Orbit Utility
Is it possible to "hitch" a ride on an asteroid on a preexisting 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.
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, nonTerrestial 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.
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:
Earth Resonant
Every two years, cycler will rendezvous with Earth.
EarthMars Synodic
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, nonTerrestial 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:
Object  Peri helion  Aphe lion  Semi major  Focus  Eccen tricity  Semi minor  Semi latus  Incli nation  Period 

q  Q  a  c  e  b  L  i  P  
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  Observed  Observed  (q+Q)/2  a  q  c / a  √(a^{2}c^{2})  b^{2}/a  Observed  √a^{3} 
Earth Resonant
Every two years, cycler will rendezvous with Earth.
Object  Peri helion  Aphe lion  Semi major  Focus  Eccen tricity  Semi minor  Semi latus  Incli nation  Period 

q  Q  a  c  e  b  L  i  P  
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  Observed  Observed  (q+Q)/2  a  q  c / a  √(a^{2}c^{2})  b^{2}/a  Observed  √a^{3} 
EarthMars Synodic
Object  Peri helion  Aphe lion  Semi major  Focus  Eccen tricity  Semi minor  Semi latus  Incli nation  Period 

q  Q  a  c  e  b  L  i  P  
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  Observed  Observed  (q+Q)/2  a  q  c / a  √(a^{2}c^{2})  b^{2}/a  Observed  √a^{3} 
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