Transfer Orbits
If time aplenty is on your hands;
then, a simple transfer will do.
If you're not in any hurry;
then, you don't need any fuel.
Mankind currently orbits to other planets.
It takes no fuel, but it takes a while.
Gforce propulsion will be much quicker.
It will take some fuel.
It takes no fuel, but it takes a while.
Gforce propulsion will be much quicker.
It will take some fuel.


G
 = 
6.67428 × 10^{11} m^{3}/(kgsec^{2})


M_{☉}
 = 
1.98892 × 10^{30} kg

μ_{☉}
 = 
G × M_{☉}

μ_{☉}
 = 
6.67428 ×10^{11} m^{3} /(kgsec^{2})×1.9889 × 10^{30} kg

μ_{☉}
 = 
1.32746 × 10^{20} m^{3} /sec^{2 }= 39.49 AU^{3}/yr^{2}

√(μ_{☉})
 = 
1.15 × 10^{10} √(m^{3}) /sec = 6.28 √(AU^{3})/yr

Hohmann Transfer (HT) is a special case TO; it is considered to be the most fuel efficient. TE uses HT model to conveniently compute some typical TO durations. Furthermore, TE uses some assumptions to further simplify this process. These are discussed later in this chapter.
TE settles on HT for "typical transfer times". For more on orbits and on HTs, see book, Orbital Motions, by A. E. Roy.
Table 2: Typical TO Distances
Axis
tricity
Axis
Time
Axis
Distance
**Earth
1 AU
D E P A R T U R E O R B I T
Mars
1.52 AU
0.21
1.26 AU
1.23 AU
3.91AU
Jupiter
5.2 AU
0.63
3.10 AU
2.41 AU
8.72 AU
Saturn
9.51 AU
0.81
5.26 AU
3.08 AU
13.53 AU
Uranus
19.18 AU
0.9
10.09 AU
4.40 AU
24.45 AU
Neptune
30.06 AU
0.94
15.53 AU
5.30 AU
36.45 AU
Kuiper Belt
40 AU
0.95
20.50 AU
6.40 AU
47.71 AU
a_{D}+1
2
2
π√(a_{τ}^{2 }+ b_{τ}^{2})
√2
 **Assume ships depart Earth's orbit with radius of 1 AU.
 Semimajor axis, a_{D}, of destination orbit is observed.
 TO eccentricity [e_{TO }= (a_{D}1)/(a_{D}+1)] per formula (12.3) on page 355, Orbital Motion by A. E. Roy.
 Transfer Orbit (TO) is the highly elliptical orbit which connects orbit of Earth with orbit of destination planet.
 TO period is the time needed for space vessel to depart orbit of Earth, go to orbit of destination planet; then, return to orbit of Earth.
 Of course, only half this time is needed to transit from Earth orbit to destination orbit (Transfer Time).
 TO semimajor axis, [a_{TO }= (a_{D}+1)/2] is average of Earth's a (1.0 AU) and a_{D}, destination semimajor axis.
 TO semiminor axis, b_{TO }, is computed via a common elliptical identity.
 Transfer distance (d_{TO}) is approximated by one half of TO elliptical circumference (C_{TO} ≈ 2 * π * √((a_{TO}^{2}+b_{TO}^{2})/2).)
Destination  Transfer Orbit  

SemiMajor Axis  Period  Circum  Aphelion  Perihelion  Min Velocity  Ave Velocity  Max Velocity  
a_{D}  P  C_{T}  Q_{T}  q_{T} 
v_{min}
 v_{ave}  v_{max}  
Earth  1 AU  3.16x10^{7} sec  9.40x10^{8} km  1 AU  1 AU 
29.81 k/s
 29.8 k/s 
29.81 k/s
 
Mars  1.52 AU  4.47x10^{7} sec  1.17x10^{9} km  1.52 AU  1 AU 
21.54 k/s
 26.24 k/s 
32.75 k/s
 
Jupiter  5.2 AU  1.73x10^{8} sec  2.61x10^{9} km  5.2 AU  1 AU 
7.43 k/s
 15.15 k/s 
38.61 k/s
 
Saturn  9.51 AU  3.81x10^{8} sec  4.05x10^{9} km  9.51 AU  1 AU 
4.22 k/s
 10.65 k/s 
40.11 k/s
 
Uranus  19.18 AU  1.01x10^{9} sec  7.32x10^{9} km  19.18 AU  1 AU 
2.14 k/s
 7.23 k/s 
41.11 k/s
 
Neptune  30.06 AU  1.93x10^{9} sec  1.09x10^{10} km  30.06 AU  1 AU 
1.38 k/s
 5.65 k/s 
41.48 k/s
 
Kuiper Belt  40 AU  2.93x10^{9} sec  1.43x10^{10} km  40 AU  1 AU 
1.04 k/s
 4.87 k/s 
41.65 k/s
 
Observed  2π√(a_{T}^{3}) √μ  2π√(a_{T}^{2}+b_{T}^{2}) √2  a_{T} + c_{T}  a_{T}  c_{T} 
 C_{T} P 

v_{τ} =  √[μ_{☉}( 
2
r_{τ}   
1
a_{τ}  )] 

μ_{☉}
 = 
132,712,440,018 km^{3}/sec^{2 }


μ_{☉}  = 
132,712,440,018 km^{3}
sec^{2}  × 
AU
149,597,870.7 km 

μ_{☉}  = 
887.128 km^{2}
sec^{2}  × 
AU


√μ_{☉}  = 
29.785 km
sec  × 
√AU


v_{τ} = 29.785 km/sec ×  √[AU( 
2
r_{τ}   
1
a_{τ}  )] 

v_{τ} ≈ 30 kps ×  √[ 
2
r_{τ}   
1
a_{τ}  ] 

Minimum Velocity happens at TO's farthest point from Sun (aphelion, Q_{τ}).
Each TO aphelion is semimajor axis of destination orbit (a_{Dest}).
v_{τ} ≈ 30 kps ×  √[ 
2
Q_{τ}   
1
a_{τ}  ] 

v_{τ} ≈ 30 kps ×  √[ 
2
q_{τ}   
1
a_{τ}  ] 

v_{τ} ≈ 30 kps ×  √[ 
2
  
1
a_{τ}  ] 

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