ASTE 520 Spacecraft Design Section 00, Part 1
ASTE 520
Spacecraft Design Fall 2019
Mike Gruntman
Department of Astronautical Engineering Viterbi School of Engineering University of Southern California Los Angeles
1997–2019 by Mike Gruntman 1/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Spacecraft Design, Fall 2019 (set of notes on spacecraft design) Mike Gruntman, 2019
Copyright 1997–2019 by Mike Gruntman
All rights reserved
No part of these materials may be reproduced, in any form or by any means, or utilized, in any form or by any means, by any information storage and retrieval system, without the written permission of the author.
1997–2019 by Mike Gruntman 2/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
ASTE 520 Spacecraft Systems Design Required for Astronautical Engineering Regardless of your engineering or science major (electrical, mechanical, aerospace, systems, computer, etc. or physics, astronomy, chemistry, math, etc.) and regardless of your job function (research, development, design, test, manufacturing, management, marketing, etc.) – if you work or plan/desire to work in the space/defense industry or in government space R&D centers or in space operations, then ... This is a course (on space systems) that you must take. ASTE520 focuses on fundamentals of space systems. It will help you to put into perspective your area of specialization and enable professional communications with other subsystem specialists. The course is popular at USC. It is among largest graduate space systems and space technology courses in the United States, with more than 1260 students during the last 12 years alone.
Academic year 2019–2020 ASTE520 Spacecraft Systems Design is offered only in the fall (2019) semester. Fall 2019 Class enrollment is unlimited.
For students enrolled in the class: Course materials for ASTE-520 Spacecraft Systems Design will be posted on the class web site at DEN around August 14, 2018. Spring 2020 Course not offered
ASTE-520 public web site ( http://astronauticsnow.com/aste520/ ) provides information on the syllabus, textbooks, and much more.
1997–2019 by Mike Gruntman 3/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Spacecraft Design – ASTE 520 Thursday, 6:40–9:20 p.m., OHE-100D Fall 2019
W&E&P New SMAD HW Class Date Subject NS (L&W SMAD) Due Chapters Organization of the class. 0 1 Aug 29 History of rocketry and space (self study). 1 1 Universe, galaxy, solar system. 2
2 Sep 05 Space environment. 3 7 (8) 1,2
3 Sep 12 Orbital mechanics. 4 9 (5,6,7) 3,4,5
Orbital mechanics. 4 8,9 (5,6,7) 4 Sep 19 6, 7, 8 Space mission geometry. 5 8,9 (5,6,7)
Space mission geometry. 5 8 (5) 5 Sep 26 9,10,11 Attitude determination and control (ADC). 7 19 (10,11)
6 Oct 03 Attitude determination and control (ADC). 7 19 (10,11) 12,13,14
7 Oct 10 MID-TERM 7:00–9:00 p.m., on campus, room TBD
Spacecraft and mission design overview. Fall Fall 1,3,4,6,14,24,28,29 Facilities. Operations. Reliability 6 recess recess (1,3,4,10,14,15,19) watch lecture NS-6; no HW; 2-day recess
8 Oct 24 Spacecraft Propulsion. 8 18 (17,18) 15,16,17,18
Launch systems. 9 26,27 (17,18) 9 Oct 31 19,20 Communications 10 16,21 (10,11,13)
16,21 10 Nov 07 Communications 10 21,22 (10,11,13,16)
11 Nov 14 Electric Power systems. 11 21 (10,11) 23,24,25
12 Nov 21 Thermal control. 12 22 (10,11) 26,27
13 Dec 05 Structures and mechanisms 13 22 (10,11) 28,29,30
Dec 12 FINAL EXAM 7:00–8:30 p.m., on campus, room TBD
W&E&P New SMAD = Wertz, Everett, Puschell, The New SMAD L&W SMAD = Larson and Wertz, SMAD NS = Notes Section; HW = homework
1997–2019 by Mike Gruntman 4/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
ASTE 520 Spacecraft Design Notes Contents
Section 0, Part 1 and Part 2 spacecraft distribution at Organization of the Class LEO and GEO atmospheric drag syllabus atomic oxygen contents spacecraft charging organization of the class trapped radiation homework, exams, grading radiation shielding books, WWWeb resources South Atlantic Anomaly survey cosmic rays micrometeoroids and debris Section HW Home Work problems Section 04 Orbital Mechanics Section 01 Brief History motion in gravitational field two-body approximation Section 02 elliptical orbit Universe, Galaxy, solar system circular and escape velocities Universe classical orbital elements Galaxy hyperbolic excess velocity solar system orbital transfers planets Hohmann transfer coordinate systems orbit plane change time rocket launch
launch sites Section 03 transfer to geostationary Space environment orbit
gravity-assist maneuvers Sun orbit perturbations solar cycle sun-synchronous orbit plasma Molniya orbit solar wind eclipses Earth’s magnetic field geosynchronous orbit atmosphere Hill’s equations ionosphere libration points magnetosphere The "Big One" geomagnetic storm
1997–2019 by Mike Gruntman 5/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Section 05 control loop Mission geometry attitude representation disturbance torques coordinate systems magnetic torques satellite horizon gravity gradient / stability swath aerodynamic torques satellite ground track solar radiation torques coverage, repeating orbits attitude measurements satellite constellations magnetometers sun sensors horizon sensors star sensors Section 06 inertial-measurement units Spacecraft and Mission Design (IMU) Overview. Operations. Reliability gyroscopes (rate,
integrating, laser) space missions angular momentum national security space nutation damping mission life cycle spacecraft attitude control reviews configurations (spin, mission design and reaction wheels, .....) planning stability of spinners flowdown of requirements examples of problems technology readiness levels (bang-bang, precession (TRL) control) system engineering reaction/momentum wheel learning from mistakes example of problems science payloads (reaction/momentum ground stations wheels) Deep Space network (DSN) Global Positioning System STDN, SGLS, SLR (GPS) TDRSS reliability mean time between failures (MTBF) failure rates Section 08 Spacecraft Propulsion
propulsion requirements Section 07 rocket classification Spacecraft Attitude Determination thrust, specific impulse, total and Control impulse rocket equation ADCS multistaging configuration constraints
1997–2019 by Mike Gruntman 6/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
flow parameters in the Section 10 nozzles Communications nozzles and diffusers thrust coefficient, frequency bands characteristic velocity evolution of nozzles communications satellites rocket heat transfer and communications cooling architecture propellant feed systems constraints on spacecraft propellant tanks systems propellant expulsion decibel language liquid propellants electromagnetic waves monopropellants and EIRP monopropellant thrusters antenna gain gelled propellants antennas solid rockets beam patterns grains polarization loss thrust vector control digital vs. analog electric propulsion Nyquist’s and Carson’s electrostatic thrusters rules field emission electric sampling and digitization propulsion (FEEP) and data rate colloid thrusters Shannon’s Law resistojet origins of noise arcjet noise figure magnetoplasmadynamic link design thrusters (MPD) effect of atmosphere, rain, snow modulation encoding BER Section 09 multiplexing Launch Systems multiple access techniques data compression launch issues telemetry functions selection process transponder Atlas family telemetry data formats Delta family) data handling ground data systems
space computer systems
1997–2019 by Mike Gruntman 7/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Section 11 Electric Power Systems Section 13 Structures and Mechanisms requirements classification purposes subsystem elements sources of structural loads design drivers launch orbital effects acoustic environment bus voltage basics of mechanics of solar cells materials solar arrays flexible-body dynamics batteries finite-element analysis nickel-cadmium batteries materials nickel-hydrogen batteries primary structure power processing examples spacecraft weight estimate tests and simulations
Section 12 Thermal Control
thermal control problems environmental loads blackbody concept Planck and Stefan- Boltzmann Laws thermal control components coatings multilayer insulation (MLI) heaters louvers heat pipes radiators thermal analysis thermal testing
1997–2019 by Mike Gruntman 8/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
ASTE 520 Spacecraft Design Fall 2019 Home Work Schedule
Problem due date 1 09/05/2019 2 09/05/2019 3 09/12/2019 4 09/12/2019 5 09/12/2019 6 09/19/2019 7 09/19/2019 8 09/19/2019 9 09/26/2019 10 09/26/2019 11 09/26/2019 12 10/03/2019 13 10/03/2019 14 10/03/2019 no HW – midterm exam 10/10/2019 15 10/24/2019 16 10/24/2019 17 10/24/2019 18 10/24/2019 19 10/31/2019 20 10/31/2019 21 11/07/2019 22 11/07/2019 23 11/14/2019 24 11/14/2019 25 11/14/2019 26 11/21/2019 27 11/21/2019 28 12/05/2019 29 12/05/2019 30 12/05/2019
1997–2018 by Mike Gruntman 9/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Mike Gruntman, Professor of Astronautics tel. 213–740–5536 Department of Astronautical Engineering, MC–1192 [email protected] University of Southern California http://astronauticsnow.com Los Angeles, California 90089–1192 Mike Gruntman was graduated (M.Sc.) from the Department of Aerophysics and Space Research of the Moscow Physical-Technical Institute in 1977 and received his Ph.D. in physics from the Space Research Institute (IKI) of the USSR Academy of Sciences in 1984. He received specialized training in servicing liquid rocket engines. Dr. Gruntman actively worked on the development of space technology, in particular novel instrumentation for laboratory and space applications, and conducted research in experimental and space physics. He has been especially active in the development of imaging photon-counting detectors for ground and space telescopes. Gruntman excelled in the study of the neutral components of space plasmas and developed new instrumentation for detection of energetic neutral atoms. He was a visiting scientist at the FOM-Institute for Atomic and Molecular Physics in Amsterdam. In March 1990, Gruntman joined the University of Southern California (USC), where he initially worked on reduction and evaluation of the data from Pioneer 10/11 spacecraft and actively participated in sounding rocket and space instrument development programs. He worked on the sounding rocket payload integration and testing at White Sands Missile Range. Dr. Gruntman is Professor of Astronautics at USC since 1993. (He is also Professor of Aerospace Engineering and Professor of System Architecture Engineering.) He was/is Principal Investigator and Co-Investigator in theoretical and experimental programs funded by NASA and Air Force; he is Co-Investigator on current NASA missions TWINS and IBEX. His interests include astronautics, space mission and spacecraft design, satellite technologies, rocket and spacecraft propulsion, space sensors and instrumentation, local interstellar medium, heliospheric and magnetospheric physics, space plasmas and environment, particle and photon analyzers and detector systems, ion and neutral particle beams, atomic collisions, and particle interactions with surfaces. Dr. Gruntman authored and co-authored 300+ scholarly publications, including four books, in various fields of astronautics, space technology, space physics, space and laboratory sensors and instrumentation, history of rocketry, spacecraft, and missile defense, and space education. He presented results of his research at numerous international and national scientific and technological conferences and symposia and at scientific seminars at leading American and foreign research institutions and universities. Prof. Gruntman taught/teaches courses in astronautics, spacecraft design, spacecraft propulsion, and space sciences. He also teaches short courses on space technology for government and industry. Prof. Gruntman is the founder of the USC Astronautics Program that today offers BS, BS Minor, MS, Engineer, and PhD degrees and Graduate Certificate in Astronautical Engineering. In August 2004, Dr. Mike Gruntman was appointed the first (founding) Chairman (2004–2007) of the new space-engineering-focused (unique for American universities) academic unit in the USC Viterbi School of Engineering, known today as the Department of Astronautical Engineering (ASTE). He again served as the department chairman from 2016-2019. Gruntman is Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA) and he served as Vice Chair (elected) for Education of the Los Angeles Section of AIAA from 1996–1998. He is Member (Academician) of the International Academy of Astronautics (IAA). Dr. Gruntman is also a member of the American Physical Society (APS) and American Geophysical Union (AGU). Gruntman is a recipient of NASA’s Group Achievement Awards (2000, 2001, 2011) and the USC School of Engineering Exceptional Service Award (1999). He served (elected) on the USC Engineering Faculty Council in 1996–1998, 1998–2000, 2008–2010, 2011–2012, 2014–2016. In 2006, his AIAA-published book on history of rocketry and spacecraft received an award from the International Academy of Astronautics. Mike served (2001–2003) on the editorial board of the world leading journal on experimental techniques and scientific instrumentation, the Review of Scientific Instruments. He reviews manuscripts for several scientific journals, for book publishers, and for NASA. He organized (convened) sessions at major scientific conferences (AGU, COSPAR). Dr. Gruntman served/serves on advisory panels on science and technology programs at NASA Headquarters, NASA centers, and in other government agencies.
1997–2018 by Mike Gruntman 10/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Department of Astronautical Engineering (ASTE)
In August 2004, Dean of Engineering (later USC President) Prof. Max Nikias established a new independent academic unit to take the full advantage of rapidly growing opportunities in space exploration and technology. This Department of Astronautical Engineering (ASTE) offers the full set of degrees (BS, BS Minor, MS, Engineer, PhD, and Graduate Certificate) in Astronautical Engineering. USC’s Master’s program in Astronautical Engineering (MS ASTE) is highly visible nationally and internationally and among largest in the United States. More information on history of Astronautical Engineering at USC – see next page. ASTE is responsible for VSoE programs in astronautics and space technology, concentrating on meeting the educational and research needs of interest to the space and defense industries, government research and development centers, and academia. Please contact Astronautics faculty and ASTE Senior Administrator Ms. Dell Cuason (RRB–228; tel. 213–821–5817; [email protected]) and Student Adviser Ms. Nicole Valdez (RRB–223; tel. 213–821–4234; [email protected]) should you have any questions. Dell and Nicole are your contact persons for all questions regarding class registration and admission or transfer to the program in astronautical engineering.
Student Transfer to Degrees in Astronautical Engineering (code ASTE)
Transfer Process – Viterbi Engineering Students Please refer to VSOE change of major form and contact ASTE Student Services Director Ms. Nicole Valdez (RRB–223; tel. 213–821–4234; [email protected]) for further details of the process.
Transfer Process – Non-Engineering Students Transfer to a program in Astronautical Engineering, Code ASTE, requires a non-engineering student to file the USC application for graduate admission to the program in Astronautical Engineering. Processing of the application does not require re-submission of supporting documents (e.g., transcripts) that have been previously submitted to USC. Check with ASTE Student Services Director Ms. Nicole Valdez (RRB–223; tel. 213–821–4234; [email protected]).
Restrictions Transfer to a program in Astronautical Engineering, Code ASTE, cannot be requested during the first semester of student studies at USC.
Questions? Please contact ASTE Senior Administrator Ms. Dell Cuason (RRB–225; tel. 213–821–5817; [email protected]).
More information on MS ASTE – http://astronauticsnow.com/msaste/ Frequently asked questions – http://astronauticsnow.com/msaste/faq.html
1997–2018 by Mike Gruntman 11/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
USC Astronautics program history, focus, rationale, and organization
Article in Acta Astronautica v. 103, 92–105, 2014
Abstract
Ten years ago in the summer of 2004, the University of Southern California established a new unique academic unit focused on space engineering. Initially known as the Astronautics and Space Technology Division, the unit operated from day one as an independent academic department, successfully introduced the full set of degrees in Astronautical Engineering, and was formally renamed the Department of Astronautical Engineering in 2010. The largest component of Department’s educational programs has been and continues to be its flagship Master of Science program, specifically focused on meeting engineering workforce development needs of the space industry and government space research and development centers. The program successfully grew from a specialization in astronautics developed in mid-1990s and expanded into a large nationally-visible program. In addition to on-campus full-time students, it reaches many working students on-line through distance education. This article reviews the origins of the Master’s degree program and its current status and accomplishments; outlines the program structure, academic focus, student composition, and enrollment dynamics; and discusses lessons learned and future challenges.
Article download http://astronauticsnow.com/2014aste.pdf
Article on the Master’s ASTE degree online (DEN) http://astronauticsnow.com/2018aste.pdf
1997–2018 by Mike Gruntman 12/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Admission Requirements for Graduate Degrees in Astronautical Engineering – Code ASTE
The Department of Astronautical Engineering (ASTE) of the USC Viterbi School of Engineering offers degrees in astronautical engineering, code ASTE. The admission to MS ASTE is based on the totality of applicant's record which includes GPA, GRE, and two letters of recommendation. Required items: Send To Application Office of Grad. and Int’l Admission Official Transcript(s) Office of Grad. and Int’l Admission General Record Exam Office of Grad. and Int’l Admission TOEFL (international students only) Office of Grad. and Int’l Admission Recommendation Letters Office of Grad. and Int’l Admission Application All applications should be submitted on-line at http://www.usc.edu/admission/graduate/apply/ Official Transcript(s) The University requires official transcripts from the accredited colleges or universities the applicant has attended. The MS Degree Program in Astronautical Engineering (Code ASTE) requires a minimum GPA of 3.0. General Record Exam The Department of Astronautical Engineering requires the general GRE. The GRE must be taken within five years of the application date. USC’s ETS school code is 4852. Applicants taking the GRE should use this code to ensure official submission of test scores. English Language Proficiency for International Applicants In addition to the general admission criteria listed above, international students whose first language is not English are required to take the TOEFL or IELTS examination to be considered a candidate for admission. There is no minimum TOEFL or IELTS score required for admission to the Viterbi School. For possible exemption from additional language requirements, you must achieve an Internet Based TOEFL (iBT) score of 90, with no less than 20 on each section or an IELTS score of 6.5, with no less than 6 on each band score. For more details on English Proficiency Criteria for the University of Southern California, please visit https://www.usc.edu/admission/graduate/international/englishproficiency.html. Recommendation Letters Please provide two professional letters of reference (former instructors, supervisors, professional colleagues, advisors, etc.) to be filed through the on-line application process. Mailing addresses, if needed Office of Graduate and International Admission University of Southern California, Los Angeles, CA 90089-0911 Department of Astronautical Engineering ASTE Graduate Program, 854 W. Downey Way University of Southern California, Los Angeles, CA 90089-1192 Application deadline: 15 June for fall; 1 October for spring; 1 February for summer. Please note that verification and processing of materials by the Office of Graduate and International Admission may take four to six weeks. Transfer to Astronautics Program and other Questions: Please contact ASTE Senior Administrator Ms. Dell Cuason (RRB–225; tel. 213–821–5817; [email protected]) and visit http://astronautics.usc.edu.
1997–2018 by Mike Gruntman 13/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Integrity
Academic integrity of all students participating in this course is of the fundamental importance for this instructor and is one of the most important components of the University rules and regulations. Students who violate University standards of academic integrity are subject to disciplinary sanctions, including failure in the course and suspension from the University. Since dishonesty in any form harms the individual, other students and the University, policies on academic integrity will be strictly enforced. I expect you will familiarize yourself with Section 11, Behavior Violating University Standards in Scampus. HomeWork, Exams, etc. are individual efforts
Academic Conduct Plagiarism – presenting someone else’s ideas as your own, either verbatim or recast in your own words – is a serious academic offense with serious consequences. Please familiarize yourself with the discussion of plagiarism in SCampus in Section 11, Behavior Violating University Standards https://scampus.usc.edu/1100-behavior- violating-university-standards-and-appropriate-sanctions/. Other forms of academic dishonesty are equally unacceptable. See additional information in SCampus and university policies on scientific misconduct, http://policy.usc.edu/scientific-misconduct/. Discrimination, sexual assault, and harassment are not tolerated by the university. You are encouraged to report any incidents to the Office of Equity and Diversity http://equity.usc.edu/ or to the Department of Public Safety http://capsnet.usc.edu/department/department-public- safety/online-forms/contact-us. This is important for the safety whole USC community. Another member of the university community – such as a friend, classmate, advisor, or faculty member – can help initiate the report, or can initiate the report on behalf of another person. The Center for Women and Men http://www.usc.edu/student-affairs/cwm/ provides 24/7 confidential support, and the sexual assault resource center webpage [email protected] describes reporting options and other resources. Support Systems A number of USC’s schools provide support for students who need help with scholarly writing. Check with your advisor or program staff to find out more. Students whose primary language is not English should check with the American Language Institute http://dornsife.usc.edu/ali, which sponsors courses and workshops specifically for international graduate students. The Office of Disability Services and Programs http://sait.usc.edu/academicsupport/centerprograms/dsp/home_index.html provides certification for students with disabilities and helps arrange the relevant accommodations. If an officially declared emergency makes travel to campus infeasible, USC Emergency Information http://emergency.usc.edu/ will provide safety and other updates, including ways in which instruction will be continued by means of blackboard, teleconferencing, and other technology.
1997–2018 by Mike Gruntman 14/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Some Useful Math (1) Taylor’s theorem
Let f x be analytic at a. Then
23 n xa xa xa n xa fx fa f' a f '' a f ''' a ... f a ... 1! 2! 3!n !
An alternative form hh23 h hn fa h fa f' a f '' a f ''' a ... fn a ... 1! 2! 3!n !
Maclaurin’s series: the special case when a 0 xx23 x xn fx f 0 f ' 0 f '' 0 f ''' 0 ... fn 0 ... 1! 2! 3!n !
Examples (elementary functions):
n nn112 nn n n! 1xnxx 123 x ... xk ... , x 1 2! 3!nk ! k !
xx23 exx 1 ... 2! 3! xxx234 ln 1xx ... , x 1 234 xxx357 sinxx ... 3! 5! 7! xxx246 cosx 1 ... 2! 4! 6! xx3521762 x 7 x 9 tanxxx tg ... , x 3 15 315 2835 2 1xx24 5 61 x 6 1385 x 6 secxx 1 ... , cosx 2! 4! 6! 8! 2 xx3513 135 x 7 sin1 xxx arcsin ... , x 1 62452467 xxx357 tan1 xxx arctan ... , x 1 357
1997–2018 by Mike Gruntman 15/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Some Useful Math (2) Indefinite integral of functions dF() x F () x C Integration by parts udv uv vdu
Change of variables Consider two sets of coordinates, or variables, xyz,, and uvw,, such that xxuvw ,, , yyuvw ,, , and zzuvw ,, . As an example, think about uvw,, being a set of spherical coordinates R,, . Then the function Fxyz,, would be
F xyz,, F xuwz ,, , yuwz ,, , yuwz ,, Guvw ,, and xyx,, F x,, y x dxdydz G u ,, v w dudvdw uvw,, xyz uuu xyx,, xyz where the Jacobian determinant is uvw,, vvv xyz www
Plane oblique triangle
ABC 180
abc Law of sines: sinABC sin sin
Law of cosines: ca222cos abCb 2
1997–2018 by Mike Gruntman 16/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Some Useful Math (3) Vector relations
Consider a set of mutually perpendicular unit vectors ijk,, pointing along the three mutually perpendicular axes X ,,YZ .
Consider vector Viabcj k . Its magnitude is V V abc222 .
Consider vectors Vi111abcj 1k and Vi222abcj 2k .
Scalar (dot) product. The scalar (or dot) product of these two vectors is
VV12 VV 21 VV 1 2cos V 12 V cos
where is the angle between vectors V1 and V2 . Expressed through vector components, the scalar product is
VV1 2 VV 2 1aa 12 bb 12 cc 12 .
Vector (cross) product. The vector (or cross) product of two vectors V1 and V2 is
VV12 VV 21 VV 12sin u where is the angle between vectors V1 and V2 and the unit vector u is
perpendicular to both V1 and V2 . Expressed through vector components, the vector product is ijk
VV12 VV 21abc 111 bcbc 122112211221 i caca j abab k
abc222
Consider now three vectors a , b , and c. abc bca cab abcbaccab
1997–2018 by Mike Gruntman 17/18 2019-Fall_ASTE-520_Sec-00_part-1_no_pswd ASTE 520 Spacecraft Design Section 00, Part 1
Some Useful Math (4) Vector relations Differential operator (del) ijk x yz
Gradient. The gradient of a scalar function x,,yz grad ijk x yz
Divergence. The divergence of the vector finction A ,
AAx,,yz Axyz xyz ,, i A xyz ,, j A xyz ,, k A
A A A divAA x y z x yz
Curl. The curl, or rotation, of the vector finction A ijk AAAAAA curlAA rot A zzyy i xx j k xyz y z z x x y AAAxyz
Laplacian 222 2 div grad x222yz
Gauss’s theorem (divergence theorem) divAAndV da VS
Stoke’s theorem Aldcurlda An CS
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