Revisiting Hansen’s Ideal Frame Propagation with Special Perturbations—1: Basic Algorithms for Osculating Elements

Abstract

A review of the basic Hansen’s ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge.

1. Introduction

Special perturbation methods in orbital mechanics follow two basic approaches. On the one hand, the use of coordinates as the integration variables notably eases the implementation of the differential equations for typical celestial mechanics and astrodynamics problems and provides the solution in a form useful for ephemeris prediction. The classical Encke and Cowell methods are usually programmed in Cartesian coordinates [1,2,3,4], but more sophisticated approaches such as those based on KS or projective coordinates are customary, too [5,6,7,8]. On the other hand, the use of “elements” (or “parameters”) as the integration variables for perturbation problems—whose variations remain constant in the simplifications of the unperturbed, integrable model and which are commonly obtained as different combinations of its basic integrals—allows for more stable propagation of the initial state. However, this is at the cost of more involved programming, which may need the inclusion of a variety of routines to transform the elements into coordinates in order to obtain the ephemeris, when required.

For perturbed Keplerian motion, classical implementations of the variation-of-parameters method rely on the use of elliptic orbital elements or related nonsingular variables [9,10,11,12,13], yet alternative vectorial implementations may enjoy more symmetric formulations due to the fact that they gather the differential equations of the slowly varying elements into two vectorial equations, which, therefore, are simpler to code and faster to evaluate [14,15,16,17,18,19,20,21,22]. Because of that, the latter fit very well for automatic computing and hence have become very popular nowadays, especially regarding the long-term propagation of average dynamics [23,24,25,26,27,28,29]. In either case, the characteristics of elliptic motion require an uneven step-size distribution for optimal numerical integration, with shorter steps near periapsis and larger ones near apoapsis. While numerical step-size regulation is the aim of different numerical solvers [30,31,32,33], analytical step-size regulation by means of time-regularization strategies [6,7,8,34,35,36,37,38,39,40,41,42] allows for the use of simpler, robust, and highly efficient fixed-step integration methods. Moreover, it provides definite stabilization of the numerical integration procedure by constraining the effects of the Lyapunov-type instability—which is unavoidably attached to Keplerian dynamics—to the integration of the physical time, which now plays the role of one more dependent variable of the differential system [43,44,45].

In a mixed way, Cowell-type implementations of the perturbed Keplerian dynamics may have clear benefits from concomitant integration in Cartesian coordinates of one or more of the invariants of the Kepler problem. While the implementation of the differential equations remains analogously simple in this case, projection methods can be used to preserve the geometric properties of the perturbed motion with a minor increase in the computational burden [46,47]. Still, the integration of elements alone is generally regarded as more stable and efficient than any variant of the Cowell- or Encke-type integrations [21,48,49]. Indeed, the propagation of slowly varying elements would allow for much longer step sizes for the numerical solver than in the case of much faster evolving coordinates, in this way largely compensating for the more involved evaluation of the variations in the elements in each computation step. Note, however, that this fact does not deny the utility of coordinates for numerical integration, which show remarkable performance in the case of KS or projective methods [50,51,52]. Moreover, beyond the perturbed Keplerian dynamics, as may be the case for N-body regimes, coordinate-based methods usually provide higher stability characteristics and are customarily used.

Among the different sets of variations of parameter methods, those based on Hansen’s ideal frame [53], for which rotation under perturbations occurs along the instantaneous radius vector, enjoy unique merits for such orbital configurations in which the role played by the orbital plane is significant enough [22,54,55]. In these cases, the orbital motion can be viewed as the composition of two different effects. Namely, the slow rotation of the orbital plane and the fast rotation of the particle within the orbital plane. In spite of fast and slow effects not being decoupled, the fact that the velocity is the same when measured by Hansen’s ideal frame as when calculated by the inertial frame presents this approach as an appealing formulation. Indeed, compared to orbital or nodal frame formulations, the ideal frame brings the motion very close to the separation by removing the first derivatives of the variables related to the attitude of the orbital plane from the equations related to the motion in the orbital plane [56]. It must be noted, however, that in those cases for which the role of the orbital plane is not relevant, as happens when the perturbations in the out-of-plane direction are large, or in the case of rectilinear or almost-rectilinear orbits, other approaches may certainly be more effective [21,57,58].

The quest for higher efficiency in the numerical integration of perturbed Keplerian motion over the years has led to recurrent refinements of Hansen’s ideal frame formulations that can be efficiently implemented as operational software [59,60]. Basically, extant formulations are limited to two different, quite effective approaches. The hodographic approach pinpoints the elements describing the evolution of the eccentricity vector of the osculating orbit, or, more precisely, the coordinates of the center of the velocity hodograph, whereas the orbital approach relies on the use of projective coordinates to linearize the radial motion. While these methods clearly differ from each other on how to materialize the polar coordinates in the orbital plane, the two avoid singularities in the description of the attitude of the orbital plane by using Euler parameters and the Rodrigues rotation formula [61]. In both cases, too, the variation of the polar angle defines a finite rotation from the ideal to the orbital frame, whereas the total angular momentum is integrated from the corresponding variation equation.

Rather than using the physical time as the independent variable, popular versions of these two approaches [62,63,64,65,66] find an accurate description of the orbit geometry, as well as efficient numerical integration, by referring the integration to a fictitious time based on a Sundman-type transformation that involves some power of the orbit radius [67,68]. More precisely, the time transformation is commonly stated by a differential equation relating the variations to the physical and fictitious times by a coefficient that involves some power of the radius and a regularizing term. The case in which the exponent of the former is one plays a fundamental role in the integration of the three-body problem [67,69,70], whereas the choice of two for the exponent has been common in planetary theories after Laplace [71,72]. Additionally, raising the radius to three halves shows important advantages regarding the control of truncation errors in numerical integration [36,73]. This latter case was first examined in relation to space exploration [74,75], yet it also finds application in classical celestial mechanics problems [76]. Regarding the regularizing term, it may depend generally on any orbital variable as well as on the total energy [6,77,78,79]. This is the noteworthy case of time regularizations originally proposed in [80] (see also [39]), a variant of which has been tested in the framework of the ideal frame integration with encouraging results [65].

Alternatively, the need for redundant variables stemming from the use of Euler parameters is kept to a minimum without losing any merit of the ideal-element formulation when re-scaling the Euler parameters by the total angular momentum [81,82]. While the integration of the modulus of the angular momentum is now dispensable, this is at the cost of a slightly more complicated form of the variations of the attitude-related variables. Preliminary computations demonstrated that this simpler formulation may be, in addition, more effective in terms of computing time. However, this conclusion cannot be taken as definitive as far as it is based on the use of a single numerical solver, whereas it is well known that the performance of numerical integration of a particular problem may vary depending on the numerical solver used for propagation [83].

On the other hand, it is known that the accuracy of the solution of time-regularized differential equations may be increased when replacing the numerical integration of the physical time by the alternative integration of time elements [43,77,84,85,86]. The efficient time element proposed in [80] deserves special mention in reference to analytical perturbations based on (true anomaly) Delaunay-similar variables [87,88], but that also applies to nonsingular variables [89]. Adoption of this linear element four decades later as a time element of the ideal-frame setting showed exceptional accuracy in the numerical integration of circumterrestrial orbits [90]. The improvements to the determination of the physical time introduced with the use of a variety of existing time elements are expected to affect the different formulations presented equally. While we do not discuss this topic yet due to the lack of accessible, detailed descriptions in the literature, we provide in Appendix A the solution to the Hamilton–Jacobi equation of the Kepler problem in the extended phase space, which gives rise to a remarkable set of variables that include the time element and the total energy of the perturbation problem among them. We believe that the exercise is worth being carried out because not only does the popular, linear time element of the (true anomaly) Delaunay-similar variables emerge in the procedure [80,91], but this kind of solution relies on a set of auxiliary, non-osculating functions that are repeatedly reported in the literature [39] and, in particular, are related to the ideal-frame formulation [65,90]—whose merits when used in a special perturbation approach are discussed elsewhere. Moreover, as was expected from the Hamilton–Jacobi solution of the standard Kepler problem [92,93,94], the solution in the extended phase space is also expedited when using polar canonical coordinates instead of the usual approach in spherical coordinates [77,80,87].

The outline of the paper is as follows. In Section 2, we recall the basics of the Kepler problem, the orbital and hodographic solutions to which provide the common variables and elements used in the integration of perturbed Keplerian motion. Next, the relevant, nonsingular equations of the ideal frame formulation are presented in Section 3, where we also discuss the pros and cons of replacing the integration of the fast-evolving, in-track variable by the integration of an element. The time regularization of the ideal-element variations is considered in Section 4, where the suitability of the radial variables for integration in the fictitious-time scale is shown. The convenience of using time elements instead of integrating the physical-time variation is further elaborated in this section, too. This is followed by the presentation in Section 5 of alternative formulations based on the ideal frame concept. However, the variety of refinements and applications of Hansen’s theory that exist without constraint for the case of elliptic motion are not discussed in this review, and interested readers are referred to [42] for additional information.

2. The Orbital Plane and the Geometry of the Kepler Problem

In an inertial, space frame $(O,\mathit{i},\mathit{j},\mathit{k})$, we denote x the position vector of a mass point with respect to the origin and denote

$$\mathit{X}=\frac{\mathrm{d}\mathit{x}}{\mathrm{d}t},$$

(1)

its velocity, with t denoting the physical time. Define the angular momentum vector (per unit mass) as

$$\mathit{G}=\mathit{x}\times \mathit{X}=\mathrm{\Theta }\mathit{n},$$

(2)

where $\mathrm{\Theta }=\parallel \mathit{G}\parallel$ and $\mathit{n}=\mathit{G}/\mathrm{\Theta }$. Define also the eccentricity vector

$$\mathit{e}=\frac{1}{\mu }\mathit{X}\times \mathit{G}-\mathit{u},$$

(3)

where $\mathit{u}=\mathit{x}/r$ and $r=\parallel \mathit{x}\parallel$. Note the orthogonality constraint $\mathit{G}·\mathit{e}=0$.

2.1. The Orbital Plane

The unit vector n defines the instantaneous orbital plane. Based on it, we define the moving orbital frame$(O,\mathit{u},\mathit{v},\mathit{n})$, where $\mathit{v}=\mathit{n}\times \mathit{u}$. Let us also consider a reference frame $(O,{\mathit{u}}^{*},{\mathit{v}}^{*},\mathit{n})$ and denote $\theta$ the polar angle in the orbital plane reckoning the position of the orbital frame with respect to the reference frame. That is,

$$\mathit{u}={\mathit{u}}^{*}cos\theta +{\mathit{v}}^{*}sin\theta ,$$

(4)

$$\mathit{v}={\mathit{v}}^{*}cos\theta -{\mathit{u}}^{*}sin\theta .$$

(5)

As customary after Cayley, we denote the instantaneous direction ${\mathit{u}}^{*}$ as a departure point in the orbital frame [95,96].

From the theorem of the moving frame (Coriolis theorem)

$$\mathit{X}={\left(\frac{\mathrm{d}\mathit{x}}{\mathrm{d}t}\right)}_{\omega }+\mathit{\omega }\times \mathit{x},$$

(6)

where the time derivative is computed in the moving frame, which rotates with angular velocity $\mathit{\omega }=(\mathrm{d}\theta /\mathrm{d}t)\mathit{n}$. Then, replacing $\mathit{x}=r\mathit{u}$ in Equation (6) and taking into account that u remains constant in the rotating frame, we obtain

$$\mathit{X}=\frac{\mathrm{d}r}{\mathrm{d}t}\mathit{u}+r\mathit{\omega }\times u=\frac{\mathrm{d}r}{\mathrm{d}t}\mathit{u}+r\frac{\mathrm{d}\theta }{\mathrm{d}t}\mathit{v},$$

(7)

which turns Equation (2) into

$$\mathit{G}=r^2\frac{\mathrm{d}\theta }{\mathrm{d}t}\mathit{n},$$

(8)

and yields the fundamental relation

$$\frac{\mathrm{d}\theta }{\mathrm{d}t}=\frac{\mathrm{\Theta }}{r^2}.$$

(9)

Moreover, again using the theorem of the moving frame and in view of Equation (9), we obtain

$$\frac{\mathrm{d}\mathit{u}}{\mathrm{d}t}=\mathit{\omega }\times u=\frac{\mathrm{\Theta }}{r^2}\mathit{v}.$$

(10)

Then, replacing Equations (9) and (10) in Equation (7) and denoting the radial velocity $R\equiv \mathrm{d}r/\mathrm{d}t$, we obtain the decomposition

$$\mathit{X}=R\mathit{u}+\frac{\mathrm{\Theta }}{r}\mathit{v}.$$

(11)

For illustration purposes of the orbit geometry and the different frames used in what follows, the reader is referred to Figure 1.

2.2. The Keplerian Hodograph

In general, all the introduced vectors vary with time under the action of forces according to the laws of mass point dynamics. In the particular case of Newtonian dynamics and the Kepler problem,

$$\frac{\mathrm{d}\mathit{X}}{\mathrm{d}t}=-\frac{\mu }{r^2}\mathit{u},$$

(12)

where $\mu$ denotes the gravitational parameter. Then, from differentiation of Equations (2) and (3), it is simple to check that the vectors G and e become constant, in this way comprising the needed integrals that make the Kepler problem solvable.

Using Equation (9), we rewrite Equation (12) as

$$\frac{\mathrm{d}\mathit{X}}{\mathrm{d}\theta }=-\frac{\mu }{\mathrm{\Theta }}\mathit{u},$$

(13)

which, in view of Equations (4) and (5), is trivially integrated to yield the trajectory

$$\mathit{X}=\frac{\mu }{\mathrm{\Theta }}\mathit{v}+{\mathit{K}}_0,$$

(14)

where ${\mathit{K}}_0={(K_{0,1},K_{0,2},0)}^{\mathrm{T}}$, with T denoting transposition, is an arbitrary integration vectorial constant, thus showing that the velocity hodograph of the Kepler problem is a circumference of radius $\mu /\mathrm{\Theta }$ and origin ${\mathit{K}}_0$, cf. [97]. Now, plugging Equation (14) into Equation (3), we readily obtain

$$\mathit{e}=\frac{\mathrm{\Theta }}{\mu }{\mathit{K}}_0\times \mathit{n},$$

(15)

from which

$${\mathit{K}}_0=\frac{\mu }{\mathrm{\Theta }}\mathit{n}\times \mathit{e}.$$

(16)

Therefore, the decomposition in Equation (14) splits the velocity of the Kepler problem into one component orthogonal to the radius of constant modulus $\mu /\mathrm{\Theta }$ and another (constant) component orthogonal to the eccentricity vector and modulus $e\mu /\mathrm{\Theta }$, with $e=\parallel \mathit{e}\parallel$ denoting the eccentricity of the orbit, cf. [98]. Namely

$$\mathit{X}=\frac{\mu }{\mathrm{\Theta }}\left(\mathit{v}+\mathit{n}\times \mathit{e}\right).$$

(17)

This classical decomposition is illustrated in Figure 2 in reference to the orbit.

Therefore, the velocity hodograph of the Kepler problem is determined as a function of the polar angle and three constant, scalar velocities. That is, $\mathit{X}=\mathit{X}(\theta ;C,S,{\zeta }_3)$, where the velocities

$$C=+\frac{\mu }{\mathrm{\Theta }}(\mathit{n}\times \mathit{e})·{\mathit{v}}^{*}=\frac{\mu }{\mathrm{\Theta }}\mathit{e}·{\mathit{u}}^{*},$$

(18)

$$S=-\frac{\mu }{\mathrm{\Theta }}(\mathit{n}\times \mathit{e})·{\mathit{u}}^{*}=\frac{\mu }{\mathrm{\Theta }}\mathit{e}·{\mathit{v}}^{*},$$

(19)

$${\zeta }_3=\frac{\mu }{\mathrm{\Theta }},$$

(20)

have been repeatedly pointed out as insightful parameters regarding astrodynamics applications [99,100,101]—yet their evident geometric character was not always clearly recognized [92,102,103,104]. The geometry of the velocity hodograph is illustrated in Figure 3, showing the relevant scalar and vectorial components of the velocity.

2.3. The Keplerian Orbit in Terms of the Hodographic Constants

Replacing X from Equation (11) and G from Equation (2) into Equation (3) and multiplying it scalarly by u, we readily obtain the conic equation that materializes the orbit of the Kepler problem

$$\mathit{e}·\mathit{u}=\frac{p}{r}-1,$$

(21)

where $p={\mathrm{\Theta }}^2/\mu$ is the parameter of the conic. Differentiation of Equation (21) with respect to time yields

$$-\frac{\mathrm{\Theta }}{\mu }\frac{\mathrm{\Theta }}{r^2}\frac{\mathrm{d}r}{\mathrm{d}t}=\mathit{e}·\frac{\mathrm{d}\mathit{u}}{\mathrm{d}t},$$

from which, on account of Equation (10), we obtain

$$R=-\frac{\mu }{\mathrm{\Theta }}\mathit{e}·\mathit{v}.$$

(22)

Now, plugging Equation (4) into Equation (21) and taking Equations (18)–(20) into account, we obtain

$$\frac{p}{r}=1+\frac{1}{{\zeta }_3}(Ccos\theta +Ssin\theta ),$$

(23)

thus disclosing the harmonic character of the inverse of the distance on a time scale proportional to $\theta$. Analogously, replacing Equation (5) into Equation (22), we obtain

$$R=Csin\theta -Scos\theta .$$

(24)

The Cartesian coordinates are then recovered from the canonical set of polar variables $(r,\theta ,R,\mathrm{\Theta })$ by means of standard transformations.

3. The Basic Set of Ideal Elements for Perturbed Kepler Motion

Beyond the integrable Kepler problem, the Newtonian equations of perturbed Keplerian motion are written in the form

$$\frac{\mathrm{d}\mathit{x}}{\mathrm{d}t}=\mathit{X},$$

(25)

$$\frac{\mathrm{d}\mathit{X}}{\mathrm{d}t}=-\frac{\mu }{r^3}\mathit{x}+\mathit{P},$$

(26)

where the perturbation of the Keplerian acceleration P depends, in general, on position, velocity, and time. For given initial conditions ${\mathit{x}}_0\equiv \mathit{x}\left(t_0\right)$, ${\mathit{X}}_0\equiv \mathit{X}\left(t_0\right)$, the solution to Equations (25) and (26) is unique and can always be obtained by numerical methods.

Alternatively to the numerical integration in Cartesian coordinates, the integration of elements can be advantageous. They are defined such that their variations remain constant in unperturbed Keplerian motion. Therefore, it is expected that their variations under small perturbations change only slowly for constant elements, like the semimajor axis or the epoch of periapsis passage, or almost linearly in the case of linear elements like the mean anomaly.

Obviously the angular momentum vector and the eccentricity vector are constant elements of the Keplerian motion, whose variations under perturbations are obtained from differentiation of Equations (2) and (3). For the first, taking Equation (26) into account, we obtain

$$\frac{\mathrm{d}\mathit{G}}{\mathrm{d}t}=r\mathit{u}\times \mathit{P},$$

(27)

from which, on account of the identity $\mathrm{\Theta }\mathrm{d}\mathrm{\Theta }/\mathrm{d}t=\mathit{G}·\mathrm{d}\mathit{G}/\mathrm{d}t$, we readily obtain

$$\frac{\mathrm{d}\mathrm{\Theta }}{\mathrm{d}t}=r(\mathit{P}·\mathit{v}).$$

(28)

Moreover, replacing $\mathit{G}=\mathrm{\Theta }\mathit{n}$ and $\mathit{u}\times \mathit{P}=(\mathit{P}·\mathit{v})\mathit{n}-(\mathit{P}·\mathit{n})\mathit{v}$ in Equation (27) and solving the resulting equation for the variation of n, in which Equation (28) is, in turn, replaced, we obtain

$$\frac{\mathrm{d}\mathit{n}}{\mathrm{d}t}=-\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})\mathit{v},$$

(29)

and hence, replacing Equations (10) and (29) for the variation of $\mathit{v}=\mathit{n}\times \mathit{u}$,

$$\frac{\mathrm{d}\mathit{v}}{\mathrm{d}t}=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})\mathit{n}-\frac{\mathrm{\Theta }}{r^2}\mathit{u}.$$

(30)

On the other hand, the variation of the eccentricity vector is computed using Equations (10), (26), and (27) to obtain

$$\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}=\frac{\mathrm{\Theta }}{\mu }\mathit{P}\times \mathit{n}+\frac{r}{\mu }\mathit{X}\times (\mathit{u}\times \mathit{P}).$$

(31)

Different combinations of the components of the basic vectorial integrals give rise to constant elements, too. This is the particular case of the three velocities that characterize the Keplerian hodograph in Equations (18)–(20), whose definitions only require algebraic operations between the components of the eccentricity vector and the unit vectors defining the reference frame (all of them being constant in the Keplerian approximation). It is also the case of the variables determining the orientation of the reference frame with respect to the space frame, which only involve trigonometric relations between the constant vectors of the space frame and those defining the reference frame. On the contrary, since the orbital frame evolves non-linearly in general, even in the Kepler problem, the variables related to the radial distance are not elements—as becomes clear from Equations (21) and (22). Still, they are efficiently integrated on a time scale proportional to $\theta$, a case in which the inverse of the distance is solved from a second-order, linear differential equation with constant coefficients, thus giving rise to harmonic oscillations, as follows from Equations (23) and (24). This latter case will be discussed in Section 4.

Finally, the location of the particle, either in the orbit or the hodograph, must be determined through the computation of $\theta$, which is not an element. While this is not a major concern in the integration of lower-eccentricity orbits, replacing $\theta$ by a linear element commonly brings benefits to the solution of eccentric orbits—yet this is at the expense of losing the generality of $\theta$ in the description of any kind of conic. Still, we adhere to tradition and use the nonsingular, mean distance to the departure point like the linear element providing the timing on the ellipse.

3.1. Ideal Frames

Different choices of the reference frame, like nodal or apsidal frames, may offer different advantages. However, as already mentioned in the introduction, appealing simplifications of the equations of motion arise from the choice of ideal frames, in which the velocity is the same as when measured in the inertial frame. As follows from the theorem of the moving frame in Equation (6), this means that the vectorial product of $\mathit{\omega }$ and x must vanish for an ideal frame, which happens only when the rotation of the reference frame takes place about the u axis of the orbital frame.

Imposing this condition to the reference frame is easily done by noting that the time variation of the orbital frame given by Equations (10), (29) and (30) can be written like the rotation [22]

$$\frac{\mathrm{d}}{\mathrm{d}t}(\mathit{u},\mathit{v},\mathit{n})=\mathit{\omega }\times (\mathit{u},\mathit{v},\mathit{n}),$$

(32)

with angular velocity

$$\mathit{\omega }=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})\mathit{u}+\frac{\mathrm{\Theta }}{r^2}\mathit{n}.$$

(33)

Then, choosing a reference frame that rotates with angular velocity

$${\mathit{\omega }}^{*}=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})\mathit{u},$$

(34)

turns it into ideal and makes the rotation from the reference to the orbital frame ${\mathit{\omega }}_{\mathrm{K}}=\mathit{\omega }-{\mathit{\omega }}^{*}$ to be purely Keplerian in spite of the perturbations. Therefore, on account of Equation (4), the variation of the chosen elements that materialize the reference, ideal frame must be computed from

$${\mathit{\omega }}^{*}=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})({\mathit{u}}^{*}cos\theta +{\mathit{v}}^{*}sin\theta ).$$

(35)

The attitude of the reference frame is customarily obtained from the Euler angles $(\mathrm{\Omega },I,\beta )$. More precisely, the inclination $0\le I\le \pi$ between the orbital plane n and the inertial plane k is obtained from $cosI=\mathit{n}·\mathit{k}$. The right ascension of the ascending node $\mathsf{\Omega }\in [0,2\pi )$ is reckoned from i to the direction of the ascending node $\ell =\mathit{k}\times \mathit{n}/sinI$. Finally, the angle $\beta \in [0,2\pi )$ is unambiguously determined from $\ell ·{\mathit{u}}^{*}=cos\beta$, $\ell \times {\mathit{u}}^{*}=\mathit{n}sin\beta$. The triad $(\ell ,\mathit{n}\times \ell ,\mathit{n})$ defines the nodal frame.

The variation of the Euler angles under perturbations is obtained from the usual relation [105,106]

$${\mathit{\omega }}^{*}=\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}\mathit{k}+\frac{\mathrm{d}I}{\mathrm{d}t}\ell +\frac{\mathrm{d}\beta }{\mathrm{d}t}\mathit{n},$$

(36)

from which, on account of Equation (35), the components $({\mathit{\omega }}^{*}·{\mathit{u}}^{*},{\mathit{\omega }}^{*}·{\mathit{v}}^{*},{\mathit{\omega }}^{*}·\mathit{n})$ are computed and solved for the time derivatives to obtain

$$\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})\frac{sin(\theta +\beta )}{sinI},$$

(37)

$$\frac{\mathrm{d}I}{\mathrm{d}t}=\frac{r}{\mathrm{\Theta }}(\mathit{P}·\mathit{n})cos(\theta +\beta ),$$

(38)

$$\frac{\mathrm{d}\beta }{\mathrm{d}t}=-\frac{\mathrm{d}\mathrm{\Omega }}{\mathrm{d}t}cosI,$$

(39)

which are the variations that must fulfill the reference frame to be ideal under perturbations. Note that while the initial values of $\mathrm{\Omega }$ and I are fixed from the dynamics, the freedom in choosing the initial value of $\beta$ gives rise to countless ideal frames. Usual options are to choose an initial $\beta$ such that ${\mathit{u}}^{*}\left(t_0\right)=\mathit{u}\left(t_0\right)$ [22] or such that ${\mathit{u}}^{*}\left(t_0\right)$ points to the initial pericenter [66].

3.2. HODEI Formulation

To avoid the singularities affecting Equations (37) and (39) when $I=0$, the Euler angles are conveniently replaced by the nonsingular, redundant set of Euler parameters

$${\lambda }_1=sin\frac{1}{2}Icos\frac{1}{2}(\mathrm{\Omega }-\beta ),$$

(40)

$${\lambda }_2=sin\frac{1}{2}Isin\frac{1}{2}(\mathrm{\Omega }-\beta ),$$

(41)

$${\lambda }_3=cos\frac{1}{2}Isin\frac{1}{2}(\mathrm{\Omega }+\beta ),$$

(42)

$${\lambda }_4=cos\frac{1}{2}Icos\frac{1}{2}(\mathrm{\Omega }+\beta ),$$

(43)

which are linked by the geometric constraint ${\sum }_{j=1}^4{\lambda }_j^2=1$. The variations of the Euler parameters are then computed by differentiation of Equations (40)–(43) and the consequent replacement of Equations (37)–(39). Straightforward computations yield

$$\frac{\mathrm{d}\mathit{\lambda }}{\mathrm{d}t}=\frac{r}{2\mathrm{\Theta }}(\mathit{P}·\mathit{n})\left(\begin{array}{cc}\hfill {\mathbf{0}}_{\mathbf{2}}& \hfill \mathbf{m}\\ \hfill -\mathbf{m}& \hfill {\mathbf{0}}_{\mathbf{2}}\end{array}\right)\mathit{\lambda },$$

(44)

where $\mathit{\lambda }={({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4)}^{\mathrm{T}}$, ${\mathbf{0}}_{\mathbf{2}}$ denotes the null matrix of rank 2, and, assuming vectors as columns of a matrix, $\mathbf{m}={({\mathit{u}}^{*},{\mathit{v}}^{*})}^{\mathrm{T}}(\mathit{v},\mathit{u})$. That is, from Equations (4) and (5),

$$\mathbf{m}=\left(\begin{array}{cc}\hfill {\mathit{u}}^{*}·\mathit{v}& \hfill {\mathit{u}}^{*}·\mathit{u}\\ \hfill {\mathit{v}}^{*}·\mathit{v}& \hfill {\mathit{v}}^{*}·\mathit{u}\end{array}\right)=\left(\begin{array}{cc}\hfill -sin\theta & \hfill cos\theta \\ \hfill cos\theta & \hfill sin\theta \end{array}\right).$$

(45)

Equation (44) comprises the differential equations that describe the attitude of the ideal reference frame without any singularity. That the Euler parameters are elements is obvious from the vanishing of their variations in the absence of perturbations.

Tracking the slow variation of the hodograph under perturbations requires differentiation of the velocities in Equations (18)–(20). The variation of ${\zeta }_3$ is unrelated to the ideal frame and is trivially obtained from Equation (28) as

$$\frac{\mathrm{d}{\zeta }_3}{\mathrm{d}t}=-\frac{{\zeta }_3^2}{\mu }r(\mathit{P}·\mathit{v}).$$

(46)

On the contrary, the computation of the variations of C and S must adhere to the ideal-frame definition and requires slightly more-involved computations, which we summarize from [22] (see also [82]).

From their definition in Equations (18) and (19), we may write $C{\mathit{u}}^{*}+S{\mathit{v}}^{*}=(\mu /\mathrm{\Theta })\mathit{e}$, whose differentiation in the rotating, ideal frame yields

$$\frac{\mathrm{d}C}{\mathrm{d}t}{\mathit{u}}^{*}+\frac{\mathrm{d}S}{\mathrm{d}t}{\mathit{v}}^{*}=-\frac{\mu }{{\mathrm{\Theta }}^2}\frac{\mathrm{d}\mathrm{\Theta }}{\mathrm{d}t}\mathit{e}+\frac{\mu }{\mathrm{\Theta }}{\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}},$$

where we replace the variation of $\mathrm{\Theta }$ from Equation (28) and e from Equation (3), in which X is, in turn, replaced from Equation (11). Straightforward operations yield

$$\frac{\mathrm{d}C}{\mathrm{d}t}{\mathit{u}}^{*}+\frac{\mathrm{d}S}{\mathrm{d}t}{\mathit{v}}^{*}=\frac{\mu }{\mathrm{\Theta }}{\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}}+(\mathit{P}·\mathit{v})\left[\left(\frac{r}{p}-1\right)\mathit{u}+\frac{rR}{\mathrm{\Theta }}\mathit{v}\right],$$

and hence,

$$\begin{array}{cc}\hfill \frac{\mathrm{d}C}{\mathrm{d}t}=& \frac{\mu }{\mathrm{\Theta }}{\mathit{u}}^{*}·{\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}}+(\mathit{P}·\mathit{v})\left[\left(\frac{r}{p}-1\right)\mathit{u}·{\mathit{u}}^{*}+\frac{rR}{\mathrm{\Theta }}\mathit{v}·{\mathit{u}}^{*}\right],\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}S}{\mathrm{d}t}=& \frac{\mu }{\mathrm{\Theta }}{\mathit{v}}^{*}·{\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}}+(\mathit{P}·\mathit{v})\left[\left(\frac{r}{p}-1\right)\mathit{u}·{\mathit{v}}^{*}+\frac{rR}{\mathrm{\Theta }}\mathit{v}·{\mathit{v}}^{*}\right].\hfill \end{array}$$

(48)

The derivative of the eccentricity vector in the rotating frame is now computed from Equation (31) using the theorem of the moving frame. That is, ${\left(\mathrm{d}\mathit{e}/\mathrm{d}t\right)}_{{\omega }^{*}}=\mathrm{d}\mathit{e}/\mathrm{d}t-{\mathit{\omega }}^{*}\times \mathit{e}$, where we replace Equations (3) and (34) in the cross product of ${\mathit{\omega }}^{*}$ and e to obtain

$${\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}}=\frac{\mathrm{\Theta }}{\mu }\mathit{P}\times \mathit{n}+\frac{r}{\mu }\mathit{X}\times (\mathit{u}\times \mathit{P})-\frac{r}{\mu }(\mathit{P}·\mathit{n})\mathit{u}\times (\mathit{X}\times \mathit{n}),$$

(49)

in which we further replace X from Equation (11), and $\mathit{P}=(\mathit{P}·\mathit{u})\mathit{u}+(\mathit{P}·\mathit{v})\mathit{v}+(\mathit{P}·\mathit{n})\mathit{n}$. Hence,

$${\left(\frac{\mathrm{d}\mathit{e}}{\mathrm{d}t}\right)}_{{\omega }^{*}}=\frac{\mathrm{\Theta }}{\mu }\left[(\mathit{P}·\mathit{v})\left(2\mathit{u}-\frac{rR}{\mathrm{\Theta }}\mathit{v}\right)-(\mathit{P}·\mathit{u})\mathit{v}\right],$$

(50)

which is, in turn, replaced into Equations (47) and (48) using Equation (45) to yield

$$\frac{\mathrm{d}C}{\mathrm{d}t}=(\mathit{P}·\mathit{v})\left(\frac{r}{p}+1\right)cos\theta +(\mathit{P}·\mathit{u})sin\theta ,$$

(51)

$$\frac{\mathrm{d}S}{\mathrm{d}t}=(\mathit{P}·\mathit{v})\left(\frac{r}{p}+1\right)sin\theta -(\mathit{P}·\mathit{u})cos\theta .$$

(52)

The variations of the ideal elements $(C,S,{\zeta }_3,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4)$ is what we call the HODEI formulation—for hodographic, Euler parameters, ideal-frame formulation—and is customary in the literature with minor modifications [22].1 Still, to integrate the solution for given initial conditions, we need to know $\theta$ as a function of time. This can be achieved by complementing the differential system with the variation of $\theta$, as given in Equation (9), which can be rewritten as

$$\frac{\mathrm{d}\theta }{\mathrm{d}t}=\frac{\mu }{r^2{\zeta }_3}=\frac{{\zeta }_3^3}{\mu }\frac{p^2}{r^2}.$$

(53)

3.3. The Timing on the Osculating Ellipse

Recall that the radius is replaced in terms of the ideal elements and the polar angle in the above equations using Equation (23). Thus, Equation (53) can be written as

$$\frac{\mathrm{d}\theta }{\mathrm{d}t}=\frac{{\zeta }_3^3}{\mu }{\left[1+\frac{1}{{\zeta }_3}(Ccos\theta +Ssin\theta )\right]}^2.$$

(54)

which clearly shows that, while C, S, and ${\zeta }_3$ are constant in the Keplerian case, $\theta$ evolves non-linearly with time at a rate given by the variation of the true anomaly $f=\theta -\omega$, where $\omega$ denotes the argument of the periapsis reckoned from the departure point. Therefore, $\theta$ is not an element. On the contrary, the variation of the mean distance from the departure point $F=M+\omega$, with M denoting the mean anomaly, is certainly an element of the perturbed Keplerian motion. This becomes obvious from its definition and the fact that

$$M=n(t-t_{\mathrm{P}}),$$

(55)

with constant mean motion n and time of periapsis passage $t_{\mathrm{P}}$ for pure Keplerian motion.

Let us check this fact in the context of the ideal-frame formulation, too. The variation of F in terms of the hodographic elements is thoroughly derived in Section 4 of [22], from which we directly borrow its Equation (101). Namely,

$$\frac{\mathrm{d}F}{\mathrm{d}t}=\frac{{\left(2Q\right)}^{3/2}}{\mu }+\frac{1}{1+\eta }\frac{1}{{\zeta }_3^2}\left(C\frac{\mathrm{d}S}{\mathrm{d}t}-S\frac{\mathrm{d}C}{\mathrm{d}t}\right)+2\frac{\eta }{\mu }r{\zeta }_3\left(\frac{\mathrm{d}S}{\mathrm{d}t}cos\theta -\frac{\mathrm{d}C}{\mathrm{d}t}sin\theta \right),$$

(56)

where $\eta ={(1-e^2)}^{1/2}$, whereas $Q=\mu /\left(2a\right)$ is the positive Keplerian energy, with a denoting the orbit semimajor axis. More precisely, from Equations (18) and (19), the components of the eccentricity vector in the ideal reference frame are

$$\mathit{e}·{\mathit{u}}^{*}=ecos\omega =\frac{1}{{\zeta }_3}C,\mathit{e}·{\mathit{v}}^{*}=esin\omega =\frac{1}{{\zeta }_3}S,$$

(57)

from which

$$\eta =\sqrt{1-(C^2+S^2)/{\zeta }_3^2},$$

(58)

and hence

$$Q=\frac{{\mu }^2}{2{\mathrm{\Theta }}^2}(1-e^2)=\frac{1}{2}\left[{\zeta }_3^2-(C^2+S^2)\right].$$

(59)

In the Keplerian case, in which C, S, ${\zeta }_3$, and Q are constant, Equation (56) turns into

$$\frac{\mathrm{d}F}{\mathrm{d}t}=\frac{{\left(2Q\right)}^{3/2}}{\mu },$$

(60)

thus clearly adhering to the definition of an orbital element.

Undeniably, the computation of the position from the ideal element F requires the solution of Kepler’s equation to obtain $\theta$ at each step of the integration procedure. More precisely, we replace the eccentric anomaly u in Kepler’s equation $M=u-esinu$ by the eccentric distance from the departure point $\psi =u+\omega$. After trigonometric expansions, we also replace the components of the eccentricity vector from Equation (57) and obtain Kepler’s equation in the modified form

$$F=\psi -\frac{1}{{\zeta }_3}(Csin\psi -Scos\psi ),$$

(61)

cf. [108]. Once $\psi$ has been computed from Equation (61), we use the standard relations of the ellipse $rcosf=a(cosu-e)$, $rsinf=a\eta sinu$, in which we replace the true anomaly f by $f=\theta -\omega$ and the eccentric one by $u=\psi -\omega$ to obtain

$$\frac{r}{a}cos\theta =cos\psi -\frac{1}{{\zeta }_3}\left(C-\frac{\psi -F}{1+\eta }S\right),$$

(62)

$$\frac{r}{a}sin\theta =sin\psi -\frac{1}{{\zeta }_3}\left(S+\frac{\psi -F}{1+\eta }C\right),$$

(63)

which are Equations (84) and (85) of [22] and from which the in-track angle $\theta$ is determined without ambiguity.

3.4. Initialization

The initial ideal elements needed to launch the propagation are readily obtained from the Cartesian coordinates ${\mathit{x}}_0=\mathit{x}\left(t_0\right)$, ${\mathit{X}}_0=\mathit{X}\left(t_0\right)$. Thus, we first compute $r_0=\parallel {\mathit{x}}_0\parallel$ and ${\mathit{G}}_0={\mathit{x}}_0\times {\mathit{X}}_0$, from which

$${\mathrm{\Theta }}_0=\parallel {\mathit{G}}_0\parallel .$$

(64)

Hence, ${\mathit{u}}_0={\mathit{x}}_0/r_0$, ${\mathit{n}}_0={\mathit{G}}_0/{\mathrm{\Theta }}_0$, and ${\mathit{v}}_0={\mathit{n}}_0\times {\mathit{u}}_0$. Next, the Euler parameters are obtained from the standard relations [109,110]

$${\lambda }_{4,0}=\frac{1}{2}{(1+{\mathit{u}}_0·\mathit{i}+{\mathit{v}}_0·\mathit{j}+{\mathit{n}}_0·\mathit{k})}^{1/2},\left(\begin{array}{c}{\lambda }_{1,0}\\ {\lambda }_{2,0}\\ {\lambda }_{3,0}\end{array}\right)=\frac{1}{4{\lambda }_{4,0}}\left(\begin{array}{c}{\mathit{v}}_0·\mathit{k}-{\mathit{n}}_0·\mathit{j}\\ {\mathit{n}}_0·\mathit{i}-{\mathit{u}}_0·\mathit{k}\\ {\mathit{u}}_0·\mathit{j}-{\mathit{v}}_0·\mathit{i}\end{array}\right).$$

(65)

Finally, for

$${\theta }_0=0,$$

(66)

and on account of ${\mathit{e}}_0=(1/\mu ){\mathit{X}}_0\times {\mathit{G}}_0-{\mathit{u}}_0$, we compute

$$\begin{array}{ccc}\hfill {\zeta }_{3,0}& =& \mu /{\mathrm{\Theta }}_0,\hfill \\ \hfill C_0& =& {\zeta }_{3,0}{\mathit{u}}_0·{\mathit{e}}_0\hfill & =& {\mathit{X}}_0·{\mathit{v}}_0-{\zeta }_{3,0},\hfill \\ \hfill S_0& =& {\zeta }_{3,0}{\mathit{v}}_0·{\mathit{e}}_0\hfill & =& -{\mathit{u}}_0·{\mathit{X}}_0.\hfill \end{array}$$

(67)

Ephemeris at the event $t=t_i$ are obtained from the polar variables in the orbital plane $(r,\theta ,R,\mathrm{\Theta })$, where $r\left(t_i\right)$ is computed from Equation (23), $\theta \left(t_i\right)$ is solved from Equations (62) and (63), $R\left(t_i\right)$ is obtained from Equation (24), and $\mathrm{\Theta }\left(t_i\right)=\mu /{\zeta }_3\left(t_i\right)$. Then,

$$\mathit{x}=\mathcal{M}\left(\begin{array}{c}rcos\theta \\ rsin\theta \\ 0\end{array}\right),\mathit{X}=\mathcal{M}\left(\begin{array}{c}Rcos\theta -(\mathrm{\Theta }/r)sin\theta \\ Rsin\theta +(\mathrm{\Theta }/r)cos\theta \\ 0\end{array}\right),$$

(68)

where

$$\mathcal{M}=\left(\begin{array}{ccc}1-2({\lambda }_2^2+{\lambda }_3^2)& 2({\lambda }_1{\lambda }_2-{\lambda }_4{\lambda }_3)& 2({\lambda }_1{\lambda }_3+{\lambda }_2{\lambda }_4)\\ 2({\lambda }_1{\lambda }_2+{\lambda }_4{\lambda }_3)& 1-2({\lambda }_1^2+{\lambda }_3^2)& 2({\lambda }_2{\lambda }_3-{\lambda }_1{\lambda }_4)\\ 2({\lambda }_1{\lambda }_3-{\lambda }_2{\lambda }_4)& 2({\lambda }_2{\lambda }_3+{\lambda }_1{\lambda }_4)& 1-2({\lambda }_1^2+{\lambda }_2^2)\end{array}\right),$$

(69)

is the rotation matrix from the ideal to the space frame.

Alternatively, the use of a fixed departure frame matching the orbital frame at the starting time may help in preventing the loss of significant digits along the numerical integration. This is the case for high-inclination orbits, whose inclination variations are tracked with only a few significant digits in the space frame but are integrated with full accuracy when referred to the departure frame, in which the inclination remains close to zero [22,63,64,82]. In that case, Equation (68), which now denotes the coordinates in the departure frame, is further referred to the space frame by a constant, additional rotation.

4. Time Transformation and Fictitious-Time Elements

Alternatively to the classical integration of the physical-time variations, it may be advantageous to carry out the integration in a fictitious time $\varsigma$, defined by the time transformation

$$\mathrm{d}t=c_{\alpha }{r}^{\alpha }\mathrm{d}\varsigma ,$$

(70)

where $\alpha$ is a positive, numeric constant, and the coefficient $c_{\alpha }$ may depend on the chosen set of coordinates or elements as well as on the total energy—whether it is constant or not [6,77].

4.1. Laplace’s Time-Regularization

In particular, for $\alpha =2$, the customary change stemming from the fundamental relation Equation (9)

$$\mathrm{d}t=\frac{r^2{\zeta }_3}{\mu }\mathrm{d}\theta ,$$

(71)

sets the ideal-frame variations in the convenient form

$$\frac{\mathrm{d}\mathit{\lambda }}{\mathrm{d}\theta }=\frac{1}{2}({\mathit{P}}^{*}·\mathit{n})\left(\begin{array}{cc}\hfill {\mathbf{0}}_{\mathbf{2}}& \hfill \mathbf{m}\\ \hfill -\mathbf{m}& \hfill {\mathbf{0}}_{\mathbf{2}}\end{array}\right)\mathit{\lambda },$$

(72)

$$\frac{\mathrm{d}C}{\mathrm{d}\theta }=\frac{\mu }{r{\zeta }_3}\left[({\mathit{P}}^{*}·\mathit{v})\left(\frac{r}{p}+1\right)cos\theta +({\mathit{P}}^{*}·\mathit{u})sin\theta \right],$$

(73)

$$\frac{\mathrm{d}S}{\mathrm{d}\theta }=\frac{\mu }{r{\zeta }_3}\left[({\mathit{P}}^{*}·\mathit{v})\left(\frac{r}{p}+1\right)sin\theta -({\mathit{P}}^{*}·\mathit{u})cos\theta \right],$$

(74)

$$\frac{\mathrm{d}{\zeta }_3}{\mathrm{d}\theta }=-{\zeta }_3({\mathit{P}}^{*}·\mathit{v}),$$

(75)

$$\frac{\mathrm{d}t}{\mathrm{d}\theta }=\frac{r^2{\zeta }_3}{\mu },$$

(76)

where we abbreviated ${\mathit{P}}^{*}=\mathit{P}{r}^3{\zeta }_3^2/{\mu }^2$, thus making the perturbation non-dimensional.2

The improvements to the simple and effective time-regularization given by Equation (71) result into a dramatic accuracy increase for the integration of the geometric coordinates of the orbit. However, the time transformation does not change the Keplerian dynamics and simply gathers the Lyapunov-type instabilities into the integration of the physical time from Equation (76). The errors in the timing stemming from this fact produce a lack of synchronism between the computed and the true solution that may result in analogous ephemeris errors to those obtained when the integration is carried out using the physical time as the independent variable.

We remark that the definition of elements widely applies to the integration in a fictitious time without constraint to the physical-time integration [6]. That the physical-time variable is not an element of the fictitious-time integration is readily checked from Equation (76) in the same way as we already did in Equation (54). Therefore, replacing the direct integration of the physical time by the integration of a fictitious-time element followed by the computation of the physical time from the time-element definition could bring additional benefits to the integration process [6,43,84,85]. Among the different possibilities, elements related to the time of periapsis passage $t_{\mathrm{P}}$ are natural choices for constant time elements, which we inherit from traditional formulations of the variation-of-parameters method by astronomers [10,111]. Alternatively, the integration of the physical time can be replaced by a linear time element of the form $\tau =t_{\mathrm{P}}+n\varsigma$, which in the Keplerian approximation evolves linearly in the fictitious time $\varsigma$ with frequency $n={\left(2Q\right)}^{3/2}/\mu$. In the end, the fictitious time must be recovered from its relation with the time element, which is naturally established through Kepler’s equation.

4.2. Time Elements

Based on the anomalies of the elliptic motion, different authors derived from Kepler’s equation time elements that apply for regularized equations relying on either osculating [78,79,84] or non-osculating elements [80,90,112]. In order to avoid the singularities associated with the classical anomalies, in what follows, we derive the variation equations of analogous time-elements, which are new to our knowledge, based on the mean distance to the departure point, which is nonsingular for circular orbits.

A constant time element ${\tau }_{\mathrm{cons}}$ of the fictitious-time dynamics is defined as follows. We solve the time from Equation (55) as a function of the mean anomaly and rewrite it in the form

$$t=-\frac{1}{n}(M_0+\omega )+\frac{1}{n}(M+\omega )={\tau }_{\mathrm{cons}}+\frac{\mu }{{\left(2Q\right)}^{3/2}}F,$$

(77)

where $M_0$ denotes the value of the mean anomaly at $t=0$, and

$${\tau }_{\mathrm{cons}}\equiv \frac{\mu }{{\left(2Q\right)}^{3/2}}(-F_0),$$

(78)

with $F_0$ denoting the value of F at $t=0$. Under perturbations, ${\tau }_{\mathrm{cons}}$ is no longer constant, and its fictitious-time variation is obtained by the variation-of-parameters method. That is, differentiation of Equation (77) with respect to $\theta$ yields

$$\frac{\mathrm{d}{\tau }_{\mathrm{cons}}}{\mathrm{d}\theta }=\frac{\mu }{{\left(2Q\right)}^{3/2}}\left[\frac{{\left(2Q\right)}^{3/2}}{\mu }\frac{\mathrm{d}t}{\mathrm{d}\theta }-\frac{\mathrm{d}F}{\mathrm{d}\theta }+\frac{3}{2}\frac{F}{Q}\frac{\mathrm{d}Q}{\mathrm{d}\theta }\right],$$

(79)

in which $\mathrm{d}t/\mathrm{d}\theta$ is replaced from Equation (71), the variation of the Kepler energy

$$\frac{\mathrm{d}Q}{\mathrm{d}\theta }={\zeta }_3\frac{\mathrm{d}{\zeta }_3}{\mathrm{d}\theta }-C\frac{\mathrm{d}C}{\mathrm{d}\theta }-S\frac{\mathrm{d}S}{\mathrm{d}\theta },$$

(80)

is trivially obtained from Equation (59), and, from Equation (56), $\mathrm{d}F/\mathrm{d}\theta =(r^2/\mathrm{\Theta })\mathrm{d}F/\mathrm{d}t$ results in

$$\frac{\mathrm{d}F}{\mathrm{d}\theta }=\frac{{\left(2Q\right)}^{3/2}}{\mu }\frac{\mathrm{d}t}{\mathrm{d}\theta }+\frac{1}{1+\eta }\frac{1}{{\zeta }_3^2}\left(C\frac{\mathrm{d}S}{\mathrm{d}\theta }-S\frac{\mathrm{d}C}{\mathrm{d}\theta }\right)+2\frac{\eta }{\mu }r{\zeta }_3\left(cos\theta \frac{\mathrm{d}S}{\mathrm{d}\theta }-sin\theta \frac{\mathrm{d}C}{\mathrm{d}\theta }\right).$$

(81)

It is important to note that, as pointed out in [84], at each step of the numerical integration of Equation (79), the value of Q entering the above formulas should be obtained from the numerical integration of Equation (80) rather than from direct evaluation of Equation (59), where the small errors resulting from the numerical integration of ${\zeta }_3$, etc. would be multiplied by twice their respective values, thus increasing the error in the determination of Q.

Alternatively, for a linear time element ${\tau }_{\mathrm{lin}}$, we rewrite Equation (77) as

$$t={\tau }_{\mathrm{lin}}+\frac{\mu }{{\left(2Q\right)}^{3/2}}(F-\theta ),$$

(82)

where

$${\tau }_{\mathrm{lin}}\equiv \frac{\mu }{{\left(2Q\right)}^{3/2}}(\theta -F_0),$$

(83)

evolves linearly with $\theta$ in the Keplerian case. Therefore, ${\tau }_{\mathrm{lin}}$ fits the definition of a (fictitious) time element, from which the physical time is recovered through Equation (82). Now, under perturbations, differentiation of Equation (82) with respect to the fictitious time $\theta$ yields

$$\frac{\mathrm{d}{\tau }_{\mathrm{lin}}}{\mathrm{d}\theta }=\frac{\mu }{{\left(2Q\right)}^{3/2}}\left[1+\frac{{\left(2Q\right)}^{3/2}}{\mu }\frac{\mathrm{d}t}{\mathrm{d}\theta }-\frac{\mathrm{d}F}{\mathrm{d}\theta }+\frac{3}{2}\frac{F-\theta }{Q}\frac{\mathrm{d}Q}{\mathrm{d}\theta }\right],$$

(84)

in which, again, $\mathrm{d}t/\mathrm{d}\theta$ is replaced from Equation (71), whereas the variations of the Keplerian energy Q and the mean distance to the departure point F are the same as those given in Equations (80) and (81), respectively.

Needless to say, both time elements ${\tau }_{\mathrm{cons}}$ and ${\tau }_{\mathrm{lin}}$ are free from singularities related to circular orbits since the direct computation of F from the (nonsingular) polar coordinates, which is needed at each integration step in order to evaluate Equation (84), is straightforward. Thus, proceeding analogously as we did in the previous derivation of Equations (62) and (63), we readily obtain

$$\frac{r}{a}cos\theta =cos\psi +\frac{1-\eta }{\eta }\frac{r}{a}sinfsin\omega -ecos\omega ,$$

(85)

$$\frac{r}{a}sin\theta =sin\psi -\frac{1-\eta }{\eta }\frac{r}{a}sinfcos\omega -esin\omega ,$$

(86)

from which, recalling that $R=(\mathrm{\Theta }/p)esinf$ from Equation (22), we can compute $\psi$ without ambiguity from

$$cos\psi =\frac{r{\zeta }_3^2}{\mu }{\eta }^{2}cos\theta +\frac{C}{{\zeta }_3}-\frac{\eta }{1+\eta }\frac{rR}{\mu }S,$$

(87)

$$sin\psi =\frac{r{\zeta }_3^2}{\mu }{\eta }^{2}sin\theta +\frac{S}{{\zeta }_3}+\frac{\eta }{1+\eta }\frac{rR}{\mu }C,$$

(88)

where r and R are replaced from Equations (23) and (24), respectively, and $\eta$ is from Equation (58). Finally, F is obtained by simple evaluation of Equation (61). Alternatively, the dimension of the differential system can be augmented with the variation of F in Equation (81), whose computation is, in fact, an unavoidable step in the evaluation of the fictitious-time element in Equation (84). Then, Equations (87) and (88) only need to be evaluated once at $\theta =0$ to compute the initial value $F_0$.

We remark that the computation of ephemeris from the regularized differential equations requires the interpolation of the physical time from the numerical integration.

4.3. ORBELTI Formulation

Replacing Equation (76) with either Equation (79) or Equation (84) results in a time-regularized system of elements, as desired. On the other hand, computing the orbit from this differential system requires, in addition, recovery of the radial coordinates from the hodograph velocities and the regularized time, which is done using Equations (23) and (24). However, as already mentioned in Section 3, it so happens that in the fictitious time $\theta$, both the inverse of the distance and the radial velocity are subject to harmonic oscillations, which are efficiently also integrated under perturbations. Therefore, the computation of the hodographic velocities C and S is optional, and we can abbreviate calculations by directly integrating the orbital coordinates. That is, calling $s=1/r$ and using Equations (23) and (24), we readily obtain

$$\frac{\mathrm{d}s}{\mathrm{d}\theta }=-\frac{{\zeta }_3}{\mu }R.$$

(89)

A new differentiation of Equation (89) in fictitious time after replacing R from Equation (24), taking Equations (73)–(75) into account, and again making use of Equations (23) and (24), after straightforward simplifications, yields

$$\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{\theta }^2}+s=\frac{{\zeta }_3^2}{\mu }-({\mathit{P}}^{*}·\mathit{u})s-({\mathit{P}}^{*}·\mathit{v})\frac{\mathrm{d}s}{\mathrm{d}\theta },$$

(90)

which is the equation of a perturbed harmonic oscillator—cf. Equation (15) on p. 70 of [63] with typos amended or Equation (70) on p. 145 of [82].

The differential system made of Equations (72), (75), (76) and (90), where the latter now reads $\mathrm{d}t/\mathrm{d}\theta ={\zeta }_3/\left(s^2\mu \right)$ and ${\mathit{P}}^{*}=\mathit{P}{\zeta }_3^2/\left(s^3{\mu }^2\right)$, is what we call the ORBELTI formulation—for orbital, Euler parameters, Laplace time-regularized, ideal-frame formulation. It was originally proposed in [63] (see also [64]) under the “pirq” denomination with the only change being the replacement of the fictitious-time integration of the hodographic velocity ${\zeta }_3$ with that of the angular momentum $\mathrm{\Theta }$. Variants of the ideal-frame formulation in the framework of projective decomposition using different time transformations have been proposed in [113].

5. Alternative Formulations

An elementary arrangement of the Euler parameters was later proposed in [82] and applies to both ideal-frame variants: either based on pinpointing the eccentricity vector in the ideal frame or in the linearization of the differential equation of the radial distance. The reformulation consists of replacing the standard Euler parameters ${\lambda }_i$, $i=1\dots 4$ by their modified version $g_i=\sqrt{\mathrm{\Theta }}{\lambda }_i$ (see also [81]). In this way, the integration of Equation (72) is replaced with the integration of the variations

$$\frac{\mathrm{d}\mathbf{g}}{\mathrm{d}\theta }=\frac{1}{2}({\mathit{P}}^{*}·\mathit{v})\mathbf{g}+\frac{1}{2}({\mathit{P}}^{*}·\mathit{n})\left(\begin{array}{cc}\hfill {\mathbf{0}}_{\mathbf{2}}& \hfill \mathbf{m}\\ \hfill -\mathbf{m}& \hfill {\mathbf{0}}_{\mathbf{2}}\end{array}\right)\mathbf{g},\mathbf{g}={(g_1,g_2,g_3,g_4)}^{\mathrm{T}};$$

(91)

cf. Equations (65)–(68) of [82].

The evaluation of Equation (91) is just slightly more demanding than that of the variations of the standard Euler parameters in Equation (72). Still, it remains competitive in a numerical integration scheme because it makes dispensable the integration of the variation of the total angular momentum in Equation (28), which now becomes a derived quantity, namely $\mathrm{\Theta }={\sum }_{i=1}^4{g}_i^2$. Therefore, the dimension of the differential system is reduced to a minimum while retaining all the merits of the ideal-frame, nonsingular formulation. Remark that at each integration step, the computational time in practical scenarios will primarily be absorbed by the evaluation of the physical model originating the perturbing acceleration rather than by the evaluation of the rest of the equation terms specific to the formulation.

Additionally, it is well-known that the incorporation of the total energy in the time transformation can be beneficial in the stabilization of the integration. This approach, which was originally taken in [77] and then followed by different authors [39,65,80], leads to the use of non-osculating elements as well as useful (non-osculating) time elements. When this latter approach is taken, the time elements discussed in Section 4 are trivially upgraded by replacing the Keplerian energy by the total energy in the trail of the time element introduced originally in [80] and re-derived here in Appendix A for completeness.

6. Conclusions

The basic approaches to the Hansen ideal-frame formulation of perturbed Keplerian motion have been briefly revisited with an emphasis on the use of osculating elements. For time-regularized formulations, we dealt fundamentally with a Sundman-type transformation that turns the true distance from the departure point—the fast, in-track variable of the ideal-frame approach—into the fictitious time of the differential equations. The use of the latter stems naturally from the classical, physical-time variation equations, which are efficiently reformulated into fictitious time when using this time transformation. More sophisticated time transformations involving the total energy and relying on the use of non-osculating variables were originally presented in the context of the canonical set of Scheifele’s (true anomaly) Delaunay-similar variables, whose thorough derivation is provided as an Appendix and which we prefer to refer to as Hill-similar variables.

In the process of time-regularizing the ideal-frame variations, we derived a constant and a linear, nonsingular, fictitious-time element that are new to the best of our knowledge and may be useful for increasing the position accuracy of numerical solutions of perturbed Keplerian motion. In this task, we found notable help in the mastery of Deprit, who provided the physical-time variation of the mean distance from the departure point in terms of the ideal elements for the first time and whose seminal paper on the topic should be mandatory reading for both professors and students involved in classical celestial mechanics and astrodynamics courses.

While the discussed ideal-element formulations are mostly equivalent from a dynamical point of view and hence should be amenable for analogously efficient implementation, we did not enter the topic of which one should be preferred in the propagation of orbits pertaining to specific dynamical configurations. Certainly, the feasibility of applying simplification strategies in the transformation of an algorithm into software code is a facet of special perturbation methods that cannot be ignored in the selection of an orbit-propagation approach for particular purposes. The technicalities involved in coding the different algorithms as well as the mandatory tests that may validate them when combined with existing numerical solvers are delayed to a sequel.