Case Study: Designing a Satellite-Servicing Rendezvous

"Getting to the right orbit is the easy part. Arriving at the right place in that orbit, at the right time, next to something else moving at eight kilometers a second — that is rendezvous."

Executive Summary

In this case study you design a mission rather than audit one. A servicing spacecraft — a small robotic vehicle built to refuel or repair an aging satellite — has been delivered to a low parking orbit. Its target is a client satellite a few hundred kilometers higher, in the same plane but well ahead along the orbit. Your job is to plan the complete maneuver sequence that brings the two together: a Hohmann transfer to match altitude, a phasing campaign to close the along-track gap, and a final approach — then to roll it all into a delta-v budget with margin.

This is one tier harder than the launch-site audit of Case Study 1 because it composes the chapter's maneuvers: transfer, phasing, and rendezvous, each feeding the next, with a timing constraint that ties them together. Every number is built from vis-viva (Ch. 6) and Kepler's third law (Ch. 8); the scenario is representative of real rendezvous profiles (Tier 3 — a constructed but realistic mission).

Skills applied

• Designing a Hohmann transfer between two circular orbits (§10.2).
• Designing a phasing campaign with Kepler's third law to hit a moving target (§10.5).
• Keeping a phasing orbit physically safe (above the atmosphere) by choosing the number of revolutions.
• Rolling maneuver legs into a delta-v budget with margin (§10.6, the Checkpoint, and Ch. 3/29).

Background

The starting state. The servicer is in a circular parking orbit at $400\ \text{km}$ altitude, $r_1 = 6{,}771\ \text{km}$, inclination matched to the target (it was launched into the target's plane on purpose — Section 10.4 taught us never to fix the plane later). Its speed is $v_{c1} = 7.673\ \text{km/s}$.

The target. A client satellite in a circular orbit at $700\ \text{km}$ altitude, $r_2 = 7{,}071\ \text{km}$, same plane, moving at $v_{c2} = \sqrt{\mu/r_2} = 7.508\ \text{km/s}$. At the moment the servicer is ready to begin, the target is $45^\circ$ ahead along the orbit.

The goal. Arrive co-orbiting with the target at $700\ \text{km}$, at nearly zero relative velocity, ready for the final delicate approach. We will size every burn and total the delta-v.

Phase 1: Match the plane (cost: zero, by design)

The single most expensive thing we could do — a plane change (§10.4) — we have already avoided by launching into the target's plane. This is the first design decision, and it is worth stating explicitly because it is the one that saves the most: had the target been in a plane $10^\circ$ away, matching it would have cost $2(7.673)\sin 5^\circ \approx 1.34\ \text{km/s}$ in LEO, dwarfing everything else in this budget. Plane match: $0\ \text{km/s}$ — the cheapest maneuver is the one you designed away.

Phase 2: Hohmann transfer to the target's altitude

Raise the servicer from $400$ to $700\ \text{km}$ with a standard two-burn Hohmann transfer (§10.2). The transfer ellipse has $$a_t = \frac{6{,}771 + 7{,}071}{2} = 6{,}921\ \text{km}.$$ The perigee (departure) speed is $v_p = \sqrt{\mu(2/6{,}771 - 1/6{,}921)} = \sqrt{60.13} = 7.755\ \text{km/s}$, so the first burn is $$\Delta v_1 = 7.755 - 7.673 = 0.082\ \text{km/s} = 82\ \text{m/s (prograde)}.$$ The apogee (arrival) speed is $v_a = \sqrt{\mu(2/7{,}071 - 1/6{,}921)} = \sqrt{55.14} = 7.425\ \text{km/s}$, so the second burn is $$\Delta v_2 = 7.508 - 7.425 = 0.083\ \text{km/s} = 83\ \text{m/s (prograde)}.$$ Hohmann total: $0.164\ \text{km/s}$. The transfer takes $t = \pi\sqrt{6{,}921^3/\mu} = \pi(912\ \text{s}) = 2{,}865\ \text{s} \approx 48\ \text{minutes}$ (half of the transfer ellipse's period).

Design note. This altitude change is cheap — only $164\ \text{m/s}$ — because $400\ \text{km}$ and $700\ \text{km}$ are close together (ratio $1.04$). The expensive part of this mission will not be climbing; it will be arriving at the right place, which is the phasing problem next.

Phase 3: Phasing to close the $45^\circ$ gap

After the Hohmann transfer the servicer is co-altitude with the target but $45^\circ$ behind it (the target was $45^\circ$ ahead and both kept moving). To catch up, we use a phasing orbit (§10.5): drop into a slightly lower, faster orbit, lap around gaining on the target, then return.

The design choice is how many revolutions to spread the catch-up over. Doing it in one revolution would demand a large period change — and a dangerously low phasing orbit. Let us design it over 3 revolutions, gaining $15^\circ$ per revolution. Each phasing period must be short enough that the target (moving at $360^\circ$ per its period $T_2$) advances only $360^\circ - 15^\circ = 345^\circ$ while the servicer completes one full phasing lap:

import math
MU = 3.986e5   # km^3/s^2

def period(a):               return 2*math.pi*math.sqrt(a**3 / MU)
def sma_from_period(T):      return (MU * (T/(2*math.pi))**2) ** (1/3)
def vis_viva(r, a):          return math.sqrt(MU * (2/r - 1/a))

R2 = 7071.0
T2 = period(R2)                       # target orbital period
T_ph = (345/360) * T2                 # gain 15 deg per rev (45 deg over 3 revs)
a_ph = sma_from_period(T_ph)          # phasing-orbit semi-major axis
dv_each = abs(vis_viva(R2, R2) - vis_viva(R2, a_ph))   # burn at r2 (apogee of phasing orbit)
print(round(T2), round(T_ph))
print(round(a_ph, 1), round(2*a_ph - R2, 1))           # a_ph and phasing perigee radius
print(round(dv_each, 3), round(2*dv_each, 3))
# Expected output:
# 5917 5671
# 6873.4 6675.9
# 0.109 0.218

The phasing orbit has semi-major axis $a_{\text{ph}} = 6{,}873\ \text{km}$ and, since we burn retrograde at $700\ \text{km}$ (making that the apogee), a perigee of $2a_{\text{ph}} - r_2 = 6{,}676\ \text{km}$ — altitude $305\ \text{km}$, comfortably above the atmosphere. Each burn (drop in, and later circularize back) costs $0.109\ \text{km/s}$, for a phasing total of $0.218\ \text{km/s}$, and the campaign takes $3 \times 5{,}671\ \text{s} = 17{,}013\ \text{s} \approx 4.7\ \text{hours}$.

The revolution tradeoff. Had we insisted on catching up in a single revolution ($45^\circ$ in one lap), the phasing period would need to be $(315/360)T_2$, forcing a much lower, faster orbit whose perigee would graze the upper atmosphere — and costing far more delta-v. Spreading the catch-up over more revolutions keeps the phasing orbit gentle and cheap, at the price of time. This is exactly why real cargo vehicles take a day or more to reach the ISS: patience is fuel.

Phase 4: Final approach and proximity operations

With the gap closed to a few kilometers, the servicer stops thinking in whole orbits and switches to relative-motion guidance — the slow, looping approach along the "V-bar" (velocity direction) governed by the relative dynamics we flagged in Section 10.5 and formalize with orbit determination in Chapter 13. These maneuvers are small: a series of nudges totaling, on a mission like this, on the order of $30\ \text{m/s}$, flown at centimeters per second in the final meters to a soft capture. We carry $0.030\ \text{km/s}$ as a proximity-operations allowance.

Phase 5: The delta-v budget and sanity check

Roll up the legs, exactly as dv_budget will in the Mission Design Checkpoint:

Sanity check. Two independent features confirm the plan. First, the phasing cost ($0.218\ \text{km/s}$) is larger than the altitude change ($0.164\ \text{km/s}$) — which is correct and instructive: on a rendezvous, arriving at the right place and time often costs more than getting to the right orbit. Second, every burn is small (tens to low hundreds of m/s) because both orbits are low and close together; nothing here approaches the kilometers-per-second of a GEO transfer or a plane change, so a total under $0.5\ \text{km/s}$ is exactly the right order of magnitude for a same-plane LEO rendezvous.

Discussion Questions

1. The phasing campaign cost more delta-v than the altitude transfer. Explain physically why closing an along-track gap can be more expensive than changing orbital altitude.
2. Why did we burn retrograde (dropping lower) to catch a target that was ahead of us? Tie your answer to the threshold concept of Chapter 6.
3. We spread the catch-up over 3 revolutions. What are the two competing pressures that set the "right" number of revolutions?
4. Suppose the servicer had been launched into a plane $3^\circ$ off the target's. Estimate what the plane match would have cost in LEO, and compare it with the entire rest of this budget. What does that tell you about the importance of launch timing?

Your Turn: Extensions

• Option A (harder design). The target has drifted $2^\circ$ in inclination since launch (a real effect — perturbations, Chapter 12). Add a plane-change contingency to your budget: compute $2v\sin(1^\circ)$ at the $700\ \text{km}$ orbital speed ($v = 7.508\ \text{km/s}$) and compare it with the Hohmann altitude change. Which is bigger, and what does that reveal about the relative cost of altitude versus plane?
• Option B (computation). Redo the phasing over 6 revolutions instead of 3 (gain $7.5^\circ$ per rev). Recompute $a_{\text{ph}}$, the perigee altitude, and the total phasing delta-v, and compare the delta-v and time with the 3-revolution plan. Quantify the "patience is fuel" tradeoff. Hand-trace; do not run.
• Option C (your mission). If your mission (Track A/B/C/D) includes a rendezvous or proximity operation — a lunar-orbit rendezvous, an asteroid approach, a docking — sketch its phasing plan and add a rendezvous line (transfer + phasing + proximity allowance, with margin) to your MDR delta-v budget.

Key Takeaways

1. A rendezvous is a composition of maneuvers: match the plane (ideally for free, at launch), transfer to the target's altitude (Hohmann), phase to close the along-track gap, then creep in on proximity ops.
2. Phasing exploits "higher is slower": to catch a target ahead, drop into a lower, faster, shorter-period orbit; spread the catch-up over enough revolutions to keep the phasing orbit safely above the atmosphere.
3. Arriving at the right place can cost more than reaching the right orbit — here the phasing ($0.218\ \text{km/s}$) beat the altitude transfer ($0.164\ \text{km/s}$).
4. The whole same-plane LEO rendezvous fits in under $0.5\ \text{km/s}$ — a reminder that, once you are in the right plane, orbit-to-orbit work near Earth is cheap compared with the plane changes and GEO transfers that dominate other missions.