Construction2

The older rendezvous page, Construction1, started from a launch time and an exact associated longitude; calculations were complicated and the radial velocity at capture was large.

A better approach is to start with the capture angle at the construction orbit, which defines an exact radius and radial velocity. From there, find a launch orbit with an apogee, launch angle, and launch time that arrives at the construction orbit with the same radius and arrival velocity. The tangential velocity will be different, on the order of 100 m/s, but we can accommodate that with a "passive net capture system".

So, given the construction orbit, choose an angle from apogee \theta . Construction orbits have low perigees and very high apogees, far beyond GEO, so the apogee velocity is small and the angular velocity very small near apogee. Given a desired arrival time t_{ac} referenced from t_{ac} = 0 at apogee, we can estimate \theta given the apogee radius r_{ac} and apogee velocity v_{ac}

The construction orbit perigee should be well above LEO; assume an altitude of 2000 km above the equatorial radius 6378 km, thus r_{pc} = 8378 km. The construction orbit semimajor axis is defined from the orbit period, which should be a multiple N of an 86164.0905 second sidereal day. From that, we can compute the apogee radius, velocity, and angular velocity \omega_{ac} , then iterate from the desired arrival time to the exact capture angle.

period

semimajor

apogee

apogee V

ang.freq.

1 hr angle

Intercept

N

P_c

a_c

r_{ac}

v_{ac}

\omega_{ac}

radians

r_i

v_{ic}

sdays

seconds

km

km

km/s

radians/sec

est.

exact

km

km/s

1

86164.1

42164.17

75950.34

1.02118

1.3445E-5

0.048403

0.048557

75591.05

-0.19988

2

172328.2

66931.45

125484.89

0.63056

5.0250E-6

0.018090

0.018104

125341.34

-0.07978

3

258492.3

87705.01

167032.01

0.47745

2.8584E-6

0.010290

0.010294

166948.27

-0.04653

4

344656.4

106247.05

204116.10

0.39241

1.9225E-6

0.006921

0.006922

204058.99

-0.03173

5

430820.5

123288.78

238199.56

0.33721

1.4157E-6

0.005096

0.005097

238157.13

-0.02357

6

516984.5

139223.02

270068.04

0.29802

1.1035E-6

0.003973

0.003973

270034.76

-0.01849

7

603148.6

154291.59

300205.17

0.26851

8.9442E-7

0.003220

0.003220

300178.07

-0.01506

Note that the estimate is pretty good, off by 0.000154 radians for one hour delay and the 1 sidereal day construction orbit; that corresponds to an earth rotation of 2.11 seconds, or about 1 kilometer of distance. Good for capability estimation, too much error for accurate aiming from the loop, which will need precise (millisecond) timing.