The bizarre effects of Special Relativity, introduced by Albert Einstein in 1905, are manifest as time dilation, length contraction, and varying mass. Thus, as an object moves faster, time (t) passes more slowly for it, its length in the direction of motion (l) shrinks, and its mass (m) increases. 
At the velocity of light, time would stand still, length in the direction of motion would shrink to zero, and mass would become infinite.
These distortions occur so that the velocity of light will always appear to be a constant (c), regardless of relative motions and one's own inertial frame of reference (i.e. coordinates at a constant velocity). In Einstein's view, this simply preserves the absolute universality of the laws of nature, since the velocity of light turns out to be an artifact of Maxwell's Equations for electromagnetic interactions. Maxwell's Equations, and so the velocity of light, are equally valid for every inertial frame of reference (the original "Galilean" form of Relativity), which means however it is that one is moving, as long as one is moving at a constant velocity (which means the same speed and direction), the velocity of light (in a vacuum) will be a constant.
The change in mass itself explains why ordinary objects cannot attain the velocity of light: They would have an infinite mass there and so would need an infinite force to accelerate themselves to that velocity. This circumstance is not always appreciated, even by great science fiction writers like Robert Heinlein, who has one character in a story ask why we can't go faster than the velocity of light and the answer is given that "we don't know" but "we'll see when we get there." Another science fiction story speculates that a ship hitting the velocity of light would be bounced back into the past.
As it happens we must ask where the mass comes from. The Conservation of Mass means that mass cannot just come out nothing (except with transient quantum effects). So if your mass approaches infinity while your velocity approaches the velocity of light, where does the mass come from? Well, notice that as you approach the velocity of light, your acceleration begins to slow down. Velocity is kinetic energy; and the power you use to accelerate towards the velocity of light means that you are expending energy, much of which goes into the higher velocity that you achieve. But the energy accomplishes less as acceleration declines towards zero. How can that be? Well, since there is Conservation of Energy, the energy cannot just become nothing. So we must accept that the energy drains away into the mass that you notice has been increasing. The energy does not become nothing; and the mass does not increase from nothing. Without that energy for acceleration, the force that causes your acceleration attenuates. You cannot get to the velocity of light. If you could reach the velocity of light, all energy would go directly to mass, which thus could become infinite. What does this all mean? Well, it means E = mc2. Energy and mass can go back and forth. So Einstein's perhaps most famous discovery is an artifact of the velocity of light as a constant.
The original formulae for the transformation of coordinates, from one frame of reference moving past another in the x axis [as given in The Universe and Dr. Einstein, by Lincoln Barnett, Mentor, 1948, 1950, 1957, footnote pp. 54-55, and Michael Berry, Principles of cosmology and gravitation, Cambridge U Press, 1976, pp. 35] are at left. These are the "Lorentz Transformations," proposed prior to Einstein by the Dutch physicst H.A. Lorentz to account for the anomaly of the Michelson-Morley experiment in 1881, where the velocity of light had not varied regardless of the direction in which it was measured. 
The form of the equations is for the x coordinate, and time (t). The y and z coordinates are unaffected. The equation is also given for the addition of velocities (v). Lorentz did not know, however, why this effect had occurred, so these were just ad hoc mathematical descriptions. Einstein provided the reason. Simpler equations for the length (l) contraction for the object, the dilation for a unit of time (t), and for the increase in the mass (m) of the moving object are all given at right [versions given in Physics, The Foundation of Modern Science, Jerry B. Marion, John Wiley & Sons, 1973, pp. 197-205].
The Lorentz Transformations are not mathematically very difficult, but they do not transparently relate space and time to each other, and they do not relate to any intuitive sense of why this all would happen. Another equation that does provide a better sense of things, called the "Separation Formula," is given at right, where s is the "separation" or "proper time," which is the elapsed time for a moving object, while Δt, Δx, Δy, and Δz are the changes in the coordinates in time and space as an object moves [cf. Michael Berry, Principles of cosmology and gravitation, Cambridge U Press, 1976, pp. 48-49, and Roger Penrose, The Emperor's New Mind, Oxford U Press, 1989, pp. 195-196]. At first this may not seem like an improvement over the previous equations. But, as with many equations, it can be simplified.
First of all, the Greek delta, which indicates the change in the coordinates, can be left out, as in the first equation at left, giving us variables for the movement in each of four dimensions. Then we should take into account that the Separation Formula is really an extension of the Pythagorean Theorem. The basic Pythagorean Theorem is in two dimensions (x2+y2), but it can be generalized into three dimensions (x2+y2+z2). In four dimensions we get an anomaly, since all the dimensions of space are added together, while they are all subtracted from the dimension of time (t2-((x2+y2+z2)/c2)). This all by itself is revealing, since it answers the question whether time is treated exactly like a dimension of space in Relativity. It isn't. Since what concerns us is the relation between time and space, the separation formula can be simplified by replacing the x2+y2+z2 term with its simple equivalent, r2 (r = "radius"). This can then be further simplified by picking the right units. The velocity of light can be set equal to one with the choice of light years (LY) and years (y). The velocity of light is, indeed, one light year per year (that is the definition of a light year). 
With all those simplifications, the Separation Formula ends up as a very simple equation indeed: s2=t2-r2. Although the velocity of light term has been eliminated, it should be remembered that its units of velocity are there and that the "separation" comes out in units of time.
Curiously enough, a full Pythagorean formula, s2=t2-(x2+y2+z2/1), has integer solutions using Pythagorean Triples, as we can see in the diagram at right.
With the simplified equation, we can inspect some Relativistic effects. The graph at left, 
in vertical units of years and horizontal units of light years, shows two trips in space-time. The red path is an object (a spaceship) moving at 60% of the velocity of light -- it travels 3 light years in 5 years. The blue path is the movement of light itself -- 5 light years in five years. The green line is a stationary object in this inertial frame of reference -- we could think of it as the Earth, from which the spaceship travels 3 light years away. We discover from the Separation Formula, that while 5 years have elapsed on Earth, the ship has arrived at its destination in only 4 years, according to its clocks. Meanwhile, the ray of light experiences no elapsed time -- it's separation is zero (25-25=0). That is called "light-like" separation, in comparison to the "time-like" separation of the other object. It can be seen from this that the longest path in space-time is the shortest separation. On the other hand, what if an object had gone faster than light? If so, the r term would be larger than the t term, resulting in a negative number, and the separation would be the square root of that number. The separation would therefore be an imaginary number (√-1 = i). That is called "space-like" separation, and it signals us that such motion is impossible. You can't get there from here in space-time, though this does not tell us why -- we need to know about the change in mass for that. If the light ray is rotated around the vertical axis, this would generate a cone, a "light cone," which defines the area in space-time accessible from the point at the origin.
The previous graph gives rise to a paradox. If the spaceship is moving at a constant velocity away from the Earth, then it has its own inertial frame of reference, in relation to which it is the Earth that is moving away from the ship. Thus, time should be passing more slowly on Earth than in the ship. The graph at right 
illustrates this situation. The red line represents the ship, at "rest" in its own inertial reference frame, experiencing the passage of 4 years. At a distance of 3 light years, the Earth (green line) would only have experienced the passage of 2.65 years (s=√(16-9)=√7=2.65). This is peculiar, since back on Earth, everyone has experienced the passage of 5 years (purple line). What this paradox illustrates is that in Special Relativity simultaneity is relative. In the simultaneous space for the spaceship, 2.65 years have elapsed on Earth and 5 years there would be in the future, while in the simultaneous space for Earth, defined by the thin diagonal purple line to the spaceship, 5 years have passed. A graph of simultaneous space for Earth, placed on the inertial frame of reference for the spaceship, would be systematically distorted, a "Poincaré motion," so that simultaneous events in one frame of reference will be in the future or the past for another [cf. Roger Penrose, The Emperor's New Mind, pp. 199-200].
The paradox of simultaneity can be solved by reuniting the objects in a single frame of reference. For this to be done, of course, at least one object must experience an acceleration. This provides the simple rule for Special Relativity: The "absolute" time we end up with is determined by the unaccelerated frame of reference. 
Thus, in the graph at right, we are back to the Earth's frame of reference, since the spaceship has turned around (which is a change in velocity and so an acceleration) and returned to the Earth. It returns at 60% of the velocity of light again and so experiences a separation of 4 years. Overall, 10 years have elapsed on Earth while only 8 years have elapsed for the spaceship. Simultaneity is reconciled. Longer trips at higher velocities produce more dramatic differences. But if the spaceship did not return to Earth, there would be nothing special about the Earth's inertial frame of reference and simultaneity between the ship and Earth would not be reconciled.
The only truly absolute frame of reference, velocity, and time in all of this is that of light. The only absolute time is none -- "light-like" separation. This sounds like nothing less than Platonism, where that which truly exists is eternal and unchanging. At the velocity of light, no time will pass at all for the entire history of the universe, which sounds like Socrates imagining that, if death is like dreamless sleep, then "all of eternity will be no more than a single night."
There is actually another paradox in the first diagram above (reproduced at left): If, on the spaceship, it seems that 4 years have passed, then the spaceship, to travel 3 light years in 4 years, had to have been going 75% of the velocity of light (3LY/4y), not 60% (3LY/5y). The 60%c velocity is only the way the velocity would appear in the Earth's inertial frame of reference. However, this answer will not do; for if the spaceship went 4 light years in 5 years, 
or 80% of the velocity of light (4LY/5y), as in the diagram at right, only 3 years would have passed for the ship (√(52 - 42) = 3), which would mean that the ship went 4 light years in 3 years, at a velocity then of 167% of the velocity of light (4LY/3y). But the whole point of Special Relativity is that it can't go faster than light, however we measure it. What should handle this, however, is that as the spaceship is moving, the distance (4 LY), as measured in the Earth's frame of reference, contracts (using the equation l = lo√(1-v2/c2)): 4 √(1-(0.8)2) = 4√(1-0.64) = 4√0.36 = 4(0.6) = 2.4. The ship sees the distance it traverses as only 2.4 light years -- at high velocities the universe literally becomes smaller, at least in the direction of motion. 2.4 light years in 3 years is, indeed, 80% of the velocity of light (2.4LY/3y = 0.8LY/y). Similarly, the ship that goes 3 light years in 5 years (60%c), experiences that distance as 2.4 light years also, which also confirms the velocity of 60%c (2.4LY/4y = 0.6LY/y).
The equation at right is a version of the Separation Formula combined with terms for mass and the Gravitational Constant, in polar coordinates, stated as derivatives (d) of separation (s), times (t), distance (r), and angle (θ and φ). I have not seen the formula in print but copied it off the television screen of the cable show The Universe. I hope I haven't made any mistakes. The image could have been clearer. Since separation is stated in units of time, I'm not sure the units of this come out quite right.
Historic Equations in Physics and Astronomy
Einstein's Equivalence Principle
Philosophy of Science, Physics
Being more of a spectator on mathematics than a participant, my favorite historic equations are those that the ignorant layman (like me) can play with just by plugging in the right values and seeing what happens. As it does happen, this doesn't get one very far with the more recent and more important equations, which often feature complex operators whose use requires a mathematical education in itself and whose solutions are often still matters of contemporary exploration and controversy. Here I give both kinds, and the mathematically literate are welcome to sneer at my rudimentary understanding of the more sophisticated material. What I can now do, however, is refer the reader to a recent examination of historic equations by the specialists themselves: It Must be Beautiful, Great Equations of Modern Science, edited by Graham Farmelo [Granta Books, London, New York, 2002, 2003]. Here we have the likes of Roger Penrose and Steven Weinberg doing what I would really like to be doing here. Farmelo, however, has not chosen an essay about Kepler's Laws, which I think are more my speed and that I think I explain rather well.
A more recent book that is similar in purpose to my treatment is Archimedes to Hawking, Laws of Science and the Great Minds Behind Them, by Clifford A. Pickover [Oxford University Press, 2008]. Pickover gives equations, but his more general focus is on "laws of nature." Many of these laws involve very simple equations, and they are often swamped with supplemental material about the discoverers and discoveries. Towards the end of the book, when we get to Maxwell's or Schödinger's Equations, we tend to get less discussion and not much at all about the way in which the equations work. Nevertheless, Pickover's treatment can be very illuminating, as with his treatment of Coulomb's Law.
Constants for these equations are given in the table at "Physical Constants."
Equations for Kepler's Laws: Kepler's Second Law is that the radius from the focus to a planet sweeps out equal areas in equal times. The mathematics of this is still not easy to deal with. Given the mean angular motion (n), which would be 360o divided by the period (περίοδος) of the orbit (in radians: 2π/p), Calculating the eccentric anomaly from the mean anomaly is difficult because the equation M = E - e*sin E (for ellipses) cannot just be solved for E. Instead, we can write the equation as M + e*sin Em = E and, starting with Em = M, solve the equation over and over, substituting each new approximation of E for Em. This permits calculating E as accurately as desired. With the eccentric anomaly determined, the true anomaly (θ) can be calculated directly. Equations are given at left for circular, elliptical, parabolic, and hyperbolic orbits. For circular and parabolic orbits, the true anomaly can be calculated directly either from the mean anomaly or just from the time since perihelion (t - T). The hyperbolic equations work much like the elliptical ones, usually just with some opposite signs and hyperbolic functions.
The diagram at right gives the conventions for the physical characteristics of a planetary orbit. The celestial equator is the Earth's equator projected onto the sky. The apparent path of the Sun in the sky, however, the ecliptic, is at an inclination (ε) to the equator. At the ascending node there is also the angle of inclination of the oribt (i) to the ecliptic. Angles of inclination are small for the planets but can be very large, up to 90o, for asteroids and comets. If the motion of the object is retrograde, i.e. from East to West, the angle of inclination will be larger than 90o. Once on the plane of the orbit, the angle of longitude is measured to the East until the perihelion (perigee, etc.) point is reached. That angle is the "argument of perihelion" (ω). The longitude of the ascending node and the argument of perihelion can be added together for the "longitude of perihelion" (ϖ), but the two angles are not measured in the same plane, so the angle may differ from the true angle of separation between the Vernal Equinox and the perihelion. A similar caution holds for the "true longitude" of an object, which is the sum of the longitude of perihelion and the true anomaly (L = θ + ω + Ω).
It is noteworthy that Kepler's confidence in the mathematical nature of the universe was Platonic in inspiration, derived from the revival of Plato by Renaissance scholars and ultimately from the Platonism of Mistra in Romania. Kepler's hope to match planetary orbits up with the Platonic Solids, however, was not successful.
Isaac Newton and the Earls of Portsmouth
Bode's Law: Stated by J. Bode in 1778 but discovered by J. Titius in 1766 -- and so now frequently called the "Titius-Bode Law." A simple numerical sequence that produces the mean distance from the Sun in Astronomical Units for most of the major planets. The construction of the series begins with 0 and 3; 3 is then doubled as many times as desired for subsequent terms in the series; 4 is added to each number; and then each number is divided by 10. The results are shown in the following table below, up to the twenty-first term.
The period of the orbit can be calculated using Kepler's Third Law (p2 = a3), where the mean distance (a) is in Astronomical Units and the period (p) is in Earth years. The results were very close to the planets known at the end of the 18th century, up to Uranus (discovered in 1781), with one conspicuous gap: There was no planet corresponding to B5. However, on New Year's Eve of the year 1801 the first asteroid, Ceres, was discovered, orbiting at exactly 2.8 AU. Neptune and Pluto, discovered later (1846 & 1930) did not fit Bode's Law, as can be seen from the actual physical data in the table below, although Neptune and Pluto bracket B9.
Karl Popper enjoys this example of Hegel's cluelessness in science, but Hegelian apologists try to defend Hegel by saying that his numerology was meant as a parody of Bode's Law, as though it was Bode rather than Hegel who was "unscientific." But Bode's Law was "scientific" just because it fit, and it made what turned out to be a dramatic prediction. Why it fit can now be explained by the conditions of the gravitational stablity of the Solar System, while Hegel, of course, was just a fool.
The discovery of Ceres was, as it happened, only the beginning. By the late 1980's, 5000 "Minor Planets" (a term officially dropped in 2006, but still in the name of the Minor Planet Center) had been catalogued and named. That seemed like a lot; but now there are well over 250,000 of them that have been assigned a Minor Planet Number -- with over 535,000 objects that have been observed and registered, pending calculation and confirmation of their orbits -- the numbering has now about reached that point. Only 20,000 or so have been named, with names that have become increasingly whimsical and peculiar. Minor Planet 4535 Adamcarolla is even named after comedian Adam Carolla.
At the high end, four asteroids have been promoted to "Dwarf Planets"; and one former Planet, Pluto, has been infamously demoted down to that novel status. The Dwarf Planets are numbered in the Minor Planet sequence. The debate about the status of Pluto has gone on long enough that the number 10000 was once suggested for it; but the whole business was so long delayed that Pluto, discovered in 1930, now bears the ignominy of being only Minor Planet 134340, which in no way distinguishes it from fellow Dwarf Planets 136108 Haumea, 136472 Makemake, and 136199 Eris, none of which was discovered before 2003. The opportunity missed here is that Pluto could retroactively be made Minor Planet Zero. Many have wondered why there seems to have been a positive animus against Pluto. How did it offend these people?
On January 1, 2019 the "New Horizons" spacecraft (launched 2006), which had previously flown by Pluto (July 14, 2015), passed by a Kuiper object that was discovered by a search instituted just so it could be reached for a fly-by. This was identified in 2014 and assigned the Minor Planet Number 486958, which says a lot about the number of objects currently identified. This object was informally assigned the name "Ultima Thule" (although it is less "ultimate" than Makemake, Eris, and Sedna -- albeit now the most distant object visited by a spacecraft). When it finally was imaged, "Ultima Thule" turned out to be dumb-bell shaped, with two small spheroids pushed together. Serendipitously, these parts could then be distinguished as "Ultima" and "Thule." The whole object, however, has now been named Arrokoth. For some reason, the discovers solicited a name from the Pamunkey Indian Tribe, descendants of the Powhatan Nation. "Arrokoth" is said to mean "Sky," or "Cloud," taken off an old word list, not from a living language. Only 14 words of the Pamunkey language survive.
Meanwhile, the images of Pluto rocked the astrophysical community. Its surface was not dead and cratered like the Moon, but young, uncratered, and apparently geologically active. Some hints of this had been gathered from the moons of Jupiter and Saturn, but here there seemed to be a whole "cryo-geology" where water ice, nitrogen, and methane are sometimes as active at temperatures close to Absolute Zero as water and magma are on Earth. A large, heart-shaped "glacier" on Pluto appears to be a sometimes boiling lake of nitrogen. Jaws dropped. This has now been named the "Tombaugh Regio," after the discoverer of Pluto, Clyde Tombaugh (1906-1997), some of whose ashes were carried on the New Horizons spacecraft.
This may reopen the debate about Pluto. It is not just the "largest Kuiper object," but a body of remarkable and general interest. The argument becomes more forceful to "grandfather" it in as the planet it had long been since its discovery. It clearly has a unique status, both historically and scientifically. And its discovery as long ago as 1930 now is explained, not only because it is on the near edge of the Kuiper Belt, but because of the very bright areas on its surface. The whole issue over Pluto may make us wonder if it was just a matter of the International Astronomical Union just jerking people around, as we've seen with often arbitrary changes of other traditonal names. Otherwise, we must ask again why the astronomical community seems to have some kind of hostility towards Pluto. Perhaps somebody didn't like the long-lived Clyde Tombaugh.
A recent discovery in the Kuiper Belt (2018 AG37) was given the whimsical name "Farfarout." This was because, even though its semi-major axis was a modest 80.2 AU, it was currently near its Aphelion, at 132.7 AU, which was further out than Sedna was at the time. This made it the most distant object observed.
Beyond the most distant observed objects is the hypothetical Ort Cloud. Nothing has been observed in this Cloud, but it is postulated as the source of the comets, periodic and non-periodic, that drop into the inner solar system. The orbits of such comets can be calculated, and their Aphelia put them at Ort Cloud distances -- if they even have closed orbits, but hyperbolic orbits instead that send them into deep space.
The life of a meteor, however, is brief. After ceasing to be a meteoroid, it quickly ceases to be a meteor. If it is not entirely burned up in the atmosphere, the parts that fall to earth then become meteorites and enter the realm of geology and mineralogy as much as that of astronomy or astrophysics. If the meteoroids are large enough, blast effects and craters on the ground can result. A full asteroid, of kilometer sizes, hitting the Earth can, of course, have major geological consequences, with the ends of both the Mesozoic and Paleozoic Eras dated, the former with some certainty, the latter tentatively, to asteroid impacts.
The peculiar circumstance that modern meteorology is not the study of meteoroids, meteors, or meteorites, but of the weather, is due to the Greek word, μετέωρος, metéôros, simply meaning "up in the air." Both meteors and the weather, including clouds, are, indeed, "up in the air." Bode is the astronomer who originally suggested the name of the planet Uranus. Little did he know the trouble this would later cause. When I was a child, I remember pronouncing "Uranus" as you-rânus and thinking nothing about it. However, in 1974 a movie came out called The Groove Tube which had a sequence that derived considerable scatological humor from pronouncing "Uranus" as "your anus." For some reason, this seems to have cut astronomers to the quick; and it wasn't long before I began to see people like Carl Sagan pronouncing the planet as yúrin-us.
I can't see this as a particular improvement, since it sounds like "urine us," which has an equally, if not more, unpleasant ring to it. So the whole business has become a fiasco. If I ever attend an astronomy lecture where the speaker starts saying "Urine-us," I may raise my hand and say, "Please stop saying 'urine'; it is 'triggering' me." We might take a poll whether people like anuses better than urine.
My own modest proposal would be to return to something like the Greek or Latin pronunciation of the original god: Οὐρανός, Oó-ran-os. This avoids both urine and the anus, although, really, people should not be so up tight about either. We should say you-rânus just to push back. Carl Sagan reacted to the whole business the way Bevis and Butt-head would have.
For two of the issues about the outer planets, we have the awkward situation of the Periodic Table of the Elements. The elements 92, 93, and 94 are named, respectively, Uranium, Neptunium, and Plutonium. These are, of course, named, in sequence, after Uranus, Neptune, and Pluto. This enshrines Pluto as a planet. There are no elements named after Minor Planets -- although we could say that the element Cerium (58), named after the goddess Ceres, goes with the asteroid, now Dwarf Planet, Ceres (1).
Historic Equations
in Physics and Astronomy
Kepler's First Law is that the orbits of the planets are ellipses, with the Sun at one focus. At right is a general equation for a conic section (circle, ellipse, parabola, or hyperbola) in polar coordinates, together with commonly used physical dimensions for the orbits of planets, asteroids, comets, etc. The "major axis" (2a) is the longest line that can be drawn in an ellipse; the "minor axis" (2b) is the shortest line that can be drawn through the center. The shortest distance from the focus to the curve (q) is "perihelion" for a planet, "perigee" for an object in orbit around the earth, or, in general, "periapsis," "closest approach," for any kind of orbit. The longest distance from a focus to the curve (Q) is "aphelion" for a planet, "apogee" for an object in orbit around the earth, or, in general, "aphapsis," "furthest approach," for any kind of orbit. The distance across the curve, through the focus, at right angles to the major axis (2d), is the "latus rectum". The periapsis (q) and semi-latus rectum (d) are physical dimensions of any conic section, not just ellipses -- though a hyperbola might be thought of as an ellipse with a negative major axis and an imaginary minor axis. The "eccentricity" (e) defines the shape of the curve: e=0 is a circle; 0<e<1 is an ellipse; e=1 is a parabola; and 1<e is a hyperbola. Finally, the angle made by the radius (from focus to orbiting object) with the point of periapsis is the "true anomaly" (θ).
and the time elapsed since the "epoch of perihelion" (T, a benchmark time when the planet was at perihelion), the "Mean Anomaly" (M) can be calculated, which would give the angle with perihelion if the planet had been moving with a uniform angular speed. With the mean anomaly in hand, it is not the true anomaly that is calculated first, at least for elliptical and hyperbolic orbits, but the "eccentric anomaly" (E), which is the point on a superscribed circle that corresponds to the point where the planet is on its curve. This is illustrated in the diagram at left.
The point where the Sun crosses the celestial equator and enters the northern hemisphere is the Vernal Equinox, providing the benchmark to celestial longitude, which is then measured along the ecliptic to the East, the direction in which the Sun and planets move against the background of the stars. The first physical feature of an orbit is the Ascending Node, the point where the object (planet, asteroid, comet, etc.) crosses to the north of the ecliptic. The "longitude of the ascending node" (Ω) is thus the celestial longitude of that point, measured East from the Vernal Equinox.
Kepler's Third Law is that the square of the period of a planet's orbit is directly proportional to the cube of the semi-major axis. By the same token, the square of the mean angular motion is inversely proportional to the cube of the semi-major axis. However, the equation for the period can be considerably simplified by choosing the right units. If (Earth) years and Astronomical Units (the semi-major axis of the Earth's orbit) are chosen, then the rest of the equation can be put equal to 1, which means that the Third Law can simply be written: p2 = a3. This was useful in Kepler's day when the true physical distances, let alone the Gravitational Constant, were unknown. It is still useful today. See discussion of the word "period" (Greek περίοδος) under the note for the Olympic Games.
Newton's Equation for Gravity: The force exerted between any two bodies with mass -- "G" is the gravitational constant; "m1" and "m2" are the masses of the two bodies; and "r" is the distance between them. Below the equation for force is the equation for the acceleration of gravity produced by a single body. The acceleration of gravity on the surface of the Earth is 9.8 m/s2 (aE = g).
Coulomb's Law: The force exerted between two electrical charges -- "k" is the electrostatic force constant; "q1" and "q2" are electrical charges; and "r" is the distance between the charges. Electrical charges can be positive (+) or negative (-). Opposite charges ("+" & "-") result in a positive (attactive) force; like charges ("+" & "+" or "-" & "-" ) result in a negative (repulsive) force. The second version is the way I see this equation written now, with the "permittivity of empty space" (ε) used instead of the electrostatic force constant. For long I had not seen it explained what the "permittivity of empty space" is supposed to mean. Factoring out 4π must have some relevance to space. Indeed, this is where Pickover's Archimedes to Hawking is of great value. The "permittivity" is "an electrical property of the medium that surrounds the two charges." The value of "k, sometimes known as Coulomb's constant, is approximately equal to 9 x 109 N.m2/C2" for empty space [p.154, boldface added]. The permitttivity is different for different materials, and Pickover helpfully gives a list of several values [p.155].
The possible physical reason for Bode's law may be seen in the relationship of the periods. Beginning with Mercury, which orbits the Sun in about three months, we approximate all the periods up to Ceres simply by doubling. What this suggests is that the orbits are in gravitational step with each other -- a harmonic resonance. That is, they reinforce rather than disturb each other's orbits. When we get out to where there are greater distances between the planets, around Neptune and Pluto, the relationship breaks down. Nevertheless, the philosopher Hegel believed that Platonic numerology, rather than gravity, was the issue; and he tried proving in his doctoral dissertation that there could be no planet at B5 -- an argument published on the very eve of the discovery of Ceres on 1 January 1801.
Meteoroids, Meteors, and Meteorites
At the low end, asteroids trail off into meteoroids, which more or less by definition are simply objects that are too small to be permanently catalogued and assigned numbers. The informal cut-off used to be a size of 10 meters; but now some objects smaller than 10 meters have been numbered, so the informal cut-off is down to 1 meter. Another informal definition of meteoroids could be that they are objects too small to be noticed until they hit the atmosphere and are observed burning in the air. Then they become proper meteors.
Pronouncing Uranus
| The Moons of Uranus | |||||
|---|---|---|---|---|---|
| The Inner Moons | |||||
| order from planet | order of discovery | Name | size | period | play |
| 1 | VI | Cordelia | 40km | +0.3d | Lear |
| 2 | VII | Ophilia | 44km | +0.4d | Ham |
| 3 | VIII | Bianca | 51km | +0.4d | Shrew |
| 4 | IX | Cressida | 80km | +0.5d | Troi |
| 5 | X | Desdemona | 65km | +0.5d | Othel |
| 6 | XI | Juliet | 95km | +0.5d | R&J |
| 7 | XII | Portia | 136km | +0.5d | Venice |
| 8 | XIII | Rosalind | 72km | +0.6d | Like |
| 9 | XXVII | Cupid | 18km | +0.6d | Timon |
| 10 | XIV | Belinda | 91km | +0.6d | Rape |
| 11 | XXV | Perdita | 31km | +0.6d | Winter |
| 12 | XV | Puck | 163 | +0.8d | Dream |
| 13 | XXVI | Mab | 25km | +0.9d | R&J |
| The Major Moons | |||||
| 14 | V | Miranda | 472km | +1.4d | Temp |
| 15 | I | Ariel | 1158km | +2.5d | Temp |
| 16 | II | Ubriel | 1169km | +4.1d | Rape |
| 17 | III | Titania | 1577km | +8.7d | Dream |
| 18 | IV | Oberon | 1523km | +13.5d | Dream |
| The Irrgegular Moons | |||||
| 19 | XXII | Francisco | 22km | -267d | Temp |
| 20 | XVI | Caliban | 42km | -580d | Temp |
| 21 | XX | Stephano | 32km | -677d | Temp |
| 22 | XXI | Trinculo | 18km | -749d | Temp |
| 23 | XVII | Sycorax | 157km | -1288d | Temp |
| 24 | XXIII | Margaret | 20km | +1661d | Much |
| 25 | XVIII | Prospero | 50km | -1978d | Temp |
| 26 | XIX | Setebos | 48km | -2225d | Temp |
| 27 | XXIV | Ferdinand | 20km | -2790d | Temp |
Meanwhile, Uranium enshrines the old pronunciation of Uranus. Thus, the element is you-rân-i-um, not yúri-ni-um or yúrin-i-um -- "urine-ium." The latter are even hard to say. All the talking heads spouting "urine-us" in documentaries and lectures should be challenged to explain the pronunciation of Uranium, or try to change its pronounciation into "urine" something.
Uranus was discovered by Sir William Herschel (1738-1822) in 1781. It had previously been seen, even by the Greeks, but mistaken for a star, since it moves so slowly. The first moons, Titania and Oberon, were also discovered by Herschel, in 1787. They were not named until 1852, by Herschel's own son, John Herschel (1792-1871). Although Titania and Oberon were discovered first, they are numbered in a group of the other earliest discoveries, beginning with Ariel.
The members of the group of the Inner Moons don't start being discovered until 1986. The most recent there, Cupid and Mab, weren't discovered until 2003. The outer group, the Irregular Moons, don't start being discovered until 1997, and the most recent, again, in 2003. Most of the Irregulars have retrograde motion (shown with a negative period). They look like captured asteroids. They also are distant from the planet and have very long periods, all but one longer than a year, up to more than seven years.
The names are the fun part here. Almost all the names come from plays, the most from Shakespeare's The Tempest, indicated with the abbreviation "Temp" -- in fact all of the retrograde Irregular Moons. Next we get the play from which Titania and Oberon themselves are taken, A Midsummer Night's Dream, with an abbreviation "Dream."
Other Shakespeare plays are King Lear ("Lear"), Hamlet ("Ham"), The Taming of the Shrew ("Shrew"), Troilus and Cressida ("Troi"), Othello ("Othel"), Romeo and Juliet ("R&J"), The Merchant of Venice ("Venice"), As You Like It ("Like"), Much Ado About Nothing ("Much"), The Winter's Tale ("Winter"), and Timon of Athens ("Timon").
One other source is not by Shakespeare but a poem by Alexander Pope, The Rape of the Lock ("Rape"). The name "Ariel," found in The Tempest, is also in Pope's poem. As more moons are discovered, probably to be placed in the Irregular group, there are other names available from The Tempest. Miranda obviously led the way for this.
An interesting feature of Planck's equation concerns the units. The factor on the right of the equation involves a power of e, and the result should be a dimensionless number. Thus, all the units in the superscript of e should cancel out. What we see are the terms hc/λkT. The units of Planck's Constant are J*s, Joule-seconds, which resolves as s*kg*m2/s2 = kg*m2/s. These are units of angular momentum, which is a curious feature when we consider Bohr's application of the constant to the orbitals of atoms. The whole set of constants then resolves as (J*s)*(m/s)/m*(J/K)*K = JsmK/smJK = Jm/mJ = 1. I have thought this sort of thing was a bit of fun since first learning to do it in Mr. Falb's chemistry class in High School.
While Planck's Law introduces Planck's Constant, in 1900 Max Planck clearly had no clue what this was going to mean and what it would lead to in physics. Planck's discovery is often presented as the discovery of Quantum Mechanics in all its bizarre and curious details, or at least as of some fundamental feature of it all; but I suspect that Planck had no intimation of such things at the time. It is not clear to me what he thought that the Constant meant that, empirically, he found it necessary to introduce into the equation. And there is an important point there. Planck fiddled with the equation until it worked. It was not founded on some deep theory or insight about physics. Einstein made the first leap in that respect five years later.
Wien's Law: The wavelength at which Black Body radiation for a given temperature peaks. -- Under Physical Constants I have recorded the "second radiation constant" as 0.01438786 m*K. In the given formula, we divide by 5 and get 0.002877572 m*K. Dividing by temperature in Kelvins (K) then get us units in meters for wavelength.
As I see now, Wien's Law is often given by eliminating the step of division by 5 and listing a single number as "Wien's (Displacement) Constant." I see this given with no more accuracy than 2.898 x 10-3 m*K, or 2.8977719 mm*K.
The "second radiation constant" is supposed to be calculated from hc/k, where h is Planck's Constant (J*s), c the velocity of light (m/s), and k = 5.8789232 x 1010 Hz/K (=1/sK). I need to check if this multiplies out correctly.
2.898 x 10-3 m*K, of course is equal to 0.0028977719 m*K. This is a little different from 0.002877572 m*K, which would be a value from my older sources (actually, from back in the 1970's).
If the surface temperature of the Sun is 5,778 K, then the maximum wavelength of the Sun's Black Body radition, that we get from Wien's Law, would be 5015.2 x 10-10 m or 5015.2 Å. With slightly different values, of course, like my old information, the result will be slightly different.
I still have never seen a natural language explanation of what a tensor is. A vector is easy. It's a direction. But a tensor involves two dimensions, which could mean a surface, but, as I said, I have seen no explanation of this. A similar issue intrigues me with respect to Planck's Constant, which is in units of J*s, or Joule-seconds, but which resolves as s*kg*m2/s2 = kg*m2/s. These are actually units of angular momentum, which means ordinary momentum, kg*m/s, times the radius, m, from the center of motion. While factor of the radius is then lost into the mathematial expression, it remains of physical significance: as the ballet dancer pulls in her arms, the radius of rotation decreases, and the speed of rotation increases. Angular momentum itself has a vector, given by the "right hand rule," which means that if you make a fist with the thumb up, the curled fingers signify the direction of rotation, with the thumb indicating the direction of the vector. The effect of this physics, remarkably, is why a bicycle remains upright. A force (like gravity) applied against the vector is displaced 90 degrees. The effects can be examined with a gyroscope.
All of quantum mechanics becomes a matter of angular momentum by way of Bohr's hydrogen atom. But does this have anything to do with tensors? I do not know; but it does intrigue me that the question of rotation enters into the debate about the metaphysics of space. Leibniz argued that all motion was (absolutely) relative and that motion, strickly speaking, only exists in relation to other objects. In empty space there can be no motion. Samuel Clarke, arguing for Newton, countered that rotation could certainly be detected even in empty space, because its affects are apparent without reference to any other objects. Partisans of Leibniz (like Ernst Mach), perhaps not understanding that Leibniz did not believe space to even exist, exulted that Einstein's Relativity meant that he was right and that Clarke and Newton were wrong.
However, Special Relativity is about velocity alone. Rotation is not a velocity. It is an acceleration, since the path followed by the parts of a rotating body do not follow a straight line. Acceleration, in turn, is a matter of General, not Special, Relativity, where things are actually not quite as Relative. The Twin who departs to Alpha Centauri, and then returns, is the one who has not aged as much, and the physical situation cannot be interpreted such that the Twin remaining on Earth is the one who has moved. This is because the traveling Twin is the one who has accelerated, breaking the Relativistic symmetry of the motion [note].
In turn, Einstein's Equivalence Principle is that acceleration and gravity (or inertial and gravitating mass) are physically equivalent and identical. And so we get the curious but logical result that rotation is gravity (somehow). Does this then in fact link angular momentum and tensors? I wonder.
How rotation is the equivalent of gravity we can see with some simple considerations. Although rotation is an acceleration, it is definitely one of a peculiar sort. While Ancient and Mediaeval physics held that a force was necessary to keep an object moving, the Modern view since Galileo has been that inertia keeps an object moving but a force is necessary to accelerate (or decelerate) it. A rotating object in a vacuum, however, does not need a force to keep rotating -- the Earth has gradually slowed from tidal friction, but it has otherwise rotated for hundreds of millions, in fact billions, of years. To be sure, the speed of rotation is not changed by the acceleration of its parts. This acceleration is entirely a matter of direction, in which the vector of the velocity changes but not the scalar speed.
Next, we notice that something on the surface of a rotating object may be thrown off. When a small merry-go-round is spun up, one must cling to it to stay on. The feeling that something is pulling you off is what we call "centrifugal force," but the curious thing is that this is not a force. You are pulled off the merry-go-round because the mass of your body, with its inertia, wants to continue in a straight line -- the line tangent to the circular motion of the merry-go-round. The true force, the centripetal force, is what holds the merry-go-round together and what your hands and arms exert to cling to the equipment. The sensation, however, is that one exerts a force to hold on simply to counter the force that pulls away. The fictitious nature of centrifugal force should remind us of Einstein's principle that gravity is itself a fictitious force. A body in free fall is accelerated without sensations either of movement or weight. Space itself is moving, and it carries the falling body into the surface, which then resists further movement. An analogous situation can be arranged with rotation. Now when we return to Einstein's Equivalence Principle, it is evident that the centrifuge does not produce "artificial gravity." It produces what physically will count as the real thing. If it looks like a duck, walks like a duck, and quacks like a duck, it's a duck. Some Metaphysics of Angular Momentum and Gravity
Relativity and the Separation Formula
The Clarke-Leibniz Debate (1715-1716)
Three Points in Kant's Theory of Space and Time
Hubble's Law: The Red Shift from the Doppler Effect is how we actually know about the radial velocity of recession, and distance, of galaxies. The Red Shift (z) is the change in a wavelength of radiation (Δλ) from the object divided by what the original wavelength was (Δλ/λ). This can be related to velocity (u) as: (z+1)2=(c+u)/(c-u) or u/c=(z2+2z)/(z2+2z+2), where the velocity of light, c=1. Distance (s) then depends on Hubble's Law: u=sH, where the Hubble Constant (H) is now taken to be 75 km/s/MPC (it is somewhere between 50 and 100 km/s/MPC). 1/H is the Hubble Time, which for the given H is 13.04 Gy. c/H, the Hubble Radius, is thus 13.04 GLY. At low values, z is virtually identical to u/c. 0.023c and z=0.023 is about the point where the two values begin to diverge. u/c cannot be larger than 1, but z can be any number up to infinity (at the velocity of light).
The June 2010 edition of Sky & Telescope reports the Hubble Constant to be 70.4 +/- 1.4 km/s/MPC [p.14]. They also report the age of the universe to be 13.75 +/- 0.11 years. It used to be that the Hubble Time would be larger than the age of the unvierse; but if the expansion of the universe is accelerating, as is now believed, then the Hubble Time is smaller than the age of the universe.
Edwin Hubble proved that spiral nebulae are external galaxies, vindicating the speculation of Immanuel Kant. At the same time, he discovered the expansion of the universe -- all this while working at the new Mt. Wilson Observatory up above Pasadena, California. In popular presentations of the history of science, I have now noticed it often said that Hubble was the first to imagine that there were external galaxies. This not only ignores Kant but it betrays complete ignorance of the intense debate about the matter all through the 19th century, culminating in a debate between Harlow Shapley and H.D. Curtis at the National Academy of Sciences in 1920. Today Mt. Wilson holds the radio and television transmission towers for much of the Los Angeles Basin and is visible, on a clear day, from the San Fernando Valley.
Schrödinger's Equation: Schrödinger, That whole side of the equation is all equivalent to the "Hamiltonian" of the state vector of the wave function. Penrose explains (p. 288) that the "classical Hamiltonian" represents the total energy of the system but that the "quantum Hamiltonian" substitutes partial differential operators with respect to the momentum for the simple occurrence of momentum in the original Hamiltonian. Thus, as in Maxwell's equations above, a great deal of the mathematics of this equation is presupposed by the symbolism. The simplicity of the equation thus conceals a level of mathematical sophistication that is rarely explained, in any way, to the lay public. Penrose's own effort to intelligibly present the details of much of this stuff, although limited and not always successful, is therefore exemplary, and in stark contrast to Stephen Hawking's A Brief History of Time, which only contained, on the publisher's advice, one equation (E = mc2).
The detailed biography of Dirac reviewed here, The Strangest Man, The Hidden Life of Paul Dirac, Mystic of the Atom, does not discuss the Dirac Equation, or even show a full form of it; but it does have considerable material of interest about Dirac and his science.
Peat thanks Bell himself (before his untimely death) for reading the manuscript, so this may be more than the typical scientific popularization. The actual equations here predict results consistent with "local reality," i.e. what Einstein wanted, with the possiblity of quantum states predetermined by "hidden variables." The two detectors (A and B) catch the "correlated" particles that are the issue in the EPR Paradox, i.e. they may have opposite spins, but which is which is undetermined, in reality as well as in knowledge, until one of them is observed. Then the spin of the other is instantaneously fixed, violating the velocity of light limitation of Special Relativity.
In the second form of the equation, the probabilities should range between negative 2 and postive 2. (Since probability ranges up to 1, which is certainty, the three positive and three negative terms of the first equation cannot add up to more than 0.) However, the predictions of "non-local" quantum mechanics are going to be different. Peat does not give equations because he says that they are different for every angle. When there were experimental tests of the equations in 1982, they were at angles where a quantum correlation of 2.70 was predicted [pp.117-118]. The experimental result was 2.697, much larger than the "local reality" prediction and very close to the quantum prediction (or >2.682 and <2.712).
Thus, Quantum Mechanics violates Special Relativity, and indeterminacy is established for particles when they cannot be observed or their states inferred from observations, i.e. the wave function completely specifies the reality. The Realists, like Einstein or Bell himself, were not going to like this, but, on the other hand, it still allows a Realism for those who take the wave function to be real, as did de Broglie or as is possible in a Kantian Quantum Mechanics.
Philosophy of Science, Physics
Philosophy of Science, Geology and Astronomy
These relationships go back to Euler's Theorem: 
Maxwell's Equations: In The Emperor's New Mind (p. 186) Roger Penrose shows us Maxwell's equations, though I can't say that I understand much of the mathematics. Penrose says that the "curl" and "div" functions are "certain combinations of partial derivative operators, taken with respect to the space coordinate." The two equations on the left relate the rate of change (the partial derivative with respect to time) of the electric field (above) and the magnetic field (below) to changes in the magnetic field and electric current (above) and to changes in the electric field (below). At least the symmetry between time on the left and space on the right is evident, though I can't say that the meaning and beauty of the equations is otherwise obvious to the non-mathematician (like me). The equations on the right Penrose says are versions of the inverse square law (above) for the electric field and, for the magnetic field, the fact that there are no isolated magnetic poles (below). Paul Dirac predicted magnetic monopoles, but none have yet turned up. Penrose doesn't explain what I think is an important aspect of equations like this: the units. Return to "Relativity and Separation Formula".
Planck's Law: The energy of Black Body radiation for a given temperature and wavelength. A "black body" is called that because it does not reflect any radiation; it only emits radiation because of its temperature. Stars are natural black bodies, though the effect can be duplicated by heating a box with only a small hole in it. The light that comes out of the box from the hole is black body radiation. -- "h" is Planck's Constant; "kB" is Boltzmann's Constant; "c" is the velocity of light; "λ" is the wavelength in meters; "T" is the temperature in Kelvins; "e" is the base of natural logarithms, Napier's Constant; and "B(λ,T)" is the "spectral radiance" or the power per volume per steradian (the solid angle corresponding to the radian), for a given wavelength and given temperature.
The units of the final solution of the equation are also of interest. When I first saw this equation, and cited it here, the final units were simply energy. But that is not what we get from the internal units of the equation: hc2/λ5 gives us units as J*s*m2/s2/m5. This resolves to J/s/m3, where J/s are units of power, namely Watts, so that we can restate the result as W/m3. I find this stated, however, as Watts per meter cubed per steradian [/sr], where a steradian is the solid angle corresponding to the radian (i.e. a circle is 2π radians in circumference). This leaves me curious where the angular measure has come from in the equation. Perhaps the term at right, which is dimensionless, actually contributes this measure.
c2 is the "second radiation constant"; and T is the temperature in Kelvins (degrees Kelvin).
Stefan-Boltzmann Law: Power emitted by a Black Body per unit area for a given temperature. Given the temperature and size of a star (and so its surface area), its total power could be calculated. -- "σ" is the Stefan-Boltzmann Constant; and T is the temperature in Kelvins. Boltzmann is also associated with "Boltzmann's Constant" in an equation for entropy, S = k ln W. This may be called "Boltzmann's Law," although Boltzmann did not actually write the equation, despite it being inscribed on his grave.
Einstein's Equation: The upper equation at right is Einstein's field equation for gravity. Roger Penrose explains and discusses it in detail in It Must be Beautiful, Great Equations of Modern Science (pp.180-212), along with the Separation Formula and other things. As I understand it, whereas a "vector" is a quantity with a direction (one dimension), like velocity, a "tensor" is quantity expressed in two dimensions (a "scalar" quantity is dimensionless). It is nice to read that when Einstein got interested in tensor calculus, he "had to enlist the help of his colleague Marcel Grossmann to teach him" (p.199). This is the basis of General Relativity, where space-time curvature, Black Holes, the Big Bang, and the whole business comes from. The second equation at right is Einstein's Equation with the addition of the "cosmological constant." The negative signs in both equations indicate the attractive nature of gravity. The positive sign on the cosmological constant makes it repulsive. Einstein famously said that this was the "biggest mistake" of his life, adding a repulsive force in order to get a static universe; but it now turns out that there may very well be a cosmological constant, one big enough to make the expansion of the universe accelerate.
If the merry-go-round is surrounded by a solid surface, the rider does not need to hold on but will fetch up against the inner side of that surface. You will feel weight against it. The merry-go-ground has become a centrifuge. It used to be common in science fiction that space stations and even space ships would be spun up to create artificial gravity within them. For some reason, it later became the norm in science fiction, e.g. both Star Trek and Firefly, that artificial gravity be induced in some static way, even though no one has a clue how that would work and rotation is a very simple mechanical business. Since it is now well understood that prolonged weightlessness is damaging to the human body, space flight cannot wait on the imaginary static gravity and will require recourse to the original expedient. This is actually what we see in the recent movie Interstellar [2014], although one of the astronauts is portrayed as being made dizzy by the spin of the spacecraft, where in fact it is weightlessness rather than rotation that tends to induce nausea.
So if you have weight, whether this is against the deck of your accelerating spaceship, or against the exterior wall of your centrifuge, or on the surface of your planet, you have physically the same thing. 
According to Einstein, anyway. And if we are looking for something to be physically the second dimension of our tensor, you definitely have it with our rotation, where speed and radius combine in kg*m2/s. Of course, we still have the peculiarity of rotation, that a force is unnecessary (in space) to maintain the rotation or induce the phenomenon of weight. We do not need to keep firing a rocket to maintain the sensation of gravitational acceleration. If this rotational gravity has nothing to do with tensors, well, then let the physicists explain it in natural language, which I have not seen them do. Perhaps their escape is the Sin of Galileo -- they don't need to explain it; you just don't understand the math. It is more like, of course, they understand the math, as math, but cannot say what it means conceptually.
Roger Penrose discusses Schrödinger's Equation in The Emperor's New Mind (p. 288). This is the deterministic equation for the undisturbed wave function
, ψ, in quantum mechanics. The application of the equation ends when the wave function is "disturbed" by observations, or even inferences, that can determine the location of particles. Then the square of the wave function is interpreted, as by Heisenberg, as the probability distribution of where the particle can be found.
like Einstein, didn't like that indeterministic aspect of quantum mechanics: Indeed, he said "I don't like it, and I wish I'd never had anything to do with it." Here the rate of change (the partial derivative with respect to time) applies to the "state vector," |〉, of the wave function. The "state vector" notation is discussed by Penrose on page 257. This is multiplied by the imaginary (i) "reduced" Planck's Constant (ℏ). (My non-mathematician's impression is that imaginary numbers often occur in equations of periodic functions, but Penrose does not discuss the significance of the imaginary number here.) [note]
Dirac's Equation: The equation at right is Paul Dirac's equation for the electron. Frank Wilczek explains and discusses it in detail in It Must be Beautiful, Great Equations of Modern Science (pp.132-160, with explanation of the terms in the equation in an appendix, pp.268-270). Dirac's equation reconciles Schrödinger's Equation with Relativity in the description of the electron. Wilczek mentions that the spin of electrons (up or down, right or left) was derived by Dirac in a natural way from the equation, which also resulted in the prediction of anti-particles, which shortly thereafter were actually observed. The way the equation is written, however, the variable x is simply said to take four values, with the electron and positron each in two spin states. This makes it look like the four values are written into the equation, rather than derived from it. So I must have missed something. I also understand that Dirac predicted the existence of particles that are magnetic monopoles, i.e. have a magnetic charge that is only North or South. These have not been observed, and Wilczek doesn't seem to mention them.
Bell's Theorem, Bell's Inequality: The equations at right are versions of John Bell's equations to test the Einstein-Podolsky-Rosen (EPR) Paradox. The form at top is Bell's own, published in 1964. Below it is a slightly rewritten form. The equations are given and discussed, with the EPR Paradox and the attendant issues, in Einstein's Moon, Bell's Theorem and the Curious Quest for Quantum Reality, by F. David Peat [Contemporary Books, Chicago, 1990, pp.111-112].
Copyright (c) 1998, 2002, 2003, 2005, 2009, 2010, 2012, 2013, 2014, 2015, 2016, 2019, 2020, 2021, 2022, 2024 Kelley L. Ross, Ph.D. All Rights Reserved
Historic Equations in Physics and Astronomy, Note 1
For instance, the equations at right for sine and cosine functions, which are periodic, contrast with the equations for hyperbolic sine and hyperbolic cosine functions, which are not periodic. The results of all the equations are real numbers; but for the sine and cosine we have imaginary powers of Napier's Constant (e), and, for the sine function, these imaginary powers are divided by the imaginary number itself. Also, the sine equation can alternatively be written: 
(because: i-1 = -i).
. How these equations with imaginaries can be evaluated with real results is examined at the Imaginary Powers of Napier's Constant, where the process is set out step by step. I would have liked to see that as a student.