Tuesday, June 03, 2014

Hohlman Transfer

Of course, we currently use non-powered spacecraft to orbit throughout the Solar System. They routinely travel between planets via "transfer orbits", which take months and years. Furthermore, interplanetary trips must be timed to accommodate planetary positioning; such synchronization necessities can add considerable delay. Thus, human transport via transfer orbit is infeasible, and practical payloads are limited to Artificial Intelligence (AI) devices.

First, consider current spacecraft capabilities for interplanetary travel. Far from a straight line, typical flight path is more like a semi-ellipse. Kepler and Newton initiated our current knowledge of orbital mechanics which demonstrate that non-propelled objects within the Solar System must fly in elliptical orbits around the Sun.
We are currently unable to propel a spacecraft for entire interplanetary trips; instead, we must satisfy ourselves with well planned "fuel burns". Between these "burns", spacecraft orbits the Sun. An orbit that connects two planetary orbits is called a transfer orbit.
Thus, consider possible path of vehicle traveling from Earth to Mars. Following diagram shows three relevant orbits.
  1. Earth's Orbit is nearly circular. Diagram assumes prior launch from Earth's surface; vehicle orbits Earth for a while; then, follows path shown on diagram.
  2. Destination Orbit. Most planetary orbits are nearly circular; diagram shows orbit of Mars as an example.
  3. Transfer Orbit is a highly eccentric orbit which connects two planetary orbits. A simple, well planned transfer orbit would require following events.
    • Enter Transfer  Orbit (*Δ v= 2.947 km/sec): burst of fuel burn to leave Earth orbit and enter transfer orbit. (*Ref: page 357, On Motion by AE Roy)
    •  Exit Transfer Orbit (*Δ v= 0.396 km/sec): burst of fuel burn to exit transfer orbit and enter Mars orbit. (*Ref: page 357, On Motion by AE Roy)
Careful planning is required to ensure that spacecraft arrives near destination planet at completion of orbit transfer. For example, above diagram shows that Mars very far from destination point when vehicle initiates transfer orbit; during the 259 days of zero-g flight, Mars and Earth both travel through considerable portions of their respective orbits.

Hohmann Transfer Orbits. A Hohmann Transfer is a special case of a transfer orbit discovered by Walter Hohmann in 1928. Strictly speaking, a pure Hohmann transfer would exit Earth's orbit at its perihelion (nearest point to Sol) and enter destination orbit at aphelion (farther point from Sol). Typical transit times for relevant Hohmann Transfer orbits can be approximated via an online Hohmann Transfer calculator.

Perihelion, q, is orbit's closest point to Sol. Since we're now considering Hohman Transfers from Earth to outer planets, HT's perihelion must be on Earth's orbit.  Thus, qHT equals Earth's semimajor axis, aE. Thus,
qHT = aE = 1 AU
Aphelion, Q, is orbit's farthest point from Sol. Thus, HT's Q must be destination orbit's semimajor axis, aD. Thus,
QHT = aD
HT's focus, c, is half of difference of aphelion and perihelion:
cHT = .5×(QHT  - qHT) = .5×(aD - 1)
HT's semimajor axis is average of HT's aphelion and perihelion:
aHT = .5×(QHT + qHT) = (aD + 1) / 2
Thus, HT's eccentricity is focus divided by semimajor axis.
eHT = (a -1)/(aD+1)
To determine HT orbital period, recall Kepler's Third Law, square of orbital period is proportional to cube of orbit's semimajor axis. .5×(aD + 1).
THT2 ∝ aHT3 = [(aD+1)/2]3

Hohman Transfer Examples
Destination OrbitTransfer Orbit
Eccen-tricity **time  
e (ratio)aD (AU) T (Yrs) e (ratio) t (Yrs)
Mars0.09341.521.87.21.71 yr
Jupiter0.04835.2011.86.682.73 yr
Saturn0.05609.5429.47.816.05 yr
Uranus0.046119.1884.00.9116.03 yr
Neptune0.009730.06164.81.9430.61 yr
2π a3/2

aD - 1

aD + 1

*   Assume departure from Earth's orbit.
**Time from Earth Orbit to Destination. See A. E. Roy, On Motion, pg. 354.

kg sec
Sol's Mass
1.99x1030 kg

= G×MSol
Convert to
AUs and Yrs
μSol=39.5 AU³

= G×MSol

×(aD +1)3/2

tt = .5 T=(aD +1)3/2

Quicker and more flexible transfer orbits can be accomplished; however, Hohman Transfers are the most energy efficient and are easily calculated.
  1. HT transit times give us a rough order of magnitude of required travel times for constant velocity flights between planets.
  2. Orbital flights can be extremely complicated. Flight planners has used "flybys" and "gravity assists". (Recall: Pioneer and Voyager missions.)
SUMMARY: Transfer orbits are very fuel efficient, but they are too limited and too slow.
(Analogy: Consider today's maritime environment. Sailships are certainly much more energy efficient then a modern powered vessel. However, there are about 165,000 ocean going vessels (over 100,000 tons); virtually every single one is powered; owners clearly prefer greater control versus energy efficiency.)
Traveling between planets at constant velocity will take months and years. This might work for robots and other AI devices; it probably would not work so well for humans and other biologics.


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