Subject headings: celestial mechanics—astrometric error—space situational awareness—geeky spook fun

The XML version of this document is available on the web at
http://www.alpheratz.net/murison/papers/LongitudinalErrorDegradation/LongitudinalErrorDegradation.xml

The PDF version of this document is available on the web at
http://www.alpheratz.net/murison/papers/LongitudinalErrorDegradation/LongitudinalErrorDegradation.pdf

1  Motivation

Suppose, by whatever means, we determine the position of a planetary satellite (artificial or natural) by astrometric observations from an orbital platform. The position coordinates will be in error, in general with some coordinate components worse off than others. We are interested here in the degradation of those errors with time, assuming no further observations (for the time being). In particular, since the motion of an orbiting body is mainly in the azimuthal, or longitudinal, direction, we will examine the degradation with time of the initially-determined errors in the longitudinal coordinate. If we assume that the orientation of the satellite orbit plane is known, we can perform this calculation analytically by reducing it to a much simpler two-dimensional problem.

In actual practice, all parameters (orbital elements) of the satellite orbit will be determined from the observations. The approximations we make here, making use of a somewhat unrealistic mathematical device, nevertheless allow us to accomplish the specific goal of deriving an analytic approximation for the degradation with time of the precision ${\sigma }_{\theta }\left(t\right)$ of the longitudinal angle $\theta$. As a byproduct, we will also derive the satellite (or target) semimajor axis $a$ and mean motion $n$, in terms of the parallax angle $\psi$ resulting from a set of observations constituting a normal point, as well as upper bounds on the errors in $a$ and $n$.

2  Definitions and Useful Relations

Let two ideal point masses (central body and satellite) be $M$ and $m$, and define the mass parameter

The angular momentum of this gravitational two-body problem per reduced mass $\frac{Mm}{M+m}$ is

where $\overrightarrow{r}$ is the position vector of $m$ with respect to $M$, and $\overrightarrow{v}=\frac{d\overrightarrow{r}}{\mathit{dt}}=\dot{\overrightarrow{r}}$. One can show that, for elliptical orbits, the magnitude is

The magnitude of the relative position vector is, in terms of the true anomaly $\theta$ (see Figure 1),

where

is the semilatus rectum. For elliptical orbits, $p=a(1-e^2)$. Another useful relation is

which follows from the conservation of angular momentum.

Next, Kepler's third law may be stated in the form

where $n=\frac{2\pi }{T}$, $T$ being the orbital period, is the mean motion.

Finally, it will be useful to note a certain half-angle formula for the tangent. Since

and

we obtain, using first (9) and then (8), the identity

As an aside, a second identity then follows from multiplying numerator and denominator of (10) by $1-\cos \psi$,

3  Simplified Orbit Geometry and Formal Errors of the Parallax Angle and the Longitude

3.1  Error in the Parallax Angle in Terms of the Observation Error

Let the single-observation measurement error be ${\sigma }_{\phi }$, where $\phi$ is, or is related to, the observed parallax angle (perhaps projected onto some axis on the plane of the sky) between the target and a background star. Then the error ${\sigma }_{\psi }$ in the projected parallax angle $\psi$ (see Figure 1) is

where $\eta$ is a coefficient whose magnitude depends on the details of the observations, being generally proportional to (and in fact roughly equal to) the number of observations that determine a normal point.

The particular parallax angle we are interested in here lies in the orbit plane of the target satellite. Hence, we refer to it as the projected parallax angle. The true measured parallax angle is determined from two or more single observations and in general makes some angle $\zeta$ on the sky with respect to the target orbit plane. The errors on $\psi$ will be larger than the true measurement errors ${\sigma }_{\phi }$; hence ${\sigma }_{\psi }$ should be interpreted as an upper bound. We will assume here, as a calculational device, that we are able to project the measured parallax angle onto the target orbit plane, that the orientation of the target orbit plane is known, and that projected angle, $\psi$, is the fundamental quantity used in the calculations. This will allow us to derive analytically an expression for the degradation with time of the errors ${\sigma }_{\theta }$ in the longitudinal coordinate.

3.2  Orbit Geometry and the Error in the Longitudinal Coordinate

For simplicity, let us assume that the observing platform is on a circular orbit of radius $b/2$. Thus, $b$ is the maximum possible baseline for the projected parallax angle $\psi$. The observation target is on some orbit described by the orbital elements $\{a,e,\iota ,\mathrm{\Omega },\omega ,\theta \}$, where $\theta$ is the true anomaly. See Figure 1. We shall use "true anomaly" and "longitudinal coordinate" interchangeably.

Figure 1:  Orbital elements $\{\mathrm{\Omega },\iota ,\omega \}$ of the satellite orbit (blue curve), maximum baseline $b$ of the observing platform (red), observing platform orbit (purple curve), and projected parallax angle $\psi$. The projection of the observation chord of length $\gamma b$ along the baseline is $\gamma b\cos \zeta \mathrm{.}$ The distance from the chord projection midpoint to the target is $d$, which is determined by the normal-point measurement of the parallax $\psi$. The triangles involving $d$, $r$, and the half-baseline $b/2$ are magnified and illustrated in Figure 2. This geometry enables us to approximate analytically the error ${\sigma }_{\theta }\left(t\right)$.

In order to quickly obtain a result, let us further assume the target object ("satellite") is also on a circular orbit, of constant radius $r\left(\theta \right)=a$. Then, using (7), we have

Let $t_0$ be the time at which the target position is determined (i.e., the time associated with the normal point resulting from a set of two or more closely-spaced observations). Then the true anomaly at subsequent times reduces to the simple linear relation

For a noncircular target orbit, $\theta$ and $t$ are related via the Kepler equation, which is transcendental and therefore must be solved iteratively. We will consider eccentric orbits in another paper. With the much simpler eq. (14), the formal error in $\theta \left(t\right)$ follows from propagation of errors, which yields

with the error ${\sigma }_n$ in the mean motion yet to be determined. It is not strictly true that ${\sigma }_n$ is independent of ${\sigma }_{{\theta }_0}$, so that in fact there will be a cross term in (15). We shall blissfully ignore it. Thus, from (15) we see that the error in the longitudinal coordinate initially has the value ${\sigma }_{{\theta }_0}$ associated with the normal point at $t_0$, grows quadratically with time for a short while, then asymptotically grows linearly with time. The explicit relations for (15) are developed below in section 5.3.

4  Calculation of the Target Orbit Elements from the Parallax Angle

4.1  Geometry

Let the baseline for the normal point observations be some chord of length $\gamma b$, $\gamma \le 1$. The projection onto the maximum baseline $b$ is then $\gamma b\cos \zeta$, where $\zeta$ is the angle between the chord and the maximum baseline in the target orbit plane. (See Figure 1.)

Figure 2:  The triangles from Figure 1 involving $d$, $r$, and the half-baseline $b/2$.

As a further simplification, assume that the observation platform orbit plane is perpendicular to the line segment of length $d$ between the chord projection midpoint on the baseline and the target at the time(s) of position determination. (That is, $d$ is parallel to the observation platform angular momentum vector.) Furthermore, assume that the position normal point is determined quickly enough relative to the orbit period timescales that the relative orientation of $b$ and $\overrightarrow{r}$ does not change appreciably over the course of the normal point observations. This last assumption will often be quite unreasonable in practice, but for now we adopt it in order develop simplified equations that are designed to be a starting point for more realistic calculations later. The approximate case developed here can be used for conservative, back-of-the-envelope purposes. The in-plane parallax angle $\psi$ with respect to the background stars, corresponding to the (projected) chord, is then (Figure 2)

4.2  Orbit Radius

As we can see from Figure 2, the length of the perpendicular segment $d$, which we may use as an intermediary for calculating the target orbit radius, is

Define $ϵ\equiv \frac{b}{2a}$, the ratio of the platform and target orbit radii. Then

Hence, making use of (10), we have, from (16) and (18), the result

which gives the target orbit radius in terms of the observed parallax angle projected onto the target orbit plane.

4.3  Mean Motion

From (7), the mean motion is $n=\sqrt{\mu /a^3}$. Substitute (19) for $a$ and we have the mean motion in terms of the projected parallax angle,

Upon expanding on $\psi$ and on $ϵ$, we obtain

5  Calculation of the Target Orbit Element Errors

5.1  Orbit Radius Error

The error ${\sigma }_a$ of the determination of the target orbit radius $a$, due to the imperfect knowledge of the projected parallax angle $\psi$ from the normal point observations, is

From (19), we discover the simple result

Hence,

Now, in reality, the errors ${\sigma }_{\psi }$ must be viewed as constituting an approximate upper bound to the actual measurement errors ${\sigma }_{\phi }$. Hence, the errors in semimajor axis calculated from (24), as well as the corresponding errors in mean motion derived in the next section, are likewise an approximate upper bound.

5.2  Mean Motion Error

Propagation of errors for ${\sigma }_n$ yields

From (13), we have

Making use of (26), and (23), we therefore have, simply,

5.3  Longitude Error, and its Degradation over Time

Let us return to consideration of the error in the longitude coordinate from section 3.2. Putting (27) into (15) yields

Use (13) for $n$ and then (19) for $a$ (or, directly, (20) for $n$) to get

Expanding on $\psi$ and on $ϵ$, we find

Thus, near the time of the normal point, the error in the longitudinal coordinate grows quadratically with time, while asymptotically it grows linearly with time. See Figure 3.

Figure 3:  Illustrating the short-term (${\sigma }_{\theta }\propto t^2$, blue curve) and long-term (${\sigma }_{\theta }\propto t$, black line) behavior of the longitudinal coordinate error (red curve).

6  A Numerical Example

Suppose the observing platform is in a 900 km Sun-synchronous orbit ($r\approx 7271\mathrm{km}$, $T\approx 1.7\mathrm{hr}$) about the Earth, and that the target object is another satellite in geosynchronous orbit ($r\approx 42240\mathrm{km}$). Thus, $b\approx 14540\mathrm{km}$ and $ϵ\approx 0.172\approx \frac{16}{93}$, while $\mu \approx 3.99\times {10}^{14}\frac{m^3}{s^2}$. Assume a measurement error ${\sigma }_{\psi }\approx 1\mathrm{mas}\approx 4.85\times {10}^{-9}\mathrm{rad}$, and that ${\sigma }_{{\theta }_0}={\sigma }_{\psi }$. Furthermore, let $\mathrm{\gamma }\cos \zeta =0.1$. Then $\psi \approx 6.2\times {10}^{-4}\mathrm{rad}\approx 127\mathrm{arcsec}$, and the mean motion is $n=15\mathrm{arcsec}/\sec$.

The error in the calculated target orbit radius is, from (24), ${\sigma }_a\approx 333m$. The error in the calculated mean motion is, from (27), ${\sigma }_n\approx 1.2\times {10}^{-4}\mathrm{arcsec}/\sec$. Figure 4 is a plot of the resulting error ${\sigma }_{\theta }$ in the longitudinal coordinate.

Figure 4:  Longitudinal coordinate error ${\sigma }_{\theta }$ for the numerical example, showing the dependence on time and on the geometric factor $\mathrm{\gamma }\cos \zeta$. The vertical scale is mas. "g factor" is $\mathrm{\gamma }\cos \zeta$.