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Preface

This book includes more than 160 homework and exam problems that were given, could have been given, or should have been given (had the time allowed) in the first half of a graduate-level introductory astronautics "boot camp" course on fundamentals of space systems at the University of Southern California (USC).

Typically, each week of the course covers the fundamentals of one spacecraft subsystem or technical area. The problems in this book primarily concentrate on mission-related subjects such as the solar system and coordinate systems, space environment and interactions, the Earth and the gravitational field, orbital mechanics, and mission geometry. Therefore, the title is "Fundamentals of Space Missions." The follow-up publication will include problems covered in the second half of the course, focusing on spacecraft-related subsystems, including rocket dynamics, spacecraft and rocket propulsion, attitude determination and control, space communications, spacecraft thermal control, and electric power.

The space engineering master's program at USC is among the largest in the country [1]. More than 2,200 graduate students took this course over the last 27 years. The students in this program have reflected the technical diversity of the engineering workforce in the space industry and in government research and development centers. Students' bachelor's degrees have been in all imaginable flavors of engineering (astronautical, aerospace, mechanical, electrical, chemical, civil, computer, etc.) or in hard sciences such as physics, astronomy, chemistry, or mathematics.

In the past, a bachelor's degree was sufficient to have a life-long technical career in the aerospace industry. This is no longer the case. Today, a master's degree has become the "terminal degree" for successful work in a technical field.

About one-half of the students in this graduate course have studied full-time on campus. The other half worked full-time as engineers in the space enterprise and came back to school with years, sometimes decades, of "real-world" experience to pursue part-time a master's degree in astronautical engineering, an important step in advancing their standing in the profession. These students continued to work, and their employers, particularly the legacy aerospace and defense companies and government research and development centers, often supported their studies financially. Many such students took classes online through the school's distance education network [2]. Some of the working full-time engineers had advanced in their careers to become managers of engineering projects, and much of the science and engineering fundamentals had understandably faded away.

Consequently, this introductory course and the homework problems constitute a "rocket-science boot camp" for a diverse student population-one that is studying full-time or part-time. It provides scientific and engineering basics of space systems and prepares students for specialized coursework in various areas of space technology.

The content of the course and the homework problems were also influenced by my observation of student deficiencies in foundational physics. I noticed that the deficiencies depend very little on where students took their undergraduate coursework, be it in a community college or in an institution of higher learning that describes itself as "elite." These deficiencies will certainly increase in the future because of the growing use of harmful "holistic" practices in student admissions and the destructive pressure for non-merit "equity" hiring of science and engineering faculty in universities.

This publication is not a textbook and it is not a substitute for indispensable coursework in orbital mechanics, spacecraft dynamics, rocket propulsion, space environment, space communications, spacecraft power systems, thermal control, and other areas. The compiled problems could serve, however, as an aid to build up foundations in introductory graduate and undergraduate courses in astronautics. Various continuous-education programs could also benefit from the book.

All problems include detailed solutions. Consequently, one could consider them a how-to guide for making basic estimates in planning and designing space missions and systems. The ability to do such calculations is an indispensable skill for a system engineer working on a spacecraft and space mission, a major spacecraft subsystem, or a payload. The system engineer must comfortably navigate various engineering areas, make top-level estimates, and intelligently communicate with subsystem and payload specialists working in various technical domains.

A few problems are discussed in significant detail. Many engineering students have never been and will never be adequately exposed to some foundational concepts and effects. My experience over many years of teaching short courses on the fundamentals of space systems to government and industry supports this observation.

While several problems may look simple, they serve the exceptionally important purpose of developing a feel for the "right numbers": the range of realistic parameters to expect in practical applications. In addition, such exercises firm up a systematic approach to solving problems and making estimates.

Specific comments on the material in the book.

Fundamentals of Space Missions – List of Problems

1. Universe, Stars, and Solar System

1.1  Light year
1.2  Parsec
1.3  Sun's galactic motion
1.4  Interstellar wind
1.5  Interstellar Probe
1.6  Apparent and absolute magnitudes of stars
1.7  Difference in apparent magnitude
1.8  Apparent and absolute magnitudes of the Sun
1.9  Apparent and absolute magnitudes of α-Centauri A
1.10  Apparent magnitude of the Sun as seen from a departing starship
1.11  Two brightest stars for navigation
1.12  The number of square degrees in the sky
1.13  Star mappers
1.14  Visual magnitude of a satellite in orbit
1.15  GEO satellite visible by the unaided eye
1.16  Sphere of influence and patched-conic orbits
1.17  Sphere of influence of the Moon

2. Coordinate Systems, Time, Angles

2.1  Inertial reference frames
2.2  Subtended plane angle
2.3  Earth as viewed from geostationary and GPS orbits
2.4  Angular diameters of the Sun, Earth, and Moon
2.5  Solid and plane angles
2.6  Vernal equinox vector
2.7  Earth at perihelion
2.8  Vernal equinox precession
2.9  Sidereal year and tropical year
2.10  Mean solar day and sidereal day
2.11  Orbit eccentricity and apparent solar day
2.12  Geostationary satellite orbital period
2.13  Julian day
2.14  Apparent rotation period of the Sun
2.15  Spherical and Cartesian systems of coordinates
2.16  Geocentric celestial system of coordinates
2.17  Heliocentric ecliptic system of coordinates
2.18  Rotation matrices
2.19  Transformation between celestial and ecliptic systems of coordinates
2.20  Sirius and Canopus for space navigation
2.21  Orientation of the Earth's orbit with respect to the Galaxy
2.22  Angle and distance between two Voyager spacecraft
2.23  Distance between two points on the globe
2.24  Distance between North Korea and the United States
2.25  Distance between Iran and the United States

3. Space Environment and Interactions

3.1  Radiation flux density and the solar constant
3.2  Solar radiation flux densities at the Earth's orbit
3.3  Solar radiation flux densities at Mars
3.4  Solar radiation pressure force for normally inci-dent radiation
3.5  Solar radiation pressure force for non-normally in-cident radiation
3.6  Solar radiation pressure force for non-normally in-cident radiation
3.7  Solar radiation pressure on a foil in interplanetary space
3.8  Solar sail
3.9  Solar radiation pressure torque on a spacecraft
3.10  Equal normal and lateral solar radiation pressure forces
3.11  Solar electromagnetic radiation (light) and solar wind for electric power generation and exerting force on spacecraft
3.12  Time to reach 1 AU and interstellar boundary of the solar system
3.13  Wavelength of the peak of solar radiation
3.14  Wavelength of the peak of spacecraft thermal emission
3.15  Airburst of extraterrestrial bodies
3.16  Atmosphere scale height
3.17  Atmosphere scale heights at sea level and ISS altitude
3.18  Atmospheric drag and satellite velocity loss
3.19  Brushing up integration
3.20  Maxwell-Boltzmann distribution
3.21  Thermal velocities of atoms, ions, and electrons in low Earth orbit
3.22  Debye length (radius)
3.23  Debye length in low Earth orbit and interplanetary space
3.24  Spacecraft floating potential
3.25  Plasma frequency
3.26  Plasma frequency in the ionosphere
3.27  Gyrofrequency (cyclotron frequency) and gyroradius (Larmor radius)
3.28  Radiation (Van Allen) belt electrons and protons
3.29  Drift velocity
3.30  Drift velocities of radiation belt particles
3.31  Area density and shielding thickness of aluminum
3.32  Solar radiation pressure on interplanetary dust

4. Gravitational Field and Earth

4.1  Gauss's law for gravity
4.2  Gauss's law in the differential form
4.3  Spherically symmetric body and point mass
4.4  Thin spherical shell and point mass
4.5  Semimajor and semiminor axes and eccentricity of the Earth ellipsoid
4.6  Spinning Earth
4.7  Heights of mountain peaks
4.8  Geopotential of the Earth
4.9  Gravitational force due to the Earth's oblateness
4.10  "Pear-like" shape of the Earth
4.11  Zonal harmonic (2,0) and principal moments of inertia of the Earth
4.12  Mass, moment of inertia, and normalized moment of inertia of a sphere with a uniform density
4.13  Radius of the Earth's dense core\
4.14  Precession period of the Earth's spin axis

5. Basics of Orbital Mechanics

5.1  Two-body equation of motion
5.2  Conservation of energy
5.3  Orbital velocity and period in low Earth orbit (LEO)
5.4  Escape velocity
5.5  Properties of Deimos
5.6  Spacecraft in orbit around the Earth with an orbit radius equal to that of the Moon
5.7  Conservation of angular momentum
5.8  Trajectory equation: Kepler's first law
5.9  Energy and vis-viva equations
5.10  Orbit eccentricity
5.11  Kepler's second and third laws
5.12  Radial and transverse velocities in orbit
5.13  Orbit with a satellite radial velocity larger than its transverse velocity
5.14  Flight-path angle
5.15  Earth orbit
5.16  Perihelion of orbits of Neptune and Pluto
5.17  Maximum altitude for a given radial velocity\
5.18  Comet and interstellar body
5.19  Eccentric anomaly and true anomaly
5.20  Position in orbit as a function of time. Kepler's equation. Mean anomaly
5.21  Hyperbolic escape velocity and C3
5.22  C3 for launch from aphelion and perihelion of the Earth orbit
5.23  Orbits around the barycenter
5.24  Motion of stars in α-Centauri system
5.25  Lagrange points
5.26  Trajectory deflection in planetary flyby
5.27  Body capture radius for the Earth
5.28  Poynting-Robertson effect (simplified)

6. Orbital Elements and Maneuvers – I

6.1  Determine orbital characteristics
6.2  Orbit inclination change
6.3  Inclination change at apogee and perigee
6.4  Geostationary satellite north-south stationkeeping
6.5  Determine orbital characteristics
6.6  Determine orbital characteristics
6.7  Lowering satellite apogee
6.8  State vector and classical orbital elements
6.9  Velocity increment at a wrong point
6.10  Velocity vector change at apogee
6.11  Convert classical orbital elements to the state vector
6.12  Two-line element format
6.13  Flight time between two points in an orbit
6.14  Coplanar intercept of a geostationary satellite
6.15  Orbiting object breakup and Gabbard diagram

7. Orbital Elements and Maneuvers – II

7.1  Hohmann transfer
7.2  Hohmann transfer and C3 to Mars
7.3  Missions to planets
7.4  Graveyard orbit
7.5  Repositioning of a geostationary satellite
7.6  Bi-elliptic transfer
7.7  Getting close to a central body (Sternfeld transfer)
7.8  Deorbiting satellites
7.9  Transfer from low Earth orbit to geostationary orbit
7.10  Deployment to geostationary orbit from various launch sites
7.11  Transfer to Sun-Earth Lagrange point L1
7.12  Rotation of the line of apsides in a one-burn maneuver
7.13  Rotation of the line of apsides in a two-burn maneuver

8. Orbit Evolution and Common Orbits

8.1  Atmospheric drag and satellite drag paradox
8.2  Atmospheric drag in low Earth orbit
8.3  Atmospheric drag effect on the International Space Station
8.4  Atmospheric drag in prograde and retrograde orbits
8.5  Atmospheric drag and solar radiation pressure
8.6  Regression of nodes and rotation of apsides
8.7  Sun-synchronous orbit (simplified)
8.8  Maximum altitude of a circular sun-synchronous orbit
8.9  Sun-synchronous orbit on Mars
8.10  Molniya orbit (simplified)
8.11  Ground track shift (simplified)
8.12  Repeat ground track orbits (simplified)
8.13  Magic orbit (simplified)
8.14  Nodal period

9. Mission Geometry

9.1  Satellite viewing conditions
9.2  Satellite communication (visibility) time
9.3  Geostationary orbit coverage
9.4  Elevation angle to a GEO satellite
9.5  Communication time with a satellite in a sun-synchronous orbit
9.6  Apparent angular rate of satellite motion
9.7  Groundtrack azimuth
9.8  Launch azimuth and orbit inclination
9.9  Overflight of two sites on the ground in the same orbit
9.10  Eclipse duration

Appendix A1. List of problems

Appendix A1. List of problems

Appendix A2. Physical constants

Appendix A3. Prefixes, Greek alphabet, units

Appendix A4. Spaceflight constants

Appendix A5. Some useful mathematics

Appendix A6. Acronyms and abbreviations

Appendix A7. Bibliography

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