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The Earth orbital period is (to first order) a function of the standard gravitational parameter μ = 398600.44 km³/s² and the semimajor axis a. The Earth orbital period is (to first order) a function of the standard gravitational parameter μ = 398600.44 km³/s² and the semimajor axis $ a $.  $ T ~=~ \sqrt{ a^3 / \mu } $
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$ T ~=~ { \large \sqrt{ a^3 / \mu } } $

The semimajor axis is half the sum of the perigee and apogee radius:

$ 2 a ~=~ r_a + r_p $
The semimajor axis is half the sum of the perigee and apogee radius:  $ 2 a ~=~ r_a + r_p $

Sidereal Orbits

Launch loop vehicle mass is limited; however, with high precision launch it is "relatively" easy to rendezvous vehicles to connect into larger assemblies. For example, launching rocket/heatshield packages, followed later by cargo packages to complete a vehicle for apogee plane change and subsequent aerobraking-assisted orbit changes.

The launch loop will rotate under the perigee of a previous launch once per sidereal day, every 86164.089 seconds, about 4 minutes shorter than a solar day. Launch times should be scheduled for exact integer multiples of these orbit times. Otherwise, the loop will not be exactly positioned at the perigee time.

The actual rendezvous time will be modified by the nonlinear gravity field of the Earth due to the equatorial bulge, and the gravity of the Sun, Moon, and Jupiter. These perturbations are not included here, but for a real system they can be predicted and computed to millimeter accuracy. There will also be less-predictable perturbations due to residual atmospheric gas drag (a function of solar UV activity), spacecraft out-gassing, light pressure, and other small effects. The following is for rough estimates only.


The Earth orbital period is (to first order) a function of the standard gravitational parameter μ = 398600.44 km³/s² and the semimajor axis a . T ~=~ \sqrt{ a^3 / \mu }

The semimajor axis is half the sum of the perigee and apogee radius: 2 a ~=~ r_a + r_p

So, we can rearrange to compute the sum r_a + r_p :

r_a + r_p = { \large \left( 2 \mu T^2 / \pi^2 \right)^{1/3} }

If D is the number of sidereal days, t = 86141.09 \times D . Filling in the numbers,

r_a + r_p = 84313.33 \times D^{2/3} km

Here's a table of r_a + r_p in kilometers versus D in sidereal days :

D

r_a + r_p

D

r_a + r_p

D

r_a + r_p

S. days

kilometers

S. days

kilometers

S. days

kilometers

1

84313.33

10

391347.81

45

1066689.25

2

133839.07

15

512810.70

50

1144307.94

3

175378.79

20

621225.93

55

1219377.10

4

212456.28

25

720868.83

60

1292202.00

5

246533.67

30

814036.25

65

1363029.16

6

278396.48

35

902142.08

70

1432061.28

8

337253.32

40

986134.69

75

1499467.59

The average radius of the Moon is around 385,000 km, and beyond 1.5 million kilometers, solar tidal forces are stronger than the Earth's gravity. These very slow orbits cannot be easily estimated. On the other hand, the multi-body tidal effects can be exploited to pump energy into or out of long period orbits; a subject for more adept orbit calculators than I am. Belbruno's Fly Me to the Moon describes these "chaotic" effects for lay audiences.

MoreLater

StellarDayOrbits (last edited 2018-03-29 07:10:33 by KeithLofstrom)