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in general , observed astrophysical objects are characterized by a non - spherically symmetric distribution of mass and by rotation . in many cases , like ordinary planets and satellites , it is possible to neglect the deviations from spherical symmetry and the frame dragging effect , so that the gravitational field can be described by the exterior schwarzschild solution . in fact , the three classical tests of general relativity make use of the schwarzschild spacetime in order to describe gravity within the solar system@xcite . in the case of strong gravitational fields , however , the deviation from spherical symmetry and the rotation become important and must be taken into account , at least to some extent . the first metric describing the exterior field of a slowly rotating slightly deformed object was found by hartle and thorne@xcite in 1968 . alternative methods were proposed independently by fock and abdildin@xcite and sedrakyan and chubaryan@xcite . only recently , it was shown that in fact all these approaches are equivalent from a mathematical point of view@xcite . at the level of the interpretation of the parameters entering the metric used in each approach , certain differences can appear which could make a particular approach more suitable for the investigation of certain problems . for the purpose of the present work , it is convenient to use the hartle - thorne formalism which leads to an approximate metric describing , up to the first order in the quadrupole and the second order in the angular momentum , the exterior gravitational field of a rotating deformed object . we will use in this work the hartle - thorne metric in the form presented by bini et al.@xcite which in geometrical units is given by @xmath0dt^2 \nonumber\\ & & + \left(1-\frac{2{m}}{r}\right)^{-1}\left[1 - 2k_2p_2(\cos\theta)-2\left(1-\frac{2{m}}{r}\right)^{-1}\frac{j^{2}}{r^4}\right]dr^2 \nonumber\\ & & + r^2[1 - 2k_3p_2(\cos\theta)](d\theta^2+\sin^2\theta d\phi^2)-4\frac{j}{r}\sin^2\theta dt d\phi\ , \end{aligned}\ ] ] where @xmath1 here @xmath2 is legendre polynomial of the first kind , @xmath3 are the associated legendre functions of the second kind determined as @xmath4 , \nonumber\\ q_{2}^{2}(x)&=&(x^{2}-1)\left[\frac{3}{2}\ln\frac{x+1}{x-1}-\frac{3x^{3}-5x}{(x^{2}-1)^2}\right],\end{aligned}\ ] ] and the constants @xmath5 , @xmath6 and @xmath7 are the total mass , angular momentum and mass quadrupole moment of the rotating object , respectively ( for more details see refs .   and ) . is related to the mass quadrupole moment defined by hartle and thorne@xcite through @xmath8 . ]    the approximate kerr metric@xcite in boyer - lindquist coordinates can be obtained from the above hartle - thorne metric after setting @xmath9 and making a coordinate transformation implicitly given by @xmath10\ , \nonumber\\ \theta&=&\theta+\frac{a^2}{2r^2}\left(1+\frac{2m}{r}\right)\sin\theta\cos\theta.\end{aligned}\ ] ]    the kerr metric is important to investigate the physical processes taking place around rotating black holes , i.e. , the source with probably the strongest possible gravitational field . the role of rotation is essential in the physics of accretion disks and energy extraction from a black hole . moreover , depending on the direction of the rotation , the radius of the accretion disk can be larger or smaller with respect to the schwarzschild case . the situation changes when one involves compact objects such as white dwarfs , neutron stars and quark stars as they have additional parameters to be taken into account . the combination of the strong field with the quadrupolar deformation of the source plays a pivotal role when one considers the motion of test particles . there exist many exact solutions that include a quadrupole parameter . the importance of the quadrupole moment in the astrophysical context has been emphasized in several works@xcite . however , most analysis must be performed numerically due to the complexity of the exact metrics . the advantage of considering the hartle - thorne approximate solution is that several physical quantities can be calculated analytically which facilitates their study . we will prove below that this is possible for a particular set of geodesics . in this work , we are interested in studying the motion of test particles in the hartle - thorne spacetime . therefore , we will perform both analytical and numerical analysis of the timelike and lightlike geodesic equations . in particular , we are interested in comparing the effects of the quadrupole and angular momentum parameters within the approximation allowed by the hartle - thorne metric . in this work , we will make use the timelike normalization condition @xmath11 which for equatorial circular geodesics is equivalent to @xmath12 sometimes the following convenient notations are used for the four - velocity of equatorial circular geodesics @xmath13 where @xmath14 is the normalization factor and @xmath15 is the orbital angular velocity . using the fact that the hartle - thorne solution possesses two killing vector fields @xmath16 and @xmath17 , which determine two constants of motion , from the geodesic equations for equatorial circular orbits ( @xmath18 and @xmath19 , we obtain @xmath20 where a comma indicates partial differentiation . then , a straightforward computation yields @xmath21 ^ 2-\left[\frac{d\phi(s)}{ds}\right]^2\right\}-\frac{2m^2(r-2m)}{r^3}\frac{dt(s)}{ds}\frac{d\phi(s)}{ds}j \nonumber\\ & & \qquad\qquad\qquad+\left\{a_{1}(r)\left[\frac{dt(s)}{ds}\right]^2+a_{2}(r)\left[\frac{d\phi(s)}{ds}\right]^2\right\}j^2 \nonumber\\ & & \qquad\qquad\qquad\qquad\qquad\qquad + \left\{a_{3}(r)\left[\frac{dt(s)}{ds}\right]^2+a_{4}(r)\left[\frac{d\phi(s)}{ds}\right]^2\right\}q = 0 , \end{aligned}\ ] ] where we introduced the dimensionless quantities @xmath22 and @xmath23 and new functions defined as @xmath24 this expansion in terms of the quadrupole and angular momentum parameters can be used to derive analytical expressions for the parameters that characterize the orbits of the test particles .      starting from the @xmath26 component of the geodesic equation and using the fact that @xmath26 and @xmath27 are both constant on circular equatorial orbits , one easily derives the following expression for the angular velocity : @xmath28 performing an analysis similar to the one carried out in the previous section , we finally obtain @xmath29,\ ] ] where @xmath30^{-1}[48m^7 - 80m^6r+4m^5r^2 + 42m^4r^3\nonumber\\ & & -40m^3r^4 - 10m^2r^5 - 15mr^6 + 15r^7]-f(r ) , \nonumber\\ f_{3}(r)&=&-\frac{5(6m^4 - 8m^3r-2m^2r^2 - 3mr^3 + 3r^4)}{16m^2r(r-2m)}+f(r ) , \nonumber\\ f(r)&=&\frac{15(r^3 - 2m^3)}{32m^3}\ln\frac{r}{r-2m}.\end{aligned}\ ] ]    this expression for the orbital angular velocity of a test particle at the equatorial plane can be used to determine the mass shedding limit of the source in general relativity@xcite . moreover , in x - ray astronomy , the orbital angular velocity is associated with the upper frequency of the quasiperiodic oscillations@xcite . the analytic expression obtained here leads to results that are in agreement with those obtained by using pure numerical methods .      in the case of lightlike geodesics , we can use the expression for the norm of the corresponding 4-velocity to calculate its components . evaluating @xmath32 and @xmath33 directly from the hartle - thorne line element , we then obtain for the orbital angular velocity : @xmath34,\ ] ] where @xmath35      the specific energy per unit mass @xmath37 is usually used to estimate the radius of marginally ( innermost ) bound orbits of test particles . thus , this radius determines the stable region which is essential for the formation of accretion disks . evaluating @xmath38 from the line element , one obtains @xmath39 @xmath40,\ ] ] where @xmath41^{-1}[144m^8 - 144m^7r-28m^6r^2 + 122m^5r^3\nonumber\\ & & + 184m^4r^4 - 685m^3r^5 + 610m^2r^6 - 225mr^7 + 30r^8]+h(r ) , \nonumber \\ h_{3}(r)&=&\frac{5(r - m)(6m^3 + 20m^2r-21mr^2 + 6r^3)}{16mr(r-2m)(r-3m)}-h(r ) , \nonumber\\ h(r)&=&\frac{15r(8m^2 - 7mr+2r^2)}{32m^2(r-3m)}\ln\frac{r}{r-2m}.\end{aligned}\ ] ]      as we will see below , the specific angular momentum for test particles per unit energy @xmath43 is crucial for the determination of the marginally ( innermost ) stable orbits of test particles forming accretion disks . calculating @xmath44 and @xmath38 , we obtain the following analytic expression for @xmath43 @xmath45 @xmath46,\ ] ] where @xmath47^{-1}[96m^8 - 112m^7r-8m^6r^2 + 72m^5r^3\nonumber \\ & & -18m^4r^4 - 220m^3r^5 + 260m^2r^6 - 105mr^7 + 15r^8]-g(r ) , \nonumber \\ g_{3}(r)&=&\frac{5(6m^4 - 22m^2r^2 + 15mr^3 - 3r^4)}{16m^2 r(r-2m)}+g(r ) , \nonumber \\ g(r)&=&\frac{15(2m^3 + 4m^2r-4mr^2+r^3)}{32m^3}\ln\frac{r}{r-2m}.\end{aligned}\ ] ]      the normalization condition @xmath48 gives the photon orbit , @xmath49 , where @xmath50 is the photon four - momentum and for circular orbits @xmath51 . indeed the normalization condition @xmath52 gives the orbital angular velocity for the photon @xmath53 , but @xmath54 remains arbitrary . to determine the photon orbit , @xmath49 , first one should use the above expression for @xmath53 and then evaluate the four - acceleration @xmath55 . for a circular geodesic @xmath56 , and only from this condition one can determine @xmath49 . note , alternatively it is also convenient to use the condition @xmath57 to find @xmath49 . moreover , in order to determine the radius of the innermost ( marginally ) bound circular orbits @xmath58 one should use the condition @xmath59 . in addition , the condition @xmath60 allows one to find the radius of the innermost stable circular orbits , @xmath61 . here we used the methods of perturbation theory and the results of these calculations are : @xmath62,\\ r_{mb}&=&4m\left[1\mp\frac{1}{2}j+\left(\frac{8033}{256}-45\ln2\right)j^2+\left(-\frac{1005}{32}+45\ln2\right)q\right],\\ r_{ms}&=&6m\left[1\pm\frac{2}{3}\sqrt{\frac{2}{3}}j+\left(-\frac{251903}{2592}+240\ln\frac{3}{2}\right)j^2+\left(\frac{9325}{96}-240\ln\frac{3}{2}\right)q\right ] . \qquad\end{aligned}\ ] ]      finally , we mention that using the hartle - thorne line element it is possible to derive the radial and vertical fundamental frequencies.@xcite the application of these frequencies to the observed quasiperiodic oscillations from the low - mass x - ray binaries has been considered on the basis of the relativistic precession model.@xcite all the epicyclic frequencies have been derived in previous works.@xcite with the method proposed in this work we obtained equivalent results after applying the redefinition @xmath63 or @xmath64 . it is convenient to investigate the motion of test particles numerically in the hartle - thorne spacetime as the full set of equations is cumbersome even for the equatorial plane . we select different values for the parameters of the source and initial conditions for test particles to consider all types of trajectories . the results of the numerical integration of timelike geodesics are shown in figs . [ plotq]-[ploth2 ] .    [ cols= " < , > " , ]     in fig . [ plotq ] ( left panel ) , we analyze the motion of a test particle in the field of a static and deformed source . as one can see , the quadrupole parameter generates different deviations from the schwarzschild spacetime , depending on its value and sign . the frame dragging effect is illustrated in fig . [ plotq ] ( right panel ) for a spherically symmetric source . the drag strengthens as the test particles approach the source . the influence of the frame dragging effect on the circular motion has considered in ref .   for one revolution of the test particle . by analyzing the behavior of the test particles with the certain initial conditions , it is possible to select the values of @xmath65 and @xmath66 in order to recover circular geodesics , i.e. , the effects caused by the deformation of the source can be balanced by its rotation and vice - versa ( see ref .  ) . unbound orbits are shown in fig [ ploth1 ] ( left panel ) , where we change only the initial angular component of the velocity @xmath67 . the remaining quantities are fixed for the sake of comparison . for different values of @xmath66 , we obtain the plot of fig . [ ploth1 ] ( right panel ) , whereas in fig . [ ploth2 ] ( left panel ) we present the results for different values of @xmath65 .    due to this interplay between the initial conditions of the test particles and the parameters of the source , one can construct all kind of geodesics . an example of a double loop trajectory , which is missing in classical physics , is shown in fig . [ ploth2 ] ( right panel ) . from here we conclude that the parameters of the geodesic motion can be used to determine the main parameters of the source such as @xmath68 , @xmath65 and @xmath66 . in this work , we have explored geodesics in the hartle - thorne spacetime both analytically and numerically . we considered the geodesics on the equatorial plane and investigated the role of the quadrupole parameter , as well as the frame dragging effect on the motion of test particles . we investigated bounded and unbounded orbits varying the initial conditions of the test particles and the main parameters of the source . we conclude that using different combinations of both initial conditions and main parameters , one can generate many different geodesic curves .    in all our computations we used the methods of perturbation theory . our results have the same order of approximation as the harte - thorne solution . namely , we derived the expressions for the orbital angular velocity , energy , orbital angular momentum for test particles and orbital angular velocity for photons . in turn , with the help of these expressions we obtained the radii of the innermost bound , innermost stable and photon orbits . all the analytic expressions obtained here and in refs .   and are in agreement with the results of refs .   and , if one redefines @xmath64 . we briefly discussed some applications of our theoretical results in the astrophysical context . in fact , the results presented in this paper can be applied to study the physics of accretion disks , the motion of test particles near the source and the epicyclic frequencies ; all these aspects are of high relevance and importance in relativistic astrophysics and x - ray astronomy . for instance , using epicyclic frequencies and quasiperiodic oscillation data , one can test the strong field regime of general relativity , determine the parameters of the gravitational source and test the equations of state of compact objects .    in a future work , we expect to apply the analytic expressions we obtained in the present work in the context of observational astrophysics . furthermore , the investigation of the stability of the geodesics and the structure of the accretion disks are crucial to understand the physical properties of the hartle - thorne spacetime . we expect to perform such an analysis in a future work by applying the procedure shown , for instance , in ref .  . this work was supported by the ministry of education and science of the republic of kazakhstan grants no . 3101/gf4 ipc-11/2015 and no . 1597/gf3 ipc-30 .
we investigate equatorial geodesics in the gravitational field of a rotating and deformed source described by the approximate hartle - thorne metric . in the case of massive particles , 
 we derive within the same approximation analytic expressions for the orbital angular velocity , the specific angular momentum and energy , and the radii of marginally stable and marginally bound circular orbits . 
 moreover , we calculate the orbital angular velocity and the radius of lightlike circular geodesics . 
 we study numerically the frame dragging effect and the influence of the quadrupolar deformation of the source on the motion of test particles . 
 we show that the effects originating from the rotation can be balanced by the effects due to the oblateness of the source .