Saturday, October 13, 2012

Orbit Utility

Is it possible to "hitch" a ride on an asteroid on a pre-existing orbit.
GOAL: Determine orbits which contribute best utility toward transiting between orbits of Earth and Mars. 
ASSUME: Nearest point of orbit to the Sun (perihelion, q) must be inside Earth's orbit.
Farthest point (aphelion, Q) must be outside orbit of Mars.
By definition, the semimajor axis, a, is the average of q and Q.
a = (q + Q)/2
We need an orbit which intersects orbits of Earth and Mars in an useful way. Thus, cycler's perihelion must be less then Earth's perihelion (approximately 1.0 AU); and cycler's aphelion must exceed Mars's aphelion (1.52 AU).

Interplanetary Space is three dimensional.  Since we use Earth's orbital plane (Ecliptic) as a reference, it's easy to picture an orbit in two dimensional terms (i.e., X and Y).  However, non-Terrestial objects will have a Z, a third dimension.
EXAMPLE: Compare orbits of Earth and Mars; orbit of Mars is inclined 1.8° to the Earth's orbital plance, the Ecliptic. Since Mars has an aphelion (Q) of 1.52 AU; Z of Mars could be as much as 4,698,982.67 km above/below the Ecliptic.
7,800,311.2 km = 1.66 AU (149,597,870.7 km /AU) × sin(1.8°)
Z = R × sin(i)  = Q × sin(i)
Assume orbital velocity of 24 km/sec, then 3.76 days away but at lateral distance.
If inclination is exactly opposite of Mars, this could double distnce away and time to get there.
Inclination of cycler orbit must be small. Even small inclination can result in distances which make rendezvous challenging if not impossible.

Mars Synchronous

Assume: minimum "a" for a useful transfer orbit is about same as a of Mars, 1.52 AU.
Recall Third Law of Kepler: Same "a" means same period of orbit.
Thus, an asteroid with exact "a" of Mars orbit would have same period, 1.88 years. A quick search on NASA's web site quickly determined 2004 MR1 as most likely candidate.
semimajor axis, a:
intersection potential:
inclination, i:

ObjectPeri-
helion
Aphe-
lion
Semi-
major
FocusEccen-
tricity
Semi-
minor
Semi-
latus
Incli-
nation
Period
 qQacebLiP
Earth
0.983 AU
1.017 AU
1.00 AU
0.017 AU
0.017
1.000 AU
1.000 AU
0.00005°
1.000 yr
Mars
1.382 AU
 1.666AU
1.524 AU
0.142 AU
0.093 
1.517 AU
1.510 AU
1.85°
1.881 yr
2004 MR1
0.955 AU
2.090AU
1.522 AU
0.568 AU
0.373 
 1.413AU
1.311 AU
1.8°
1.880 yr
Given ObservedObserved(q+Q)/2a - qc / a(a2-c2)b2/aObserveda3



Earth Resonant
Every two years, cycler will rendezvous with Earth.




ObjectPeri-
helion
Aphe-
lion
Semi-
major
FocusEccen-
tricity
Semi-
minor
Semi-
latus
Incli-
nation
Period
qQacebLiP
Earth
0.983 AU
1.017 AU
1.00 AU
0.017 AU
0.017
1.000 AU
1.000 AU
0.00005°
1.000 yr
Mars
1.382 AU
1.666AU
1.524 AU
0.142 AU
0.093
1.517 AU
1.510 AU
1.85°
1.881 yr
(2007 UC6)
0.639 AU
2.540 AU
1.59 AU
 0.95 AU
 0.60
1.27 AU
1.02 AU
1.75°
2.00 yr
Given ObservedObserved(q+Q)/2a - qc / a(a2-c2)b2/aObserveda3


Earth-Mars Synodic





ObjectPeri-
helion
Aphe-
lion
Semi-
major
FocusEccen-
tricity
Semi-
minor
Semi-
latus
Incli-
nation
Period
qQacebLiP
Earth
0.983 AU
1.017 AU
1.00 AU
0.017 AU
0.017
1.000 AU
1.000 AU
0.00005°
1.000 yr
Mars
1.382 AU
1.666AU
1.524 AU
0.142 AU
0.093
1.517 AU
1.510 AU
1.85°
1.881 yr
(2005 GJ8)
0.793 AU
2.740 AU
1.77 AU
0.98 AU
0.551
1.27 AU
1.23 AU
3.4°
2.35 yr
Given ObservedObserved(q+Q)/2a - qc / a(a2-c2)b2/aObserveda3

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