Case Study: Designing an All-Electric GEO Communications Satellite

"The cheapest kilogram to orbit is the one you never launch."

Executive Summary

In the first case study we took a flown mission apart. Here we build one on paper — and we build it the way the modern satellite industry actually does. A geostationary communications satellite must reach GEO, hold its slot for fifteen years, and dispose of itself responsibly at the end. The traditional answer is a chemical apogee engine and years of chemical station-keeping. The modern answer, pioneered by platforms like Boeing's 702SP, is to do all of it electrically: raise the orbit with a Hall thruster, keep station with the same thruster, and deorbit with it too. We will size both designs from a common on-station mass, compare their launch masses, and discover why the all-electric satellite launches at roughly one-third the mass of its chemical twin — and what it pays for that in time.

Skills applied

• Building a mission delta-v budget from orbit-raising, station-keeping, and disposal legs (§20.6).
• Sizing propellant top-down through the rocket equation for both electric and chemical options (§20.1, §20.6).
• Reading thrust and transfer time from a thruster's power and $I_{sp}$ (§20.5).
• Judging a design by launch mass and transfer time — the two numbers that decide a comsat's economics.

Background

The requirement

• On-station mass (bus + payload, no propellant): $m_{\text{station}} = 2{,}000\ \text{kg}$.
• Drop-off: the launch vehicle delivers the satellite to a $400\ \text{km}$ LEO; the satellite makes its own way to GEO. (This is the extreme, most-demanding case; we revisit the industry's shortcut in Phase 5.)
• Delta-v legs the satellite's own propulsion must supply: 1. Orbit-raising, LEO → GEO. 2. Station-keeping, $50\ \text{m/s}$ per year for a 15-year life (Chapter 12). 3. Disposal, a small boost to a graveyard orbit at end of life (Chapter 9).

The design choice

We will size two propulsion options against identical requirements:

• All-electric: a Hall thruster at $I_{sp} = 1{,}800\ \text{s}$ ($v_e = 17{,}652\ \text{m/s}$) for every leg.
• Chemical baseline: a bipropellant system at $I_{sp} = 320\ \text{s}$ ($v_e = 3{,}138\ \text{m/s}$) for every leg.

Everything the electric option does, it does at nearly six times the specific impulse — and, as we will see, that ratio is the whole story.

Phase 1: The delta-v budget

Each leg's delta-v (Tier 2/3 — representative figures):

Note the orbit-raising line: the electric spiral costs more delta-v ($4.59$ vs $3.85\ \text{km/s}$, the 19% low-thrust penalty we derived in §20.6), while station-keeping and disposal are the same small burns for both. If delta-v were the bill, electric would already be losing. It is not the bill — propellant mass is — so we press on.

Phase 2: Size the all-electric satellite (top-down)

Because each leg must carry the propellant for every later leg, we size from the end of life backward, exactly as we sized stages from the payload down in Chapter 3. For each leg the propellant is $m_p = m_{\text{after}}\left(e^{\Delta v/v_e} - 1\right)$, where $m_{\text{after}}$ is the mass the satellite still has after that leg is done. Using $v_e = 17{,}652\ \text{m/s}$ throughout:

Disposal (last, $\Delta v = 11\ \text{m/s}$, ending at $2{,}000\ \text{kg}$): $$m_p = 2{,}000\left(e^{11/17{,}652} - 1\right) = 2{,}000\,(0.000623) = 1.2\ \text{kg}.$$ Mass before disposal: $2{,}001.2\ \text{kg}$.

Station-keeping ($\Delta v = 750\ \text{m/s}$, ending at $2{,}001.2\ \text{kg}$): $$m_p = 2{,}001.2\left(e^{750/17{,}652} - 1\right) = 2{,}001.2\,(0.04340) = 86.9\ \text{kg}.$$ Mass at start of station life (end of orbit-raising): $2{,}088.1\ \text{kg}$.

Orbit-raising ($\Delta v = 4{,}590\ \text{m/s}$, ending at $2{,}088.1\ \text{kg}$): $$m_p = 2{,}088.1\left(e^{4{,}590/17{,}652} - 1\right) = 2{,}088.1\,(0.29696) = 620.1\ \text{kg}.$$

Totals: xenon $= 1.2 + 86.9 + 620.1 = 708\ \text{kg}$; launch-to-LEO mass $= 2{,}000 + 708 = \mathbf{2{,}708\ \text{kg}}$.

Phase 3: Size the chemical baseline (same method)

Now the identical calculation with $v_e = 3{,}138\ \text{m/s}$ and the Hohmann orbit-raising delta-v:

Disposal ($11\ \text{m/s}$): $m_p = 2{,}000\,(e^{11/3{,}138}-1) = 7.0\ \text{kg}$. Mass before: $2{,}007.0\ \text{kg}$.

Station-keeping ($750\ \text{m/s}$): $m_p = 2{,}007.0\,(e^{750/3{,}138}-1) = 2{,}007.0\,(0.2700) = 541.8\ \text{kg}$. Mass before: $2{,}548.8\ \text{kg}$.

Orbit-raising ($3{,}850\ \text{m/s}$): $m_p = 2{,}548.8\,(e^{3{,}850/3{,}138}-1) = 2{,}548.8\,(2.410) = 6{,}143.8\ \text{kg}$.

Totals: propellant $= 7.0 + 541.8 + 6{,}143.8 = 6{,}693\ \text{kg}$; launch-to-LEO mass $= 2{,}000 + 6{,}693 = \mathbf{8{,}693\ \text{kg}}$.

We can package the whole calculation in a reusable sizing function:

import math

G0 = 9.80665  # standard gravity, m/s^2

def leg_propellant(mass_after, dv, isp):
    """Propellant (kg) for one delta-v leg, given the mass carried AFTER the leg."""
    ve = isp * G0
    return mass_after * (math.exp(dv / ve) - 1)

def size_from_top(m_station, legs):
    """legs in TIME order as (dv_ms, isp_s); build backward from the on-station mass.
    Returns (total_propellant_kg, launch_mass_kg)."""
    m, total = m_station, 0.0
    for dv, isp in reversed(legs):
        p = leg_propellant(m, dv, isp)
        total += p
        m += p
    return total, m

if __name__ == "__main__":
    m_station = 2000.0
    electric = [(4590, 1800), (750, 1800), (11, 1800)]  # Hall, Isp 1800 s
    chemical = [(3850, 320),  (750, 320),  (11, 320)]   # bipropellant, Isp 320 s
    pe, le = size_from_top(m_station, electric)
    pc, lc = size_from_top(m_station, chemical)
    print("all-electric: propellant", round(pe), "kg, launch to LEO", round(le), "kg")
    print("chemical    : propellant", round(pc), "kg, launch to LEO", round(lc), "kg")
    print("electric launch mass is", round(le / lc * 100), "% of chemical")
    # Expected output:
    # all-electric: propellant 708 kg, launch to LEO 2708 kg
    # chemical    : propellant 6693 kg, launch to LEO 8693 kg
    # electric launch mass is 31 % of chemical

Phase 4: Compare — and see where the mass went

The all-electric satellite launches at 2,708 kg — about 31% of the chemical satellite's 8,693 kg, saving nearly $6{,}000\ \text{kg}$, or roughly two-thirds of the launch mass, for exactly the same on-station capability. Almost all of that saving is in the orbit-raising leg: the chemical apogee burn needs a mass ratio of $e^{3850/3138} = 3.41$, so it must carry more than $6\ \text{tonnes}$ of propellant just to climb to GEO, while the electric spiral needs only $620\ \text{kg}$ of xenon despite paying a higher delta-v. The rocket equation's exponential, so cruel to the chemical vehicle, is nearly toothless against a $17{,}652\ \text{m/s}$ exhaust.

💡 Intuition: The exponential $e^{\Delta v/v_e}$ is where all the mass lives. For the chemical raise, $\Delta v/v_e = 3850/3138 = 1.23$ — a mass ratio of 3.4, so most of the vehicle is propellant. For the electric raise, $\Delta v/v_e = 4590/17652 = 0.26$ — a mass ratio of just 1.30, so propellant is a thin skin on the satellite. Six times the exhaust velocity moves you from the steep part of the exponential to the gentle part, and that is the whole ballgame.

This is not a paper exercise. Because two $2.7$-tonne all-electric satellites weigh less together than one $8.7$-tonne chemical satellite, operators can launch two comsats on a single rocket — precisely the economics that made all-electric platforms commercially popular. A dramatically lighter satellite means a smaller, cheaper launch, or a shared ride.

Phase 5: The catch — time, and how the industry beats it

Nothing is free. Size the orbit-raising thruster and time it. A $4.5\ \text{kW}$ Hall thruster at $I_{sp} = 1{,}800\ \text{s}$ and $\eta = 0.55$ makes $$F = \frac{2\eta P}{v_e} = \frac{2(0.55)(4{,}500)}{17{,}652} = 0.280\ \text{N},$$ and its mass flow is $\dot m = F/v_e = 1.59\times10^{-5}\ \text{kg/s}$. Expelling the $620\ \text{kg}$ of orbit-raising xenon then takes $$t = \frac{620}{1.59\times10^{-5}} = 3.9\times10^{7}\ \text{s} \approx 452\ \text{days} \approx 15\ \text{months}.$$ Fifteen months of spiraling before the satellite earns a dollar — versus a chemical transfer of a few hours. With two thrusters running, the raise halves to about $226\ \text{days}$ (roughly seven months), but even that is a long time to fly through the Van Allen belts (accumulating radiation dose) before service.

This is why real all-electric comsats do not spiral all the way from LEO. They launch to a geostationary transfer orbit instead — an ellipse whose apogee already reaches GEO altitude — so the electric system need only raise perigee and trim inclination, an orbit-raising delta-v of roughly $1.8\ \text{km/s}$ rather than $4.59$. That cuts the electric raise to about six months and the xenon to a couple of hundred kilograms, while still launching far lighter than a chemical bird. The design lesson is the one that runs through the whole book: you trade delta-v, propellant, launch mass, and time against one another, and the best design depends on which of those your mission can most afford to spend.

🔧 Engineering Reality: Our even, single-$I_{sp}$ design is deliberately simple. Real all-electric platforms may use higher-$I_{sp}$ gridded ion thrusters (some run near $3{,}500\ \text{s}$), several thrusters at once, and carefully optimized transfer profiles that thrust only near perigee. Those refinements change the numbers by tens of percent — they do not change the headline, which is that high specific impulse turns a $6$-tonne propellant load into a few hundred kilograms.

Discussion Questions

1. The electric orbit-raising leg costs more delta-v than the chemical one, yet needs one-tenth the propellant. Explain the apparent paradox using the exponent $\Delta v/v_e$.
2. Station-keeping was $750\ \text{m/s}$ for both designs, but the electric option's station-keeping propellant ($87\ \text{kg}$) was far less than the chemical option's ($542\ \text{kg}$). Why does high $I_{sp}$ help even more, proportionally, on the small station-keeping leg than you might expect?
3. If the launch vehicle instead dropped the satellite at GTO (electric raise $\approx 1.8\ \text{km/s}$), re-estimate the orbit-raising xenon for the all-electric design. Roughly how much lighter does the satellite get?
4. The all-electric satellite spends ~6–15 months raising its orbit before service. Name two mission costs of that delay (financial and physical) that might make an operator choose chemical after all.

Your Turn: Extensions

• Option A (analysis). Redo Phases 2–4 with a gridded ion thruster at $I_{sp} = 3{,}500\ \text{s}$ instead of the Hall's $1{,}800\ \text{s}$. How much more xenon does the higher $I_{sp}$ save — and what does it cost in orbit-raising time (at the same power)?
• Option B (computation). Extend size_from_top to also return each leg's propellant separately, and print a table for both designs. (Do not run it — hand-trace and add # Expected output:.)
• Option C (your mission). If your mission is Track A (GEO comsat), run this sizing for your on-station mass and record the all-electric xenon load, launch mass, and orbit-raising time in your MDR. If Track B/C/D, adapt the leg list to your mission's delta-v budget.

Key Takeaways

1. Size top-down through the rocket equation, one leg at a time, because each leg carries the propellant for every later leg.
2. High $I_{sp}$ collapses the launch mass. The all-electric comsat launches at ~31% of its chemical twin (2,708 vs 8,693 kg) for the same capability — the difference between one launch and two, or a big rocket and a small one.
3. The electric spiral costs more delta-v but far less propellant, because $v_e$ sits in the exponent; $6\times$ the exhaust velocity beats a 19% delta-v penalty many times over.
4. The price is time. Months of orbit-raising is why operators launch to GTO rather than spiral from LEO — trading a little launch performance to shorten the wait to revenue service.