25
(we’ll have to verify later how high), we can’t rely on anything fragile, but rather we
should launch something very compact. Rocks are then the first thing we should think of.
There could be also the possibility of sending powdery material like regolith, if we could
smelt a part of it in order to create a hardened crusty shell able to contain the rest; however
the energy required would be too high and the machines capable of that too complex to
operate safely without human assistance. Rocks, then: especially volcanic glass, which is
easier to reduce in order to get oxygen, as suggested by personnel from the Colorado
School of Mines (picture above).
If we think of the payload as a rock, we necessarily have to introduce in the mission’s
concept a lunar mobile robot, like a rover, able to select adequate rocks from the surface
and deliver them to the launcher. About the payload’s mass: this parameter is conditioned
both by economical and by technical issues. On one side it would be better to shoot as
much mass as possible at once, rather than having tiny chunks to be launched and caught
more frequently. On the other side, centrifugal forces would preferably have smaller
payloads for not burdening too much the structures involved. Burdens which depend from
other parameters, especially the wire’s length. A careful analysis is needed at this regard
for reaching an optimum, and this task is beyond my work. However, on a mere intuitive
basis, we can set the payload’s mass on the 20 kg.
Let’s talk about destination of the launch. For what we have said in the first part, the best
thing would be to have directly lunar material delivered to on-orbit processing facilities;
for example, directly to the ISS. With the launcher’s characteristics mentioned above,
though, it is clear that this is impossible: we would have a massive payload without any
kind of control heading directly towards LEO, which contains thick constellations of
super-expensive satellites and an international space station, and its relative speed would
be in the order of the km/s. A minor error in the shooting parameters, or in the lunar
gravity field model, and there would be a catastrophe.
Instead, we could aim at something more distant from Earth, such as the geostationary
orbit. If we are not aiming directly to a manufacturing facility (and we are not) we
necessarily have to introduce a third element in the mission scenario: a retrieval satellite, or
a “catcher” satellite, if we want to use baseball terminology. Its first purpose is to
successfully intercept the payload in a safe position: that is, in a position in which, if the
catching operation fails, there will be no further harm. Second, it shall bring the payload
safely to the manufacturing facility. In this context we see that this robot -satellite balances
the lack of direct control on the payload.
26
Fig. F1.2: position of the five libration points in the Moon-Earth system
There is another interesting target for the launch, and that is the libration point placed
between the Earth and the Moon. From space mechanics theory, we know that, between
any two sources
of gravitational
field mutually
influenced,
there are five
equilibrium
points, called
libration or
Lagrange points
[“Basic Physics
of the Solar
System”, M.
Blanco, W.
McCuskey];
these points
exist also in the
Moon-Earth
system (see figure). Their physical interpretation is better understandable if we remain in a
rotating frame centered on Earth and rotating with the same angular speed of the Moon,
and is explainable with a balance of centrifugal and gravitational forces. L1, L2 and L3 are
points slightly unstable: if a mass is placed there, with a small perturbation the body will
slowly drift away from them. L4 and L5, placed 60° forward and backwards the Moon-
Earth axis, are on the contrary stable points: their explanation is less intuitive than for L1,
L2 and L3 as involves Coriolis forces. The L1 point is of particular interest for our
purposes, because of its position. It is a point of equilibrium because, for a mass placed
there, it is true that:
Earth’s gravitational force = Moon’s gravitational force + centrifugal force.
It is a point of unstable equilibrium because if the mass falls from there towards the Moon
or towards the Earth, it will be more and more accelerated away. The picture on the side
shows the effective potential of both centrifugal and gravitational forces of the Earth
27
(right) and of the Moon (left). The L1 point is placed where the colors melt. The cross right
next to the Moon represents
instead the equilibrium point
between the gravitational forces
only. On the x axis is shown the
distance between the Earth and the
Moon, normalized to 1.
The L1 point falls about at 84% of
the Earth-Moon distance. On one
side it allows less shooting
precision from David, being it
closer to the Moon. On the other
the catcher satellite will have to
cover about seven times the
distance from LEO than if it was
placed on a GEO. The question
now is: would it still be convenient to launch towards the L1 point? The answer comes
looking at the diagram below, which shows clearly the energy required (potential+kinetic)
Fig. F1.3: effective potential of lunar and terrestrial
gravitation plus centrifugal forces
Fig. F1.4: energy required for accessing orbits in the Earth-Moon system
28
for entering different orbits in the Moon-Earth system:
We have to change our way of looking at distances, in the space environment: presenting
no friction, theoretically an infinite distance can be covered with just a small initial
impulse. For the same reason, there is no “free downhill”: for example, going from LEO to
GEO theoretically requires exactly the same amount of energy than from GEO back to
LEO, because there is no dissipation mechanism (such as friction) to eliminate redundancy
of energy: engines must be used in both cases. Having said that, it is now clear that it is not
the distance between two points that matters in space, from an economical point of view,
rather the difference of potential energy separating them.
Now, looking at the energy diagram above, if we imagine our manufacturing facilities
parked in GEO it is evident the tremendous advantage we gain launching from the Moon
rather than from the Earth; and we must remember that the movement of payload to GEO
for commercial communications satellites is the largest current space launch business. If
the SMF are instead in a LEO we can see that we don’t gain much, in terms of energy.
Unless we introduce a dissipating mechanism again, such as aerobraking: if the catcher
satellite is properly structured, it can execute one or more fly-by in the upper layers of our
planet’s atmosphere, losing the excess of kinetic energy in the form of heat. At this regard
we have to remember the abundance of silicon in the Moon’s soil, which is excellent for
producing heat shields.
For the same reason we don’t gain much, energetically speaking, positioning the catcher
satellite on the GEO rather than on the L1; actually, aiming at the GEO, the risk of
accidentally hitting something valuable is higher. It is worth, however, to consider both
possibilities.
2- Preliminary verifications
The next step in the conception of the launcher is to verify that it can actually work. Of
course this is the hardest part of the project, as it involves so many different aspects:
tether’s dynamic, structural resistance, space dynamics of the launch, concept of the lunar
rover and of the catcher satellite, economics...; but the only way of doing it is to carefully
29
examine each one of them. For example, my supervisor M.me B. Escudier, at the Sup’Aero
of Toulouse, suggested that one of the main problems could be sensibility of the launch
from the initial conditions, especially speed and launch direction. This aspect regards space
dynamics of the launch, and it is both of vital importance for correct functioning of the
launcher and it is not so much dependent from other factors: as a matter of fact, if the
launcher requires a too high precision in the launch parameters (i.e. current technology
does not allow it) all the rest is not important because this idea of launcher is just not
feasible. Thus I decided that this could be the aspect on which my final project could focus.
But before that there are surely some other factors that can easily be checked and which are
as well necessary (but not sufficient) conditions for making David work.
First of all, for instance, let’s see how big are the centrifugal accelerations involved.
Having to deal with a wire, length in first approximation does not represent a severe
problem, as it can be coiled. We’ll have to verify its mass later, of course. Let us think that
the tether has a length of 2000 meters, for example. Acceleration is now:
2000
2400
22
r
v
a
c
2880 = about 290g,
having raised the speed to 2.4 km/s. It is still a high acceleration, but nonetheless surely
withstandable by a rock.
We can say that this first verification was successful, and easy to be tested. Now we would
like to see how massive would be the wire: if it turns out to be too heavy, again this kind of
launcher is unfeasible. As we don’t know how the wire will deploy from the central
element during the phase of angular acceleration, we can make this test just in one
particular moment: when the tether is fully deployed and is ready for launch. In this case
we imagine the wire to be completely parallel to centrifugal direction, rotating with the
same angular speed. The angular speed depends from the wire’s length and so do the forces
at stake, and also the wire’s mass. Centrifugal forces vary linearly from the center (where
they are zero) to the extremity, where they reach the maximum. To compute the section of
the wire we must consider that each part must be able to sustain the rest of the wire: thus
we’ll have a bigger root and a thinner extremity. It is clear that if we see that the root is so
big that we can no longer talk of a wire, the advantage of coiling it is no longer present and
the idea loses its attractiveness. Since the section of the wire depends from the mass of
itself, which depends from the section again, the best way of solving this problem is to start
30
calculating the section from the extremity, where acceleration is known (given the wire’s
length) and so is the mass (only the mass of the payload + the detachment system); from
here we go towards the root. Of course we can’t do the math by hand, we need a
computing help. I chose to use the software Matlab and in particular its tool Simulink,
which provides a powerful and intuitive graphics. It is able of integrating equations and is
turns out to be very useful for our purposes if we integrate in space instead of time: at each
step the mass of the wire is known, so are the forces and thus the new section of the wire.
Of course we must know the tether’s mechanical properties; my expertise doesn’t cover
specifically materials, but for the task it has to fulfill it seemed to me that fibers are suited
most. With a quick search on the internet I arrived to choose Kevlar 49, although I am sure
there could be more specific ones. Below is a table with the main characteristics of Kevlar
49:
Nominal Properties Chart
Table 1
Kevlar
29
Kevlar
49
Dacron
T68
Steel
Wire
Glass
S
Graphite
HT
Density g/cc 1.44 1.45 1.38 7.74 2.50 1.50
lb/in
3
0.052 0.052 0.050 0.280 0.09 0.054
Denier/Filaments 1500/1000 380/258 1300/1000 Fil. Fil. Fil.
Tensile Strength
psi x 10
3
MN/m
2
GPD (tenacity)
500 (400)
3,450
20-22
525 (400)
3,620
22.4
105 (80)
550
4.5 (8)
600 (500)
4,140
3.9
650 (500)
4,480
12
500 (350)
3,440
16
Tensile Modulus
psi x 10
6
MN/M
2
x 10
3
GPD (stiffness
9.1
63
480
19
131
1004
1.5
10
21
30
208
—
12.6
87
—
35
240
—
Elongation % 3.6 2.75 (2.4) 15 1.1 (10) 3 1.9
Dielectric Constant 3.4 3.4 4 — 4.5 5
Loss Tangent 0.005 0.005 .01 — .014 2.5
Specific T.S. (in) 10 (8) 10 (8) 2 2.1 7.2 9.3
Melt Point ºF 800 ºF
450 ºC
800 ºF
450 ºC
482 ºF
250 ºC
2550 ºF
1400 ºC
1540 ºF
840 ºC
6600 ºF
3650 ºC
Specific Modulus (in) 1.75 3.6 .3 1.1 1.4 6.5
(Note): This table oversimplifies the properties with the use of single number filament properties. All have ranges of strengths, densities
and statistical distributions of all properties. They are commercially available materials but yarn and composite properties will tend to be
lower (in parenthesis). © 2001 GME, Inc.
Fig. F2.1: Mechanical properties of different materials
31
Of particular interest is the tensile strength. In my program I used a value of 3.2 Gpa as
safe factor. In the next page I show the Simulink model I used for simulation. It is very
intuitive, it is not difficult to recognize operations executed by each block. For example it
can be seen the formula for obtaining the mass:
Where A is the cross section and Υ is the material’s density. A can be obtained as:
s
F
A ,
where F is the traction force and s the tensile strength.
Thanks to this scheme we can compute tether’s mass and cross section depending from the
length. We have also to consider that a longer wire will allow a smaller angular velocity
for launch, and this will positively influence precision. For example, for a wire of length
2000 meters and launch speed of 2.35 km/s we get the graph at page 33.
In first approximation, I have neglected lunar gravity and considered only centrifugal
forces; this approximation is not so bad after all if we consider the enormous difference
between the two values of acceleration: they are comparable only near the root. As regards
the mass, the total wire’s mass is to be read at the end of the graph (since the computer
operates a sum): in this case, about 120 kg. The cross section, instead, is representative for
the whole wire and it is given in square centimeters: as expected it is thinner at the
extremity (less than 0.2 cm ²) and thicker at the root, almost 0.6 cm ² (remember that in the
program the zero is the extremity; on the x axis is represented the length in meters). These
data are very reassuring, as the tether is really a wire and the weight is well within the
range of a space mission. Of course we still have to consider the weight of the central
element and of the wire for the counterweight, but our purpose was to see if the order of
magnitude is acceptable and it clearly is.
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