Falling in Space
Erik Vermaat
A comprehensive primer on the technique of space travel, i.e. how we travel from A to B in space.
1. 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?”.
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 of
Newton’s first law of motion .
Weightless in space. Image credit: NASA
So 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).
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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.
2. 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 𝑉𝐶2 = 𝐺
𝑀𝐸
𝑅
where G is the Universal Gravitational Constant, ME 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.
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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.
When an object is moving in an elliptical orbit it has a velocity that is always directed along the
tangent to the orbit at every point. It also has an
acceleration in the direction of the focal point where the
central mass is. This acceleration is due to the
gravitational attraction between the object and the
central mass.
Image credit: Wikipedia
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.
3. 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.
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To learn more about conic sections, circles and ellipses, etc. watch these Kahn Academy video
lectures:
Introduction conic sections
Introduction ellipses
Introduction parabolas
Introduction hyperbolas
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 initial 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 the
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.
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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 VC 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.
Image from http://www.baen.com/rendezvous
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.
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Undocking and descent
Undocking from the ISS and controlled descent back to Earth is basically the same procedure in
reverse. Nothing is done “manually” but this procedure is carefully prepared and carried out with
support from both space and ground crews. A good explanation of this can be seen in the third of the
ESA movies linked below.
As a conclusion of this section watch these three excellent ESA videos explaining 1. Soyuz launch, 2.
rendezvous and docking and 3. Undocking, reentry and landing.
https://www.youtube.com/watch?v=AVvgpKt5uCA&list=PLbyvawxScNbsoD_tGlw8kWCw3S5htiVKZ&
index=1
https://www.youtube.com/watch?v=M2_NeFbFcSw&index=2&list=PLbyvawxScNbsoD_tGlw8kWCw3
S5htiVKZ
https://www.youtube.com/watch?v=l7MM9yoxII&index=3&list=PLbyvawxScNbsoD_tGlw8kWCw3S5htiVKZ
4. 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.
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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.
Tip: If you really want to “get your hands dirty” try this practical lesson in calculating launch windows
for Mars.
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.
Image from https://solarsystem.nasa.gov/basics/grav/run.html
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 here and here.
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5. EXAMPLES OF ACTUAL DEEP SPACE TRAJECTORIES
To illustrate the reality of inter-planetary space travel based on the principles we discussed above,
we show here a few of the actual missions that have been carried out successfully.
CASSINI to Saturn
Links
https://saturn.jpl.nasa.gov/resources/1776/
https://saturn.jpl.nasa.gov/mission/gravity-assists/
https://en.wikipedia.org/wiki/Gravity_assist#The_Cassini_probe_.E2.80.93_multiple_gravity_assists
This trajectory shows multiple flyby’s requiring a favourable configuration of planets, in particular for
the last gravity assist past Jupiter. Launch windows are usually quite specific when gravity assist
manoeuvers are needed. It took Cassini 6.7 years to reach its destination Saturn. Since its arrival in
2004 most of Cassini’s manoeuvring has been done with further gravity assist at Saturn’s moon Titan.
The graph depicts the speed increases (∆V’s) during the trajectory towards and while at Saturn. After
mid 2004 (arrival at Saturn) the frequent ∆V’s are due to flyby’s at Titan. This programme would not
have been possible with Cassini’s own propellant. The highly successful mission will continue until
the end of 2017 when Cassini will be steered into Saturn.
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JUNO to Jupiter
Links
http://spaceflight101.com/juno/juno-mission-trajectory-design/
https://www.missionjuno.swri.edu/orbit/
The Juno trajectory took it initially beyond Mars’ orbit and then after a deep space manoeuvre back
for an Earth flyby. Then it followed the outer cruise until Jupiter orbit insertion five years after
launch.
After 20 months of science at Jupiter the mission is scheduled to end in February 2018 when Juno
will be steered into the gas giant.
The insertion
manoeuvre occurred at
the spacecraft’s closest
approach to Jupiter, and
slowed it enough to be
captured by Jupiter’s
gravity into a 53.5-day
orbit. In this way the
spacecraft saved fuel
as compared to going
directly into the 14-day
orbit required for the
science mission.
Images credit: NASA
Juno is in a highly eccentric, polar orbit over Jupiter and passes very close to the planet at its closest
approach (jovelion). Juno needs to get extremely close to Jupiter to make very precise measurements
required by the mission. This orbital path carries the spacecraft repeatedly through hazardous
radiation belts, which are similar but much stronger than the Earth’s Van Allen belts.
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NEW HORIZONS to the Kuiper Belt
Links
http://pluto.jhuapl.edu/Mission/Where-is-New-Horizons/index.php
http://pluto.jhuapl.edu/Mission/The-Path-to-Pluto/Mission-Timeline.php
Although there were backup launch opportunities in February 2006 and February 2007, only the first
twenty-three days of the January 2006 window permitted a Jupiter flyby. Any launch outside that
period would have forced the spacecraft to fly a slower trajectory directly to Pluto, delaying its
encounter by 2–4 years. Fortunately, after some delays, the spacecraft could successfully launch on
19 January 2006 in a direct path towards the outer solar system with a heliocentric speed of 16.3
km/s, which is above solar escape velocity. It used the Atlas V551 rocket with five solid fuel rocket
boosters and a third stage added to reach the necessary speed. This third stage is now also in a solar
system escape trajectory and must have crossed Pluto’s orbit approximately in October 2015.
The only gravity assist of New Horizons was at Jupiter in February 2007 which increased its speed to
23 km/s. It flew past the Pluto-Charon system in July 2015. Data transmission of the Pluto encounter
observations was finally completed in October 2016.
New Horizons’ position on 3 November 2016. Credit: http://pluto.jhuapl.edu
New Horizons is healthy and now on its way further through the Kuiper Belt. Its mission has been
extended to 2021 with a close flyby of Kuiper Belt Object 2014 MU69 and to perform more distant
observations of another couple of dozen objects.
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MESSENGER to Mercury
Links
https://www.youtube.com/watch?v=Ownzbb1mKxs
http://ccar.colorado.edu/asen5050/projects/projects_2004/park/
MESSENGER was the first spacecraft to go into orbit around Mercury in 2011. This is not because
there was little interest in this planet, but primarily because it is so difficult to do this. A direct Kepler
orbit to Mercury would be a heliocentric ellipse with perihelion in Mercury’s orbit. But the
spacecraft’s speed at that point would be very much faster than Mercury itself and directly braking
into an orbit insertion is impossible.
The actual 6.6 year trajectory has been one of the most complex ever flown, with six gravity assist
flyby’s at Earth (1), Venus (2) and Mercury itself (3).
Image credit: NASA/JPL
Notice the colour code in the diagram, indication the various phases. After each flyby the spacecraft
gets into a narrower orbit around the Sun. In the final phase MESSENGER is almost in the same orbit
as Mercury, and thus will have a small relative velocity with that planet. See also this animation.
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GRAIL to the Moon - We can also make it really difficult to save energy
Links
http://moon.mit.edu/design.html
https://www.youtube.com/watch?v=Cf7jaWWwu3o
The Apollo spacecraft took only three days to reach the Moon in a direct approach. The twin GRAIL
mission launched in 2011 took a very indirect approach taking 3.5 months to save energy.
Image credit: NASA
Grail trajectory to the Moon passed by the L1 Lagrangian point between the Sun and Earth at low
speed, before swinging back towards the Moon. The twin spacecraft arrived at low relative speed
with respect to the Moon to also save energy with the slowing down.
Orbit insertion was performed separately for each of the two spacecraft. They needed to be in
specific relative orbits around the Moon for their mission. The precise gravity mapping involved
microwave ranging systems between the two spacecraft who therefore had to be in mutual visibility
during the entire science phase at a very low 50 km orbit.
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6. 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.
The 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.
Image credit http://dawn.jpl.nasa.gov
The graph shows DAWN’s trajectory from launch,
which included a gravity assist past Mars. It also
shows the extended periods that the ion-thrusters
were operating (thrusting). Watch an interview
with Marc Rayman, Dawn Chief Engineer at JPL
discussing the advantage of the Ion Thruster for
the Dawn mission here.
A “traditional” trip to Mars such as the MSL
mission discussed above takes about 9 months.
Then you will have to stay on Mars for another
three months before Earth is at the correct
position relative to Mars to allow you to go home. An on-board propulsion system could shorten the
flight time to Mars significantly (although it forces the system to stay longer on Mars for return
missions). This would also have the important advantage that astronauts will not be exposed to
harmful radiation and micro-meteorites while in space for too long. On-board propulsion systems are
therefore much at the forefront of current research in space technology.
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.
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7. 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).
8. 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.
"The future is not there waiting for us. We create it by the power of imagination”
Vilayat Inayat Khan
Note:
If any of the links in this resource are not working anymore please notify [email protected].
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