Physics & Orbital Mechanics of Commercial Crew Rendezvous
When SpaceX launches NASA Crew missions (such as Crew-14 carrying commander Kayla Barron and pilot Chris Birch aboard Falcon 9), the spacecraft is inserted into a lower parking orbit (~210 × 310 km) inclined 51.6° to match the ISS orbital plane. Because lower orbits travel faster, Dragon naturally catches up to the ISS via Keplerian phasing.
1. Keplerian Catch-Up (Phasing Rate)
From the Vis-viva equation $v = \sqrt{\mu (2/r - 1/a)}$ and period $T = 2\pi\sqrt{a^3/\mu}$, an insertion orbit at 260 km mean altitude completes an orbit in ~89.7 minutes versus the ISS's ~92.9 minutes. This generates a phase advance of ~12.4° per orbit until the spacecraft reaches the co-elliptic rendezvous gate.
2. Co-Elliptic & Hohmann Transfers
Sequential burns (Phase burn NC1, Height adjust NH, Co-elliptic burn NC2, and Final circularization NSR) raise perigee and apogee to establish an orbit parallel to the ISS roughly 10–15 km below. This preserves constant line-of-sight geometry and prevents accidental collision during lighting handoffs.
3. Clohessy-Wiltshire (Hill's Frame)
Within 50 km, navigation transitions to the target-centered Hill coordinate frame: +V-bar aligned along the orbital velocity vector, +R-bar pointed towards the center of Earth, and +H-bar cross-track. Relative motion yields characteristic elliptical loops governed by coriolis and gravity gradient coupling.
4. Keep-Out Zone (KOZ) & Waypoints
Dragon approaches along the V-bar through designated waypoints: Waypoint 0 (Approach Ellipsoid at 4 km), Waypoint 1 (400 m outside the Keep-Out Sphere), and Waypoint 2 (220 m). Final ingress is completed on the docking axis into International Docking Adapter (IDA) forward or zenith ports.