In this document, I derive the equations of motion for an ion-thruster powered spacecraft and use numerical methods to calculate its optimal trajectory to Saturn. I did this work within 48 hours for the University Physics Competition in 2020.
This unit carry information of Acceleration Due to the Gravity (g), Satellite and Planetary Motion and Gravitational Field, Potential Energy, Kinetic Energy and Total energy of the satellite. in each section, there is an example so as you could be able to manipulate those equations that are associated with this unit. Also, there is problem practice so as to straighten the understanding of this module.
This unit carry information of Acceleration Due to the Gravity (g), Satellite and Planetary Motion and Gravitational Field, Potential Energy, Kinetic Energy and Total energy of the satellite. in each section, there is an example so as you could be able to manipulate those equations that are associated with this unit. Also, there is problem practice so as to straighten the understanding of this module.
A model for non-circular orbits derived from a two-step linearisation of the ...Premier Publishers
Β
In the Solar System most orbits are circular, but there are some exceptions. The paper addresses results from a two-step linearisation of the Kepler laws, to model non-circular orbits, at Newtonian gravity and other interactions with adjacent bodies. The orbit will then be characterised by a generalised eccentricity and a secondary frequency denoted L-frequency, ΟL (and considered proportional to the angular velocity). The path will be that of a circle, superimposed by small vibrations with the L-frequency. Hereby, the amplitude corresponds to an eccentricity, such that the radius varies, with time. When the ratio between the L-frequency and angular velocity is a non-integer, βperihelionβ moves. Bounds are derived and resulting orbits are generated and visualized.
For the integer ratio 2, results are compared with an ellipsoidal, and a tidal wave. For a non-integer ratio, the orbit is related to data for Mercury. Methods for detecting and measuring the secondary frequency are discussed, in terms of transfer orbits in Spaceflight dynamics.
Digital Library of GLT Saraswati Bal Mandir. Gravitation is a natural phenomenon by which all physical bodies attract each other. It is most commonly experienced as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped.
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
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As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the KΓ‘rmΓ‘n Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
A model for non-circular orbits derived from a two-step linearisation of the ...Premier Publishers
Β
In the Solar System most orbits are circular, but there are some exceptions. The paper addresses results from a two-step linearisation of the Kepler laws, to model non-circular orbits, at Newtonian gravity and other interactions with adjacent bodies. The orbit will then be characterised by a generalised eccentricity and a secondary frequency denoted L-frequency, ΟL (and considered proportional to the angular velocity). The path will be that of a circle, superimposed by small vibrations with the L-frequency. Hereby, the amplitude corresponds to an eccentricity, such that the radius varies, with time. When the ratio between the L-frequency and angular velocity is a non-integer, βperihelionβ moves. Bounds are derived and resulting orbits are generated and visualized.
For the integer ratio 2, results are compared with an ellipsoidal, and a tidal wave. For a non-integer ratio, the orbit is related to data for Mercury. Methods for detecting and measuring the secondary frequency are discussed, in terms of transfer orbits in Spaceflight dynamics.
Digital Library of GLT Saraswati Bal Mandir. Gravitation is a natural phenomenon by which all physical bodies attract each other. It is most commonly experienced as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped.
Why Does the Atmosphere Rotate? Trajectory of a desorbed moleculeJames Smith
Β
As a step toward understanding why the Earth's atmosphere "rotates" with the Earth, we use using Geometric (Clifford) Algebra to investigate the trajectory of a single molecule that desorbs vertically upward from the Equator, then falls back to Earth without colliding with any other molecules. Sample calculations are presented for a molecule whose vertical velocity is equal to the surface velocity of the Earth at the Equator (463 m/s) and for one with a vertical velocity three times as high. The latter velocity is sufficient for the molecule to reach the KΓ‘rmΓ‘n Line (100,000 m). We find that both molecules fall to Earth behind the point from which they desorbed: by 0.25 degrees of latitude for the higher vertical velocity, but by only 0.001 degrees for the lower.
A paper which analyses the motion of a satellite launch vehicle, a rocket, from the moment it is launched till when it is placed into orbit. The paper contains derivations for equations for thrust, mass, mass loss, distance, velocity, burnout time and burnout velocity
The Optimization of the Generalized Coplanar Impulsive Maneuvers (Two Impulse...paperpublications3
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Abstract: The orbit transfer problems using impulsive thrusters have attracted researchers for a long time [3]. One of the objectives in these problems is to find the optimal fuel orbit transfer between two orbits, generally inclined eccentric orbits. The optimal two-impulse orbit transfer problem poses multiple local optima, and classical optimization methods find only local optimum solution. McCue [7] solved the problem of optimal two-impulse orbit transfer using a combination between numerical search and steepest descent optimization procedures. The transfer of satellites in too high orbits as geosynchronous one (geostationary), usually is achieved firstly by launching the satellite in Low Earth Orbit (LEO) (Parking orbit), then in elliptical transfer orbit and finally to the final orbit (Working orbit). The three steps process is known as Hohmann transfer. The Hohmann transfer which involves two circular orbits with different orbital inclinations is known as nonβcoplanar Hohmann transfer. If both orbital planes are aligned the Hohmann transfer is known as coplanar what is further considered in this paper. In terms of propellant consumptions the Hohmann transfer is the best known transfer to be applied when transferring between elliptical coplanar orbits. For transfer between elliptical coplanar orbits, the given information usually consists of the altitude of perigee and apogee of the initial and the altitude of perigee and apogee of the final orbits. The velocity to be applied into two orbit points in order to attain the dedicated final orbit is analyzed.
The aim of this paper is compare between three types of coplanar impulsive transfer (two impulses, three impulses and one tangent burn) and conclude about the velocity changes for these types under relation between initial low Earth altitudes and final orbit. For the relation between initial orbit altitudes and final orbit altitude, the velocities to be applied in process of Hohmann transfer are simulated. From respective simulations, the velocity variations on dependence of this relation are derived. And the time of flight is considered too. The problem of spacecraft orbit transfer with minimum fuel consumption is considered, in terms of testing numerical solutions.
This presentation covers the two robots I designed for the VEXU robotics competition in 2016. I built and programed the robots with a team, and we competed in the regional VEXU robotics competition.
Research proposal: Thermoelectric cooling in electric vehicles KristopherKerames
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This research proposal describes the theory behind thermoelectric cooling (TEC) in the context of electric vehicle thermal management systems, and describes the experimental setup and error analysis required to study TEC in that context.
This is one of the presentations I gave with my team, in the 2020-2021 school year, to demonstrate the final design of the Titan Rover senior design project that I co-led. Titan Rover is a legacy senior design project. Its main objective is to create a version of the Mars Rover for the University Rover Challenge.
This document includes multiple volumes from the critical design review of the Titan Rover senior design project that I led. Each volume covers different subsystems of the rover. Volumes are organized as follows,
Pages 1-26: System Overview
Pages 27-61 : Technical Volume 1, Robotics sub-system
Pages 62-170: Technical Volume 2, Mobility sub-system
Pages 171-202: Technical Volume 3, Chassis sub-system
Pages 203-234: Technical Volume 4, Life-Detection sub-system
This critical design review reflects work completed on the Titan Rover under my leadership throughout the 2020-2021 school year at California State University, Fullerton.
Student information management system project report ii.pdfKamal Acharya
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Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologistβs survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Courier management system project report.pdfKamal Acharya
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It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
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AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with ο¬y-by-wire ο¬ight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
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This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
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Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Event Management System Vb Net Project Report.pdfKamal Acharya
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In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named βEvent Management Systemβ is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
β’ Remote control: Parallel or serial interface.
β’ Compatible with MAFI CCR system.
β’ Compatible with IDM8000 CCR.
β’ Compatible with Backplane mount serial communication.
β’ Compatible with commercial and Defence aviation CCR system.
β’ Remote control system for accessing CCR and allied system over serial or TCP.
β’ Indigenized local Support/presence in India.
β’ Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
β’ Remote control: Parallel or serial interface
β’ Compatible with MAFI CCR system
β’ Copatiable with IDM8000 CCR
β’ Compatible with Backplane mount serial communication.
β’ Compatible with commercial and Defence aviation CCR system.
β’ Remote control system for accessing CCR and allied system over serial or TCP.
β’ Indigenized local Support/presence in India.
Application
β’ Remote control: Parallel or serial interface.
β’ Compatible with MAFI CCR system.
β’ Compatible with IDM8000 CCR.
β’ Compatible with Backplane mount serial communication.
β’ Compatible with commercial and Defence aviation CCR system.
β’ Remote control system for accessing CCR and allied system over serial or TCP.
β’ Indigenized local Support/presence in India.
β’ Easy in configuration using DIP switches.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
Β
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Immunizing Image Classifiers Against Localized Adversary Attacks
Β
Optimal trajectory to Saturn in ion-thruster powered spacecraft
1. Problem A:β¨Optimal Trajectory to Saturn for Ion-Thruster
Powered Spacecraft
Kristopher Kerames
November 2020
Figure 1. Spacecraft Propelled by an Ion Thruster [2]
2. Kerames 2
Abstract
Ion propulsion uses too low a thrust to be able to perform Hohmann transfer like most rocket-
powered spacecraft do. Instead of solving for a Hohmann transfer trajectory, a system of second
order partial differential equations were derived to define the required motion of the spacecraft
from Eulerβs laws of motion. Numerical methods were used to solve those equations. A constant
force of thrust was used throughout the entire trajectory. It was found that the most fuel-efficient
trajectory involved using a constant thrust of 0.1N, despite that the spacecraft was capable of
0.4N of thrust. This trajectory took 25.9 years, and only used 2,085kg of fuel. A 7.5 year
trajectory was the shortest one calculated and could be achieved using only about 330kg more
fuel than the most fuel-efficient trajectory. Some of the assumptions made in the development of
the solution prevented the final trajectory from being as efficient as it could be. The main ways
in which the solution may have been improved would be by using elliptical rather than circular
orbits, and using a gravity assist maneuver.
3. Kerames 3
Problem Definition
In order to travel from Earth to Saturn, a propulsion system is needed. The problem states
that an Ion Thruster spacecraft is in a circular orbit around the Earth with an orbital period of 90
minutes and needs to enter a circular orbit around Saturn with an orbital period of 40 hours. The
spacecraft has relatively low thrust in comparison to chemical propellants. The ion thrusters can
produce 400mN of thrust and have a specific impulse of 4000 seconds. The minimum amount of
fuel required, duration and control strategy for the thruster need to be analyzed to reach Saturn in
the most fuel efficient manner possible.
Analysis
Assumptions
Several assumptions were made in this analysis. The first assumption is that Saturn and
Earth orbit in the same plane. It was found that Saturnβs and Earthβs orbital planes only varied by
2.485Β° In reality, planetary orbits are elliptical however, in order to simplify the problem, the
assumption was made that planetary orbits are circular due to their low eccentricity. Earthβs
eccentricity is 0.0167 and Saturnβs eccentricity is 0.0565 according to the Goddard Space Flight
Center [1]. Another assumption is that the Sun was fixed and did not move with respect to time.
The Sun was used as a fixed reference point. Furthermore, the assumption was made that
standard gravity is 9.8 m/s2 Standard gravity is used in our specific impulse calculations and by
convention is 9.8m/s2
. The next assumption is that the trajectory is free from obstructions and
gravity other than that of Earth and Saturn. For instance, it was assumed that the spacecraft
would not get hit by an asteroid in the asteroid belt and that the spacecraft would be launched at
a position where gravitational influences from other planets would be small along its trajectory.
The effects of general relativity were also disregarded. The final assumption made is that each
planet can be approximated as a point particle, containing all of the planetβs mass, positioned at
their center of mass.
General Approach
The general approach is to use Eulerβs laws of motion to write a governing equation for
acceleration of the spacecraft. This equation is a non-linear, two-dimensional differential
equation. Thus, a numerical method was used to calculate the time it would take to reach Saturn
with the speed that results in an orbital period of 40 hours. Namely, Eulerβs method was used.
This governing equation accounts for the varying phase angle between the planets, and varying
thrust. The magnitude of thrust will remain constant throughout the trajectory, but will be varied
in each iteration. By leaving these quantities variable, the launch point and thrust force that
produce the most efficient trajectory can also be found. The most efficient way to use fuel in a
low-power thruster is to minimize thrust while keeping it constant. For this reason, FT will be
varied [6]. Given the thrust of 400mN (0.4N) that the spacecraft is capable of, the thrust will be
varied from 0β0.4N.
Generally, Hohmann transfer is used to launch rockets to other planets. However, a large
amount of thrust is required during Hohmann transfer in order to propel the spacecraft to escape
velocity, then later slow it down causing it to trace the orbit of the destination planet. The ion
thruster produces relatively low thrust. Therefore, the thruster will need to be left on for a longer
duration, rather than applying quick pulse of thrust that propels the spacecraft to escape velocity.
4. Kerames 4
Therefore, escape velocity will not need to be reached. Instead, the spacecraft will accelerate
continuously, spiraling away from the Sun until it reaches the final desired position and final
speed, Vdest. The spacecraft will be launched at the point where its tangential velocity, with
respect to Earthβs center, is in the same direction as the tangential velocity of Earthβs center. This
is the point where the craft is moving fastest with respect to the Sun, and its kinetic energy is
maximized.
Derivation of Governing Equation
Figure 2. Free Body Diagram of the Spacecraft.
The specific impulse in seconds can be defined by the following equation:
(1)
where πΉ! is the thrust force in Newtons (N),β¨π" = 9.8 π/π 2 is standard gravity. πΌ#$ is the
specific impulse measured in seconds. π
Μ is the mass flow rate in kg/s.
By dividing both sides of the equation above by π
Μ , the exhaust velocity, π£%& is obtained:
(2)
The thrust force can be expressed as the product of the mass flow rate, mΜ and exhaust velocity:
(3)
πΉ
β! = βπ" β πΌ#$ β π
Μ
π£%& = πΌ#$ β π"
πΉ
β! = βπ
Μ β π£%&
5. Kerames 5
After substituting equation (2) into (3) and plugging in the given values for πΌ#$ and π", the mass
flow rate can be expressed as:
(4)
Where m(t) represents mass as a function of time. In order to find m(t), we integrate the mass
flow rate over time:
(5)
In addition, the rocket equation including external forces and constant thrust is:
(6)
Where a(t) is acceleration as a function of time, and β πΉ is the sum of external forces on the
rocket, which also vary with time. From equations (3) and (6), we get:
(7)
Where π
4β#'(π‘) is the position vector of the spacecraft with respect to the Sun.
Aside from thrust, the external forces on the spacecraft are only the forces of gravity due to Sun,
Earth, and Saturn.β¨The equation for the force due to gravity is:
(8)
Where:β¨G=6.674x10-11
Nm2
/kg2
, M is the mass of the gravitational body, r is the magnitude of
the position vector of the gravitational body with respect to the spacecraft, and πΜ is the unit
vector corresponding to r. The masses for M will be defined as follows:
β’ Mass of Earth: Me = 5.9724 E24 kg
β’ Mass of Saturn: Ms = 568.34 E24 kg
β’ Mass of Saturn: Msun = 1.9885 E30 kg
[3] [4] [5]
To obtain the position vectors of the planets with respect to the moving spacecraft, we will
define those vectors in terms of their positions in a fixed Cartesian coordinate system where the
Sun is fixed at the origin.
π
Μ =
ππ(π‘)
ππ‘
=
βπΉ
β!
39,200
> ππ
((*)
,"""
= >
βπΉ
β!
39,200
ππ‘
*!
"
π(π‘) = 5000 β
πΉ
β!
39,200
@ πΉ
β β π
Μ β π£%& = π(π‘) β π
β(π‘)
π
β(π‘) =
π-
π
4β#'(π‘)
ππ‘-
=
β πΉ
β β πΉ
β!
π(π‘)
πΉ
. = πΊ
ππ(π‘)
π-
πΜ
6. Kerames 6
The planetsβ average distances relative to the Sun (semi-major axis) are defined as follows:
β’ Earthβs semi-major axis Re = 149.6 E9 m
β’ Saturnβs semi-major axis: Rs = 1,433.5 E9 m
First, the positions of each object relative to the Sun must be defined:
Figure 3. Angle Diagram. Figure is not to scale.
π
4β# , π
4β% , π
4β#/0, and π
4β#' are the position vectors of Saturn, Earth, Sun, and the spaceship with
respect to the Sun, respectively. Let ΞΈi = 0Β° be the position of the Earth when the spacecraft is
launched. π is the phase angle between Earth and Saturn. Ξπ is the change in angle of
Saturn over some time interval.
The angular speeds of Earth and Saturn about the Sun are Οe and Οs, respectively. These can be
related to ΞΈ and Ξπ with:
π
4β# and π
4β% can be rewritten in terms of ΞΈ and Ξπ. The position vectors become:
(9)
π
4β#
π
4β%
Οet= ΞΈ and Οst= Ξπ
π
4β% = π % cos(Ο%π‘) π€Μ + π % sin(Ο%π‘) π₯Μ
π
4β# = π # cos(Ο#π‘ + Ο) π€Μ + π # sin(Ο#π‘ + Ο) π₯Μ
π
4β#/0 = 0π€Μ + 0π₯Μ
π
4β#' = π₯π€Μ + π¦π₯Μ
7. Kerames 7
Earthβs angular speed of revolution is
Οe = 2 * Ο / T e
Similarly, Saturnβs revolution speed is,
Οe = 2 * Ο / T s
Whereβ¨T e = 365.25 days is the period of Earthβs rotation around the Sun, and T s = 29.46 years
is the period of Saturnβs rotation around the Sun.
Now, position vectors of the gravitational bodies with respect to the spacecraft can be developed.
These will be used to describe the forces acting on the spacecraft.
Figure 4. Position Vector Diagram (not to scale). The spacecraft is drawn in a moving, body-
fixed coordinate system.
π
4β%/#' , π
4β#/#' , and π
4β#/0/#' are the position vectors of Earth, Saturn, and the Sun with respect to
the spacecraft, respectively. They can be written in terms of the other vectors:
π
4β#
π
4β%
π
4β#'
π
4β#/#'
π
4β%/#'
8. Kerames 8
(10)
By plugging in equation set (9) into equation set (10) we get equations for the position vectors,
with respect to the craftβs body-fixed coordinate system, in terms of x and y:
(11)
The net forces on the craft can now be written in terms of these vectors. In general, from
equation (8),
πΉ
. = πΊ
ππ(π‘)
π-
πΜ
r can be calculated as eπ
4βe, where π
4β is a general variable to represent each of the position vectors
in equation set (10). πΜ can be calculated as
2
3β
52
3β5
Thus, Fg for each gravitational body becomes:
(12)
Where eπ
4βe = fπ &
-
+ π 6
-
β eπ
4βe
8
= hπ &
-
+ π 6
-
i
8/-
π
4β%/#' = π
4β% β π
4β#'
π
4β#/#' = π
4β# β π
4β#'
π
4β#/0/#' = βπ
4β#'
π
4β%/#' = (π % cos(Ο%π‘) β π₯)π€Μ + (π % sin(Ο%π‘) β π¦) π₯Μ
π
4β#/#' = (π # cos(Ο#π‘ + Ο) β π₯) π€Μ + (π # sin(Ο#π‘ + Ο) β π¦) π₯Μ
π
4β#/0/#' = βπ₯π€Μ β π¦π₯Μ
πΜ
π-
=
π
4β
eπ
4βe
Γ· eπ
4βe -
=
π
4β
eπ
4βe
8
πΉ
. = πΊππ(π‘)
π
4β
eπ
4βe
8
9. Kerames 9
Equation (7) can now be rewritten as:
(13)
As this equation is two-dimensional, it must be split into two partial differential equations in the
x and y directions. If the spacecraft is always firing its ion thruster in the direction of its velocity,
then πΉ
β&! and πΉ
β6! can be defined as:
(14)
Complete Set of Governing Equations
By plugging in equations (14), (11), and (5) into (13), the final set of governing equations
becomes:
(15)
π-
π
4β#'(π‘)
ππ‘-
=
β πΉ
β + πΉ
β!
π(π‘)
=
πΊπ%π(π‘)
π
4β%/#'
eπ
4β%/#'e
8 + πΊπ#π(π‘)
π
4β#/#'
eπ
4β#/#'e
8 + πΊπ#/0π(π‘)
π
4β#/0/#'
eπ
4β#/0/#'e
8 + πΉ
β!
π(π‘)
= πΊ k
π%
eπ
4β%/#'e
8 π
4β%/#' +
π#
eπ
4β#/#'e
8 π
4β#/#' +
π#/0
eπ
4β#/0/#'e
8 π
4β#/0/#'l +
πΉ
β!
π(π‘)
πΉ
β&! = πΉ!
ππ₯
ππ‘
f(
ππ₯
ππ‘
)- + (
ππ¦
ππ‘
)-
πΉ
β6! = πΉ!
ππ¦
ππ‘
f(
ππ₯
ππ‘
)- + (
ππ¦
ππ‘
)-
π"
π₯
ππ‘"
= πΊ &
π#(π # cos(Ο#π‘) β π₯)
0π
1β#/%&0
' +
π%(π % cos(Ο%π‘ + Ο) β π₯)
0π
1β%/%&0
' +
π%()(βπ₯)
0π
1β%()/%&0
'5 +
πΉ
11β
π₯π
5000 β
πΉ*
39,200
π"
π¦
ππ‘"
= πΊ &
π#(π # cos(Ο#π‘) β π¦)
0π
1β#/%&0
' +
π%(π % cos(Ο%π‘ + Ο) β π¦)
0π
1β%/%&0
' +
π%()(βπ¦)
0π
1β%()/%&0
'5 +
πΉ
11β
π¦π
5000 β
πΉ*
39,200
10. Kerames 10
Results
MATLAB was used to perform Eulerβs method using the governing equations. A
convergence study was conducted and revealed that a 300-second time increment was sufficient
for convergence of results. The minimum amount of fuel necessary to arrive at Saturn with the
desired velocity is 2,085kg of fuel. This is equal to 41.7% of the rocketβs initial mass. The
magnitude of constant thrust required was 0.1N. The phase angle between the Earth and Saturn
was -13Ο/30 radians. It took 25.9 years to arrive at Saturn given these conditions. Alternative
conditions resulted in trajectories as short as 7.5 years using 0.4N of thrust, and consuming
2,415kg of fuel. A graph of the optimal trajectory can be seen below (Fig. 5). The speed of the
craft increases as it approaches Saturn indicating that it is using the planetβs momentum to
increase its speed (Fig. 6). The distance of the craft from the Sun increases at an accelerated rate
over time as spirals get larger (Fig. 7).
Figure 5. Graph of most efficient trajectory. The orange dot is the sun. The yellow dot is Saturn.
The blue line traces the trajectory of the spacecraft.
11. Kerames 11
Figure 6. Speed of the craft over time.
Figure 7. Distance of the craft from the Sun over time.
12. Kerames 12
Conclusion
The most efficient trajectory was found to use only 2,085kg of fuel. However another
trajectory was found to last 7.5 years using only 330kg more fuel than the most efficient
trajectory did. This shorter trajectory may be more practical, as time constraints may outweigh
the cost of fuel in some contexts. Results confirmed that keeping the ion thrusters on a low thrust
for the entire duration of the trip was the most efficient way to travel.
Results may have been improved by avoiding some of the assumptions that were made.
Using elliptical orbits, rather than circular, would have improved the accuracy of results.
Depending on how the elliptical orbits are oriented, the shortest distance between them is
considerably shorter than that of circular orbits. This could have reduced the overall distance of
the trip, and saved fuel. Another thing that would have made the most material impact on results
would have been to use a gravity assist maneuver. Mars and Jupiter are positioned such that they
could be used for this maneuver. That would have allowed the spacecraft to use some of those
planetsβ angular momenta in order to speed itself up without using additional fuel. By adding the
forces due to Marβs and Jupiterβs gravity to the governing equations, and performing more phase
angle iterations in MATLAB, the optimal trajectory using gravity assist could be found.
13. Kerames 13
References
[1] Bombardelli, C., BaΓΉ, G. & PelΓ‘ez, J. Asymptotic solution for the two-body problem
with constant tangential thrust acceleration. Celest Mech Dyn Astr 110, 239β256
(2011). https://doi-org.lib-proxy.fullerton.edu/10.1007/s10569-011-9353-3 β¨
[2] Dunbar, B. (n.d.). Ion Propulsion: Farther, Faster, Cheaper. Retrieved November 08,
2020, from http://www.nasa.gov/centers/glenn/technology/Ion_Propulsion1.html β¨
[3] Earth Fact Sheet. (n.d.). Retrieved November 08, 2020, from
https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html β¨
[4] Planetary Fact Sheet. (n.d.). Retrieved November 08, 2020, from
https://nssdc.gsfc.nasa.gov/planetary/factsheet/ β¨
[5] Saturn Fact Sheet. (n.d.). Retrieved November 08, 2020, from
https://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html β¨
[6] S. X. staff, βResearcher calculates optimal trajectories to Mars and Mercury for a
spacecraft with electric propulsion,β Phys.org, 12-Sep-2019. Available:
https://phys.org/news/2019-09-optimal-trajectories-mars-mercury-spacecraft.html.
[Accessed: 08-Nov-2020]. β¨
14. Kerames 14
Appendix
MATLAB Code:
%Author: Team 124
%This script uses Euler's method to solve a governing equation for the
%trajectory of a spacecraft, powered by an ion thruster, traveling
%from Earth to Saturn.
clc; clear; close;
%The following lines store the required constants.
Re=149.6e9; %[m] Average distance from Sun to Earth
Rs=1433.5e9; %[m] Average distance from Sun to Saturn
Vsrev=9680.678558; %[m/s] Velocity with which Saturn revolves around Sun
we=1.991e-7; %[rad/s] Angular velocity of Earth about the Sun
ws=6.7584e-9; %[rad/s] Angular velocity of Saturn about the Sun
Me=5.9724e24; %[kg] Mass of Earth
Ms=568.34e24; %[kg] Mass of Saturn
Msun=1.9885e30; %[kg] Mass of Sun
Vdest=21509.10786; %[m/s] Desired velocity when entering Saturn's orbit
G=6.674e-11; %[Nm^2/kg^2] Gravitational constant
phi=(-78)*pi/180:-pi/3500:-15175*pi/35000; % [rad] This varies the phase
angle, phi,
%for each trajectory.
FT=0.1:0.1:0.4; %0:0.1:0.4; [N] This varies the constant thrust for each
trajectory
dt=300; %[s] This is the time increment of ten days
tend=25.9*365.25*24*3600; %[s] This is the max possible time for spaceship's
trajectory
t=0:dt:tend; %[s] Varies time
%Turns variables into matrices with as many elements as time steps to
%store values from each iteration.
x=zeros(1,length(t));
y=zeros(1,length(t));
Vx=zeros(1,length(t));
Vy=zeros(1,length(t));
Vmag=zeros(1,length(t));
xddot=zeros(1,length(t));
yddot=zeros(1,length(t));
FTx=zeros(1,length(t));
FTy=zeros(1,length(t));
Msc=zeros(1,length(t));
Rmag=zeros(1,length(t));
for l=FT
%The following are initial conditions
X0=1.4961e11; %[m] X component of initial position
Y0=0; %[m] Y component of initial position
Vx0=0; %[m/s] X component of initial velocity
Vy0=we*Re+7740.584; %[m/s] Y component of initial velocity. Made up of
%tangential speed of Earth with respect to Sun plus tangential speed of
%spacecraft with respect to Earth.
FTx0=0; %[N] x component of initial thrust
FTy0=l; %[N] y component of initial thrust
Msc0=5000; %[kg] Initial mass of spacecraft
Rmag0=(X0^2+Y0^2)^(1/2);
15. Kerames 15
%Stores initial values in matrices
x(1)=X0;
y(1)=Y0;
Vx(1)=Vx0;
Vy(1)=Vy0;
FTx(1)=FTx0;
FTy(1)=FTy0;
Msc(1)=Msc0;
Vmag(1)=Vy0;
Rmag(1)=Rmag0;
%The following for loop iterates through different values of t to find the
%one that maximizes final mass.
for k=phi
for j=2:length(t)
if Rmag(j)<Rs
n=(j-2)*dt; %This is the time elapsed
Rxsunsc=-x(j-1); %x coordinate of Sun (sun) position with respect to
spacecraft (sc)
Rysunsc=-y(j-1); %y coordinate of Sun (sun) position with respect to
spacecraft (sc)
Rxesc=Re*cos(we*n)-x(j-1); %x coordinate of Earth (e) position with respect
to spacecraft (sc)
Ryesc=Re*sin(we*n)-y(j-1); %y coordinate of Earth (e) position with respect
to spacecraft (sc)
Rxssc=Rs*cos(ws*n+k)-x(j-1); %x coordinate of Saturn (s) position with
respect to spacecraft (sc)
Ryssc=Rs*sin(ws*n+k)-y(j-1); %y coordinate of Saturn (s) position with
respect to spacecraft (sc)
Mag3Rsunsc=(Rxsunsc^2+Rysunsc^2)^(3/2); %Cubed magnitude of position of
%Sun (sun) with respect to spacecraft (sc)
Mag3Resc=(Rxesc^2+Ryesc^2)^(3/2); %Cubed magnitude of position of
%Earth (e) with respect to spacecraft (sc)
Mag3Rssc=(Rxssc^2+Ryssc^2)^(3/2); %Cubed magnitude of position of
%Saturn (s) with respect to spacecraft (sc)
%The following equation is the governing equation for motion in
%x-direction. xddot is the x component of acceleration of the spacecraft.
xddot(j-1)=G*(Me*Rxesc/Mag3Resc+Ms*Rxssc/Mag3Rssc+Msun*Rxsunsc/Mag3Rsunsc)...
+FTx(j-1)/Msc(j-1);
%Similarly, the equation in the y-direction is below.
yddot(j-1)=G*(Me*Ryesc/Mag3Resc+Ms*Ryssc/Mag3Rssc+Msun*Rysunsc/Mag3Rsunsc)...
+FTy(j-1)/Msc(j-1);
%The following are equations for velocity, position, and thrust respectively.
Vx(j)=Vx(j-1)+xddot(j-1)*dt;
Vy(j)=Vy(j-1)+yddot(j-1)*dt;
x(j)=x(j-1)+Vx(j-1)*dt;
y(j)=y(j-1)+Vy(j-1)*dt;
Vmag(j)=(Vx(j)^2+Vy(j)^2)^(1/2);
Msc(j)=Msc0-l*n/39200;
Rmag(j)=(x(j)^2+y(j)^2)^(1/2);
FTx(j)=l*Vx(j-1)/Vmag(j-1);
FTy(j)=l*Vy(j-1)/Vmag(j-1);
end
end
plot(x, y); %Plots spacecraft's trajectory around sun, and final position of
Saturn.
hold on