Paul Hatherly explores the physics behind visiting the Red Planet
Last Christmas was a dramatic period in the history of the exploration of Mars, with no fewer than five spacecraft attempting to orbit or land on the Red Planet. Was this coincidence or a planned strategy by the parties concerned? Or was it all down to the immutable laws of celestial mechanics?
Let's look at some of the basic physics behind getting to Mars in the first place and the strategies used on arrival.
Activity: Look at the timing of past launches to Mars. Do they occur randomly, or is there some pattern?
Curriculum references: impulse, Kepler's laws, law of gravity (A-level physics); Newton's laws of motion, Kepler's laws, law of gravity (GCSEphysics and GCSE astronomy).
Getting to Mars
Contrary to popular conceptions, space travel is not as simple as frequently depicted on film or television. We do not have exotic engines capable of propelling us directly from Earth to Mars in a matter of days or hours when the two planets are closest (Figure 1a).
What we do have are chemical rockets capable of burning for a short time and providing us with a modest amount of the currency of space flight - Fv (read as "delta-vee"), the change in velocity given to our spacecraft following an engine burn. These rockets provide enough impulse to place our spacecraft in an elliptical orbit with its closest point to the Sun (the perihelion) intersecting the Earth's orbit, and its furthest point (aphelion) intersecting Mars' orbit (figure 1b). Such an orbit is referred to as a Hohmann transfer orbit (HTO). Once in the HTO, our spacecraft coasts along the orbit with its path governed by Kepler's laws, themselves consequences of Newton's laws of motion and the law of gravity.
Activity: Why do rockets work in space? Think about this in terms of Newton's Laws (GCSE). Investigate forms of spacecraft propulsion other than chemical rockets (A-level).
How long does it take?
Our limited capability means that getting to Mars is not quick. Figure 2 shows how to obtain the dimensions of the HTO. Knowing the semi-major axis, a, of the orbit (half the length of the long axis in astronomical units (AU) - the mean radius of the Earth's orbit) we can obtain the period of an orbit, P (in years), using Kepler's third law (a3 = P2, so P = Ca3).
For our transfer orbit, Figure 2 shows: 2a = RE + RM = 1 + 1.52AU = 2.52AU, so a= 1.26AU.
The period of the Hohmann transfer orbit is, then, PHTO=C1.263 = 1.41 years (for comparison, PE = 1 year, PM = 1.87 years).
PHTO is the time for a complete orbit of the HTO. Since we only want to get to Mars, half of this time is sufficient, hence the time to get to Mars is about 0.71 years, or about 8.5 months.
Activity: Find the semi-major axes of other planets - eg Jupiter - and calculate the Hohmann transfer orbit periods to these planets (GCSE).
Curriculum reference: Elliptical orbit (GCSE astronomy).
When can we go? (1)
Ever been clay pigeon shooting? If so, you will know that you can guarantee missing a moving target by aiming your gun at it. By the time the shot reaches where the target was, it's gone. The same is true of interplanetary flight. It's no good launching into a trajectory that will take you from Earth to where Mars is now - in 81Z2 months, Mars will be almost on the other side of its orbit.
When can we launch? Since the flight time is 81Z2 months, we have to wind the clock back and see where Mars was 81Z2 months before arrival. Figure 3 shows this situation. A Martian year is 1.87 Earth years, and 81Z2 months is 0.71 Earth years. So, the flight time is 0.711.87 = 0.38 Martian years.
In this time, Mars travels 0.38 x 360 LESS THAN OR EQUAL TO 137x around its orbit. In other words, we must launch our spacecraft when Mars is positioned 137x ahead of the arrival point.
From the geometry of Figures 2 and 3, we can see that the perihelion and aphelion of the transfer orbit are 180x apart. So, our spacecraft travels through 180x between leaving Earth and arriving at Mars. This means that we can launch when the angle between the Earth and Mars is 180x - 137x = 43x.
To go to Mars, we just need to work out when Mars is ahead of the Earth by 43x.
Activity: Using the transfer orbit times calculated previously for other planets, work out the Earth-planet launch angles for other planets (GCSE).
When can we go? (2)
We have worked out what the configuration of Mars and Earth must be to be able to use a HTO. How frequently does this occur? Every year or so? Or is it a rare event? To answer this question, we need do little more than watch the hands on a clock. How long does the minute hand take to go round once? Simple: one hour (Figure 4a). Now imagine you're a fly on the hour hand. At midnight, you are covered by the minute hand. How long before you're covered again? It's not one hour, because, in the meantime, the hour hand has moved a bit. Rather, it's a bit more than an hour - 1 hour 5.45 minutes (Figure 4b).
The same is true of planets. Imagine Earth and Mars are in a configuration suitable for a launch. Time passes. The Earth goes round the Sun in one year - the sidereal period. However, in one year, Mars has moved on, so Earth has to catch up to repeat the same configuration. This extra time gives us the idea of the synodic period - the time between equivalent configurations. For Earth and Mars, the synodic period is about 26 months.
So, every 26 months an opportunity - or "window" - Jlasting several weeks opens, enabling us to launch missions to Mars.
Activity: Using the argument above, show that the Earth-Mars synodic period S is given by: 1S = 1PE - 1PM. What are the synodic periods for the other planets? Think carefully about Venus and Mercury (A-level).
Now we understand that the flurry of activity around Mars had nothing to do with international co-operation (although there was plenty) and all to do with our limited propulsion technology and the laws of physics.
Curriculum reference: for Earth, the sidereal day is four minutes shorter than the synodic day (GCSE astronomy).
We've launched when Earth and Mars are placed correctly. We have enough Fv to put us in the right HTO. We've waited many months while our spacecraft has coasted along its orbit to Mars. Now comes the make-or-break moment.
What happens depends on the type of mission. Only two need to concern us: orbiters and landers. These have very different strategies for achieving their goals.
First, orbiters. Our spacecraft has to be captured by Mars' gravity. To do this, it needs an engine burn to slow it relative to Mars. (Note the italics - we may actually have to increase speed to match the speed of Mars because our transfer orbit may take us slightly beyond Mars, hence our speed will be slower). Usually, this initial orbit is no good for science: too high, too elongated or in the wrong orientation (or all three). We therefore need to shape the orbit, which we can do in one of two ways.
First, by use of further engine burns. For example, a retro-burn at periapsis (the closest approach to Mars) will result in a lowering of the apoapsis (furthest distance), thus making the orbit more circular. The same effect can be achieved by dipping the spacecraft slightly into the atmosphere at periapsis and using air resistance (Figure 5). Although this so-called aerobraking technique is risky, the benefits are enormous. For example, less fuel is required, saving mass that can be used for more instruments.
For landers, the requirements are even more gruelling. Somehow, the lander has to slow down from orbital speeds of 10s of kms to a standstill and still be operating at the end of it.
The most obvious technique is to use rockets to slow down and undergo a controlled landing. Although this technique was successful with the Viking landers of the 1970s, the level of control and reliability required of the engines has proved too problematic. Again however, the atmosphere of Mars comes to our rescue. The technique of using airbags to "bounce" down raised many eyebrows when proposed, but was amply demonstrated by the Pathfinder mission in 1997, and again with the Spirit and Opportunity rovers which landed in January 2004.
The technique is deceptively simple, as illustrated in Figure 6 for the Spirit lander. The lander, encased in a heatshield, enters the Martian atmosphere from its transfer orbit at a speed of about 5kms and an altitude of about 130km. Atmospheric friction heats the shield to about 1500x and reduces the speed to about 400ms in about four minutes. Now, at about 9km above the surface, a parachute is deployed and the heatshield jettisoned. Shortly afterwards, the lander is lowered on a tether and starts monitoring its altitude by radar. About eight seconds before touchdown, bags around the lander inflate, rockets slow the descent further, and finally, at a height of about 10m, the tether is cut and the lander falls the rest of the way. The airbags cushion the landing, and the lander bounces and rolls, for perhaps as far as a kilometre. The airbags are then deflated, the lander opens and deploys its instruments, and science begins.
Activities: Investigate parachute designs. What design will be most suitable for a lander? Remember, you want good deceleration and stability, with minimal swaying or spinning (GCSE physics). What is the average deceleration experienced by the lander in this period? Remember, acceleration = (change in velocity)(time taken) (GCSE and A-level physics). Investigate the effect of atmospheric pressure on the drag of a parachute. You may find the result surprising (A-level physics).
Curriculum references: relative speeds, projectiles (A-level).
So we have got to Mars, but the challenge is only just beginning. Our orbiters have to remotely collect data; our landers have to get moving and also commence their work. Crucially, both have to get the information back to Earth, else what was the point? Assuming success, enough material will be returned to keep scientists busy for years, answering questions like: Was there once water on Mars? How does the atmosphere behave? Are the rocks like anything on Earth? And, with an eye on the future, are there resources available to support a manned mission? If, however, the missions fail, a long, heartbreaking two-year wait for the next launch window begins.
Activity: Think about restrictions on when communications can take place (GCSE physics).
Curriculum reference: telecommunications (GCSE physics).
Dr P A Hatherly works at the Department of Physics, University of Reading.
Much of the information in this article can be supplemented from reliable web-based resources, a number of which are given here (opposite page).
Unless indicated otherwise, material is suitable for all levels.
* NASA Mars Rovers: marsrovers.jpl.nasa.govhomeindex.html
* Mars Express: www.esa.intspecialsMars_Expressindex.html
* Education Resources (up to GCSE): www.marsquestonline.orgresourcesindex.html
* Orbits (Kepler's laws, Hohmann orbits and synodic periods - A-level) From University of Reading Part 1 module "Exploring the Universe", P.A.
Hatherly: www.rdg.ac.ukphysicsnetunits1ph1005bhomelecturesLecture_6_04_filesf rame.htm
Communications (Deep Space Network): http:marsrovers.jpl.nasa.govmissioncommunications.html