Uniform formulation for orbit computation: the intermediate elements

Abstract

We present a new method for computing orbits in the perturbed two-body problem: the position and velocity vectors of the propagated object in Cartesian coordinates are replaced by eight orbital elements, i.e. constants of the unperturbed motion. The proposed elements are uniformly valid for any value of the total energy. Their definition stems from the idea of applying Sundman’s time transformation in the framework of the projective decomposition of motion, which is the starting point of the Burdet–Ferrándiz linearisation, combined with Stumpff’s functions. In analogy with Deprit’s ideal elements, the formulation relies on a special reference frame that evolves slowly under the action of external perturbations. We call it the intermediate frame, hence the name of the elements. Two of them are related to the radial motion, and the next four, given by Euler parameters, fix the orientation of the intermediate frame. The total energy and a time element complete the state vector. All the necessary formulae for extending the method to orbit determination and uncertainty propagation are provided. For example, the partial derivatives of the position and velocity with respect to the intermediate elements are obtained explicitly together with the inverse partial derivatives. Numerical tests are included to assess the performance of the proposed special perturbation method when propagating the orbit of comets C/2003 T4 (LINEAR) and C/1985 K1 (Machholz).

Appendices

Avoiding secular terms, alternative formulation

When the total energy is negative ($\mathscr {E}<0$), secular terms that appear in the derivatives of the intermediate elements (Eqs. 66–73) may deteriorate the accuracy of the predicted state in long-term propagations. The proposed formulation can be modified in order to overcome this drawback. The idea is inspired by the regularised methods presented in Stiefel et al (1967, chap. 1).

By setting $\beta =1$, we can rewrite Eq. (7) as

$$\begin{aligned} r''=2\bar{\mathscr {E}} r+\mu +r(rF_{r}+2\mathscr {E}_K-2\bar{\mathscr {E}}), \end{aligned}$$

where $\mathscr {E}_K=\mathscr {E}-\mathscr {U}$ is the Keplerian energy and $\bar{\mathscr {E}}$ is the value taken by $\mathscr {E}$ at the initial time of the propagation. The quantity $\bar{\mathscr {E}}$ is a constant which is fixed by the initial position and velocity of the particle. Let us introduce

$$\begin{aligned} \bar{\alpha }=-2\bar{\mathscr {E}},\qquad \alpha =-2\mathscr {E},\qquad \delta \alpha =\bar{\alpha }-\alpha . \end{aligned}$$

The solution of $r''=2\bar{\mathscr {E}} r+\mu$ is given by

$$\begin{aligned} r=r_0u_0(\chi ;\bar{\alpha })+\sigma _0u_1(\chi ;\bar{\alpha })+\mu u_2(\chi ;\bar{\alpha }), \end{aligned}$$

(100)

where we have introduced the universal functions (see Eq. 11):

$$\begin{aligned} u_{n}(\chi ;\bar{\alpha })=\chi ^n\sum _{k=0}^{\infty }(-1)^{k} \frac{(\bar{\alpha }\chi ^2)^k}{(n+2k)!},\quad n\in {\mathbb {N}}. \end{aligned}$$

The derivatives of $u_n$ with respect to $\chi$ do not contain secular terms. Following the same steps as in Sects. 2.3 and 2.4, we obtain

$$\begin{aligned} r_0'&= -ru_1(rF_r-2\mathscr {U}+\delta \alpha ),\\ \sigma _0'&= ru_0(rF_r-2\mathscr {U}+\delta \alpha ),\\ t_0'&= ru_2(rF_r-2\mathscr {U}+\delta \alpha ). \end{aligned}$$

Then, by substituting in Eq. (24) the expression of r in (100) we arrive at the Gaussian equation

$$\begin{aligned} \tan \frac{\nu }{2}=\frac{c\,u_1\left( \frac{1}{2}\chi ;\bar{\alpha }\right) }{r_0 u_0\left( \frac{1}{2}\chi ;\bar{\alpha }\right) +\sigma _0 u_1\left( \frac{1}{2}\chi ; \bar{\alpha }\right) }. \end{aligned}$$

Differentiation with respect to $\chi$ yields

$$\begin{aligned} \nu '=\frac{2}{d}\Bigl [\frac{c}{r}(d-r_0)+\frac{r}{c}(rF_r-2\mathscr {U}) (\bar{\alpha }r_0u_2-\sigma _0u_1)-\frac{\alpha 'r}{2c}(r_0u_1+\sigma _0u_2)\Bigr ], \end{aligned}$$

where

$$\begin{aligned} d=2r_0+r\delta \alpha \,u_2. \end{aligned}$$

The expressions of $r_0'$, $\sigma _0'$, $t_0'$, $\nu '$ reported above do not contain secular terms for $\bar{\mathscr {E}}<0$. Thus, we can select $r_0$, $\sigma _0$, $\alpha$, $t_0$ and four Euler parameters exactly as we did in Sect. 2.6, for the elements of a formulation of the perturbed two-body problem, which will be free of secular terms. The case $d=0$ does not introduce additional singularities to those affecting the intermediate elements (see Sect. 2.6).¹⁴ Finally, we note that the same relation as in (83) for $c^2$ does not hold anymore, and we have to use instead

$$\begin{aligned} c^2=r_0(2\mu -r_0\bar{\alpha })-\sigma _0^2+\delta \alpha \,r^2. \end{aligned}$$

Partial derivatives of $\iota _5$, $\iota _6$, $\iota _7$, $\iota _8$ with respect to position and velocity at the initial time

We show a possible way of deriving formulae (98) and (99). We recall that $\mathbf{e}_r$, $\mathbf{e}_{\nu }$, $\mathbf{e}_z$ are the unit vectors of the LVLH reference frame (see Eqs. 5). From the following relation for the angular momentum vector

$$\begin{aligned} \mathbf{r}\times {\dot{\mathbf{r}}}=h\mathbf{e}_z, \end{aligned}$$

we obtain

$$\begin{aligned} h\frac{\partial {\mathbf{e}}_z}{\partial {\mathbf{r}}}= V-\mathbf{e}_z\frac{\partial h}{\partial {\mathbf{r}}},\qquad h\frac{\partial {\mathbf{e}}_z}{\partial {\dot{\mathbf{r}}}}= R-\mathbf{e}_z\frac{\partial h}{\partial {\dot{\mathbf{r}}}}, \end{aligned}$$

(101)

where V, R are the skew-symmetric matrices defined by

$$\begin{aligned} V(1,2)&=v_3,&V(1,3)&=-v_2,&V(2,3)&=v_1,\\ R(1,2)&=-r_3,&R(1,3)&=r_2,&R(2,3)&=-r_1, \end{aligned}$$

with $r_i=\mathbf{r}\cdot \mathbf{e}_i$, $v_i={\dot{\mathbf{r}}}\cdot \mathbf{e}_i$, $i=1,2,3$, and

$$\begin{aligned} \frac{\partial h}{\partial {\mathbf{r}}}={\mathbf{e}}_z^TV,\qquad \frac{\partial h}{\partial {\dot{\mathbf{r}}}}={\mathbf{e}}_z^TR. \end{aligned}$$

By inserting in Eqs. (101) the expression of $\mathbf{e}_z$ as a function of I, $\varOmega$, that is

$$\begin{aligned} \mathbf{e}_z=\mathbf{e}_1\sin \varOmega \sin I-\mathbf{e}_2\cos \varOmega \sin I+\mathbf{e}_3\cos I, \end{aligned}$$

we find

$$\begin{aligned} \frac{\partial \varOmega }{\partial {\mathbf{r}}}&= -\frac{1}{p\sin I}(\cos L+e\cos \omega ){\mathbf{e}}_z^T,&\frac{\partial \varOmega }{\partial {\dot{\mathbf{r}}}}&= \frac{r\sin L}{h\sin I}\,{\mathbf{e}}_z^T, \end{aligned}$$

(102)

$$\begin{aligned} \frac{\partial I}{\partial {\mathbf{r}}}&= \frac{1}{p}(\sin L+e\sin \omega ){\mathbf{e}}_z^T,&\frac{\partial I}{\partial {\dot{\mathbf{r}}}}&= \frac{r}{h}\cos L\,{\mathbf{e}}_z^T, \end{aligned}$$

(103)

where $L=\omega +f$ is the argument of latitude.

Then, from the relation

$$\begin{aligned} \cos L=(\mathbf{e}_r\cdot \mathbf{e}_1)\cos \varOmega +(\mathbf{e}_r\cdot \mathbf{e}_2)\sin \varOmega , \end{aligned}$$

we can write

$$\begin{aligned} \frac{\partial {L}}{\partial {\mathbf{r}}}=-\frac{\partial {\varOmega }}{\partial {\mathbf{r}}}\cos I+ \frac{1}{r}{\mathbf{e}}_{\nu }^T,\qquad \frac{\partial {L}}{\partial {\dot{\mathbf{r}}}}=-\frac{\partial {\varOmega }}{\partial {\dot{\mathbf{r}}}}\cos I, \end{aligned}$$

(104)

where we have used

$$\begin{aligned} \frac{\partial {{\mathbf{e}}_r}}{\partial {{\mathbf{r}}}}=(I_d-\mathbf{e}_r{\mathbf{e}}_r^T), \end{aligned}$$

and $I_d$ is the $3\times 3$ identity matrix.

The Euler parameters $\iota _5$, $\iota _6$, $\iota _7$, $\iota _8$ at the initial time $t_*$ are written in terms of L, $\varOmega$, I by means of Eqs. (58), in which we set $\varPsi =L$. Then, these expressions are differentiated with respect to $\mathbf{r}$, $\dot{\mathbf{r}}$, and taking into account (102), (103), and (104), we obtain formulae (98) and (99).

Partial derivatives of $\iota _1',\ldots ,\iota _8'$ with respect to intermediate elements

Let us recall that ${{\varvec{\iota }}}=(\iota _1,\ldots ,\iota _8)^T$. We define

$$\begin{aligned} \mathcal{K}_n= & {} (rF_r-2\mathscr {U})\frac{\partial (ru_n)}{\partial {{\varvec{\iota }}}}+ ru_nF_r\frac{\partial r}{\partial {{\varvec{\iota }}}}-\frac{1}{2}(\sigma P_r+hP_{\nu }) \frac{\partial b_n}{\partial {{\varvec{\iota }}}}\\&+\frac{b_n}{4}\frac{\partial \iota '_3}{\partial {{\varvec{\iota }}}} +ru_n\Bigl (r\frac{\partial F_r}{\partial {{\varvec{\iota }}}}- 2\frac{\partial \mathscr {U}}{\partial {{\varvec{\iota }}}}\Bigr ),\quad n=1,2,4,5, \end{aligned}$$

where

$$\begin{aligned} u_1=-U_1,\quad u_2=U_0,\quad u_4=U_2,\quad u_5=\frac{\iota _2U_1-\iota _1\iota _3U_2}{c\iota _1}, \end{aligned}$$

and

$$\begin{aligned} b_1&= -\iota _1{\tilde{U}}_2-\iota _2{\tilde{U}}_3-2\mu U_2^2,\\ b_2&= \iota _1(2\chi +{\tilde{U}}_1)+\iota _2{\tilde{U}}_2+\mu ({\tilde{U}}_3-4U_3),\\ b_4&= \iota _1({\tilde{U}}_3-4U_3)+2\iota _2U_2^2+\mu ({\tilde{U}}_5-8U_5),\\ b_5&= \frac{1}{2\iota _1}\Bigl [\frac{r}{c}(\iota _1U_1+\iota _2U_2)-cU_3\Bigl ]. \end{aligned}$$

The desired derivatives take the form

$$\begin{aligned} \frac{\partial \iota '_n}{\partial {{\varvec{\iota }}}}&= \mathcal{K}_n,\quad n=1,2,4, \end{aligned}$$

(105)

$$\begin{aligned} \frac{\partial \iota '_3}{\partial {{\varvec{\iota }}}}&= -2\Bigl (P_r\frac{\partial \sigma }{\partial {{\varvec{\iota }}}} +P_{\nu }\frac{\partial h}{\partial {{\varvec{\iota }}}}+\sigma \frac{\partial P_r}{\partial {{\varvec{\iota }}}}+ h\frac{\partial P_{\nu }}{\partial {{\varvec{\iota }}}}\Bigr ), \end{aligned}$$

(106)

$$\begin{aligned} 2\frac{\partial \iota '_{k+4}}{\partial {{\varvec{\iota }}}}&= \frac{\alpha _k}{r}\Bigl (\frac{\partial h}{\partial {{\varvec{\iota }}}}-\frac{\partial c}{\partial {{\varvec{\iota }}}}+\frac{c-h}{r}\frac{\partial r}{\partial {{\varvec{\iota }}}}+r\mathcal{K}_5\Bigr )+\frac{r}{h}\Bigl (2F_z\frac{\partial r}{\partial {{\varvec{\iota }}}} -F_z\frac{r}{h}\frac{\partial h}{\partial {{\varvec{\iota }}}} +r\frac{\partial F_z}{\partial {{\varvec{\iota }}}}\Bigr )(\beta _kc_{\nu }+\gamma _ks_{\nu })\nonumber \\&\quad \, +\frac{r^2}{h}F_z\Bigl [(\gamma _kc_{\nu }-\beta _ks_{\nu })\frac{\partial \nu }{\partial {{\varvec{\iota }}}} +\frac{\partial \beta _k}{\partial {{\varvec{\iota }}}}c_{\nu }+\frac{\partial \gamma _k}{\partial {{\varvec{\iota }}}}s_{\nu }\Bigr ] +N\frac{\partial \alpha _k}{\partial {{\varvec{\iota }}}},\quad k=1,2,3,4, \end{aligned}$$

(107)

where $c_{\nu }$, $s_{\nu }$ denote $\cos \nu$, $\sin \nu$, respectively, and ${\alpha }_{k}, \beta _{k}, \gamma _{k}$ denote the kth component of the vectors $\alpha , \beta , \gamma$ defined below:

$$\begin{aligned} \alpha =(-\iota _8,\,\iota _7,\,-\iota _6,\,\iota _5),\qquad \beta =(-\iota _6,\,\iota _5,\,\iota _8,\,-\iota _7),\qquad \gamma =(-\iota _7,\,-\iota _8,\,\iota _5,\,\iota _6). \end{aligned}$$

The partial derivatives of r, $\sigma$, c, h, $\nu$ are reported in Sect. 3.1. Moreover, we need the following relations:

$$\begin{aligned} \frac{\partial b_1}{\partial {{\varvec{\iota }}}}&= -\Bigl ({\tilde{U}}_2,\,{\tilde{U}}_3,\,\iota _1{\tilde{U}}_4 +\frac{3}{2}\iota _2{\tilde{U}}_5+4\mu U_2U_4-\chi (\iota _1{\tilde{U}}_3+\iota _2{\tilde{U}}_4 +2\mu U_2U_3),\,\mathbf{0}_5\Bigr ),\\ \frac{\partial b_2}{\partial {{\varvec{\iota }}}}&= \Bigl (2\chi +{\tilde{U}}_1,\,{\tilde{U}}_2,\,\frac{1}{2} \iota _1{\tilde{U}}_3+\iota _2{\tilde{U}}_4+\frac{3}{2}\mu ({\tilde{U}}_5-4U_5)+\chi (b_1-2\mu U_4),\,\mathbf{0}_5\Bigr ),\\ \frac{\partial b_4}{\partial {{\varvec{\iota }}}}&= \Bigl ({\tilde{U}}_3-4U_3,\,2U_2^2,\,\frac{3}{2}\iota _1 ({\tilde{U}}_5-4U_5)+4\iota _2U_2U_4+\frac{5}{2}\mu ({\tilde{U}}_7-8U_7)\\&\quad \, -\chi [\iota _1({\tilde{U}}_4-2U_4)+2\iota _2U_2U_3+\mu ({\tilde{U}}_6-4U_6)],\,\mathbf{0}_5\Bigr ),\\ 2\iota _1c\frac{\partial b_5}{\partial {{\varvec{\iota }}}}&= (\iota _1U_1+\iota _2U_2)\Bigl (\frac{\partial r}{\partial {{\varvec{\iota }}}}-\frac{r}{c}\frac{\partial c}{\partial {{\varvec{\iota }}}}\Bigr )-cU_3\frac{\partial c}{\partial {{\varvec{\iota }}}}+\frac{1}{2}\Bigl (\frac{2}{\iota _1}(c^2U_3-r\iota _2U_2),\,2rU_2,\\&\quad \, r(\iota _1U_3+2\iota _2U_4)-3c^2U_5+\chi [c^2U_4-r(\iota _1U_2+\iota _2U_3)],\,\mathbf{0}_5\Bigr ),\\ \frac{\partial u_5}{\partial {{\varvec{\iota }}}}&= \frac{1}{c\iota _1}\Bigl (-\frac{\iota _2}{\iota _1}U_1, \,U_1,\,\frac{1}{2}[\iota _2U_3-\chi (\iota _1U_1+\iota _2U_2)],\,\mathbf{0}_5\Bigr ) +u_5c\frac{\partial c}{\partial {{\varvec{\iota }}}}, \end{aligned}$$

where $\mathbf{0}_5\in {\mathbb {R}}^5$ is a row vector of null entries. Assuming that $\mathscr {U}$ depends only on $\mathbf{r}$, t (see the remark in Sect. 2.6), we have

$$\begin{aligned} \frac{\partial \mathscr {U}}{\partial {{\varvec{\iota }}}}= \frac{\partial \mathscr {U}}{\partial {\mathbf{r}}}\frac{\partial {\mathbf{r}}}{\partial {{\varvec{\iota }}}}+ \frac{\partial \mathscr {U}}{\partial t}\frac{\partial t}{\partial {{\varvec{\iota }}}}. \end{aligned}$$

(108)

Let us denote by $\mathbf{y}$ either $\mathbf{F}$ or $\mathbf{P}$, and with $y_{\ell }$ the component of $\mathbf{y}(\mathbf{r},\dot{\mathbf{r}},t)$ along one of the directions associated to $\mathbf{e}_r$, $\mathbf{e}_{\nu }$, $\mathbf{e}_z$. Then, we can write

$$\begin{aligned} \frac{\partial y_{\ell }}{\partial {{\varvec{\iota }}}}=\mathbf{e}_{\ell }^T\frac{\partial {\mathbf{y}}}{\partial {{\varvec{\iota }}}} +{\mathbf{y}}^T\frac{\partial {\mathbf{e}}_{\ell }}{\partial {{\varvec{\iota }}}}, \end{aligned}$$

(109)

where

$$\begin{aligned} \frac{\partial {\mathbf{y}}}{\partial {{\varvec{\iota }}}}= \frac{\partial {\mathbf{y}}}{\partial {\mathbf{r}}}\frac{\partial {\mathbf{r}}}{\partial {{\varvec{\iota }}}}+ \frac{\partial {\mathbf{y}}}{\partial {\dot{\mathbf{r}}}}\frac{\partial {\dot{\mathbf{r}}}}{\partial {{\varvec{\iota }}}}+\frac{\partial {\mathbf{y}}}{\partial t}\frac{\partial t}{\partial {{\varvec{\iota }}}}. \end{aligned}$$

(110)

The matrices $\partial {\mathbf{r}}/\partial {{\varvec{\iota }}}$, $\partial {\dot{\mathbf{r}}}/{\partial {{\varvec{\iota }}}}$ are provided in Sect. 3.1, together with $\partial {\mathbf{e}}_r/\partial {{\varvec{\iota }}}$, $\partial {\mathbf{e}}_{\nu }/\partial {{\varvec{\iota }}}$, while $\partial {\mathbf{e}}_z/\partial {{\varvec{\iota }}}$ can be easily obtained from the expression

$$\begin{aligned} \mathbf{e}_z=(2\iota _6\iota _8+2\iota _5\iota _7,\,2\iota _7\iota _8-2\iota _5\iota _6, \,\iota _5^2-\iota _6^2-\iota _7^2+\iota _8^2)^T. \end{aligned}$$

(111)

Also note that

$$\begin{aligned} \frac{\partial {\mathbf{F}}}{\partial {\mathbf{r}}}= \frac{\partial {\mathbf{P}}}{\partial {\mathbf{r}}}-\frac{\partial (\nabla \mathscr {U})}{\partial {\mathbf{r}}},\qquad \frac{\partial {\mathbf{F}}}{\partial t}= \frac{\partial {\mathbf{P}}}{\partial t}-\frac{\partial (\nabla \mathscr {U})}{\partial t}. \end{aligned}$$

(112)

Finally, we have

$$\begin{aligned} \frac{\partial t}{\partial {{\varvec{\iota }}}}=\Bigl (U_1,\,U_2,\,\frac{1}{2}[\iota _1U_3+2\iota _2U_4 +3\mu U_5-\chi (\iota _1U_2+\iota _2U_3+\mu U_4)],\,1,\,0,\,0,\,0,\,0\Bigr ).\nonumber \\ \end{aligned}$$

(113)

Identities for the universal functions

We collect the identities for the universal functions introduced in Eq. (11) that we used to derive some equations of this paper (see Battin 1999, Sects. 4.5, 4.6). For simplicity, we omit the argument $\alpha$ in the universal functions. These formulae are:

$$\begin{aligned}&U_n(\chi )+\alpha U_{n+2}(\chi )=\frac{\chi ^n}{n!},\,\,\, n\in {\mathbb {N}},\\&U_0(\chi )^2+\alpha U_1(\chi )^2=1,\\&U_1(\chi )^2-U_0(\chi )U_2(\chi )=U_2(\chi ),\\&U_0(\chi )U_3(\chi )-U_1(\chi )U_2(\chi )=U_3(\chi )-\chi U_2(\chi ),\\&U_1(\chi )U_3(\chi )-U_2(\chi )^2=2U_4(\chi )-\chi U_3(\chi ), \end{aligned}$$

the double argument identities:

$$\begin{aligned} U_0(2\chi )&= U_0(\chi )^2-\alpha U_1(\chi )^2,\\ U_1(2\chi )&= 2U_0(\chi )U_1(\chi ),\\ U_2(2\chi )&= 2U_1(\chi )^2,\\ U_3(2\chi )&= 2U_3(\chi )+2U_1(\chi )U_2(\chi ),\\ U_5(2\chi )&= 2U_1(\chi )U_4(\chi )+\chi ^2 U_3(\chi )+2U_5(\chi ), \end{aligned}$$

and the differential relations:

$$\begin{aligned}&\frac{\partial U_0}{\partial \chi }=-\alpha U_1,\qquad \frac{\partial U_m}{\partial \chi }=U_{m-1}, \quad m\in \mathbb {N^+},\\&\frac{\partial U_n}{\partial \alpha }=\frac{1}{2}(nU_{n+2}-\chi U_{n+1}),\quad n\in {\mathbb {N}}. \end{aligned}$$