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the determination of the signal template of the radiating companion of a black hole is of great importance for the forthcoming interferometric gravitational observatories . an inspiralling neutron star or a several solar - mass black hole orbiting a massive black hole , as well as a particle constituent contributing to the radiation of the debris trapped by the black hole are notable examples of these companions . either is the case , one wants to describe the evolution of the orbit of a test particle in the neighborhood of a black hole under the influence of gravitational radiation backreaction . in a highly idealized picture , the orbit of the particle is a carter geodesic@xcite characterized by four constants of the motion , the energy , the total angular momentum , the rest - mass of the particle and the separation constant . in the spirit of perturbation theory , the radiation backreaction effects may be taken into account by evolving these constants of the motion , thus picturing the trajectory of the particle by a sequence of geodesic orbits . the history of the orbit may be divided in separate epoches . for example , the first thing that happens to a particle trapped by a nonrotating _ schwarzschild _ black hole on an arbitrary but distant orbit is that it loses eccentricity due to radiation losses , and the radius shrinks until the close region of the last stable circular orbit is approached . there one has to replace the adiabatic picture with one valid in the strong field regime .    the description of a generic orbit in the strong field region of a spinning _ kerr _ black hole is difficult ( as has been pointed out by thorne@xcite ) due to the special nature of the separation constant which lacks a straightforward geometrical interpretation . recently , progress in treating the separation constant has been reported by ori@xcite . there is a considerable ease in the description of special orbits such as equatorial or quasicircular , and in fact , several papers have obtained the signal templates for such special orbits . in order that any of these special orbits can be credited as significant contributor to the gravitational signal , one has to make sure that the carter constant evolves to the envisaged special values . several authors have recently published results on radiative losses by systems of spinning masses . kidder , will and wiseman@xcite formulate the problem by using the lagrangian ( proposed by barker and oconnell@xcite ) for bodies with finite masses and spins . they and kidder @xcite compute the spin - orbit and spin - spin contributions to the momentary radiative loss of energy , linear and angular momenta . in the test particle limit their results have subsequently been used by shibata@xcite , in his study of equatorial orbits , and by ryan@xcite . by computing the backreaction forces , ryan obtains the power and angular momentum losses for circular and generic orbits .    in our work , we have taken up the evolution of the orbit in the far field region of the black hole . the trajectory of the idealized , nonradiating particle is described by the lense - thirring picture@xcite . this picture emerges in the test particle limit of the scenario described in @xcite . the spin @xmath5 of the black hole is fixed to the @xmath6 axis . the orbit is viewed approximately as an ellipse precessing in a plane which rotates about @xmath7 . the motion is characterized by three constants : the energy @xmath8 , the magnitude @xmath9 and spin projection @xmath10 of the orbital angular momentum , and by two slowly changing angle parameters @xmath2 and @xmath11 . the lense - thirring approximation is linear in the spin terms , thus the ( additive ) schwarzschild effects may be neglected in this treatment .    in an earlier paper@xcite , we have computed the radiation losses from the mass quadrupole tensor in the lense - thirring approximation , by the orbit smoothing method of landau and lifshitz@xcite . however , one may be rightly concerned about the degradation of accuracy of this computation , as a price of the relative simplicity introduced by the smoothing method .    in the present paper , we drop the orbit smoothing , and we include the velocity quadrupole tensor @xmath12 in the description of the radiative losses . in the test particle limit , this ( formally ) corresponds to the 1pn approximation@xcite . the rest of the paper is organized as follows . in sec . 2 we describe the orbital motion in the lense - thirring approximation . in terms of a set of new angular variables ( related to euler rotations ) , the equations of motion take a particularly simple form : all time derivatives depend on the radial variable @xmath13 alone .    in sec . 3 , we explore the advantages of having a pure radial equation of motion . we find the turning points at @xmath14 which allow us to parameterize the radial motion by trigonometric functions . integrations over the time can be replaced by integrations over the parameter , which are straightforward in principle , but can be messy in practice . we evade such complications by finding suitable parametrizations . the first parameter @xmath15 we introduce is a generalization of the eccentric anomaly of the keplerian orbits . it is employed for computing the half period defined as the time spent by the particle between consecutive turning points . for the purpose of averaging the radiative losses , and for determining the change in the new angular variables , a generalized true anomaly parameter @xmath16 is properly defined . in terms of this parameter , when integration by the residue theorem is performed , the only pole of the integrands will be in the origin .    in sec . 4 , we compute the rates of change of the constants of motion due to radiation backreaction , to first order in @xmath7 , by employing the radiative multipole tensors of kidder , will and wiseman@xcite , which originate in the blanchet - damour - iyer formalism@xcite . as we are interested in the cumulative effects of the secular motion , we compute averaged losses over the period of radial motion . we give the losses in an unambiguous manner in terms of the constants of motion @xmath17 and @xmath10 . as has been pointed out by ryan @xcite , the definition of the orbit parameters @xmath18 and @xmath19 is subject to ambiguities . in sec . 5 , we define the semimajor axis and eccentricity . with advantages explained in the main text , our definitions differ both from ryan s and the ones used in the lense - thirring paper . we rewrite all losses in terms of our orbital parameters @xmath20 and @xmath19 .    in the concluding remarks , we complete the characterization of the changes caused by gravitational radiation by giving the change in the @xmath3 angle parameters over one orbit . we then compare our results with those of other authors . we wish to make some comments on the expansion procedure we have adopted . earlier works employ either the expansion parameter @xmath21 or the inverse @xmath22 of the speed of light . since we eliminate @xmath23 and @xmath24 by use of the first integrals of the motion , we choose to retain @xmath25 and @xmath26 for bookkeeping of the orders in our formalism . the correction terms in the lagrangian , in the equations of motion and radiation losses will carry an extra @xmath27 factor . in our computation , we keep only the terms containing the spin @xmath0 from among these correction terms , thus we may as well choose @xmath0 as our expansion parameter . our treatment of radiation backreaction effects can be generalized beyond the lense - thirring approximation . in a follow - up paper@xcite we will apply our averaging method and parameterization for computing the backreaction effects on a bound system of two finite masses . we consider the orbit of a small particle of mass @xmath28 about a body with mass @xmath29 and angular momentum vector @xmath30 . in the lense - thirring approximation@xcite , the lagrangian of the system has the form@xcite @xmath31 where @xmath32 , a dot denotes @xmath33 and the term describing ( perturbative ) rotation effects is @xmath34 the cartesian coordinates of the test particle are @xmath35 , with origin chosen at the black hole . hence the orbital momentum @xmath36 and runge - lenz vector @xmath37 satisfy the equations of motion , respectively @xmath38 @xmath39 here @xmath40 is the momentum of the orbiting mass . note that the angular momentum @xmath41 undergoes a pure rotation about the spin with angular velocity : @xmath42    from ( [ l ] ) and ( [ a ] ) , the vector @xmath43 can be written in terms of the basis @xmath44 in the plane with normal @xmath41 : @xmath45 where @xmath46 . inserting this expression of @xmath47 in ( [ adot ] ) , the direction @xmath48 of the runge - lenz vector is found to rotate about a linear combination of the vectors @xmath7 and @xmath41 : @xmath49\times{{\bf a}\over a } \ .\label{averst}\ ] ] thus the direction vector rotates about @xmath7 with the angular velocity @xmath50 ( as does @xmath41 ) and about @xmath41 with the angular velocity : @xmath51    although the orbital momentum and the runge - lenz vector are not conserved ( unlike in the zeroth order in @xmath7 ) , here also there are constants of motion . because of the fact that the potential is axisymmetric and stationary , the orbital momentum component @xmath10 and the energy @xmath52 are conserved . since the motion of @xmath41 is a pure rotation , its magnitude @xmath9 is also constant : @xmath53 the constants of the motion are @xmath54}-{g \mu m\over r}\label{e}\\ \label{eq : lz } l_z&=&\mu r^2\sin^2\theta\ \dot\varphi-2\frac{g\mu}{c^2r}s\sin^2\theta \label{lz } \\ \label{eq : l2 } l^2&=&{\mu^2}r^4 ( \dot\theta^2+\sin^2\theta\ \dot\varphi^2)-4{g\mu^2s\over c^2}r\sin^2\theta\ \dot\varphi   \ . \label{l2 } \\end{aligned}\ ] ] ( here we are using polar coordinates , @xmath55 , @xmath56 , @xmath57 ) . as a consequence , the angle @xmath58 subtended by @xmath41 and @xmath7 is also conserved : @xmath59 the first integrals of the equations of motion follow from ( [ eq : e])-([eq : l2 ] ) by simple algebra ( we need not recourse to the hamilton - jacobi formalism ) . thus the equation of the radial motion takes the first - order form , decoupled from the angular degrees of freedom : @xmath60 the equations for the angular variables @xmath61 , however , are coupled among themselves and with the @xmath13 motion : @xmath62    there is an alternative description of the motion in the lense - thirring approximation which bears a more intimate geometrical relation to the picture provided by perturbation theory . we shall introduce this formalism by performing time - dependent euler rotations about the origin so as to place the kepler ellipse in its momentary orientation : @xmath63 here @xmath64 is the initial position of the particle on the plane perpendicular to @xmath41  ( although @xmath41 is different from @xmath65 , the condition @xmath66 is fulfilled ) . a rotation about the cartesian @xmath67 and @xmath6 axes is given , respectively , by    @xmath68    the coordinates of the particle are written in terms of the euler angles @xmath69 and @xmath70 as @xmath71    hence we get the relations    @xmath72    which allow us to express the time derivatives of the new variables :    @xmath73    note that @xmath74 and @xmath75 carries the interpretation of angular velocity of rotation about @xmath7 and in the plane perpendicular to @xmath41 respectively . an important feature of the new coordinates @xmath76 is that their derivatives appear linearly in the equations of motion ( [ psit ] ) and ( [ phit ] ) . this is not the case with the polar coordinates ( [ polar ] ) because the root of @xmath77 can not be extracted . the geometrical interpretation of the angle variable @xmath78 assures that it is monotonously changing . we choose @xmath70 to increase with the motion evolving . there is a freedom in the precise way one introduces the three euler - angle variables in place of the polar angles . even though the equations of motion are made simple by use of the angular variables ( [ newvar]-[newvar2 ] ) , the description of the momentary plane of orbit is tied to a different choice of the euler angles @xmath79 , where the initial @xmath6 axis is chosen along @xmath80 rather than along @xmath41 . these angular variables appear ( with the appropriate change in notation ) in the lense - thirring paper . the angle @xmath81 is not a constant , and the resulting equations are less simple . the primed angles will not be employed in our computations , and we relegate their discussion to the appendix . we now proceed with the solution of the equations of motion . first consider the radial equation ( [ radial ] ) . for the turning points @xmath14 , we get the cubic equation @xmath82 in the no - spin limit , @xmath83 , the solutions are ( the unphysical ) @xmath84 and @xmath85 where @xmath86 is the length of the runge - lenz vector to zeroth order : @xmath87    when a small spin term is present , the roots of ( [ cubic ] ) will have a slightly different form from ( [ rpm ] ) : @xmath88 and @xmath89 , where @xmath90 and @xmath91 are assumed to be small . note that the constants @xmath17 and @xmath10 will not coincide with their zeroth order limits . the constant @xmath86 is still defined by ( [ a0 ] ) , although it is not identical with the length @xmath92 of the runge - lenz vector ( which is not constant ) . writing the cubic equation as @xmath93 , we get    @xmath94r+2e\mu[-r^2(\epsilon_++\epsilon_- + \epsilon)+r(\epsilon_+r_- + \epsilon_-r_++\epsilon(r_++r_- ) ) -\epsilon r_+r_-]=0\ .\ ] ]    the first term is the zeroth - order part of the original equation , written in terms of the roots , thus it equals @xmath95 . inserting the chosen form of the roots we get @xmath96 the orbit is supposed to lie at large distance , thus the first root will not occur .    using the above results , a parameter @xmath15 generalizing the eccentric anomaly of the kepler orbits is readily introduced : @xmath97 such that @xmath98 is at @xmath99 and @xmath100 , respectively . from here : @xmath101 for equatorial orbits , this parametrization has been employed by shibata@xcite . expressing @xmath102 from ( [ radial ] ) and linearizing in @xmath0 , the expression for @xmath103 is found : @xmath104    integration of ( [ txi ] ) from 0 to @xmath105 gives the orbital period : @xmath106    however the parametrization ( [ eq : xi ] ) is inconvenient for carrying out the averaging of the losses , because it yields a complicated array of poles of the integrands . hence in the sequel we will , in place of ( [ xi ] ) , find a parametrization @xmath107 satisfying the following two criteria : @xmath108 where @xmath109 are constants . that is , we want to keep the properties of the true anomaly parametrization of the kepler orbit . property ( b ) generalizes kepler s second law for the area@xcite . the unique parametrization satisfying both ( a ) and ( b ) is :    @xmath110    hence @xmath111r^2\sin\chi\ ] ] and to first order in @xmath0 : @xmath112 \ . \label{tchi}\ ] ]    we will need also the relation of the type @xmath113 . this can be obtained from integration of ( [ psit ] ) : @xmath114    the constant of integration @xmath2 measures the angle subtended by the semiminor axis ( @xmath115 ) and the node line at @xmath116 . in a perturbative picture , the orientation of the momentary ellipse of the orbit is reinterpreted at each revolution , and the value of the integration constant is subject to a corresponding shift . our parametrization ( [ chi ] ) makes especially simple the integration over one period of all expressions @xmath117 containing in the denominator no @xmath70 dependence and no other @xmath13 dependence than @xmath118 , with @xmath119 a positive integer . note that ( [ omegal ] ) , ( [ psit ] ) and ( [ phit ] ) have this property . further , as will be shown in the next section , the instantaneous radiative losses of the constants of motion share this property . in all of these cases one has : @xmath120 where the @xmath13 dependence of the denominator of the integrand is especially simple : @xmath121 . the _ average _ over one period is @xmath122 with @xmath123 given by eq . ( [ period ] ) . the integrals may conveniently be evaluated by use of the residue theorem . one introduces the complex variable @xmath124 , and the integral is taken over the unit circle in the @xmath125 plane . the value of the integral is given by the sum of the residues of the poles inside the circle . in all relevant cases our parametrization assures that the only pole is at @xmath126 . simple examples are the integrations of ( [ phit ] ) , ( [ psit ] ) and ( [ omegal ] ) : @xmath127 here ( [ dephi ] ) is the precession angle of the orbital angular momentum @xmath128 about @xmath7 . the interpretation of the variable @xmath70 as the polar angle in the plane perpendicular to @xmath41 is in accordance with the identical first order terms in ( [ eq : depsi ] ) and ( [ eq : shift ] ) . the ellipse is found to counterrotate ( for @xmath129 . the periastron shift is a combination of ( [ eq : dephi ] ) and ( [ eq : shift ] ) . both @xmath130 and @xmath131 will be unchanged in the averaged - motion approximation of landau and lifsitz @xcite . our angles are related to the polar angle @xmath132 for equatorial orbits ( @xmath133 ) by @xmath134 . this relation helps comparison with shibata s @xcite expression ( 2.19 ) for the periastron shift @xmath135 . ryan@xcite has introduced a parametrization @xmath136 which , when rewritten in terms of the constants of motion @xmath17 and @xmath10 ( employing his definitions for the orbital parameters ) , will read :    @xmath137    this differs from our parametrization ( [ chi ] ) by the last @xmath138 term , which shows that ryan s parameter @xmath139 and our @xmath16 reach the turning points simultaneously . in fact both ( [ xi ] ) and ( [ chi ] ) can be supplemented by arbitrary terms involving a @xmath140 squared factor , and they still will satisfy ( a ) . however none of these parameters , and among those ryan s , will satisfy the generalized second kepler law ( b ) . the mass and current quadrupole tensors of a test particle with coordinates @xmath141 are @xmath142 @xmath143^{stf } +         { \frac { 3 \mu}{2 m } } ( r_i s_j)^{stf } \ , \ ] ] where stf means symmetrization and trace removal in the free indices . the rates of the energy and the angular momentum losses in the test particle 1pn approximation are , respectively@xcite ,    @xmath144    @xmath145    evaluating the right - hand sides by use of ( [ radial ] ) , ( [ psit ] ) and ( [ phit ] ) and dropping higher - order terms in @xmath0 , the total power is @xmath146    this is independent of the angles @xmath70 and @xmath147 . the loss in @xmath9 is evaluated by @xmath148 and it has the form @xmath149 the momentary loss in @xmath10 is @xmath150\sin^2\iota+3\,r\dot r \ , l\mu\,\sin2\psi\sin^2\iota \left ( 2\,gm\mu^{2}r-3\,l^{2}+6\,e\mu\,r^{2}\right ) \nonumber\\ & & -6\,g^{2}m^{2}\mu^{4}r^{2}+22\,gl^{2}m\mu^{2}r-24\,\mu^{3}er ^{3}gm-36\,\mu\,er^{2}l^{2 } \ } \ . % -{\frac { 8\,g^{3}l^{3}m^{2}\cos\iota}{45\,\mu\,r^{6}c^{7 } } } \ .\end{aligned}\ ] ]    after parameterizing by @xmath16 and averaging over the period @xmath123 as described in the previous section , we obtain @xmath151    among these losses averaged over one period of the radial motion , the only quantity depending on the initial angle @xmath2 is @xmath152 . this is to be interpreted such that the rate of loss of the angular momentum component @xmath10 varies with the position of the periastron . in ( [ eq : avelosses ] ) , the terms with @xmath153 average to zero when the precession time scale is short compared to the radiation reaction time scale . ( for a more extensive discussion of this subject , _ cf . _ @xcite . ) the major axis @xmath18 and the eccentricity @xmath19 of the kepler motion are constants and together with @xmath58 they determine the full set of the constants of motion @xmath17 and @xmath10 . in the lense - thirring picture of a perturbed kepler motion @xmath17 and @xmath10 contain first - order terms , and the corresponding constants @xmath18 and @xmath19 should also contain first - order terms . there is an ambiguity in how to introduce these orbit parameters , as noted by ryan@xcite . he chooses these parameters by referring to the quasicircular orbits in the kerr metric . this method is passing when wishing to approximate carter orbits . however , there appears to be no reason for invoking phenomena tied to black holes when describing arbitrary weakly bound spinning bodies . in fact , we want to keep our treatment general enough to cover all axially symmetric gravitational fields with first order contributions from the spin - orbit interaction term in the lagrangian ( [ delag ] ) . thus we choose as the definition of the semimajor axis @xmath18 and eccentricity @xmath19 : @xmath154 which implies :    @xmath155    these relations can be inverted : @xmath156\ .\label{ee}\\end{aligned}\ ] ]    the averaged losses in terms of @xmath18 and @xmath19 then read : @xmath157    differentiating ( [ iota ] ) one has : @xmath158\ .\ ] ] those terms in the brackets which are zero - order in @xmath0 , cancel . the averaged losses of the ellipse parameters can be obtained by differentiating ( [ aa ] ) and ( [ ee ] ) and from ( [ eq : aviota ] ) : @xmath159            \ .\end{aligned}\ ] ]    let us take an example of a binary system contaning a massive black hole with mass @xmath160 and spin @xmath161 , and where the companion s orbit has the semimajor axis @xmath162 km , _ i.e. _ , one - hundredth of the hulse - taylor system . then the increase in the angle @xmath58 per loss of @xmath163 ( a dimensionless quantity ) is @xmath164 for a circular orbit . the enhancement factor for an eccentric orbit is @xmath165    the enhancement factor is monotonously increasing ( fig.1 ) from @xmath166 to make the change in @xmath58 comparable to that in @xmath167 only when the eccentricity approaches the unit value to one part in @xmath168 . the kepler orbits are completely characterized by the energy @xmath8 , the angular momentum vector @xmath41 and the runge - lenz vector @xmath169 at the initial time @xmath170 . these are six independent constants of the motion . in the lense - thirring picture , the constants of the motion alone do not suffice for the full description of the orbit . we complete the characterization by specifying the euler angles @xmath2 and @xmath11 of the periastron . their evolution over one period , under the influence of the spin ( disregarding radiation backreaction ) is given in ( [ dephi ] ) and ( [ depsi ] ) . taking into account the gravitational radiation backreaction , the shifts ( [ depsi ] ) and ( [ dephi ] ) will vary from one period to another , as the constants @xmath9 and @xmath10 evolve . one may interpret these as the radiation - induced changes in the parameters @xmath2 and @xmath11 : @xmath171 @xmath172 where @xmath173 and we have used that in zeroth order @xmath174 . the total shift in the euler angle @xmath2 of the periastron point is given by the sum of ( [ eq : depsi ] ) and ( [ eq : dpsi ] ) , and likewise for @xmath11 . we continue with some comments on the radiative losses in the constants of motion . the terms independent of @xmath0 agree with the results of peters and mathews@xcite . the terms linear in the spin @xmath0 in our averaged losses agree with ryan s corresponding results , when the parameters @xmath18 , @xmath19 and @xmath58 are converted to his parameters . in the present paper , the energy and angular momentum losses have been directly computed . this is to be contrasted with ryan s method where the backreaction forces have been used . the perfect agreement achieved underscores the reliability of these computations . when @xmath133 , the averaged losses in @xmath9 are equal to the averaged losses in @xmath10 up to first order in @xmath0 . hence the equatorial plane of the orbits is stable with respect to radiation losses . as we have shown in the previous section , the inclination of the orbit can not change significantly in the adiabatic era . hence we conclude that it is insufficient to restrict the computation of signal templates from a black hole  test particle system to equatorial carter orbits .    in the limiting case of circular orbits , @xmath175 , there is no distinguished point on the ellipse . the dependence of @xmath176 on @xmath2 vanishes . as expected from the axial symmetry , the radiative losses , for arbitrary excentricity , do not depend on the initial angle @xmath11 .    in the present paper , we have developed a toolchest for the computation of radiation losses , which include a new parametrization of the orbit , new angular variables and the application of the residue theorem for obtaining time averages . currently we are employing these procedures with the inclusion of finite mass effects.@xcite this work has been supported by otka no . t17176 and d23744 grants . the algebraic packages reduce and maplev have been used for checking our computations . introduce three angle variables @xmath177 and @xmath178 , where @xmath179 is the angle of rotation about @xmath80 , bringing the position of the particle to the node ; @xmath81 is the angle subtended by @xmath80 and @xmath7 and @xmath178 is the angle of precession of @xmath80 about @xmath7 . the angle @xmath179 is the _ argument of the latitude _ , @xmath81 the _ inclination _ and @xmath178 the _ longitude of the node _ unlike @xmath58 , the angle @xmath81 is not constant . let us express @xmath81 in terms of @xmath58 and @xmath70 . computing the @xmath6 component of @xmath80 in two different ways we get the two sides of the equation @xmath180 with the time derivatives of polar angles ( [ polar ] ) after linearization in @xmath0 , the previous equation gives      here we are using that the primed angles are equal , to zeroth order in @xmath0 , with the corresponding unprimed ones . taking the time derivative of @xmath81 we get @xmath182 with the new time - dependent euler rotations the coordinates of the particle become @xmath183 and the time derivatives of the polar angles are @xmath184 @xmath185 inserting these equations in ( [ polar ] ) , we get the changes of the primed euler angles in terms of the unprimed ones @xmath186 @xmath187 integrating these equations , we get the new angle variables and their shifts over one period @xmath188 note that these changes in the primed variables are equal to the shifts ( [ dephi ] ) and ( [ depsi ] ) of the unprimed variables .    99 b.carter , phys . rev .  * 174 * , 1559 ( 1968 ) k.s.thorne , private communication a.ori , phys * d55 * , 3444 ( 1997 ) l.kidder , c.will and a.wiseman , phys . rev .  * d47 * , 4183 ( 1993 ) b.m.barker and r.f.oconnell , gen.rel.gravitation * 11 * , 149 ( 1979 ) the computations of @xcite have been extended to the 5/2pn approximation by l.kidder , phys . * d52 * , 821 ( 1995 ) m.shibata , phys . rev .  * d50 * , 6297 ( 1994 ) f.ryan , phys * d52 * , r3159 ( 1995 ) f.ryan , phys . rev .  * d53 * , 3064 ( 1996 ) h. thirring and s. lense , phys . zeitschr . * 19 * , 156 ( 1918 ) , english translation : gen.rel.gravitation ,  * 16 * 727 ( 1984 ) l.d.landau and e.m.lifsitz , the classical theory of fields , pergamon , oxford , 1975 l. . gergely and z.perjs , in _ proceedings of the virgo conference , eds . i.ciufolini and f.fidecaro , world scientific , singapore , 1997 _ l.blanchet and t. damour , ann.inst.henri poincar ,  * a50 * , 377 ( 1989 ) t. damour and b. iyer , ann.inst.henri poincar ,  * 54 * , 115 ( 1991 ) l..gergely , z.perjs and m.vasth , in preparation . the parametrization of the kepler orbit by the angle @xmath16 is close in spirit to ptolemy s idea of the _ equant_. both the equant and the orbit parameter are means for setting up a rule for determining the angular velocity of the orbiting body . peters and s.mathews , phys . 131 * , 435 ( 1963 ) p.c . peters , phys . rev .  * 136 * , b1224 ( 1964 )
the gravitational radiation backreaction effects are considered in the lense - thirring approximation . 
 new methods for parameterizing the orbit and for averaging the instantaneous radiative losses are developed . to first order in the spin @xmath0 of the black hole , both in the absence and in the presence of gravitational radiation , a complete description of the test - particle orbit 
 is given . 
 this is achieved by two improvements over the existing descriptions . 
 first , by introducing new angle variables with a straightforward geometrical meaning . 
 second , by finding a new parametrization of a generic orbit , which assures that the integration over a radial period can be done in an especially simple way , by applying the residue theorem . 
 the instantaneous gravitational radiation losses of the system are computed using the formulation of _ blanchet , damour and iyer(1989)_. all losses are given both in terms of the dynamical constants of motion and the properly defined orbital elements @xmath1 and @xmath2 . 
 the radiative losses of the constants characterizing the lense - thirring motion , when suitably converted , are in agreement with earlier results of _ kidder , will and wiseman(1993 ) , ryan(1996 ) and shibata(1994)_. in addition , the radiative losses of two slowly changing orbital elements @xmath3 are given in order to complete the characterization of the orbit . 
 # 1@xmath4#1