Intersecting Orbits Rendezvous Method

Starting Conditions:

• Space Station: Circular Orbit
  • a = 7000 km
  • v = 7.546 km/s
  • P = 5828 s
• Shuttle: Circular Orbit near
  but not directly under the Station. (exact position to be determined)
  • radius 6990 km
  • v = 7.552 km/s
  • P = 5772 s

The Shuttle is moving faster and will pass the Station in its present orbit. We know that adding velocity to the shuttle will raise it into a higher, slower orbit which will can cross the orbit of the Station.
What velocity will

1. raise the Shuttle past the Station orbit
2. give the Shuttle a velocity matching that of the Station at the cross over.

Determine a neww elliptical orbit for the Shuttle with v=7.542 km/s at a distance of r=7000 km from the center of the Earth. The orbit's perihelion with be its present circular orbit distance r_peri = 6990 km

• So a = 1/[2/r - v²/k_E] = 6999.1 k
• eccentricity = 1 - r_peri/a = 0.001286
• r_aphelion = (1 + 0.001286) 6999.1 = 7008.1 km
  The shuttles orbit rises to 8 km above that of the Station.
• the angle, φ, from perihelion at r = 7000 is given by cos(φ) = [1 - a (1-e²)/r]/e
  So... φ = 96.5°
• The time of flight to this point from perihelion is 1559 s = 26 minutes.
  This was determined by calculating the eccentric anomaly and then the mean anomaly.
• v_c = 7.547 km/s
  Note that this new orbit has a mean velocity very similar to that of the Station's.
• v_peri = v_c √[(1+e)/(1-e)] = 7.557 km/s

In the same time the Station will revolve (1559/5828) 360° = 96.3° about the Earth.

In order to rendezvous precisely, the Shuttle should

1. increase its velocity by 7.557-7.552 = 0.005 km/s
2. when the Station is 0.2° ahead of it. (24 km)