808, ∆߮ ൌ  ߨ   
(c) ܷଵൌ 0.62334, ∆߮ ൌ 3/ߨ2  
(d) ܷଵൌ 0.64720, ∆߮ ൌ 2ߨ  
 
48Fig. 10. Trajectories of light that terminated                                 
(a) ܷଵൌ 0.8333, ߮ ଵൌ 3.8345  
(b) ܷଵൌ1 , ߮ ଵൌ ߨ 
(c) ܷଵൌ 10, ߮ ଵൌ 0.78133  
 
49