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First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet

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The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably close to reality. The approach involves two essential steps. The first one concerns a new approximate mathematical model that admits a closed-form solution with respect to a set of new variables. The second step is the determination of an infinitesimal contact transformation that allows to navigate between the new and the original variables. This contact transformation is obtained in exact form, and afterwards a Taylor series approximation is proposed in order to make all the computations explicit. The aforementioned transformation accommodates both perturbations, improving the accuracy of the orbit predictions by one order of magnitude with respect to the case when the atmospheric drag is absent from the transformation. Numerical simulations are performed for a low Earth orbit starting at an altitude of 350 km, and they show that the incorporation of drag terms into the contact transformation generates an error reduction by a factor of 7 in the position vector. The proposed method aims at improving the accuracy of analytic orbit propagation and transforming it into a viable alternative to the computationally intensive numerical methods.

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  2. Due to the presence of significant uncertainties in the atmosphere (influence of the solar activity, day/night density variations, the atmospheric bulge, winds), realistic long-term propagations in the presence of drag cannot be addressed with deterministic tools (see Dell’Elce and Kerschen 2014).

  3. The critical inclination is a resonance of the dynamical system (see Lara 2015b) and therefore cannot be removed.

  4. Deprit was not the first one to use this approach. Hori (1966) proposed a slightly different version a few years before.

  5. They are uniquely defined up to a translation on the real numbers axis that is an integer multiple of \(2\pi \). But since only their trigonometric functions \(\sin \) and \(\cos \) are involved in computations, no restrictions need to be imposed.

  6. A transformation is said to be canonical if its Jacobian is a symplectic matrix, i.e. it obeys the condition in Eq. (19).

  7. Whittaker variables are singular for equatorial orbits, but this does not pose a problem for the case where only even zonal harmonics are taken into consideration, which is the case of the present paper.

  8. Elementary transformations between different sets of coordinates are also required at some steps of the algorithm, but these coordinates are present in the analytic expressions, making this requirement obvious,.

  9. These values were inspired by the prospective QARMAN satellite, a CubeSat mission developped by the von Karman Institute of Fluid Dynamics in Brussels and the University of Liège.

  10. The argument of perigee drift was omitted, since it is incorporated in the argument of latitude, depicted in Fig. 2c.

  11. Consequently, the orthogonal right-handed frame associated to the Poincaré canonical variables is found to be the same as for the equinoctial ones (see Battin 1999, Problem 10-10, p. 494).


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Correspondence to Vladimir Martinusi.



1.1 Lie series approximation for \(\overline{L}\)

Although the variable \(\overline{L}\) has an explicit expression, depicted in Eq. (10a), the numerical implementation of the complex error function, as well as its inverse, rise the problem of dealing with differences of very large numbers. It would be much more convenient to have an accurate approximation for \(\overline{L}\), and this is obtained by writing the solution to Eq. (8d) with the help of Lie series. For a complete description of the expansion mechanism, see Steinberg (1984).


$$\begin{aligned} \lambda =\sqrt{\mu \alpha };\quad \beta =\frac{\lambda }{\overline{L}_{0}} \end{aligned}$$

The IVP to solve is rewritten as:

$$\begin{aligned} \dot{\overline{L}}=-\mu C_{0}\exp \left( \frac{\overline{L}_{0}-\mu r_{0}}{ \mu \alpha }\right) \exp \left( -\frac{\overline{L}^{2}-\overline{L}_{0}^{2} }{\lambda ^{2}}\right) ,\quad \overline{L}\left( t_{0}\right) =\overline{L}_{0} \end{aligned}$$

and the change of variable:

$$\begin{aligned} z=\frac{\overline{L}^{2}-\overline{L}_{0}^{2}}{\lambda ^{2}} \end{aligned}$$

is performed. Denote

$$\begin{aligned} \xi _{0}=\overline{L}_{0}\frac{2\mu }{\lambda ^{2}}C_{0}\exp \left( \frac{ \overline{L}_{0}-\mu r_{0}}{\mu \alpha }\right) \end{aligned}$$

The IVP in z is:

$$\begin{aligned} \dot{z}=-\xi _{0}\sqrt{z\beta ^{2}+1}\exp \left( -z\right) ,~z\left( t_{0}\right) =0 \end{aligned}$$

Consider D to be the differential operator, defined for any arbitrary function \(\phi \):

$$\begin{aligned} D\phi =-\xi _{0}\sqrt{z\beta ^{2}+1}\exp \left( -z\right) \frac{\partial \phi }{\partial z} \end{aligned}$$

and define its formal powers as:

$$\begin{aligned} D^{k+1}\phi =D\left( D^{k}\phi \right) ,\;k\ge 1 \end{aligned}$$

Consider the formal exponential operator:

$$\begin{aligned} e^{\left( t-t_{0}\right) D}\phi =\phi +\sum _{k=1}^{\infty }\frac{\left( t-t_{0}\right) ^{k}}{k!}\left( D^{k}\phi \right) \end{aligned}$$

Then the solution to IVP (50) is expressed as:

$$\begin{aligned} z\left( t\right) =\left. e^{\left( t-t_{0}\right) D}z\right| _{z=0}, \end{aligned}$$

and explicitely:

$$\begin{aligned} z\left( t\right) =\sum _{k=1}^{\infty }\frac{\left( t-t_{0}\right) ^{k}}{k!} \left( \left. D^{k}z\right| _{z=0}\right) \end{aligned}$$

By taking into account the expression in Eq. (51) of the operator D,  the formal powers \(\left. D^{k}z\right| _{z=0}\) are expressed as:

$$\begin{aligned} \left. D^{1}z\right| _{z=0}= & {} -\xi _{0} \\ \left. D^{2}z\right| _{z=0}= & {} \xi _{0}^{2}\left( \frac{1}{2}\beta ^{2}-1\right) \\ \left. D^{3}z\right| _{z=0}= & {} \xi _{0}^{3}\left( 2\beta ^{2}-2\right) \\ \left. D^{4}z\right| _{z=0}= & {} \xi _{0}^{4}\left( -\beta ^{4}+9\beta ^{2}-6\right) \\ \left. D^{5}z\right| _{z=0}= & {} \xi _{0}^{5}\left( -13\beta ^{4}+48\beta ^{2}-24\right) \\ \left. D^{6}z\right| _{z=0}= & {} \xi _{0}^{6}\left( \frac{13}{2}\beta ^{6}-137\beta ^{4}+300\beta ^{2}-120\right) \\ \left. D^{7}z\right| _{z=0}= & {} \xi _{0}^{7}\left( 176\beta ^{6}-1422\beta ^{4}+2160\beta ^{2}-720\right) \\ \left. D^{8}z\right| _{z=0}= & {} \xi _{0}^{8}\left( -88\beta ^{8}+3365\beta ^{6}-15354\beta ^{4}+17640\beta ^{2}-5040\right) \\ \left. D^{9}z\right| _{z=0}= & {} \xi _{0}^{9}\left( -4069\beta ^{8}+57628\beta ^{6}-175752\beta ^{4}+161280\beta ^{2}-40320\right) \end{aligned}$$

Then \(\overline{L}=\overline{L}\left( t\right) \) is evaluated based on Eq. ( 49):

$$\begin{aligned} \overline{L}\left( t\right) =\overline{L}_{0}\sqrt{\beta ^{2}z\left( t\right) +1} \end{aligned}$$

1.2 Evaluation of \(I_{1,2,3}\) integrals

In the form they are written in Eq. (13), the integrals \(I_{1,2,3}\) are not suitable for an efficient series expansion approximation, given the considerable length of the interval on which the integration is performed (i.e., for a good approximation, a large number of terms are required in the truncated series expansion). For a rapid convergence, and also to avoid differences between very large numbers (since \(L=\sqrt{\mu \alpha }\) has the same order of magnitude of the angular momentum). It would therefore be ideal to transform them to integrals from 0 to some value z that has the smallest possible absolute value, so that the accuracy of the series expansion is maximum.

First, note that \(I_{1,2,3}\) may be rewritten as:

$$\begin{aligned} I_{k}=\exp \left( \frac{\overline{a}_{0}-r_{0}}{\alpha }\right) \int \limits _{ \overline{L}_{0}}^{\overline{L}}q_{k}\left( u\right) \exp \left( \frac{u^{2}- \overline{L}_{0}^{2}}{\mu \alpha }\right) \text {d}u,\quad k=1,2,3 \end{aligned}$$


$$\begin{aligned} q_{1}\left( u\right) =u^{-3},\quad q_{2}\left( u\right) =\left( u+\overline{G} _{0}-\overline{L}_{0}\right) ^{-3}u^{-4},\quad q_{3}\left( u\right) =\left( u+ \overline{G}_{0}-\overline{L}_{0}\right) ^{-4}u^{-3} \end{aligned}$$

To this end, denote

$$\begin{aligned} \lambda =\sqrt{\mu \alpha },\quad z=\frac{\overline{L}^{2}-\overline{L}_{0}^{2} }{\lambda ^{2}},\quad w_{k}\left( y\right) =\frac{q_{k}\left( \sqrt{\lambda ^{2}y+\overline{L}_{0}^{2}}\right) }{\sqrt{\lambda ^{2}y+\overline{L}_{0}^{2} }} \end{aligned}$$

and perform the change of variable:

$$\begin{aligned} y=\frac{u^{2}-\overline{L}_{0}^{2}}{\lambda ^{2}}\Rightarrow \text {d}u=\frac{ \lambda ^{2}}{2}\frac{\text {d}y}{\sqrt{\lambda ^{2}y+\overline{L}_{0}^{2}}} \end{aligned}$$

The integrals become:

$$\begin{aligned} I_{k}=\frac{\lambda ^{2}}{2}\exp \left( \frac{\overline{a}_{0}-r_{0}}{\alpha }\right) \int \limits _{0}^{z}w_{k}\left( y\right) \exp \left( y\right) \text {d}y \end{aligned}$$

By performing a Taylor series expansion for \(\exp \left( y\right) ,\) the integrals become:

$$\begin{aligned} I_{k}=\frac{\lambda ^{2}}{2}\exp \left( \frac{\overline{a}_{0}-r_{0}}{\alpha }\right) \sum \limits _{N=0}^{\infty }\left[ \frac{1}{N!}\int \limits _{0}^{z}y^{N}w_{k}\left( y\right) \text {d}y\right] \end{aligned}$$

and written separately:

$$\begin{aligned} I_{1}= & {} \frac{\lambda ^{2}}{2}\exp \left( \frac{\overline{a}_{0}-r_{0}}{ \alpha }\right) \sum \limits _{N=0}^{\infty }\left[ \frac{1}{N!} \int \limits _{0}^{z}\frac{y^{N}}{\left( \lambda ^{2}y+\overline{L} _{0}^{2}\right) ^{2}}\text {d}y\right] \\ I_{2}= & {} \frac{\lambda ^{2}}{2}\exp \left( \frac{\overline{a}_{0}-r_{0}}{ \alpha }\right) \sum \limits _{N=0}^{\infty }\left[ \frac{1}{N!} \int \limits _{0}^{z}\frac{y^{N}}{\left[ \sqrt{\lambda ^{2}y+\overline{L} _{0}^{2}}+\overline{G}_{0}-\overline{L}_{0}\right] ^{3}\left( \lambda ^{2}y+ \overline{L}_{0}^{2}\right) ^{\frac{5}{2}}}\text {d}y\right] \\ I_{3}= & {} \frac{\lambda ^{2}}{2}\exp \left( \frac{\overline{a}_{0}-r_{0}}{ \alpha }\right) \sum \limits _{N=0}^{\infty }\left[ \frac{1}{N!} \int \limits _{0}^{z}\frac{y^{N}}{\left[ \sqrt{\lambda ^{2}y+\overline{L} _{0}^{2}}+\overline{G}_{0}-\overline{L}_{0}\right] ^{4}\left( \lambda ^{2}y+ \overline{L}_{0}^{2}\right) ^{3}}\text {d}y\right] \end{aligned}$$

The evaluation of \(I_{1,2,3}\) has been simplified to definite integrals that may be expressed explicitly through elementary functions.

1.3 Jacobian for coordinate transformations

The Jacobian of the transformation from Delaunay to Whittaker elements is:

$$\begin{aligned} {\mathcal {J}}_{\mathcal {D}}^{{\mathcal {W}}}=\frac{\partial \left( r,\theta ,\nu ,R,\varTheta ,N\right) }{\partial \left( l,g,h,L,G,H\right) }=\left[ \begin{array}{cccccc} \dfrac{p\sigma }{\eta ^{3}} &{} 0 &{} 0 &{} \dfrac{r\left( \kappa ^{2}+\kappa -2e^{2}\right) }{Le^{2}} &{} \dfrac{\eta p\kappa }{Le^{2}} &{} 0 \\ &{} &{} &{} &{} &{} \\ \dfrac{\left( 1+\kappa \right) ^{2}}{\eta ^{3}} &{} 1 &{} 0 &{} \dfrac{\sigma \left( 2+\kappa \right) }{Le^{2}} &{} -\dfrac{\eta \sigma \left( 2+\kappa \right) }{Le^{2}} &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ &{} &{} &{} &{} &{} \\ \dfrac{L\kappa \left( 1+\kappa \right) ^{2}}{\eta ^{2}p} &{} 0 &{} 0 &{} \dfrac{ \eta \sigma }{pe^{2}}\left[ 1+2\kappa -\sigma ^{2}\right] &{} -\dfrac{\sigma \left( 1+\kappa \right) ^{2}}{pe^{2}} &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array} \right] \end{aligned}$$

The Jacobian of the transformation from Poincaré to Delaunay variables is:

$$\begin{aligned} {\mathcal {J}}_{\mathcal {P}}^{\mathcal {D}}= & {} \frac{\partial \left( l,g,h,L,G,H\right) }{\partial \left( Q_{1},Q_{2},Q_{3},P_{1},P_{2},P_{3}\right) }\\= & {} \left[ \begin{array}{cccccc} 1 &{} \dfrac{P_{2}}{Q_{2}^{2}+P_{2}^{2}} &{} 0 &{} 0 &{} -\dfrac{Q_{2}}{ Q_{2}^{2}+P_{2}^{2}} &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} -\dfrac{P_{2}}{Q_{2}^{2}+P_{2}^{2}} &{} \dfrac{P_{3}}{Q_{3}^{2}+P_{3}^{2}} &{} 0 &{} \dfrac{Q_{2}}{Q_{2}^{2}+P_{2}^{2}} &{} -\dfrac{Q_{3}}{Q_{3}^{2}+P_{3}^{2} } \\ &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} -\dfrac{P_{3}}{Q_{3}^{2}+P_{3}^{2}} &{} 0 &{} 0 &{} \dfrac{Q_{3}}{ Q_{3}^{2}+P_{3}^{2}} \\ &{} &{} &{} &{} &{} \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} -Q_{2} &{} 0 &{} 1 &{} -P_{2} &{} 0 \\ &{} &{} &{} &{} &{} \\ 0 &{} -Q_{2} &{} -Q_{3} &{} 1 &{} -P_{2} &{} -P_{3} \end{array} \right] \end{aligned}$$

The Jacobian of the transformation from Whittaker to Poincaré elements may now be explicitly derived as:

$$\begin{aligned} {\mathcal {J}}_{\mathcal {P}}^{{\mathcal {W}}}=\frac{\partial \left( r,\theta ,\nu ,R,\varTheta ,N\right) }{\partial \left( Q_{1},Q_{2},Q_{3},P_{1},P_{2},P_{3}\right) }={\mathcal {J}}_{\mathcal {D}}^{ {\mathcal {W}}}{\mathcal {J}}_{\mathcal {P}}^{\mathcal {D}} \end{aligned}$$

Since it involves only elementary computations, that are nevertheless leading to relatively long expressions, the full matrix form of \({\mathcal {J}} _{\mathcal {P}}^{{\mathcal {W}}}\) will not be displayed. In any case, it is obtained by the multiplication of two explicit matrices.

1.4 Transformation from Poincaré to Cartesian coordinates

The Poincaré canonical variables are mentioned and used in very few Astrodynamics/Orbital Mechanics textbooks, but they lack a comprehensive approach, and to our knowledge, a straightforward transformation from these variables to Cartesian is absent from the literature. The closest approach is found in Battin (1999), but it is made for the equinoctial variables, that are nonsingular but not canonical.

The transformation is determined here in the most direct way possible. It is known that the inertial position and velocity vectors are expressed in the perifocal frame \(\left\{ P\right\} \) as:

$$\begin{aligned} {\mathbf {r}}^{\left\{ P\right\} }=r\left[ \begin{array}{c} \cos f \\ \sin f \\ 0 \end{array} \right] ;\quad {\mathbf {v}}^{\left\{ P\right\} }=\frac{\mu }{G}\left[ \begin{array}{c} -\sin f \\ e+\cos f \\ 0 \end{array} \right] \end{aligned}$$

and their corresponding column matrices in the inertial frame originated in the attraction center are:

$$\begin{aligned} {\mathbf {r}}^{\left\{ IN\right\} }= & {} {\mathbf {R}}\left( {\mathbf {i}}_{3},h\right) {\mathbf {R}}\left( {\mathbf {i}}_{1},i\right) {\mathbf {R}}\left( {\mathbf {i}} _{3},g\right) {\mathbf {r}}^{\left\{ P\right\} } \\ {\mathbf {v}}^{\left\{ IN\right\} }= & {} {\mathbf {R}}\left( {\mathbf {i}}_{3},h\right) {\mathbf {R}}\left( {\mathbf {i}}_{1},i\right) {\mathbf {R}}\left( {\mathbf {i}} _{3},g\right) {\mathbf {v}}^{\left\{ P\right\} } \end{aligned}$$

where the orthogonal matrices \({\mathbf {R}}_{h},\) \({\mathbf {R}}_{i},\) \({\mathbf {R}} _{g}\) are:

$$\begin{aligned} {\mathbf {R}}\left( {\mathbf {i}}_{3},h\right)= & {} \left[ \begin{array}{ccc} \cos h &{} -\sin h &{} 0 \\ \sin h &{} \cos h &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] ,\quad {\mathbf {R}}\left( {\mathbf {i}}_{1},i\right) =\left[ \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} \cos i &{} -\sin i \\ 0 &{} \sin i &{} \cos i \end{array} \right] ,\\ {\mathbf {R}}\left( {\mathbf {i}}_{3},g\right)= & {} \left[ \begin{array}{ccc} \cos g &{} -\sin g &{} 0 \\ \sin g &{} \cos g &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] \end{aligned}$$

If a new fixed frame \(\left\{ \mathcal {P}\right\} \) is defined such that the position vector has the expression:

$$\begin{aligned} {\mathbf {r}}^{\left\{ \mathcal {P}\right\} }=r\left[ \begin{array}{c} \cos \left( f+g+h\right) \\ \sin \left( f+g+h\right) \\ 0 \end{array} \right] =r\left[ \begin{array}{c} \cos \left( Q_{1}+f-l\right) \\ \sin \left( Q_{1}+f-l\right) \\ 0 \end{array} \right] , \end{aligned}$$

it will follow (by elementary computations) that the velocity in the same frame \(\left\{ \mathcal {P}\right\} \) is:

$$\begin{aligned} {\mathbf {v}}^{\left\{ \mathcal {P}\right\} }=\frac{2\mu }{\sqrt{4P_{1}-2\left( Q_{2}^{2}+P_{2}^{2}\right) }}\left[ \begin{array}{c} \dfrac{Q_{2}}{2P_{1}}\sqrt{4P_{1}-\left( Q_{2}^{2}+P_{2}^{2}\right) }-\sin \left( Q_{1}+f-l\right) \\ \dfrac{P_{2}}{2P_{1}}\sqrt{4P_{1}-\left( Q_{2}^{2}+P_{2}^{2}\right) }+\cos \left( Q_{1}+f-l\right) \\ 0 \end{array} \right] \end{aligned}$$

Consequently, the inertial counterparts of \({\mathbf {r}}^{\left\{ \mathcal {P} \right\} },\) \({\mathbf {v}}^{\left\{ \mathcal {P}\right\} }\) are:

$$\begin{aligned} {\mathbf {r}}^{\left\{ IN\right\} }= & {} \mathbf {Ar}^{\left\{ \mathcal {P}\right\} } \\ {\mathbf {v}}^{\left\{ IN\right\} }= & {} \mathbf {Av}^{\left\{ \mathcal {P}\right\} } \end{aligned}$$

The matrix \(\mathbf {A}\) is found to beFootnote 11:

$$\begin{aligned} \mathbf {A}={\mathbf {R}}\left( {\mathbf {i}}_{3},h\right) {\mathbf {R}}\left( \mathbf { i}_{1},i\right) {\mathbf {R}}\left( {\mathbf {i}}_{3},-h\right) \end{aligned}$$

Define U such that:

$$\begin{aligned} U=\sqrt{4P_{1}-2\left( Q_{2}^{2}+P_{2}^{2}\right) -\left( Q_{3}^{2}+P_{3}^{2}\right) }=\sqrt{2G\left( 1+\cos i\right) } \end{aligned}$$

Then the explicit expression of matrix \(\mathbf {A,}\) in Poincaré variables, is:

$$\begin{aligned} \mathbf {A}=\frac{1}{U^{2}+Q_{3}^{2}+P_{3}^{2}}\left[ \begin{array}{ccc} U^{2}-Q_{3}^{2}+P_{3}^{2} &{} -2P_{3}Q_{3} &{} -2{{ UQ}}_{3} \\ -2P_{3}Q_{3} &{} U^{2}+Q_{3}^{2}-P_{3}^{2} &{} -2{{ UP}}_{3} \\ 2{{ UQ}}_{3} &{} 2{{ UP}}_{3} &{} U^{2}-Q_{3}^{2}-P_{3}^{2} \end{array} \right] \end{aligned}$$

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Martinusi, V., Dell’Elce, L. & Kerschen, G. First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet. Celest Mech Dyn Astr 127, 451–476 (2017).

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