Subject headings: celestial mechanics—orbit determination—SSA—methods: analytical

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1  Introduction

Orbit determination from astrometric observations is in general an "exceedingly difficult problem" (I. Newton, Principia Book III, Props. 40 & 41 — see [3], [4]). Over the centuries, several approaches for initial determination and subsequent orbit improvement — two very different processes — have been developed (and discarded), the most popular in the modern era being the straightforward method of Laplace and the elegant but somewhat more-complicated method of Gauss. Orbit improvement, by nature dealing with small quantities, can be linearized and thus is amenable to linear least squares methods. Orbit determination, on the other hand, is highly nonlinear, hence the exceeding difficulty. Thus, neither classical method, nor their many variations, seem particularly attractive, especially for gaining analytical insight into the characterization of the orbital parameter errors.

Our primary interest is the orbit parameter errors, not so much the determination of the parameters themselves. Thus, here we make an attempt to bypass classical orbit determination and look for an approach that can instead make use of certain approximations to the relative position and transverse velocity vectors. Given the modern possibility for spectroscopic instruments in space, we also investigate what may additionally be gained from radial velocity observations.

For reference, the rest of this section states without proof certain elementary two-body orbit relations we shall later find useful, in this and subsequent work. In Section 2 we write the distance and velocity vectors of a target body with respect to an orbiting observation platform. Section 3 deals with a complication in performing series expansions, due to the Kepler equation. In Section 4 we get down to business and approximate the relative (i.e., orbit-orbit) distance and velocity vectors which are allowed by certain assumptions such as small eccentricities, inclinations, and ratio of orbit radii. In Section 5 we are ina position to accomplish our goal: propagate the observational errors through the equations and attempt to characterize certain target orbit parameter errors analytically.

1.1  Two-Body Position Vector

An unperturbed two-body elliptical orbit radius vector (see Fig. 1) can be written in the form

where $E\left(t\right)$ is the eccentric anomaly, $a$ is the semimajor axis, $e$ the orbital eccentricity, $\iota$ the orbital inclination, $\mathrm{\Omega }$ the longitude of the ascending node, and $\omega$ the argument of pericenter.

Figure 1:  The two-body radius vector $\overrightarrow{r}$, and its geometric relation to the true anomaly $\theta$ and the eccentric anomaly $E$.

The position/shape vector is

and the orbit orientation matrix that acts upon $\overrightarrow{q}$ (and thus orients it in 3-space) is

where

is the composition of orthogonal rotation matrices

which individually rotate a vector counterclockwise around the respective x or z coordinate axis of an inertial frame. (Alternatively, these rotate the coordinate frame clockwise about the respective axes.) This formulation cleanly separates the orbit scale $a$, the shape of the orbit plus the position of the particle at a given time $\overrightarrow{q}\left(t\right)$, and the orientation of the orbit in space $\mathbf{𝒬}\left(\mathrm{\Omega },\iota ,\omega \right)$. See Fig. 2.

Figure 2:  The orbital elements $\{\mathrm{\Omega },\omega ,\iota \}$ orient a two-body orbit in space. $\overrightarrow{h}$ is the angular momentum vector.

The matrix $\mathbf{𝒬}\left(\mathrm{\Omega },\iota ,\omega \right)$ is orthogonal, since it is composed of orthogonal matrices. Hence, $\mathbf{𝒬}{\left(\mathrm{\Omega },\iota ,\omega \right)}^{\top }\mathbf{𝒬}\left(\mathrm{\Omega },\iota ,\omega \right)=\mathbf{ℐ}$, where $\mathbf{ℐ}$ is the unit matrix. The magnitude of the radius vector can therefore formally be derived as

where we have made use of the fact that

where $\mathrm{𝒜}$ and $\mathrm{ℬ}$ are matrices or a vector and a matrix. From the definition (2), $q^2={(1-e\cos E)}^2$. Thus, in terms of the eccentric anomaly, the scalar radius is especially simple:

1.2  Relating True and Eccentric Anomalies to Each Other and to Time

The two independent angular variables $\theta \left(t\right)$ and $E\left(t\right)$, where $\theta \left(t\right)$ is the true anomaly, are functions of time through the Kepler equation (e.g. Danby [2], Colwell [1])

where $n$ is the mean motion $n=2\pi /T$ (with $T$ the orbital period), also given by $G(M+m)=n^2{a}^3$, and $\tau$ is a time of pericenter passage. The Kepler equation is transcendental and hence cannot be inverted to solve for $E\left(t\right)$ in closed form. Solving the Kepler equation is an entire subject in and of itself (see e.g. [1]).

Transformations between true and eccentric anomalies can be performed via the standard geometric relations

and their inverses

1.3  Two-Body Velocity Vector

Assume the orbits are confocal unperturbed Newtonian two-body orbits. Then the eccentric anomalies are the only time-dependent variables, and from (1) the velocity vector is

where it can be shown that

by (9). Thus, we write

where

2  The Orbit-Orbit Relative Position and Velocity Vectors

2.1  Relative Position Vector

Consider a target orbit with orbital elements (interchangably, orbital parameters) $\{a_g,e_g,{\iota }_g,{\omega }_g,{\mathrm{\Omega }}_g,{\tau }_g\}$, where ${\tau }_g$ is a time of pericenter passage and the others are the usual orbital parameters; and a platform, or observer's, orbit with elements $\{a_p,e_p,{\iota }_p,{\omega }_p,{\mathrm{\Omega }}_p,{\tau }_p\}$. The instantaneous distance vector $\overrightarrow{R}\left(t\right)$ from observer to target is then a simple subtraction of two-body position vectors,

where the subscript $g$ corresponds to the target and the subscript $p$ refers to the observing platform. If we scale by the semimajor axis of the target orbit, letting

then

2.2  Magnitude of the Relative Position Vector

Recalling (7), we note that the magnitude of the relative distance vector (19) is

where

is a symmetric matrix and ${\mathbf{𝒬}}_k=\mathbf{𝒬}\left({\mathrm{\Omega }}_k,{\iota }_k,{\omega }_k\right)$, ${\overrightarrow{q}}_k=\overrightarrow{q}\left(E_k\left(t\right)\right)$. But

Hence, we arrive at the useful result

Performing the matrix multiplications, we find that we can write

where the coefficients $B_k$, which contain only orbit orientation parameters which are constant in time (for Keplerian two-body orbits), are of the form

with

2.3  On Transverse Observed Quantities

The observed quantities are, transversally, two astrometric angles specifying position on the sky and, radially, perhaps a radial velocity or even a ranging measurement. Let the two astrometric angles be denoted $\{\beta ,\phi \}$, where $\phi$ is an azimuthal coordinate and $\beta$ is the complement of a polar axis angle (Fig. 3).

Figure 3:  The astrometric angles $\{\beta ,\phi \}$.

These could, for example, be apparent right ascension and declination. The apparent astrometric angles are then related to the Cartesian components of the relative distance vector by

The unknown parameters are the orbital elements of the target, $\{a_g,e_g,{\iota }_g,{\omega }_g,{\mathrm{\Omega }}_g,{\tau }_g\}$. Upon substituting eqs. (2) and (3) for the corresponding target and platform orbital bodies, we see that the relative distance vector (17) becomes complicated. It is apparent from (1)-(3), (10), (17), and (29) that the inversion to recover the target orbital elements from observed astrometric angles is nonlinear and thus, generically, analytically intractable.

2.4  Relative Velocity Vector

The relative velocity vector is $\overrightarrow{V}\left(t\right)\equiv \frac{\mathrm{d}\overrightarrow{R}\left(t\right)}{\mathrm{d}\mathit{t}}$, which by (17) becomes

Now, the mean motion is related to the semimajor axis by $n=\sqrt{\frac{G(M+m)}{a^3}}$ (i.e., Kepler's third law). If we assume the target and platform masses are negligible compared to the central mass (e.g., Earth), we therefore can write

Thus, making use of (14), we can write (30) in dimensionless form as

where time is scaled by the orbital period of the observing platform and distance is scaled by the semimajor axis of the target orbit. We obtain the $\frac{\partial \overrightarrow{q}}{\partial E}$ from (16). The instantaneous radial velocity can be calculated from

either by using (17) and (32) directly or, equivalently, by differentiating (23) with respect to time. Thus,

with $\frac{\partial {\overrightarrow{q}}_x}{\partial E_x}$ from (16).

2.5  Remark on Radial Velocity Observations

In reality, passive radial velocity measurements actually measure the Doppler shift of spectral lines with respect to their rest wavelengths, not necessarily the relative orbit-orbit radial velocity directly. For objects in Earth orbit, those measurements will be contaminated by, among other things, the Doppler signal due to the relative motion between the target and the Sun at the time of observation. The Earth-Sun signal certainly needs to be included in the radial velocity equation at some point. However, its importance depends on orbit type and viewing geometry, and even then for the most part we need not worry too much about it for present purposes because the current analysis concerns only how the (target orbit) parameter errors behave in comparisons where the Earth-Sun signal (mostly) cancels out in the comparisons.

For reference, the magnitude of the two-body radial velocity (with respect to the orbit focus) is, from (9) and (14),

where

is the magnitude of the circular velocity at orbital radius $a$. The eccentricity of Earth's orbit is $0.0167$. Thus, the Earth-Sun radial velocity magnitude peaks twice per year with a maximum value of $497\mathrm{m}/\mathrm{s}$. The circular velocity of a geosynchronous orbit ($a=42,241\mathrm{km}$) has magnitude $3072\mathrm{m}/\mathrm{s}$. The circular velocity at Sun-synchronous low-Earth orbit ($a=7271\mathrm{km}$) has magnitude $7400\mathrm{m}/\mathrm{s}$.

For looking at objects in geosynchronous orbit the Earth-Sun signal changes very slowly compared to target orbital periods — much less the time spans associated with clusters of observations — and targets are likely viewed in the anti-Sun direction, so the relative phases within an observation set are essentially always the same. Hence the Earth-Sun Doppler contamination is a (nearly) constant offset which for current parameter error characterization purposes we can without consequence assume has already been removed from the "observations" (along with a gazillion other effects we'd have to worry about in the real world). In addition, at the other extreme which covers transfer orbits the relative radial velocity will in most cases swamp the Earth-Sun signal, so the latter is a small quantity we can also safely ignore for now (as well as, again, assume has been removed from the error analysis "data"). Thus, for the present analysis, we shall consider only the uncontaminated two-body orbit-orbit relative velocity.

3  On Series Expansions and the Incorporation of Time into the Expansions

Let us define a bit of nomenclature. Consider series expansions to order $n$ on the $m$ small quantities $u=\{u_1,u_2,\dots ,u_m\}$ (for this study, $u=\{\alpha ,{\psi }_p,{\iota }_g,e_g,e_p\}$), where polynomials $f$, in small-parameter monomials, of the form

result. The polynomial sum is over a finite number of m-tuples $\beta =({\beta }_1,{\beta }_2,\dots ,{\beta }_m)$. The order of a monomial ${\zeta }^{\beta }$ is the sum of the orders of the small parameters $u$ that form the monomial. We will denote this by $\left|\beta \right|={\beta }_1+\cdots +{\beta }_m$. For example, ${\alpha }^2{e}_g$ is a monomial of order $\left|\beta \right|=\sum _{j=1}^m{\beta }_j=3$. The coefficients $A_{\beta }$ of the monomials, due to the complexity of the equations we are considering, will at times be quite messy.

Performing simultaneous series expansions of eqs. (19), (23), (32) and (33) on the small quantities to a given order $n$ is relatively straightforward, if tedious. However, there is a complication. Due to the fact that orbital position is not a linear function of time, one must take into account the Kepler equation (10) that relates time to orbital position. As we shall see, this creates difficulties. Since, depending on circumstances, the Kepler equation can contain a small quantity (the orbital eccentricity), it must be incorporated into the equations that are to be expanded in a series, wherever the eccentric anomaly $E\left(t\right)$ appears, and before the series expansion is performed. Furthermore, the velocity requires time derivatives and thus the incorporation of (14) for $\frac{\mathrm{d}\mathit{E}}{\mathrm{d}t}$, which itself is a function of $E\left(t\right)$ and contains the eccentricity and must therefore also be included before the series expansion takes place.

Now recall the Kepler equation (10). Assuming small eccentricity, one can successively approximate $E\left(t\right)$, starting with $E_0\left(t\right)=n(t-\tau )\equiv M\left(t\right)$, to easily develop a series expansion. Thus, for example, one finds

which can also be rearranged into an equivalent Fourier expansion,

Fortunately, to first order we just have

which suffices much of the time — but not, as we shall discover, when dealing with the velocity vector.

For use with the velocities, where $\frac{\mathrm{d}}{\mathrm{d}t}E\left(t\right)$ makes an appearance, the expansion of (14) is

Upon inclusion of the expansion (38) of $E\left(t\right)$, this becomes the useful result

4  Approximations for the Case of a Dominant Transverse Motion

Equations (19), (23), (32) and (33) in all their gory detail are for the general case less than illuminating, so let us assume certain parameters to be small and then look for better analytic utility and hence some useful insight. Suppose we adopt the case of a target object in high orbit (e.g., geosynchronous) and the observing platform in low orbit. In such a case the transverse relative motion dominates over the relative radial velocity for almost all oribital positions. Furthermore, we can assume a Sun-synchronous observation platform orbit to maximize the advantages for observing geosynchronous objects. Thus, the eccentricities are small, the inclination of the target orbit is small, the inclination of the platform orbit is near ${\iota }_p\approx \frac{\pi }{2}$, and the ratio of semimajor axes $ϵ\equiv a_p/a_g$ is small. As we shall see below in Section 4.2, radial velocity observations do not contribute at first order. In other circumstances — for example highly eccentric target orbits, or transfer orbits from LEO to GEO or from suborbital to orbital — the roles can be reversed and the radial velocity observations are of primary importance. (See Paper II.)

4.1  The Relative Distance Vector and Vector Magnitude

To first order in the small quantities $\{\alpha ,{\psi }_p,{\iota }_g,e_g,e_p\}$, where ${\psi }_p\equiv \frac{\pi }{2}-{\iota }_p$, we find, upon substitution of (38) into (23) and subsequent series expansion, a simple result for the magnitude of the relative distance vector:

Similarly, the unit relative distance vector $\hat{R}\left(t\right)\equiv \frac{\overrightarrow{R}\left(t\right)}{\left|\overrightarrow{R}\left(t\right)\right|}$, calculated from (19), becomes

at zeroth order and

at first order. We will assume the platform orbital parameters, $\{a_p,e_p,{\iota }_p,{\omega }_p,{\mathrm{\Omega }}_p,{\tau }_p\}$, including associated errors, are all known quantities. The unknown parameters, whose behavior we wish to characterize, are thus $\{\alpha ,e_g,{\iota }_g,{\omega }_g,{\mathrm{\Omega }}_g,{\tau }_g\}$.

Unfortunately, we see in eqs. (43)-(45) that several of the unknown parameters, namely the longitude of the ascending node, the argument of pericenter, and — via $E_g\left(t\right)$ and the Kepler equation — the mean motion (or, equivalently, the semimajor axis, since $G(m+M)=n^2{a}^3$) are the arguments of trig functions, making it very difficult to solve for the unknowns analytically. We can expand the trig terms and isolate individual sines and cosines of ${\mathrm{\Omega }}_g$ and ${\omega }_g$ and ${\tau }_g$, but the terms involving $n_{g}t$ appear to be a hopeless obstacle. This is very disappointing. However, if we assume a mean motion and give up for the moment solving for it, then in principle we should still be able to characterize analytically the other parameters and, more importantly, their associated errors. Thus, all is not lost and we shall continue with our calculations.

4.2  The Relative Velocity Vector

We now calculate the relative velocity vector $\overrightarrow{V}\left(t\right)=\frac{\mathrm{d}\overrightarrow{R}\left(t\right)}{\mathrm{d}t}$ to several orders of expansion. We go beyond first order because, as we shall see below, the appearance of the six orbital parameters does not occur as a complete set at first order but is distributed to terms beyond first order.

One further complication requires attention. From (31) we see that, roughly speaking, since the target orbit semimajor axis is large compared to that of the platform, the azimuthal motion $\frac{\mathrm{d}}{\mathrm{d}t}{E}_g\left(t\right)$ of the target body around the gravitational center becomes a small quantity. Thus, for the platform body we have directly from (41) that

while for the target body we instead have

which explicitly contains the small-parameter multiplier ${\alpha }^{3/2}$ from (31). Upon substitution of (38) for $E_g\left(t\right)$ in (47), one must be careful to not make the mistake of treating $n_g$ as the small parameter $n_p{\alpha }^{3/2}$ inside the arguments of the trig functions of (47) and expanding on them, since the orbital phase $n_g\left(t-{\tau }_g\right)$ is not, in general, small.

To first order, then, we find, after substitution of (38) for $E\left(t\right)$ and (42) for $\frac{\mathrm{d}}{\mathrm{d}t}E\left(t\right)$ into (32) and subsequent series expansion,

Notice that, upon multiplying both sides by $n_p{a}_g$, (48) is just the two-body velocity vector of the platform to this level of approximation. In other words, at this order of approximation it is as if the target object is essentially motionless compared to the relatively rapid movement of the observing platform. Note that this expression is accurate to $O\left({\zeta }^{|3/2|}\right)$, where $\zeta$ is any monomial of the small parameters. It appears that this also means that any available radial velocity observations do not contribute to parameter estimation at the first order, but rather at the $O\left({\zeta }^{|3/2|}\right)$ level and, as we shall now show, beyond. In fact, the next term in the series expansion is

There are no unknown small parameters other than $a_g{\alpha }^{3/2}=a_p\sqrt{\alpha }$, but at this order of approximation the node and pericenter parameters of the target orbit come into play, as well as ${\tau }_g$ via $E_g\left(t\right)$ and the Kepler equation (10).

We find that the second-order terms are

which introduce no further unknown parameters, so we must keep going. The next level of approximation, consisting of $O\left({\zeta }^{|5/2|}\right)$ terms, results in

where we finally begin to address the target orbit inclination ${\iota }_g$ and eccentricity $e_g$, which were the remaining unknowns. Thus, in this transverse-dominant case, radial velocity observations will contribute at the third and fifth half-orders $O\left({\zeta }^{|3/2|}\right)$ and $O\left({\zeta }^{|5/2|}\right)$ of approximation. This is in distinct contrast to the astrometric observations, which contribute to the solution of all six unknowns at the first order.

4.3  Magnitude of the Relative Radial Velocity

Upon substitutions and expansions as in Section 4.2, we find that (33) for the relative radial velocity magnitude yields

at first order. Notice that $e_g$ and ${\iota }_g$ are not present. Further parameters appear at second order:

Here, we have $e_g$ and ${\iota }_g$, as well as ${\psi }_p$. There are no $O\left({\zeta }^{|3/2|}\right)$ terms, but $O\left({\zeta }^{|5/2|}\right)$ terms do appear.

5  Analytical Error Propagation

We are finally in a position to make the attempt to characterize the parameter errors analytically. Consider some function $f\left(\overrightarrow{b}\right)$ of $M$ parameters $b=\left\{b_1,\dots ,b_M\right\}$ with associated parameter errors ${\sigma }_b^2=\left\{{\sigma }_{b_1}^2,\dots ,{\sigma }_{b_M}^2\right\}$, with ${\sigma }_b^2$ the variances of Gaussian error distributions. Assuming the parameter error distributions are uncorrelated, the error distribution of $f$ is then

where the sum is over the parameters $b$. We may apply this propagation of errors to the unit relative distance vector. Write the unit relative distance vector as a function of known parameters $s=\left\{e_p,{\iota }_p,{\omega }_p,{\mathrm{\Omega }}_p,{\tau }_p\right\}$ and unknown parameters $u=\left\{\alpha ,e_g,{\iota }_g,{\omega }_g,{\mathrm{\Omega }}_g,{\tau }_g\right\}$. The connection to the observed quantities is then

with $\overrightarrow{R}\left(s,u;t\right)$ given by (17). Separately, propagation of errors yields

where we abuse vector notation to indicate the three equations corresponding to the three components of $\hat{R}\left(s,u;t\right)$. But from the rhs of (55) (using (54)) we also have

Hence we may equate the "components" of (56) with those of (57), resulting in the set of three equations

We may therefore hope to make some progress in characterizing a few of the unknown-parameter errors by solving (58) for three of the ${\sigma }_u^2$ and assuming values of the parameters.

Let us solve eqs. (58) for the three most useful variances: ${\sigma }_{\alpha }^2$, ${\sigma }_{e_g}^2$, and ${\sigma }_{{\iota }_g}^2$. After some laborious calculation, we find the result