Week 1 — Orbital Mechanics

00 — Overview

Why orbit matters to a communication engineer

An orbit is not merely where a satellite is parked. It controls distance and changing geometry, which in turn influence propagation delay, free-space path loss, coverage footprint, visibility duration, relative motion, Doppler shift, handover, ground-station tracking, and constellation design.

This week does not derive all those communication effects. It builds the two-body mechanics needed to reason about them later.

One of the four fundamental forces in nature is gravity. Gravity is the force responsible for the motion of galaxies, stars, and planets in the universe. Launching an artificial satellite and keeping it in a stable orbit around Earth depends on the forces acting on it.

For the practical work of launching and tracking satellites, the original lecture proceeds with a classical Newtonian treatment. It notes that a general-relativistic approach is unnecessary for this orbital-mechanics treatment, while relativity may still be invoked where synchronization and signal-processing accuracy require it.

From the original lecture notes · Topic 1, pp. 1–2 →

Orbit consequence	Communication question it creates
Greater distance	How much delay and path loss must the link tolerate?
Rapid apparent motion	How will terminals track, hand over, and manage Doppler?
Coverage footprint	What region can see the spacecraft at a useful elevation angle?

01 — Orbital motion

Falling around Earth

A satellite is continuously accelerated toward Earth by gravity while its tangential velocity carries it forward. With the right velocity, Earth's curved surface falls away beneath it: the spacecraft keeps missing Earth.

The handwritten lecture describes this orbital condition as a delicate balance between Earth's gravitational pull and a “centrifugal pushing force due to rotation.” That wording records the historical teaching treatment. In the Earth-centered inertial frame used for the modern course model, no outward centrifugal force acts on the satellite: gravity supplies the inward centripetal acceleration that continuously bends the satellite's inertial motion into an orbit.

From the original lecture notes · Topic 1, p. 1 →

Gravity bends the path; inertia carries the spacecraft onward. The strength of Earth's gravity is conveniently represented by the gravitational parameter

\[\mu=GM\]

Using kilometres and seconds throughout this page,

\[\mu_E=398600.4418\ \text{km}^3/\text{s}^2,\qquad R_E\approx6378\ \text{km}\]

02 — Kepler's laws

Three laws, one coherent motion

First law — shape

A bound two-body orbit is an ellipse with the central body at one focus. A circle is the special case with zero eccentricity.

Second law — changing speed

The radius vector from Earth to the satellite sweeps equal areas in equal times. Near perigee the satellite covers a longer arc and moves faster; near apogee it covers a shorter arc and moves slower.

Third law — period

The square of period is proportional to the cube of semi-major axis:

\[T^2\propto a^3,\qquad T=2\pi\sqrt{\frac{a^3}{\mu}}\]

The original sequence connects this proportionality to the gravitational parameter by writing

\[a^3=\frac{\mu}{(2\pi/T)^2}=\frac{\mu}{n^2},\qquad n=\frac{2\pi}{T}.\]

Modern orbital terminology calls \(n\) the mean motion. It is the constant average angular rate associated with the orbital period, not the satellite's instantaneous angular velocity along an eccentric orbit. The handwritten treatment calls \(n\) the angular speed in radians per second; that historical wording remains available through the linked source.

From the original lecture notes · Topic 1, pp. 2–3 →

Concept check

Where is a satellite in an elliptical orbit fastest?

Near perigee Near apogee

03 — Orbit geometry

Size, shape, perigee, and apogee

The semi-major axis \(a\) sets the ellipse's overall size. Eccentricity \(e\) sets its shape: \(e=0\) is circular, while increasing \(e\) elongates a bound ellipse.

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

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

The handwritten ellipse construction also introduces the semi-minor axis \(b\) and defines eccentricity geometrically:

\[e=\frac{\sqrt{a^2-b^2}}{a},\qquad a\ge b,\qquad 0\le e<1.\]

Its limiting cases are explicit: \(a=b\) gives \(e=0\), a circle, while \(b=0\) gives \(e=1\), the degenerate line limit.

From the original lecture notes · Topic 1, p. 4 →

Here \(r_p\) and \(r_a\) are distances from Earth's center. Altitude \(h\) is measured above Earth's surface:

\[r=R_E+h\]

Semi-major axis \(a\) 14,000 km Eccentricity \(e\) 0.35

Perigee radius

Apogee radius

Perigee altitude

Apogee altitude

Orbit and Earth are scaled consistently within the diagram. Earth is enlarged only when its true scale would be too small to identify; the label reports that state.

Concept check

For an elliptical orbit, is Earth at the center of the ellipse?

Yes, always No, Earth lies at one focus

04 — Orbital elements

Six numbers describe a Keplerian orbit

The classical orbital elements separate size, shape, orientation, and instantaneous position. We use them conceptually here; coordinate transformations come later.

\(a\) · Semi-major axisOverall size

\(e\) · EccentricityShape

\(i\) · InclinationTilt relative to the equator

\(\Omega\) · RAANDirection of the ascending node in the reference plane

\(\omega\) · Argument of perigeePerigee direction within the orbital plane

\(\nu\) · True anomalyCurrent angular position measured from perigee

05 — Velocity & period

Higher circular orbit, slower motion, longer period

For a circular orbit, \(e=0\), \(r=a\), and speed is constant:

\[v=\sqrt{\frac{\mu}{r}},\qquad T=2\pi\sqrt{\frac{r^3}{\mu}}\]

Increasing radius weakens the required centripetal acceleration. Circular speed decreases, but the path is larger; together these effects make period increase strongly.

For an ellipse, speed at any radius follows the vis-viva equation:

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

At perigee, \(r\) is smallest and speed is largest. At apogee, \(r\) is largest and speed is smallest—exactly the behavior summarized by Kepler's second law.

Circular-orbit altitude \(h\) 500 km

Orbital radius

Orbital speed

Orbital period

Orbit context

Reference markers at 500 km and 35,786 km are comparisons, not boundaries for every orbit classification.

Concept check

If circular-orbit altitude increases, what happens?

Speed decreases and period increases Speed and period both decrease Speed decreases but period stays fixed

06 — LEO, MEO & GEO

Orbit class becomes system architecture

LEO

Low delay and lower path loss, but rapid apparent motion, shorter visibility, tracking and handover, and often many satellites for continuous coverage.

MEO

Intermediate distance, delay, footprint, and apparent motion. Navigation constellations are familiar MEO examples.

GEO

A circular, equatorial, prograde orbit whose period matches Earth's sidereal rotation. It appears fixed to an Earth observer.

Derive GEO rather than memorize it

Use Earth's sidereal rotation period, approximately \(T_{\text{sidereal}}=86164\ \text{s}\), rather than exactly 24 solar hours. Rearranging Kepler's third-law relation gives

\[a=\left[\mu\left(\frac{T}{2\pi}\right)^2\right]^{1/3}\]

Insert \(\mu_E=398600.4418\ \text{km}^3/\text{s}^2\) and \(T=86164\ \text{s}\).
Compute \(a\approx42164\ \text{km}\) from Earth's center.
Subtract equatorial Earth radius: \(h=a-R_E\approx42164-6378\approx35786\ \text{km}\).

Concept check

Does an orbit with a period equal to Earth's sidereal rotation automatically qualify as geostationary?

Yes No

07 — Interactive Homework Lab

From Orbital Data to Orbital Motion

Work through three historical Homework #1 data sets. Each activity follows the same classroom rhythm: given data → predict → calculate → check → visualize → discuss.

Given data

Epoch: 30 September 2020 21:37:15 UTC

Eccentricity: 0.0002792
Inclination: 28.4708°
Perigee height: 536 km
Apogee height: 540 km
RAAN: 342.2031°
Argument of perigee: 142.1847°
Mean anomaly: 20.7941°

Step 1

Altitude versus orbital radius

Calculate the radii from Earth's center using the course Earth-radius convention.

Perigee radius \(r_p\) km Apogee radius \(r_a\) km

Expected: \(r_p=6914\ \text{km}\), \(r_a=6918\ \text{km}\). Accepted tolerance: ±2 km.

Step 2

Predict the orbit shape

With \(e=0.0002792\), what do you expect the HST orbit to look like? Almost circular Clearly elliptical

\(a=6916\ \text{km}\), \(b\approx6915.9997\ \text{km}\). Their difference is only about \(0.00027\ \text{km}\).

Step 3

Reason toward orbital period

Which orbital period seems reasonable for HST? 24 hours 12 hours About 1.5 hours 27 days

Use \(n=\sqrt{\mu/a^3}\), \(T=2\pi/n\), and \(N=86400/T\). Prediction tolerance is conceptual.

Step 4

Visualize and connect forward

The 2D orbital-plane preview above is available now. Scientifically valid 3D and ground-track views will connect here when the advanced orbital laboratory is integrated.

3D OrbitAdvanced visualizer integration point

Ground TrackAdvanced visualizer integration point

Given data

Epoch: 01 October 2020 12:28:09 UTC

Eccentricity: 0.0001294
Inclination: 51.6443°
RAAN: 182.3318°
Argument of perigee: 100.3462°
Revolutions/day: 15.48834285
Mean anomaly: 259.7834°

Calculate

From orbital frequency to orbit size

Use \(T=86400/N\), \(n=2\pi/T\), and \(a=(\mu/n^2)^{1/3}\), then recover the apsides and heights.

Period \(T\) s Semi-major axis \(a\) km Perigee radius \(r_p\) km Apogee radius \(r_a\) km Perigee height \(h_p\) km Apogee height \(h_a\) km Semi-minor axis \(b\) km

Expected with \(R_E=6378\ \text{km}\): \(T\approx5578.389\ \text{s}\), \(a\approx6798.272\ \text{km}\), \(r_p\approx6797.392\ \text{km}\), \(r_a\approx6799.152\ \text{km}\), \(h_p\approx419.392\ \text{km}\), \(h_a\approx421.152\ \text{km}\), and \(b\approx6798.272\ \text{km}\). Accepted tolerances: ±3 s and ±2 km.

Predict

Ground-track latitude

Compared with HST, which ground track should reach farther north and south? HST ISS Approximately the same

Given data

Semi-major axis: 384,748 km
Apogee radius: ≈406,375 km
Perigee radius: ≈356,790 km
Earth–Moon correction: \(1.012\mu_E\)

Predict and calculate

How long is a lunar orbit?

HST takes roughly 1.5 hours. What order of magnitude do you expect for the Moon? A few hours Tens of days Several years

Calculated period days

\(T=2\pi\sqrt{a^3/(1.012\mu_E)}\approx2.36094\times10^6\ \text{s}\approx27.3257\ \text{days}\). Accepted tolerance: ±0.08 day.

Interpret

Orbital versus synodic period

Orbital period ≈ 27.3 daysSynodic lunar cycle ≈ 29.5 days

Why is the synodic period longer?

Investigate measured data

Two estimates of eccentricity

\[e_a=\frac{r_a}{a}-1,\qquad e_p=1-\frac{r_p}{a}\]

From apogee \(e_a\) From perigee \(e_p\)

\(e_a\approx0.05621\), \(e_p\approx0.07267\). Using both apsides together would give \((r_a-r_p)/(r_a+r_p)\approx0.06497\), a third estimate rather than a correction.

Reveal the calculated comparison after attempting the HST and ISS activities, or use Instructor Mode during a live solution.

Quantity	HST	ISS
Semi-major axis	Hidden	Hidden
Approximate altitude	Hidden	Hidden
Eccentricity	0.0002792	0.0001294
Inclination	28.4708°	51.6443°
Period	Hidden	Hidden
Revolutions/day	Hidden	15.48834285

Discuss together

Which satellite is closer to Earth?
Which completes more revolutions per day, and why?
Which orbit reaches higher terrestrial latitudes, and why?
Can you visually distinguish either orbit from a circle?
Which parameter mainly explains the north/south ground-track difference?

A synchronized multi-satellite 3D/ground-track view is intentionally deferred to the future advanced orbital laboratory.

08 — MATLAB lab

Turn orbital relations into experiments

All examples use kilometres and seconds, require no specialist toolbox, and are intentionally compact enough to modify.

Example 1 — Circular orbit from altitude

                % Circular orbit: use km and seconds consistently
                  Re_km = 6378;
                  mu_km3_s2 = 398600.4418;
                  h_km = 500;
                  r_km = Re_km + h_km;
                  v_km_s = sqrt(mu_km3_s2 / r_km);
                  T_s = 2*pi*sqrt(r_km^3 / mu_km3_s2);
                  fprintf('Radius: %.1f km\n', r_km)
                  fprintf('Speed: %.3f km/s\n', v_km_s)
                  fprintf('Period: %.2f min\n', T_s/60)

Example 2 — Period versus altitude

                % Circular-orbit period over a useful altitude range
                  Re_km = 6378;
                  mu_km3_s2 = 398600.4418;
                  h_km = linspace(160, 40000, 800);
                  r_km = Re_km + h_km;
                  T_hr = (2*pi*sqrt(r_km.^3 / mu_km3_s2))/3600;
                  plot(h_km, T_hr, 'LineWidth', 1.8)
                  hold on
                  xline(35786, '--', 'GEO altitude');
                  grid on
                  xlabel('Altitude above Earth surface (km)')
                  ylabel('Orbital period (hours)')
                  title('Circular-orbit period versus altitude')

Example 3 — Elliptical orbit with Earth at a focus

                % Polar ellipse with true anomaly measured from perigee
                  a_km = 14000;
                  e = 0.35;
                  nu = linspace(0, 2*pi, 721);
                  r_km = a_km*(1-e^2) ./ (1 + e*cos(nu));
                  x_km = r_km .* cos(nu);
                  y_km = r_km .* sin(nu);
                  plot(x_km, y_km, 'LineWidth', 1.8)
                  hold on
                  plot(0, 0, 'o', 'MarkerSize', 10, 'MarkerFaceColor', [0.1 0.45 0.8])
                  axis equal; grid on
                  xlabel('x from Earth focus (km)')
                  ylabel('y from Earth focus (km)')
                  title('Elliptical orbit: Earth is at one focus')

Example 4 — Vis-viva around the ellipse

                % Speed around the same ellipse using vis-viva
                  mu_km3_s2 = 398600.4418;
                  a_km = 14000;
                  e = 0.35;
                  nu = linspace(0, 2*pi, 721);
                  r_km = a_km*(1-e^2) ./ (1 + e*cos(nu));
                  v_km_s = sqrt(mu_km3_s2 .* (2./r_km - 1/a_km));
                  plot(rad2deg(nu), v_km_s, 'LineWidth', 1.8)
                  grid on
                  xlabel('True anomaly (degrees)')
                  ylabel('Orbital speed (km/s)')
                  title('Vis-viva speed around an elliptical orbit')

Download Example 1 Download Example 2 Download Example 3 Download Example 4

09 — Check yourself

Reason across the whole story

1 · Shape

What does \(e=0\) mean?

The orbit is circular The spacecraft escapes Earth

2 · Period

Which parameter primarily determines two-body orbital period?

Semi-major axis Inclination True anomaly

3 · Geometry

If \(r_p\) and \(r_a\) are known, how is \(a\) obtained?

\(a=r_a-r_p\) \(a=(r_p+r_a)/2\)

4 · Engineering reasoning

Why can GEO simplify a ground terminal while making the radio link harder?

Fixed pointing is possible, but distance increases delay and path loss GEO removes delay and path loss

10 — Week summary

From gravity to communication consequences

Gravity bends inertial motion into an orbit. Bound two-body motion forms an ellipse with Earth at one focus. Semi-major axis controls period, eccentricity controls shape, and vis-viva explains why an elliptical satellite moves fastest at perigee. Circular-orbit altitude changes both speed and period, while GEO emerges directly by matching the sidereal rotation period.

Geometry
Radius is measured from Earth's center; altitude is measured from its surface.

Period
For a given central body, \(T\) is controlled by \(a\).

Speed
Higher circular orbits move more slowly; elliptical speed peaks at perigee.

Systems
Orbit choice shapes coverage, delay, loss, tracking, handover, and constellation needs.

Return to course overview

Why orbit matters to a communication engineer

Falling around Earth

Three laws, one coherent motion

First law — shape

Second law — changing speed

Third law — period

Concept check

Size, shape, perigee, and apogee

Concept check

Six numbers describe a Keplerian orbit

Higher circular orbit, slower motion, longer period

Concept check

Orbit class becomes system architecture

LEO

MEO

GEO

Derive GEO rather than memorize it

Concept check

From Orbital Data to Orbital Motion

Hubble Space Telescope

Given data

Altitude versus orbital radius

Predict the orbit shape

Reason toward orbital period

Visualize and connect forward

International Space Station

Given data

From orbital frequency to orbit size

Ground-track latitude

Earth's Natural Satellite

Given data

How long is a lunar orbit?

Orbital versus synodic period

Two estimates of eccentricity

HST and ISS

Discuss together

Turn orbital relations into experiments

Example 1 — Circular orbit from altitude

Example 2 — Period versus altitude

Example 3 — Elliptical orbit with Earth at a focus

Example 4 — Vis-viva around the ellipse

Reason across the whole story

1 · Shape

2 · Period

3 · Geometry

4 · Engineering reasoning

From gravity to communication consequences