HOHMANN ORBITS ============ John Pazmino NYSkies Astronomy Inc email@example.com www.nyskies.org 2001 October 25 initial 2012 July 28 current
Introduction ---------- 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 trajectories. 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.
History ----- 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 planets.
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.
. . . . . . # # . . # # . . #. . #. . #. . #. . A O . F . #. . #. . #. . #. . # # . . # # . . . . . .
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 . . . . . . # # . . # # . . #. . #. . #. . #. . A O . # . #. . #. . #. . #. . # # . B # # . . . . . .
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
. . . . . . # # . . # # . . #. . #. . #. . #. . # O . # . #. . #. . #. .C #. . # # . . # # . . . . . 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. . . . . . . # # . . # # . . #. . E #. . #. . #. . # O . F . #. . #. . #. . #. . # # . . # # . . . . . .
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. . . . . . . # # . . # # . . # . #. . #.G . #. . # O . # . #. . #. . #. . #. . # # . H # # . . . . . .
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. . . . . . . # # . . # # . . #. . #J . #. I . #. . # O . # . #. . #. . #. . #. . # # . . # # . . . . . .
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.