19-681

# The Long-Term Forecast of Station View Periods for Elliptical Orbits111©
2019 copyright. all rights reserved.

Andrew J. Graven Undergraduate, Department of Mathematics, College of Arts and
Sciences, Cornell University, Ithaca, NY, 14853, United States Martin W. Lo
Technologist, Mission Design and Navigation Section, Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, CA, 91109, United
States

###### Abstract

In a previous paper, using ergodic theory, Lo [1] derived a simple definite
integral that provided an estimate of the view periods of ground stations to
satellites. This assumes the satellites are in circular orbits with non-
repeating ground tracks under linear $J_{2}$ perturbations. The novel feature
is that this is done without the propagation of the trajectory by employing
ergodic theory. This accelerated the telecommunications mission design and
analysis by several orders of magnitude and greatly simplified the process. In
this paper, we extend the view period integral to elliptical orbits.

##  1 Introduction

When planning the placement of ground stations and the trajectories of the
satellites communicating with them, it’s often important to know how
frequently and for what duration the two will be able to directly communicate.
This problem amounts to answering the question: for what ratio of total flight
time can we expect the satellite to have line of sight communication with it’s
ground station? Both in the short and long term, the standard method of
calculating this statistic is propagating the satellite trajectory over a
number of orbits, and averaging the visibility time. For shorter term
missions, this sort of trajectory propagation is feasible. However, if the
determination of longer term trends is needed, this technique of trajectory
propagation becomes impractical due to excessive run time. For example,
running the trajectory propagation of a single satellite for 5 integration
years on Matlab’s ode45 takes $\approx 20$ minutes. When this propagation is
needed for a large number of potential satellite trajectories and ground
station placements, say fifty potential satellite trajectories and ten
potential ground station locations, the required compute time is about a week.

An alternative way to determine this average satellite-ground station
visibility ratio is through the use of dynamical systems and statistical
mechanics. A key aspect of statistical mechanics is the determination of the
long term statistical behavior of a dynamical system, in this case, satellite
trajectories. Using this approach, Lo[1] was able to derive a simple definite
integral which yields the expected satellite-ground station visibility ratio
to a high degree of speed and accuracy. However, this came with the caveat
that the formula is only valid for non-periodic, circular, orbits. Employing a
similar technique to that originally used by Dr. Lo, the satellite-ground
station visibility formula has been extended to the much wider class of non-
periodic elliptical orbits.

Each of the mathematical symbols used throughout the paper are explained in-
line and collected in the Notation section at the end of the paper.

##  2 The Long Term Ground Station View Period Ratio, $\boldsymbol{\rho}$

In both the circular and elliptical cases, the integral yields an
approximation of the long-term station view period ratio, $\rho$. This $\rho$
is the ratio of the total time a satellite spends in sight of its ground
station to total flight time. Specifically

$\rho=\lim\limits_{T\to\infty}\dfrac{P(T)}{T}$ (1)

Where, $T$ is the total flight time, and $P(T)$ is the time spent with line of
sight between the satellite and the ground station. So, for a given flight
time $T$, the total line of sight visibility time should be approximated by:

$V(T)=\rho T$ (2)

Thus, given that the limit exists we would expect that as $T$ grows, $V(T)$
should become a better approximation for $P(T)$. That is,

$\lim\limits_{T\to\infty}|V(T)-P(T)|=0$ (3)

Thus, for sufficiently long time periods, we may obtain a good approximation
of $P(T)$ using $\rho$ alone.
For example, for an elliptical orbit with
$a=10,000.14\text{km},\;e=.2,\;i=28.5\degree,\;\lambda=0\degree$, (where $a=$
the major axis, $e=$ the eccentricity, $i=$ the inclination of the orbit and,
$\lambda=$ the latitude of the ground station), we have:

$P(1000\text{ days})=258.7599\text{ days}$ $V(1000\text{ days})=258.7937\text{
days}$ $\text{Percent Error}=.0131\%$

Where $P(T)$ was computed by direct orbit propagation, then summing over all
view periods in that time interval. $V(T)$ was computed as in Eq. (2). $\rho$
was computed as the ratio: $\dfrac{P(T)}{T}$, by direct orbit propagation for
$T=6000$ days (or, approximately 16 years), beyond which the changes to $\rho$
for larger $T$ became negligible.

##  3 The Linear $\boldsymbol{J_{2}}$ Model of Orbits

As a result of the rotation of the Earth about its axis, it is not a perfect
sphere, but an oblate spheroid. Because of this, the equations of motion
governing the motion of an object orbiting it are perturbed. This perturbation
can be quantified through the use of spherical harmonics, yielding a sequence
of coefficients $J_{2},J_{3},...$ the normalized zonal harmonic gravitational
coefficients. $J_{2}$ is by far the most significant of the coefficients, with
$J_{2}=1.083\cdot 10^{-3}$ and $J_{3}=-2.5\cdot 10^{-6}$ (these are the values
for Earth).

###  3.1 The Nonlinear $\boldsymbol{J_{2}}$ Model

Neglecting higher order terms, under the $J_{2}$ perturbation the equations of
motion which govern the motion of a body of negligible mass orbiting a planet
are as follows[2]

$\displaystyle\ddot{x}$ $\displaystyle=-\dfrac{\mu
x}{r^{3}}(1-J_{2}\dfrac{3R_{B}^{2}}{2r^{2}}(\dfrac{5z^{2}}{r^{2}}-1))$ (4)
$\displaystyle\ddot{y}$ $\displaystyle=\dfrac{y}{x}\ddot{x}$
$\displaystyle\ddot{z}$ $\displaystyle=-\dfrac{\mu
z}{r^{3}}(1+J_{2}\dfrac{3R_{B}^{3}}{2r^{3}}(3-\dfrac{5x^{2}}{r^{2}}))$
$\displaystyle r$ $\displaystyle=(x^{2}+y^{2}+z^{2})^{\frac{1}{2}}$

(Where $R_{B}$ and $\mu$ are the equatorial radius and gravitational parameter
of the body respectively)
Note: This is in the non-inertial, body fixed rotating frame (rotating at the
rate of the body’s rotation about its axis).

###  3.2 The Linear $\boldsymbol{J_{2}}$ Model

A good approximation to these equations of motion may be found if we first
pass into the state-space, $\Gamma$, of orbital elements such that

$\displaystyle\Gamma$ $\displaystyle=M\times\Omega\times\omega\text{, where}$
(5) $\displaystyle M$ $\displaystyle=\text{Mean Anomaly,}$
$\displaystyle\Omega$ $\displaystyle=\text{Longitude of the Ascending Node,}$
$\displaystyle\omega$ $\displaystyle=\text{Argument of the Periapsis.}$

Then, linearizing these elements with respect to time, we get the following
formulae approximating their time evolution [2]

$\displaystyle M(t)$ $\displaystyle=M_{0}+\dot{M}t$ (6)
$\displaystyle\Omega(t)$ $\displaystyle=\Omega_{0}+\dot{\Omega}t\text{, mod
}2\pi$ $\displaystyle\omega(t)$ $\displaystyle=\omega_{0}+\dot{\omega}t\text{,
mod }2\pi$

Where their time derivatives are given by

$\displaystyle\dot{M}$
$\displaystyle=\sqrt{\dfrac{\mu}{a^{3}}}[1+3J_{2}\dfrac{R_{B}^{2}}{4a(1-e^{2})^{\frac{3}{2}}}(3\cos^{2}(i)-1)]$
(7) $\displaystyle\dot{\Omega}$
$\displaystyle=-\dfrac{3\sqrt{\mu}J_{2}R_{B}^{2}\cos(i)}{2a^{\frac{7}{2}}(1-e^{2})^{2}}+\Omega_{B}$
$\displaystyle\dot{\omega}$
$\displaystyle=\dfrac{3\sqrt{\mu}J_{2}R_{B}^{2}(4-5\sin^{2}(i))}{4a^{\frac{7}{2}}(1-e^{2})^{2}}$

##  4 The Ergodic Theory of Satellite Coverage

A dynamical system is considered to be ergodic if its trajectory repeatedly
visits almost everywhere (in the measure-theoretic sense) in its state space
[3]. The Birkhoff-Khinchin Theorem states that if a a system is ergodic then
for every $f\in\mathcal{L}^{1}$ (where $\mathcal{L}^{1}$ is the set of
functions which are absolutely Lebesgue integrable) defined on the state space
of that system, the time mean of $f$ is approaches to the space mean of $f$ as
the run time of the system grows.[3] [4] That is, if $\vec{x}(t)$ is the state
of a system, $S$, at a time $t$, then if $f\in\mathcal{L}^{1}$:

$\lim_{T\to\infty}\dfrac{1}{T}\int\limits_{t=0}^{T}f(\vec{x}(t))dt=\int\limits_{S}fd\mu$
(8)

In the case of satellite trajectories, we’re interested in computing the time
mean for the satellite in the region of space in which the satellite has line
of sight communication with its ground station. More specifically, we want to
compute the mean time that the satellite spends in the intersection of the
region of visibility of the ground station, $V$, with the spatial state space,
$S$, of the satellite, where $S$ is defined as the 3-dimensional region of
space that the satellite can travel. Note that $V\cap S$ is the only region in
which the satellite and ground station can feasibly communicate. This is
because in the volume outside $V$, the line of sight communication isn’t
possible; and outside of $S$, the satellite cannot travel. We define this
volume, $V^{*}=V\cap S$ as the region of feasible visibility.

(a) Spatial State Space of Satellite in Elliptical Orbit

(b) Cross Section of Spatial State-Space

(c) Intersection of Spatial State Space with Ground Station Visibility Region

Figure 1: Spatial State Space Visualizations

Figure 1(a) provides the visualization of the geometry of $S$ for a satellite
in an elliptical orbit. Figure 1(b) shows a 2D cross section of the spatial
state space, where $R_{p}$ is the periapsis and $R_{a}$ is the apoapsis.
Figure 1(c) shows $V^{*}=V\cap S$ (in red) for a low latitude ground station.
Note that for circular orbits, $S$ and $V^{*}$ reduce to 2-dimensional
surfaces.

Now, applying the Birkhoff-Khinchin Theorem, we should expect a relationship
between the ratio: $\frac{\text{Vol}(V^{*})}{\text{Vol}(S)}$ and the mean time
spent in that region. However, there are complications resulting from the fact
that the mean density of the position of the satellite in this spatial state
space is not uniform. More specifically, the satellite spends a
disproportionately large amount of its flight time at extreme latitudes and
near the apoapsis of its trajectory. Thus, there isn’t a direct correspondence
between the volume of a region of the satellite’s state space and the time it
spends there. That is, the spatial state space of the satellite is not
ergodic. The bias of more extreme latitudes is shown below with the space
tracks of a circular orbit:

Figure 2: Space Tracks of Satellite in Circular Orbit

However, the linearized orbital element state space, $\Gamma$, of the
satellite defined in Eq. (LABEL:OrbEls) is ergodic. This is because the flow
(trajectory) is linear in time, hence the trajectory spends equal time in each
region of $\Gamma$. Thus,
$\frac{\text{Vol}(V^{*}_{\Gamma})}{\text{Vol}(\Gamma)}$ should approximate the
ratio of time the satellite spends in $V^{*}$. Where $V^{*}_{\Gamma}$ is the
region of visibility for the ground station in $\Gamma$.

##  5 The View Period Ratio Integrals

In each of the cases below, we compute
$\rho=\frac{\text{Vol}(V^{*}_{\Gamma})}{\text{Vol}(\Gamma)}$. However, the
geometry of the region of integration in the space of orbital elements,
$\Gamma$, is sufficiently complicated that direct integration in this space is
difficult. To resolve this, we can pull the integral back into the original
space, $S$, where the geometry is much simpler. Let $\Phi:S\rightarrow\Gamma$
be a change of variables from $\Gamma$ to $S$, then the volume of the region
in Gamma can be computed as an integral in S with the volume element
$|\text{Det}[D\Phi]|$. This yields the desired formulae below.

###  5.1 View Period Ratio for Circular Orbits

$\displaystyle\rho(i,\lambda_{s})$
$\displaystyle=\int\limits_{V^{*}}\dfrac{\cos(\lambda)}{2\pi^{2}\sqrt{\sin^{2}(i)-\sin^{2}(\lambda)}}dLd\lambda$
(9) $\displaystyle\lambda_{s}$ $\displaystyle=\text{Latitude of Ground Station
(Degrees)}$ $\displaystyle i$ $\displaystyle=\text{Inclination of Orbital
Plane (Radians)}$ $\displaystyle\lambda$ $\displaystyle=\text{Latitude}$
$\displaystyle L$ $\displaystyle=\text{Longitude (Radians)}$

###  5.2 View Period Ratio for Elliptical Orbits

$\displaystyle\rho(a,e,i,\lambda_{s})$
$\displaystyle=\dfrac{1}{2\pi^{3}}\int_{V^{*}}\dfrac{r\cos(\lambda)}{a\sqrt{\sin^{2}(i)-\sin^{2}(\lambda)}\sqrt{a^{2}e^{2}-(a-r)^{2}}}dLd\lambda
dr$ (10) $\displaystyle a$ $\displaystyle=\text{Major Axis (Kilometers)}$
$\displaystyle e$ $\displaystyle=\text{Eccentricity}$ $\displaystyle i$
$\displaystyle=\text{Inclination of Orbital Plane (Degrees)}$
$\displaystyle\lambda_{s}$ $\displaystyle=\text{Latitude of Ground Station
(Degrees)}$ $\displaystyle r$ $\displaystyle=\text{Radius from Center of Body
(km)}$ $\displaystyle\lambda$ $\displaystyle=\text{Latitude (Radians)}$
$\displaystyle L$ $\displaystyle=\text{Longitude (Radians)}$

##  6 Numerical Verification

###  6.1 Evaluating the Integral

In this section, we restrict our focus to the numerical evaluation of $\rho$
in the elliptical case. A fast technique for computing the integral for
circular orbits is given by Lo[1]. Furthermore, the elliptical formula may be
used to compute the ratio for circular orbits by setting the eccentricity,
$e<<1$.

As a result of the irregular shape of $V^{*}$ and multiple singularities in
the integrand, accurately evaluating the definite integral over $V^{*}$ has
proven to be a difficult problem. Several different potential techniques for
evaluating the integral are discussed below.

####  6.1.1 Monte Carlo:

Due to it’s speed and ease of implementation, early on Monte Carlo Integration
was method of choice for evaluating the integral. For early verification of
the formula’s accuracy it was a great tool, producing decently accurate
results. Unfortunately, by the nature of Monte Carlo, it is difficult to get
beyond two to three or so decimal places of accuracy. Which is problematic if
the ratio being calculated begins in the second decimal place, as many did. It
is possible to increase accuracy by increasing the number of trials but this
comes at the cost of the calculation becoming impractically long. Monte Carlo
was eventually abandoned as a method of evaluation for these reasons.

####  6.1.2 Direct Integration:

Another technique which was investigated is direct definite integration over
the region using casework, with a different case for each distinct shape of
the region. The shape of the region can be affected by many factors, such as
whether it intersects the edges of the state space, and the shape of the
region of visibility. Initial results for this technique were good. Where the
casework method worked, it worked well, yielding five to six decimal places of
accuracy in less than a second. However, there were many cases for which the
region was simply too complex to admit a comprehensible definite integral form
(such as for nearly polar ground stations). In addition, this technique
wouldn’t be very portable to other geometries of ground station visibility
regions due to the amount of case work required.

####  6.1.3 Direct Integration with Indicator Function:

Another option is integration over the entire state space with an indicator
function on the region of interest. This was discovered to be by far the most
successful of the three methods. It retained the versatility of the Monte
Carlo method, being capable of effectively integrating over even quite
pathological volumes. While also maintaining the level of accuracy offered by
the casework definite integration method. This was at the cost, however, of
speed. Although much faster than Monte Carlo, integrating over such a large
region when only a small portion of it is actually non-zero forces the
numerical integrator to compute a very large number of function evaluations.
Below, a brief explanation/derivation of the indicator function method is
given. Taking Eq. (10):

$\displaystyle\rho(a,e,i,\lambda_{s})=\dfrac{1}{2\pi^{3}}\int_{V^{*}}\dfrac{r\cos(\lambda)}{a\sqrt{\sin^{2}(i)-\sin^{2}(\lambda)}\sqrt{a^{2}e^{2}-(a-r)^{2}}}dLd\lambda
dr$

The idea is to integrate over the entire spatial state space, defined in part
2, multiplying the integrand by the indicator function, $\mathbf{1}_{V}$, on
the region of visibility of the ground station, $V$:

$\displaystyle\rho(a,e,i,\lambda_{s})=\dfrac{1}{2\pi^{3}}\int_{S}\dfrac{r\cos(\lambda)\mathbf{1}_{V}(r,\lambda,L)}{a\sqrt{\sin^{2}(i)-\sin^{2}(\lambda)}\sqrt{a^{2}e^{2}-(a-r)^{2}}}dLd\lambda
dr$ (11)

Which becomes:

$\displaystyle\rho(a,e,i,\lambda_{s})=\dfrac{1}{2\pi^{3}}\int\limits_{r=a(1-e)}^{a(1+e)}\int\limits_{\lambda=-i}^{i}\int\limits_{L=0}^{2\pi}\dfrac{r\cos(\lambda)\mathbf{1}_{V}(r,\lambda,L)}{a\sqrt{\sin^{2}(i)-\sin^{2}(\lambda)}\sqrt{a^{2}e^{2}-(a-r)^{2}}}dLd\lambda
dr$ (12)

It was discovered that applying the following change of variables simplifies
the integral considerably:

$\displaystyle r(\theta)$ $\displaystyle=a(1-e\sin(\theta))$ (13)
$\displaystyle\lambda(\alpha)$ $\displaystyle=\sin^{-1}(\sin(i)\sin(\alpha))$

We get:

$\displaystyle\rho(a,e,i,\lambda_{s})=\dfrac{1}{2\pi^{3}}\int\limits_{\theta=-\frac{\pi}{2}}^{\frac{\pi}{2}}\int\limits_{\alpha=-\frac{\pi}{2}}^{\frac{\pi}{2}}\int\limits_{L=0}^{2\pi}\mathbf{1}_{V}(r(\theta),\lambda(\alpha),L)(1-e\sin(\theta))dLd\alpha
d\theta$ (14)

This integral clearly contains far fewer operations per function evaluation.
In addition, this form of the formula has no singularities, which reduces
integration time significantly.

###  6.2 Generating the Test Data

For the purposes of testing the formula, the elevation angle of the line of
sight for the ground station was set to zero. That is, the satellite was
assumed to be able to communicate with the ground station if it was above the
plane tangent to the planet at the location of the ground station.

After selecting the set of points to test (in next section), a method of
comparing the results generated via Eq. (10) to the real world (or some good
approximation of the real world) was needed. Generating the formula data was
relatively fast, only taking a couple of hours to sequentially compute ratios
for 6000 data points. The method of choice for generating the ”real world”
data was the classical method mentioned above, integrating the trajectory of
the satellite with a numerical ordinary differential equation solver.

However, this approach proves problematic because of how slow it is. Using the
Matlab implementation of ode45, run for 6000 simulation days, it
conservatively took an hour for the computation to complete. Which would
amount to computing 8000 data points taking on the order of a year to complete
($\sim$333 days), which is not an acceptable run time. Luckily, another
research group in the same department was working on parallel numerical ode
integration using GPUs. The GPU group was able to produce an integrator which
would work well for integrating these satellite trajectories. This brought
down the compute time down from a year to several hours.

###  6.3 Verification of Results

Figure 3 below shows the residual error between the Ergodic results and the
integrated (direct orbit propagation) values for Earth. Each data point was
randomly sampled from the below sample space (Monte Carlo):

$\displaystyle 100+R<$ $\displaystyle a<50000+R$ $\displaystyle.001<$
$\displaystyle e<.95$ $\displaystyle 0<$ $\displaystyle i<\pi$
$\displaystyle-\dfrac{\pi}{2}<$ $\displaystyle\lambda<\dfrac{\pi}{2}$
$\displaystyle R$ $\displaystyle=\text{Radius of planet (km)}$ $\displaystyle
a$ $\displaystyle=\text{Major Axis (km)}$ $\displaystyle e$
$\displaystyle=\text{Eccentricity}$ $\displaystyle i$
$\displaystyle=\text{Inclination angle of orbital plane (radians)}$
$\displaystyle\lambda$ $\displaystyle=\text{Latitude of ground station
(radians)}$

With the restriction on each data point that:
1\. The eccentricity not be so large that the trajectory would intersect the
planet.
2\. That the ground station latitude is no larger than the inclination angle.
3\. The trajectory is non-periodic.

Figure 3: Histogram of Percent Error

| Max percent error | = $1.22$
---|---|---
| Min percent error | = $8.38\times 10^{-8}$
| Mean percent error | = $.0658$
| Median percent error | = $.0239$

Next, a more systematic method of sampling was used, with each feasible point
in the below range of values for the inputs included.
Altitude $\in\\{100,200,300,400,500,1000,2000,5000,10000,25000,50000\\}$
(kilometers) Eccentricity $\in\\{.01,.05,.1,.2,.3,.4,.5,.6,.7,.8,.9\\}$
Inclination $\in\\{5,10,15,...,80,85\\}$ (degrees) GS Latitude
$\in\\{0,5,10,...,85,90\\}$ (degrees) With the restriction on each data point
that:
1\. The eccentricity not be so large that the trajectory would intersect the
planet.
2\. The trajectory is non-periodic.

Figure 4: Histogram of Absolute Error

The above chart is a histogram of the absolute error between the ergodic and
numerical results for Earth. It is everywhere less than .01, which implies
that the ergodic result will be off by, at most 1 percentage point. Moreover,
the mean absolute error is .00058. So, on average, the theoretical result will
be off by .058 percentage points.

(a) Percent Error Unsorted

(b) Percent Error Sorted by Ratio Size

Figure 5: Percent Error

The percent error numbers on Fig. 5(a) make the situation look far worse. In
places, percent error is as high as 60%. This is a result of the fact that if
the ratio is sufficiently small, even very small error ends up being a
considerable percentage of that ratio. Fig. 5(b) is sorted by ratio size from
smallest to largest (with the ratio magnitude in orange and percent error in
blue). Clearly, the results with very large percent error are restricted to
the smallest ratios, where percent error is expected to be larger.

##  7 Conclusion

The fact that very large portion of satellite trajectories are periodic would
seem to impose a severe restriction on the usefulness of this formula.
However, it has been shown that the formula still tends to give reasonably
good results for periodic orbits. In addition, the exact result may be
computed for any periodic orbit by propagating the trajectory for one full
period, which is much cheaper than long-term orbit propagation. Another issue
encountered was the fact that percent error blows up when the visibility ratio
becomes sufficiently small. This may be somewhat problematic. However,
trajectories where this sort of issue happens are relatively rare in practice.
For example, low altitude trajectories (100-150 km) tend to lead to small
ratios where this problem occurs. But such low Earth orbits are uncommon in
practice due to atmospheric drag among other issues (the ISS stays at least
300 km above the Earth and still has to make relatively large corrections due
to drag). In general, configurations with small ratios will often be of little
interest in comparison to high ratio configurations where the percent error is
small.

Another issue with the visibility ratio for elliptical orbits is that it
requires a volume integral which can be very slow to compute; whereas for
circular orbits, the visibility ratio is given by a simple definite integral
that can be quickly integrated numerically. To solve this problem, we plan to
implement a GPU based algorithm to compute the volume integral. We expect this
to reduce the computation time by several orders of magnitude. By implementing
a GPU-based function to compute the visibility ratio of elliptical orbits such
as in Matlab or Python, this would provide engineers and designers with a fast
and powerful tool for designing telecommunications systems and network in real
time.

##  8 Acknowledgments

This research was carried out in part at the Jet Propulsion Laboratory,
California Institute of Technology, and was sponsored by Caltech Summer
Undergraduate Research Fellowship Program and the National Aeronautics and
Space Administration. This work was also supported in part by the Hummer-
Tuttle gift to Professor Al Barr through the Caltech Division of Engineering
and Applied Science. Thanks to Ryan Burns, Brian Anderson, Max Zhan, Tom
Gorordo and Sam Blazes for assistance with GPU programming, data generation,
and visualization.

##  9 Notation

$\rho$ = | Satellite-Ground Station Visibility Ratio
---|---
$T$ = | Total Flight Time
$P(T)$ = | Total Line of Sight Time over $T$
$V(T)$ = | Expected Line of Sight Time over $T$ (given $\rho$)
$a$ = | Major Axis (Kilometers)
$e$ = | Eccentricity
$i$ = | Inclination of Orbital Plane (Degrees)
$\lambda_{s}$ = | Latitude of Ground Station (Degrees)
$L$ = | Longitude (Degrees)
$R_{B}$ = | Radius of Body (Kilometers)
$\mu$ = | The Standard Gravitational Parameter of the Body ($\text{km}^{3}\text{s}^{-2}$)
$M$ = | Mean Anomaly (Radians)
$\Omega$ = | Longitude of the Ascending Node (Radians)
$\omega$ = | Argument of the Periapsis (Radians)
$J_{2}$ = | The Body’s Oblateness Parameter
$\mathcal{L}^{1}$ = | $\\{f|(\int_{S}|f|d\mu)<\infty\\}$
$S$ = | Spatial State Space of Satellite
$V$ = | Spatial Region of Visibility for Ground Station
$V^{*}$ = | The Region of Feasible Visibility
$\Gamma$ = | The Orbital Elements State Space of the Satellite
$V_{\Gamma}$ = | Region of Visibility for Ground Station in $\Gamma$
$V^{*}_{\Gamma}$ = | The Region of Feasible Visibility in $\Gamma$

## References

  * [1] M. W. Lo, “The Long-Term Forecast of Station View Periods,” tech. rep., Jet Propulsion Laboratory, Pasadena, California, 1994.
  * [2] Bate, Mueller, and White, Fundamentals Of Astrodynamics.  Dover Publications, Inc., New York, 1971.
  * [3] V. Arnold, Ergodic Problems of Classical Mechanics.  New York: Addison-Wesley, 1989.
  * [4] Y. G. Sinai, Introduction to Ergodic Theory.  Princeton University Press, Princeton N.J., 1976.