Case Study: Budgeting Fifteen Years of Station-Keeping for a GEO Comsat

"A geostationary satellite is not parked. It is flown, every day, for fifteen years, just to stay still."

Executive Summary

In Case Study 1 we reverse-engineered a mission from its orbit. Here we do the forward job: given a requirement, we design the perturbation-fighting budget that will size a spacecraft's propellant, decide its propulsion technology, and cap its useful life. Our subject is a Track-A geostationary communications satellite that must hold its assigned slot — a small box in longitude and latitude — for fifteen years against the luni-solar and Earth-shape perturbations of this chapter. We will build its station-keeping delta-v budget line by line, convert it into propellant mass through the rocket equation, and confront the decision every real operator now faces: chemical or electric propulsion. The answer will turn out to be worth hundreds of kilograms — which, on a satellite, is worth tens of millions of dollars. This is the calculation you will run on your own Track-A mission.

Skills applied

• Identifying the dominant perturbation for a given orbit and sizing its correction (§12.5–12.6).
• Converting a luni-solar inclination drift into a north–south plane-change delta-v (Ch. 10, §12.5).
• Building a multi-year station-keeping budget and rolling it into a propellant mass via the rocket equation (Ch. 3).
• Running a chemical-versus-electric propulsion trade and reasoning about the lifetime it buys (theme 4).
• Adding a disposal maneuver and a sanity check against real industry figures.

Background

The requirement

• Orbit: geostationary — circular, equatorial ($i = 0^\circ$), $a = 42{,}164\ \text{km}$, speed $v = 3{,}075\ \text{m/s}$.
• Slot tolerance: the satellite must stay within $\pm 0.05^\circ$ in both longitude (east–west) and latitude (north–south) of its assigned $\text{longitude}$ — a box a fixed ground antenna can point into without tracking.
• Design life: $15\ \text{years}$ of service, plus a responsible end-of-life disposal.
• Spacecraft: wet mass $m_0 = 3{,}500\ \text{kg}$ at the start of station-keeping life.

The two enemies

From §12.5, a GEO satellite faces two perturbations that push it out of its box, and they cost very different amounts to fight:

1. North–south (latitude): the Sun and Moon, orbiting near the ecliptic rather than Earth's equator, torque the orbital plane and drive its inclination up by about $0.75$–$0.95^\circ$ per year. Correcting inclination is a plane change, which Chapter 10 showed is brutally expensive because it fights the full orbital velocity.
2. East–west (longitude): Earth's equator is not a perfect circle — it is slightly elliptical (the tesseral harmonic $J_{22}$) — so a satellite is pulled toward one of two stable longitudes ($\approx 75^\circ\text{E}$ and $\approx 105^\circ\text{W}$) and drifts away from the two unstable ones. Correcting longitude is a cheap in-track nudge.

Phase 1: Size the north–south budget

The dominant cost. To keep inclination near zero against a drift of $\Delta i = 0.85^\circ$ per year, the satellite must periodically perform a plane change that removes that accumulated inclination. From Chapter 10, the delta-v to change an orbital plane by an angle $\Delta i$ at speed $v$ is $\Delta v = 2v\sin(\Delta i/2) \approx v\,\Delta i_{\text{rad}}$:

$$ \Delta v_{\text{NS}} = 2\,(3{,}075)\sin\!\left(\frac{0.85^\circ}{2}\right) = 2(3{,}075)\sin(0.425^\circ) = 2(3{,}075)(0.007418) = 46\ \text{m/s per year.} $$

Round to $\sim 48\ \text{m/s/yr}$ to allow for the worst of the $18.6$-year lunar cycle. Over the $15$-year life:

$$ \Delta v_{\text{NS,total}} = 48 \times 15 = 720\ \text{m/s.} $$

💡 Intuition: why north–south dwarfs everything. A plane change is the most expensive maneuver in orbital mechanics because it must rotate the entire velocity vector without changing its length — you fight all $3{,}075\ \text{m/s}$ of orbital speed to move the plane a fraction of a degree. That is why one degree of inclination per year costs about $50\ \text{m/s}$, while drifting a degree in longitude costs almost nothing. The geometry of §12.5's luni-solar torque, filtered through Chapter 10's plane-change penalty, is what makes north–south keeping consume roughly $90\%$ of a GEO satellite's fuel.

Phase 2: Size the east–west budget and disposal

East–west keeping fights the gentle longitudinal pull of Earth's triaxiality (and a small contribution from solar radiation pressure, which nudges the eccentricity). Near a well-chosen slot this is small — on the order of $2$–$4\ \text{m/s}$ per year. Take $3\ \text{m/s/yr}$:

$$ \Delta v_{\text{EW,total}} = 3 \times 15 = 45\ \text{m/s.} $$

At end of life the satellite must vacate the crowded, valuable geostationary ring. De-orbiting into the atmosphere from GEO is prohibitively expensive (it would cost more delta-v than reaching GEO), so the standard is to boost a few hundred kilometers into a graveyard orbit (🔗 Ch. 9), which costs only about $\Delta v_{\text{disp}} = 11\ \text{m/s}$. The total station-keeping-plus-disposal budget is

$$ \Delta v_{\text{total}} = 720 + 45 + 11 = 776\ \text{m/s} \approx 0.78\ \text{km/s.} $$

🔗 Connection: this is the number Chapter 3 wrote on credit. Back in Chapter 3 we sketched a comsat delta-v budget with "$\sim 0.05\ \text{km/s}$ per year of station-keeping" and an "$\sim 0.01\ \text{km/s}$ disposal burn," summing to about $0.75\ \text{km/s}$ over its life. We have now derived that line from the actual perturbations — luni-solar torque and Earth's triaxiality — instead of quoting it. The budget line and the physics finally meet.

Phase 3: Convert the budget to propellant mass

The station-keeping delta-v is only meaningful once it becomes a mass, because that mass competes with payload and must be launched. Invert the rocket equation of Chapter 3: $m_p = m_0(1 - e^{-\Delta v/v_e})$.

import math
G0 = 9.80665

def propellant_mass(m0, dv, isp):
    ve = isp * G0
    return m0 * (1 - math.exp(-dv / ve))

dv = 48*15 + 3*15 + 11     # north-south + east-west + disposal, m/s
print("budget (m/s):", dv)
print("chemical  Isp=300 s:", round(propellant_mass(3500, dv, 300)), "kg")
print("electric  Isp=1500 s:", round(propellant_mass(3500, dv, 1500)), "kg")
# Expected output:
# budget (m/s): 776
# chemical  Isp=300 s: 811 kg
# electric  Isp=1500 s: 180 kg

The hand-check for the chemical case: $v_e = 300 \times 9.807 = 2{,}942\ \text{m/s}$, so $m_p = 3{,}500\,(1 - e^{-776/2{,}942}) = 3{,}500\,(1 - e^{-0.2638}) = 3{,}500\,(1 - 0.7682) = 811\ \text{kg}$. For the electric case, $v_e = 1{,}500 \times 9.807 = 14{,}710\ \text{m/s}$, so $m_p = 3{,}500\,(1 - e^{-776/14{,}710}) = 3{,}500\,(1 - 0.9486) = 180\ \text{kg}$.

Phase 4: The chemical-versus-electric trade

Lay the two options side by side:

Electric propulsion cuts the propellant from $811$ to $180\ \text{kg}$ — a $630\ \text{kg}$ saving — at the cost of perhaps $100\ \text{kg}$ of extra thrusters and solar-array power and a more complex system. The net gain, several hundred kilograms on a $3{,}500\ \text{kg}$ satellite, can become additional transponders (revenue) or extra propellant for a longer life. This is exactly why the geostationary comsat industry has largely switched to electric station-keeping over the past decade, accepting the slower burns (electric thrusters take hours, not minutes) in exchange for the mass.

🔧 Engineering Reality: the "inclined orbit" retirement trick. When a satellite's north–south propellant runs low near end of life, operators often stop correcting inclination and let it grow — spending fuel only on the cheap east–west keeping. The satellite then traces a small daily north–south figure-eight in the sky, which large steerable ground antennas can track even though fixed dishes cannot. Because north–south keeping was $\sim 90\%$ of the budget, abandoning it can extend a satellite's working life by years on the remaining fuel. It is a direct, money-making consequence of the fact that the luni-solar perturbation of §12.5 is the expensive one.

Phase 5: Lifetime, sanity check, and the mass verdict

Notice what the budget has quietly done: it has turned the abstract perturbations of this chapter into a hard limit on the mission. A satellite carries a fixed propellant load; every year of station-keeping spends part of it; when it runs out, the satellite can no longer hold its slot and must be retired — even if every other system still works perfectly. Station-keeping propellant, not electronics, is what usually ends a healthy geostationary satellite's life. That is theme four, mass is the enemy, in its starkest commercial form: the perturbations of §12.2–12.5, converted through the rocket equation, set the number of years the business has to earn back the satellite.

Sanity checks. Our $776\ \text{m/s}$ over 15 years matches the "$\sim 50\ \text{m/s}$ per year" figure quoted across the satellite industry and the $\sim 0.75\ \text{km/s}$ we budgeted in Chapter 3. The chemical propellant fraction, $811/3{,}500 \approx 23\%$, is right in the band real GEO comsats carry. And the north–south share, $720/776 \approx 93\%$, reproduces the well-known "north–south is about 90% of the budget" rule of thumb. Every number lands where the industry says it should — the sign that the physics of this chapter genuinely describes how satellites are flown.

Discussion Questions

1. North–south keeping costs about $16\times$ more than east–west, yet both hold the satellite to the same $\pm 0.05^\circ$ box. Explain, in terms of the type of maneuver each requires, why they cost so differently.
2. If the design life were cut from $15$ to $10\ \text{years}$, would the chemical propellant fall by exactly one-third? Compute both and explain any difference (hint: the rocket equation is not linear).
3. Electric thrusters produce milli-newtons and take hours to complete a correction. Why is that acceptable for station-keeping but not for the original launch to orbit?
4. A competitor proposes skipping north–south keeping entirely from day one to save fuel. What capability does the satellite lose, and which customers would refuse it?

Your Turn: Extensions

• Option A (design). Redo the budget for a $3\ \text{year}$ small GEO satellite and a $20\ \text{year}$ flagship. Plot propellant mass versus design life for both chemical and electric, and identify the life at which electric's mass saving exceeds its $\sim 100\ \text{kg}$ dry-mass penalty.
• Option B (computation). Extend the code to a function sk_budget(years, ns, ew, disp, m0, isp) that returns both the delta-v and the propellant mass, and tabulate the four combinations of {chemical, electric} × {10 yr, 15 yr}. (Do not run it; hand-trace and add # Expected output:.)
• Option C (your mission). For your Track-A comsat, choose a design life and propulsion type, compute the station-keeping propellant, and enter it as a line in your MDR's mass and delta-v budgets. Note explicitly which perturbation dominates your number.

Key Takeaways

1. North–south (luni-solar) keeping dominates a GEO budget — about $50\ \text{m/s}$ per year and $\sim 90\%$ of the total, because correcting inclination is a full-speed plane change.
2. A 15-year budget is $\sim 0.78\ \text{km/s}$ (north–south + east–west + graveyard disposal), the number Chapter 3 quoted and this chapter derives.
3. Perturbations become propellant becomes lifetime. Through the rocket equation, the budget is $\sim 23\%$ of a chemical satellite's mass, and running out of it — not hardware failure — usually ends the mission.
4. Electric propulsion saves hundreds of kilograms ($811 \to 180\ \text{kg}$ here) by raising $I_{sp}$, which is why the industry adopted it and why your Track-A design should seriously consider it.