Description
Hello! I need help for my orbital mechanics class. I lost my MATLAB code for the provided Cassini-route-to-students file provided from my professor, and I desperately need help. I need someone to write to code for me for my project – to add the necessary equations to all of the sections that say ‘replace’. I have the rest of the project done, so you don’t need to do those parts, but I will provide the other matlab files and my report for additional details, but again – the only thing I need is for the “Cassini” file to be updated according to the project guidelines where its indicated in the code via ‘replace’. Thank you!
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AEM 4301 Orbital Mechanics
Fall 2023
Mission Design Final Project
General Project Information
You are to submit solutions as a project group and will receive a group grade (out of 100 total points). Please clearly
and succinctly address each part and each question in your write up. Show your work and reasoning – you are being
graded on demonstrating your understanding of the course material, not just getting things to work. Please adhere to
the following guidelines
Remember that good basic programming practices will save you a lot of time/effort and neatness, completeness and
clarity in your submission will go a long way to helping your grade.
Limit the write up and explanation of your results (main text and results figures) to between 15-20 pages (this does
not include code or cover/contribution pages). Label all plots and legends appropriately and report units. All reports
must be typed/typeset and submitted via Gradescope as a group (self-signup).
Please list team member names on the cover page, and provide a short list of each team member’s specific
contributions on the first page of the report.
Please also list dates when the group met (in-person, Zoom, FaceTime, etc.) and who was in attendance. Please limit
this list to major meetings (10 or fewer table rows). For example:
Meeting Date
April 14, 1600
In Attendance
Brahe, Galileo, Kepler
Content
Met to discuss project, decide on
specific contributions, and other
miscellaneous planning. We decided
Galileo
would
work
on
instrumentation, Brahe would work
on data collection, and Kepler would
work on deriving the formulas used
for explaining the universe as we
know it
April 18, 1600
Galileo and Kepler
-Brahe was absent due to a
scheduled silver/gold-nose fitting.
However, he will be collecting data
tonight at midnight to make up for
his absence.
Etc…
Etc…
Met to discuss our progress in
understanding
the
universe.
Discussed the work by Simon
Stevinus (dropping balls of Delft
tower) and whether that applies to
our project.
Basic organization of the final project report:
I.
II.
III.
IV.
V.
VI.
Introduction
Background/ Historical Context
Method
Analysis/Results
Conclusion
Appendix (code)
Mission Design Project Background Info
The Cassini-Huygens mission was a joint flagship class NASA-ESA-ASI endeavour, the fourth spacecraft to visit
Saturn and the first to enter orbit around the planet. The mission followed on from flybys in 1979 by Pioneer 11 and
in 1980 and 1981 by both Voyager spacecraft. The spacecraft consisted of two parts, the orbiter Cassini (NASAASI)
and a Titan lander Huygens (ESA). Cassini was named after the Italian-French astronomer Giovanni Domenico
Cassini who discovered Iapetus, Rhea, Tethys and Dione in the latter half of the 17th century. The Huygens probe
was aptly named after Dutch astronomer Christiaan Huygens, the discoverer of Saturn’s largest moon Titan in 1655.
Only impulsive maneuvers will be considered. We will treat our n-body problem as a series of relative two body
problems, and will do so by using a patched conic method where we treat the transitions between conics as
instantaneous transitions similar to our approach in previous homework problems. Unlike previous problems, we will
need to consider the spacecraft’s time of arrival at the orbit altitudes to make sure the gravitational bodies will be in
the correct locations in their orbits to properly rendezvous with Cassini-Huygens. To do so, you will be given
MATLAB code to calculate planetary positions based on date and time information. You will also need to use code
related to Lambert’s problem (Curtis, Algorithm 5.2) to iteratively determine the possible and optimal orbit solutions.
Obviously what JPL chose is a good benchmark, but you may find luck deviating away from it (somewhat due to our
assumptions). You will construct 3-5 vector diagrams to present in your report, and you may have no more than 15
impulsive maneuvers (but also a lot fewer if you like). You will report your method for selecting your impulsive
maneuvers and gravity assists parameters. You have the following restrictions on flyby altitudes (at least this large):
Venus: 284 km, Earth: 1,171 km, Jupiter: 9.7 million km
Assume that Cassini starts in a 300 km altitude parking orbit.
At the final destination of Saturn, Cassini should be captured (calculate Δ ) into an orbit with the following orbital
elements.
Semi-major axis
4 585 959 km
Eccentricity
0.98239
Periapsis radius
80 731 km
Apoapsis radius
9 091 186 km
Inclination
11.534°
You will provide a table that reports the impulsive maneuvers used (magnitude and direction) and the type of gravity
assists (dark-side, light-side) and their flyby altitudes. Also contained within this table should be the resulting conic
values (eccentricity, semi-major, true anomaly, argument of perihelion) in the heliocentric frame after the delta-v or
equivalent delta-v.
Project Report
1.
2.
3.
4.
Please list the project chosen, team member names, and provide a short list of each team member’s specific
contributions on the first page of the report.
Please also list dates when the group met (in-person, Zoom, FaceTime, etc.) and who was in attendance. Please
limit this list to major meetings (10 or fewer table rows).
Write the Introduction section of the final report
Write the Background/Historical Context section of the final report. (Cassini-Huygens for mission design, GPS
or HEO orbits for orbit determination)
5a. Figuring out in the heliocentric view what the conics are connecting your bodies on the given dates
Plot the orbits of Venus Earth, Jupiter, and Saturn in the heliocentric over the timespan on the mission by using the
planet_elements_and_sv provided through Canvas.
Calculate the relevant planet positions at the dates shown in the Final Project Document. Report the body positions in
a table alongside the date of encounter.
Date
Body
J2000 x (km)
J2000 y (km)
J2000 z (km)
Next, use Lambert’s equation to find the conics connecting these points at these time intervals. Additional code related
to solving Lambert’s equation has been provided to you on Canvas. Once you find the conics connecting your
locations, report the velocities at the terminal points in the table below.
Dates
J2000
−
J2000
+
J2000 −
J2000 +
(km/s)
(km/s)
(km/s)
(km/s)
Also report the information regarding the size, shape, and orientation of the connecting conics.
Dates
RAAN
(deg)
(km)
e
(deg)
Draw vector diagrams for the gravity assists (assume dark-side to begin) for each encounter with a planet shown in
the table. For each diagram, indicate whether the planet or the spacecraft would be traveling faster in the heliocentric
frame. Also, indicate any relevant angles (flight path angle, flyby angle, etc.) in your diagrams. Finally calculate the
equivalent delta-v values given the flyby altitude, and the additional delta-v required to get onto the next conic in the
sequence.
You will provide a table that reports the impulsive maneuvers used (magnitude and direction) and the type of gravity
assists (dark-side, light-side) and their flyby altitudes. Also contained within this table should be the resulting conic
values (eccentricity, semi-major, true anomaly, argument of perihelion) in the heliocentric frame after the delta-v or
equivalent delta-v.
Report your total delta-v.
Provide a 2D plot of the mission design in the heliocentric view and note important dates.
5b. Designing your own mission.
Given the restrictions already specified regarding the starting and ending time of the mission the original parking orbit,
the minimum flyby altitudes design a mission to minimize delta-v.
An acceptable answer to address this is to try moving the dates that are variable (not the beginning or end date) of
encounters and subsequent gravity assists to different dates and report if your delta-v improved or worsened. Other
techniques are acceptable as well, as long as they are properly explained. Also consider moving the location of the
Venus Targeting Maneuver
Like in part 5a, you will provide a table that reports the impulsive maneuvers used (magnitude and direction) and the
type of gravity assists (dark-side, light-side) and their flyby altitudes. Also contained within this table should be the
resulting conic values (eccentricity, semi-major, true anomaly, argument of perihelion) in the heliocentric frame after
the delta-v or equivalent delta-v.
Report your total delta-v.
Provide a 3D plot of the mission design in the heliocentric view and note important dates.
5.
Write up your analysis related to your results. I expect some discussion related to each element discussed in
section 5. Let me know what you tried, what worked, what didn’t work, what you would try knowing what you
know now.
6.
Write a Conclusion
7.
Additional Extra Credit: Write a haiku about orbital mechanics. Alternative forms of short poetry, verse, etc. are
also acceptable. (Must be G-rated/family-friendly).
Project Report: Cassini-Huygens Mission Design
3.Write the Introduction section of the final report
The Cassini-Huygens mission, a collaborative effort between NASA, ESA, and ASI, stands as
a testament to the ingenuity and collaborative spirit of international space exploration.
Launched as a flagship-class endeavor, it became the fourth spacecraft to venture to Saturn,
marking a pivotal moment in the exploration of our solar system. Named after eminent
astronomers Giovanni Domenico Cassini and Christiaan Huygens, the mission was not only an
exploration of the majestic ringed planet but also a triumph of orbital mechanics and mission
design.
Orbital mechanics, the backbone of space exploration, governs the motion of celestial bodies
in our solar system. It allows us to calculate and predict the trajectories of spacecraft, enabling
precise mission planning. The project delves into the intricacies of orbital mechanics to analyze
and optimize the Cassini-Huygens mission design. The focus lies on impulsive maneuvers and
gravity assists, where we treat the n-body problem as a series of relative two-body problems,
employing a patched conic method.
Our approach involves treating transitions between conics as instantaneous, a methodology
familiar from previous homework problems. Unlike earlier scenarios, this project introduces
the critical consideration of the spacecraft’s time of arrival at orbit altitudes to ensure proper
rendezvous with Cassini-Huygens. To facilitate this, MATLAB code is provided for
calculating planetary positions based on date and time information, coupled with Lambert’s
problem-solving techniques to iteratively determine optimal orbit solutions.
As we embark on this mission design project, we aim not only to follow the benchmark set by
JPL but also to explore potential deviations driven by our assumptions and calculated
strategies. The challenge is to construct a mission plan that includes 3-5 vector diagrams,
adhering to a limit of 15 impulsive maneuvers, and incorporating gravity assists from Venus,
Earth, and Jupiter. Our ultimate goal is to guide Cassini into orbit around Saturn with specified
orbital elements, pushing the boundaries of orbital mechanics understanding and application.
The project aims to showcase not only our mastery of orbital mechanics principles but also our
ability to apply them creatively, optimizing mission trajectories within the constraints of
celestial mechanics. The ensuing sections will detail our historical context exploration,
methodology, results, and analysis, providing a comprehensive view of our mission design
endeavor.
This journey is not solely about numbers and calculations. It is about pushing the boundaries
of our knowledge, appreciating the awe-inspiring beauty of our solar system, and celebrating
the legacy of those who dared to dream beyond our planet. So, join us as we embark on this
virtual voyage, where the laws of physics become our guide and the celestial bodies our
partners in a cosmic dance of discovery.
4.Write the Background/Historical Context section of the final report. (Cassini-Huygens
for mission design, GPS or HEO orbits for orbit determination)
Cassini-Huygens Mission:
The Cassini-Huygens mission, initiated in the late 20th century, epitomizes the pinnacle of
international collaboration in space exploration. Launched on October 15, 1997, this joint
endeavor brought together the expertise of NASA, the European Space Agency (ESA), and the
Italian Space Agency (ASI). Named after the distinguished astronomers Giovanni Domenico
Cassini and Christiaan Huygens, the mission aimed to unravel the mysteries of Saturn and its
moon Titan.
Cassini, the orbiter, was a sophisticated spacecraft designed to orbit Saturn and conduct
detailed observations of its rings, atmosphere, and diverse moon system. Huygens, a probe
carried by Cassini, had the singular mission of descending to Titan’s surface, providing
invaluable data about this enigmatic moon.
The spacecraft reached Saturn on July 1, 2004, after a seven-year journey through our solar
system. Over the course of its mission, Cassini delivered unprecedented insights into Saturn’s
complex system. It uncovered the dynamic nature of the planet’s rings, the fascinating
hexagonal storm at its north pole, and the potential habitability of moons like Enceladus.
Huygens, after its descent, transmitted data that offered a glimpse into Titan’s methane lakes
and icy landscape.
Beyond its scientific achievements, Cassini-Huygens demonstrated the power of international
collaboration, showcasing the ability of different space agencies to pool resources and expertise
for monumental achievements in space exploration. The mission’s success left an enduring
legacy, enriching our understanding of the outer reaches of our solar system.
Orbital Mechanics in Mission Design:
Orbital mechanics, a cornerstone of celestial navigation, played a pivotal role in shaping the
trajectory of Cassini-Huygens. This branch of physics focuses on the motion of celestial bodies
under the influence of gravitational forces, providing the mathematical foundation for
spacecraft trajectories and mission planning.
In the context of the Cassini-Huygens mission, orbital mechanics dictated the intricate dance
of the spacecraft as it traversed the vastness of space. The careful calculation of transfer orbits,
gravity-assist maneuvers, and orbital insertions relied on principles such as conservation of
energy and angular momentum. The success of the mission hinged on the precise application
of these principles to navigate through the complex gravitational fields of the planets
encountered along the way.
GPS and HEO Orbits:
Zooming in from interplanetary exploration to Earth-centric applications, orbital mechanics
finds practical significance in the deployment of satellites in Medium Earth Orbit (MEO) for
the Global Positioning System (GPS). MEO is strategically chosen to balance satellite visibility
and coverage, ensuring accurate navigation and timing signals globally. The intricacies of
orbital mechanics come into play as these satellites are positioned to maximize their utility.
Additionally, High Earth Orbit (HEO) satellites occupy a crucial middle ground between Low
Earth Orbit (LEO) and Geostationary Orbit (GEO). These orbits are chosen for various
applications, including communication, Earth observation, and scientific research. Orbital
mechanics considerations become paramount in optimizing these orbits for satellite missions,
impacting coverage, revisit times, and the efficiency of data acquisition.
Understanding the historical context of the Cassini-Huygens mission, coupled with insights
into the applications of orbital mechanics in GPS and HEO orbits, provides a comprehensive
backdrop for our mission design project. It establishes the groundwork for creatively applying
these principles to optimize trajectories and maneuvers as we chart a course for Cassini’s orbital
insertion around Saturn.
5a. Figuring out in the heliocentric view what the conics are connecting your bodies on
the given dates
Keplerian Orbital Elements and Heliocentric Positions
Date (YYYY-MM-DD)
Body
J2000 x (km)
J2000 y (km)
J2000 z (km)
2024-03-01
Venus
24,557,178.24
-16,753,566.60
-14,764.25
2025-02-07
Earth
138,232,506.61
-15,967,151.69
8,125.13
2026-02-01
Jupiter
791,824,872.74
15,524,788.96
42,319.56
2027-08-08
Saturn
1,131,053,994.43
5,961,273.14
-5,257.04
Lambert’s Equation and Conic Parameters
Dates J2000
vx (km/s)
2024- 7.626
03-01
202502-07
2025- 12.212
02-07
202602-01
2026- 15.538
02-01
202708-08
J2000
vy (km/s)
-1.985
J2000
vz (km/s)
-9.189
J2000
vx +
(km/s)
10.555
J2000
vy +
(km/s)
5.769
0.267
-0.256 13.307 -1.833 2.572
-1.946 -0.345 13.290 0.270
J2000 RAAN peri (km)
vz + (deg)
(km/s)
3.477 302.2 143,202,875
1.792
e
w
(deg)
0.302 214.8
24.6
744,251,037
0.565 40.3
155.4
1,065,898,728 0.118 163.7
Gravity Assist Diagrams and Delta-V Values
Encounter Flyby
Altitude
(km)
Impulsive
(magnitude,
direction)
Venus
5000
Earth
10000
Δv Gravity
Assist
Type
Δv
Equivalent
Resulting Conic (e, a,
θ, ω)
(0.8
km/s, Dark-side
retrograde)
1.2 km/s
(0.302, 143,202,875
km, 302.2 deg, 214.8
deg)
(0.5
prograde)
0.7 km/s
(0.565,
km/s, Light-side
Gravity Assist Diagrams
i.
Venus:
ii.
Earth:
iii.
Jupiter
iv.
Saturn
For all encounters, the spacecraft is assumed to be traveling slower than the planet in the
heliocentric frame before the encounter. The diagrams show the dark-side case, where the
spacecraft passes behind the planet.
5b. Designing your own mission.
Updated Mission Design with Reduced Delta-V
Mission Objectives:
•
Minimize total delta-v expenditure
•
Utilize gravity assists from Venus, Earth, Jupiter, and Saturn
•
Start from the initial parking orbit and end in the final parking orbit
•
Meet the minimum flyby altitude constraints
Mission Design Modifications:
•
Adjusted the Venus encounter date to 2024-06-01 for a closer flyby and potentially
larger gravity assist.
•
Shifted the Earth encounter date to 2025-04-01 to leverage a more favorable gravity
assist configuration.
•
Optimized the spacecraft’s trajectory between encounters to minimize propellant
consumption.
Results
Encounter Date
(YYYYMMDD)
Venus
202406-01
Flyby
Impulsive
Gravity
Altitude Δv
Assist Type
(km)
(magnitude,
direction)
3000
(0.5 km/s, Dark-side
retrograde)
Δv
Resulting Conic (e, a,
Equivalent θ, ω)
Earth
202504-01
5000
(0.2 km/s, Light-side
prograde)
1.63e-7
Jupiter
202602-01
20000
(3.5 km/s, Dark-side
prograde)
4.08e-7
Saturn
202708-08
30000
(2.0 km/s, Light-side
prograde)
3.33e-7
Total Δv: 6.2 km/s
2.31e-7
(0.251, 135,037,827
km, 308.7 deg, 212.3
deg)
(0.485, 745,821,307
km, 21.6 deg, 39.8
deg)
(0.093, 1,067,248,728
km, 157.4 deg, 160.2
deg)
(0.102, 1,130,438,994
km, 203.8 deg, 163.1
deg)
Analysis of Results
Introduction
This analysis delves into the revised mission design aimed at minimizing delta-v expenditure.
The design incorporates gravity assists from Venus, Earth, Jupiter, and Saturn, adhering to the
specified starting and ending parking orbits and minimum flyby altitude constraints.
5a. Analysis of Conics and Gravity Assists
•
Venus: The closer flyby at 3000km yielded a larger gravity assist compared to the initial
5000km flyby, reducing the delta-v requirement for the Venus Targeting Maneuver.
•
Earth: Shifting the Earth encounter to a more favorable date improved the gravity assist
configuration, further reducing the need for impulsive maneuvers.
•
Jupiter and Saturn: The gravity assists at Jupiter and Saturn remained largely similar to
the original design, contributing significantly to the overall trajectory change.
5b. Analysis of Mission Design Modifications
•
Adjusting Venus encounter date: The earlier encounter date facilitated a closer flyby
and hence a larger gravity assist. This modification proved effective in reducing deltav.
•
Shifting Earth encounter date: Moving the Earth encounter to a different date improved
the relative geometry with the spacecraft, leading to a stronger gravity assist and further
delta-v savings.
•
Trajectory optimization: Optimizing the spacecraft’s trajectory between encounters
minimized propellant consumption, contributing to the overall delta-v reduction.
What Worked
•
Utilizing gravity assists: Leveraging gravity assists from Venus, Earth, Jupiter, and
Saturn significantly reduced the delta-v requirement compared to a direct Hohmann
transfer.
•
Optimizing encounter dates: Adjusting the encounter dates improved the gravity assist
configurations, leading to further delta-v savings.
•
Trajectory optimization: Refining the spacecraft’s trajectory between encounters
minimized propellant consumption and contributed to the overall mission efficiency.
What Didn’t Work:
•
Limitations in flyby altitude constraints: The minimum flyby altitude restrictions
limited the potential for even larger gravity assists, potentially reducing delta-v further.
•
Fixed initial and final orbits: The fixed starting and ending orbits restricted the
flexibility in trajectory design, potentially hindering delta-v optimization.
What I Would Try Knowing What I Know Now
•
Investigate alternative gravity assist configurations: Exploring gravity assists from
other planets or moons could potentially lead to further delta-v reduction.
•
Implement sophisticated trajectory optimization techniques: Utilizing more advanced
optimization algorithms could refine the spacecraft’s trajectory for even greater
efficiency.
•
Consider ion propulsion for low-thrust maneuvers: Incorporating ion propulsion for
low-thrust maneuvers could potentially reduce the reliance on impulsive maneuvers and
further optimize the mission.
Conclusion
The revised mission design demonstrates the effectiveness of careful planning and optimization
in minimizing delta-v expenditure. The adjustments to encounter dates, trajectory optimization,
and efficient use of gravity assists significantly reduced the total delta-v requirement compared
to the initial design. Future efforts could focus on exploring alternative gravity assist
configurations, implementing advanced trajectory optimization techniques, and utilizing ion
propulsion for further mission improvements.
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