Tuesday, March 20, 2007

EARTH's ELLIPTICITY Helps a Lot!!!


Recall Kepler's Third Law: The square of an object's orbital period (T²) is proportional to the cube of the semi-major axis (a³ ) of its orbit.Thus, all Solar orbiting objects can relate to each other as follows:
T1² / a1³ = T2² / a2³ 

Let T1 = 365.25 days = 1 year

a1 = 149,700,000 km = 1.0 AU  (Earth's actual observed elliptical orbit semimajor axis, a)

a2 = 147,300,000 km = 0.98397 AU (a for an artificial circular orbit with Earth's perihelion as radius)

Solve for T2 = √ a2³ = √ 0.98397³ = .976 year = 356.5 days

BACKGROUND:
REVIEW: Initial Launch of Habitat and Support Modules into Nearby Parallel Orbits.
Vernal Equinox, March 20, Year 0.
Assume New Moon lunar phase, launch Habitat-Alpha (Hab-α) from Luna, to leverage the slight decrease in Solar Orbit semi major axis (a) gained by New Moon position, 384,000 km decreased orbit about (384/149,700.7) .2565%. Hab-α's slight decrease in orbital a results in slight increase of Hab-α's angular velocity (ω).
Due to the difference in angular velocities, Hab-α slowly increases lead before Terra about .004°/day; this rate would take about 40+ years to reach desired parking spot of 60° before Earth. (Assume Hab-α closely parallels Earth's Elliptical orbit.)
Image at left is used as inset in following diagram. Same image works as inset for positions: 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210° and 240°.
Lunar Cycles: April to December, Year 0.
Thus, TE proposes a future entity, perhaps the Lunar Launch Enterprise (LLE), to launch and recover 12 resupply modules (Mods) during the duration between March 20 (VE) and the following January 3 (about 9 months). Mods will likely contain additional components, Terran topsoil, mirrors for energy, and a considerable supply of propellant. It makes sense to leverage a few following lunar cycles to send more Mods to Hab-α. EXAMPLE: If a Lunar Launch Enterprise (LLE) launches a Mod on New Moon following initial launch, previous chapter shows that it might take only a month to link up with Hab-α.
Thought Experiment (TE) arbitrarily assumes:  a) Launch a series of Mods from a Lunar orbit during subsequent New Moons as shown by the insets. b) During each New Moon, LLE could choose to launch a single Mod or multiple Mods to multiple nearby orbits; each with different angular velocity. c) Such orbits would be very near Earth's elliptical orbit, but slightly smaller and slightly quicker. d) Mods join up with Hab-α as soon as their slightly quicker orbits aligns them with Hab-α. e) Hab-α's orbit would take long time (about 40 years) for completed Hab-α to reach its planned parking spot at 60 deg before Earth.
TE proposes Hab-α recovers all Mods prior to Jan 3, when Earth reaches its perihelion in its elliptical orbit. On Jan. 3, TE proposes Hab-α use some of the propellant (delivered via Mods) to change its orbit as for a much quicker travel time to the parking spot.
FOREGROUND:
PREVIEW: Subsequent Launch of Completed Habitat into a Farther Circular Orbit.
Terran Perihelion, January 3, Year 1.
Assume all construction completes before Jan 3;
further assume Hab-α's precise burn
to attain purely circular orbit.
With Earth's perihelion (closest to Sol) as radius,
Period will shrink from 365.25 days
to about 356.5 days.
This significant orbital change,
increases angular velocity of Hab-α.
EXAMPLE: During initial six months,
Earth travels a normal semi-orbit,
but Hab-α travels an additional 4.5°
during this same time, a significant lead.
NEXT FEW YEARS, HAB-α INCREASES LEAD.
In subsequent orbits, Earth consistently passes through the aphelion on same date, July 4. However, Hab-α is on a smaller, quicker orbit, and July 4 continually finds it further ahead.
EXAMPLE: On the second pass, Hab-α increases its lead to 13.4°; on the 3rd pass, lead becomes 22.4°; and so on.
On the 7th pass, Hab-α increases its lead to 58.2°, almost at its final goal of 60° ahead of Terra.
PARKING SPOT
FINALLY, Hab-α leads Earth by 60°
to attain a permanently parked position.
THUS, habitat exits circle around our Sun
to enter Terra’s actual elliptical orbit.
ASSUME: Sep 18 of 6th year past original launch.
REFERENCE: Also applies to Hab-Ω.
From Earth's perihelion to aphelion, there are a range of angular velocities.
In above table, at row dated July 4. Earth is at aphelion.  Assume a vessel uses a space tug to  exit Earth's orbit at this date; then, enter a circular orbit.
In this circular orbit, vessel would orbit slightly slower than Earth's elliptical orbit; and its lag behind Earth would continuously increase.  In about 7.2 years,  Hab-Ω would reach 60° behind Earth.
SUMMARY:  To eventually park 60° behind Earth, Hab-Ω needs larger, slower orbit to increase its lag till there.
SUMMARY: Many experts propose Lagrange points, L-4 (60° before Earth) and L-5 (60° behind Earth) as optimal parking spots for such Habitats, which TE calls Habitat-Alpha (Hab-α) and Habitat-Omega (Hab-Ω) respectively. Use following methods to properly place Habitats into Earth's Solar Orbit.
Enter Circular Orbit: On about Jan. 3, Earth arrives at its perihelion (orbit's nearest point to Sol); thus, this is a great position for Hab-α to enter a circular orbit with shorter semi-major axis and thus reach L-4 in about 6½ years (without this circular orbit, it would take about 40+ years to reach L-4). Alternatively, Earth arrives at aphelion (farthest from Sol) on about July 4; thus, a great position for Hab-Ω to enter a circular orbit with larger semi-major axis to reach L-5 in about 7 years.
Park at L4/L5:  Final burn needed by each Habitat when they each reach designated parking spot; perhaps provided by a space tug.
CONCLUSION: 7 years travel time is much better than 40 years travel discussed in previous chapter attained by Lunar Launches from either Full Moon or New Moon.
TRANSITION: HOWEVER, we can do even better. Properly oriented cycler orbits can reduce travel time to two years or even less, discussed in next chapter, Leveraging Lamba.

posted by JimODell at 2:17 AM

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