Case Study 2: Designing a Communications System for the Far North

"The right engineering answer depends on where you stand — literally." — a lesson from §9.4

Executive Summary

Where the first case study audited an existing system, this one asks you to design one. A northern nation — call its service region the band of latitudes from $60^\circ$ to $80^\circ\ \text{N}$, including remote Arctic settlements — wants continuous, reliable television and broadband relay. The obvious answer, geostationary orbit, quietly fails this requirement, because from the far north a geostationary satellite sits almost on the horizon. In this study we prove that GEO fails, then design the orbit that works: a Molniya orbit that loiters high over the north for hours at a time. We size its semi-major axis, eccentricity, and apogee; explain its peculiar $63.4^\circ$ inclination; and work out how many satellites a continuous service needs. Finally we run a trade study — Molniya versus Tundra versus a heroic GEO — and justify a recommendation. This is orbit selection as a real, defensible engineering decision (§9.6).

Skills applied

• Geostationary coverage geometry and its high-latitude failure (§9.3, §9.4).
• Designing a highly elliptical / Molniya orbit from a period requirement (§9.4).
• Kepler's second law as a coverage tool — apogee dwell (§9.4, Chapter 8).
• Constellation sizing and hand-over timing; a structured trade study (§9.6, forward to Chapter 29).

Background

A communications satellite is only useful to a ground station when it is high enough above the local horizon to clear terrain, buildings, and atmospheric noise — engineers usually demand a minimum elevation angle of at least $5^\circ$–$10^\circ$, and more is better. For a user near the equator, a geostationary satellite rides high overhead and easily clears this bar. But our users are at $60^\circ$–$80^\circ\ \text{N}$, and that changes everything: a satellite parked over the equator, tens of thousands of kilometers to the south, appears very low in their sky. Our design must put a satellite high over the north, and keep one there at all times. Let us first quantify the problem, then solve it.

Phase 1 — The requirement, stated in numbers

• Service region: latitudes $60^\circ$–$80^\circ\ \text{N}$ (with Arctic outposts approaching the pole).
• Availability: continuous — a usable satellite in view $24$ hours a day, $365$ days a year.
• Minimum elevation: at least $\sim 20^\circ$ desired for a robust high-latitude link (terrain and low winter sun angles make marginal elevations unreliable).
• Ground segment: steerable antennas are acceptable (they can track a moving satellite and hand over between satellites), so we are not forced into a stationary-in-the-sky solution.

That last point is the crucial freedom: because our antennas can track, we are not chained to GEO, and we can consider orbits where the satellite moves — as long as one is always usably high in the north.

Phase 2 — Why geostationary orbit fails

Compute the elevation angle of a geostationary satellite (radius $42{,}164\ \text{km}$) as seen from a ground station on the same meridian, at a range of northern latitudes. For a station at latitude $L$ on the satellite's meridian, the Earth-central angle to the sub-satellite point equals $L$, and $$\tan(\text{elevation}) = \frac{\cos L - R_\text{eq}/r_s}{\sin L}.$$

import math
Re, rs = 6378.0, 42164.0
k = Re / rs
for L in [55, 65, 70, 75, 80]:
    g = math.radians(L)
    el = math.degrees(math.atan((math.cos(g) - k) / math.sin(g)))
    print("lat", L, "-> GEO elevation", round(el, 1), "deg")
# Expected output:
# lat 55 -> GEO elevation 27.3 deg
# lat 65 -> GEO elevation 16.7 deg
# lat 70 -> GEO elevation 11.5 deg
# lat 75 -> GEO elevation 6.4 deg
# lat 80 -> GEO elevation 1.3 deg

The verdict is damning. At $65^\circ\ \text{N}$ a geostationary satellite is only $17^\circ$ above the horizon — and that is the best case, on the satellite's own meridian; a station east or west of it sees the satellite even lower. By $75^\circ\ \text{N}$ it is a marginal $6^\circ$, and past $81^\circ$ it drops below the horizon entirely and cannot be seen at all. In the real world of mountains, forests, and buildings, a $6^\circ$ satellite is routinely blocked. GEO does not meet the requirement. We must abandon the equatorial belt.

Phase 3 — Designing the Molniya orbit

The fix, invented by Soviet engineers facing exactly this geography, is to give up the circle and place a slow, high apogee over the northern hemisphere. We design the orbit in four decisions.

Decision 1 — Period. Choose a period of half a sidereal day ($43{,}082\ \text{s}$), so the ground track repeats daily and the apogee returns to the same northern longitude every orbit. Invert Kepler's third law for the semi-major axis.

Decision 2 — Perigee and eccentricity. Put the perigee low, at $1{,}000\ \text{km}$ altitude, over the southern hemisphere; the apogee then rises high over the north.

import math
MU, R = 3.986e5, 6371.0
T = 43082.0                                   # half a sidereal day
a = (MU * T**2 / (4 * math.pi**2)) ** (1/3)
r_p = R + 1000.0                              # perigee 1000 km up
r_a = 2*a - r_p                               # apogee radius
e = (r_a - r_p) / (r_a + r_p)
v_a = math.sqrt(MU * (2/r_a - 1/a))           # apogee speed (vis-viva)
print("semi-major axis (km):", round(a))
print("apogee altitude (km):", round(r_a - R))
print("eccentricity:", round(e, 3))
print("apogee speed (km/s):", round(v_a, 2))
# Expected output:
# semi-major axis (km): 26562
# apogee altitude (km): 39381
# eccentricity: 0.722
# apogee speed (km/s): 1.55

Our satellite reaches an apogee of $\sim 39{,}400\ \text{km}$ — higher than GEO — directly over the north, and by vis-viva it crawls through that apogee at just $1.55\ \text{km/s}$, versus roughly $9.6\ \text{km/s}$ at perigee. That slowness is the entire design: by Kepler's second law (equal areas in equal times) the satellite dwells near apogee, hanging high over the northern service region for hours, and races through the southern perigee in minutes.

Decision 3 — Apogee placement. Set the argument of perigee to $270^\circ$ so that apogee occurs at the northernmost point of the orbit — over the service region, not the southern hemisphere.

Decision 4 — Inclination $63.4^\circ$. Here is the subtle one. Earth's equatorial bulge would ordinarily make the orbit's perigee (and thus apogee) rotate slowly around the orbit over months, dragging our carefully placed apogee away from the north. At the critical inclination of $63.4^\circ$, this rotation cancels exactly, and the apogee stays frozen over the northern hemisphere for years. (The J2 effect that causes and cancels this drift is derived in Chapter 12; for the design we simply adopt $63.4^\circ$ as the inclination that holds the apogee in place.)

Phase 4 — Sizing the constellation

One satellite is not enough: for the hours it spends near the southern perigee, it is useless to the north. We must fly several, staggered in time so that as one dives away toward perigee, the next is already climbing into its northern apogee dwell.

A Molniya satellite is operationally usable — high enough over the service region — for roughly 8 hours of each $\sim 12$-hour orbit (the arc around apogee). To guarantee continuous coverage:

period_h = 11.97          # orbit period, hours
usable_h = 8.0            # usable arc around apogee, hours
import math
n_min = math.ceil(period_h / usable_h)
print("minimum satellites (coverage ratio):", n_min)
print("operational choice with margin:", 3)
print("stagger between satellites (h):", round(period_h / 3, 2))
# Expected output:
# minimum satellites (coverage ratio): 2
# operational choice with margin: 3
# stagger between satellites (h): 3.99

The bare coverage ratio ($12/8$) says two satellites could in principle overlap, but phasing two so that their $8$-hour dwells never leave a gap is fragile — any drift opens a hole. The robust, standard answer is three satellites, staggered about $8$ hours apart in their orbits (equivalently, apogees passing over the region every $\sim 8$ hours), so that at every instant at least one satellite is comfortably within its northern dwell, with overlap for clean hand-overs. This is exactly the architecture the Soviet Molniya network used (with a fourth satellite added for on-orbit spare and margin).

Phase 5 — The trade study and recommendation

Three architectures can serve the far north. Score them against the requirement.

The Tundra orbit — a geosynchronous ($24$-hour) highly elliptical orbit, also at $63.4^\circ$, tracing a lazy figure-eight over one region — is a real alternative: with a $24$-hour period, two or three satellites can cover a region, and each satellite's apogee returns to the same spot daily. It trades a smaller constellation and a tidy repeating track for a higher, more expensive orbit and a longer transfer. Modern satellite-radio broadcasters use Tundra orbits for exactly this reason.

Recommendation. For a dedicated far-north service demanding high elevation and $24/7$ availability, a three-satellite Molniya constellation ($63.4^\circ$, perigee $\sim 1{,}000\ \text{km}$, apogee $\sim 39{,}400\ \text{km}$, apogee over the service longitude) is the defensible choice: it is proven, sized modestly, and puts a satellite nearly overhead in the Arctic where GEO cannot reach. If the operator also serves customers at mid-latitudes and values a simpler ground track, a Tundra system is the fallback. Either way, the geometry of §9.4 — not preference — chose the orbit. A launch from a high-latitude site (itself near $63^\circ$) is a bonus: launching due into the $63.4^\circ$ inclination wastes no delta-v fighting the site's latitude, an interaction we develop in Chapter 30.

Discussion Questions

1. Our GEO elevation numbers assumed the ground station sits on the satellite's meridian. Explain why a station at a different longitude sees the GEO satellite lower still, and why this strengthens the case against GEO for a wide-longitude service.
2. Why is the Molniya perigee deliberately placed over the southern hemisphere? What would go wrong if the apogee were over the south instead?
3. The design uses a period of exactly half a sidereal day. What operational advantage does the resulting daily-repeating ground track give the operator, and how does it relate to the GPS repeat in Case Study 1?
4. A colleague proposes using just two Molniya satellites to save money. Describe the specific failure mode (when and where does coverage drop out?) and why three is the safer number.

Your Turn: Extensions

• Option A (redesign). Re-run Phase 3 with a perigee altitude of $500\ \text{km}$ instead of $1{,}000\ \text{km}$. How do the eccentricity and apogee altitude change, and what is the trade (higher apogee dwell versus a faster, lower, more radiation- and drag-exposed perigee)?
• Option B (Tundra). Design the Tundra alternative: repeat Phase 3 with a full sidereal-day period ($86{,}164\ \text{s}$) and a milder eccentricity of $e = 0.27$. Compute the semi-major axis, apogee, and perigee, and argue how many satellites a Tundra system needs.
• Option C (your mission). If your Mission Design Track serves a specific region or latitude band (or if you invent a regional-service variant of it), decide whether a GEO, Molniya, or Tundra architecture fits, and record the decision and its one-line justification in your MDR.

Key Takeaways

• Geostationary orbit fails the far north: its elevation angle falls below usable limits past $\sim 70^\circ$ latitude and reaches zero near $81^\circ$. Geometry, not engineering effort, is the barrier.
• A Molniya orbit solves it by placing a slow, high apogee over the north; Kepler's second law turns that slowness into hours of dwell — coverage bought purely by orbit shape.
• The $63.4^\circ$ critical inclination freezes the apogee over the north by cancelling J2-driven apsidal drift (the mechanism is Chapter 12's; the design just adopts the value).
• Continuous service needs a three-satellite constellation, staggered so their apogee dwells overlap.
• Orbit selection is a trade study: Molniya versus Tundra versus GEO, scored against a real requirement — the first genuine design decision of the mission, exactly as §9.6 argues.