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_i

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.

Other tables can be constructed for other vehicle capture times.


Starting from the capture time T seconds after construction orbit apogee, we can compute the capture angle L radians to the west of apogee.


Given the launch orbit perigee ( r_{pl} = 6378 km + 80 km loop altitude), the intercept radius r_i, and the intercept radial velocity v_i, we can algebraically compute the launch orbit perigee r_{al} and the angle between intercept and launch perigee L . From those parameters, we can compute the semimajor axis a_l, the eccentricity e_l, the period P_l, then the launch time and launch angle.

The radius and the radial velocity at angle L are given by:

r_i = { \Large { { ( 1 - e_l^2 ) a_l } \over { 1 + e_l \cos( L ) } } } ~~~~~~~~~~~~~~~~~~~~~~~~~ v_i = { \Large \sqrt{ \mu \over { ( 1 - e_l^2 ) a_l } } } \sin( L )

Let's define two new known parameters in terms of the known parameters r_{pl}, r_i, and v_i$ ,

\alpha \equiv { \Large { { v_i^2 r_pl } \over \mu } } ~~~~~~~~~~~~~ hence ( e_l \sin( L ) )^2 = \alpha ( 1 - e_l )

\beta \equiv { \Large { r_pl \over r_i } } ~~~~~~~~~~~~~ hence e_l \cos( L ) = \beta ( 1 - e_l ) - 1

This is enough to create a quadratic equation:

( e_l \sin( L ) )^2 + ( e_l \cos( L ) )^2 ~=~ e_l^2 ~=~ ( \beta - \beta e_l -1 )^2 + \alpha ( 1 - e_l )

0 = ( \beta^2 - 1 ) e_l^2 + ( 2 \beta - 2 \beta^2 - \alpha ) e_l + ( \beta^2 - 2 \beta + \alpha + 1 )

Define the coefficients of the quadratic polynomial:

A \equiv ( \beta^2 - 1 ) ...... B \equiv ( 2 \beta - 2 \beta^2 - \alpha ) ....... C \equiv ( \beta^2 - 2 \beta + \alpha + 1 )

And solve for e_l :

e_l = { { -B + \Large { \sqrt{ B^2 ~-~ 4 A C } } \over { 2 A } } }

The other root of the equation yields e_l = 1 which is nonphysical .

Here's the result of a spreadsheet

mu km3/s3

398600.44

re km

6378.00

sday s

86164.09

prime

launch

constr

solution

period s

83238.21

83229.63

86164.09

alpha

2.0389E-4

omega rad/s

7.5484E-5

7.5492E-5

7.2921E-5

beta

8.5156E-2

altitude km

80.00

80.00

2000.00

A

-0.9927484

perigee km

6458.00

6458.00

8378.00

B

0.15560584

apogee km

75950.34

75944.68

75950.34

C

0.83714253

semimajor km

41204.17

41201.34

42164.17

radical

3.34850072

v0 km/s

5.787

5.787

5.139

good e+

-0.8432575

eccentricity e from apogee

-0.84327

-0.84326

-0.80130

check

0.0E+0

v perigee

10.66631

10.66628

9.25746

junk e-

1.00000

v apogee

0.90695

0.90701

1.02118

check

0.0E+0

intercept theta radians

0.00000

0.02299

0.02724

intercept radius km

75950.34

75836.85

75836.85

E eccentric anomaly

0.00000

0.07881

0.08199

time(theta) sec

0.000

879.447

900.000

tangent velocity km/s

0.90695

0.90830

1.02271

radial velocity km/s

0.00000

-0.11218

-0.11218

perigee time

-41619.10

-40735.37

-42182.04

Construction2 (last edited 2019-04-20 11:52:44 by KeithLofstrom)