The 22nd of June 2007
A project by Torsten Nielsen (20042596)
for the exam in
Space mission analysis and Design
(Rumforskning og Rumfart)
Professor: Hans Kjeldsen
Introduction
In this
project we have designed a mission to bring back helium-3 from the Moon. The
mission has been named: Drheeme (pronounced like dreame) and is an acronym for
‘Danish Return of HElium Extracted on the Moon to Earth’. The main focus has been
on the astrodynamics involved, and the application of the rocket equation to
simulate realistic changes in velocity and consumption of fuel. In the
calculation of the trajectory of our spacecraft – named Torma – we have included
the gravitational forces from the Earth and the Moon. The forces from the Sun has
been excluded because only the differential force / the tidal force of the Sun
would have any influence on the trajectory, and since it is inversely
proportional to the distance cubed, the effect is only about 20% of the effect
felt from the Moon. The force from the Sun is not negligible as it holds the
complete Earth-Moon system in its rotation, but it cannot perturbate the
trajectories, and as we work with a coordinate system that is fixed to the barycenter
of the Earth-Moon system we can disregard it.
Other
perturbations that are not included are the forces exerted by the Suns
radiation pressure and the non-spherical geometry of the Earth. We have however
included the elliptical trajectory of the Moon and the motion of the Earth
around the barycenter.
Apart from
the simplification of the physics involved we have also had to assume that a
whole industry for mining the He-3 on the Moon was in place.
We have
developed the MatLab program in the paradigm of Object-oriented programming
(OOP)[1].
That’s why we use concepts like ‘classes’, ‘methods’, and ‘objects’ to describe
the code later on.
The Part ‘The
road to Drheeme’ can be skipped when reading this report since it is only
included to build the framework around our mission. It is complete fictional
but can work as a motivation for the mission.
Figure 1:
Just a nice shoot of the mission. The red circle is the equatorial plane. The
trajectory of the Moon is seen as the yellow line and the white line is the
trajectory of Torma.
The road to Drheeme
Probably no one had
expected the visionary outcome of the 2007 G8 meeting in
This decision would turn
out to provide the biggest boost in the space industry since the space race in
the 60s. The reason for this was that scientists soon discovered that the
fastest way of making fusion power available and commercially profitable was to
use helium-3 in the reactors. This isotope of helium has a number of advantages
over deuterium and tritium or other candidates normally considered for fusion.
Here we shall list three. First of all the process doesn’t create fast neutrons
that will damage the reactor over time. Secondly it is very efficient because
the result of the process is a fast moving proton that can be converted directly
into electricity, meaning that you don’t have go to through the process of
converting the fast moving products into heat and then into electricity. The
third advantage, the one that made it easy to sell to the public, is that the
pollution from this reaction is minimal.
Unfortunately one of the
major disadvantages is that He-3 is a very rare isotope here on Earth, but on
the Moon however it’s more common as it comes from the Sun and is trapped in
the lunar dust - the regolith. Consequently a project was started to build a
lunar mining facility to extract the valuable He-3 and the other available raw
materials[2].
The following is a timeline of the
events that took place before the birth of Drheeme.
2007:
The G8 meeting in
2015:
Alsvid and Arvak[3]
were launched. They were the two moon missions that brought the materials to
build the first solar furnace on the Moon.
2015:
The mission: Mundilfari [4].
This mission took the robot of the same name to the Moon. It was this robot
that constructed the first solar furnace and subsequently worked as a digger
robot to bring regolith (moon dust) to the furnace.
In the
next years more solar furnaces with higher capacities were built from the already
extracted material and the extraction accelerated.
2020:
Construction of LUFF and LUPP: From the extracted materials the construction of
the LUnar Fusion Facility and the LUnar Processing Plant started. This new
fusion powered extraction plant was needed to get the He-3 extraction up to
speed.
2030:
Production starts in the new facilities: LUFF and LUPP
2035:
The launch of Drheeme (pronounced dreame)
Overview of the mission
Figure 2:
An overview of the whole mission. The trajectories of the Moon and of Torma are
seen, the yellow and white lines respectively.
Below is a
description of what is going on during the mission and the approximate mission
timestamps for the events. Many of the consideration we have been through and
the choices we have made are also described below in the context where they fit
in. A log file from one of our simulation runs is enclosed as a textfile with the report. To better visualize the mission and the different
stages in it, take a look at the
movie made from the simulation, while reading
this section.
00:00:00:00:
The mission starts with Torma circling the Earth in a LEO of 186 km. It has
been placed in this parking orbit by a launch vehicle that we haven’t designed
but just assume exists. Torma weighs at mission start with fuel and all is 40,200
kg, so it would have to be a quite powerful launch vehicle to put it into LEO. The
launch will have taken place from a platform in the sea, a sea launch, to give
Torma’s orbit the same inclination as the Moons orbit.
00:02:00:00:
Two hours in the mission the program starts looking for the possibility of a
lunar rendezvous. It is critical to the success of the mission that Torma’s
main engine is fired at the correct time to lift the orbit and start the
journey to the Moon. The correct time is when the Earth-Moon-Torma system has
the desired geometry, in other words when the phase angle between Torma’s orbit
and the orbit of the Moon is correct. For a rendezvous with another spacecraft
you would lift the orbit to this other spacecrafts orbit but for making a
rendezvous with the Moon we found that it is wiser to choose an orbit lower
than that of the Moon and behind the Moon. This is because of the huge
gravitational disturbance of the spacecrafts orbit once it gets close to the
Moon. We calculate the phase angle from the description and equations in
section ‘6.3.3 Orbit Rendezvous’[5]
and then we add our own phase in order to get into the new orbit behind the
Moon. A leading target rendezvous.
00:02:15:00:
Some time later the desired geometry is in place, and the TormaDrive[6]
is fired for approximately 50 s to apply the delta-v needed for the Hohmann
transfer from LEO to an orbit of 310,000 km (from the center of the Earth). We
find the delta-v from the formulas in section ‘6.3.1 Coplanar Orbit Transfers’[7].
02:20:30:00:
When we achieve the wanted distance to the Earth (i.e. radius in the new
orbit.), the TormaDrive is fired again to apply the delta-v needed to go from
the Hohmann transfer orbit to a circular orbit with the present radius. Torma
starts the orbit behind the Moon but because the orbit is lower, than that of
the Moon, Torma will eventually catch up. At the same time it is being
accelerated by the gravity of the Moon and that will lift its orbit, so with the
right phase angle behind the Moon and the right height of the circular orbit,
Torma will rendezvous with the Moon. By using the gravitational pull of the
Moon to lift our orbit we save fuel and this was an important reason for
choosing this trajectory.
09:06:00:00:
After about a week chasing the Moon we are finally there. The relatively long
transit time is not a problem in this mission because Torma is an unmanned
spacecraft. When we get within 162 km of the Moons surface we spend the last
fuel we have to go into a circular orbit. While in this orbit a transport ship
from the Moon mining site will rendezvous with Torma and start the transfer of
fuel and the valuable cargo of He-3. We expect it to take a little more than
two days to make the transfer. The time is not critical as circling the Moon is
a free ride.
11:10:00:00:
The transfer is complete. We took on board 5,000 kg of He-3 and 54,800 kg fuel.
Now we start looking for the correct geometry of the Torma-Earth-Moon system to
go into a Hohmann transfer that can bring us home. This is when the spacecraft
is on opposite side of the Moon from the Earth and the three are on a line.
This time we aim directly for the target instead of a lower orbit where we
could take advantage of the gravitational pull. The reason for this is mainly
that now we have fuel in abundance.
11:11:30:00:
When we find the geometry is right we fire the engine and start the journey
back to the Earth. Even though we only have to apply a delta-v of about
1 km/s it
still takes a lot of time and fuel because of the great mass of our fully
tanked spacecraft.
14:11:00:00:
About three days later we reach the minimum distance to Earth in our present
orbit and we fire the engine again to go into an orbit around the Earth, which
we circle until the time is right to start the descent.
When the
spacecraft is in the right position to start the descent we go into a transfer
orbit to take us to the surface. However when we reach the top of the
atmosphere (100 km) we spend the rest of the fuel to brake and get the velocity
below 8 km/s, which is the velocity that Torma have been designed to be able to
dissipate in the atmosphere. From this point on the descent is completely
unpowered[8],
except for the electrical circuits working inside the spacecraft of course.
Our mission
ends with this last burst from the TormaDrive and we haven’t looked much into
how the physics of flying, braking, and landing in the atmosphere works. We
have just noticed that this is the way that the Space Shuttle of today makes
its descent and so we consider it to be a realistic scenario.
The Trajectory Calculation
The central
part of the program is the orbit calculation algorithm that we worked with
during the course. It calculates the new position and velocity of an object from
the current values. Basically it finds the total force on the object from all
the other nearby objects, and then calculates the acceleration that this force
will lead to.
Using the
algorithm, the program first makes an initial guess on the new position based
on the current position and velocity. This guess of course would only be
correct if no forces were acting on the object. The guess is only used to find
the approximate position of the object during this time step that is currently
being calculated. This position is taken to be an average between the two, and
is used to calculate the forces acting of the object and thereby the
acceleration. When the acceleration is known we can find the new velocity and
finally use this to also calculate the new position[9].
In our
simulation we have defined a class MassiveObject and a class Spacecraft. It is
possible in the program to make as many instances of these two classes as
needed. We only work with two instances of MassiveObject, the Earth and the
Moon, and one of class Spacecraft, Torma. All massive objects feel the
gravitational attraction of all the other massive objects and move according to
that. The spacecrafts however doesn’t contribute to any gravitational forces,
we have ignored the gravitation from the spacecrafts since it is insignificant
compared to that from the massive objects, but it still moves according to the
gravitational potential of all the massive objects.
The
algorithm mentioned above is implemented into the doStep() method of the class
TheUniverse. This is a class that has a list of all the massive objects and a
list of all the spacecrafts in the program. The method name doStep() signifies
that it makes the time go one time step forward. When you call the method
doStep() it calculates the new positions and velocities of all the objects in
the two lists. It does so by going through all the objects one by one and
calculating the effect that all the massive objects have on it, as explained
above in the discussion about the algorithm.
The results
of the repeated calls of this method are the trajectories of Earth, Moon, and
spacecraft as seen in the animation we made of the mission. Actually in the
animation you don’t see the Earth moving. We decided to keep the Earth still
when plotting the results, mainly because it makes the whole picture more
confusing and also because the Earth moves very little compared to the Moon and
the spacecraft. This is clearly seen in the figures below.
|
|
|
Figure 3: To
the left the trajectories of Earth and Moon. To the right a zoom-in on the
Earth trajectory.
In the end
of the animation, when we plot the last part of the return to earth, we have
moved the Earth to its final position and keep there.
Figure 4:
The trajectory of the Earth is seen as the blue line. The green shows how Torma
starts circling the Earth, leaves it, and returns 15 days later to the new
position of the Earth.
The Propulsion system
The Torma
spacecraft is equipped with one TormaDrive as its main engine. This engine is
only used in space and is not fired during the initial launch from Earth. To
place Torma in the LEO, where our mission starts, other and more powerful solid
rocket boosters are used. A detailed description of these lift-off rockets are
beyond the scope of this project. Nevertheless we are certain that lifters with
the needed power are possible to make, since already today we have available
the Saturn V[10] launcher capable of
lifting 118 metric tons into LEO and the Ares V[11],
now under development, will be able to lift 130 metric tons.
Liquid
oxygen (LOX) and hydrogen (LH2) are the fuels used by the TormaDrive. The main
reason for this choice of fuel is that it gives us the possibility to refuel
the spacecraft while in the lunar orbit[12].
The
TormaDrive is a sophisticated power plant, which has the capability of
providing a thrust of 1.8 MN with an exhaust velocity of 4.4 km/s in vacuum.
This gives a specific impulse of approximately
, which is sufficient for making the velocity changes that we
need during the mission reasonably fast. It can be seen in the mission log how
much time each of the velocity changes took.
From the
thrust and the exhaust velocity we can calculate the fuel flow rate because
thrust = exhaust Velocity * fuel flow rate.
This is quite a bit more than your average
everyday car!
Our
simulation needs one more important property of the engine and that is the
weight, which is 3,200 kg. We need to have the weight of the spacecraft in
order to be able to calculate the effect our engine has when we try to change
velocity. The heavier the spacecraft the more engine power is needed to change
the velocity.
Before we
see how the engines parameters have been programmed in the simulation lets have
a look at how rocket propulsion works. Below is a figure, which tries to show
the principles.
Figure 5: It
is the conservation of momentum that makes spacecrafts go. When you push
something away in one direction you will receive a push in the opposite
direction.
Imagine that
you have a spacecraft with mass m0 and that your engine burns and
with high velocity exhausts propellant with mass mp. Afterwards your
spacecrafts total mass will be mf = m0 - mp,
and because of the conservation of momentum you will have received momentum
(i.e. velocity):
.
This is the
instantaneous change in velocity but since no engine has a thrust that is
infinite we always have to integrate this to find the velocity change over
time. The solution is the rocket equation
, where ve
is the exhaust velocity (velocity of propellant).
The
equation tells us the total velocity change for a journey involving any number
of maneuvers. It doesn’t matter how many times you changed the velocity or how
long time it took to change it. This gives the totals and for that reason it is
very convenient for estimating how much fuel is needed during a mission. If you
know the sum of all the velocity changes your spacecraft has to do, then this
equation can tell you how much fuel you need. From the rocket equation you get
the following
, and then the amount of fuel is: 
.
This was
used to calculate the needed fuel to go from the LEO to the moon orbit and to
go back home. Our initial simulation gave us the delta-v needed, we already
knew the exhaust velocity and the final mass is just that of the spacecraft
since we don’t bring any additional cargo to the moon. From the moon however we
will bring 5,000 kg of He-3.
Fuel needed
to travel to the moon:
And to go
from the moon with our cargo of 5,000 kg of He-3:
These
values have been input into the final program and every time we use the engine
we also use some of this fuel. The mass of the fuel also makes the velocity
changes take some time. For example in the beginning of the mission when we go
from the LEO into a Hohmann transfer to lift the orbit. We need to apply a
delta-v of 3.1 km/s and with our spacecraft mass of 13,200kg + 29,945kg =
40,145 kg it takes our engine 51s to apply the needed power.
Now let’s
turn to how the propulsion system is made in the simulation. The Spacecraft
class has the attributes of mass, fuel, thrust and flow rate and then it has
the method ‘fireThruster()’[13].
We call this method with a parameter that is the new wanted velocity, and then
the method has the responsibility of making the change. In the progress of
changing the velocity the spacecraft will spend fuel, which is also calculated
by this method.
First it
calculates the maximum velocity change that can be made within the ongoing time
step. Of course if the time step (our dt in the program) is set high the engine
might be able to apply all the wanted velocity change within just one step.
Most of the time however we run the simulation with time steps of only a couple
of seconds and then the engine needs more than one step to apply the change.
If the
maximum change possible is not big enough to give the wanted change, the
program applies the maximum change and goes one time step further and then it
can apply another change. Like this we get a gradual and realistic change of
velocity.
The maximum
change is calculated in the following way:
dv = exhaustVelocity*flowrate*dt/mf
equivalent
to
,
because the
mass of the propellant mp is equal to the flow rate multiplied by
the time step.
In the end,
after a number of time steps and small velocity changes, the wanted velocity is
within reach of the current velocity. We then just change the current to the
wanted velocity, but probably we didn’t need to apply the maximum change in
this last step, and thereby using as much fuel as in the other time steps, so
we have to calculate how much fuel we actually did spend.
dv_needed = newVelocity - currentVelocity;
fuel_spent = dv_needed*m0/(dt*(dv_needed-exhaustVelocity));
We can then subtract this amount from the fuel
supply of the spacecraft. Like this we all the way through the mission have an
exact knowledge of how much fuel is left. This was important to integrate into
the program to give us an idea about whether our chosen trajectory was
realistic or not – realistic concerning fuel consumption that is.
Radiation exposure calculated with Spenvis[14]
It’s
important for a mission to know how much radiation the spacecraft will be
exposed to during its journey. This has an impact on the choice of components
for the craft and the needed shielding.
To find the
amount of radiation that Torma would be exposed to during the 15 days in space,
we entered its trajectory into the online Spenvis program. The journey was
broken down into 6 stages in order to be able to input them into Spenvis. The
different stages are described below.
|
|
# of
orbits |
Altitude
(km) |
inclination |
Perigee
(km) |
Apogee
(km) |
|
1: LEO |
1 |
200 |
74.96 |
- |
- |
|
2:
Hohmann |
0.5 |
- |
74.96 |
6564 |
309000 |
|
3: LLCO
(Lower than lunar circular orbit when chasing the Moon) |
0.2 |
- |
74.96 |
311000 |
370000 |
|
4: Lunar
Orbit |
0.1 |
380000 |
74.96 |
- |
- |
|
5: Leave
Moon |
3 days |
- |
74.96 |
47000 |
380000 |
|
6:
Hohmann transfer to Earth atmosphere |
0.5 |
- |
74.96 |
100 |
47000 |
When
inputting the data into Spenvis we split the segments in 2 missions. The first
four segments in one mission – going to the Moon, and the last two in a mission
for going home.
First we
look at the amount of protons from the Sun that we receive on the way to the
Moon. While circling the Earth Torma only receives radiation when over the
poles. Then later as the orbit is lifted in the Hohmann transfer, the radiation
increases since the atmosphere no longer protects us.
Figure 6:
Protons from the Sun received on the way to the Moon.
The next
figures shows the total amount of radiation received on the trip back from the
Moon. Note that the radiation from protons is a factor 105 lower than
the total radiation in the Van Allen belt that is why most of the dots don’t
show any radiation. Torma only receives a significant amount of radiation while
doing the last Hohmann transfer to enter the atmosphere and thereby going
through the Van Allen belt.
Figure 7:
Total radiation received on the way home from the Moon.
From
Spenvis we also got graphs of the integrated dose for the two missions as
function of the shield thickness. This showed us that if we protect the more sensitive
components of our spacecraft – for instance the main computer - with 2mm of
aluminum the total radiation absorbed is only 15 rad, which is quite low
compared to other satellite missions. This is mainly because of the short
mission duration of only 15 days.
Figure 8:
The amount of radiation absorbed in Silicon as function of the shield
thickness.
The Subsystems
Below is a very short description of the most
important subsystems on the spacecraft with the exception of the propulsion
system that is described in a section of its own.
Main computer:
The CPU we have chosen for the main computer is
the Proton200K[15] developed by Space Micro.
It is capable of doing 900 MFLOPS (million floating-point operations per
second) at a Single Event Upset (SEU) rate of 1E-4 unrecoverable errors/day
using only 5-7 Watts of power. And important for space missions is that it also
has a total dose tolerance of greater than 100 krad (Si), meaning that we will
be able to fulfill many mission before needing to be replaced.
Attitude control:
This system makes sure we keep the right
orientation in space and controls the rotation of the spacecraft. It’s critical
to have the correct orientation when we fire the engines, otherwise we will not
end up where we wanted to go. We make use of four control moment gyroscopes
(CMG) mounted in a formation like a tetrahedron, so that we still have full
control of the spacecraft even if one of the CMGs stop working. The reason,
that we use CMGs instead of reaction wheels, is that they are much more power
efficient.
Power:
The electrical power needed on the mission is
supplied by fuel cells that convert hydrogen and oxygen into electricity. This
way of providing electricity was chosen because we have plenty of hydrogen and
oxygen since that is also the fuel for the rocket engine. It’s difficult to
make an accurate power budget for the mission because we don’t have a thorough
knowledge of our subsystems. A rough estimate could however be like the
following:
|
Spacecraft
subsystem: |
Avg. (W) |
Peak (W) |
|
Thermal control |
0 |
0 |
|
Attitude control |
50 |
100 |
|
Onboard computer |
10 |
20 |
|
Communication |
20 |
40 |
|
Propulsion |
20 |
30 |
|
Total: |
100 |
190 |
An average power consumption of 100W gives a
total consumption of 
for the whole mission.
The reaction: H2 + ˝O2
→ H2O gives 241.8 kJ/mol which means we will need 538 mol or
542g H2 and 4300 g O2 to produce the energy needed. Furthermore
fuel cells only have a 17% conversion efficiency for liquid hydrogen, so we
will really need 3.2 kg H2 and 25 kg of O2. This is an
insignificant amount compared to the amount the engine uses.
Communication:
The bandwidth needed for communication is
minimal. We only need to be able to send new commands to the spacecraft and
receive system status reports and position information. We are talking a
bandwidth of less than 1 kbit/s.
Thermal control:
The job of keeping Torma under thermal control
is expected to be solved purely by passive systems, like connecting the hot
components and the exterior of the spacecraft with a copper rod. It will be necessary
to connect it to more than one side of the spacecraft, since Torma is not
always facing the same side towards the Sun. Therefore it’s also necessary to
be able to connect and disconnect the rods according to the orientation of the
spacecraft.
Conclusion:
When starting this project we didn’t expect it
to reach the extent that is has, but working with it we found that it grew on
us and a desire to make it as realistic as possible emerged. This is the reason
for also programming the rocket equation or consumption of fuel into the
program even though it wasn’t originally our intent. The result is a trajectory
calculation and fuel consumption that we are confident in. These parts are the
most realistic in the project and working through them has given us a greater
understanding of astrodynamics and the problems one encounters when applying it
to solve real problems. The story surrounding the mission is however not as
realistic. First of all He-3 fusion is probably not the best candidate for
fusion. Secondly the investments to make all this happen are vast. On the other
hand building this framework around the mission, and trying to make it as plausible
as possible, has compelled us to lookup facts and read about former missions,
and through this we have expanded our general knowledge of how space missions
previously have been designed. Examples of this is finding out what kind of
rocket power is realistic, i.e. how great a mass we can expect to be able to
bring into LEO, and the fact that the Space Shuttle lands unpowered after
entering the atmosphere with a velocity of 8 km/s.
Writing your conclusion you would always like
to be able to present some breathtaking results and discoveries. That is going
to be difficult for our project, since we didn’t really make any new
breakthroughs. Sufficient it is to say that we have made a program able to more
or less realistically simulate a spacecrafts trajectory to the Moon and back.
Besides it being very instructive to work with
this project it has also been both interesting and entertaining.