# Section 1.1 - Basic Sciences: Physics

## Physics as a Subject

Physics is the study of how the Universe behaves in its component parts, which includes matter, energy, forces, motion, space, and time. It is an experimental science in that experiments and observations are used to discard inaccurate ideas in favor of ones that match reality. Ideas are loosely graded in quality as hypotheses, theories, principles, and laws, based on how firmly and widely they have been tested by observation. No such idea is considered final or absolute truth. They are always subject to revision or replacement by new observations and experiments, but many ideas have been tested for so long and in so many ways, we can confidently rely on them in engineering projects.

In this section we discuss key physical principles that relate to space projects. These principles are only a subset of physics as a whole. We first look at them in ideal terms individually. However realistic design work has to consider less than ideal conditions, such as friction or perturbing additional forces, and also uncertainties in how well we know any measured property. This difference represents the difference between physical laws and practical engineering.

Physical principles are usually expressed as algebraic formulas and geometric relationships. 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 space projects. As noted in the introduction to Part 1, the reader should have an adequate understanding of mathematics if they want to actually use these formulas themselves.

It is not our purpose here to include an entire physics textbook, but rather a summary of the most important relationships that apply to space systems. For more detail on physics in general, you can refer to one of the following sources:

## Units and Coordinates

#### Units

In order to obtain the correct results from a formula, a consistent set of units and method of measuring physical quantities such as position in those units is necessary. For example, adding two feet to three meters to get five of something does not produce a meaningful result because the units are different. 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 in English). 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 gravities being a multiple of Earth's surface gravity. It is convenient to express acceleration effects on humans in this way, relative to the Earth normal value. It should always be recognized as a convenience, and converted to SI units when doing calculations by it's standard ratio of 1 gravity = 9.80665 m/s2. Physical quantities include both the numerical value and the units, and units must be carried through properly when doing calculations.

The base SI units are not defined in terms of other units, but rather by a description of how to measure them from nature, or by a physical artifact. These base units are currently the Meter 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. Efforts are underway to define these base units in terms of constants of nature, but they are not complete. 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 times seconds-2. Many of the derived units are named after famous physicists, but these named units are identical 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 -24 to +24. Among the more common are kilo, indicating 103 or 1000, and milli- indicating 10-3 or 0.001.

#### Position

In modern physics there is no absolute or preferred reference frame in the universe. Therefore position is measured relative to a starting point known as the Origin, which is given a value of zero. Altitude on Earth is measured relative to sea level in the direction opposite the local gravity direction. On bodies without an ocean to serve as a reference, an average ellipsoid based on the shape of the planet or satellite is defined as zero altitude. On gas giants, which do not have a visible solid or liquid surface, it is based on pressure. On near-spherical bodies, 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 called the Zero Point. Units of Degrees, which are 1/360th of a circle measure the location relative to the zero point.

Objects in space are generally moving in relation to each other in paths defined by gravitational forces. When the paths are purely the result of gravity they are called Orbits and are measured by six parameters called Orbital Elements from which you can calculate position at a given time. When absolute position is more useful, it can be measured 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. Most often the reference plane is that which contains the Earth's orbit around the Sun, known as the Ecliptic. 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'.

## Motion

Displacement is the 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) indicates change in those values, then velocity v is given by the following formula:

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

Acceleration is the rate of change in velocity, or the second derivative of position, with respect to time. Thus acceleration a is:

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

The horizontal bar over v and a in the respective formulas indicates velocity is directional. A value such as this when stated with both a magnitude 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. Vector algebra is somewhat different and more complex than simple algebra.

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

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

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

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

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.

## Forces

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 sufficiently accurate to use. Examples where classical mechanics starts to be insufficiently accurate 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.

### Newton's Three Laws

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 it via the property of Inertia. The relationship of forces to the motion they create is known as Dynamics. Forces are vectors, having magnitude and direction, and multiple forces act as the vector sum of the component forces. This also means single forces can be decomposed 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. The three laws are:

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

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

This is contrary to common earthly experience where friction acts to stop objects in motion.

Objects moving freely in the vacuum of space demonstrate this Law more clearly since it is a frictionless environment. 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 typically are not accelerating we can deduce there is no net force acting on them. To put it another way, the sum of the forces is zero. Since gravity acts to pull the building down, there must be an equal force from the bedrock acting to hold it up. Applying this idea to every structural component of the building is a powerful way to determine the necessary design of those components - at each point where components connect, the forces must sum to zero, therefore you can calculate the forces which a particular component must withstand.

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

$\vec F = m \vec a$.

The arrows above the symbols indicate they are vectors, meaning a quantity in a particular direction. Thus a force in a given direction produces an acceleration in the same direction. Manipulating this simple formula has very wide ranging use in space systems. Given any two of the values, we can find the third. Summing across time, we can find total velocity change.

Since mass has units of kilograms, and acceleration has units of meters per second squared, then by the above formula force has units of kilogram-meter per second squared, which is called a Newton (abbreviated "N") in the SI system of units, named after the scientist. The force which the Earth exerts on a falling apple is coincidentally, given stories about the scientist and falling apples, about 1 Newton. The force of gravity on an object is referred to as Weight. Since most humans live where that force causes an approximately equal (within 0.2%) acceleration, we often confuse weight with mass. They are proportional, but they have different units. On another planet, the same object would have a different weight. Weight does not disappear when in orbit - aboard the Space Station the force of the Earth's gravity is only 10% 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 force of gravity. Thus the difference between the forces 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 pressing against other parts of your body. This pressure is what you experience as "weight".

The product of mass times velocity is called momentum and given the symbol p since mass already uses the letter m. Force also equals the change of momentum with respect to time, since acceleration is the change in velocity with respect to time and we are just adding the multiplier of mass:

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

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, where forces are the combined action of many particles, we observe the dual action as for every force there is an equal and opposite reaction force. Where the subscripts indicate the force of object A on object B and object B on object A:

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

Therefore a body can never move itself by applying forces only to itself, because the reaction force would cancel it out. You cannot lift 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 a rocket engine applies a great deal of force to expel gases in one direction, and the gas applies a reaction force in the opposite direction, which moves a vehicle.

Multiplying both sides of the above formula by units of time, and subtracting the left side from the right side, we obtain the sum of momentum (mass times velocity) changes 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, and conservation in the physics sense means a value which does not change. It is found to be conserved both for linear and rotational motion. The latter is referred to as Angular Momentum. Thus 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 23 microseconds per year. But since angular momentum is conserved, slowing the Earth's rotation means the Moon increases its angular momentum. This increases the size of its orbit by a measurable amount (3.8 cm/year)

### The Forces of Nature

There are only four fundamental forces responsible for all known motion in the Universe. These are the gravitational, electromagnetic, weak nuclear, and strong nuclear forces. These forces interact via 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.

#### Gravity

Gravitons, are the hypothetically postulated particles which should carry the gravitational force. 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. As a result the total gravitational field surrounding an object remains the same at any distance. The area of a sphere surrounding an object is 4 times pi times the radius squared. Thus the gravitational field per unit area decreases with the square of the radius r. The rates of graviton production and absorption are both proportional to the mass of an object. So between any two objects the total gravitational force depends on the product of the two masses. Since the first object emits a number depending on it's mass M, and then the second object absorbs some of them according to it's second mass m. The rate of graviton production and absorption is measured as a universal constant G which applies to every object in the Universe, as far as we know:

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

Gravity always acts to attract two objects to each other, in other words reduce the distance, therefore 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 formula

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

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

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

When restrained from accelerating, such as when you stand on the surface of the Earth, you experience the downward force as weight. Your mass does not change according to what object you are standing on, but the acceleration does, due to the object's different mass and radius. Therefore your weight will be different on other planets and satellites.

When unrestrained from accelerating, also known as free fall, then the parts of your body, or for instance if riding in vehicle, all will accelerate at the same rate. They do not have any acceleration relative to each other, sometimes called Zero gravity, but incorrectly since nowhere in the Universe there is truly no gravitational field. So in the case of unrestrained acceleration "free fall" is the more correct term, and "zero apparent gravity" will provide a better indication as to express how it appears to a human relative to their surroundings.

The local acceleration at the Earth's surface has a standard value of 9.80665 meters per second squared, but actually varies slightly depending on location (increases at the poles and decreases in the meridian, due to the changes in distance to the planet's center). This is given the symbol g, and accelerations are sometimes stated as multiples of standard Earth gravity to give an impression of how humans would experience it, but for calculation purposes meters/second squared should be used to avoid unit errors. Similarly weight should not be confused with mass. Weight is in reference to the local gravity field, while mass is the more correct unit to use at any location.

#### Electromagnetism

Photons, the particles which carry the electromagnetic force, behave similarly to gravitons in that they do not decay as they travel, and obey an inverse square field law. 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, positive charges are not larger or higher than negative ones, but they have the property that like charges repel each other, and unlike charges attract. The electromagnetic force is found by the formula

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

Where F is the force, k(e) is fixed value called Coulomb's constant, q1 and q2 are the electric charges, 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 thus the force acts to decrease distance. Coulomb's constant, where C is charge in units of 1 Coulomb = 1 mole of elementary charges is

$k_\mathrm{e} = 8.987 \times 10^9 \ \mathrm{N \cdot m^2 / C^{2}}$

Elementary charges are those on a single electron or proton, and are always observed as integer multiples of those charges, never fractions. Charges are additive by simple arithmetic, with negative charges canceling the fields of positive charges. Since unlike charges attract each other, they tend to annihilate 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 always have large amounts of gravity.

Moving electric charges create a magnetic field, including the imputed spin of the charge from elementary particles. Materials with aligned atomic spins can have a static magnetic field. A steady flow of electric charges is called a Current, and also creates a 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:

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

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 metal core.