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 bi-elliptic 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 bi-elliptic 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 bi-elliptic 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 semi-major 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 semi-major 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 semi-major axis of the transfer orbit is calculated in equation (13) using the average distance of both planets form the Sun.

 

                                (13)

 

Using both the semi-major 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:

 

 

 

 

 

 

 

 

Bi-elliptic Hohmann Transfer:

As previously discussed, a bi-elliptic 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 bi-elliptic 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: Bi-elliptic Hohmann Transfer [5, page 226]

 

Similar to the Hohmann transfer, this bi-elliptic Hohmann transfer will use the Sun as the reference point.  Equations (20) through (33) outline the Bi-elliptic Hohmann transfer process between Earth and Mars.  The return bi-elliptic Hohmann transfer to Earth will follow this same set of equations relative to the transfer being initiated at Mars. 

 

For the bi-elliptic 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 semi-major 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 semi-major 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 bi-elliptic Hohmann transfer equations can be used in reverse to calculate the requirements of the return bi-elliptic transfer Hohmann.

 

The ∆V needed to complete this bi-elliptic Hohmann transfer can now be calculated using this range of velocities, see equations 29-32.

 

                  (29)

 

                 (30)

 

                    (31)

 

                        (32)

 

The transfer time for this transfer orbit is calculated using the following equation.

 

        (33)

 

Results from Bi-elliptic 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 bi-elliptic Hohmann transfer will use the Sun as the reference point.  Equations (20) through (33) outline the Bi-elliptic Hohmann transfer process between Earth and Mars.  The return bi-elliptic 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 gravity-assist', 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 de-assist, .

 

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 bi-elliptic Hohmnn transfer will compared in Table 2.  For this case there is not a significant increase in the ∆V required to complete the bi-elliptic Hohmann transfer.  Table 2 also shows the possible ∆V’s that can be achieved by combining maneuvers.

 

Table 2: Hohmann verses Bi-elliptic ∆V

Compare Hohmann and Bi-elliptic ∆V’s

Transfer Out Transfer Type

Hohmann

Bi-elliptic

Return Transfer Type

Hohmann

11.1870

11.1875

Bi-elliptic

11.1875

11.1880

 

There is a much more significant difference in the time of transfer between the Hohmann and bi-elliptic 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 Bi-elliptic Transfer Times

Compare Hohmann and Bi-elliptic Transfer Durations

Transfer Out Transfer Type

Hohmann

Bi-elliptic

Return Transfer Type

Hohmann

517.7342

861.2413

Bi-elliptic

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

Bi-elliptic 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 bi-elliptic 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 Bi-elliptic 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, 3rd edition, David A. Vallado

[9] Fundamentals of Astrodynamics and Applications, 4th 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/5325-scientists-revisit-mars-sample-return-plans.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