Orbital Mechanics, Coverage and Constellation Design

Introduction

Satellite-system architecture begins with orbital geometry. Orbit altitude and inclination determine motion, visibility, propagation delay, radiation exposure and launch requirements. Orbital planes and phasing determine whether coverage is continuous or intermittent. Elevation masks, antenna constraints and terrain reduce the usable portion of the geometric footprint, while traffic distribution and spectrum reuse determine whether nominal coverage translates into useful capacity.

This chapter develops a design chain from Keplerian motion to constellation-level decisions. Two-body equations provide the baseline, after which perturbations, Earth-fixed geometry, eclipses, footprints, latency, temporal coverage and capacity constraints are introduced. The resulting models support preliminary design, but high-confidence engineering decisions require numerical propagation, link simulation and statistical demand analysis.

Learning Objectives

After studying this chapter, the reader should be able to:

• apply Kepler’s laws, orbital elements and the vis-viva equation;
• calculate satellite position and period for circular and elliptical orbits;
• explain the effects of Earth oblateness, drag and third-body perturbations;
• calculate range, azimuth, elevation and visibility using correct geometry;
• determine elevation-limited footprints and approximate coverage areas;
• estimate eclipse duration and propagation latency;
• interpret Walker and related constellation models;
• distinguish geometric coverage from availability, capacity and quality of service;
• formulate preliminary constellation-sizing and resource-allocation decisions.

1. Keplerian Orbital Mechanics

1.1 The two-body model

The classical two-body model assumes that Earth is spherically symmetric, the satellite has negligible mass relative to Earth, and no other forces act. The satellite position vector \(\mathbf r\) then satisfies

\[

\ddot{\mathbf r}=-\frac{\mu}{r^3}\mathbf r,

\]

where \(r=\|\mathbf r\|\) and \(\mu=GM_E\) is Earth’s gravitational parameter. For Earth,

\[

\mu\approx 398600.44\ \text{km}^3\text{s}^{-2}.

\]

Using kilometres consistently is essential when this value of \(\mu\) is used.

The specific mechanical energy is

\[

\varepsilon=\frac{v^2}{2}-\frac{\mu}{r}=-\frac{\mu}{2a},

\]

which gives the vis-viva equation

\[

v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}.

\]

Here, \(a\) is the semimajor axis. For a circular orbit, \(a=r\) and \(v=\sqrt{\mu/r}\).

Kepler’s third law gives the period of an elliptical orbit:

\[

T=2\pi\sqrt{\frac{a^3}{\mu}}.

\]

The equation applies to an unperturbed bound orbit with \(a>0\).

1.2 Classical orbital elements

A Keplerian orbit is described by six classical elements:

1. Semimajor axis, \(a\): orbit size.

2. Eccentricity, \(e\): orbit shape; \(0\le e<1\) for an ellipse.

3. Inclination, \(i\): angle between the orbital plane and Earth’s equatorial plane.

4. Right ascension of the ascending node, \(\Omega\): orientation of the ascending-node direction.

5. Argument of perigee, \(\omega\): angle from the ascending node to perigee.

6. True anomaly, \(\nu\): satellite position measured from perigee.

For an ellipse,

\[

r=\frac{p}{1+e\cos\nu},\qquad p=a(1-e^2).

\]

Perigee and apogee radii are

\[

r_p=a(1-e),\qquad r_a=a(1+e),

\]

so that

\[

a=\frac{r_p+r_a}{2},\qquad e=\frac{r_a-r_p}{r_a+r_p}.

\]

Altitude must not be confused with geocentric radius: approximately \(r=R_E+h\), where a spherical mean-equatorial radius \(R_E\approx6378\) km is adequate for preliminary calculations.

1.3 Position propagation

Time is related to orbital position through Kepler’s equation:

\[

M=E-e\sin E,

\]

where \(M=n(t-\tau)\) is mean anomaly, \(E\) is eccentric anomaly, \(\tau\) is perigee-passage time, and

\[

n=\sqrt{\frac{\mu}{a^3}}

\]

is mean motion. Kepler’s equation is normally solved iteratively. True anomaly follows from

\[

\tan\frac{\nu}{2}=\sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}.

\]

In the perifocal frame,

\[

\mathbf r_{pf}=\frac{p}{1+e\cos\nu}

\begin{bmatrix}\cos\nu\\ \sin\nu\\0\end{bmatrix}.

\]

Rotations through \(\omega\), \(i\) and \(\Omega\) transform this vector into an Earth-centred inertial frame. Earth rotation must subsequently be applied to obtain Earth-fixed longitude and ground tracks.

Worked Example 1: Elliptical-orbit parameters

A satellite has perigee altitude 500 km and apogee altitude 2000 km. Using \(R_E=6378\) km,

\[

r_p=6878\ \text{km},\qquad r_a=8378\ \text{km}.

\]

Therefore,

\[

a=7628\ \text{km},\qquad e=\frac{1500}{15256}=0.0983.

\]

The period is

\[

T=2\pi\sqrt{\frac{7628^3}{398600.44}}\approx6630\ \text{s}=110.5\ \text{min}.

\]

Using vis-viva, the perigee and apogee speeds are approximately

\[

v_p=7.98\ \text{km/s},\qquad v_a=6.55\ \text{km/s}.

\]

The larger perigee speed is consistent with conservation of angular momentum.

2. Operational Orbit Classes

Low Earth orbit (LEO), commonly several hundred to about 2000 km altitude, provides low propagation delay and comparatively small path loss, but each satellite has a limited footprint and moves rapidly relative to users. Medium Earth orbit (MEO) offers larger footprints and longer visibility at the cost of greater delay and radiation exposure.

A geostationary satellite has a circular, equatorial, prograde orbit with a sidereal period equal to Earth’s rotation period. Its radius is approximately 42,164 km, corresponding to an altitude of about 35,786 km. Geosynchronous orbits share the period but need not be circular or equatorial.

Highly elliptical orbits, including Molniya-type orbits, remain near apogee for long intervals. Their dwell time can favour high-latitude service that is geometrically difficult from geostationary orbit.

3. Perturbations and Orbit Maintenance

Real orbits depart from Keplerian motion. Important perturbations include Earth’s nonspherical gravity field, atmospheric drag, solar-radiation pressure, and lunar and solar gravity.

3.1 Earth oblateness

Earth’s second zonal harmonic, \(J_2\approx1.08263\times10^{-3}\), produces secular changes in the ascending node and argument of perigee. To first order,

\[

\dot\Omega=-\frac{3}{2}J_2n\left(\frac{R_E}{p}\right)^2\cos i,

\]

\[

\dot\omega=\frac{3}{4}J_2n\left(\frac{R_E}{p}\right)^2(5\cos^2i-1).

\]

These averaged equations assume that \(J_2\) dominates the disturbance and that osculating short-period variations are not the principal interest.

A sun-synchronous orbit uses nodal precession of approximately \(+0.9856^\circ\) per mean solar day so that the orbital plane retains nearly constant local solar time. Since positive nodal precession requires \(\cos i<0\), most LEO sun-synchronous orbits are retrograde.

Worked Example 2: Approximate sun-synchronous inclination

For a circular 700 km orbit, \(r=p=7078\) km and \(n\approx0.001060\) rad/s. Setting the first-order nodal rate equal to \(+0.9856^\circ/\text{day}\) gives

\[

\cos i\approx-0.1425,

\]

and hence

\[

i\approx98.2^\circ.

\]

Operational design would refine this result using the selected geopotential model and the desired mean-orbit convention.

3.2 Atmospheric drag and other disturbances

Drag is significant in LEO and depends on atmospheric density, relative velocity, spacecraft area, mass and drag coefficient. Density varies strongly with altitude, solar activity and geomagnetic conditions. Drag reduces orbital energy and generally lowers semimajor axis; eccentric orbits are affected most strongly near perigee.

Solar-radiation pressure can alter eccentricity and attitude, while third-body gravity becomes important for high-altitude and long-duration missions. Station keeping consumes propellant or electric-propulsion lifetime, so constellation economics must account for orbit raising, collision avoidance, disposal and replenishment.

4. Ground Geometry and Look Angles

4.1 Coordinate-based calculation

Reliable operational software should calculate look angles from vectors rather than rely exclusively on spherical special cases. Let \(\mathbf r_s\) and \(\mathbf r_g\) be satellite and ground-station position vectors in the same Earth-fixed frame. The line-of-sight vector is

\[

\boldsymbol\rho=\mathbf r_s-\mathbf r_g,

\qquad \rho=\|\boldsymbol\rho\|.

\]

For geodetic latitude \(\phi\) and longitude \(\lambda\), project \(\boldsymbol\rho\) onto local east, north and up unit vectors to obtain components \((\rho_E,\rho_N,\rho_U)\). Then

\[

El=\sin^{-1}\left(\frac{\rho_U}{\rho}\right),

\qquad

Az=\operatorname{atan2}(\rho_E,\rho_N),

\]

with azimuth normalized to the required convention. An oblate-Earth model should be used when precision matters because geodetic and geocentric latitude differ.

4.2 Spherical-Earth visibility relationship

For a spherical Earth, let \(\psi\) be the geocentric angle between the ground station and the satellite subsatellite point, and let \(r_s\) be satellite radius. The slant range is

\[

\rho=\sqrt{r_s^2+R_E^2-2r_sR_E\cos\psi}.

\]

The elevation angle satisfies

\[

\sin El=

\frac{\cos\psi-R_E/r_s}

{\sqrt{1+(R_E/r_s)^2-2(R_E/r_s)\cos\psi}}.

\]

The denominator depends explicitly on \(\psi\); omitting that dependence gives incorrect elevations.

If ground and subsatellite geocentric latitudes are \(\phi_g\) and \(\phi_s\), with longitude difference \(\Delta\lambda\), then

\[

\cos\psi=\sin\phi_g\sin\phi_s+

\cos\phi_g\cos\phi_s\cos\Delta\lambda.

\]

For an equatorial ground station and an equatorial GEO satellite, \(\psi=|\Delta\lambda|\) only after reducing the longitude difference to the appropriate central angle.

Worked Example 3: GEO elevation at 30° longitude separation

An equatorial ground station observes a GEO satellite separated by \(30^\circ\) in longitude. Here \(r_s=42164\) km, \(R_E=6378\) km and \(\psi=30^\circ\). Thus

\[

\sin El=

\frac{\cos30^\circ-6378/42164}

{\sqrt{1+(6378/42164)^2-2(6378/42164)\cos30^\circ}}

\approx0.819.

\]

Therefore,

\[

El\approx55.0^\circ.

\]

The corresponding slant range is approximately 36,780 km.

5. Elevation Masks, Footprints and Visibility

A satellite is geometrically visible above a minimum elevation \(El_{min}\) if \(\psi\le\psi_{max}\). Solving the elevation geometry gives

\[

\boxed{\psi_{max}=\cos^{-1}\left(\frac{R_E}{r_s}\cos El_{min}\right)-El_{min}}.

\]

Angles must use consistent units. At zero elevation this reduces to the horizon result

\[

\psi_{max}=\cos^{-1}(R_E/r_s).

\]

An equivalent expression is

\[

\cos\psi_{max}=\frac{R_E}{r_s}\cos^2El_{min}

+\sin El_{min}\sqrt{1-\left(\frac{R_E}{r_s}\cos El_{min}\right)^2}.

\]

For a spherical Earth, the footprint area is the area of a spherical cap:

\[

A_f=2\pi R_E^2(1-\cos\psi_{max}).

\]

This is a geometric access area, not necessarily a communications service area. Antenna beamwidth, link margin, terrain, interference, regulation and gateway access may impose tighter boundaries.

Worked Example 4: GEO footprint with a 10° elevation mask

For GEO, \(R_E/r_s=6378/42164\approx0.15127\). With \(El_{min}=10^\circ\),

\[

\psi_{max}=\cos^{-1}(0.15127\cos10^\circ)-10^\circ

\approx71.4^\circ.

\]

The corresponding spherical-cap area is approximately

\[

A_f\approx1.74\times10^8\ \text{km}^2,

\]

or about 34% of Earth’s total surface. Actual usable coverage is smaller because low-elevation paths experience greater atmospheric attenuation, clutter and antenna-tracking difficulties.

5.1 Coverage is temporal

A static footprint does not establish continuous service. Relevant metrics include:

• access duration: length of a visibility interval;
• revisit time: interval between usable accesses;
• coverage multiplicity: number of simultaneously visible satellites;
• availability: fraction of time service requirements are met;
• handover rate: frequency of satellite or beam changes;
• dilution of precision: satellite geometry quality for navigation.

These metrics require propagation over time, including Earth rotation and perturbations. Worst-case performance may occur at specific latitudes or between adjacent ground tracks.

6. Eclipse Analysis

Spacecraft enter eclipse when Earth blocks sunlight. Eclipses affect battery sizing, thermal cycles, power-limited throughput and payload availability.

Under a cylindrical-shadow approximation, appropriate for preliminary Earth-orbit calculations, let \(\beta\) be the angle between the Sun direction and the orbital plane. For a circular orbit, eclipse is possible when

\[

|\beta|<\sin^{-1}(R_E/r_s).

\]

If this condition is met, the eclipse half-angle is

\[

\alpha=\n\cos^{-1}\left(

\frac{\sqrt{r_s^2-R_E^2}}{r_s\cos\beta}

\right),

\]

and the eclipse duration is

\[

t_e=\frac{T\alpha}{\pi},

\]

where \(\alpha\) is in radians. The model neglects the finite angular diameter of the Sun and therefore does not distinguish penumbra from umbra.

For GEO at \(\beta=0\), \(\alpha=\sin^{-1}(R_E/r_s)\approx8.70^\circ\), giving a maximum cylindrical-shadow duration of approximately 69 minutes. GEO eclipses cluster around the equinox seasons.

7. Propagation Delay

Free-space propagation time is distance divided by the speed of light:

\[

t_p=\frac{L}{c},\qquad c\approx299792\ \text{km/s}.

\]

For a single transparent satellite link between two ground terminals,

\[

t_{one-way}\approx\frac{\rho_1+\rho_2}{c}+t_{processing}+t_{network}.

\]

Round-trip time approximately doubles the forward propagation path but may include different routes and additional processing. A GEO path has a best-case ground-to-satellite-to-ground distance near 71,572 km for two terminals close to their respective nadir directions, corresponding to about 239 ms one way before processing. A network round trip commonly exceeds 500 ms.

LEO reduces each access-link delay, but a long route through multiple inter-satellite links may not always be shorter than a terrestrial fibre route. Optical fibre propagates more slowly than vacuum, yet routing distance, queueing and gateway placement remain decisive.

8. Constellation Models and Design

8.1 Walker notation

A Walker-delta constellation is commonly written

\[

i:T/P/F,

\]

where \(i\) is inclination, \(T\) is the total number of satellites, \(P\) is the number of equally spaced orbital planes, and \(F\) is an integer phasing parameter. Each plane contains \(T/P\) satellites, requiring \(T\) to be divisible by \(P\). Adjacent-plane phasing is commonly represented by a relative anomaly shift of \(F(360^\circ/T)\), subject to the adopted Walker convention.

Walker-star arrangements are often used for near-polar systems, while regional systems may use inclined planes, elliptical orbits or asymmetric distributions. Walker models provide orderly initial architectures but are not automatically optimal for nonuniform demand.

8.2 Area-ratio sizing bound

The fractional area of Earth inside one ideal spherical footprint is

\[

f_A=\frac{A_f}{4\pi R_E^2}=\frac{1-\cos\psi_{max}}{2}.

\]

An optimistic lower bound for single global coverage is therefore

\[

N_{LB}=\frac{1}{f_A}=\frac{2}{1-\cos\psi_{max}}.

\]

For coverage multiplicity \(m\), the area-counting bound becomes

\[

N_{LB,m}=\frac{2m}{1-\cos\psi_{max}}.

\]

This is not a deployable constellation-sizing formula. It assumes perfect nonoverlapping tiling, ignores orbital-plane geometry, temporal gaps, polar overlap, handover constraints, outages and capacity. Since equal spherical caps generally cannot tile Earth without overlap or gaps, the actual number is higher.

8.3 Numerical coverage analysis

A defensible constellation evaluation propagates every satellite across a representative horizon and samples relevant ground locations. At each time and location, it should test:

1. elevation and line-of-sight constraints;

2. beam steering and antenna limits;

3. link margin and weather availability;

4. gateway or inter-satellite routing availability;

5. required coverage multiplicity;

6. handover continuity and minimum access duration.

Grid weighting must match the objective. An equal-angle latitude–longitude grid overweights polar regions; equal-area cells or population-weighted demand points are preferable.

9. Capacity-Aware Constellation Design

Geometric coverage answers whether a satellite can be seen. Capacity analysis asks how much traffic can be delivered with acceptable quality.

For beam \(b\), an idealized upper bound is

\[

C_b=B_b\log_2(1+\mathrm{SINR}_b),

\]

where \(B_b\) is allocated bandwidth and \(\mathrm{SINR}_b\) includes thermal noise and cochannel interference. Practical throughput is lower because of coding gaps, pilots, guard bands, protocol overhead, scheduler inefficiency and fading margins.

A useful system model distinguishes at least four bottlenecks:

• service-link capacity: user-beam spectrum, power and antennas;
• feeder-link capacity: gateway spectrum and weather availability;
• onboard processing: switching, routing and digital payload limits;
• network capacity: inter-satellite links and traffic routing.

If \(x_{sbu}(t)\) denotes the fraction of beam resources from satellite \(s\), beam \(b\), assigned to user region \(u\), a simplified allocation model imposes

\[

\sum_u x_{sbu}(t)\le1,

\]

along with power, bandwidth, interference and steering constraints. Served demand must satisfy

\[

y_u(t)\le d_u(t),

\qquad

y_u(t)\le\sum_{s,b}x_{sbu}(t)C_{sb}(t).

\]

Objectives may maximize weighted served demand, revenue, minimum regional service, or resilience. These objectives can conflict. Maximizing total throughput may divert resources toward dense profitable markets, while universal-service obligations favour geographic fairness.

Coverage and capacity therefore require different constellation characteristics. Additional satellites can improve spatial frequency reuse and reduce congestion even when geometric coverage is already continuous. Conversely, a large footprint may spread finite power over too many users.

10. Illustrative Applied Case: Preliminary LEO Broadband Architecture

Background

This fictional case considers a proposed LEO broadband system at 1200 km circular altitude. The preliminary architecture uses a 10° elevation mask, six planes and twelve satellites per plane, for a total of 72 satellites.

Problem

The design team must assess whether the architecture is plausible in terms of footprint, temporal coverage, latency and aggregate capacity. The assessment is screening-level rather than a final design certification.

Analysis

The orbital radius is

\[

r_s=6378+1200=7578\ \text{km}.

\]

The period is approximately

\[

T=2\pi\sqrt{\frac{7578^3}{398600.44}}\approx109.4\ \text{min}.

\]

The elevation-limited central angle is

\[

\psi_{max}=\cos^{-1}\left(\frac{6378}{7578}\cos10^\circ\right)-10^\circ

\approx24.0^\circ.

\]

Thus,

\[

f_A=\frac{1-\cos24.0^\circ}{2}\approx0.0433.

\]

The ideal area-ratio lower bound is about

\[

N_{LB}\approx23.1,

\]

or 24 whole satellites. This does not demonstrate that 24 satellites can provide continuous global service. The selected 72-satellite architecture provides design margin for overlap and temporal geometry, but continuity must still be verified through propagation at the chosen inclination and phasing.

The edge-of-coverage slant range can also be written as

\[

\rho=-R_E\sin El+

\sqrt{r_s^2-R_E^2\cos^2El},

\]

which gives approximately 3130 km at \(El=10^\circ\). A worst-case same-satellite path between two users both near the footprint edge is therefore approximately 6260 km, corresponding to about 20.9 ms one-way free-space propagation. Processing and routing increase this value.

Assume each satellite supports 16 simultaneously active beams, each with 250 MHz effective bandwidth and an average net spectral efficiency of 2 bit/s/Hz. At a 70% scheduling utilization, the notional capacity per satellite is

\[

C_s=16(250\times10^6)(2)(0.70)=5.6\ \text{Gbit/s}.

\]

The nominal orbital aggregate is 403.2 Gbit/s. This figure is valid only if interference is already represented in the spectral-efficiency assumption and feeder links, onboard processing and demand distribution can support all beams concurrently.

Approach and Recommended Solution

The 72-satellite Walker candidate should be subjected to a time-domain simulation covering multiple orbital precession cycles or a justified representative interval. The design should be evaluated using population-weighted demand points, gateway weather diversity, handover constraints and failure scenarios. Capacity should be optimized jointly with beam hopping, gateway assignment and inter-satellite routing.

Outcome

The architecture is plausible for further study but cannot be approved from footprint area alone. The latency is attractive compared with GEO, while capacity depends more strongly on spectrum reuse and gateway infrastructure than on satellite count by itself.

Learning Points

• Area-ratio calculations provide lower bounds, not final constellation sizes.
• Elevation masks simultaneously affect coverage, range and link quality.
• Continuous geometric access does not guarantee sufficient throughput.
• Demand distribution, gateway diversity and failures must be included early.

11. Design Challenges, Governance and Contemporary Practice

Constellation design is a multidisciplinary optimization problem. Lower altitude improves latency and link budget but requires more satellites, increases drag sensitivity and accelerates handovers. Higher altitude enlarges footprints but increases delay, free-space loss and radiation exposure. Inclination must reflect service latitude rather than simply maximize geographic reach.

Regulatory design includes spectrum coordination, power-flux-density limits, equivalent power-flux-density constraints, national landing rights and International Telecommunication Union filings. A filing does not itself guarantee interference-free operation or market authorization.

Large constellations also create externalities. Operators must address collision probability, conjunction response, reliable post-mission disposal, casualty risk from re-entry, optical and radio astronomy impacts, and equitable access to orbital and spectrum resources. Reliability analysis should consider correlated launch failures, space-weather events and common software faults rather than treating satellite failures as independent.

Contemporary systems increasingly employ digital transparent processors, regenerative payloads, electronically steered arrays, optical inter-satellite links and beam hopping. Machine-learning methods can assist demand forecasting, anomaly detection and resource scheduling, but safety-critical orbit determination and collision avoidance require validated models, uncertainty bounds and human-governed escalation procedures.

Best practice is to preserve traceability from assumptions to decisions: document reference frames, Earth models, time standards, elevation masks, force models, traffic maps and outage criteria; validate analytic results against independent propagators; and perform sensitivity and Monte Carlo analyses before committing to deployment.

Chapter Summary

Keplerian mechanics establishes the relationship between orbital size, shape, velocity and period. Classical orbital elements describe orbit orientation and satellite position, while perturbations—especially Earth’s \(J_2\) term and atmospheric drag—govern long-term evolution and station-keeping requirements.

Look angles are most reliably calculated from Earth-fixed vectors. Under a spherical approximation, elevation depends on both satellite radius and ground-to-subsatellite central angle. The elevation mask determines the limiting footprint angle, but a footprint represents only instantaneous geometric access. Continuous service requires temporal simulation of visibility, overlap, handovers and failures.

Eclipse geometry affects spacecraft energy storage and payload operation, while slant range determines propagation delay. Walker constellations provide structured plane and phasing models, although simple footprint-area ratios yield only optimistic lower bounds. Postgraduate-level constellation design must integrate coverage, link quality, spectrum reuse, gateway capacity, routing, demand, resilience, regulation and environmental responsibility.

Key Terms

Semimajor axis: Half the longest diameter of an elliptical orbit and the principal measure of orbit size.

Eccentricity: Dimensionless measure of orbital departure from circularity.

Inclination: Angle between an orbital plane and Earth’s equatorial plane.

True anomaly: Angular satellite position measured from perigee.

Mean motion: Average angular rate in a Keplerian orbit.

Vis-viva equation: Relationship among orbital speed, radius, semimajor axis and gravitational parameter.

Perturbation: Force or gravitational effect that causes departure from ideal two-body motion.

Nodal precession: Secular rotation of an orbit’s ascending-node direction.

Sun-synchronous orbit: Orbit whose nodal precession maintains approximately constant local solar time.

Subsatellite point: Point on Earth directly below the satellite in a geocentric sense.

Elevation angle: Angle of the satellite line of sight above the local horizon.

Elevation mask: Minimum elevation accepted for operational access.

Footprint: Earth-surface region satisfying a specified geometric or service criterion.

Coverage multiplicity: Number of satellites simultaneously meeting access requirements at a location.

Revisit time: Time between successive usable access intervals.

Beta angle: Angle between the Sun direction and the orbital plane.

Walker constellation: Structured constellation defined by inclination, satellite count, plane count and phasing.

Beam hopping: Dynamic reassignment of payload beams or resources among service regions.

Spectral efficiency: Net information rate delivered per unit bandwidth.

Review Questions

Short Answer Questions

1. State the assumptions underlying the two-body orbital model.

2. Distinguish geosynchronous orbit from geostationary orbit.

3. Why are most LEO sun-synchronous orbits retrograde?

4. What is the operational purpose of an elevation mask?

5. Distinguish geometric coverage from communications capacity.

Long Answer Questions

1. Derive the vis-viva equation from conservation of specific orbital energy and discuss its engineering applications.

2. Explain how the six classical orbital elements define an orbit and satellite position.

3. Discuss the effects of \(J_2\), atmospheric drag, solar-radiation pressure and third-body gravity on satellite operations.

4. Develop the spherical-Earth relationships among central angle, slant range and elevation, stating their assumptions and limitations.

5. Critically compare LEO, MEO, GEO and highly elliptical orbits for broadband communications.

Analytical / Application Questions

1. A circular satellite operates at 1500 km altitude with a 15° elevation mask. Calculate its period, limiting central angle, footprint fraction and ideal single-coverage satellite-count lower bound.

2. Formulate a simulation procedure to test whether a proposed Walker constellation provides continuous dual coverage over latitudes from 20° to 60°.

3. Develop a capacity-allocation model for a multibeam constellation with nonuniform regional demand, feeder-link limitations and a minimum-service fairness constraint.

Multiple Choice Questions with Answers

Question 1

A satellite has perigee radius 7000 km and apogee radius 9000 km. What is its semimajor axis?

A. 1000 km

B. 7000 km

C. 8000 km

D. 16000 km

Correct Answer: C. 8000 km

Question 2

For an equatorial ground station and a GEO satellite separated by 30° in longitude, the approximate elevation angle is:

A. 25°

B. 46°

C. 55°

D. 71°

Correct Answer: C. 55°

Question 3

Using a spherical Earth and a 10° elevation mask, the limiting central angle for a GEO satellite is approximately:

A. 61.4°

B. 71.4°

C. 81.4°

D. 90.0°

Correct Answer: B. 71.4°

Question 4

Why are most LEO sun-synchronous orbits retrograde?

A. Retrograde motion eliminates atmospheric drag.

B. A negative cosine of inclination produces the required eastward nodal precession.

C. Retrograde motion makes the orbit geostationary.

D. It prevents all eclipses.

Correct Answer: B. A negative cosine of inclination produces the required eastward nodal precession.

Question 5

Under the cylindrical-shadow model, eclipse is possible in a circular orbit when:

A. The beta angle is smaller in magnitude than arcsin(RE/rs).

B. The inclination equals zero.

C. The elevation angle exceeds 90°.

D. The orbital period equals one day.

Correct Answer: A. The beta angle is smaller in magnitude than arcsin(RE/rs).

Question 6

In Walker notation i:T/P/F, the quantity P represents:

A. Orbital period

B. Payload power

C. Number of orbital planes

D. Number of gateways

Correct Answer: C. Number of orbital planes

Question 7

At 1200 km altitude and a 10° elevation mask, the footprint fraction is approximately 0.0433. What is the ideal area-ratio lower bound for single global coverage?

A. 12 satellites

B. 24 satellites

C. 48 satellites

D. 72 satellites

Correct Answer: B. 24 satellites

Question 8

If two users communicate through one satellite and each user-to-satellite slant range is 3130 km, what is the approximate one-way free-space propagation delay?

A. 10.4 ms

B. 20.9 ms

C. 41.8 ms

D. 239 ms

Correct Answer: B. 20.9 ms

Question 9

Which resource can limit delivered constellation throughput even when service-link coverage is continuous?

A. Feeder-link capacity

B. Semimajor axis notation

C. True-anomaly origin

D. Earth's calendar year

Correct Answer: A. Feeder-link capacity

Question 10

Why is an area-ratio estimate based on footprints at a 10° elevation mask insufficient for final constellation sizing?

A. It does not use any elevation constraint.

B. It assumes ideal spatial tiling and ignores temporal gaps, overlap and operational constraints.

C. It is applicable only to geostationary satellites.

D. It always overestimates propagation delay.

Correct Answer: B. It assumes ideal spatial tiling and ignores temporal gaps, overlap and operational constraints.