Case Study: Designing a Far-Side Lunar Relay at Earth–Moon L2

"The far side of the Moon has one permanent disadvantage: it can never see home."

Executive Summary

In the first case study we took apart a mission that already flies. Here we design one. A lander is bound for the far side of the Moon — the hemisphere that, because the Moon keeps one face to Earth, never has a direct line of sight to home. No relay, no mission: the lander could gather data and never send a bit of it back. Our job is to place a communications relay that can see both the far-side lander and the Earth at the same time, continuously, for years. We will show that the Earth–Moon L2 point is the natural home for such a relay, size its halo orbit so the Moon never blocks the Earth link, estimate its station-keeping cost, and choose how to deliver it. This is exactly the design China flew as Queqiao ("Magpie Bridge") to enable the Chang'e-4 far-side landing in 2019 — and we will build the logic from scratch.

Skills applied

• Locating the Earth–Moon L2 point from the Hill radius (§15.3).
• Sizing a halo orbit from a line-of-sight geometry constraint (§15.4).
• Estimating station-keeping delta-v at an unstable point (§15.4).
• Weighing a direct vs. a low-energy delivery (§15.5).
• Connecting a Lagrange-point mission back to the delta-v budget of Chapter 3.

Background

The Moon is tidally locked: its rotation period equals its orbital period, so the same near side always faces Earth and the far side always faces away. A lander on the far side is permanently over the horizon from every antenna on Earth. The classic fix — a relay satellite in low lunar orbit — fails, because such a satellite spends half of every orbit behind the Moon itself, giving only intermittent coverage. What we want is a relay that hangs beyond the Moon, high enough to see over its limb to the far-side surface on one side and past its limb to Earth on the other, without ever being occulted. That is precisely the vantage an Earth–Moon L2 halo provides.

Phase 1: Requirements

• Coverage: continuous, simultaneous line of sight to (a) a lander near the center of the far side and (b) an Earth ground station.
• Lifetime: at least $5$ years.
• Constraint: the Moon (radius $1{,}737\ \text{km}$) must never block the relay-to-Earth link.
• Delivery: get the relay to its orbit within a reasonable time and modest delta-v.

Phase 2: Locate Earth–Moon L2

L2 lies about one Hill radius beyond the Moon, on the far side from Earth (§15.3):

def hill_radius(m1, m2, R):
    return R * (m2 / (3.0 * m1)) ** (1.0 / 3.0)

R_em = 384400.0                       # km, Earth-Moon distance
d_moon = hill_radius(5.972e24, 7.342e22, R_em)   # L2 distance beyond the Moon
d_earth = R_em + d_moon                          # L2 distance from Earth
print(f"EM L2 is {d_moon:,.0f} km beyond the Moon")
print(f"EM L2 is {d_earth:,.0f} km from Earth")
# Expected output:
# EM L2 is 61,514 km beyond the Moon
# EM L2 is 445,914 km from Earth

So the relay will hover about $61{,}500\ \text{km}$ past the Moon, some $446{,}000\ \text{km}$ from Earth. Sanity check: that is a modest fraction ($16\%$) of the Earth–Moon distance — comfortably inside the Earth–Moon system, as a Hill-radius distance should be — and it places the relay well above the far-side surface, with the Moon between it and Earth. That last fact is the whole design problem: from L2 exactly, the Moon sits squarely on the line to Earth and blocks the link. We must not sit on L2.

Phase 3: Size the halo so the Moon never blocks Earth

Because the relay cannot sit on the axis, we fly it in a halo orbit about L2 — a loop that keeps it swung off to the side, always with a clear sightline past the Moon's limb to Earth. How large must the loop be? The relay's lateral offset from the Earth–Moon line, projected to the Moon's distance, must exceed the Moon's radius:

R_moon = 1737.0        # km
# Earth->relay sightline crosses the Moon's distance at a fraction (R_em/d_earth)
# of the relay's lateral offset y; require that projected offset to clear the Moon:
#   y * (R_em / d_earth) > R_moon   ->   y > R_moon * d_earth / R_em
y_min = R_moon * d_earth / R_em
print(f"minimum halo lateral amplitude to clear the Moon: {y_min:,.0f} km")
# Expected output:
# minimum halo lateral amplitude to clear the Moon: 2,015 km

The geometry demands a lateral amplitude of at least about $2{,}000\ \text{km}$ just to peek past the limb — but that is a bare minimum, giving a grazing sightline and terrible coverage of the far-side surface. For robust, continuous coverage of both the lander (which needs the relay high in its sky) and Earth, we size the operational halo far larger — an amplitude of tens of thousands of kilometers.

(Real Queqiao flew a halo with amplitude of tens of thousands of km and a period of about two weeks — attributed figures of exactly this size, which is a reassuring check on our sizing.) The larger halo also keeps the relay clear of the awkward on-axis geometry at all times, not just on average.

Phase 4: Station-keeping

Earth–Moon L2 is unstable, like every collinear point — and because the Earth–Moon system is far more dynamic than the Sun–Earth system (a much larger mass parameter, $\mu^{*} \approx 0.012$, and strong solar perturbations), the drift is faster and the upkeep costlier than JWST's few m/s per year.

sk_rate = 20.0     # m/s per year, illustrative EM-L2 halo budget (Tier 3)
years = 5
print(f"5-year station-keeping delta-v ~ {sk_rate*years:.0f} m/s")
# Expected output:
# 5-year station-keeping delta-v ~ 100 m/s

On the order of $100\ \text{m/s}$ over five years — still tiny beside the kilometers per second to get to lunar space in the first place, but an order of magnitude more than a Sun–Earth libration mission, and a real line item in the relay's propellant budget. We fold it into the delta-v budget begun in Chapter 3: trans-lunar injection, then a Moon-flyby-and-capture into the L2 halo, then $\sim 100\ \text{m/s}$ of station-keeping over the mission.

Phase 5: Delivery — direct vs. low-energy

Two ways to get the relay to its L2 halo:

• Direct (fast). Trans-lunar injection ($\sim 3.1\ \text{km/s}$ from LEO, Chapter 3's delta-v map), a lunar swing-by, and insertion into the halo — a few days, at the cost of the insertion burn. This is what Queqiao did, reaching its halo in under a week via a lunar flyby.
• Low-energy (patient). A weak-stability-boundary path (§15.5) that loops out toward the Sun–Earth L1 region and drifts into the Earth–Moon L2 neighborhood, ballistically captured with almost no insertion burn — months instead of days, but propellant saved. For a relay that must be in place before the lander arrives, schedule usually wins, so we baseline the direct route and hold the low-energy option in reserve for a follow-on or a fuel-limited spare.

🔧 Engineering Reality: A relay is critical infrastructure — if it fails, the lander is deaf, so its reliability drives the design (Chapter 32). It must reach its halo and commission before the lander launches, its antennas must cover both links with margin, and it needs enough propellant for station-keeping plus contingency for the whole lander mission — you cannot refuel it $446{,}000\ \text{km}$ away. The Lagrange-point orbit is elegant; making it a dependable bridge for years is the engineering.

Discussion Questions

1. Why does a relay in low lunar orbit fail to give continuous far-side coverage, while an Earth–Moon L2 halo succeeds? Frame your answer in terms of occultation by the Moon.
2. We required a minimum halo amplitude of $\sim 2{,}000\ \text{km}$ but chose tens of thousands. What does the extra amplitude buy, and what does it cost?
3. Earth–Moon L2 station-keeping ($\sim 20\ \text{m/s/yr}$) is much larger than Sun–Earth L2's ($\sim 3\ \text{m/s/yr}$). Give a physical reason rooted in the mass parameter and the environment.
4. Why must the relay be delivered and checked out before the lander launches, and how does that argue for the direct delivery over the low-energy one?

Your Turn: Extensions

• Option A (design). Your lander is now near the far side's south pole, not its center. Argue whether an Earth–Moon L2 halo still gives good coverage, and whether you might prefer a different orbit family (e.g., a larger halo, or a highly elliptical "frozen" lunar orbit). Sketch the geometry.
• Option B (computation). Write earth_moon_L2(mass_earth, mass_moon, R) returning the L2 distance from the Moon and from Earth, and min_halo_amplitude(R_moon, d_earth, R) returning the Moon-clearing amplitude. Hand-trace both for the values above; do not run them. Then explore how min_halo_amplitude changes if the Moon were half its radius.
• Option C (your mission). If your MDR track involves the Moon (Track B), add this relay as a supporting asset: note its orbit, its coverage constraint, its station-keeping line in your delta-v budget, and its delivery choice. If your track does not, write one sentence on why a lunar relay is irrelevant to it.

Key Takeaways

1. A far-side lander needs a relay, and Earth–Moon L2 is its natural home — the one perch that sees the far side and Earth at once, beyond the Moon, without going behind it.
2. The halo must be sized to a geometry constraint: its lateral amplitude must exceed the Moon's radius projected to the relay's distance ($\gtrsim 2{,}000\ \text{km}$ minimum; tens of thousands for robust coverage).
3. Living at an unstable Earth–Moon point costs $\sim 20\ \text{m/s/yr}$ — modest, but an order of magnitude more than a Sun–Earth halo, because the Earth–Moon system is more dynamic.
4. Delivery is a time-vs-fuel trade (direct lunar-flyby insertion vs. a low-energy ballistic capture); for critical relay infrastructure that must be in place first, the fast route usually wins.