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the exosphere is the upper layer of any planetary atmosphere : it is a quasi - collisionless medium where the particle trajectories are more dominated by gravity than by collisions . above the exobase , the lower limit of the exosphere , the knudsen number @xcite becomes large , collisions become scarce , the distribution function can not be considered as maxwellian anymore and , gradually , the trajectories of particles are essentially determined by the gravitation and radiation pressure by the sun . the trajectories of particles , subject to the gravitational force , are completely solved with the equations of motion , but it is not the case with the radiation pressure @xcite .    in the absence of radiation pressure , we can distinguish three types of trajectories for the exospheric particles :    * the escaping particles come from the exobase and have a positive mechanical energy : they can escape from the gravitational influence of the planet with a velocity larger than the escape velocity . these particles are responsible for the jeans escape @xcite . they can also be defined as crossing only once the exobase , * the ballistic particles also come from the exobase but have a negative mechanical energy , they are gravitationally bound to the planet . they reach a maximum altitude and fall down on the exobase if they do not undergo collisions . they cross the exobase twice , * the satellite particles never cross the exobase . they also have a negative mechanical energy but their periapsis is above the exobase : they orbit along an entire ellipse around the planet without crossing the exobase . the satellite particles result in their major part from ballistic particles undergoing few collisions mainly near the exobase ( @xcite ) . thus , they do not exist in a collisionless model of the exosphere . the radiation pressure disturbs the conics ( ellipses or hyperbolas ) described by the particles under the influence of gravity . the resonant scattering of solar photons leads to a total momentum transfer from the photon to the atom or molecule @xcite . in the non - relativistic case , assuming an isotropic reemission of the solar photon , this one is absorbed in the sun direction and scattered with the same probability in all directions . for a sufficient flux of photons in the absorption wavelength range , the reemission in average does not induce any momentum transfer from the atom / molecule to the photon . the momentum variation , each second , between before and after the scattering imparts a force , the radiation pressure . @xcite proposed to analyze its effect on the structure of planetary exospheres . in particular , they highlighted analytically the  tail " phenomenon at earth : the density for atomic hydrogen , which is sensitive to the lyman-@xmath0 photons , is higher in the nightside direction than in the dayside direction in the earth corona . nevertheless , their work was limited only to the sun - planet axis , with a null component assumed for the angular momentum around the sun - planet axis . we thus generalize here their work to a full 3d calculation , in order to investigate the influence of the radiation pressure on the trajectories ( this paper ) , as well as the density profiles and escape flux ( following works ) . this problem is similar to the so - called stark effect @xcite : the effect of a constant electric field on the atomic hydrogen s electron . its study can be transposed to celestial mechanics in order to describe the orbits of artificial and natural satellites in the perturbed ( e.g. by the radiation pressure force ) two - body problem . a recent description of the stark effect solutions was already given by @xcite . however , they give the analytical solutions of all trajectories only in the 2d case ( and for bounded trajectories in the 3d case ) , and their formulas have some issues as will be discussed later . another analytical study proposed by @xcite uses the weierstrassian formulations to solve the motions for bounded and unbounded trajectories and to find periodic motions . also , the motion can be approached numerically by developing the equations of motion in taylor series but this leads to some issues for high eccentricities @xcite . @xcite compared recently these methods and their computing efficiencies .    in this paper , based on the same formalism as @xcite , we provide for the first time the complete exact 3d solutions of the stark effect ( and its celestial mechanics analogue ) for any initial condition and for both bounded and unbounded trajectories . the first section describes the formalism used , before the sections [ model]/[section3]/[time ] provide the equations of motion and time . we then discuss about circular orbits in section [ circular ] , while a comparison with previous works is given in section 6 , before we conclude in section [ summary ] . in this work , we decide to study the effect of the radiation pressure on atomic hydrogen in particular . nevertheless , this formalism can be applied to any species subject to this force or to the interplanetary dust . we model the radiation pressure by a constant acceleration @xmath1 coming from the sun . according to @xcite , this acceleration depends on the line center solar lyman-@xmath0 flux @xmath2 , in @xmath3 photons.@xmath4.s@xmath5.@xmath5 : @xmath6    in spherical coordinates , the hamiltonian of one hydrogen atom can be written : @xmath7 with @xmath8 the distance from the planet , @xmath9 the solar angle , @xmath10 the angle with respect to the ecliptic plane , @xmath11 , @xmath12 and @xmath13 the conjugate momenta . @xmath14 represents the gravitational potential and @xmath15 the potential energy from the radiation pressure acceleration @xmath1 . an example of trajectory of a @xmath16 atom subject to the radiation pressure is given in the figure [ trajectoire ] . this problem is similar to the classical stark effect @xcite : a constant electric field ( here the radiation pressure ) is applied to an electron ( here an hydrogen atom ) attached to a proton ( here the planet ) . both systems are equivalent because the force applied by the proton ( the planet ) to the electron ( the hydrogen atom ) , i.e. the electrostatic force , varies in @xmath17 as the gravitational force from the planet on the hydrogen atom . thus , we adopt the same formalism as @xcite and use the parabolic coordinates . we use the transformation : @xmath18 with @xmath19 and @xmath20 always positive . consequently , the hamiltonian becomes : @xmath21 independent from @xmath22 and @xmath10 .    according to hamiltonian canonical relations , we have : @xmath23      in this new system of coordinates , we study this hamiltonian . first , @xmath24 is independent from @xmath22 explicitly . @xmath24 is a conserved quantity along the time and corresponds to the mechanical energy @xmath25 of the system . moreover , as @xmath24 is independent from @xmath10 , according to canonical relations : @xmath26    thus , @xmath27 is another constant of the motion . once @xmath25 and @xmath13 defined , the equation [ hamiltonuw ] can be rewritten : @xmath28    the left hand side is a function dependent only on @xmath19 and @xmath29 , the right hand side depends only on @xmath20 and @xmath30 . as both functions are equal and independent , they are equal to a constant @xmath31 , a separation constant : @xmath32    the motion possesses three constants : @xmath25 , @xmath31 and @xmath13 . the equation [ a ] allows to express @xmath29 ( respectively @xmath30 ) as a functions of @xmath25 , @xmath31 , @xmath13 and @xmath19 ( respectively @xmath20 ) :    @xmath33    with @xmath34      we have already introduced the hamiltonian @xmath24 of the system . we can extend the approach according to hamilton - jacobi equations : @xmath35 where @xmath36 is the hamilton s principal function or action . this function depends on initial conditions ( as @xmath37 , @xmath38 , @xmath39 and @xmath40 ) and the actual position of the particle ( as @xmath19 , @xmath20 , @xmath10 and @xmath22 ) . as previously demonstrated , @xmath41 and @xmath27 are constants . thus , @xmath42 and leads to : @xmath43\ ] ] with @xmath44 the part of the action independent from @xmath40 , @xmath22 , @xmath39 and @xmath10 . moreover , the action @xmath44 can be separated into two parts : one written as a function of @xmath19 and @xmath29 coordinates , the other one with @xmath20 and @xmath30 . by definition , according to the hamilton - jacobi equations , we have : @xmath45 with @xmath29 ( resp . @xmath30 ) a function only of @xmath19 ( resp . @xmath20 ) , assuming @xmath25 , @xmath31 and @xmath13 values already fixed by initial conditions . then , we can separate again the action , leading to : @xmath46=\mathcal{s}_{u}[u , e , a , p_{\phi}]+\mathcal{s}_{w}[w , e , a , p_{\phi}]\ ] ] @xmath47 according to the equation [ pupw ] , we have the following relations : @xmath48 with @xmath49 @xmath50 and @xmath51 are effective potentials applied in @xmath19 and @xmath20 directions ( represented in the figure [ potentiel ] ) . these potentials play key roles for the motion because they constrained the motion in @xmath19 and @xmath20 directions independently . for the motion of the particle , we must respect two conditions : @xmath52 and @xmath53 . these conditions are analogous to : @xmath54     ( corresponding to @xmath55 , blue line ) and @xmath56 ( corresponding to @xmath57 , red line ) for a set of @xmath25 , @xmath31 and @xmath27 values . the motion is possible only in the area where the potential is below the mechanical energy @xmath25 , represented by the black horizontal line . the different roots of @xmath58 and @xmath59 are displayed and correspond to the intersection of the potentials with the horizontal black line . @xmath60 is the dimensionless value referring to @xmath61 ( cf . section [ dimensionless_section ] , table [ dimensionless ] ) . notice that @xmath57 will cross only once the energy level @xmath62 for too high or too low @xmath62 values . ] these both conditions are more restrictive than the usual @xmath63 where @xmath64 is the potential energy . @xmath58 is a polynom of degree 3 with @xmath65 . this polynom possesses three roots , whose one is real at least . as @xmath66 , one of these roots is real negative , according to intermediate value theorem since @xmath67 . nevertheless , the motion occurs for positive @xmath19 values , and we know this motion exists . it implies there is an interval in @xmath68 such as @xmath69 ( otherwise , there is no motion in @xmath19-direction , no possible physically , cf . eq . [ conditions ] ) . to comply with this last condition , both other roots are real and positive . in summary , @xmath58 has three real roots : one negative and two positive . we call each root @xmath37 , @xmath61 and @xmath70 such as @xmath71 and the @xmath19-motion is restricted to @xmath72 $ ] ( @xmath73 $ ] in term of dimensionless quantities , as can be seen in the figure [ potentiel ] ) . @xmath59 is a polynom of degree 3 with @xmath74 . this polynom possesses three roots , whose one is real at least . as @xmath75 , one of these roots is real positive , according to intermediate value theorem since @xmath74 . nevertheless , the motion occurs for positive @xmath20 values . we have restrictions on both other roots : they must be both real positive , both real negative or both complex conjugates .    in the case where the three roots are real positive , we call each root @xmath38 , @xmath76 and @xmath77 such as @xmath78 and the motion is restricted to @xmath79\cup[w_{0};+\infty[$ ] ( as can be seen in the figure [ potentiel ] with the dimensionless quantities , cf . section [ dimensionless_q ] ) .    in the case with one positive root and both other complex or real negative , we call each root @xmath38 ( the positive one ) , @xmath76 and @xmath77 ( keep the same order as previously defined if they are real ) such as the motion is restricted to @xmath80 only ( @xmath81 in term of dimensionless quantities ) .      each constant value of @xmath19 or @xmath20 defines a paraboloid in three dimensions . for each interval , constrained by fixed values of @xmath19 and @xmath20 , the motion will be contained between the paraboloids defined by these limit values as shown in the figure [ restriction1 ] . right panel : the red area corresponds to the allowed region for the w - motion . the regions are limited by paraboloids ( parabolas here by projection ) . the sun location is shown with a yellow circle.,title="fig:"]-motion . right panel : the red area corresponds to the allowed region for the w - motion . the regions are limited by paraboloids ( parabolas here by projection ) . the sun location is shown with a yellow circle.,title="fig : " ]    for the @xmath19-motion , this is always limited by two paraboloids defined by @xmath82 and @xmath83 as shown by the blue area in the figure [ restriction1 ] ( left panel ) . similarly , for the @xmath20-motion , there are two cases : the motion is constrained between one paraboloid ( @xmath84 ) and the infinity or between two paraboloids ( @xmath85 and @xmath86 ) . both cases are represented by the red area in the figure [ restriction1 ] ( right panel ) . as previously demonstrated , the motion is constrained in specific areas of the 3d space . the motion is always between two paraboloids due to restrictions on @xmath19 but it can be constrained between two other paraboloids or only one regarding @xmath20 . thus , the motion is constrained by four paraboloids or three , opened to infinity as shown on the figure [ restriction2 ] by the green area .    . the final allowed region is in green . right panel : representation in green of the trajectory in the figure [ trajectoire ] in the @xmath87 frame at earth.,title="fig : " ] . the final allowed region is in green . right panel : representation in green of the trajectory in the figure [ trajectoire ] in the @xmath87 frame at earth.,title="fig : " ]    nevertheless , there is at this point no other information on the exact motion of the particle . as shown in the figure [ restriction2 ] ( right panel ) , the particle seems to explore all the area when its motion is restricted by four paraboloids . this is simply an observation . how can we prove that without any information on the exact motion of the particle ? according to the poincar recurrence theorem , if the dynamical trajectory of the autonomous system evolves in a finite volume of the phase space , then in any domain , as small as it could be , there are at least two points which belong to the same trajectory . here , the motion is constrained in space with the four paraboloids but also in velocity because @xmath29 and @xmath30 are finite values and @xmath27 is constant . all the positions in this part of the phase space can belong to the same trajectory . this is also linked to the kam theory : here , the perturbation ( the radiation pressure ) affects the periodic motion ( the ellipse , bounded trajectories ) but it can remain quasi - periodic . as we will see , the global motion is not periodic but @xmath19 and @xmath20 motions possess their own period according to another parameter . the global motion can be periodic only if all periods are commensurable . this is an important result because if the particle belongs to an area crossing the exobase and if this area is closed in the phase space , along a finite time , the particle will again cross the exobase . we can now define the ballistic and satellite particles as follows : both populations have no elliptic trajectories due to the radiation pressure , but they evolve in a closed domain . depending on their constants of the motion , we can easily determine whether they cross ( i.e. the domain crosses ) the exobase or not , corresponding to ballistic and satellite particles respectively . escaping particles are in the case where the initial value of @xmath20 is higher than the highest real root of @xmath59 and their available area is opened to the infinity . thus , the theorem can not be applied here . even if the restriction area crosses the exobase , the particle may come from infinity , come close to the exobase and go away without crossing the exobase . we need , in order to identify escaping particles ( that cross the exobase and go to the infinity ) , to track along the time the particle to know if these particles come from the exobase or not . thus , it is necessary to solve their trajectory along the time .      for convenience , as usual in fluid mechanics , we define characteristic quantities . we decide to write all equations with dimensionless parameters . first , for distance , we define : @xmath88 this characteristic value was introduced by @xcite and defines the limit distance where the radiation pressure overwhelms the gravitation of the planet . then , we rewrite : @xmath89 where @xmath90 and @xmath91 are the dimensionless quantities associated to @xmath19 and @xmath20 . the energy @xmath25 is adimensionned with the use of the characteristic energy @xmath92 . for @xmath29 and @xmath30 , we express them in units of @xmath93 , whereas @xmath27 is expressed in units of @xmath94 . finally , we choose @xmath95 as the unit for @xmath31 and the unit for the adimensionless time @xmath96 is derived from the units of @xmath97 and @xmath98 . we summarize these changes in table [ dimensionless ] .        ' '' ''    & & +    ' '' ''    & & +    ' '' ''    @xmath19    @xmath20&@xmath99 & @xmath90   @xmath91 +    ' '' ''    @xmath29    @xmath30&@xmath100 & @xmath101   @xmath102 +    ' '' ''    @xmath13&@xmath103 & @xmath104 +    ' '' ''    @xmath25&@xmath92&@xmath62 +    ' '' ''    @xmath31&@xmath95&@xmath105 +    ' '' ''    @xmath50  @xmath51&@xmath92&@xmath55  @xmath57 +    ' '' ''    @xmath22&@xmath106&@xmath107 +    we express all the previous equations as a function of these new quantities :    @xmath108    @xmath109    @xmath110 @xmath111 with @xmath112 : @xmath113 the jeans parameter at the distance @xmath114 . we can also reduce the equations of the motion . we introduce the dimensionless time @xmath96 . in this part , we give implicit expressions for the dynamical trajectories of the particles , under the influence of both gravity and radiation pressure . such expressions were already given for the 2d case with @xmath116 and partially generalized to the 3d case in @xcite with jacobi elliptic functions ( only bounded ) @xcite and in @xcite with the weierstrass functions for bounded and unbounded trajectories . this last work dealt with the dynamical trajectories of artificial satellites but it can be applied to exospheric species subject to the radiation pressure . we propose here corrections as well as a better way to give  simple " expressions for dynamical trajectories . we also provide expressions for unbounded trajectories that are missing in the literature . in the same way , we introduce our notations for the next papers to be published , where the influence of the radiation pressure on the density profiles and escape flux will be investigated .    according to the previous part , we have different restrictions on the motion and thus , we must distinguish the cases , that will correspond to the different types of possible trajectories . we may thus define the ballistic / satellite / escaping populations based on the roots of the @xmath58 and @xmath59 polynoms . the trajectories are not elliptic or hyperbolic at all but one can keep their basic definitions : crossing twice the exobase for ballistic particles , orbiting but not crossing the exobase for satellite particles , coming from the exobase and escaping to the infinity for escaping particles . the number of real roots is a key parameter for the analytical resolution of the trajectories . the previous equations of the motion , expressed as a function of time , must be rewritten as a function of a new variable @xmath117 ( the subscript 0 is necessary to distinguish from @xmath96 ) defined as : @xmath118    until we know the expression of @xmath119 and @xmath120 , we can not express solutions as a function of time . the system of equations leads now to : @xmath121    all solutions , trajectories and time , are parametrized according to the variable @xmath117 without physical sense . we solve the differential equation describing the @xmath19-motion : @xmath122 with @xmath123 , @xmath60 and @xmath124 being real roots . first , we set @xmath125 . the motion occurs always with @xmath126 and @xmath127 to justify this change . we obtain : @xmath128    now , we set @xmath129 . @xmath130    finally , we have : @xmath131 we define @xmath132 as : @xmath133 @xmath134    the final equation is :    @xmath135    the solution of this equation is :    @xmath136    @xmath137 is a jacobi elliptic function and @xmath138 depends on initial conditions .    the final expression for @xmath90 is : @xmath139 with @xmath140 and @xmath141 other jacobi elliptic functions . these functions are @xmath142-periodic ( see appendix , eq . [ goodformule ] ) . to determine @xmath138 , we preferably choose the second expression for the sign . the initial conditions give us @xmath143 , then : @xmath144 where @xmath145 is defined as the reciprocal function of @xmath141 : @xmath146 with @xmath147 the incomplete elliptic integral of the first kind ( see appendix , eq . [ goodformule ] ) . the sign depends on @xmath148 : @xmath149 if @xmath148 is positive ( resp . negative ) , @xmath90 increases ( resp . decreases ) and @xmath150 increases ( resp . decreases ) too . then , @xmath138 is positive ( resp . negative ) . this solution corresponds to the solution @xmath151 given by @xcite . the @xmath90-motion is always constrained . the characteristics of the motion / particle ( ballistic , satellite or escaping ) is thus determined by conditions on @xmath152 .      for the @xmath20-motion , we need to solve the differential equation : @xmath153 @xmath154 is a positive real root but we must distinguish between various cases for the other roots : @xmath155 and @xmath156 may be real or complex conjugate roots . we must consider two cases : @xmath157 and @xmath158 . the second one occurs mathematically and not only physically when @xmath156 and @xmath155 are positive . [ [ a - for - w_-ww _ ] ] a ) for @xmath159 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    this case occurs for bounded trajectories ( ballistic or satellite trajectory ) . here , this is the same treatment as for the @xmath90-motion . after setting the two following substitutions @xmath160 and @xmath161 , we obtain a similar solution :    @xmath162    the final expression for @xmath91 is @xmath163 the initial conditions giving us @xmath164 and then leads to : @xmath165    the sign depends on @xmath166 : @xmath167 if @xmath166 is positive ( resp . negative ) , @xmath91 increases ( resp . decreases ) and @xmath168 increases ( resp . decreases ) too . then , @xmath169 is negative ( resp . positive ) . this solution corresponds to the solution @xmath170 given by @xcite . [ [ b - for - ww_0 ] ] b ) for @xmath157 + + + + + + + + + + + + + + + + + + + + + + + + + + + +    first , the motion occurs only for @xmath171 ( corresponding to escaping particles ) . then , we set @xmath172 . we obtain :    @xmath173    now , we set @xmath161 .    finally , we have : @xmath174 we define @xmath175 as : @xmath176 @xmath177    the solution of this equation is @xmath178 @xmath179 is a jacobi elliptic function and @xmath169 depends on the initial conditions . the final expression for @xmath91 is : @xmath180    these functions are @xmath181-periodic and are defined on @xmath182 . these functions diverge at @xmath183 but this is not an issue : the motion can diverge with respect to @xmath117 , but , according to the time @xmath96 and the integration ( see section [ time ] ) , the @xmath20-motion remains continuous for @xmath184 as    @xmath1850;4k(k_w)[\longleftrightarrow \mathcal{\tau}\in\mathbb{r}\ ] ]    let us assume the initial conditions provide @xmath164 , then : @xmath186 with @xmath187    the sign depends on @xmath166 : @xmath167    if @xmath166 is positive ( resp . negative ) , @xmath91 increases ( resp . decreases ) and @xmath188 increases ( resp . decreases ) too . then , @xmath169 is positive ( resp . negative ) . this solution corresponds to the solution @xmath189 given by @xcite . these expressions are useful only for three real roots . a last case remains : only one real positive root . for any initial conditions , the particle will be escaping if @xmath59 has only one real root . the solution for this case is more complex compared with the previous solutions . as done by @xcite , we could apply some transformations and obtain a new expression , that would be a combination of @xmath140 and @xmath141 . nevertheless , we propose a simpler way to determine the expression with the knowledge of all roots , even imaginaries . we start from the equation [ wdiff ] with @xmath155 and @xmath156 not real : @xmath190    after the separation of variables , we obtain : @xmath191    now , we apply the procedure proposed in @xcite ( p. 597 ) in the case where we have only one real root . first , we define : @xmath192    and also @xmath193    according to @xcite ( p. 597 ) , the left hand side corresponds to : @xmath194    with @xmath147 the elliptic function of first kind ( defined in the appendix [ appendixa ] ) and @xmath195    according to the equation [ equation60 ] : @xmath196    with @xmath197 the jacobi amplitude defined as @xmath198    thus @xmath199    finally , one can simplify the expression based on @xcite ( equation 22.6.18 ) :    @xmath200    this function is @xmath201-periodic and is defined on @xmath202 . this function diverges at @xmath203 but this does not impact our result : the motion can diverge with respect to @xmath117 , but , according to the time @xmath96 and the integration ( see section [ time ] ) , the @xmath20-motion remains continuous for @xmath184 as    @xmath204 - 2k(k_{w_0});2k(k_{w_0})[\longleftrightarrow \mathcal{\tau}\in\mathbb{r}\ ] ]    as usual , according to the initial conditions , we define @xmath205 as @xmath206 with @xmath207    if @xmath102 is positive ( resp . negative ) then @xmath205 is negative ( resp . positive ) . this solution corresponds to the solution @xmath151 given by @xcite but our derived solution is less complex to use ( with less time computing ) . we gave analytical formulations for the different kinds of trajectories , expressed implicitly . now , since we have all expressions of the trajectories as a function of @xmath117 , we can express the real time @xmath96 ( or @xmath22 ) : @xmath208    here , we use mathematica and maple to derive the primitives . it is necessary to be very careful since these different programs can have different definitions of the elliptic functions for example . to avoid these issues , we remind at each use the definition employed . the first part of the integral gives : @xmath209 \end{array } \label{tu}\ ] ] with @xmath210 , @xmath25 the incomplete elliptic function of the second kind ( see appendix , eq . [ goodformule ] ) . the second part of the integral is more complex because we have different expressions according to the number of roots and the initial conditions . in the case of three real roots , if @xmath211 then @xmath212 \end{array } \label{tw1}\ ] ] with @xmath213 . if @xmath214 with three real roots then @xmath215\\\\ -\dfrac{\beta}{\lambda_a}\left[\dfrac{{\mathrm{cn}}(\beta(\mathcal{t}_0-\mathcal{t}_w),k_w)}{{\mathrm{sd}}(\beta(\mathcal{t}_0-\mathcal{t}_w),k_w ) } -\dfrac{{\mathrm{cn}}(-\beta\mathcal{t}_w , k_w)}{{\mathrm{sd}}(-\beta\mathcal{t}_w , k_w)}\right ] \end{array } \label{tw2}\ ] ]    finally , in the case of only one real root , the time equation is given by : @xmath216\\\\ + \dfrac{\gamma}{\lambda_a}\left[\dfrac{{\mathrm{sn}}(2\gamma(\mathcal{t}_0-\mathcal{t}_{w_0}),k_{w_0}){\mathrm{dn}}(2\gamma(\mathcal{t}_0-\mathcal{t}_{w_0}),k_{w_0})}{1+{\mathrm{cn}}(2\gamma(\mathcal{t}_0-\mathcal{t}_{w_0}),k_{w_0 } ) } -\dfrac{{\mathrm{sn}}(-2\gamma\mathcal{t}_{w_0},k_{w_0}){\mathrm{dn}}(-2\gamma\mathcal{t}_{w_0},k_{w_0})}{1+{\mathrm{cn}}(-2\gamma\mathcal{t}_{w_0},k_{w_0})}\right ] \end{array } \label{tw3}\ ] ] with @xmath217 .    for both last cases , @xmath218 belongs to @xmath2190;4k(k_w)[$ ] ( respectively @xmath219 - 2k(k_{w_0});2k(k_{w_0})[$ ] ) for the equation [ tw2 ] ( resp . for the equation [ tw3 ] ) nevertheless , when @xmath218 tends to @xmath201 ( resp . @xmath220 ) , the integration diverges and @xmath96 too . the @xmath20-motion occurs on an subset of @xmath221 with respect to @xmath117 but on @xmath221 entirely with respect to @xmath96 . the transformation of @xmath96 into @xmath117 is bijective because the integrand @xmath222 is strictly positive . to complete the description of the motion as a function of time , it is also necessary to solve the evolution of the angle @xmath10 , obeying to : @xmath223 as already done in the previous part , we separate into two integrals . the first part still gives : @xmath224 with @xmath225 the incomplete elliptic integral of the third kind ( see appendix , eq . [ goodformule ] ) .    in the three real roots case for @xmath59 , if initially we have @xmath211 : @xmath226 or if initially @xmath214 : @xmath227    and for the last case with one real root : @xmath228    this last formula is only available for @xmath229 - 2k(k_{w_0});2k(k_{w_0})[$ ] , range where it is continuous . with the solutions previously derived , we can know the exact motion of a bounded or unbounded particle as a function of the time such as given in figure [ sample ] . it is clear that even a bounded trajectory the motion has no periodicity at all ( especially for the @xmath10 motion ) . nevertheless , it could be interesting to focus on stable bounded orbits and search for periodic motions ( as in @xcite ) , and thus investigate in particular the circular stable orbits for spacecrafts @xcite or the possible positions for satellite particles produced by collisions in the exosphere @xcite . thus , we dedicate this section to the conditions to obtain such orbits .    for a specific set of initial conditions , it can be possible to obtain circular orbits . this orbit occurs when , on the one hand , the attraction of the planet projected along the @xmath230-axis is equal to acceleration due to the radiation pressure : @xmath231    in dimensionless quantity , this will be expressed by : @xmath232    on the other hand , it is also necessary that the centrifugal force induced by the rotation around the @xmath230-axis is equal to the acceleration around the planet in the perpendicular plane to the @xmath230-axis . thus , we obtain the secondary equality : @xmath233    in dimensionless quantity , this will be expressed by : @xmath234    combining these two equations , we obtain : @xmath235    we need to study the polynom @xmath236 with @xmath237 $ ] . depending on the @xmath238 values , we have zero , one or two solutions for @xmath239 . indeed , @xmath240 so that according to the rolle theorem , there is @xmath2410;1[$ ] with @xmath242 . here , @xmath1 is equal to 8/9 . nevertheless , if @xmath243 is positive then @xmath239 does not have solutions and inversely , if @xmath243 is negative then we have two solutions . this value is : @xmath244    a critical maximum value of @xmath245 thus exists to allow for circular orbits and is @xmath246{3}}\ ] ]    above this value , we can not find any bounded trajectories : there is no any equilibrium point and thus no circular orbits ( stable or not ) . for lower values of @xmath247 , we have two solutions for @xmath239 : one stable and one unstable as shown by @xcite and theirs plots of the equipotentials . these two solutions correspond respectively to the stable point around which the equipotentials are closed and to the saddle point , which is the last limit where one can find closed equipotentials , and is the only point where two equipotentials can cross . as long as @xmath248 , these both specific points exist : they have the same @xmath90 . physically , the potential @xmath57 has two extrema as plotted in figure [ potentiel ] . when @xmath247 reaches @xmath249 , the local minimum goes to the right and the local maximum goes to the left at the same location @xmath250 . for higher @xmath247 values , @xmath56 have no extremum anymore : the potential is strictly decreasing and the particles are unbounded ( escaping ) . thus , the bounded particles , satellite and ballistic particles , have @xmath248 . the distinction between them thus depend on if they cross or not the exobase . the critical values are given in @xmath251 coordinates ( @xmath252 is @xmath253 for us , in comparison with @xcite ) . in dimensionless unities and in @xmath254 using the equations [ centrifuge ] then [ r ] , the critical orbit is : @xmath255 or in @xmath256 coordinates : @xmath257    the real positive roots of the polynomial [ polynom ] combined with the equality [ r ] give the positions of the circular orbits ( two coordinates are necessary ) allowed to spacecraft or particles under the influence of both gravity and radiation pressure . the knowledge of the exact trajectories of particles or satellites under the influence of gravity and radiation pressure needs the calculation of the spatial coordinates , i.e. the @xmath258 motions , as well as the time evolution . we summarize all needed equations in table [ summary ] .        ' '' ''    & & & +    ' '' ''    & & & +    ' '' ''    & & & & +    ' '' ''    & [ umotion](l2d ) & & & +    ' '' ''    & & [ wmotion1](l2d)&[wmotion2](l2d)&[wmotion3](s ) +    ' '' ''    & [ tu](l2d)&[tw1](l2d ) & [ tw2]*new*&[tw3]*new * +    ' '' ''    & [ phiu](l3d)&[phiw1](l3d ) & [ phiw2]*new*&[phiw3]*new * +    ' '' ''    & @xmath259&@xmath260&@xmath261&@xmath262 +    the @xmath90-motion is provided by the equation [ umotion ] . the @xmath91-motion is provided by the equations [ wmotion1 ] , [ wmotion2 ] or [ wmotion3 ] . the @xmath10-motion is provided by @xmath263 $ ] . the time equation is provided by @xmath264 $ ] . all the expressions are functions of @xmath117 that is not the real time . we thus have implicit expressions as a function of time . the function @xmath265 is bijective but can not be inversed analytically , a numerical inversion is needed to derive the real time . plane ( upper left panel ) , of the time as a function of @xmath117 ( upper right panel ) , of the motion in polar coordinates @xmath266 ( lower left panel ) and the @xmath267 coordinates as a function of the time ( red for @xmath90 , blue for @xmath91 , lower right panel ) . @xmath90 and @xmath91 do not show any periodicity because their periods are not commensurable ( i.e. the ratio is a rational number ) with @xmath117 and thus with the time @xmath96,title="fig : " ] plane ( upper left panel ) , of the time as a function of @xmath117 ( upper right panel ) , of the motion in polar coordinates @xmath266 ( lower left panel ) and the @xmath267 coordinates as a function of the time ( red for @xmath90 , blue for @xmath91 , lower right panel ) . @xmath90 and @xmath91 do not show any periodicity because their periods are not commensurable ( i.e. the ratio is a rational number ) with @xmath117 and thus with the time @xmath96,title="fig : " ] +   plane ( upper left panel ) , of the time as a function of @xmath117 ( upper right panel ) , of the motion in polar coordinates @xmath266 ( lower left panel ) and the @xmath267 coordinates as a function of the time ( red for @xmath90 , blue for @xmath91 , lower right panel ) . @xmath90 and @xmath91 do not show any periodicity because their periods are not commensurable ( i.e. the ratio is a rational number ) with @xmath117 and thus with the time @xmath96,title="fig : " ] plane ( upper left panel ) , of the time as a function of @xmath117 ( upper right panel ) , of the motion in polar coordinates @xmath266 ( lower left panel ) and the @xmath267 coordinates as a function of the time ( red for @xmath90 , blue for @xmath91 , lower right panel ) . @xmath90 and @xmath91 do not show any periodicity because their periods are not commensurable ( i.e. the ratio is a rational number ) with @xmath117 and thus with the time @xmath96,title="fig : " ]    besides , our 3d solutions can be easily applied to the 2d case . indeed , in the 2d case , @xmath268 and thus , one of the roots for each polynomial @xmath58 and @xmath59 is null : it could be @xmath269 or @xmath60 for @xmath58 ( if @xmath270 , there is no possible motion ) and any roots of @xmath59 . we precise that in this case the @xmath10-motion is not important because the motion is planar . compared with @xcite , our formulations are first developed for the 3d case and can be used easily for the 2d case , whereas @xcite gave only the methodology to obtain the 3d solutions based on 2d ones but not the expressions , which apparently leads to complex expressions . @xcite provided also the exact formulas for bounded and unbounded trajectories using the weierstrass functions but this formulation is also difficult particularly because of the need to use the inverse weierstrass function , not implemented in all computer softwares and the need to work with complex values ( e.g. the complex logarithm function ) . in this paper , we solved the motion for the bounded and unbounded trajectories in the 3d case ; we provide the exact formulas for all cases , as well as the definitions used in [ appendixa ] . we also highlight in table [ summary ] which solutions are simpler compared with @xcite , which are completely new and which were only provided in the 2d case by @xcite . moreover , beyond the new exact solutions given in this paper , the derivation of our solutions based on jacobi elliptic functions allows a good computing time and accuracy . @xcite compared three types of solutions for the stark effect : two exact ones , proposed by @xcite ( jacobi elliptic functions ) and @xcite ( weierstrass elliptic functions ) , and a numerical one by @xcite ( based on taylor series ) . they compared the cpu time , the number of calls for each analytic elliptic function and the accuracy between @xcite and @xcite . even if we do not agree with the number of evaluations of each jacobi elliptic function mentioned by @xcite ( e.g. to call @xmath140 and @xmath141 is similar to call @xmath197 then @xmath271 and @xmath272 : @xmath273 and @xmath274 ) , two arguments show our solutions are efficient in terms of cpu time and accuracy : first , the solutions expressed in terms of jacobi elliptic functions ( such as in this paper or by @xcite ) are more efficient than weierstrass elliptic functions ( used by @xcite ) ; second , several solutions given above are less complex to implement than those by @xcite , e.g. equation [ wmotion3 ] . also , the analytical formulations are preferable for long duration motions . we determined analytically the trajectories of the particles or spacecraft under the influence of both planetary gravity and stellar radiation pressure . we thus provide for the first time the complete exact solutions of the well - known stark effect ( effect of a constant electric field on the atomic hydrogen s electron ) with jacobi elliptic functions , for both bounded and unbounded orbits . these expressions may be implemented for modeling spacecraft or particles trajectories : instead of solving the equation of the motion , based on differential equations , with numerical methods such as the runge - kutta method where one cumulates errors along the time , it is here possible to obtain precise expressions of the motion with only periodic errors , due to the precision on the evaluation of the elliptic functions used . in particular , we provide the analytical conditions for stable circular orbits . moreover , we discuss about the possible issues inherent to the formalism used and the importance of being extremely careful with the routines implemented .    the formalism used here will allow us in a next paper to generalize the work by @xcite to derive the exact neutral densities and escape flux in planetary exospheres , under the influence of both gravity and stellar radiation pressure . this is important for understanding the atmospheric structure and escape of planets in the inner solar system , as well as the atmospheric erosion during the early ages where the radiation pressure ( and uv flux ) of the sun was extreme . this work was supported by the centre national dtudes spatiales ( cnes ) . in this paper , we use the three incomplete elliptic integrals @xmath147 , @xmath25 and @xmath225 : @xmath275        these expressions are not exactly @xmath25 , @xmath147 and @xmath279 . they agree with the previous formulas [ goodformule ] in the range @xmath280-\pi/2;\pi/2[$ ] . @xcite did not precise which formulations they used . according to theirs formulas and results , they used the left - hand side of the equations [ badformule ] . this may be a problem for bounded trajectories : for @xmath281 , the integrals [ badformule ] are not continuous contrary to [ goodformule ] . depending on the computer software , the routines and the definitions used for these functions , the results can show some issues ( e.g. no continuous motion ) .    16 natexlab#1#1url # 1`#1`urlprefix[2][]#2 [ 2]#2 , , . . , . , , , , , , . . , . , . . , . , , . . , . , , , . . , . , , . , . . , . , . . , , , , . edition . , , , . . , . edition . , . .
the planetary exospheres are poorly known in their outer parts , since the neutral densities are low compared with the instruments detection capabilities . 
 the exospheric models are thus often the main source of information at such high altitudes 
 . we present a new way to take into account analytically the additional effect of the radiation pressure on planetary exospheres . in a series of papers , we present with an hamiltonian approach the effect of the radiation pressure on dynamical trajectories , density profiles and escaping thermal flux . 
 our work is a generalization of the study by @xcite . in this first paper 
 , we present the complete exact solutions of particles trajectories , which are not conics , under the influence of the solar radiation pressure . 
 this problem was recently partly solved by @xcite and completely by @xcite . 
 we give here the full set of solutions , including solutions not previously derived , as well as simpler formulations for previously known cases and comparisons with recent works . 
 the solutions given may also be applied to the classical stark problem @xcite : we thus provide here for the first time the complete set of solutions for this well - known effect in term of jacobi elliptic functions . 
 exosphere , radiation pressure , stark effect , trajectories