Introduction

The design and development of space systems is a part of engineering in general, which in turn relies on the knowledge base of mathematics and the sciences. Therefore the first part of this book is a review of the fundamentals of these fields. We will pay the most attention to the parts that apply to space systems, but we by no means cover the whole of these fields of knowledge. A non-technical reader can get a general idea of the concepts and projects presented in this book, but it is mainly aimed at people who want a deeper understanding of, or to actually work with future space projects. To do that, a proper foundation in mathematics and the sciences is needed at a secondary education (high school) graduate/first year university science or engineering level. If you do not have such a background, there are open source textbooks available online, such as those from the CK-12 Foundation, as well as video lectures from the Khan Academy, and traditional books and classes.

Our discussion in this book is thus at an introductory engineering level. It is not a complete survey on any topic. In many cases there is simply too much detail to fit it all. In others the technical level is too advanced, and, in the case of some future methods, the ideas have not been fully developed yet. Other books, articles, and materials are linked throughout the book, especially in the References section, and also in our online Library. Readers are encouraged to delve deeper into any topics that interest them. The next few sections will give a more detailed summary of the background of mathematics and science, and how it relates to engineering and key design principles for space systems.

Mathematics

The importance of mathematics to science and engineering can be summarized in one sentence:

Our Universe appears to follow mathematical laws.

By that we mean mathematical formulas and calculations produce results which match what we see when we look at the real world. This is a very powerful circumstance, because we can do the calculations before we look, even before something exists, and thus predict the future. Why mathematics works so well in describing reality is a philosophical question to which we don't have a good answer. This was pointed out by Eugene Wigner in 1960 in an article entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences (which is also discussed in a Wikipedia article). Regardless of why, it does work in practice. That allows us, among other things, to design systems that will work as intended.

The correspondence of mathematical predictions to the real world is not just a general one. In many cases it can be astoundingly exact. One of the earliest examples of prediction is the motion of the Sun, Moon, and planets in the sky. Even in ancient times people were able to predict where they would be in the future. Those predictions allowed knowing useful things, like when to plant crops, because of the linkage of the Earth's motion around the Sun to the seasons. Nowadays we can predict the motion of objects in the Solar System to fractional parts per million accuracy. An example of this accuracy was the 2012 landing of the Curiosity rover on Mars within 2 km of the intended location, after a trip of 566 million km. This could not have been done without predicting both the spacecraft trajectory and the future location of the landing point on a moving and rotating planet to 4 parts per billion. Further examples of using mathematics in design are all around you. Every tall building and bridge relies on the simple mathematical relationship that the strength be greater than the sum of all the loads. When you design such structures you calculate the strength, and calculate the loads, and then make sure the first is greater than the second. Proof that this method works is that tall buildings and bridges rarely fall down.

Like other engineering fields, space systems engineering relies on using such formulas and calculations. They are derived either from the sciences or practical experience and measurements within engineering. We present many of these formulas and calculations in this book. Therefore as a minimum you should understand the following mathematics topics (links are provided to introductory textbooks):

• Algebra - How to manipulate algebraic formulas and how to obtain a numeric answer given input values, the relationship of formulas and functions to graphs, and exponents and polynomials.
• Geometry - The types of geometric shapes and angles, and how the dimensions of two and three dimensional shapes (perimeter, area, and volume) are calculated.
• Trigonometry - Basic trigonometric functions and graphing them, vectors, and polar coordinates.
• Probability and Statistics - The ideas of averages, random error, distributions, and regressions.

More advanced topics, such as Mathematical Analysis, Calculus and beyond are helpful in understanding how the formulas are derived, or in solving the more complex problems in engineering. They are mostly not needed for an introductory level book such as this one.

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The predictions are often embodied in the results of mathematical formulas and calculations which relate to the real world. It is pursued partly on the basis that knowledge about the Universe satisfies human interest and curiosity. It also often turns out that the knowledge is useful in some practical way. We do not know in advance what knowledge will turn out to be useful, so scientists as a group study everything. Knowledge is a seamless whole, but from its history, and for the purposes of teaching and study, it is conventionally divided into branches according to the object of study. The ones most relevant to space systems include:

• Physics - This is the study of the forces and interactions of matter and energy, the results of those interactions, and the fundamental laws and components of which things are made.
• Astronomy - This is the study of objects and phenomena outside the Earth's atmosphere. This is the same location as all space systems operate in, so is highly relevant. Planetary Science in particular studies condensed objects orbiting stars.
• Chemistry - This is the study of matter at the atomic, molecular, and larger scales as far as how they react and their physical properties.

Other fields besides these three will also prove useful depending on the type of project. At least a basic understanding of these areas of science is needed to work with space systems, since engineering of those systems is derived from that knowledge. Beyond the branches of science, you should have an understanding of the Scientific Method, by which ideas are generated, experiments and observations are made to test those ideas, and thus they are validated or rejected. Peer review, statistics, and repeatability are among the methods used to ensure observations and conclusions are reliable. Absolute truth is never reached in science, merely increasing confidence in a given explanation, which is known as a Theory. Sufficiently well tested ideas join the body of knowledge considered settled, but they are always subject to revision and new ideas are constantly proposed.

Engineering

Mathematics and science are developed for their own sake and for their ability to predict the future. Engineering then applies accumulated knowledge, from the sciences and from experience, towards useful ends by designing, building, and operating systems to perform intended functions. When the systems are complex, a method called Systems Engineering is used across an entire project to organize and optimize the resulting design. This method can coordinate the work of thousands of people. Systems Engineering is described in more detail later in Part 1.

The total of accumulated engineering knowledge is too vast for any single person to know more than a small part of it. Therefore engineering in general is divided into major fields of specialization, each of which has it's own training path. It starts with a common basis in science and mathematics, then concentrates on particular areas of application, such as Mining, Chemical, Mechanical, and Electrical Engineering. Working engineers often further specialize their study and experience. They, or the organizations they work for, are called on as needed for each project. This is more efficient than keeping full time staff for every possible subject area. The specialists who are called on also have more experience in their area from having worked on many similar projects. Since the teams working on a project are not permanent, how you manage their interaction then becomes important. Project organization is also covered later in this part of the book.

Aerospace Engineering is the specialty field within which space systems fall. Space systems are projects which happen to operate in the space environment in the same way that ships and airplanes happen to operate in the water or through the air. Although the particular environment imposes differences in how things are designed, they all rely on the same base of knowledge in subjects like mechanics, materials science, and thermodynamics. So a complex project will use engineers from many of the specialty areas such as Mechanical, Chemical, and Electrical engineering, as well as Aerospace Engineers specialized in the methods and environments that apply to space. We will identify the other specialties later in Part 1 of this book, but will concentrate on the methods that apply to space. There are many existing books and articles about the other specialties for those who are interested.

This book is aimed at an introductory university engineering level reader. If you have no prior background in engineering or in space systems in particular, you may want to start with Engineering - an Introduction by the CK-12 Foundation. You can get additional background from some of the book-length and website references in the References section, the JPL Basics of Space Flight, Glenn Research Center Beginner's Guide to Rockets, and Mark Prado's Permanent and Robert Braeunig's Rocket and Space Technology

Design Principles

Through training and experience, engineers develop a sense of what will work or not, and how to optimize a design. Partly this is through broad principles that apply in their specialty. We note a few of the more important ones that apply to space systems here. These and others will appear throughout the book and we will try to highlight them:

• Earth vs Space - On Earth, transport involves friction of various kinds, and most things are moving slowly in relation to each other. Therefore energy and cost are proportional to distance, but not time. Space is a nearly frictionless medium, and things are moving at relatively high velocity with respect to each other. So difficulty and cost are more related to kinetic and potential energy, which governs the paths you follow. It also depends on the time you start, since your destination does not stay in the same relative location, rather than absolute distance.

• Non-Linearity - Many of the formulas and variables related to space systems have values raised to a power or an exponential. So the difficulty of a task does not have a one-to-one relation to the magnitude of the desired goal. This is called a non-linear system. Understanding the direction and amount of the non-linearity is important, as this can greatly help or hinder a given task. One of many examples is atmospheric pressure, which decreases exponentially with altitude, thus decreasing aerodynamic drag proportionally.

• Uncertainty and Margins - Although some values, like the orbit of a planet, are known quite accurately, no physical parameter is known with absolute accuracy. Anything built by humans will deviate by some amount from the ideal item embodied in the design drawings. The natural environment can fluctuate over time, and be uneven from measured averages. So all engineering designs need to account for the uncertainties in the physical data they are based on and production variations. One method to do this is to introduce Design Margins above the expected conditions that are larger than the uncertainties. How much margin to use is based on cost, experience, and the use to which the design is put. For example, a passenger airplane would generally have higher margins than a drone with no crew, even though both are aircraft.