*A comprehensive primer on the technique of space travel, i.e. how we travel from A to B in space*

An extended version of this article can be downloaded as an off-line resource in pdf-format

**WEIGHTLESS**

Why are astronauts in the International Space Station (ISS) weightless? “Because there is no gravity up there” you often hear. “Astronauts and scientists themselves often talk about zero-gravity, don’t they?”.

Newton’s first law of motion .

A satellite orbit in its simplest form can be compared with uniform circular motion, like when you sling a weight around on a string. This circular motion requires a constant force towards the centre of the circle. For a satellite, including of course the ISS, gravity of the Earth provides that “centripetal” force. If there was no gravity up there, the ISS would take of in a tangential straight line and disappear into space because ofSo there definitely is gravity up there demonstrated by the fact that the ISS nicely continues to travel in its orbit. And this holds for any satellite, even the natural satellite we have: the Moon. She has been “up there” for at least 4 billion years.

**Let us do some experiments**

When a skydiver jumps out of a plane at high altitude it is advisable to have a parachute, but keep that folded up for a while. She falls down at increasing speed and experiences weightlessness. A disturbing influence here is the wind and air drag she experiences. Astronauts don’t have that of course, but otherwise the situation is quite similar.

Modify the experiment by putting the skydiver in a box and dropping the whole box out of the plane (This is a thought experiment. DON’T DO THIS AT HOME). Now the skydiver will not feel any wind and will be almost weightless inside the box. Almost, because the box itself experiences the air drag and therefore falls a little slower than the skydiver. She will feel a very slight weight force towards the bottom (in the direction of falling).

Let us now look at a thought experiment that was proposed in 1687. Isaac Newton published his *Philosophiæ Naturalis Principia Mathematica* often just referred to as *Principia*, in which he explains his ground braking theory of gravity (among other things).

The image we show here of a canon on top of a mountain is from a later popularised version of the Principia. The idea is to fire the cannon, which is supposedly well above the atmosphere, with increasing charge and thus initial speed of the cannon ball. The latter will fall down to Earth at increasing distance, but there will be a speed at which the cannon ball will never hit the ground; it will continue to fall around the Earth. In practice this is impossible because of the Earth’s atmosphere and mountains that are not that tall, but as a thought experiment it is quite illustrative.

As a matter of principle the cannon ball, orbiting the Earth will come back to the same spot where it left the cannon, therefore if this was possible, the gunner would himself be struck by the projectile he had fired a while ago.

Look at the animation of this experiment

Gradually increase the firing speed and see what happens. At which minimum speed will the cannon ball come back to point V in the diagram? (At higher speeds the cannon ball will disappear from Earth. We will come back to that below).

This experiment illustrates that a satellite orbit actually is a perpetual free fall in the gravity field of the central body, the Earth in our case. But, as Newton realised, this holds for all orbital motion in space, e.g. the motion of the Moon around the Earth and of the planets around the Sun.

**So why are astronauts in the ISS weightless?**

Because they are in a constant free fall motion around the Earth. And the space station itself and everything else in it is in that same free fall. The Earth’s gravity is the very reason they are moving that way. The term “zero-gravity” is therefore misleading. It is much more accurate to say “weightless” because while everything in orbit is accelerating in the direction of the Earth’s centre (like the cannon ball), there is no weight force like you would experience while standing on the surface of the Earth.

Let us now talk a bit more about orbits and how to move in space.

**ORBITAL MOTION**

**Some basic principles**

Without going into the mathematics [go here for that] the velocity of a spacecraft in a circular orbit around the Earth is given by V_{C}^{2}=(GxM_{E})/R where G is the Universal Gravitational Constant, M_{E} is the mass of the Earth and R is the distance between the centre of the Earth and the centre of the spacecraft. We show this formula to draw some important conclusions about orbital motion.

- The velocity decreases with increasing altitude above the Earth. Hence higher satellites move slower than satellites in lower orbits.
- Velocity is independent of the mass of the spacecraft. This is a very important conclusion because it means that e.g. at the altitude of the ISS any object will move with exactly the same orbital velocity no matter how heavy or light. So the ISS itself (500 tons) will move exactly the same as any of the astronauts inside (or outside) or as any other object, say a paperclip inside the spacecraft.
- We can calculate that at low orbit (marginally above the Earth’s atmosphere) the velocity is about 7.9 km/s.
- With a little more calculation (knowing that the circumference of a circle is 2πR, we can calculate the altitude of a geostationary satellite (42,240 km) that has an orbital period of 24 hours, i.e. it orbits at the same angular velocity as the Earth itself. These satellites are extremely important in particular for telecommunication as they are always in the same position in the sky as seen from the surface of the Earth.

**Elliptic orbits**

Johannes Kepler (1571 – 1630) was the first to realise that planetary orbits in general are ellipses. The circle we suggested above is just a special case of an ellipse with zero eccentricity. Kepler formulated his famous three laws (in about 1618) that describe orbital motion.

It was one of Newton’s great accomplishments that he could mathematically prove Kepler’s laws, what Kepler had never been able to do, because Kepler did not have access to the required mathematics, in particular Calculus.

**gravitational attraction** between the object and the central mass.

**Another animation**

Now get a feeling for the elliptic orbit and Kepler’s Laws by spending some time with this great animation.

(Choose “Newtonian features” to switch on the v and a vectors). Show ellipse features, e.g. centre, focal points, semi-major and –minor axis, eccentricity, etc. Also note that the ellipse becomes a circle at zero eccentricity.

**EARTH ORBIT**

Getting a payload into Earth orbit requires a frustratingly large amount of energy. This has nothing to do with orbital motion, but is because we also need to launch a large amount of rocket fuel and a whole structure of a rocket to make it happen. Once we are in orbit and the spacecraft has the necessary velocity (speed and direction) it follows a freefall orbit which generally is elliptic as we saw. You do not need any further energy to maintain that orbit. The launch of a spacecraft is thus the propelled trajectory designed to bring the spacecraft into the required initial conditions of free fall. And the velocity (direction and speed) once in orbit determines where the spacecraft will go.

Let us now go back to Newton’s animation of the canon at the mountain top. You may already have found that at the minimum speed of 6.84 km/s the cannon ball just falls around the entire Earth (it is actually closer to 7.9 km/s as we stated above). If you now stepwise increase the initial speed you will see that the orbit becomes a little more elliptical as it extends further at the opposite side of the Earth. However the cannon ball keeps coming back to the canon where it took off.

There is however a speed with which the cannon ball does not come back but disappears into space. The minimum speed with which this happens is called the Escape Velocity. The shape of that orbit is no longer an ellipse but a related shape called parabola. Beyond escape velocity the orbit becomes a so-called hyperbola.

The diagram below shows the conditions for the various types of orbit. For Earth the escape velocity is just above 11 km/s.

To get a feeling for the possible variations in orbit have a play with this animation that in particular shows how the initial velocity vector defines the orbit (you must drag your mouse to choose the intial vector). Once the object is launched, it follows a freefall orbit that can either be an ellipse, a parabola (special case) or a hyperbola.

Before we go into deep space let us first look at spacecraft in Earth orbit.

**Modifying Earth orbit**

For reasons of fuel efficiency actual space missions very often start with bringing the spacecraft a circular low Earth orbit (LEO) the blue circle in the diagram. Then we can increase the altitude in two steps with the so-called Hohmann Transfer.

At one point P we burn a booster rocket for a short time. The orbit then becomes an ellipse where the point we started from is the perigee, the closest point to Earth (and the radius of the original circular orbit is perigee distance). At the furthest point A (apogee) in the new orbit, the yellow ellipse, we burn the booster again for a short time and if we do this precise the orbit becomes another circle (the red orbit) but now with radius equal to the distance to the apogee.

**Note on apsides.** *In an ellipse the closest point to the central body’s focal point is called the periapsis and the furthest point the apoapsis. For Earth orbit these are called perigee and apogee and for a planetary orbit around the Sun perihelion and aphelion. More here.*

The ∆V’s in the figure are the increments in the velocity that we apply by the boosters. Very little energy is needed for these manoeuvres and they are far more efficient than bringing the spacecraft directly into the higher circular orbit. At typical satellite altitude a 1m per second speed increase will raise the orbit by about 3 km. During these two short burns the spacecraft is being accelerated but otherwise it continues in a free fall around the Earth. The same technique but in opposite direction is used the bring a spacecraft back to Earth. The burns are then called retroburns.

**Note:** *If we burn the boosters in a direction perpendicular to the velocity of the spacecraft we do not change the orbit itself, but we change the tilt of the orbit in space (its inclination). In contrast to the previous manoeuvers, a change in inclination requires a lot of energy.*

**Orbit maintenance**

Over longer periods an Earth satellite will experience some orbital decay (loss of orbital energy) due to surface forces such as solar radiation pressure, atmospheric drag, etc. This results in a lowering of orbital altitude (and actually increase in orbital speed; see the formula for V_{C} above). In the case of the ISS the ground crew initiates regular short booster buns to bring the orbit back to its nominal altitude. This video is an excellent illustration of such a manoeuver and in particular shows how the spacecraft temporarily goes from free fall to an accelerated motion, although everything that is loose inside wants to continue its free fall.

**Rendezvous**

While we can now launch our spacecraft into pre-designed orbits, the second most important technique in space travel is the rendezvous. This is French for “getting together” or “meeting up”. In space travel it is the ability to let a spacecraft catch up and dock with another spacecraft that is already in orbit. Even in the early days of space travel, e.g. for the Apollo programme in the 1960’s and 1970’s, rendezvous manoeuvres have been absolutely essential both above Earth and in Lunar orbit.

First of all the orbits of the two spacecraft must have the same inclination (tilt with respect to the Earth’s equatorial plane) which sets critical conditions for the launch time and location of the visiting spacecraft. You basically get only one chance to do it right and if you would miss the target, there is practically no way to try again from orbit, because of the impossible fuel requirements for such major orbit corrections.

Secondly for rendezvous with an object in Earth orbit, e.g. the ISS, the visitor is first brought into an intermediate orbit a few km below the target. Of course the timing needs to be right so that not only the orbits are close but that both objects themselves are also at the right position in their orbit.

The next stage of the rendezvous involve small ∆V’s to gradually bring the visitor higher up to the target. This generally requires several orbits to gradually reach the same orbit. Finally the two spacecraft, once in the same orbit, need to be brought together. This is quite counter-intuitive because when the visitor trails the target it will need to slow down to get closer after one orbit and when it leads the target it needs to speed up to meet after one orbit. Such “proximity operations” are calculated using appropriate software to determine the correct burns for a successful rendezvous.

Most Sci-Fi movies and TV series on space suggest that these manoeuvres can be carried out manually by pilots who have clear view of the target, but reality is a lot more complex.

For more on the space rendezvous technique go here or see astronaut Buzz Aldrin’s work.

**LEAVING EARTH**

Once our spacecraft has exceeded escape velocity it is leaving Earth in a hyperbolic orbit as we saw above. But that is with respect to Earth’s gravity. Now we must look at the bigger picture and see that the gravity field of the Sun becomes important because the spacecraft is actually now in an elliptic orbit around the Sun just like Earth itself. Actually for precise calculations we must also include the gravity fields of most of the planets, in particular the largest planet Jupiter. But we will ignore these effects here for simplicity and continue to work with Kepler orbits about one central body.

The simplest way to get e.g. to Mars is similar to the Hohmann transfer technique we discussed above. The diagram shows the actual trajectory of the Curiosity mission (Mars Science Laboratory, MSL) that was launched in October 2011. You can see that MSL was launched into an elliptic orbit around the Sun, that has its aphelion just touching the orbit of Mars. Once the MSL got to that position Mars obviously needed to be there too. This sets critical conditions on the time of the launch from Earth and requires a just favourable common configuration of Earth and Mars. Such configuration happens only once in about every 2.1 years.

Once MSL reaches Mars, its aphelion velocity is slower than Mars in its orbit so the spacecraft must speed up. Then in order to go into orbit around Mars it needs to slow down to be caught in Mars’ gravity field.

It must be noted that the gravity field of the Sun is dominant during most of the trajectory, but as the spacecraft approaches Mars, the gravity field of that planet has increasing effect on the spacecraft’s orbit. Hence this seemingly simple trajectory is already highly complex and requires careful planning and calculations as well as in-flight orbital corrections at critical points. An overriding issue always is the fuel economy of inter-planetary travel as it is prohibitively expensive, and from some point impossible, to take large amounts of propellant into space.

**Gravity Assist**

Many missions that have been carried out would never have been possible without an important technique for saving propellant. This is the technique of gravity assist that could be referred to as “stealing a bit of orbital energy from a natural Solar system object”.

Compare this in its simplest form with an elastic ball that bounces off a wall. If the collision is elastic the ball will bounce off with the same speed but opposite direction as when it came in. Now assume that the wall itself is moving towards the incoming ball. The ball now bounces off the wall with the sum of the incoming speed plus twice the speed of the wall. If the wall is very massive in comparison to the mass of the ball, there will be no noticeable change in the speed of the wall. When we do this with spacecraft passing by (of course not bouncing off) say a planet, the spacecraft can gain a lot of speed that in effect is “stolen” from the planet. The image shows the principle of a spacecraft performing a gravity assist at Jupiter.

This technique is also used to slow a spacecraft down, which is important when travelling to the inner part of the Solar system or when going into orbit around a planet or moon. Read more about this important technique

**ACCELERATED SPACEFLIGHT**

Above we discussed spacecraft that are essentially in a free fall motion in gravity. The only orbital corrections are performed by short bursts of on-board booster rockets or through flyby’s past planets for gravity assist. Electric propulsion systems for spacecraft have been imagined for as long as rockets in general have been. But lifting a payload off the Earth into space can still only be done with chemical propulsion systems that work with the burning (oxidation) of solid or liquid fuel. We cannot give a full account in this article of non-chemical rocket propulsion systems, (find a good overview here) but want to emphasise that there are good options to use electrical systems while the spacecraft is in space.

The chief advantage of such systems is that they can provide a thrust over long periods of time and with very efficient use of propellant, although the thrust itself is far less than that of chemical systems. This means that such propulsion systems can provide extended periods of acceleration during the space flight, making it more complicated than non-propelled spaceflight, but also making it a lot more economical.

DAWN mission was launched in September 2007 and visited the asteroid 4Vesta and subsequently went into orbit around the dwarf planet 1Ceres. This mission would have been impossible without the ion thrusters that DAWN had on board. This propulsion system also saved the mission a lot of propellant (fuel and oxidiser) that conventionally would have been used, as it only used a relatively small amount of Xenon gas as propellant.

The

**No propellant at all?**

Arguably the most interesting recent invention that has first been made by British engineer Roger Shawyer is the “impossible” **EM Drive**. Many scientists have claimed that this idea would violate conservation of momentum, but after years of study and experimentation by several research groups, the idea persists and claims are made that the Cannae drive, which is different but similar to Shawyer’s design, will soon be tested in orbit on a Cubesat. These drives do not require any propellant but only electric energy and in principle employ radiation pressure from microwave radiation leaving an antenna.

**TO THE STARS**

The New Horizons spacecraft that flew by the Pluto and Charon dwarf planet system is one of the fastest spacecraft ever launched. It continues to fly at a speed of about 13 km/s and will eventually leave the Solar system, just like the Voyager spacecraft are doing. If New Horizons would fly towards the closest star system Alpha Centauri (which it isn’t) it would take about 80,000 years to get there.

So if we ever want to stand a chance to travel to the nearest stars in a reasonable time (as compared to a human lifetime) we will need on-board propulsion systems with very long life time.

A deep-space probe with a mass of 10,000 kg based on the Cannae drive discussed above, claims to reach a distance from Earth of 0.1 lightyears within 15 years and 0.5 lightyears within 33 years (ref http://cannae.com/deep-space-probes/).

Alternatively cosmologist Stephen Hawking and others have suggested a fleet of mini spacecraft that could make the journey to Alpha Centauri in 20 years, using light sails that are laser powered from Earth orbit (project Stars shot).

**EPILOGUE**

It will be very interesting to see how space travel will evolve throughout this century but do not forget that gravity dictates the motion of all objects in the Universe and the gravitational freefall orbit will always be the primary principle of getting from A to B in space. Space based propulsion systems can provide an important add-on to make space travel more economical and/or time saving.