John Pazmino
 NYSkies Astronomy Inc
 2001 October 25 initial
 2012 July 28 current
    I was favored to give a presentation to the New York chapter of the 
National Space Society on Saturday 14 July 2001. The meeting was held 
in New York University in the very room that the late Junior Astronomy 
Club held its sessions in the 1950s and 1960s. The talk was 'Finding 
Mars in our sky' and included some material on orbits and 
    One viewgraph showed the path of a spaceship from Earth to Mars. 
The path was an ellipse with perihelion at Earth and aphelion at Mars. 
I explained that this was both a trajectory for the spaceship and also 
[half of] the orbit of an Amor class of asteroid. 
    A couple guests recognized this path as a Hohmann trajectory, 
prevalently proposed as the flight path between planets. Yet few 
seemed to know how this Hohmann path works. I tried elaborating during 
the postlecture Q&A and also during the supper I went to with the 
chapter members. It was pretty hopeless without some number work.  
    Right off of the bat I found out when I got home that this peculiar 
orbit is called both a Hohman and a Hohmann path. Note that there is 
either one or two 'n's in the name. This swing of spelling occurs in 
works on space flight, astrodynamics, and publications of various 
space groups and agencies! The one and correct spelling is with two 
'n's, Hohmann. It's pronounced 'HOH-mann', not HA-mann' or 'HO-mann'. 
    Walter Hohmann was an engineer and architect in Germany who in the 
1910s got interested in space travel. It may surprise some readers 
that modern concepts of travel in outer space were first bantered 
around in the early 20th century! In this period, mostly in Europe, 
there sprang up circles of space travel enthusiasts who dreamed of 
actually making voyages to the other planets. 
    Hohmann wrote an article on possible flight paths from Earth to 
Mars and Venus. He found that of the many ways to get from Earth to 
Mars (or back!) one path had the least expenditure of energy. You 
needed the least amount of rocket burn or load of fuel to travel over 
this one path as compared to all others. This is the path now named 
for him, the Hohmann path. 
    He finished his piece in 1916 right smack in the middle of World 
War I. His native Germany was fully embroiled in the war. He felt this 
was not the time to publish his work because rockets had clear and 
strong military value. He put his work away. 
    In 1924 he revisited the problem of flying to Mars and elaborated 
his path. With the war well over he published 'The attainability of 
celestial bodies' in 1925. This was a beefed up version of his earlier 
paper and included a full scenario for sending a two-man crew to Mars. 
    His book considered many flight paths, notably those which were 
tangent to one, but not the other, planet's orbit. The ideal Hohmann 
path is tangent to both orbits at its own aphelion and perihelion. 
    As fate would dictate, Germany was the first country to seriously 
consider space flight as a spinoff of its activity in World War II. 
The United States and Russia captured many German rocket scientists 
after the war. Hohmann himself was killed in a wartime bombing of his 
town in 1944. Sadly, unlike the accolades heaped on other space 
advocates of his era, Hohmann received very little major fame. 
The energy problem
    A spaceship once it leaves Earth is utterly on its own devices 
until it reaches its target. It must take with it all the necessaries 
for the entire flight. One major need is fuel, the source of energy to 
propel the craft. In space science 'energy' is more or less synonymous 
with 'fuel'. 
    Fuel, liquid oxygen and liquid hydrogen as examples, is very bulky 
and heavy. Such ugly fluids can be dangerous if they leak or ignite 
during the voyage. The less fuel the ship can get away with, the 
overall better off the mission is. 
    The motivation to exploit the Hohmann trajectory is that in the 
ideal situation it requires for a flight between two planets the least 
amount of fuel. The craft is smaller, lighter, safer. The down side is 
that the path is the longest in duration for a direct route between 
the planets. The crew spends appallingly long spells en route. 
    For an uncrewed vehicle the length of the journey is not too 
critical. With the early craft being small anyway, with little luxury 
to carry extra fuel, the Hohmann orbit was a perfect path to the 
The concept of Hohmann orbits
    In the figure below I sketch out a Hohmann path between Earth and 
Mars. The Hohmann path nests between the two planet orbits and is 
itself merely an other elliptical heliocentric orbit. 
    The astronomer will recognize this as the typical path of an Amor 
asteroid, Named for the prototype Amor itself, it has its perihelion 
at Earth's orbit and aphelion near Mars. Not so far away as the main 
belt of asteroids, but substantially farther out than Earth's orbit. 
Amors are potentially dangerous to Earth because the chance exists, 
however slim, that Earth and asteroid would meet together where the 
two orbits touch. This asteroid is a potential Earth-smasher. 
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    The proportions are exaggerated. The inner complete circle is the 
orbit of Earth. The outer complete circle is the Mars orbit. The 
ellipse touching the two is the Hohmann orbit. I made the planet 
orbits with '.'; Hohmann, '#'; Sun, 'O'. 
    The trick is to send the rocket from Earth at A such that it 
arrives at F when Mars also arrives at F. At first it looks like Mars 
is in superior conjunction from Earth's eye, but this is very 
misleading in most simple diagrams. Earth and Mars are placed in the 
locations where they coincide with the spacecraft location. There is 
motion going on which a static sketch can not properly show. 
    Assuming this path is actually achieved, it works out that the 
energy or fuel needed to insert the ship into this path at Earth and 
fetch it from the orbit at Mars is the least of all flightpaths 
between the two. If the craft is not captured by Mars it'll continue 
round the orbit and return to its perihelion. But it will NOT -- as 
some explanations have it -- reach Earth again. Earth moved on 
downrange and is no where near the craft. 
    As a transfer path, only one half of the ellipse is traversed. The 
other half is simply never used. Also, in the usual scheme of the 
solar system, full advantage is taken of the orbital speed of the 
planets for the ship's own speed. The Hohmann path is traversed in the 
same counterclockwise sense as the planets. In the figure above, for a 
move from Earth to Mars the lower half of the Hohmann orbit is used. 
For a move from Mars to Earth, it's the upper half. 
Actual Hohmann orbits 
    The Hohmann orbit is an idealism in spaceflight and no ship flies 
along it exactly. There were many approximations of Hohmann flight in 
the early years of the world's space program but for the most part 
this path is used mainly for changing elevation in Earth orbit. 
    The typical flightpaths more nearly resemble the vonPirquet 
trajectory, explained later, where the ellipse overlaps the home and 
target planet orbits. These are used nowadays only for missions to 
Earth's adjacent planets, Venus and Mars. For other planets, a much 
more complex path is taken, involving gravity assists and resists 
along the way. 
Orbit adjustment near planet 
    If in the diagram above we put the Earth in place of the Sun and 
leave out the planets, we got two orbits around Earth with a transfer 
between them. The usual situation is for an Earth satellite to be 
placed in low-Earth orbit, the inner one in the figure, first from 
liftoff. Then a rocket burn sends it via a Hohmann trajectory into a 
higher orbit, such as a geosynchronous one. A second burn at the upper 
end of the trajectory, the apogee of the Hohmann orbit, sets the 
satellite into the higher orbit. 
    Conversely, the higher orbit may be the arrival orbit of a 
spaceprobe around, say, Mars. The Hohmann transfer brings the probe to 
a low orbit for closeup monitoring of the planet. Mars is the central 
body. The rocket is fired to enter the Hohmann orbit at apoareon and 
again to settle into the low orbit at periareon. (You probably never 
knew there were such words!) 
    A third plausible use of Hohmann transfer near a planet is to 
retrieve a high-orbit satellite for return to the surface. The Space 
Shuttle as an example orbits the Earth about 500 kilometers above the 
ground. It can not directly fetch a high-orbit craft. The craft is 
lowered to the Shuttle's elevation by a Hohmann transfer trajectory 
and the Shuttle can then capture it. 
Trajectory versus orbit 
    Space science litterature call paths in gravity either a trajectory 
or an orbit. The terms are rather much the same. The trajectory term 
comes from rocketry and ballistics on Earth while orbit comes from 
astronomy. Both are freefall paths in gravity and are essentially the 
same thing. 
    In particular, the theory of orbital mechanics apply to rocket 
trajectories in space as well as they do for natural planets, comets, 
and asteroids. In the case of the Hohmann path between Earth and Mars, 
it is a glatt Kepler orbital ellipse. Even tho only half is used -- 
the vehicle quits the orbit to land on Mars -- it is a true freefall 
orbit from Earth up until Mars. 
Properties of a Hohmann orbit
    Consider the Earth-Mars Hohmann path. Immediately we see that its 
perihelion is at Earth and its aphelion is at Mars. Altho Hohmann 
orbits apply between any pair of planets, the rest of this paper deals 
only with that between Earth and Mars. Hence, we can write, with 
figures from astronomy references: 
    q = 1.000 AU, Earth-Sun mean distance
    Q = 1.524 AU, Mars-Sun mean distance
    A = q+Q = (1.000 AU)+(1.524 AU) = 2.524 AU, major axis 
    a = A/2 = (2.524 AU)/(2) = 1.262 AU, semimajor axis 
    We astronomers use the astronomical unit for distances within the 
solar system. One AU is nearly enough 149.6 million kilometers, the 
mean or average distance of Earth from the Sun. This is sometimes 
called the Earth's orbit radius, a bit misleading being that the orbit 
is not at all circular. 
    From Kepler's theory we can compute the excentricity and the period 
of this Hohmann orbit. 
    e = (1)-(q / a) =(1) - ((1.000) / (1.262)) = 0.2076 
    e = (Q / a)-(1) = ((1.524) / (1.262)) - (1) = 0.2076 
I checked my work by banking off of both the aphelion and perihelion 
distances. The two answers should be, and are, the same. 
    The period of the orbit comes directly from the semimajor axis 
    P = a ^ (1.5) = (1.262) ^ (1.5) = 1.418 yr 
Because we are going only one way to Mars, the flight takes but half 
of this period or 0.709 year or 258.96 day. 
Planet motion
    The daily motion and orbital speed come from Kepler theory, too. 
    n = (360 deg) / P = (360 deg) / (1.418 yr) = 253.88 deg/yr 
      = 0.695 deg/day
Note well that in all work I use EARTH years and days and NOT those of 
Mars. Altho this is a standard practice in astronomy I do see 
astronautical works converting times into those on other planets. 
    The mean orbital speed is
    V = 2 * pi * a / P
      = 2 * pi * (1.262 AU) / (1.418 yr) 
      = 5.592 AU/yr 
This is tough to visualize; speed is usually expressed in Km/s, so 
    V = (5.592 AU/yr) * (149.6e6 Km/AU) / (31.56e6 sec/yr) 
      = (5.592 AU/yr) * (4.740 Km.yr/AU.sec)
      = 26.51 Km/s 
    The speed at the Hohmann orbit's perihelion and aphelion are 
specially important. From Kepler theory, we got 
    Vperi = V * sqrt((1 - e) / (1 + e)) 
          = (26.51 Km/s) * sqrt((1.2076) / (0.7924)) 
          = 32.71 Km/s 
    Vapo = V * sqrt((1 + e)/(1 - e)) 
         = (26.51 Km/s) * (sqrt((0.7924) / (1.2076)) 
         = 21.47 Km/s 
    I collect these properties here for easier reference
    property         value
    ---------------  -----------------
    perihelion dist  1.000 AU, at Earth
    aphelion dist    1.524 AU, at Mars
    semimajor axis   1.262 AU
    excentricity     0.2076
    orbital period   1.418 yr = 517.92 day
    oneway time      0.709 yr = 258.96 day
    daily motion     0.695 deg/day
    orbital speed    26.51 Km/s
    perihelion speed 32.71 Km/s
    aphelion speed   21.47 Km/s
Properties of Earth and Mars
    I could go thru the same computation for Mars and Earth properties 
but it's just as well to copy them off from astronomy references. I 
set them out here without derivation 
    property         Earth       Mars
    ---------------  -------     -------
    semimajor axis   1.000 AU    1.524 AU
    excentricity     0.00        0.00
    inclination      0.00 deg    0.00 deg
    orbital period   1.000 yr    1.881 yr
    daily motion     0.986 d/d   0.524 d/d
    orbital speed    29.78 Km/s  24.13 Km/s
    Because I'm merely showing how Hohmann orbits work, I let Earth 
and Mars run in circular orbits. The distance from the Sun and orbital 
speed are the same all around the orbit. 
    In addition to the use of circular orbits for Mars and Earth I 
crank in a few other simplifications. The most important of these is 
the neglect of local gravity near Earth and Mars. I allow that the 
spaceship is already a ways from Earth having completed its lift from 
the ground to some parking orbit. By the same token I neglect the 
gravity near Mars. The spaceship ends its trip in some high parking 
orbit around the planet. 
    I neglect the inclination between Mars and Earth. It's slight any 
way but in a real mission itinerary we have to include a transfer 
between orbit planes. This calls for more fuel. I further miss out 
midcourse corrections, which will in a real case be required no matter 
how careful we inject the craft into the Hohmann trajectory. 
Position and coordinates
    Here's the part that throws most people who try to walk thru the 
mechanism of Hohmann orbits -- or just alignments of the planets in 
general. The Earth and Mars are in continuous motion around the Sun. 
In the diagram above where we are looking from the north pole of the 
Earth's orbit thru the ecliptic plane, to the south pole, the motion 
or revolution is counterclockwise. 
    The location of the planets are specified by the angle they stand 
on around the Sun, the heliocentric ecliptic longitude. We could 
pretend that the launch from Earth occurs when the Earth is in 
longitude zero. Because there probably was never an occasion when 
Earth and Mars were properly lined up for a Hohmann transfer when 
Earth was in longitude zero, I think it's best to deal with a relative 
longitude scale around the Sun. I zero this scale on the Earth at the 
start of the mission and bank all longitudes off of this zero. 
    In the diagram zero is left from the Sun; 90 degrees, down; 180 
degrees, right, where Mars will be when we arrive; 270 degrees, up. 
    When I have to cite the angle between Earth and Mars I mean the 
angle as seen from the Sun, that may be the difference of the two 
heliocentric longitude. A positive difference or offset is counted off 
counterclockwise around the Sun; negative, clockwise. 
Launch from Earth
    We have to launch the ship at a moment such that it arrives at the 
180-degree point when Mars also arrives there. Since the oneway trip 
to Mars takes 0.709 year, or 258.96 day, we must have Mars at launch 
0.709 year BEHIND this point. We lead Mars with our craft by this 
aamount so that target and craft meet at the same place and time. 
    This 180-degree point is sometimes called the superior conjunction 
point, but this is very misleading. Mars is NOT in superior 
conjunction at launch, at arrival, or at any time during either the 
outbound or inbound flight. 
    We must roll back Mars by 258.96 days of his own daily motion to 
put it at the proper heliocentric angle relative to Earth. 
    Mlaunch = (-258.96 day) * (0.524 deg/day)
            = -135.7 deg 
This is the clockwise angle from the 180-deg mark. From the zero point 
the clockwise angle is 
    Mlaunch = (180 deg) + (-135.7 deg)
            = +44.3 deg 
Mars at launch has to be near point B in the diagram here. Earth is at 
A, where the Hohmann orbit touches it. The 44.3 deg is angle AOB 
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    With Earth behind Mars but running faster, an opposition of Mars 
is coming up. The launch has to go off some days BEFORE opposition. 
Earth is gaining on Mars because its daily motion is greater. The 
relative angular speed between the two planets is the difference in 
their daily motions 
    n[M/E] = (0.986 deg/day) - (0.524 deg/day) 
           = 0.462 deg/day 
At this rate it'll take
    Topp = (44.3 deg) / (0.462 deg/day) 
         = 95.89 days 
to achieve opposition. The launch must take place 96 days BEFORE this 
opposition. Such a relation makes it easy to plan the flight. Mars 
oppositions are well known in advance. There was an opposition of Mars 
on 13 June 2001. If a Hohmann ship were to go to Mars in 2001, it 
would have to set off on its way on March 8th, 96 days earlier. 
Arrival at Mars
    En route to Mars the craft sees Earth and Mars continue to 
circulate around the Sun. The ship left Earth before opposition by 96 
days. Opposition comes and goes. Earth overtakes Mars and pulls ahead. 
    Opposition of Mars occurs when Mars and Earth are near points C and 
D in their orbits, below. This happens when Earth advances in its 
orbit by 
    Mopp = (0.986 deg/day) * (95.89 day)
         = 94.5 deg. 
    I go thru the same computation based on Mars, to check my maths. 
Remember, at the start of the flight Mars already had a head start of 
44.3 degrees! 
    Mopp = (0.524 deg/day) * (95.89 day) + (44.3 deg) 
         = 94.5 day 
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                   .    .  D . 
    The spaceship arrives at Mars 258.96 days after launch. Mars is 
then at the 180-deg point. Earth is, at arrival, some where else in 
its orbit. It moved around 
    MEarth = (0.986 deg/day) * (258.96 day) 
           = 255.3 deg 
or (255.3 deg) - (180 deg) = 75.3 deg AHEAD of Mars. Earth at arrival 
is near E in the diagram below with Mars at F. Angle FOE is 75.3 deg. 
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Return to Earth 
    A completely parallel analysis will bring the spaceship from Mars 
back to Earth. With the oneway flight time of 258.96 days, we must 
back off Earth by 
    MEarth = (0.986 deg/day) * (-258.96 day) 
           = -255.3 deg 
           = 104.7 deg 
Earth is at the launch from Mars some 75 degrees BEHIND him. 
Dwell at Mars
    After arriving at Mars and doing our thing there we want to return 
on the next opportunity. We sure can not return immediately, like we 
just rounded Mars and start heading back on the inbound arm of the 
Hohmann path. We have to wait until the planets are lined up. 
    We found that at arrival at Mars Earth was 75.3 degrees ahead of 
Mars. At launch from Mars Earth was 75.3 degrees behind Mars. We must 
wait for Earth to circulate around the Sun and and get to the proper 
launch position. 
    The arc to be covered by Earth as seen from Mars is the far sector 
of Earth's orbit between 75.3 deg ahead and 75.3 degrees behind Mars. 
    Adwell = (360 deg) - (75.3 deg) - (75.3 deg)
           = 209.4 deg 
Earth must move thru this arc with the relative daily motion of 0.462 
deg/day. The time to do so is 
    Tdwell = (209.4 deg) / (0.462 deg/day)
           = 453.25 day 
We got a LONG wait before we can come home! 
    While we are on Mars, Earth passes thru the superior conjunction 
as seen from Mars. It's also Mars's superior conjunction from Earth's 
eye. Because of the symmetry of the positions of Earth and Mars at the 
start and end of the dwell we don't have to go thru all the calcs to 
find this conjunction. It happens at the midpoint of our stay on Mars, 
or at day 226.62 of our stay. This is also day 485.58 since the 
mission began. 
    The superior conjunction takes place at longitude
    MMars = (0.524 deg/day) * (485.58 day) + (44.3 deg) 
          = 298.7 deg 
          = (0.524 deg/day) * (226.62 day) + (180.0 deg) 
          = 298.6 deg 
    MEarth = (0.986 deg/day) * (485.58 day) + (0 deg) 
           = 118.8 deg 
           = (0.986 deg/day) * (226.62 day) + (255.3 day) 
           = 118.7 deg 
    These are quite 180 degrees apart, which is how the planets's 
longitudes differ at superior conjunction. 
    During the stay on Mars Earth and Mars continue to revolve around 
the Sun. We can figure this by either adding to the positions at the 
beginning of our stay on Mars or counting from the start of the entire 
trip, 712.21 days ago. 
    MEarth = (0.986 deg/day) * (453.25 day) + (255.3 deg) 
           = 342.2 deg 
           = (0.986 deg/day) * (712.21 day) 
           = 342.2 deg 
    MMars = (0.524 deg/day) * (453.25 day) + (180 deg) 
          = 57.5 deg 
          = (0.524 deg/day) * (712.21 day) 
          = 57.5 deg 
    In both cases I tossed out the whole lap around the Sun of 360 
degrees. The difference of these two is the offset of Earth from Mars 
when we start on the return arm of the trip. 
    A = (342.2 deg) - (57.5 deg)
      = -75.3 
which is exactly what we figured using the speed of Mars relative to 
Earth' Earth is 75.3 degrees behind Mars at the beginning of our 
return trip. The planets are at G and H below with angle GOH being 
75.3 deg. 
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Arrival at Earth
    Because Earth is behind Mars when we started off for Earth and 
Earth is running faster than Mars, an other opposition takes place 
during our flight. It happens at 
    Topp =  (75.3 deg) / (0.462 deg/day) 
          = 162.99 days 
after we leave Mars or on day 875.20 since leaving Earth. Let's 
doublecheck this. 
    MEarth = (0.986 deg/day) * (875.20 day) + (0 deg) 
           = 142.9 deg 
           = (0.986 deg/day) * (162.99) + (342.2 deg) 
           = 142.9 deg 
    MMars = (0.524 deg/day) * (875.20 day) + (44.3 deg) 
          = 142.9 deg 
          = (0.524 deg/day) * (162.99 day) + (57.5 deg) 
          = 142.9 deg 
    When we arrive back at Earth, after a 258.96 day Hohmann flight or 
971.17 days elapsed mission time, Mars rounded to 
    MMars = (0.524 deg/day) * (258.96 day) + (57.5) 
          = 193.2 deg 
          = (0.524 deg/day) * (971.17 day) + (44.3) 
          = 193.2 deg 
Earth rounded to 
    MEarh = (0.986 deg/day) * (258.96 day) + (342.2 deg)
          = 237.5 deg 
          = (0.986 deg/day) * (971.17 day) + (0 deg) 
          = 237.6 deg 
Mars ends up (237.5 deg)-(193.2 deg) = 44.3 deg behind Earth. The 
planets stand at points I and J below. This 44.3 deg is angle JOI. 
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The round trip schedule
    I gather here all of the steps in the journey to and from Mars 
here. Day is the total mission elapsed time. 
    event                day    heliE  heliM  comments
    -----------------  -------  -----  -----  --------
    launch from Earth     0.00    0.0   44.3
    opposition           95.89   94.5   94.5  en route to Mars
    arrive at Mars      258.96  255.3  180.0
    superior conj       485.58  118.8  298.7  during stay on Mars
    launch from Mars    712.21  342.2   57.5
    opposition          875.20  142.9  142.9  en route to Earth
    arrive at Earth     971.17  237.5  193.2
Energy budget
    The Hohmann orbit allows for passage between two planets with the 
least possible consumption of fuel or expenditure of energy. I do not 
include here the lifting of the spacecraft from Earth or Mars and 
placement into the Hohmann trajectory. Once the ship is far from Earth 
or Mars, a couple hundred thousand kilometers in either case, it is 
reasonably free of the local gravity field. This distance is the order 
of the Moon's orbit, a tiny fraction of the Earth-Marsseparation. 
    Ideally the launch at either end will take advantage of the 
orbital speed of the planet and only an increment of speed, plus or 
minus, is needed. The amount of energy or fuel allocated to the 
Hohmann portion of the entire mission is really very slight. 
    At Earth launch the ship is already moving with Earth orbital 
speed, 29.78 Km/s. The perihelion speed for the Hohmann orbit is 32.71 
Km/s. To inject into the Hohmann orbit the ship needs an increment of 
(32.71)-(29.78) = 2.93 Km/s. 
    Mars orbital speed is 24.13 Km/s; the Hohmann aphelion speed at 
Mars is 21.47 Km/s. A second increment of speed is needed to pace Mars 
of (24.13)-(21.47) = 2.66 Km/s. 
    For the return trip we must at launch from Mars slow the ship by 
2.66 Km/s. This lets it drop into the inbound arm of the Hohmann 
orbit. At Earth we must againslow the ship, now by 2.93 Km/s, to match 
Earth's orbit speed. 
Minimum fuel for Hohmann path 
    In order to see why the Hohmann path requires the least fuel of 
all paths between Earth and Mars we must look at how rockets work. 
Rockets of the sort now and in the foreseeable future operate by 
expelling mass. This expulsion reacts against the body of the rocket 
to propel it forward. The mass consists of exhaust gases from burning 
fuel within the rocket's engine. 
    Without going into derivations or proof, we have from rocketry 
litterature the following formulae 
 (thrust on rocket) = (exhaust speed) * (fuel mass burned/second) 
 (thrust) * (duration of burn) = (rocket mass) * (change of speed) 
                               = (exhaust speed) * (fuel mass burned) 
 (change of speed) = (fuel mass burned) * (exhaust speed)
                     / (rocket mass) 
    These are not strictly correct because as we burn fuel the mass of 
the entire rocket decreases but the sense of the relation is not 
seriously altered. 
    In order to move away from Earth orbit into an orbit that reaches 
Mars we must add velocity to the rocket. The rocket, by other means 
beyond this article, lifted off of Earth and is travelling parallel to 
Earth at the Earth's orbital speed. We can from this moment send the 
rocket to Mars along any direction away from Earth. But the one 
direction that takes full advantage of the initial orbital speed is 
that parallel to Earth. This direction requires the least change of 
speed, called delta-V in rocketry. 
    From the above formulae the change of speed is proportional to the 
mass of fuel expelled from the rocket. Hence, to reduce the amount of 
fuel to burn, and to carry on the rocket, we seek a path to Mars that 
calls for the least delta-V. 
    A similar argument applies at Mars. We seek an approach path near 
Mars that requires the least delta-V to bring the rocket in step with 
Mars. In both cases the path of least delta-V, and of least mass of 
fuel to carry with us, is that parallel to the planet's motion or, 
what amounts to the same thing, tangent to the planet's orbit. 
    If we build a path touching tangently at the two plants's orbits, 
we have the path between the two planets needing the least delta-V at 
both ends. This is what Walter Hohmann figured out in 1916, the 
Hohmann trajectory or orbit. 
Launch windows 
    Note that for the transfer to Mars along a Hohmann path we can not 
arbitrarily pick a launch date. We must wait until the planets are 
properly aligned. In the example here the launch must take place 96 
days before the next opposition. Flights can and were launched at 
other offsets from opposition but they travelled along nonHohmann 
paths. They spent more fuel and energy than the Hohmann minimum. the 
reason for choosing a nonHohmann path are many but once picked you 
have to provide for the extra energy and fuel. 
    Since oppositions occur every synodic cycle, the launch window also 
occurs once in that cycle. For Mars and Earth this cycle is 
    Psyn = (PMars) * (PEarth) / ((PMars) - (PEarth)) 
         = (1.881) * (1.000) / ((1.881) - (1.000))
         = 2.135 yr
         = 779.81 day 
    A launch window for a Hohmann flight to Mars from Earth comes once 
in 780 days! The same interval stands between launches from Mars back 
to Earth, in Earth days. 
Guido vonPirquet
    While Walter Hohmann's trajectory, tangent to the orbits of the 
two planets at either end of the flight, is indeed the most economical 
of energy, it does require a long duration of flight. If we allow for 
some extra fuel burn, we can shorten the flight time. In the limit we 
can use a brute force path, a radial straight run from Earth to Mars 
near Mars's opposition under continuous thrust. 
    Guido vonPirquet lived in Austria-Hungary, with correspondence 
with other rocketeers. He was a colleague in the space travel circles 
with Walter Hohmann. He was an engineer and inventor of considerable 
acclaim. He, for example, first designed an ion rocket powered by 
solar energy and showed the huge mass savings by sending 
interplanetary missions from an Earth-orbiting space station in the 
stead of from the Earth's surface. 
    In vonPirquet's and Hohmann's era there was major concern for the 
endurance of a crew to traverse interplanetary space. But there had to 
a balance between the longest but cheapest trajectory of Hohmann and 
the shortest but dearest radial path. vonPirquet came up with a 
compromise trajectory. It used only a modest extra amount of fuel, but 
shaved the flight time to about 2/3 that of a Hohmann path. 
    He worked out this solution graphicly, weighing the price to pay 
in fuel and that for food and oxygen for the crew. (Life on a 
spaceship was pretty naive in the 1920s and 1930s.) If the path were 
part of an ellipse that slightly overlapped the two planets's orbits, 
a substantial savings of time could be realized with only a small 
increase in fuel. 
    He presented his work in articles for the magazine 'The rocket' 
from 1923 thru 1929. In 1926 he further elaborated this ideas in 
sections of 'The feasibility of space travel'. Other parts were 
written by, among others, Hohmann, Ley, and Oberth. 
vonPirquet trajectory 
    The vonPirquet orbit or trajectory has its aphelion just outside 
the orbit of Mars (to stick with the planet pair I'm using here) and 
its perihelion just inside the Earth's orbit. The heliocentric angle 
of the segment of this ellipse between launch from Earth and arrival 
at Mars is 120 degrees. 
    The angle of 120 degrees was an engineering judgement based on the 
shallow angle of intersection of the vonPirquet path with the planet 
orbits. This happens, in light of modern orbit calculations, to be a 
surprisingly good solution altho analyticly it is not the very best.
    The extra delta-V is in the radial direction to veer the rocket 
out of Earth's orbit at departure or to veer it into Mars's orbit at 
arrival. The lion's share of delta-V remains tangential, or parallel, 
to the orbits. 
    There is not a single unique trajectory, like for the Hohmann 
case. Each planet pair has a family of vonPirquet flightpaths. I give 
here the parms for one typical solution for the Earth-Mars journey 
    property         value 
    ---------------  -----------------
    perihelion dist  0.962 AU, inside Earth 
    aphelion dist    1.563 AU, outside Mars 
    semimajor axis   1.263 AU 
    excentricity     0.238 
    orbital period   1.419 yr = 518.44 day 
    oneway time      0.473 yr = 172.81 day 
    daily motion     0.694 deg/day 
Amazing bit of history
    The Soviet Union's Venera 1 was launched toward Venus on 12 
February 1961. Alas, radio contact with the probe was lost en route.  
It continued on its way to sail inertly past Venus. 
    Almost entirely missed from news about the mission was that Venera 
1 flew along a vonPirquet, not a Hohmann, trajectory. The Pravda 
newspiece on 26 February 1961 presented a chart of Venera 1 and its 
glatt vonPirquwt flightpath from Earth to Venus. 
    Now for the amazing bit. That chart is a straight reproduction, 
like a photocopy, of vonPirquwt's own original one published in 1928, 
over thirty years earlier! The Soviets dubbed in Russian labels and 
inserted new dates! 
Table of flights to Mars
    I give here all the flights to Mars, but probably will not update 
them in future revisions of this article. I count as completed flights 
those that actually made it all the way to Mars, even if the overall 
mission bombed out. The fate of missions that didn't make it to Mars 
is noted for 'arrive' and 'time' in place of completion data. 
    The 'depart' is for launch from Earth. Typicly the launch is a few 
hours before the insertion into transmartian flight. The  'arrive' is 
not precisely defined in space science. It may be the initiation of 
Mars imaging or telemetry, insertion into circummartian orbit, release 
of a lander, closest approach for a flyby, or a calculated intercept 
for a probe which died in flight. The  arrival dates here may differ 
slightly from those cited elsewhere. 
 flight     depart       opposition   lead  arrive       time
 -------    -----------  -----------  ----  -----------  ----
 Marsnik 1  1960 Oct 10  1960 Dec 30   81d  died in launch
 Marsnik 2  1960 Oct 14  1960 Dec 30   77d  died in launch
 Sputnik 22 1962 Oct 24  1963 Feb 04  103d  died in Earth orbit
 Mars 1     1962 Nov 03  1963 Feb 04   93d  1963 Jun 19  228d 
 Sputnik 24 1962 Nov 04  1963 Feb 04   92d  died in Earth orbit
 Mariner 3  1962 Nov 05  1963 Feb 04   91d  died in Earth orbit
 Mariner 4  1964 Nov 28  1965 Mar 09  101d  1965 Jul 14  228d 
 Zond 2     1964 Nov 30  1965 Mar 09   99d  1965 Aug 06  249d
 Zond 3     1965 Jul 18  1967 Apr 15  ---   test of Mars probe
 Mariner 6  1969 Feb 25  1969 May 31   95d  1969 Jul 29  154d
 Mariner 7  1969 Mar 27  1969 May 31   65d  1969 Aug 03  129d
 Mars 1969A 1969 Mar 27  1969 May 31   65d  died in launch
 Mars 1969B 1969 Apr 02  1969 May 31   58d  died in launch
 Mariner 8  1974 May 08  1971 Aug 10   94d  died in launch
 Cosmos 419 1971 May 10  1971 Aug 10   92d  died in Earth orbit
 Mars 2     1971 May 19  1971 Aug 10   82d  1971 Nov 27  191d 
 Mars 3     1971 May 28  1971 Aug 10   74d  1971 Dec 02  188d 
 Mariner 9  1971 May 30  1971 Aug 10   72d  1971 Nov 03  157d 
 Mars 4     1973 Jul 21  1973 Oct 25   96d  1974 Feb 10  204d 
 Mars 5     1973 Jul 25  1973 Oct 25   92d  1974 Feb 12  202d 
 Mars 6     1973 Aug 05  1973 Oct 25   81d  1974 Mar 12  219d 
 Mars 7     1973 Aug 09  1973 Oct 25   77d  1974 Mar 09  212d 
 Viking 1   1975 Aug 20  1975 Dec 15  117d  1976 Jun 21  305d 
 Viking 2   1975 Sep 09  1975 Dec 15   97d  1976 Aug 09  335d 
 Phobos 1   1988 Jul 08  1988 Sep 28   82d  died in flight
 Phobos 2   1988 Jul 12  1988 Sep 28   78d  1989 Jan 30  202d
 Observer   1992 Sep 25  1993 Jan 07  104d  1993 Aug 21  330d
 Global Sur 1996 Nov 07  1997 Mar 17  130d  1997 Sep 12  309d 
 Mars 96    1996 Nov 16  1997 Mar 17  122d  died in Earth orbit
 Pathfinder 1996 Dec 04  1997 Mar 17  103d  1997 Jul 04  212d
 Nozomi     1998 Jul 03  1999 Apr 24  ---   via Belbruno path 
 Climate Or 1998 Dec 11  1999 Apr 24  134d  1999 Sep 23  286d
 Polar Land 1999 Jan 03  1999 Apr 24  111d  1999 Dec 03  183d
 Odyssey    2001 Apr  7  2001 Jun 13   67d  2001 Oct 23  199d
 Mars Exp   2003 Jun  2  2003 Aug 28   87d  2003 Dec 19  200d
 Spirit     2003 Jun 10  2003 Aug 28   79d  2004 Jan  4  208d
 Opport'y   2003 Jul  8  2003 Aug 28   51d  2004 Jan 25  201d
 Mars Recon 2005 Aug 21  2005 Nov  7   78d  2006 Mar 10  262d
 Phoenix    2007 Aug  4  2007 Dec 28  146d  2008 May 25  295d
 Phobos-G   2011 Nov  8  2012 Mar  3  116d  died in Earth orbit
 Curiosity  2011 Nov 26  2012 Mar  3   98d  2012 Aug  6  254d
    Of the 41 attempts to reach Mars, 12, almost 1/3, were flat out 
failures. Some of the others were failures after reaching Mars, like 
Polar Lander and Climate Orbiter. Mars Express was a partial failure 
because its lander craft Beagle 2 was lost after release. The overall 
failure rate for the whole history of Mars exploration is quite 50%. 
    Altho just about every trip to Mars was launched 95ish days before 
the next opposition and took 200ish days to arrive, none pursued a 
pure Hohmann trajectory. For the most part the craft was sent into an 
orbit that intersected Mars orbit at a shallow angle, with aphelion 
some distance outside Mars orbit. This made for a faster trip. The 
path was more like a vonPirquet orbit. 
    A few took much longer paths because they were deliberately sent 
to Mars in a roundabout route. Mars Observer was first sent inbound 
from the Earth to its own orbit's perihelion. It then slided outside 
Earth orbit, reached out to Mars, and went beyond Mars to aphelion. It 
finally dipped inbound to intersect Mars. 
    Nozomi followed a Belbruno trajectory which took over four years 
to complete. This flight involved swings past other inner planets 
before spiraling out to Mars.
    Despite these deviations from the glatt Hohmann path, its imprint 
is there, with the first actual attempt coming some 44 years after 
Walter Hohmann first laid out its principles. 
Human flights to Mars
    It has been technicly feasible to send a human crew to Mars since 
the mid 1980s. At the year 2003 the earliest attempt to send a crewed 
flight to Mars is postulated for the 2020s. Yet there persists a dull 
understanding of the duration of any human flight which I hope you now 
can appreciate. 
    The crew must be sustained and protected in flight and on Mars for 
a minimum of, uh, over 2-1/2 YEARS! A large fraction of this time is 
the forced stay on Mars of 1-1/4 YEARS until Earth lines up for the 
return arm of the Hohmann transfer. 
    Amazingly many space activists miss this crucial consideration for 
human trips to Mars. They blithely believe we can go there, spend a 
week or so, and scoot home. Perhaps they have the Apollo syndrome, 
remembering the couple days that astronauts spent on the Moon before 
heading home. 
    In any case, the logistics of sustaining even a two-person crew are 
horrendous. Mars has utterly no resources immediately available or 
adaptable to feed, clothe, house, comfort the crew. The ship must 
bring from Earth every thing needed to survive and perform for a span 
that excedes by an order the alltime accumulated man-years spent on 
the Moon. 
    What's more, there is no hope of replenishing the Mars station. 
The next opportunity for a flight comes 779 days after that of the 
human flight. This time already excedes the travel to and the stay on 
Mars, foreclosing any reasonable means of provisioning the station 
with followup missions. 
    If we insist that the crew remain on Mars for one provision flight 
to arrive, and then depart at the second opportunity, we're in for a 
LONG ride. Let the station operate thru TWO rounds of launch windows, 
for TWO synodic periods. That gives the chance on the first synodic 
cycle to send the replenishment flight and the crew returns on the 
second cycle. 
    The supply flight will arrive at the station 2 years 10 months 
after the crew left Earth. Then the crew must wait a further 1 year 3 
months for the next chance to depart for Earth. The return flight 
takes over 8 months. The crew has to keep alive and well for fully 4 
years 10 months!! 
    The schedule of such a voyage with one supply flight is given here 
in integer days and decimal years 
    event                day  year
    -------------------  ---  ----
    launch from Earth      0  0.00
    arrive at Mars       259  0.71
    skipped departure    712  1.95
    supply launch        779  2.13
    supply arrives      1038  2.83
    depart from Mars    1491  4.08
    arrive at Earth     1750  4.79
    There is one obvious way over this barrier, but it would never and 
never win favor in the United States. Just send the humans on a oneway 
trip. When their chores on Mars are finished, after a few days or 
weeks, they chew the poison pills.