Phasing Orbits

The Phasing Problem

You and your target are in the same circular orbit, but you're behind.

The naive approach: Burn toward them? Result: You'll just raise your orbit and fall even further behind!

The orbital reality: In the same circular orbit, you both have the same period. You'll maintain the same relative position forever.

To catch up, you must temporarily change your period by changing your orbit altitude.

Lower orbit = shorter period = you catch up Higher orbit = longer period = you fall behind

This is the essence of orbital rendezvous!

Phasing Orbit Mathematics

To catch up with target ahead by angle θ:

Step 1: Calculate how many orbits to take Choose n (number of orbits in phasing)

Step 2: Calculate required phasing orbit period T_phasing = T_target × n/(n+1) to catch up T_phasing = T_target × n/(n-1) to slow down

Step 3: Calculate phasing orbit altitude from period (Kepler's 3rd Law)

Step 4: Execute Hohmann transfer to phasing orbit

Step 5: Coast n orbits

Step 6: Return to target orbit at rendezvous

Example: Target ahead by 90° (1/4 orbit) - Use 3 phasing orbits - T_phasing = T_target × 3/4 = 0.75 × T_target - This means a lower altitude

The Rendezvous Strategy

Classic ISS Rendezvous Approach:

1. Launch into same orbital plane (or do plane change first)

2. Establish in lower parking orbit (~100 km below ISS) - You're moving faster, catching up ~25 km per orbit

3. Wait for proper phasing (typically 2-3 days of orbits) - Calculate when you'll be properly positioned

4. Hohmann transfer up to ISS altitude - Times to arrive ~1 km below and behind ISS

5. Final approach in steps - Multiple small burns closing distance - Constant relative velocity monitoring

6. Docking - <10 cm/s approach speed - Precise attitude control

This is how every Soyuz, Dragon, and Cygnus reaches ISS!

Practical Tips

Choosing n (number of phasing orbits): - More orbits = smaller orbit change = less delta-V - Fewer orbits = faster rendezvous = less time - Typical choice: 2-5 orbits

For catching up: - Go lower/faster - Periapsis drop is most efficient - Monitor closure rate

For slowing down: - Go higher/slower - Raise apoapsis - Let target catch up to you

Common mistakes: - Burning toward target (doesn't work in orbit!) - Not accounting for target's motion - Forgetting to return to target altitude - Approaching too fast at the end