ASEN 5050 Final
Project
Kerry Fessenden
Minimizing the
∆V Requirements and Transfer Times for a Mars Sample Return Mission
Abstract:
Scientific
exploration missions to Mars via flybys, orbiters, landers, and rovers have
allowed for the collection of important information about the Martian surface,
environment, and atmosphere. The first
flyby mission to Mars, Mariner 4, was launched on November 28, 1964. Landers have successfully collected
information from the surface of Mars starting with Viking 1’s landing in
1976. The landers and rovers deployed
since Viking 1 have successfully taken images of increasing quality, analyzed
samples, and explored the terrain of the surface of Mars. The information collected during these
missions has then been sent back to Earth for analysis. Much more in depth research could be done by either sending humans and equipment to the surface of Mars
to conduct tests or by returning samples from Mars to Earth for testing. Scientists would learn new information from
both of these types of missions. With
the current space program, a manned mission to Mars is not feasible in the near
future; however, a mission to collect and return samples from Mars could be
completed in the next decade. Aspects of
a sample return mission that have not been completed before include: launching
the samples from the surface of Mars, initiating a return transfer to return
samples to Earth orbit, and, finally, safely returning the samples to Earth’s
surface. The analysis of two transfer
methods, a Hohmann transfer and a bielliptic Hohmann transfer show no
significant difference in ∆V requirements. However the Hohmann transfer has a significantly
shorter transfer time making it the better choice for a sample return mission
to Mars.
Introduction:
Traveling to Mars has been of
interest to the United States, as well as other countries, for many years. Some experimentation has already been
completed on these subjects through the previous flyby, orbiter, and lander and
rover missions; however, a sample return mission to Mars will allow for more in
depth experimentation on samples from the surface, environment, and atmosphere
of the planet Mars.
Limiting factors of a sample return mission still
being assessed include: assembling a launching system on the surface of Mars,
launching from the surface of Mars, and getting enough fuel to Mars for the
launch and return transfer orbit. These
are all mission critical issues that will need to be solved before a Mars
sample return mission will be able to be completed.
This paper will analyze three different
methods to complete a transfer from Earth orbit to an orbit around Mars and the
returning transfer orbit from Mars to Earth orbit, a mission that has not yet
been completed. A comparison will be made between the ∆V and mission
duration for the methods analyzed.
Methods for comparison will include a Hohmann transfer, a bielliptic
Hohmann transfer and the patched conic method using a Hohmann transfer. Additionally, possible gravity assist
opportunities for the Earth, Mars transfer will be analyzed.
Background:
Mariner 4 was the first
successful mission to Mars launched by the United States on November 28,
1964. The completion of the Mariner 4
mission occurred as it flew past Mars on July 14, 1965 [1]. Since this first success, the United States
has continued sending spacecraft to Mars successfully completing flyby,
orbiter, and lander and rover missions.
Other countries have attempted to send spacecraft to Mars, but they have
not been successful. The United States
launched the Mars Science Laboratory (MSL) on November 26, 2011, which is due
to arrive at Mars in August of 2012 [1].
The two most important applications
of returning samples from Mars are the ability to examine the samples for
possible research benefit and the ability to apply such knowledge to future
manned missions. The research value of
examining samples returned from the surface of Mars is currently unknown. Many additional tests can be performed on the
samples in labs on Earth then those performed on the surface of Mars. Successfully returning samples from Mars will
be the first step towards successfully sending humans to Mars and returning
them safely to Earth.
In 2008 the MIT Space, Policy,
and Society Research Group wrote a paper assessing the future of human
spaceflight. The paper looked at the
importance of sending humans into space and the implications of doing so. One topic discussed was the use of remote and
robotic science missions and their inability to act autonomously. The paper determined that “remote and robotic
science missions have yielded astonishing new discoveries on and about our
solar system and beyond. These vehicles
have generated proof of water ice on Mars, detected organic material venting
from a moon of Saturn, and led to discoveries of “exoplanets”
outside our solar system. Despite their
technology, none of these missions are “automatic” – each is controlled by, and
sends data to, human beings on Earth [6].” The need for human involvement in
these missions emphasizes the importance of humans in all science
missions.
NASA is currently collaborating
with the European Space Agency on a proposed launch in 2018 for a potential
sample return mission. Sample return
missions from Mars have not been completed yet because of the complexity of
such a mission. NASA is also making long
term plans for Mars sample return missions after 2020, also in collaboration
with the European Space Agency. These
missions include sending an orbiter to Mars in 2022 and a Mars Science Return
lander in 2024. This series of missions
could allow for the return of Mars samples to Earth by 2027. While these plans are still in the design
phase it is promising that this collaboration is in the future for both space
agencies [8].
Foundation:
Currently in the United States,
the NASA Jet Propulsion Laboratory (JPL) conducts the majority of the research
and design for interplanetary missions.
Table 1 shows the Mars missions
that JPL has helped to plan with their respective launch and arrival
dates. The Launch and arrival dates have
been provide for the comparison of transfer durations.
Table 1: Classification of Mars Missions
Planned by JPL
Classification of Mars Missions Planned by JPL 

Mission Name 
Date Launched 
Date arrived at Mars 
Duration (days) 
Mission Type 
Mariner 4 
11/28/1964 
7/14/1965 
228.00 
Flyby 
Mariner 6 
2/24/1969 
7/31/1969 
157.00 
Flyby 
Mariner 7 
3/27/1969 
8/5/1969 
131.00 

Mariner 9 
5/30/1971 
11/13/1971 
167.00 
Orbiter 
Viking 1 
8/20/1975 
6/19/1976 
304.00 
Orbiter/Lander 
Viking 1 
9/9/1975 
8/7/1976 
333.00 
Orbiter/Lander 
Mars Observer 
9/25/1992 

Failed 
Orbiter 
Mars Global Surveyor 
6/7/1996 
9/12/1997 
462.00 
Orbiter 
Pathfinder 
12/4/1996 
7/4/1997 
212.00 
Lander/Rover 
Climate Orbiter 
12/11/1998 

Failed 
Orbiter 
Polar Lander/Deep Space 2 
1/9/1999 

Failed 
Orbiter 
Phoenix 
8/4/2007 
5/25/2008 
295.00 
Lander/Rover 
2001 Mars Odyssey 
4/7/2001 
10/24/2001 
200.00 
Orbiter 
2003 Mars Exploration Rovers (Spirit) 
6/10/2003 
1/3/2004 
207.00 

2003 Mars Exploration Rovers (Opportunity) 
7/7/2003 
1/24/2004 
201.00 

Mars Express 
6/2/2003 
12/23/2003 
204.00 
Orbiter 
Mars Reconnaissance Orbiter 
8/12/2005 
3/10/2006 
210.00 
Orbiter 
Mars Science Laboratory 
11/26/2011 


Lander/Rover 
The Phoenix Mission departed
Earth on August 4, 2007 and arrived at Mars on May, 25, 2008. The
duration of this mission transfer was 295 days.
This calculation does not take into account the time of day for launch
and arrival, but for the purpose of transfer comparison this is accurate
enough.
The following figure, Figure 1,
shows the transfer mission profile for the Phoenix mission. This mission profiles shows the spacecraft
passing through more than 180° before reaching Mars. The figure also shows the portion of the
transfer that ranges from 0° to 180°.
This 180° transfer started 27 days after the spacecraft was launched
from Earth meaning the 180° transfer, which is equivalent to the distance of a
Hohmann transfer lasted 268 days.
Figure
1: Phoenix Mission Profile [7]
Fundamental
Concepts:
The transfer methods used to
calculate interplanetary transfer between Earth and Mars uses a few important
assumptions. One assumption is that the
Earth and Mars lay in the same ecliptic plane eliminating the need for maneuvers
to adjust the plane or inclination of the object being transferred. The second assumption made is that the Earth
and Mars are in perfectly circular orbits around the Sun. The final assumption made is that there are
no additional perturbations to the object as it is transferring between the two
planets.
Hohmann transfers are also known
as the minimum energy transfer method. A
Hohmann transfer is a transfer orbit that uses two maneuvers. The first maneuver initiates the transfer
from first planet and the second maneuver takes place after the spacecraft has
traveled 180° arriving at the second planet to adjust the speed of the
spacecraft for an insertion into an orbit around Mars.
A bielliptic Hohmann transfer
completes two Hohmann transfers before reaching the second planet. During the first transfer orbit, the
spacecraft travels 180° to a secondary orbit.
This second orbit, which is reached after the first maneuver, must be
larger than the desired final orbit.
During the second transfer orbit, the spacecraft leaves the secondary
orbit and again travels 180° before arriving at the second planet.
The most common concept used to
complete an interplanetary transfer that includes the gravitational effects of
the planets is the concept of patched conics.
For the transfer between Earth and Mars the method of patched conics
uses two different conic sections, in this instance, the Earth’s conic section
and Mars’ conic section. The patched
conic approximation used in combination with the Hohmann transfer method yields
more accurate results then using the Hohmann transfer equations alone. The increased accuracy resulting from the use
of the patched conic method is due to the inclusion of the effects of gravity
from the initial and final planet. The
gravity of the planets is accounted for by the Sphere of Influence (SOI) of the
initial and final planet in the transfer.
A planets’ SOI is the sphere surrounding the planet in which the planet
is the primary gravitational influence.
A gravity assist is a process in
which a spacecraft’s trajectory hyperbolically passes by another body. This hyperbolic passage allows the spacecraft
to change the direction of its velocity with respect to the Sun. This can be used to either increase the
spacecraft’s velocity with respect to the Sun or decrease the spacecraft’s
velocity with respect to the Sun.
The two quantities that will be
compared during this study are the ∆V needed for the transfer maneuvers
and the time duration of each transfer.
The ∆V needed for a maneuver is the calculated from the change in
velocity needed to reach a desired orbit.
(1)
Where:
T is the time of the
maneuver
m = mass of propellant
The time duration needed to
complete each transfer is calculated from the initiation of the maneuver to
leave earth to the initiation of the maneuver to insert into an orbit around
Mars.
Choosing an appropriate departure
date is important for interplanetary missions because the departure timing of
the transfer determines if the spacecraft will intercept the destination
planet. When planning the departure time
you need to take into account where the first planet will be it its’ orbit and
where the second planet will be in its’ orbit.
In a noncircular orbit, the departure and arrival velocity can be
optimized using the shape of the planets orbit.
Hohmann
Transfer:
As previously discussed a Hohmann
transfer from Earth to Mars will consist of a single transfer orbit through
180°. The returning mission will also
use only one transfer orbit through 180°.
Figure 2 represents complete Hohmann transfer from Earth to Mars, where
Earth is planet a
and Mars is planet b.
Figure
2: Hohmann Transfer [5, page 225]
Equations (2) through (19)
outline the entire Hohmann transfer process between Earth and Mars and the
return Hohmann transfer back to Earth.
All calculations are done in relation to the Sun as it is the center for
interplanetary mission.
The velocity of the spacecraft
leaving Earth (point a
in Figure 2) is calculated using equation (2) below. This equation takes into account both the
spacecraft’s orbiting altitude around the Earth and the distance of the Earth
form the Sun.
(2)
Where:
Similarly, the desired final velocity of
the spacecraft in orbit around Mars (point b
in Figure 3) is calculated in equation 3.
(3)
The semimajor axis of the
transfer orbit is calculated in equation (4) using the average distance of both
planets form the Sun.
(4)
Where:
Using both the
semimajor axis of the transfer orbit and the relative distances of Earth and
Mars, the necessary transfer velocities can be calculated.
(5)
Where:
(6)
The ∆V needed to complete this
Hohmann transfer can now be calculated using this range of velocities as shown
in equations (7)(9).
(7)
Where:
(8)
(9)
The transfer time for this transfer orbit
is calculated using the following equation.
(10)
Where:
This same set of equations can be used
for the return transfer orbit of the spacecraft from Mars to Earth.
The velocity of the spacecraft
leaving Mars (point b in Figure 3 is
calculated using equation (11).
(11)
Similarly, the desired final velocity of
the spacecraft in orbit around Mars (point b)
is calculated in equation (12).
(12)
The semimajor axis of the
transfer orbit is calculated in equation (13) using the average distance of
both planets form the Sun.
(13)
Using both the semimajor axis of the
transfer orbit and the relative distances of Earth and Mars the necessary
transfer velocities can be calculated using equations (14) and (15).
(14)
(15)
The ∆V needed to complete this
Hohmann transfer can now be calculated using this range of velocities as shown
in equations (16)(18).
(16)
(17)
(18)
The transfer time for this transfer orbit
is calculated using the following equation.
(19)
Results from Hohmann Transfer:
Bielliptic
Hohmann Transfer:
As previously discussed, a
bielliptic Hohmann transfer from Earth to Mars will consist of two transfer
orbits. Similarly, a return mission will
use two transfer orbits to travel between Mars and Earth. The following image shows a complete
bielliptic Hohmann transfer from Earth to Mars. In this image the Earth is located at point a, Mars is located at point c, and the secondary orbit is at point b.
Figure
3: Bielliptic Hohmann Transfer [5,
page 226]
Similar to the Hohmann transfer,
this bielliptic Hohmann transfer will use the Sun as the reference point. Equations (20) through (33) outline the
Bielliptic Hohmann transfer process between Earth and Mars. The return bielliptic Hohmann transfer to Earth
will follow this same set of equations relative to the transfer being initiated
at Mars.
For the bielliptic Hohmann
transfer, the radius of the second orbit can be chosen to be any radius larger
than the radius of Mars.
The velocity of the spacecraft
leaving Earth (point a
in Figure 3) is calculated using equation (20).
This equation takes into account both the spacecraft’s orbiting altitude
around the Earth and the distance of the Earth form the Sun.
(20)
The desired velocity at the secondary
orbit is calculated in equation (21).
(21)
The desired final velocity of the
spacecraft in orbit around Mars (point c
in Figure 4) is calculated in equation (22).
(22)
The semimajor axis of the firs
transfer orbit is calculated using the average distance of the Earth from the
Sun and the distance of the secondary orbit from the Sun as shown in equations
(23) and (24). For these calculations
the radius for the large ellipse has been chosen as 10,000 km past the orbit of
Mars.
(23)
(24)
Using both the semimajor axis of the
transfer orbit and the relative distances of Earth and Mars the necessary
transfer velocities can be calculated as seen in equations (25)(28).
(25)
(26)
(27)
(28)
Similar to the Hohmann transfer, the
bielliptic Hohmann transfer equations can be used in reverse to calculate the
requirements of the return bielliptic transfer Hohmann.
The ∆V needed to complete this bielliptic
Hohmann transfer can now be calculated using this range of velocities, see
equations 2932.
(29)
(30)
(31)
(32)
The transfer time for this transfer orbit
is calculated using the following equation.
(33)
Results from Bielliptic Hohmann Transfer:
Patched
Conics with Hohmann Transfer:
As previously discussed, the method of
patched conics takes into account the gravitational pull of the planets
involved in the transfer. The
gravitational pull of each planet can be quantified by the sphere of influence:
the spherical region around a planet where it is the primary gravitational
influence. The radius of the sphere of
influence can be calculated using the following equation.
(34)
Where:
After calculating the SOI, the hyperbolic
trajectory that is needed for the object to escape the initial planets SOI is
calculated. The SOI of Earth and Mars in
relation to their location to the Sun can be seen in Figure 4 and Figure 5. The hyperbolic trajectory is then calculated
to be the Hohmann transfer from the SOI of the initial planet to the final
planet. The hyperbolic trajectory can be
seen in both Figure 4 and Figure 5, Figure 5 labels this trajectory the
“Transfer Ellipse”. Finally, the
trajectory becomes the hyperbolic entry into SOI of the final planet. These three portions of the orbit are then patched
together to make one trajectory completing the patched conic method.
Figure
4: Patched Conics Method [9, page
996]
Figure
5: Patched Conics Method, example of
a Mars to Earth transfer
Similar to the Hohmann transfer,
this bielliptic Hohmann transfer will use the Sun as the reference point. Equations (20) through (33) outline the
Bielliptic Hohmann transfer process between Earth and Mars. The return bielliptic Hohmann transfer back to
Earth will follow this same set of equations relative to the transfer being
initiated at Mars.
Equations (35) through (45)
outline the process to complete a Hohmann transfer between Earth and Mars in
addition to the method of patched conics.
This transfer will again use the Sun as the reference point for
calculations. The return transfer to
Earth will follow this same set of equations relative to the transfer being
initiated at Mars.
The following equation calculates
the velocity of spacecraft in relation to the Sun at the aphelion point of the
transfer.
(35)
The velocity of the Earth relative to sun
can now be calculated in equations (36).
(36)
The minimum ∆V to leave Earth
orbit can be determined by using the difference between the spacecraft’s
velocity with respect to the Sun and the Earth’s velocity with respect to the
Sun. This is also known as the
hyperbolic excess velocity and can be calculated using equation (37).
(37)
Using the satellites altitude plus the
radius of the Earth the velocity at perigee can be calculated as seen in
equation (38).
(38)
The ∆V for the first maneuver can
then be determined using equation (39).
(39)
For insertion into an orbit around Mars,
the velocity of Mars along with the velocity of the spacecraft at insertion
must be calculated as shown in equations (40) and (41).
(40)
(41)
The hyperbolic excess velocity
can be calculated for the Mars insertion (equation 42).
(42)
Using the satellite’s altitude plus the
radius of the Mars, the velocity at perigee can be calculated as shown in
equation (43).
(43)
The ∆V for the second maneuver can
then be determined by equation (44).
(44)
The sum of the ∆V’s needed for the
two maneuvers represent to total ∆V for the transfer can be calculated in
equation (45).
(45)
The same process is used to calculate the
return transfer.
Results from Patch Conics Method using a
Hohmann Transfer:
Gravity
Assist:
A gravity assist is a common
technique used when attempting a transfer from one body to another body in the
solar system. The exploration of planets
at far distances from the Earth is possible because of the use of gravity
assists. “Several important missions
would have been impossible without this gravityassist', some missions
have actually been saved in flight by this technique [10].” A gravity
assist makes interplanetary mission possible because it allows a vehicle to
have less cable upper stages. The need
for less capable upper stages comes from the increase in velocity with respect
to the Sun obtained after a planetary flyby, gravity assist.
An example of a successful
mission that has used gravity assists to reach outer planets is the Cassini
mission. Cassini has completed planetary
flybys of multiple planetary bodies including Venus, the Moon, Earth, and
Jupiter. Cassini also completed flybys of
many other smaller bodies and the spacecraft Huygens. Each of these flybys allowed the satellite to
increase its velocity with respect to the Sun.
The plot of Cassini’s transfer orbits can be seen below.
Figure
6: Cassini Transfer Orbits
Earth and Mars have both been
used to complete gravity assist maneuvers on trajectories headed to planets
further from the Sun. A mission has not
yet been completed where a gravity assist is used during the transfer between Earth
and Mars. A gravity assist could be
completed on the transfer between Earth and Mars using Earth’s Moon as the
assisting body between the two planets.
The process for calculating a gravity assist will now be analyzed.
Figure
7: Path of a Planetary Flyby [4]
Figure 7 illustrates the
different parameters of a planetary flyby.
The planets velocity with respect to the Sun is shown as while the spacecraft’s
incoming velocity is shown as . The turning angle or the number of degrees
that the spacecraft is turned through is represented by the angle δ.
Figure 8 shows the resultant
vectors of a gravity assist, and a gravity deassist, .
Figure
8: Resultant Vectors from Gravity
Assist
To calculate the parameters of
the spacecraft’s trajectory as it is approaching for a Moon gravity assist, use
equations (35) through (39) from the patched conic method explained
previously. Figure 8 can then be used to
determine which velocities should be added to find the resulting spacecraft
velocity. The velocity of the Moon and
the velocity of the spacecraft at the Moon can be calculated. For these calculations the radius of the Moon
from the sun was assumed to be the radius of the Moon from the Earth added to
the radius of the Earth from the Sun.
(46)
(47)
The hyperbolic excess velocity
can be calculated for the Moon swingby (equation 48).
(48)
Using the spacecraft’s altitude plus the
radius of the Moon, the velocity at perigee can be calculated as shown in
equation (49).
(49)
The turning angle that the spacecraft
will experience as it swings by the Moon can be calculated using equations (50)
and (51).
(50)
(51)
The change in velocity with respect to
the Sun can now be found using the turning angle and equations (52) through
(54).
(52)
(53)
(54)
Results from a Moon Gravity Assist:
Comparing the velocity of the
spacecraft entering the Moon with the Velocity at exit of the Moon shows that
there is a small negative change in velocity.
This small negative change is caused by a few factors. The first factor resulting in this decreased
velocity is the relatively small size of the moon as compared to the Earth, and
the fact that the Moon is located inside the Earth’s SOI. Another factor that contributes to the
decreased velocity is the velocity of the Moon with respect to the sun as
compared to the spacecraft’s velocity with respect to the Sun when it attempts
this gravity assist. Lower velocity of
the Moon with respect to the sun causes the spacecraft to lose some velocity. This decrease in velocity hurts the mission
because it causes the spacecraft to have a lower velocity then it needs to exit
the Earth’s sphere of influence.
For this reason there has not
been a mission departing Earth that uses the Moon for a gravity assist. There have been mission that have flown past
Earth’s Moon to complete plane change maneuvers such as the LCross
mission which complete and lunar inclination change assist after being launched
on June 18, 2009.[12]
Departure
Date:
The departure date for a transfer
from Earth to Mars is an important value to calculate. The Earth and Mars are in correct position
for the Hohmann transfer every 2.1353 years as calculated in equation (55).
(55)
Where:
The transfer time for each type
of maneuver has been described in each of the maneuver sections. Once the spacecraft reaches Mars it must wait
on the surface of Mars until Mars and Earth are in phase for the return
transfer. The phase angle for the return
transfer must be calculated before the waiting time on Mars can be found. The phase angle relies on the duration of the
transfer, this makes sense because shorter transfer
times will need to have different phase angles then longer duration transfers
(56)
(57)
Where:
Summary:
The
values of ∆V needed to complete each type of transfer and the total time
of each transfer can now be compared. First
the Hohmann and bielliptic Hohmnn
transfer will compared in Table 2. For
this case there is not a significant increase in the ∆V required to
complete the bielliptic Hohmann transfer.
Table 2 also shows the possible ∆V’s that can be achieved by
combining maneuvers.
Table 2: Hohmann verses Bielliptic ∆V
Compare Hohmann and Bielliptic
∆V’s 

Transfer Out Transfer Type 

Hohmann 
Bielliptic 

Return Transfer Type 
Hohmann 
11.1870 
11.1875 
Bielliptic 
11.1875 
11.1880 
There
is a much more significant difference in the time of transfer between the
Hohmann and bielliptic Hohmann transfers.
The variation in transfer times can be seen in Table 3. Table 3 also shows the possible transfer
times that can be achieved by combining maneuvers.
Table 3: Hohmann verse Bielliptic
Transfer Times
Compare Hohmann and Bielliptic Transfer
Durations 

Transfer Out Transfer Type 

Hohmann 
Bielliptic 

Return Transfer Type 
Hohmann 
517.7342 
861.2413 
Bielliptic 
861.2413 
1204.7484 
A
comparison can also be made between the Hohmann transfer and the Patched Conic
Hohmann transfer. Tables 4 and 5 below show this comparison. The patched conic Hohmann transfer takes into
account the gravity of each planet, explaining its increase in ∆V for the
maneuver. It is interesting to note the
small error in the transfer times compared to the more significant error in
∆V.
Table 4: Hohmann
Transfer vs Patched Conic Hohmann
Transfer ∆V
Compare Hohmann and Patched Conic
Hohmann ∆V’s 

∆V 

Return Transfer Type 
Hohmann 
11.1870 
Patch Conics 
11.2450 
Table 5: Hohmann verse Patched Conic
Hohmann Transfer Times
Compare Transfer Methods Transfer
Duration 

Transfer Duration 

Return Transfer Type 
Hohmann 
517.7342 
Patch Conics 
517.7331 
The
transfer time for these three sample transfers can now be compared to the past
US missions that have completed a transfer between Earth and Mars.
Table 6: Transfer Times for US Missions to
Mars Compared to Calculated Transfer Times
Mission 
Transfer length 
Mariner 7 
131 
Mariner 6 
157 
Mariner 9 
167 
2001 Mars Odyssey 
200 
2003 Mars Exploration
Rovers (Opportunity) 
201 
Mars Express 
204 
2003 Mars Exploration
Rovers (Spirit) 
207 
Mars Reconnaissance
Orbiter 
210 
Pathfinder 
212 
Mariner 4 
228 
Patch Conics with Hohmann 
258.866 
Hohmann 
258.8671 
Phoenix 
295 
Viking 1 
304 
Viking 1 
333 
Mars Global Surveyor 
462 
Bielliptic Hohhman 
602.3748 
Mars Observer 
Failed 
Climate Orbiter 
Failed 
Polar Lander/Deep
Space 2 
Failed 
The
Hohmann and patched conic Hohmann transfers have transfer times that lie
between the Mariner 4 mission and the Phoenix mission. As seen in Figure 1 the Phoenix mission
completed a transfer that traveled more than 180°. Figure 1 also shows Phoenix’s position 27
days after launch, which is near its original Earth orbit. From the 27 day post launch position the
Phoenix mission traveled another 180° before intercepting Mars. The 180° transfer of the Phoenix took 268
days compared to the 258.866 day transfer of the Hohmann and the 258.8671 day
transfer of the patched conic Hohmann.
Conclusion:
These results show that the Hohmann transfer is more
efficient than the bielliptic transfer both in ∆V required and Transfer
time. While there was not a significant
increase in ∆V between the two transfers, the total time to complete the
mission was more than doubled, making the Bielliptic method a less efficient
transfer.
Missions
from Table 6 that took significantly less time to complete their transfers used
more direct transfer methods, meaning the transfer took less than 180° to
complete. There are advantages and
disadvantages to completing a transfer in a shorter amount of time. If a transfer is completed in less time it
would allow the mission to have more time on the surface. However this shorter transfer time requires
more ∆V to complete. There could
be advantages in utilizing both a Hohmann type transfer and a more direct
transfer. For example the lander or
rover could be sent to Mars via a shorter transfer, increasing the payloads
time on Mars while the return vehicle could be sent on a more energy efficient
transfer.
Future work to be completed
towards the goal of a sample return mission include creating a launch vehicle
capable of reaching an orbit around Mars from the surface of Mars and
maneuvering into a transfer orbit to return the samples to Earth.
References:
Books:
[4]
Orbital Mechanics for Engineering
Students, Howard D. Curtis
[5]
Fundamentals of Astrodynamics
and Applications, 3^{rd} edition, David A. Vallado
[9]
Fundamentals of Astrodynamics
and Applications, 4^{th} edition, David A. Vallado
Papers
[6]
Space, Policy, and Society Working Group, MIT The Future of Human
Spaceflight. http://web.mit.edu/mitsps/MITFutureofHumanSpaceflight.pdf
[10] From Instrumented Comets to Grant Tours; On the History of Gravity
Assist, Gary A. Flandrot, University of Tennessee, UTSI
Websites:
[1]
http://mars.jpl.nasa.gov/technology/samplereturn/
[2]
http://www.space.com/5325scientistsrevisitmarssamplereturnplans.html
[3]
http://spaceflightnow.com/news/n1004/29mars/
[7]
http://phoenix.lpl.arizona.edu/images.php?gID=161&cID=1
[8]
http://mars.jpl.nasa.gov/programmissions/missions/future/
[11]
http://www.mtinfo.com/archfea/fa3/casmap.jpg
[12]
http://lcross.arc.nasa.gov/index.htm
Interesting: http://pubs.acs.org/subscribe/journals/ci/31/i08/html/08digregorio.html