Space Transport and Engineering Methods/Physics



&emsp;Space systems use advanced technology to operate in space and use the resources found there. A basic understanding of the sciences underlying the technology is important. This chapter is only a brief introduction. Links and references to more detailed sources are below.

&emsp;The key formulas from physics and other sciences are first presented in ideal terms. The real world is not as simple as that. Less than ideal behavior has to be accounted for, such as friction or perturbing additional forces. So do measurement uncertainties, environment variations, equipment wear, and other factors. These represent the difference between physical laws and practical engineering.

&emsp;Physical principles are usually expressed mathematically as algebraic formulas and geometric relationships, with supporting explanations to provide meaning and context. When known numerical values with proper units are inserted into these formulas, you can solve for an unknown value you wish to know. The ability to calculate unknown values is enormously useful when designing or operating any project. As noted in the introduction to Part 1, the reader should have enough understanding of mathematics to use these formulas.

1.0 - Science and its Branches
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&emsp;Scientific knowledge is an undivided whole. For study and understanding we divide into branches, but they often overlap and share ideas. Physics is the study of how the Universe behaves in its component parts such as matter, energy, forces, motion, space, and time. Astronomy is the study of objects and phenomena in the Universe at large scales, and Planetary Science in particular studies objects orbiting stars. Chemistry studies the properties and changes of matter at smaller scales and lower energies than found in stars. The many branches of the Life Sciences matter for any space projects that involve living things, especially people.

&emsp;Knowledge is gained by the Scientific Method and ideas developed to explain what we see. There are many ideas, but only one reality. So experiment and observation are used to determine which ideas best match that reality. Ideas are loosely graded in quality; as hypotheses, theories, principles, and laws. These grades are based on how firmly and widely the ideas have been tested.

&emsp;In the sciences, no idea is considered final or absolute truth. They are always subject to revision or replacement when confronted by new observations and experiments. Each new observation increases our degree of confidence in successful ideas. Many of them have been tested for so long, and in so many ways, that we rely on them routinely, even in engineering projects whose failure would be catastrophic.

2.0 - Physics
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&emsp;Most of the physics we use in engineering is well-grounded - based on things we can see, test, and use. For example, Quantum Mechanics is illustrated by spectral lines of gases, and the Theory of Relativity by observing bodies in motion through the solar system. These theories have engineering consequences in some applications, such as the design of lasers and the timing of satellite signals from the GPS Network.

&emsp;Other observations, like rotation curves and gravitational lensing of galaxies, don't yet have a good explanation or any practical use. We call what causes those observations Dark Matter, but we don't yet know what dark matter is. Physics and the other sciences are unfinished works in progress. Many parts are well understood and settled, and we can use that knowledge in projects. But around the edges are parts still being worked on, and beyond that the unknown.

&emsp;Some ideas in Theoretical Physics explore aspects of reality that are not easily observed. They need devices like large telescopes or particle accelerators to collect data, or nobody has figured out a way to test them yet. Theoretical ideas can be developed with nothing more than mathematics, a pencil, and pad of paper. So there are a lot more such ideas than firmly-tested theories.

&emsp;In some cases, it is possible to test an idea by looking for logical consequences which would not be true in competing ones. For example, measurements of the Cosmological Constant have helped narrow down theoretical ideas about the origins of the universe. We can exclude ideas that predict the wrong values.

 2.1 - Information Sources
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&emsp;For more detail on physics in general, you can refer to any of the following sources:


 * Articles linked from Wikipedia's Outline of Physics and books in varying stages of completion in the Wikibooks Physics subject heading.
 * The many articles linked from Wikipedia's Index of Physics Articles,
 * The visually outdated but still useful HyperPhysics website.
 * Many short videos and tutorial materials from the Khan Academy.
 * Open-source online textbooks like Light and Matter and Motion Mountain, and the CK-12 Physics textbook previously noted in the Part 1 page, Section 3.0.
 * Used or new printed college physics textbooks, and digital ones from the Internet Archive.

3.0 - Units and Position
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 3.1 - Units
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&emsp;To get the correct results from a formula, it is necessary to use a consistent set of units, and accurate ways to measure physical quantities such as length in those units. For example, adding two feet to three meters to get five of something does not produce a meaningful result because the units are different. You can use anything from body parts of known size to laser Interferometry for length, depending on the accuracy needed.

&emsp;The International System of Units, abbreviated from the French to SI, is the preferred system of units for engineering and scientific work. It is also known as the Metric System because the base unit of length is the "Metre" (or meter). Physical quantities include both the numerical value and the units. Units must be handled mathematically the same way as the numbers are. So in the formula F = ma (force equals mass times acceleration), m is in kg, a is in m/s2, and F is then in kg-m/s2.

&emsp;For historical reasons, some values in space systems design are reported in US customary units, but these should be converted to SI values. There are also units of convenience, such as g-force being a multiple of Earth's surface gravity. This is useful to describe the relative effects on people and equipment compared to the normal value on Earth. But it should always be recognized as a convenience, and converted to SI units for other calculations. The standard value is 1 Earth gravity (g or gee) = 9.80665 m/s2. The actual value at any point on the Earth's surface varies by about 2% from this.

&emsp;The base SI units are the Metre ("meter" in the US) for length, Kilogram for mass, Second for time, Ampere for electric current, Kelvin for thermodynamic temperature, Mole for amount of substance, and Candela for luminous intensity. Their symbols are m, kg, s, A, K, mol, and cd respectively. They are not defined in terms of other units. Since 2019 the base units, and and other units derived from them, are determined from by declaring exact numerical values for seven constants of nature, then calculating the units from those constants.

&emsp;Derived SI units are the products of powers of the base units. For example, the unit of force, called the Newton, is 1 kilogram-meter divided by seconds squared (kg-m/s2). Many of the derived units are named after famous scientists, but these named units are identical in value to the form expressed in base units. Multiples and sub-multiples of units are indicated by prefixes which indicate integer powers of ten ranging from -30 to +30. Among the more common are kilo, indicating 103 or 1000, and milli- indicating 10-3 or 0.001.

 3.2 - Position
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&emsp;In a universe of three physical dimensions, it takes three values to define a position uniquely, either distance in three axes, or a radius and two angles. These values are known as the object's Coordinates, and the types of values used make up a Coordinate System. In modern physics there is no absolute or preferred reference frame in the universe. So position is measured relative to a chosen starting point known as the Origin, which is assigned coordinate values of zero.

&emsp;For example, vertical position or Altitude on Earth is measured relative to sea level in the direction opposite the local gravity direction (up). On bodies without an ocean, an Ellipsoid based on the shape of the body is defined as zero altitude. On gas giants, which do not have a visible solid or liquid surface, altitude is referenced to pressure.

&emsp;For horizontal position Latitude and Longitude are measured relative to the points where the surface meets the axis of rotation, which are called the Poles, and a point assigned values of zero in both coordinates. On Earth, the zero point is where the Prime Meridian (zero longitude) crosses the Equator (zero latitude). This is in the Gulf of Guinea west of Africa.

&emsp;Units of Degrees, which are 1/360th of a circle, commonly measure the location relative to the zero point. This is useful for finding which parts of a body face the Sun (days and seasons), face their parent body in the case of moons, and when communications are possible. On other bodies the zero point is set arbitrarily. Very irregular smaller bodies may have multiple surfaces at a given latitude/longitude pair, in which case radius from the center of mass can be used rather than altitude. Horizontal position, or all three dimensions, can also be stated directly in meters relative to a zero point. Positive directions for each dimension define a Cartesian Coordinate System.

&emsp;Objects in space are generally moving in relation to each other on paths governed by gravitational forces. When the paths are purely the result of gravity and repeat they are called Orbits. They are defined by six parameters called Orbital Elements, from which you can calculate position at a given time. When position is more useful, it can be described in three dimensions relative to an origin, such as the center of the Sun, with axes defined relative to the stellar background. Alternately a radial distance and two angles relative to a reference plane can be used. For the Solar System, the reference plane is typically set by the Earth's orbit around the Sun, known as the Ecliptic.

4.0 - Motion
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&emsp;Kinematics is the subject that covers the motions of objects. It is important for space systems because bodies in space, orbits around them, and getting to a desired location in space all involve motion. We cannot cover the whole subject here, but to learn more there are many sources of the types noted in Section 2.1 above. Here we will cover a few key terms and formulas.

&emsp;Displacement is the net change in position. It has both amount and direction, such as "three kilometers North". Velocity is the rate of change in position per unit time. When stated without a direction and purely as an amount it has a single value in units of meters per second. Where x is position in the direction of motion, t is time, and the Greek letter delta (which looks like a triangle in its capital form) indicates the change in those values, then velocity v is given by the formula:


 * $$\bar{v} = \frac{\Delta x}{\Delta t}$$

&emsp;Acceleration is the rate of change in velocity, for which the formulas are


 * $$ \bar{a} = \frac{\Delta v_x}{\Delta t} = \frac{d^2x}{dt^2}$$

&emsp;where acceleration a is the change in velocity divided by the change in time (duration), or the second derivative (change in the changes of) position x divided by the square of the time interval.

&emsp;The horizontal bar over v and a in the respective formulas indicate the velocity is directional. A value such as this with both an amount and direction is called a Vector, while a value without a direction is called a Scalar. The direction can be given in terms of two angles, or the velocity can be expressed as components in the three (x, y, z) axes of a reference system, but either way a total of three values are required to state a velocity vector. Linear Algebra is a method for doing calculations with vectors. It is somewhat different and more complex than simple algebra.

&emsp;In accelerated motion, the velocity at any given instant is changing. We can define an Instantaneous Velocity at a particular time, and an Average Velocity over an interval. In a closed orbit the moving object returns to its starting point. So for a full orbit, the net change in position is zero, and as a vector the average velocity is also zero. If you measure the total length of the orbital path and divide by the time one orbit takes, you can obtain an average orbital speed as a positive scalar value. This illustrates how different vector and scalar values can be.

&emsp;Acceleration can also change with time. For example, the accelerating force due to gravity changes as the inverse square of distance. So a falling object will increase in acceleration as it gets closer. Under constant acceleration a in a straight line over time t we can determine change in position or distance d from:


 * $$ d = \Delta x = 1/2 \times at^2$$

&emsp;In circular motion, where v is the velocity and r is the radius, we can find the acceleration a from the following formula:


 * $$ a = \frac{v^2}{r}$$

&emsp;One use for this formula is finding a required velocity for a circular orbit from the acceleration of gravity (see under Forces below) and the radial distance from the center of the body. Note there is no centrifugal force. Gravity or a rotating structure provide an inwardly directed force to maintain circular motion of the object, but there is no outward one.

5.0 - Forces
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&emsp;The 20th century theories of General Relativity and Quantum Mechanics are more accurate predictors of how objects behave in the realms of the fast and the small, but in many cases the simpler formulas of Classical Mechanics are accurate enough to use. Examples where classical mechanics is not accurate enough include long term changes in the orbit of Mercury, which being the closest planet to the Sun, moves the fastest, and GPS navigation, which relies on extreme accuracy of the orbits of the satellites, and the effect of gravity on their signals, to determine user position. Isaac Newton formulated many of the basic ideas of classical mechanics in his Principia, published in 1687. These include his three laws of motion, conservation of momentum and angular momentum, and the law of universal gravitation.

 5.1 - Newton's Three Laws
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&emsp;These are laws in the mathematical sense, which Newton deduced from experiments performed by others before him. They involve two opposing concepts: Forces which tend to create motion, and Mass which tends to oppose changes by the property of Inertia. The relationship of forces to the motion they create is the field of Dynamics.

&emsp;Forces are vectors, having an amount and a direction, and multiple forces act as the vector sum of the component forces. This also means single forces can be decomposed mathematically into components, such as vertical and horizontal components relative to an axis system, or perpendicular (normal) and parallel components relative to a surface. Decomposition is done when it is useful in solving a problem.

&emsp;The three laws are:

&emsp;First: Inertia - A body acted on by no net force moves with constant velocity (which may be zero) and zero acceleration:


 * $$\sum \vec F_i = 0 \Rightarrow a = 0$$

&emsp;Where the Greek letter Sigma (Σ) means "sum of" and and the right double arrow &rArr; means "implies", the formula reads "The vector sum of forces i, where i = 1 to n, adding up to zero implies the acceleration is also zero. Common experience on Earth is that moving objects decelerate to a stop.  This is because hidden friction forces act to stop them.  Objects moving freely in the vacuum of space demonstrate this Law more clearly because friction is nearly absent.

&emsp;An airplane in level flight has multiple forces acting on it (gravity, lift, thrust, and drag), but if the vector sum of all the forces is zero, it will continue moving at the same altitude and velocity in the same direction. From the fact that buildings are not typically accelerating we can deduce there is no net force acting on them, or the sum of the forces is zero. Since gravity acts to pull the building down, there must be an equal force from the ground acting to hold it up. Applying this idea to every structural part of a building is a powerful way to determine the necessary design of each - at each point where components connect, the forces must sum to zero, therefore you can calculate the forces which a particular component must withstand.

&emsp;Second: Force - When a net force does act on a body of mass m, the acceleration a is related to the magnitude of the force F by the formula


 * $$ \vec F = m \vec a $$

&emsp;The force and acceleration are vectors, so a force in a given direction produces an acceleration in the same direction. Manipulating this simple formula has many uses in space systems. Given any two of the values, we can find the third. Summing across time, we can find total velocity change. Mass has units of kilograms and acceleration has units of meters per second squared. So by the above formula force has units of kilogram-meters per second squared (Newtons in SI units). The force which the Earth exerts on a medium apple is coincidentally, given stories about Newton and falling apples, about 1 N.

&emsp;The product of mass times velocity is called Momentum. It is given the symbol p since mass already uses the letter m. Force also equals the change of momentum per time interval. Acceleration is the change in velocity per time so the following formula just adds the multiplier of mass to both sides of the equation:


 * $$ \vec F = \frac{d \vec p}{dt} $$.

&emsp;Third: Reaction - Single forces do not act in isolation. At the most fundamental level the particles which carry the four forces of nature act on both the emitter and absorber of the particles. At the macroscopic level we live in forces are the combined action of many particles. So we observe the dual action as "for every force there is an equal and opposite reaction force". Where the subscripts ab and ba indicate the force of object a on object b and object b on object a, and the minus sign indicates the second force is in the opposite direction:


 * $$\vec{F}_{\mathrm ab} = - \vec{F}_{\mathrm ba}$$

&emsp;An object can never move itself by applying forces only to itself, because the reaction force would cancel it out. You cannot hold yourself above the ground, no matter how hard you try, by applying forces to your own body. A pole vaulter, however, can raise their body a considerable distance by applying force to the ground. The reaction force of the ground through the pole then acts to raise their body. Of great interest for space systems is that expanding gases in a rocket engine apply a great deal of force to to the engine, and thus the rest of the rocket, while accelerating themselves to high velocity in the other direction.

&emsp;Multiplying both sides of the above formula by units of time, and subtracting the left side from the right side, we find that sum of momentum changes (mass times velocity) is always zero. This is known as the Law of Conservation of Momentum. It is referred to as a physical law because it has never been observed to be violated. Conservation in the physics sense means a value which does not change.

&emsp;Momentum is found to be conserved both for linear and rotational motion. The latter is referred to as Angular Momentum. So the Earth would continue to rotate forever unless acted on by outside forces. Such forces do in fact act, mainly tidal forces from the Moon. So the Earth's Rotation is slowing down measurably - the day is getting longer by a small amount each year on average. But since angular momentum is conserved, slowing the Earth's rotation means the Moon increases its orbital angular momentum. This increases the size of its orbit by a measurable amount (3.8 cm/year)

 5.2 - The Forces of Nature
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&emsp;There are only four fundamental forces that we know of, responsible for all relative motion in the Universe. These are the gravitational, electromagnetic, weak nuclear, and strong nuclear forces. These forces act by their carriers, called gravitons, photons, W and Z bosons, and gluons respectively. For more detail see Fundamental Interaction. The latter two are short range forces which mostly occur within atomic nuclei, so the two that concern space projects the most are gravity and electromagnetism.

&emsp;5.2.1 - Gravity
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&emsp;Gravitons are the hypothetical particles which should carry the gravitational force. They are hypothetical because they have not yet been observed. Because gravitons never decay, their range is infinite, and the Gravitational Field of any object in the Universe affects every other object in the Universe. In practice, Gravity is most accurately described by General Relativity which treats gravity as a geometric property of four-dimensional Spacetime. That theory is complicated, but for most engineering purposes we can use the simpler "Classical" formulas based on Newton's work.

&emsp;The total gravitational field surrounding an object remains the same at any distance. The area of a sphere surrounding an object is $4πr^{2}$, where &pi; is the mathematical constant Pi and r is the radius. So the gravitational field per unit area decreases with the square of the radius. Between any two objects the total gravitational force F depends on the product of the two masses. The first object produces a field and each part of the second object is affected by it. The second object has the same effect on the first. Where M is the mass of the first object, m is the mass of the second, and G is a universal constant which applies to every object in the Universe (as far as we know):


 * $$G = 6.67 \times 10^{-11} Nm^2/kg^2$$


 * $$ F = -\frac{GMm}{r^2} $$

&emsp;Gravity always acts to attract two objects to each other, in other words reduce the distance, so the force is given a negative value. As a practical matter, since the field falls as the square of distance, objects sufficiently far away can be ignored to the extent you need to accurately calculate the total gravitational force on an object. The force acts on a line between each pair of objects and the total force is simply the sum of the individual forces accounting for the direction of each, and each is found by the above formula.

&emsp;Since force is also mass times acceleration, we can equate them and remove mass m from both sides of the equation, giving the acceleration a due to gravity of an object with mass M as


 * $$ \vec a = -\frac{GM}{r^2} $$

&emsp;The force of gravity on an object is referred to as Weight. Most people live where gravity is within 2% of the standard value, so we often confuse weight with mass. They are proportional, but they have different units. Your mass does not change according to what object you are standing on. The acceleration of gravity does due to the object's different mass and radius. So on another planet or body, you would have a different weight.

&emsp;Weight does not disappear when in orbit. Aboard a low orbit space station the force of the Earth's gravity is only 11% less than on the ground. So-called "zero gravity" is more properly described as Free Fall. The astronauts inside the Station and the Station itself are both affected by the same acceleration of gravity. So the difference between them is zero, and the astronauts do not feel their bodies pressed against anything. On Earth what you feel is parts of your body pressed against the ground or furniture, and internally pulling parts of your body down. This pressure is what you experience as "weight".

&emsp;Since the range of gravity is infinite, there is nowhere in the Universe that is truly zero gravity. There are places where the forces are balanced, and you are so far away from any massive body that your acceleration can be ignored for most, but not all purposes. Despite being 2.5 million Light Years apart, our own Milky Way Galaxy and the Andromeda Galaxy are still pulling on each other and will Collide in about 4.5 billion years. But for space systems engineering within our Solar System we can ignore anything beyond it most of the time.

&emsp;5.2.2 - Electromagnetism
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&emsp;Photons are the particles which carry the electromagnetic force. Unlike gravitons, they are easily observed whenever your eyes are open, since photons are what make up Light. They also do not decay as they travel, and obey an inverse square law of intensity with distance. In a vacuum, they travel at the Speed of Light, one of the constants of nature, because they are light.

&emsp;Where gravity is the result of mass, electromagnetic force is the result of electric charge. Unlike gravity, charge comes in two types which we call positive and negative. The names are arbitrary, but Electrons have one unit of negative charge. Positive charges are the same size, and the Proton has one unit. Like electric charges repel each other, and unlike charges attract. So protons attract electrons to produce neutral Atoms. The electromagnetic force F is


 * $$F = k_\mathrm{e} \frac{q_1q_2}{r^2}$$

where k(e) is the Coulomb Constant, q1 and q2 are two Electric Charges in units of Coulombs (symbol C), and r is the distance between them. Note the form of this equation is similar to the one for gravitational force. When both of the charges are positive, or both negative, their product is positive, and so is the force. Positive forces act to increase the distance between charges. When the charges are unlike, one positive and one negative, the product is negative, and the force acts to decrease distance. The Coulomb constant is:


 * $$k_\mathrm{e} = 8.987552 \times 10^9 \ \mathrm{N \cdot m^2 / C^{2}} $$

&emsp;Electric charges are almost always observed as integer multiples of the electron and proton charges. The exception are Quarks which combine to form particles like protons. These are never observed in isolation, so they can be ignored for space systems work. Charges are additive by simple arithmetic, with negative charges canceling the fields of positive charges.

&emsp;Since unlike charges attract each other, they tend to annihilate (convert to pure energy) if they are antiparticles, or form neutral atoms if they are protons within atomic nuclei and electrons. So large quantities of matter tend to have low net charge. Since mass is always positive, large quantities of matter will have large amounts of gravity. Despite gravity being a much weaker force on an individual basis, it tends to dominate the actions of large bodies.

&emsp;Moving electric charges create a Magnetic Field. This includes the imputed spin of the charges of elementary particles. Materials with aligned atomic spins then have a static magnetic field. A steady flow of electric charges is called an Electric Current, and also creates a magnetic field. Magnetic fields in turn affect the motion of electric charges, creating a force F, where I is the current, ℓ is the length of the wire, and B is the strength of the magnetic field in units of tesla (symbol: T).


 * $$\mathbf{F} = I \boldsymbol{\ell} \times \mathbf{B} \,\!$$

&emsp;The bold face symbols indicate these are vector values, having directions. The force is perpendicular to both the direction of the wire/current and the magnetic field. Natural magnetic fields, such as the Earth's, are assumed to be caused by electric currents within the body.