Differences between revisions 1 and 16 (spanning 15 versions)

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.

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 )

Define D = ( 1 - e_l^2 ) a_l , invert the first equation, and square the second equation. The equations become:

{ \large 1 \over r_i } = { \large { 1 + e_l \cos( L ) } \over D } ~~~~~~~~~~~~~~~~~~~~~~~~~ v_i^2 = { \large { \mu \over D } } \sin( L )^2

Move D to the numerators:

{ \large D \over r_i } = { 1 + e_l \cos( L ) ~~~~~~~~~~~~~~~~~~~~~~~~~ D v_i^2 = \mu \sin( L )^2

-  ⇤ ← Revision 1 as of 2019-03-16 17:31:06 → 
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  Editor: KeithLofstrom
  Comment:
+   ← Revision 16 as of 2019-03-17 04:38:02 → ⇥
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  Editor: KeithLofstrom
  Comment:
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-||       || period  || semimajor || perigee ||
|| N     || $ P_c $ || $ a_c $   || $r_{pc} ||
|| sdays || seconds || km        ||
+||       || period   || semimajor || apogee    || apogee V || ang.freq.     ||<-2> 1 hr angle   ||<-2> Intercept         ||
|| N     || $ P_c $  || $ a_c $   || $r_{ac}$  || $v_{ac}$ || $\omega_{ac}$ ||<-2> 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.

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:

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

Define $ D = ( 1 - e_l^2 ) a_l $,
invert the first equation, and square the second equation.  The equations become:

$$ { \large 1 \over r_i }  = { \large { 1 + e_l \cos( L ) } \over D } ~~~~~~~~~~~~~~~~~~~~~~~~~ v_i^2 = { \large { \mu \over D } } \sin( L )^2  $$

Move $ D $ to the numerators:

$$ { \large D \over r_i }  = { 1 + e_l \cos( L )   ~~~~~~~~~~~~~~~~~~~~~~~~~  D v_i^2 = \mu \sin( L )^2  $$

Diff for "Construction2"

Construction2