This is one of the webpages of Libarid A. Maljian at the Department of Physics at CSLA at NJIT.

Union County College

Division of Science, Technology, Engineering, and Mathematics

Astronomy Beyond the Solar System, Section 051

Ast 102-051

Spring 2018

First Examination lecture notes

A star system is a star with several planets orbiting that star and moons orbiting those planets. The name of our home star system is the Solar System. There is only one star in our Solar System: the Sun. We live on planet Earth, the third planet orbiting the Sun. A galaxy is a collection of billions of star systems all held to one another through their mutual gravitational attraction. The name of our home galaxy is the Milky Way Galaxy. There are roughly one hundred billion star systems that make up the Milky Way Galaxy, and the Solar System is just one of those one hundred billion star systems. Galactic groups contain a few dozen galaxies, while galactic clusters contain hundreds of galaxies. Our Milky Way Galaxy is not a member of a galactic cluster; our Milky Way Galaxy is a member of a galactic group. The name of our home galactic group is the Local Galactic Group or just the Local Group for short. The Local Group is composed of a few dozen galaxies, although most of them are small galaxies. In fact, there are only three major galaxies in the Local Group: our Milky Way Galaxy, the Andromeda Galaxy, and the Triangulum Galaxy. Galactic superclusters are enormous organizations of hundreds of thousands of galaxies. The name of our home galactic supercluster is the Laniakea Supercluster, and our Local Group is a small galactic group on the outskirts of the Laniakea Supercluster. The observable universe contains roughly one hundred billion galaxies. Assuming that each galaxy contains on average one hundred billion star systems just like our Milky Way Galaxy, then there are roughly ten sextillion star systems in the observable universe. (Please refer to the following multiplication table, where each word is one thousand times the previous word: one, one thousand, one million, one billion, one trillion, one quadrillion, one quintillion, one sextillion, one septillion, one octillion, one nonillion, one decillion. Note that this multiplication table is only correct in American English. The same words are used for different numbers in British English.) We can summarize our location in the universe with our cosmic address. Whenever anyone asks for our mailing address, we provide a list of larger and larger organizations wherein we reside. After our name comes a house number, then a street/avenue/road/boulevard (which is a collection of houses), then a municipality (which is a collection of streets/avenues/roads/boulevards), then a county (which is a collection of municipalities), then a state (which is a collection of counties), and then a country (which is a collection of states). If we were to continue, we would then provide a continent (which is a collection of countries), then a planet (which is a collection of continents), then a star system (which is a collection of planets orbiting a star), then a galaxy (which is a collection of star systems), then a galactic group or a galactic cluster (which is a collection of galaxies), then a galactic supercluster (which is a collection of galactic groups and galactic clusters), and finally a universe (which is a collection of galactic superclusters). Every person we have ever met or ever will meet and every person we have ever heard of or will ever hear of has the same cosmic address starting with planet Earth followed by the Solar System, the Milky Way Galaxy, the Local Group, the Laniakea Supercluster, and the observable universe.

One light-year is the distance light travels in one year. We must never forget that a light-year is a length of distance, not a duration of time. This is easy to forget, since a year is a duration of time. Nevertheless, a light-year is not a duration of time; a light-year is a length of distance. How far is a light-year? Light travels roughly three hundred thousand kilometers every second through the vacuum of outer space. This is extraordinarily fast by human standards. Since three hundred thousand kilometers is the length of distance light travels in one second, we multiply this by sixty to get how far light travels in one minute, since there are sixty seconds in one minute. We multiply this result by another sixty to get how far light travels in one hour, since there are sixty minutes in one hour. We multiply this result by twenty-four to get how far light travels in one day, since there are twenty-four hours in one day. Finally, we multiply this result by 365.25 to get how far light travels in one year, since there are 365.25 days in one year. The final result of this calculation is that one light-year is roughly 9.5 trillion kilometers. This is close enough to ten trillion kilometers that throughout this course we will assume that one light-year is roughly ten trillion kilometers. Our universe is so enormous that it actually takes years for light to travel from one star system to a neighboring star system. It is for this reason that telescopes are considered to be time machines. For example, when we look at a star that is one hundred light-years distant for example, we are seeing that star as it appeared one hundred years ago, since it took that long for the light to travel from there to here. The only way to know what that star looks like at this moment is to wait another one hundred years for that light to arrive here. We see a star one thousand light-years away as it appeared one thousand years ago, since it took that long for the light to travel from there to here. The only way to know what that star looks like at this moment is to wait another one thousand years for that light to arrive here. Not only are telescopes time machines, but the human eye itself is a time machine. If we look at the Sun (which we should not since that would cause permanent blindness), we see the Sun as it appeared eight minutes ago, since it takes that long for light to travel from the Sun to the Earth. The only way to know what our Sun looks like at this moment is to wait another eight minutes for that light to arrive here. If we look at the Moon, we see it as it appeared one second ago, since it takes that long for light to travel from the Moon to the Earth. The only way to know what our Moon looks like at this moment is to wait another second for that light to arrive here. When we look at the tables and chairs around us or even these words we are reading, we are seeing those tables and chairs and these words as they appeared a few nanoseconds ago, since it takes that long for light to travel from the tables and chairs and even these words to our eyes. Obviously, these time delays are so tiny in everyday life that we do not notice them at all, but they are real nevertheless. The universe is so enormous that it takes several years for light to travel from one star system to another, it takes millions of years for light to travel from one galaxy to another, and it takes billions of years for light to travel from one side of the observable universe to the other side of the observable universe. For example, our Milky Way Galaxy has a diameter of roughly one hundred thousand light-years, and the Andromeda Galaxy is more than two million light-years from our Milky Way Galaxy.

The universe is roughly fourteen billion years old. This is unimaginably old by human standards. The human mind can comprehend seconds, minutes, hours, days, weeks, months, and years of time. Ten years is called a decade. Ten decades (which is one hundred years) is called a century, which is roughly how long most humans live. Ten centuries (which is one hundred decades or one thousand years) is called a millennium. Ten millennia is almost twice as long as all of recorded human history, but it is still miniscule compared to the age of the universe. One hundred millennia is almost how long our species Homo sapiens has existed, and one thousand millennia (which is one million years) is almost how long our genus Homo has existed. Ten million years is roughly how long our family of hominids has existed. One hundred million years ago, there were no hominids; dinosaurs roamed the Earth. One billion years is almost how long our planet Earth has existed, but this is still not near the age of the universe. Ten billion years is roughly how long our Milky Way Galaxy has existed, and this is finally almost equal to the age of the entire universe. We can gain a greater appreciation for the age of the universe through the cosmic calendar. In the cosmic calendar, we pretend that the entire history of the universe fits into one calendar year. In other words, the comic calendar pretends that the creation of the universe occurred on January 01st at midnight, and the cosmic calendar pretends that the present day is December 31st at midnight. If the creation of the universe was January 01st at midnight, we must wait until September before the Earth forms! We must then wait another month (October) before the most primitive microscopic unicellular organisms appear on Earth! We must wait another month (November) before multicellular but still microscopic organisms evolve on Earth! We must wait another month (mid-December) before macroscopic but still invertebrate animals evolve on Earth! The most primitive vertebrate animals (fishes) appear on roughly December 20th, amphibians appear on roughly December 22nd, and reptiles appear on roughly December 24th. The age of reptiles, commonly known as the age of dinosaurs, lasts from roughly December 25th to roughly December 30th, when the age of mammals begins. Hominids do not appear until December 31st at roughly 05:30 p.m., and Homo sapiens do not appear until December 31st at roughly 11:52 p.m.! Finally, all of recorded human history lasts for roughly fifteen seconds, beginning on December 31st at roughly 11:59:45 p.m.! Compared with the recorded history of our species and even compared with the entire history of our species (unrecorded and recorded), the universe is unimaginably old.

Every point on the surface of the Earth can be labeled with the Geographic Coordinate System, a pair of coordinates (two numbers) called latitude and longitude. Lines of latitude are parallel to one another. The line of latitude at 0° is commonly called the equator, but we must call this line of latitude the Terrestrial Equator in this course, for reasons we will make clear shortly. The Terrestrial Equator divides the surface of the Earth into two hemispheres: the northern hemisphere and the southern hemisphere. Latitudes in the northern hemisphere are measured in degrees north, and latitudes in the southern hemisphere are measured in degrees south. The Earth’s axis of rotation pierces the Earth at two points. One of these points is at 90°N latitude and is commonly called the north pole, but we must call this line of latitude the North Terrestrial Pole in this course, for reasons we will make clear shortly. The other point pierced by the Earth’s axis of rotation is at 90°S latitude which is commonly called the south pole, but we must call this line of latitude the South Terrestrial Pole in this course, for reasons we will make clear shortly. Lines of longitude are not parallel to one another; they all begin together at the North Terrestrial Pole, they spread apart from one another until they are furthest apart from one another at the Terrestrial Equator, and they all come back together at the South Terrestrial Pole. Lines of longitude are measured from 180°W to 180°E. Every point on the surface of the Earth has a unique latitude and longitude with only two exceptions. Although the North Terrestrial Pole has the unique latitude 90°N, the North Terrestrial Pole has an undefined longitude. Although the South Terrestrial Pole has the unique latitude 90°S, the South Terrestrial Pole has an undefined longitude. These are the only two points that suffer this tragedy; every other point on the surface of the Earth has a unique latitude and a unique longitude.

The Celestial Sphere is a giant imaginary sphere that does not exist physically but is an essential concept in observational astronomy. For the purposes of observational astronomy, we assume all astronomical objects in the sky (such as stars and galaxies) are located on this Celestial Sphere, and we assume the Earth is at the center of this Celestial Sphere. Wherever we are standing on the Earth, the sky we see above us is only half of the Celestial Sphere, since the other half of the Celestial Sphere is below us where we see the ground instead. The Horizon is a giant imaginary circle that separates the ground from the sky. Above the Horizon we see the sky which is half of the Celestial Sphere, and below the Horizon we see the ground that prevents us from seeing the other half of the Celestial Sphere.

All astronomical objects (such as stars and galaxies) can be labeled with the Horizon Coordinate System, a pair of coordinates (two numbers) called altitude and azimuth. Altitude is the angle above or below the Horizon. Lines of altitude are parallel to one another. The line of altitude at 0° is the Horizon which divides the Celestial Sphere into two hemispheres: the positive-altitude hemisphere and the negative-altitude hemisphere. Altitudes in the positive-altitude hemisphere are measured in positive degrees, and altitudes in the negative-altitude hemisphere are measured in negative degrees. Astronomical objects below the horizon have negative altitudes which means we cannot see them since the ground is in the way. Astronomical objects above the horizon have positive altitudes which means we can see them in the sky (unless it is raining or cloudy!). The point on the sky directly on top of us is +90° altitude and is called the Zenith, and the point directly opposite the Zenith is −90° altitude and is called the Nadir. Lines of azimuth are not parallel to one another; they all begin together at the Zenith, they spread apart from one another until they are furthest apart from one another at the Horizon, and they all come back together at the Nadir. Lines of azimuth are measured from 0° to 360°, starting at 0° for directly north. The azimuth is 45° for directly northeast, 90° for directly east, 135° for directly southeast, 180° for directly south, 225° for directly southwest, 270° for directly west, 315° for directly northwest, and 360° for directly north (which is actually back to 0° for directly north). We will call the point on the Horizon directly north the North Point, the point on the Horizon directly east we will call the East Point, the point on the Horizon directly south we will call the South Point, and the point on the Horizon directly west we will call the West Point. Every astronomical object on the Celestial Sphere has a unique altitude and azimuth with only two exceptions. Although the Zenith has the unique altitude +90°, the Zenith has an undefined azimuth. Although the Nadir has the unique altitude −90°, the Nadir has an undefined azimuth. These are the only two points that suffer this tragedy; every other point on the Celestial Sphere has a unique altitude and a unique azimuth.

The Horizon Coordinate System is unsatisfactory for a couple of reasons. Firstly, the Horizon Coordinates (the altitude and the azimuth) of a particular astronomical object depend upon where we are standing on the Earth. An astronomical object may have a positive altitude relative to where we live on the Earth for example, but that same astronomical object will have a negative altitude relative to someone else’s different location on the Earth. The azimuth will also be different. The problem with the Horizon Coordinate System is even worse than this however. Even if we stand at one place on the Earth and never move, the Horizon Coordinates (the altitude and the azimuth) of astronomical objects will still not remain fixed because the Earth is continuously rotating from west to east. This rotation causes astronomical objects to appear to cross the Horizon in the east (rising) and cross the Horizon again in the west (setting). The Celestial Meridian (or just the Meridian for short in this course) is a giant imaginary circle that is perpendicular to the Horizon. The Meridian begins at the North Point, runs through the Zenith, runs through the South Point, runs though the Nadir, and ends back at the North Point. As the Earth rotates from west to east, astronomical objects appear to cross the Horizon in the east (rising), and they appear to have a higher and higher altitude until they reach their highest altitude when they cross the Meridian (culminating). As the Earth continues to rotate, astronomical objects then appear to have a lower and lower altitude until they cross the Horizon in the west (setting), and they eventually rise in the east again. In summary, altitudes and azimuths both continuously change as a result of the Earth’s continuous rotation. This daily apparent motion is most obvious for the Sun. The moment when the Sun crosses the Horizon in the east is called sunrise, the moment when the Sun crosses the Meridian above the Horizon is called noon, the moment when the Sun crosses the Horizon in the west is called sunset, and the moment when the Sun crosses the Meridian again but below the Horizon is called midnight. The first half of the day before the Sun crosses the meridian at noon is called “before meridian” or “ante meridiem” which is abbreviated “a.m.,” and the second half of the day after the Sun has crossed the meridian at noon is called “afternoon” or “after meridian” or “post meridiem” which is abbreviated “p.m.”

Since the Horizon Coordinates of astronomical objects continuously change as the Earth rotates, we need another pair of coordinates that remains fixed even though the Earth is rotating. The most important such coordinate system is the Equatorial Coordinate system, a pair of coordinates (two numbers) called declination and right ascension. Lines of declination are projections of lines of latitude onto the Celestial Sphere. Since lines of latitude are parallel to one another, lines of declination are also parallel to one another. The projection of the Terrestrial Equator at 0° latitude onto the Celestial Sphere is 0° declination. This is called the Celestial Equator. We now realize why we must never simply say “equator” in this course. There are two equators! The Earth’s equator is 0° latitude and is called the Terrestrial Equator, while the Celestial Sphere’s equator is 0° declination and is called the Celestial Equator. The Celestial Equator divides the Celestial Sphere into two hemispheres: the positive-declination hemisphere and the negative-declination hemisphere. Declinations in the positive-declination hemisphere are measured in positive degrees, while declinations in the negative-declination hemisphere are measured in negative degrees. The Earth’s axis of rotation pierces the Celestial Sphere at two points. One of these points is at +90° declination and is called the North Celestial Pole. We now realize why we must never simply say “north pole” in this course. There are two north poles! The Earth’s north pole is 90°N latitude and is called the North Terrestrial Pole, while the Celestial Sphere’s north pole is +90° declination and is called the North Celestial Pole. The other point pierced by the Earth’s axis of rotation is at −90° declination and is called the South Celestial Pole. We now realize why we must never simply say “south pole” in this course. There are two south poles! The Earth’s south pole is 90°S latitude and is called the South Terrestrial Pole, while the Celestial Sphere’s south pole is −90° declination and is called the South Celestial Pole. Lines of right ascension are analogous to lines of longitude. Since lines of longitude are not parallel to one another, lines of right ascension are also not parallel to one another; they all begin together at the North Celestial Pole, they spread apart from one another until they are furthest apart from one another at the Celestial Equator, and they all come back together at the South Celestial Pole. Right ascension is measured in right-ascension hour-angles, running from 00^h to 24^h. (We will clearly define the line of right ascension at 00^h in a moment.) Each right-ascension hour-angle is actually 15° of right ascension, since 360° divided by 24^h equals 15°/1^h. Every point on the Celestial Sphere has a unique declination and right ascension with only two exceptions. Although the North Celestial Pole has the unique declination +90°, the North Celestial Pole has an undefined right ascension. Although the South Celestial Pole has the unique declination −90°, the South Celestial Pole has an undefined right ascension. These are the only two points that suffer this tragedy; every other point on the Celestial Sphere has a unique declination and a unique right ascension.

As the Earth rotates, the Horizon Coordinates of astronomical objects (the altitude and the azimuth) continuously change, but the Equatorial Coordinates (the declination and the right ascension) of all astronomical objects remain fixed. (This is not exactly the truth. The Earth’s axis of rotation is slowly precessing and nutating, which cause changes in Equatorial Coordinates. Stars and galaxies are also physically moving through the universe; these motions cause additional changes in Equatorial Coordinates. Nevertheless, all these changes are so small that the naked eye does not notice them. For the purposes of this discussion, we will assume that the Equatorial Coordinates of astronomical objects are fixed as a satisfactory approximation.) Now suppose there happens to be an astronomical object (such as a star or galaxy) at either the North Celestial Pole or the South Celestial Pole. Such an object will have fixed Horizon Coordinates even though the Earth is continuously rotating. It just so happens that there is a star almost exactly at the North Celestial Pole. Therefore, this star appears to remain fixed as everything else in the sky appears to rotate around it. This star has several names: α (alpha) Ursae Minoris, Polaris, the Pole Star, or the North Star. There is no star almost exactly at the South Celestial Pole, but if there were such a star it would be called the South Star. The North Star always has an azimuth of 0° since it is directly north, and the South Star (if there were one) always has an azimuth of 180° since it is directly south. The altitude of the North Star is always equal to our latitude on Earth, while the altitude of the South Star (if there were one) is always equal to the opposite of our latitude on Earth. For example, if we lived at 40°N latitude, then the Horizon Coordinates of the North Star would be +40° altitude and 0° azimuth, while the Horizon Coordinates of the South Star (if there were one) would be −40° altitude and 180° azimuth. As another example, if we lived at 60°S latitude, then the Horizon Coordinates of the North Star would be −60° altitude and 0° azimuth, while the Horizon Coordinates of the South Star (if there were one) would be +60° altitude and 180° azimuth. If we live in the northern hemisphere, all astronomical objects (such as stars and galaxies) close enough to the North Star at the North Celestial Pole would never appear to set; they would just appear to circle around the North Celestial Pole as the Earth rotates. Also from the northern hemisphere, all astronomical objects (such as stars and galaxies) close enough to the South Star at the South Celestial Pole would never appear to rise; they would just circle around the South Celestial Pole as the Earth rotates. All of these astronomical objects are called circumpolar, since they appear to circle around the celestial poles. The situation is reversed in the southern hemisphere. If we live in the southern hemisphere, all astronomical objects (such as stars and galaxies) close enough to the South Star at the South Celestial Pole would never appear to set; they would just appear to circle around the South Celestial Pole as the Earth rotates. Also from the southern hemisphere, all astronomical objects (such as stars and galaxies) close enough to the North Star at the North Celestial Pole would never appear to rise; they would just circle around the North Celestial Pole as the Earth rotates. If we lived at the North Terrestrial Pole, our latitude would be 90°N, but this means that the North Celestial Pole is at +90° altitude which means it is at the Zenith. Also from the North Terrestrial Pole, the South Celestial Pole is at −90° altitude which means it is at the Nadir. As the Earth rotates, the entire sky would be appear to be circumpolar, with half of the sky never setting and the other half of the sky never rising. This stands to reason. At the North Terrestrial Pole, there is no east for anything to rise from, nor is there west for anything to set to. All directions are south! If we lived at the South Terrestrial Pole, our latitude would be 90°S, but this means that the North Celestial Pole is at −90° altitude which means it is at the Nadir. Also from the South Terrestrial Pole, the South Celestial Pole is at +90° altitude which means it is at the Zenith. As the Earth rotates, the entire sky would be appear to be circumpolar, with half of the sky never setting and the other half of the sky never rising. This stands to reason. At the South Terrestrial Pole, there is no east for anything to rise from, nor is there west for anything to set to. All directions are north! If we lived at the Terrestrial Equator, our latitude would be 0°, but this means that the North Celestial Pole is at 0° altitude which means it is at the North Point. Also from the Terrestrial Equator, the South Celestial Pole is at 0° altitude which means it is at the South Point. As the Earth rotates, nothing in the entire sky would be appear to be circumpolar; everything appears to rise and set. Also, the Terrestrial Equator is the only location on the Earth where the three giant circles (the Horizon, the Meridian, and the Celestial Equator) are all perpendicular to one another.

For thousands of years, humans have looked up into the sky and observed that the stars appear to be fixed relative to one another. Many cultures formed pictures from groups of stars in the sky and named them constellations. However, the modern definition of a constellation is a region of the Celestial Sphere defined by a boundary. In other words, anything on the Celestial Sphere (whether it is a star, a galaxy, a planet, or even the Sun or the Moon) is considered to be within a certain constellation if it is within the boundary that defines that constellation. The entire Celestial Sphere is divided into eighty-eight modern constellations. A group of stars that is not one of these eighty-eight modern constellations is called an asterism.

There are several circumpolar constellations worth discussing. Ursa Major (the big bear) includes seven bright stars that form the Big Dipper asterism. Two of the stars in the Big Dipper asterism can be used to find the North Star, also known as α (alpha) Ursae Minoris or Polaris or the Pole Star. This star is within Ursa Minor (the little bear) which includes the Little Dipper asterism. Cassiopeia (the queen of Aethiopia) has five bright stars shaped like the letter W. Three of the stars in the Big Dipper asterism can be used to find Arcturus, the brightest star in Boötes (the shepherd). There are a few summer constellations worth discussing. Cygnus (the swan) includes the Northern Cross asterism. The brightest star in Cygnus is Deneb. The brightest star in Lyra (the harp) is Vega. The brightest star in Aquila (the eagle) is Altair. The Summer Triangle asterism is formed by connecting Vega, Deneb, and Altair. There are several winter constellations worth discussing. Orion (the hunter) includes seven bright stars, with Betelgeuse and Rigel among them. The sword of Orion is actually the Orion Nebula, which we will discuss later in this course. The brightest star in Canis Major (the big dog) is Sirius (the dog star). The brightest star in Canis Minor (the little dog) is Procyon. The Winter Triangle asterism is formed by connecting Betelgeuse, Sirius, and Procyon. The brightest star in the constellation Auriga (the charioteer) is Capella.

For thousands of years, humans have looked up into the sky and observed that the Sun appears to wander around the Celestial Sphere. The giant circle that the Sun appears to wander around is called the ecliptic, and it takes the Sun one year to appear to take one complete journey around the ecliptic. The constellations along the ecliptic are called the zodiac constellations. If we begin with Aquarius (the water bearer), the next zodiac constellation is Pisces (the fish) followed by Aries (the ram). Next comes Taurus (the bull). The brightest star in Taurus is Aldebaran, and the Pleiades is a star cluster within the constellation Taurus. Next comes Gemini (the twins) with the two bright stars Pollux and Castor. Next comes Cancer (the crab) followed by Leo (the lion). The brightest star in Leo is Regulus. Next comes Virgo (the virgin). The brightest star in Virgo is Spica. Next comes Libra (the scales) followed by Scorpius (the scorpion). The brightest star in Scorpius is Antares. Next is Ophiuchus (the serpent bearer) followed by Sagittarius (the centaur archer) followed by Capricornus (the sea goat) followed by Aquarius, which is where we began our journey. The Sun takes one year to wander once around the ecliptic. As it does so, it appears to wander from within one zodiac constellation to another, spending approximately one month within each of these zodiac constellations.

The ecliptic intersects the Celestial Equator at two points called the equinoxes: the vernal equinox (or the spring equinox) is where the Sun appears to be on the ecliptic on roughly March 21st every year, and the autumnal equinox is where the Sun appears to be on the ecliptic on roughly September 21st every year. Both equinoxes are 0° declination since they are on the Celestial Equator. Astronomers have agreed to define 00^h right ascension by the line of right ascension that passes through the vernal equinox (the spring equinox). The autumnal equinox is 12^h right ascension since it is on the opposite side of the Celestial Sphere from the vernal/spring equinox. The furthest angle from the Celestial Equator the Sun ever wanders along the ecliptic is roughly 23½°. This occurs at two points halfway between the equinoxes called the solstices: the summer solstice is where the Sun appears to be on the ecliptic on roughly June 21st every year, and the winter solstice is where the Sun appears to be on the ecliptic on roughly December 21st every year. The summer solstice is 06^h right ascension since it is halfway from the vernal equinox at 00^h to the autumnal equinox at 12^h. The winter solstice is 18^h right ascension since it is halfway from the autumnal equinox at 12^h to the vernal equinox at 24^h (which is the same as 00^h since 24^h is all the way around the Celestial Sphere back to 00^h). The summer solstice is where the Sun has a positive maximum declination of +23½°, and the winter solstice is where the Sun has a negative maximum declination of −23½°. In summary, the Equatorial Coordinates of the vernal/spring equinox (where the Sun appears to be on the ecliptic on roughly March 21st every year) is 0° declination and 00^h right ascension, the Equatorial Coordinates of the summer solstice (where the Sun appears to be on the ecliptic on roughly June 21st every year) is +23½° declination and 06^h right ascension, the Equatorial Coordinates of the autumnal equinox (where the Sun appears to be on the ecliptic on roughly September 21st every year) is 0° declination and 12^h right ascension, and the Equatorial Coordinates of the winter solstice (where the Sun appears to be on the ecliptic on roughly December 21st every year) is −23½° declination and 18^h right ascension.

A wave is a propagating (traveling) disturbance. This implies that a wave requires a medium through which to propagate. (There can never exist a propagating disturbance if there is nothing there to disturb!) A transverse wave is a wave where the direction of the disturbance is perpendicular to the direction of propagation, while a longitudinal wave is a wave where the direction of the disturbance is parallel and antiparallel to the direction of propagation. Light is a real-life example of a transverse wave, while sound is a real-life example of a longitudinal wave. A wave can have a component of its disturbance perpendicular to the direction of propagation and another component parallel and antiparallel to the direction of propagation. In other words, a wave can be both transverse and longitudinal. Water waves that we see in the ocean are a real-life example of a wave that is both transverse and longitudinal. The maximum magnitude of a wave’s disturbance is called the amplitude of the wave, and these amplitudes occur at what we will call crests (maximum positive amplitude) and troughs (maximum negative amplitude). The distance from one crest to the next crest (which is also the distance from one trough to the next trough) is called the wavelength of the wave and is always given the symbol λ, the lowercase Greek letter lambda. (The word wavelength is misleading, since it may lead us to conclude that it is the length of the entire wave, which it is not. The wavelength of a wave is the length of only one cycle of the wave.) The frequency of a wave is the number of crests passing a point every second, and it is also the number of troughs passing a point every second. The frequency of a wave is also how many cycles or vibrations or oscillations the wave executes every second. In other words, the frequency of a wave is how frequently the wave is vibrating or oscillating, which is why it is called the frequency! A high-frequency wave is vibrating/oscillating many times every second, while a low-frequency wave is not vibrating/oscillating many times every second. We will use the symbol f for frequency, and its units are cycles per second or vibrations per second or oscillations per second. This unit is called a hertz with the abbreviation Hz. In other words, one hertz (Hz) is one cycle per second or one vibration per second or one oscillation per second. A kilohertz is one thousand hertz or one thousand cycles per second, since the metric prefix kilo- always means thousand. (For example, one kilometer is one thousand meters and one kilogram is one thousand grams.) A megahertz is one million hertz or one million cycles per second, since the metric prefix mega- always means million. On the amplitude-modulation radio band (AM radio), the radio-station numbers are kilohertz, while on the frequency-modulation radio band (FM radio), the radio-station numbers are megahertz.

The speed of a wave is a function of the medium through which it propagates. For example, the speed of sound is some speed through gases such as air, a faster speed through liquids, and an even faster speed through solids. The speed of sound through air is not even fixed; it actually changes as the temperature of the air changes. As another example, the speed of light is some speed through gases such as air, a slower speed through liquids such as water, and an even slower speed through solids such as glass. The speed of any wave is given by the equation v = f λ, where v is the speed (the velocity) of the wave. If we solve this equation for the frequency, we deduce that f = v / λ. Therefore, frequency and wavelength are inversely proportional to each other. Waves with higher frequencies have shorter wavelengths, while waves with lower frequencies have longer wavelengths.

The amplitude of any wave determines its energy. In particular, the energy of a wave is directly proportional to the square of its amplitude. Therefore, a wave with a larger amplitude has more energy, while a wave with a smaller amplitude has less energy. For example, the amplitude of a sound wave determines its loudness. A sound wave with a larger amplitude is more loud, while a sound wave with a smaller amplitude is more quiet. As another example, the amplitude of a light wave determines its brightness. A light wave with a larger amplitude is more bright, while a light wave with a smaller amplitude is more dim. The frequency of a wave is difficult to interpret physically; we must interpret the frequency of a wave on a case-by-case basis. For example, the frequency of a sound wave is its pitch, meaning that a sound wave with a higher frequency has a higher pitch while a sound wave with a lower frequency has a lower pitch. As another example, the frequency of a visible light wave is its color. In particular, a visible light wave with a high frequency is blue or violet, a visible light wave with a low frequency is red or orange, and a visible light wave with a middle frequency is yellow or green. In order starting from the lowest frequency (which is also the longest wavelength), the colors of visible light are red, orange, yellow, green, blue, indigo, and violet at the highest frequency (which is also the shortest wavelength). This is why the colors of the rainbow are in this order; a rainbow reveals the correct sequence of colors as determined by either the frequency or the wavelength. This sequence of colors can be memorized with the mnemonic roy-g-biv.

Whereas the frequency and the wavelength of a wave are constrained to one another through the equation v = f λ, no universal equation constrains the amplitude with the frequency. Therefore, a wave can have a large amplitude and a high frequency, a wave can have a large amplitude and a low frequency, a wave can have a small amplitude and a high frequency, and a wave can have a small amplitude and a low frequency. In other words, all of these combinations are physically possible. For example, a sound wave with a large amplitude and a high frequency is a loud high-pitch sound, a sound wave with a large amplitude and a low frequency is a loud low-pitch sound, a sound wave with a small amplitude and a high frequency is a quiet high-pitch sound, and a sound wave with a small amplitude and a low frequency is a quiet low-pitch sound. As another example, a visible light wave with a large amplitude and a high frequency is bright blue, a visible light wave with a large amplitude and a low frequency is bright red, a visible light wave with a small amplitude and a high frequency is dim blue, and a visible light wave with a small amplitude and a low frequency is dim red.

A wave that is a propagating (traveling) disturbance through a material medium (a medium composed of atoms) is called a mechanical wave. Sound waves and water waves are real-life examples of mechanical waves. We will call a wave that is a propagating disturbance through an abstract field medium a field wave. Light waves and gravitational waves are real-life examples of field waves. Light waves are propagating disturbances through the electromagnetic field created by charges, and gravitational waves are propagating disturbances through the gravitational field created by masses. Since light waves are propagating disturbances through the electromagnetic field, then light is actually an electromagnetic wave. The Electromagnetic Spectrum is a list of all the different types of electromagnetic waves in order as determined by either the frequency or the wavelength. Starting with the lowest frequencies (which are also the longest wavelengths), we have radio waves, microwaves, infrared, visible light (the only type of electromagnetic wave our eyes can see), ultraviolet, X-rays, and gamma rays at the highest frequencies (which are also the shortest wavelengths). All of these are electromagnetic waves. Therefore, all of them may be regarded as different forms of light. They all propagate at the same speed of light through the vacuum of outer space for example. We now realize that whenever we use the word “light” in everyday life, we mean to use the word “visible light,” since this is the type of light that our eyes can actually see. The visible part of the Electromagnetic Spectrum is actually quite narrow. Nevertheless, the visible part of the Electromagnetic Spectrum can be subdivided. In order, the subcategories of the visible part of the Electromagnetic Spectrum starting at the lowest frequency (which is also the longest wavelength) are red, orange, yellow, green, blue, indigo, and violet at the highest frequency (which is also the shortest wavelength). We now realize why electromagnetic waves just before visible light are called infrared, since their frequencies (or wavelengths) are just beyond red visible light. (In other words, infrared light is more red than red!) We also realize why electromagnetic waves just after visible light are called ultraviolet, since their frequencies (or wavelengths) are just beyond violet visible light. (In other words, ultraviolet light is more purple than purple!)

A wave detector will detect a frequency shift if the source of the wave is moving or if the detector of the wave is moving or if both are moving. This is called the Doppler Effect or the Doppler Shift. In particular, there is a higher-frequency shift if there is advancing motion (the source moves toward the detector, the detector moves toward the source, or both). Conversely, there is a lower-frequency shift if there is receding motion (the source moves away from the detector, the detector moves away from the source, or both). For example, the human ear is a detector of sound waves. Since the frequency of a sound wave is its pitch, our ears hear higher pitches as a police/ambulance/firetruck siren for example moves towards us, and our ears hear lower pitches as a police/ambulance/firetruck siren for example moves away from us. Although the human eye is a detector of light waves, the Doppler Effect for light is too small to be noticed by the naked eye. Nevertheless, astronomers use instruments to measure the Doppler Effect for light. Our instruments detect tiny higher-frequency shifts (which are also shorter-wavelength shifts) of light from stars and galaxies that move towards us; astronomers use the word “blueshift” for higher-frequency shifts (or shorter-wavelength shifts) of any type of electromagnetic wave. Conversely, our instruments detect tiny lower-frequency shifts (which are also longer-wavelength shifts) of light from stars and galaxies that move away from us; astronomers use the word “redshift” for lower-frequency shifts (or longer wavelength shifts) of any type of electromagnetic wave. By measuring these blueshifts and redshifts, astronomers can determine not only the direction of motion of stars and galaxies but in addition their speed of motion (whether fast or slow).

According to classical electromagnetic theory, radio, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays are electromagnetic waves. However, according to modern electromagnetic theory (quantum electromagnetism), radio, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays are actually composed of particles called photons. In other words, a photon is a particle of the electromagnetic field. The energy of a photon is given by the equation E_photon = h f, where h is the Planck constant, one of the fundamental physical constants of the universe. According to this equation, the energy of a photon is directly proportional to the frequency. Therefore, higher frequencies of light are actually composed of higher-energy photons, and lower frequencies of light are actually composed of lower-energy photons. Therefore, the Electromagnetic Spectrum starting with the lowest photon energy is radio, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays at the highest photon energy. Notice that ultraviolet photons have greater energy than visible photons; this is why ultraviolet causes suntans and sunburns. Also notice that X-ray photons have even greater energy, so much so that they penetrate most substances. This is why X-rays are used to take X-rays! Within the visible part of the Electromagnetic Spectrum starting at the lowest photon energy is red, orange, yellow, green, blue, indigo, and violet at the highest photon energy.

If every blue photon in the universe has more energy than every red photon in the universe, how can red light ever be brighter than blue light? Bright red light must have many more photons than dim blue light. This will ensure that the total energy of the red light is greater than the total energy of the blue light even though each red photon actually has less energy than each blue photon. In other words, the total energy of light is the number of photons multiplied by the energy of each photon. This reveals a connection between the classical wave theory of light and the modern quantum theory of light. According to the classical wave theory of light, the total energy of light is directly proportional to the square of its amplitude. According to the modern quantum theory of light, the total energy of light is directly proportional to the product of the number of photons and the frequency (since the energy of each photon is proportional to the frequency). Therefore, the square of the amplitude of light must be proportional to the product of the number of photons and the frequency.

All materials in the universe (such as solids, liquids, and gases) are composed of (made of) atoms. Atoms are composed of (made of) even smaller particles. The center of the atom is called the nucleus. (The center of anything is often called its nucleus. For example, the center of a biological cell is called the cellular nucleus. The center of an entire galaxy is called the galactic nucleus. The center of an atom should really be called the atomic nucleus, but we will often be lazy and just say nucleus.) Around the atomic nucleus are electrons. The atomic nucleus is positively charged, and the electrons are negatively charged. In fact, it is the attraction between the positive nucleus and the negative electrons that holds the atom together. (Like charges repel and unlike charges attract. In other words, positive and positive repel, negative and negative repel, and positive and negative attract.) The atomic nucleus is composed of even smaller particles: protons and neutrons. The protons are positively charged. In fact, it is because of the protons that the entire atomic nucleus has a positive charge. The neutrons have zero electrical charge. In other words, neutrons are neutral. This is why they are called neutrons!

The number of protons in the nucleus is the single most important number of the atom. It is so important that it is called the atomic number. The atomic number, which is always the number of protons in the nucleus, is so important that an atom is named solely based on its atomic number. For example, every atom in the universe with twelve protons in its nucleus is considered to be a magnesium atom. As another example, every atom in the universe with seven protons in its nucleus is considered to be a nitrogen atom. We are not saying that the number of neutrons is irrelevant, nor are we saying that the number of electrons is irrelevant. The neutrons and the electrons are quite important. We are saying that the atomic number is always the number of protons, and the name of an atom is based only upon its atomic number (the number of protons).

If we change the number of electrons, we change the charge of the atom. Why? Imagine an atom where the number of electrons balances the number of protons. Since protons are positive and electrons are negative, the atom is neutral overall. Now imagine we add extra electrons to the atom. Since electrons are negative, the atom will no longer be neutral overall; it will be negative overall. Imagine we removed electrons from the atom in the first place. Now the atom will be positive overall. A charged atom is called an ion. Therefore, changing the number of electrons results in ions. For example, consider the sodium atom with the symbol Na. The atomic number of sodium is eleven, meaning that every sodium atom in the universe has eleven protons. We will make this clear with a subscript before the atom’s symbol like this: ₁₁Na. If the sodium atom were neutral, it would have eleven electrons as well, but suppose we add three more electrons. Since electrons are negative, we now have an ion with a charge of negative three. We write the charge as a superscript after the name of the atom like this: ₁₁Na^3–. (Even though the charge is read “negative three,” the superscript is written in the strange way “3–.”) As another example, consider the aluminum atom with the symbol Al. The atomic number of aluminum is thirteen, meaning that every aluminum atom in the universe has thirteen protons. We make this clear with a subscript before the atom’s symbol like this: ₁₃Al. If the aluminum atom were neutral, it would have thirteen electrons as well, but suppose we remove two of the electrons. We now have an ion with a charge of positive two. We write the charge as a superscript after the name of the atom like this: ₁₃Al²⁺. (Even though the charge is read “positive two,” the superscript is written in the strange way “2+.”) A positive ion is called a cation, and a negative ion is called an anion.

If we change the number of neutrons, we do not get ions. Why? Neutrons are neutral. So, adding or removing neutrons does not change the charge at all. If we change the number of neutrons, what we are changing is the mass of the atom. The atomic mass of an atom is the number of protons plus the number of neutrons. (Why do we not include the electrons when calculating the mass of the atom? An electron is almost two thousand times less massive than a proton or a neutron. Thus, as far as the atomic mass is concerned, the electrons do not count. A proton and a neutron have roughly equal amounts of mass, which is why we count them equally.) When we change the number of neutrons, we are changing the atomic mass of the atom. Two atoms with the same atomic number but different atomic mass are called isotopes. Therefore, changing the number of neutrons results in isotopes. For example, consider the carbon atom with the symbol C. The atomic number of carbon is six, meaning that every carbon atom in the universe has six protons. We make this clear with a subscript before the atom’s symbol like this: ₆C. However, carbon has three isotopes: carbon-twelve, carbon-thirteen, and carbon-fourteen. An isotope is named based on its atomic mass. Thus, the numbers twelve, thirteen, and fourteen are the atomic masses of these isotopes of carbon. We make this clear with a superscript before the atom’s symbol like this: for carbon-twelve, for carbon-thirteen, and for carbon-fourteen. Notice that carbon has six protons no matter what, but the carbon-fourteen isotope has eight neutrons, since six plus eight equals fourteen. The carbon-thirteen isotope has seven neutrons, since six plus seven equals thirteen. The carbon-twelve isotope has six neutrons, since six plus six equals twelve. As another example, consider the oxygen atom with the symbol O. The atomic number of oxygen is eight, meaning that every oxygen atom in the universe has eight protons. We make this clear with a subscript before the atom’s symbol like this: ₈O. However, oxygen has three isotopes: oxygen-sixteen, oxygen-seventeen, and oxygen-eighteen. An isotope is named based on its atomic mass. Thus, the numbers sixteen, seventeen, and eighteen are the atomic masses of these isotopes of oxygen. We make this clear with a superscript before the atom’s symbol like this: for oxygen-sixteen, for oxygen-seventeen, and for oxygen-eighteen. Notice that oxygen has eight protons no matter what, but the oxygen-eighteen isotope has ten neutrons, since eight plus ten equals eighteen. The oxygen-seventeen isotope has nine neutrons, since eight plus nine equals seventeen. The oxygen-sixteen isotope has eight neutrons, since eight plus eight equals sixteen.

We can put all of this together with the following examples. Consider the neon atom with the symbol Ne. Now suppose we write ^2–. This neon atom has ten protons, eleven neutrons, twelve electrons, an atomic number of ten, an atomic mass of twenty-one, and a charge of negative two. As another example, consider the boron atom with the symbol B. (There are borons in this class!) Now suppose we write ³⁺. This boron atom has five protons, four neutrons, two electrons, an atomic number of five, an atomic mass of nine, and a charge of positive three.

The two most important atoms in this course are hydrogen and helium, since most of the atoms in the universe are hydrogen atoms, and helium atoms are the second most common atom in the universe. The symbol for the hydrogen atom is H. The atomic number of hydrogen is one, meaning that every hydrogen atom in the universe has one proton in its nucleus. We make this clear with a subscript before the atom’s symbol like this: ₁H. However, hydrogen has three isotopes: hydrogen-one which is written , hydrogen-two which is written , and hydrogen-three which is written . Hydrogen is so important that these three isotopes have additional names besides hydrogen-one, hydrogen-two, and hydrogen-three. Hydrogen-one is also called protium. It is also called “ordinary hydrogen” since most of the hydrogen atoms in the universe are this isotope. Hydrogen-two is also called deuterium. It is also called “heavy hydrogen” since it is twice as massive as “ordinary hydrogen.” (When an oxygen atom chemically bonds to two “ordinary hydrogen” atoms, the result is a molecule of “ordinary water.” When an oxygen atom chemically bonds to two “heavy hydrogen” atoms, the result is a molecule of “heavy water.”) Hydrogen-three is also called tritium. Where do the names protium, deuterium, and tritium come from? The atomic number of hydrogen is one, meaning that every hydrogen atom in the universe has one proton in its nucleus. This means that the hydrogen-one isotope (or protium or “ordinary hydrogen”) has no neutrons in its nucleus, since one plus zero equals one. In other words, its nucleus is a single proton all by itself. This is the simplest nucleus in the universe. Since the nucleus is a proton, when we put an electron around it to build the entire atom, we name the entire atom protium, since its nucleus is a proton. The hydrogen-two isotope (or deuterium or “heavy hydrogen”) has one neutron in its nucleus, since one plus one equals two. In other words, its nucleus is a proton and a neutron stuck to each other. A proton and a neutron stuck to each other is called a deuteron. Since the nucleus is a deuteron, when we put an electron around it to build the entire atom, we name the entire atom deuterium, since its nucleus is a deuteron. The hydrogen-three isotope (or tritium) has two neutrons in its nucleus, since one plus two equals three. In other words, its nucleus is a proton and two neutrons all stuck to one another. A proton and two neutrons all stuck to one another is called a triton. Since the nucleus is a triton, when we put an electron around it to build the entire atom, we name the entire atom tritium, since its nucleus is a triton. The helium atom with the symbol He has an atomic number of two, meaning that every helium atom in the universe has two protons in its nucleus. We make this clear with a subscript before the atom’s symbol like this: ₂He. Most of the helium atoms in the universe are the helium-four isotope which is written . Helium-four is also called “ordinary helium” since most of the helium atoms in the universe are this isotope. The nucleus of helium-four is composed of two protons and two neutrons, since two plus two equals four. In other words, the nucleus of helium-four is two protons and two neutrons all stuck to one another. Two protons and two neutrons all stuck to one another is called an alpha particle. To summarize, the nucleus of the protium atom is a proton, the nucleus of the deuterium atom is a deuteron, the nucleus of the tritium atom is a triton, and the nucleus of the “ordinary helium” atom is an alpha particle.

Electrons do not orbit an atomic nucleus like planets orbit the Sun. In fact, the electrons do not orbit at all; they exist in abstract quantum-mechanical states that we will not explain deeply in this course. We simply state that there are definite energy levels within an atom. Some levels are at lower energies, and other levels are at higher energies. If an electron wishes to change its energy from a lower level to a higher level, it must absorb a photon, a particle of light. However, not any photon will accomplish this transition. The energy of the photon absorbed must be exactly equal to the difference in energy between the two levels. If an electron wishes to change its energy from a higher level to a lower level, it must emit (spit out) a photon, but not any photon will accomplish this transition. The energy of the photon emitted must be exactly equal to the difference in energy between the two levels. Therefore, an atom can only absorb or emit photons of very specific energies (or very specific frequencies or very specific wavelengths). The list of all the allowed photon energies (or frequencies or wavelengths) an atom is permitted to absorb is called the absorption spectrum of the atom, and the list of all the allowed photon energies (or frequencies or wavelengths) an atom is permitted to emit is called the emission spectrum of the atom. Since different atoms have different energy levels, every atom has its own unique spectrum, different from the spectra of all other atoms. Therefore, the spectrum of an atom is rather like its fingerprint, enabling us to uniquely identify an atom. A spectacular example of this is the discovery of the Sun’s composition. How do we know which atoms compose the Sun? In the early 1800s, Joseph von Fraunhofer discovered missing wavelengths in the Sun’s light. These absorption lines are called Fraunhofer lines in his honor. By measuring the wavelengths of these absorption lines and consulting a table of absorption spectra, we can determine which atoms absorbed these missing wavelengths and thus determine the composition of the Sun. We discover that the Sun is composed of all the atoms on the Periodic Table of Elements, but not in equal amounts. Only two atoms account for close to one hundred percent of the Sun’s mass; all the other atoms on the Periodic Table of Elements account for only a tiny percentage of the Sun’s mass. What are these two elements that account for close to one hundred percent of the Sun’s mass? We discover from the Fraunhofer lines in sunlight that hydrogen atoms account for roughly seventy-five percent (three-quarters) of the Sun’s mass. What about the remaining twenty-five percent (one-quarter) of the Sun’s mass? The wavelengths of the remaining absorption lines were not found in any atom’s tabulated absorption spectrum! Apparently, one-quarter of the Sun’s mass is composed of a new atom never before discovered! This newly-discovered atom was called helium, named after Helios the personification of the Sun in ancient Greek mythology. In the early 1900s, helium was discovered on Earth as the product of certain nuclear reactions, and today we find helium everywhere on Earth (in blimps and in party balloons for example). Nevertheless, helium was first discovered from its absorption lines in the Sun’s light!

What is temperature? What do we mean when we say something is hot? What do we mean when we say something is cold? The temperature of an object is a measure of the average energy of the atoms that compose that object. In this course, we may assume that the average energy of atoms corresponds to their average speed. In other words, the atoms of a hotter object are moving relatively faster, whereas the atoms of a cooler object are moving relatively slower. There are two scales of temperature in common use: degrees fahrenheit and degrees celsius. However, neither degrees fahrenheit nor degrees celsius are acceptable units of temperature. What is wrong with these two scales? The zero is in the wrong place in both of these scales. What do we mean by this? If the temperature of an object is a measure of the average speed of its atoms, then the coldest possible temperature of our universe is the temperature at which all the atoms of an object completely stop moving. After all, there is no slower speed than not moving at all! The temperature at which all atoms completely stop moving is commonly called absolute zero. However, absolute zero temperature is not zero degrees fahrenheit nor is it zero degrees celsius. Atoms are still moving at zero degrees fahrenheit, and atoms are still moving at zero degrees celsius. There are still negative temperatures on both of these scales (commonly called temperatures below zero) where the atoms move slower still. The absolute zero of temperature when all atoms completely stop moving is exactly negative 273.15 degrees celsius or exactly negative 459.67 degrees fahrenheit. A correct unit of temperature must assign the number zero to the absolute zero of temperature. The simplest way to correct degrees celsius is to add 273.15 to all degrees celsius. What does this accomplish? Since absolute zero is negative 273.15 degrees celsius, then adding 273.15 would yield zero, and all other temperatures would be positive. The simplest way to correct degrees fahrenheit is to add 459.67 to all degrees fahrenheit. What does this accomplish? Since absolute zero is negative 459.67 degrees fahrenheit, then adding 459.67 would yield zero, and all other temperatures would be positive. When we correct the celsius scale by adding 273.15, we get correct units of temperature called kelvins. When we correct the fahrenheit scale by adding 459.67, we get correct units of temperature called rankines. To summarize, absolute zero temperature is negative 273.15 degrees celsius or negative 459.67 degrees fahrenheit on these unacceptable temperature scales, but absolute zero temperature is zero kelvins or zero rankines using acceptable units of temperature. We will use kelvins throughout this course. It is somewhat difficult growing accustomed to kelvins. For example, most humans consider 280 kelvins to be uncomfortably cold, most humans consider 300 kelvins to be a comfortable room temperature, and most humans consider 320 kelvins to be uncomfortably hot.

The Third Law of Thermodynamics states that it is impossible to cool an object to absolute zero temperature in a finite number of processes. It follows that every object in the universe has a temperature that is warmer than absolute zero. Therefore, every object in the universe has its atoms moving at some average speed. Since atoms are composed of protons, neutrons, and electrons and since protons and electrons are charged, every object in the universe radiates electromagnetic waves from its moving atoms. (The neutrons also contribute to these electromagnetic waves. Although neutrons are neutral, they still have electromagnetic properties.) The amount of energy radiated from a hot, dense object often follows the blackbody spectrum, which is a continuous spectrum with its primary radiation within a band of the Electromagnetic Spectrum determined by the temperature of the object. In particular, hotter temperatures correspond to higher photon energies (which are also at higher frequencies and shorter wavelengths), while cooler temperatures correspond to lower photon energies (which are also at lower frequencies and longer wavelengths). In other words, a hot, dense object’s primary radiation is displaced as its temperature changes. At extremely cold temperatures (close to absolute zero), objects radiate primarily in the microwave part of the Electromagnetic Spectrum. At a few hundred kelvins (such as room temperatures), objects radiate primarily in the infrared part of the Electromagnetic Spectrum. At one or two thousand kelvins, objects radiate primarily red visible light. At three or four thousand kelvins, objects radiate primarily orange visible light. At five or six thousand kelvins, objects radiate primarily yellow visible light. At roughly ten thousand kelvins, objects radiate primarily blue visible light. At hundreds of thousands of kelvins, objects radiate primarily in the ultraviolet part of the Electromagnetic Spectrum. At millions of kelvins, objects radiate primarily in the X-ray part of the Electromagnetic Spectrum. At tens of millions of kelvins, objects radiate primarily in the gamma-ray part of the Electromagnetic Spectrum. Notice how hotter temperatures displace the primary radiation to higher and higher photon energies (which are also higher and higher frequencies and shorter and shorter wavelengths), while cooler temperatures displace the primary radiation to lower and lower photon energies (which are also lower and lower frequencies and longer and longer wavelengths). This can be demonstrated by heating metal. A metal that is sufficiently hot radiates red. As the metal is made even hotter, it radiates orange. If the metal is made hotter still, it radiates yellow. This can also be demonstrated with a flame on a stovetop. At the lowest setting, the flame radiates red. At a higher setting, the flame radiates orange. At an even higher setting, the flame radiates yellow, and the hottest part of the flame is blue. The Sun is a yellow star, and from that yellow color we can correctly estimate that the surface temperature of the Sun is roughly six thousand kelvins. Stars throughout the universe that are red in color are cooler than our Sun, stars that are blue in color are hotter than our Sun, and stars that are yellow in color are approximately the same temperature as our Sun. We must emphasize that we are talking about the color that an object radiates because it is hot enough to be emitting that color. Many objects have various different colors even though they are all at room temperature, such as red ink, yellow paint, green grass, and blue jeans. These objects are not radiating these colors; these objects are reflecting these colors while absorbing all other colors. We must be careful to make a distinction between the color of an object simply because it is reflecting that color versus the color of an object because it is actually hot enough to be radiating that color. A red pen is at room temperature, while a piece of charcoal glowing red is at one or two thousand kelvins of temperature!

Consider any wave propagating in a certain medium that encounters a second medium. This wave is called the incident wave. At the boundary between the two media, a part of the wave will bounce back into the first medium while the rest of the wave will be transmitted into the second medium. The wave that bounces back into the first medium is called the reflected wave, and the wave that is transmitted into the second medium is called the refracted wave. (The meanings of the words reflection and refraction will be made clear in a moment.) Any line perpendicular to the boundary between the two media is called the normal to the boundary, since the word normal in physics and engineering means perpendicular. The angle between the incident wave and the normal is called the angle of incidence with the symbol θ₁. The angle between the reflected wave and the normal is called the angle of reflection with the symbol θ₃. The angle between the refracted wave and the normal is called the angle of refraction with the symbol θ₂. The Law of Reflection states that θ₁ = θ₃ in all cases. In other words, the angle of incidence is equal to the angle of reflection in all cases for all waves. Reflection is the bouncing of a part of a wave off of another medium with no change in angle with respect to the normal. The Law of Refraction states sin(θ₁)/v₁ = sin(θ₂)/v₂, where v₁ is the speed of the wave in the first medium and v₂ is the speed of the refracted wave in the second medium. Refraction is the bending of a wave due to a change in speed of the wave. According to the Law of Refraction, a wave is refracted (bent) toward the normal if v₂ < v₁ (if the transmitted wave propagates slower than the incident wave); conversely, a wave is refracted (bent) away from the normal if v₂ > v₁ (if the transmitted wave propagates faster than the incident wave).

Since light is a wave, light must obey the Law of Reflection and the Law of Refraction. A device that reflects light is called a mirror. A device that refracts light is called a lens. Most metals reflect light very well. Therefore, a mirror can be manufactured by coating a piece of glass with a metal (often aluminum) and polishing the metal. Any piece of glass may be regarded as a lens, since light will refract (bend) as it is transmitted from the air into the glass and will refract (bend) again as it is transmitted from within the glass back into the air. In the following discussion of mirrors and lenses, we will assume that we are in the paraxial approximation. In this approximation, all light rays incident upon a mirror or lens must be near the symmetry axis of the mirror or lens. This can be guaranteed by requiring that the mirror or lens has a large radius of curvature.

A curved mirror with its center of curvature and its focal point facing toward the light incident upon it is called a concave mirror. In this case, the focal point is said to be “in front of” the mirror. A curved mirror with its center of curvature and its focal point facing away from the light incident upon it is called a convex mirror. In this case, the focal point is said to be “behind” the mirror. In the paraxial approximation, light incident upon a concave mirror will reflect and converge at the focal point that is in front of the mirror. For this reason, a concave mirror is also called a converging mirror. Also in the paraxial approximation, light incident upon a convex mirror will reflect and diverge away from the focal point that is behind the mirror. For this reason, a convex mirror is also called a diverging mirror. A curved lens that is thicker in its middle than it is at its edge is called a convex lens. A curved lens that is thinner in its middle than it is at its edge is called a concave lens. In the paraxial approximation, light incident upon a convex lens will refract and converge at the focal point that is on the opposite side of the lens as the incident rays. For this reason, a convex lens is also called a converging lens. Also in the paraxial approximation, light incident upon a concave lens will refract and diverge away from the focal point that is on the same side of the lens as the incident rays. For this reason, a concave lens is also called a diverging lens. Notice that mirrors and lenses are completely opposite in character. A concave mirror is converging, but a concave lens is diverging. A convex mirror is diverging, but a convex lens is converging. Even the geometry of convergence or divergence (as the case may be) is opposite in character. In particular, light rays converge to a focus on the same side as the incident rays for a converging mirror, but light rays converge to a focus on the opposite side as the incident rays for a converging lens. Also, light rays diverge away from a focus on the opposite side as the incident rays for a diverging mirror, but light rays diverge away from a focus on the same side as the incident rays for a diverging lens.

A telescope is a device that collects light from a large distant object. (We must not confuse a telescope with a microscope, which is a device that collects light from a small nearby object.) When we define a telescope as a device that collects “light” from a large distant object, we mean any type of light. In other words, a telescope is a device that collects from a large distant object photons (or electromagnetic waves) from any category whatsoever of the Electromagnetic Spectrum. A telescope that collects radio is called a radio telescope. A telescope that collects infrared is called an infrared telescope. A telescope that collects ultraviolet is called an ultraviolet telescope. A telescope that collects X-rays is called an X-ray telescope. A telescope that collects gamma rays is called a gamma-ray telescope. A telescope that collects visible light is called an optical telescope. We now realize that whenever we use the word “telescope” in everyday life, we probably mean to use the word “optical telescope,” since there are other types of telescopes that collect other forms of light that our eyes cannot see. Whereas the optical telescope was invented roughly four hundred years ago, it has only been in recent decades that other types of telescopes have been built giving astronomers a more complete understanding of the universe by collecting light from stars and galaxies from across the entire Electromagnetic Spectrum.

Optical telescopes are often divided into two categories: refracting telescopes (or just refractors for short) and reflecting telescopes (or just reflectors for short). Refracting telescopes uses lenses, while reflecting telescopes use mirrors. Each of these types have their own particular advantages and disadvantages, but astronomers have agreed in recent decades that the advantages of reflectors far outweigh their advantages and that the disadvantages of refractors far outweigh their advantages. For example, different colors of light refract through a lens by different angles, causing the final image to appear blurred with color. This is called chromatic aberration. Refractors suffer from chromatic aberration, since refractors use lenses. However, reflectors do not suffer from chromatic aberration, since reflectors use mirrors. (Mirrors reflect light, and the angle of incidence is equal to the angle of reflection in all cases regardless of color.) Since astronomers have agreed in recent decades that reflectors are superior to refractors, all of the large optical telescope built in recent decades have been and continue to be reflectors. Nevertheless, the first optical telescopes ever built were small refractors. To build a primitive refracting telescope, all that is required is two lenses with different focal lengths. The lens with the smaller focal length is placed closer to the eye; this is called the ocular lens (commonly known as the eyepiece). The lens with the larger focal length is placed further from the eye; this is called the objective lens. The two lenses must be aligned with each other so that they share the same symmetry axis. The distance between the two lenses must be the sum of the two focal lengths, and the magnification of the resulting image when looking through the telescope is equal to the ratio of the two focal lengths. For example, suppose we wish to build a small refractor from two lenses, one with a three-inch focal length and another with a twelve-inch focal length. The focal length of the ocular lens is three inches, and the focal length of the objective lens is twelve inches. The distance between these two lenses must be fifteen inches, since the sum of three and twelve is fifteen. (The word sum means addition. In this example, twelve plus three equals fifteen.) Finally, everything observed through this telescope will be magnified four times, appearing to be four times larger or four times closer, since the ratio of twelve to three is four. (The word ratio means division. In this example, twelve divided by three equals four.)

Any telescope on planet Earth is called a ground-based telescope, while any telescope in outer space (almost always orbiting the Earth) is called a space telescope. Ground-based telescopes have severe limitations. Firstly, light pollution is light from human activities (such as city lights and highway lights) that adds brightness to the night sky that prevents astronomers from observing dim stars and galaxies. However, even ignoring light pollution, the Earth’s atmosphere itself is the most important factor that limits the usefulness of ground-based optical telescopes, since the Earth’s atmosphere continuously refracts the incoming light from outer space. This is why stars appear to twinkle; the atmosphere’s continuous refraction of light is so severe that even our naked eyes observe stars appearing to twinkle as a result! The situation with X-ray telescopes is much worse. The Earth’s atmosphere is opaque to X-rays. Therefore, a ground-based X-ray telescope would not collect any X-rays from stars or galaxies at all! All X-ray telescopes must therefore be space telescopes. For all of these reasons, the National Aeronautics and Space Administration (NASA) has placed a number of space telescopes in orbit around the Earth, each one covering a different band of the Electromagnetic Spectrum. These telescopes are called the NASA Great Observatories, since astronomers have gained a more complete understanding of the universe through these telescopes. The Hubble Space Telescope is the great optical telescope, placed in Earth orbit in 1990 and is still in operation. The Compton Observatory is the great gamma-ray telescope, in operation from 1991 to 2000. The Compton Observatory was replaced by the Fermi Space Telescope, placed in Earth orbit in 2008 and is still in operation. The Chandra Observatory is the great X-ray telescope, placed in Earth orbit in 1999 as is still in operation. The Spitzer Space Telescope is the great infrared telescope, in operation from 2003 to 2009. The Cosmic Background Explorer is the great microwave telescope, in operation from 1989 to 1993. The Cosmic Background Explorer was replaced by the Wilkinson Space Telescope, in operation from 2001 to 2010. Although many astronomers prefer the use of these space telescopes, it is expensive and dangerous to launch and service space telescopes. Therefore, ground-based telescopes continue to be built and used by astronomers. There are many ground-based optical telescopes much larger than the Hubble Space Telescope for example.

For thousands of years before the invention of the telescope, humans looked up into the sky and tracked the motion of the stars and the motion of a band of milk around the entire sky they called the milky way. As they watched the stars and the milky way (during the nighttime) rise in the east and set in the west and the Sun (during the daytime) rise in the east and set in the west, they concluded that the Earth is at the center of the universe, and everything in the universe moves around the Earth. The first person to question whether or not this is actually the case was Aristarchus of Samos, an ancient Greek astronomer who lived twenty-three centuries ago. Using geometry, he attempted to calculate the size of the Sun and the size of the Moon relative to the size of the Earth. He calculated that the Moon is smaller than the Earth, and he calculated that the Sun is larger than the Earth. Today, we know that his numerical results were significantly incorrect, since he made some false assumptions in his calculations. Nevertheless, we know today that the Moon is indeed smaller than the Earth, and we know today that the Sun is indeed larger than the Earth. In other words, Aristarchus’s results may not have been quantitatively correct, but they were at least qualitatively correct. Aristarchus then declared that it made no sense for the Sun to move around the Earth if it was larger than the Earth; he declared that it made more sense for the Earth to move around the Sun. Aristarchus’s Greek contemporaries persuaded him that the Earth could not be moving, since we would then see stars appear to suffer parallax. Parallax is the apparent motion of an object when in actuality the observer is moving. Since the ancient Greeks did not see stars suffer parallax, they continued to believe that the Earth is not moving and that everything else in the universe, including the Sun, moves around the Earth. Today, we of course know that Aristarchus was correct; the Earth does move around the Sun, but his Greek contemporaries were also correct: stars must appear to suffer parallax due to the Earth’s motion around the Sun. No one at the time realized how distant (far away) stars truly are. Stars are so distant that our naked eyes cannot observe the tiny parallax they appear to suffer. (The further away a star, the smaller the parallax.) Parallaxes were finally measured during the Modern Ages thanks to the invention of the telescope, but ancient humans did not have the ability to measure these tiny angles. Since the naked eye does not see the parallax of stars, ancient humans continued to believe that the Earth is the center of the universe, and that everything in the universe moves around the Earth.

For thousands of years, humans looked up into the sky and observed that all the stars in the sky appeared to remain fixed relative to one another. However, ancient humans also noticed seven objects that do not remain fixed relative to the stars or even to each another. These seven objects appeared to wander around the sky. The Greek word for wanderer is planet. The seven planets (wanderers) of ancient astronomy are the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. Today, we know that the Sun and the Moon are not truly planets, but the meaning of the word planet in ancient astronomy was wanderer, and the Sun and the Moon do indeed appear to wander around the sky. Also note that ancient humans did not understand that the Earth itself is a planet. The reason for this is obvious: we look down to see the Earth, but we look up to see the planets! We will use the word “ancient planets” for these seven objects so that we will not confuse them with the modern and correct meaning of the word planet. For thousands of years, humans observed only these seven ancient planets: the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn. For this reason, the number seven has attained almost supreme importance in many different cultures. To this day, seven is considered to be a “lucky” number. To this day, we use a calendar with seven days in a week, and each of these days is named after one of the ancient planets. This is obvious for Sunday which means day of the Sun, Monday which means day of the Moon, and Saturday which means day of Saturn. Thursday means day of Thor, who was the northern European analogue to Jupiter in Roman mythology. Friday means day of Frea, who was the northern European analogue to Venus in Roman mythology. Tuesday means day of Týr, who was the northern European analogue to Mars in Roman mythology. Wednesday means day of Odin who was the northern European analogue to Mercury in Roman mythology. Today we know that other objects appear to wander around the sky, such as Uranus, Neptune, Pluto, Eris, and Ceres for example, but these objects are too dim to be seen with the naked eye; they were not discovered until the Modern Ages after the telescope was invented. Actually, it is possible to see Uranus with the naked eye under ideal conditions. If ancient humans had discovered Uranus, then eight would be considered to be a “lucky” number instead of seven. Furthermore, we would be using a calendar with eight days in a week instead of seven, and the eighth day of the week would most certainly be called Uranusday.

For thousands of years, humans noticed that Mercury, Venus, Mars, Jupiter, and Saturn at times appear to slow down and stop and then turn around and move retrograde (backwards from their usual motion) until slowing down and stopping again before continuing with their original motion. Ancient humans also noticed that the Sun and the Moon never move retrograde. The Greco-Roman astronomer Claudius Ptolemy who lived nineteen centuries ago formulated a model of the universe to explain these motions. Ptolemy’s model of the universe was a geocentric model, meaning that it placed the Earth at the center of the universe, as all humans believed at the time. (Anyone who believes that the Earth is the center of the universe would call himself or herself a geocentrist.) According to Ptolemy’s geocentric model, the Earth is at the center of the universe, and the Moon moves on a simple circle around the Earth, since the Moon never appears to move retrograde. Next comes Mercury and Venus in this model. In order to explain their occasional retrograde motions, Ptolemy claimed that they must be moving around small circles (called epicycles) while at the same time moving around large circles (called deferents) around the Earth. Next comes the Sun, which according to Ptolemy moves on a simple circle around the Earth, since the Sun never appears to move retrograde. Next comes Mars, Jupiter, and Saturn, which Ptolemy claimed must be moving around small epicycles while at the same time moving around large deferents around the Earth to explain their occasional retrograde motions. Finally, Ptolemy claimed that the stars and the milky way were very far from the Earth and fixed relative to one another. Although we know today that Ptolemy’s geocentric model of the universe is not correct, this model nevertheless predicted the motions of the ancient planets with fair reliability. Therefore, humans at the time believed quite strongly in Ptolemy’s geocentric model of the universe. For the rest of the history of the Western Roman Empire, humans believed in Ptolemy’s geocentric model of the universe. Even after the Western Roman Empire crumbled, Europeans during the Middle Ages continued to believe in Ptolemy’s geocentric model of the universe. (Truth be told, Middle-Age Europeans believed in Ptolemy’s geocentric model not only due to its fair reliability but also fearing punishment from the Catholic Church which had adopted this model as part of its doctrine.)

Many historians agree that the Modern Ages of human history begins roughly five centuries ago due to the dramatic political, economic, social, artistic, religious, and scientific changes that occurred. The geniuses of the Renaissance for example began to create magnificent paintings, sculptures, music, and literature. The adventurers of the Age of Exploration as another example discovered and explored the American continents. The major European powers expanded their empires into the American continents as yet another example. The leaders of the Protestant Reformation questioned the doctrines and the authority of the Catholic Church as a further example. Mathematics and the sciences, astronomy in particular, is no exception to these revolutions in human in history. The Polish astronomer Nicolaus Copernicus who lived five centuries ago formulated a simpler model of the universe than Ptolemy’s geocentric model. Copernicus’s model of the universe was a heliocentric model, since it placed the Sun at the center. (Anyone who believes that the Sun is at the center of the universe would call himself or herself a heliocentrist.) According to Copernicus, Mercury and Venus move around the Sun on simple circles. Next comes the Earth, which Copernicus claimed was a planet that also moves around the Sun on a simple circle. Next comes Mars, Jupiter, and Saturn, which Copernicus claimed also move around the Sun on simple circles. If Mercury, Venus, Earth, Mars, Jupiter, and Saturn all move around the Sun on simple circles, then how did the heliocentric Copernicus model explain retrograde motion? Copernicus claimed that whenever the Earth moved passed another planet, it would appear as if the planet moved backwards when in fact the planet’s motion did not really change. Copernicus’s heliocentric model of the universe is certainly simpler than Ptolemy’s geocentric model of the universe, but it did not predict the motion of the planets in the sky any more reliably than Ptolemy’s geocentric model. Therefore, Europeans continued to believe that the Earth is the center of the universe. (Again truth be told, Europeans continued to believe in Ptolemy’s geocentric model not only due to its fair reliability but also fearing punishment from the Catholic Church. Copernicus himself waited until he was dying from natural causes before publishing his heliocentric model.)

For thousands of years before the invention of the telescope, humans built ancient observatories that used large objects to point into the sky to track the motions of stars and ancient planets. We will use the word “ancient observatory” so as not to cause confusion with modern observatories, which use telescopes. Examples of ancient observatories include Stonehenge in England and the pyramids in Egypt and Mexico. The Danish astronomer Tycho Brahe who lived more than four centuries ago built such an observatory and spent decades of his life tracking the motions of the ancient planets. He collected so much data that he hired the German mathematician Johannes Kepler to analyze the data. Tycho Brahe died shortly after hiring Johannes Kepler. Kepler then proceeded to use Brahe’s measurements to attempt to prove with certainty that Copernicus was correct, that the Earth along with the other planets do indeed move around the Sun. Kepler did not succeed until he abandoned the assumption that the planets move on simple circles. After rejecting this assumption, Kepler used Brahe’s measurements to show with superb accuracy that the planets do indeed move around the Sun. Moreover, he formulated what we today call Kepler’s three laws of planetary motion: the Law of Ellipses, the Law of Equal Areas, and the Law of Periods.

According to Kepler’s first law, the Law of Ellipses, the planets (including the Earth) move around the Sun on orbits that are ellipses. An ellipse is an elongated circle with a major axis that is longer than and perpendicular to its minor axis. Half of the major axis of any ellipse is called its semi-major axis, always denoted a; half of the minor axis of any ellipse is called its semi-minor axis, always denoted b. (The prefix semi- always means half. For example, a semicircle is half of a full circle.) Not only is the orbit of a planet around the Sun an ellipse, but the Sun is not even at the center of the ellipse; it is at one of the foci of the ellipse. (There is nothing at the other focus.) Since the Sun is at one of the foci of the elliptical orbit, there is only one point on the elliptical orbit where the planet is closest to the Sun, called the perihelion of the planet’s orbit. Also, there is only one point on the elliptical orbit where the planet is furthest from the Sun, called the aphelion of the planet’s orbit. The distance from the Sun to a planet’s perihelion is called the perihelion distance and is denoted r_perihelion; the distance the Sun to a planet’s aphelion is called the aphelion distance and is denoted r_aphelion. Notice that the sum of the perihelion distance and the aphelion distance is equal to the entire major axis of the orbit, which is twice the semi-major axis of the orbit. In other words, r_perihelion + r_aphelion = 2a. The time it takes a planet to move one complete orbit around the Sun is called the orbital period of the planet, denoted P. Notice that the orbital period of any planet is the time it takes that planet to move from its perihelion all the way around its orbit, returning to its perihelion. The orbital period is also the time it takes the planet to move from its aphelion all the way around the orbit, returning to its aphelion. According to Kepler’s second law, the Law of Equal Areas, planets sweep out equal areas in equal times. It follows from this that a planet moves faster while closer to the Sun (and fastest in fact at the perihelion) and moves slower while further from the Sun (and slowest in fact at the aphelion). According to Kepler’s third law, the Law of Periods, the square of the orbital periods of all the planets (including the Earth) around the Sun are all directly proportional to the cube of the semi-major axes of the orbits of all the planets (including the Earth) around the Sun. The orbital period of the Earth around the sun is one Earth year (1 yr). The semi-major axis of the Earth’s orbit around the Sun is one astronomical unit (1 au), roughly equal to one hundred and fifty million kilometers. Assuming we agree to measure the orbital parameters of all planets around the Sun in terms of the Earth’s orbital parameters, we may write Kepler’s third law as P² = a³, where P must be measured in Earth years and a must be measured in astronomical units. We may apply Kepler’s laws to perform simple orbit calculations. For example, consider a hypothetical planet orbiting the Sun that is six astronomical units from the Sun at its perihelion and twelve astronomical units from the Sun at its aphelion. In other words, we are given that r_perihelion = 6 au and r_aphelion = 12 au. This implies that the entire major axis of the orbit is eighteen astronomical units, since six plus twelve equals eighteen. This implies that the semi-major axis of the orbit is nine astronomical units, since half of eighteen equals nine. In other words, a = 9 au. This implies that P² = 729, since the cube of nine equals seven hundred and twenty-nine. Finally, we conclude that the orbital period of this hypothetical planet around the Sun is twenty-seven Earth years, since the square-root of seven hundred and twenty-nine equals twenty-seven. In other words, P = 27 yr for this hypothetical planet. Although Kepler deduced these three laws from Brahe’s measurements, he could not explain why any of these laws are true. Such deep questions would be answered by Isaac Newton, the person all these thousands of years of astronomical history are leading up to.

The Italian astronomer Galileo Galilei read about a new invention: the (optical) telescope. He built his own telescope after reading about this new invention, and in the year 1609 became the first person to ever make telescopic observations of the ancient planets. His discoveries were breathtaking. Galileo Galilei discovered mountains and craters on the Moon. Galileo Galilei discovered sunspots on the Sun. (Never ever observe the Sun through a telescope. Never ever observe the Sun through binoculars. Never ever observe the Sun even with the naked eye. Solar observations done incorrectly causes permanent blindness.) Galileo Galilei discovered four moons orbiting around Jupiter which were later named the Galilean Moons in his honor. (Today we know that Jupiter has more than sixty moons. Only four of them are large enough to be seen through a primitive telescope.) Galileo Galilei discovered rings around Saturn. (His telescope was too primitive to see that they are actually rings. He speculated that they were moons around Saturn.) Galileo Galilei discovered phases of Venus analogous to the phases of the Moon, such as full, half, crescent (less than half), or gibbous (more than half). Galileo Galilei discovered that the milky way is not in fact milk; it is innumerable stars sufficiently crowded together in the sky that with the naked eye all of their light blends together so as to appear to be milk. Only a telescope can produce enough magnification to reveal all of these magnificent discoveries. The phases of Venus can only be correctly explained if Venus moves around the Sun, not the Earth. Moreover, the discovery of four moons orbiting around Jupiter revealed that Jupiter is the center of its own mini-universe, further proving that the Earth is not the center of everything. For all of these discoveries and for his formulation of the scientific method, we will regard Galileo Galilei as the grandfather of modern science in this course.

The British mathematician and physicist Isaac Newton was among the most brilliant persons who have ever lived. He discovered calculus (advanced mathematics) and invented physics (the mathematical study of the equations that describe the universe) through his discovery of the three universal Laws of Motion and the law of Universal Gravitation. All of this he accomplished during the 1670s and the 1680s and published in his textbook Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) or the Principia for short. In this textbook, Newton presented what is today called the Newtonian Model of the universe. We begin our discussion of the Newtonian Model of the universe with Newton’s three universal laws of motion: the Law of Inertia, the Law of Acceleration, and the Law of Action-Reaction.

A force is a push or a pull. For most of human history, humans believed that forces cause motion. (Sadly, most humans to this day believe that force causes motion.) According to Newton’s first law of motion, the Law of Inertia, force does not cause motion. In fact, an object moves in a straight line at a constant speed when there is zero force pushing or pulling the object. If force does not cause motion, this begs the question, “What does force do? What does pushing or pulling an object accomplish?” This question is answered by Newton’s second law of motion, the Law of Acceleration. According to this law, force (pushing or pulling) causes changes in the motion of an object. This change may be speeding up the object, slowing down the object, changing the direction the object moves, or any combination of these changes. Physicists use the word acceleration to describe the rate at which an object’s motion changes, but keep in mind that this could be any of the mentioned changes. An object that is speeding up is said to be accelerating, but an object that is slowing down is also said to be accelerating. (In normal English, we would use the word “decelerating” instead.) Moreover, an object that is neither speeding up nor slowing down but only changing the direction that it moves is also said to be accelerating. Newton’s second law of motion is written mathematically as , where is the net force (the sum of all the forces) acting on the object, and is the acceleration (the rate of any change in motion whatsoever) of the object. Notice that the net force is proportional to the acceleration, meaning that stronger forces cause greater accelerations and weaker forces cause smaller accelerations. Also, m is the mass of the object. If we solve this equation for the acceleration, we conclude , and we see that the acceleration is inversely proportional to the mass of the object. In other words, larger-mass objects suffer smaller accelerations from a force (a push or a pull), while smaller-mass objects suffer larger accelerations from a force (a push or a pull). This stands to reason. For example, a baseball struck with a baseball bat will suffer a large acceleration since the baseball has a small mass, but a car struck with the same baseball bat will suffer a small acceleration since the car has a large mass. According to Newton’s third law of motion, the Law of Action-Reaction, if object A exerts a force on object B then object B must exert a force on object A that is equal in magnitude but opposite in direction. This law is often counterintuitive. For example, if a truck strikes a pedestrian, the pedestrian will be killed with body parts everywhere while the truck barely has a scratch on it. Did the truck exert a greater force on the pedestrian or did the pedestrian exert a greater force on the truck? According to Newton’s third law of motion, the Law of Action-Reaction, the answer is neither. The force that the truck exerted on the pedestrian was equal (in magnitude) to the force that the pedestrian exerted on the truck, but how can this be the case if the pedestrian was killed with body parts everywhere while the truck barely suffered a scratch? The answer is that the pedestrian had a very small mass (as compared with the truck, which had much larger mass) which caused the pedestrian to suffer a very large acceleration (as compared with the truck, which suffered a very small acceleration). When a pedestrian is killed by a moving car or truck, he or she was killed not only by the force from the car or truck; he or she was also killed by his or her small mass, resulting in a large acceleration.

The Newtonian Model of the universe is not only these three universal laws of motion but also Newton’s law of Universal Gravitation. According to Newton’s law of Universal Gravitation, everything in the universe attracts everything else in the universe. This is difficult to believe. We do not see tables and chairs and humans and cars attracting each other; we only see planets, moons, and stars causing attractions. Nevertheless, Newton’s law of Universal Gravitation is correct: everything in the universe attracts everything else in the universe. We do not notice tables and chairs and humans and cars attracting each another because gravity is by far by far by far by far the weakest force in the entire universe. It is so weak that we never notice gravitational attractions among tables and chairs and humans and cars. We only notice gravitational attractions from gargantuan objects, such as planets, moons, and stars. Even in this case gravity is noticeably weak. Every time we walk up a staircase, we are effortlessly defying planet Earth’s gravitational attraction! Newton’s law of Universal Gravitation can be written in mathematical form. Consider any two objects whatsoever, one with mass m₁ and the other with mass m₂, and suppose the distance between these two objects is d. The gravitational force (attraction) between these two objects is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them. Mathematically, F = G m₁ m₂ / d ², where F is the gravitational force (attraction) between the two objects. Also, the symbol G is called Newton’s gravitational constant of the universe, which is an example of a fundamental God-given constant of the universe. The mathematical equations that describe the universe (the laws of physics) have within them certain fixed constants (fixed numbers). These fixed constants (fixed numbers) are called the fundamental God-given constants of the universe. Most physicists agree that the three most fundamental God-given constants of the universe are the vacuum speed of light (always written with the symbol c), the Planck constant (always written with the symbol h), and Newton’s gravitational constant of the universe (always written with the symbol G). Each and every one of the fundamental God-given constants of the universe has an absolutely-fixed value, having that same value everywhere in the universe and everywhen in the universe. (The word everywhen means at all times in the past, present, and future.) Newton’s gravitational constant is roughly equal to 6.67×10^–11 (assuming we agree to measure all masses in kilograms, all distances in meters, and all times in seconds). This number is incredibly small: 10^–3 is one thousandth, 10^–6 is one millionth, 10^–9 is one billionth, and 10^–11 is even smaller than one billionth. We now realize why gravity is by far by far by far by far the weakest force in the entire universe. Whenever we calculate a gravitational force (attraction), we must multiply by Newton’s gravitational constant of the universe G, which is so incredibly small that the final answer for the gravitational attraction is incredibly weak. This is why we never notice gravitational attractions among tables and chairs and humans and cars. The only hope we have of ever feeling gravitational attractions is from gargantuan objects, such as planets, moons, and stars, and even in this case gravity is noticeably weak.

According to Newton’s law of Universal Gravitation, the gravitational force (attraction) between any two objects in the universe is directly proportional to the product of their masses. For example, if we double both masses, the gravitational force (attraction) strengthens by a factor of four, since the product of two and two is four. (The word product means multiplication.) As another example, if we triple both masses, the gravitational force (attraction) strengthens by a factor of nine, since the three times three equals nine. As yet another example, if we double one mass and triple the other mass, the gravitational force (attraction) strengthens by a factor of six, since two times three equals six. If we double only one of the masses, the gravitational force (attraction) strengthens by a factor of two, since two times one equals two. If we triple only one of the masses, the gravitational force (attraction) strengthens by a factor of three, since three times one equals three. Also according to Newton’s law of Universal Gravitation, the gravitational force (attraction) between any two objects in the universe is inversely proportional to the square of the distance between them. This means that increasing the distance between two objects weakens the gravitational force (attraction) between them, while decreasing the distance between two objects strengthens the gravitational force (attraction) between them. This stands to reason; we expect the attraction between objects to be stronger when they are closer together, and we expect the attraction between objects to be weaker when they are further apart. For example, if we triple the distance between two objects, the gravitational force (attraction) weakens by a factor of nine, since three squared equals nine. As another example, if we quadruple the distance between two objects, the gravitational force (attraction) weakens by a factor of sixteen, since four squared equals sixteen. As yet another example, if we double the distance between two objects, the gravitational force (attraction) weakens by a factor of four, since two squared equals four. If we third the distance between two objects, the gravitational force (attraction) strengthens by a factor of nine. If we fourth the distance between two objects, the gravitational force (attraction) strengthens by a factor of sixteen. If we half the distance between two objects, the gravitational force (attraction) strengthens by a factor of four. We can put all of this together with the following amusing example: if we double one mass, octuple the other mass, and quadruple the distance, the gravitational force (attraction) does not strengthen or weaken; it remains the same strength!

Isaac Newton combined his three universal laws of motion with his law of Universal Gravitation, and using calculus (which he also discovered) he proceeded to mathematically explain (what was believed at the time to be) everything that had ever been observed in the universe. This model of the universe is called the Newtonian Model of the universe, and physicists regard it as the first mathematically-correct description of the universe. We now discuss some of the greatest achievements of the Newtonian Model of the universe. Firstly, Newton explained why Kepler’s three laws of planetary motion are true, but he went even beyond this. Newton generalized Kepler’s three laws of planetary motion, showing mathematically that Kepler’s own formulation of his own planetary laws was not precisely correct.

According to Kepler’s first law as Kepler formulated it, the orbits of the planets around the Sun are ellipses. According to Newton, it is not just planets orbiting the Sun that should have elliptical orbits. If gravitation is indeed universal, then the orbit of anything around anything else (such as a moon orbiting a planet) should also be an ellipse, but Newton went even beyond this. He proved mathematically that the orbit could be a circle, an ellipse, a parabola, or a hyperbola. The total energy of the system determines the shape of the orbit. If the total energy of the system is sufficiently large, then the two objects will not remain bound to each other; they will escape from each other’s gravitational attraction. The orbit in this case will be a parabola or a hyperbola. It is for this reason that parabolic orbits and hyperbolic orbits are called unbound orbits. However, if the total energy of the system is not this large, then the two objects will remain bound to each other; they will not be able to escape from each other’s gravitational attraction. The orbit in this case will be a circle or an ellipse. It is for this reason that circular orbits and elliptical orbits are called bound orbits. Circles, ellipses, parabolae, and hyperbolae are all conic sections, the intersection of a cone and a plane. Conic sections are classified using a variable called the eccentricity. A circle is a conic section with an eccentricity equal to zero, an ellipse is a conic section with an eccentricity anywhere in between zero and one, a parabola is a conic section with an eccentricity equal to one, and a hyperbola is a conic section with an eccentricity anywhere greater than one. In summary, whereas Kepler’s formulation of his own first law states that the orbit of a planet around the Sun is an ellipse, Newton’s formulation of Kepler’s first law states that the orbit of anything around anything else is a conic section.

According to Kepler’s second law as Kepler formulated it, planets sweep out equal areas in equal times, since a planet moves faster while closer to the Sun (fastest at the perihelion) and moves slower while further from the Sun (slowest at the aphelion). According to Newton, it is not just planets orbiting the Sun that should sweep out equal areas in equal times. If gravitation is indeed universal, then anything orbiting anything else (such as a moon orbiting a planet) should also sweep out equal areas in equal times, but Newton went even beyond this. He proved mathematically that equal areas are swept in equal times because of the Conservation of Angular Momentum, another law of physics that he discovered. Ice skaters spin faster when they pull their arms in, and they spin slower when they pull their arms out. This ensures that the angular momentum of the ice skater remains conserved (remains constant). Rather like ice skaters who spin faster when they pull their arms in, planets orbiting a star (or moons orbiting a planet) speed up as they move closer to their attractor, and rather like ice skaters who spin slower when they pull their arms out, planets orbiting a star (or moons orbiting a planet) slow down as they move further from their attractor. This continuously changing speed keeps the angular momentum conserved, and Newton proved mathematically that this is why equal areas are swept in equal times. In summary, whereas Kepler’s formulation of his own second law states that planets sweep out equal areas in equal times, Newton’s formulation of Kepler’s second law states that the angular momentum of anything orbiting anything else must remain conserved.

According to Kepler’s third law as Kepler formulated it, the square of the orbital periods of all the planets around the Sun are all directly proportional to the cube of the semi-major axes of the orbits of all the planets around the Sun. According to Newton, it is not just planets orbiting the Sun where this proportionality should be true. If gravitation is indeed universal, then the square of the orbital periods should always be directly proportional to the cube of the semi-major axes of the orbits. This proportionality should also be true for moons orbiting a planet for example, but Newton went even beyond this. He proved mathematically that this proportionality actually states P² = ( 4 π² / GM ) a³, where P is the orbital period, π is roughly equal to 3.14159265358979323846264338327950288419716939937510, G is Newton’s gravitational constant of the universe, M is the total mass, and a is the semi-major axis of the orbit. With Newton’s formulation of Kepler’s third law, the units of a need not be astronomical units, and the units of P need not be Earth years. All that is required is that the units of P, G, M, and a all be consistent with one another. For example, we may use G = 6.67×10^–11 if we agree to measure P, M, and a in seconds, kilograms, and meters, respectively. Using this equation, astronomers have calculated the mass of the Sun from the orbital parameters of anything orbiting the Sun, such as planets, asteroids, and comets. Using this equation, astronomers have calculated the mass of the Earth from the orbital parameters of anything orbiting the Earth, such as the Moon and artificial satellites. Using this equation, astronomers have calculated the mass of Jupiter from the orbital parameters of the moons orbiting Jupiter. In fact, it is not an exaggeration to say that the only way astronomers can accurately calculate the mass of any object in the universe (such as a star, planet, or moon) is to use this equation. In summary, whereas Kepler’s formulation of his own third law states that P² = a³ for the planets around the Sun where P must be measured in Earth years and a must be measured in astronomical units, Newton’s formulation of Kepler’s third law states that P² = ( 4 π² / GM ) a³ for anything in orbit around anything else, where P, G, M, and a may be measured in any units that are consistent with one another.

For most of human history, humans have believed that heavier objects fall faster than lighter objects. (Sadly, most humans to this day believe that heavier objects fall faster than lighter objects.) The truth is that everything, no matter how heavy or how light, falls toward the Earth with the same acceleration near the surface of the Earth, 9.8 meters per second per second downward. (This is only the case when all non-gravitational forces such as air resistance can be ignored as compared with the gravitational force.) We can demonstrate this by dropping a heavy object and a light object at the same time from the same height, such as a textbook and a pencil. Both will hit the ground at the same time even though the book is hundreds of times heavier than the pencil! Although Galileo Galilei first demonstrated that this is true, it was Isaac Newton who explained mathematically why this is true. Actually, Newton went even beyond this; he proved mathematically that everything, no matter how light or how heavy, falls toward any planet, moon, or star with the same acceleration (ignoring all non-gravitational forces such as air resistance as usual). He discovered the following equation for this acceleration due to gravity for any planet, moon, or star in the universe: g = GM / R², where g is the acceleration due to gravity near the surface of the planet, moon, or star, G is Newton’s gravitational constant of the universe, M is mass of the planet, moon, or star, and R is radius of the planet, moon, or star. In other words, the acceleration due to gravity near the surface of any planet, moon, or star in the universe is G multiplied by the mass of the planet, moon, or star and divided by the square of the radius of the planet, moon, or star. For example, if we multiply G by the mass of the Earth and divide by the square of the radius of the Earth, the answer is 9.8 meters per second per second! As another example, if we multiply G by the mass of the (Earth’s) Moon and divide by the square of the radius of the (Earth’s) Moon, the answer is 1.6 meters per second per second, which is only one-sixth of the acceleration due to gravity near the surface of the Earth. The (Earth’s) Moon has virtually no atmosphere and therefore virtually no air resistance. Consequently, this was tested on the (Earth’s) Moon in a dramatic way. Fifty years ago, one of the astronauts on the Moon dropped a hammer and a feather, and both fell down with the same acceleration; both hit the Moon’s ground at the same time! As yet another example, if we multiply G by the mass of Mars and divide by the square of the radius of Mars, the answer is 3.7 meters per second per second, which is only one-third of the acceleration due to gravity near the surface of the Earth.

Perhaps the most brilliant of Newton’s achievements was the explanation of the tides. Sometimes the ocean is at flood tide (high tide); sometimes the ocean is at ebb tide (low tide). Why do the tides happen? Newton proved mathematically that an object will exert different gravitational forces (attractions) across another object due to the varying distances of different parts of the object. Parts of the object that are closer feel stronger attractions, while parts of the object that are further feel weaker attractions. The differences in the gravitational forces (attractions) across an object are called tidal forces, because they cause the tides in the ocean. The Moon and the Sun each exert approximately equal tidal forces on the Earth’s oceans causing them to bulge, resulting in two flood tides and two ebb tides every day. When the Earth, the Moon, and the Sun happen to form a nearly straight line (this occurs during New Moon or Full Moon), the lunar tidal force and the solar tidal force reinforce each other, causing severely high flood tides and severely low ebb tides. These are called the spring tides. When the Earth, the Moon, and the Sun happen to form a nearly right angle (this occurs during First Quarter Moon or Third Quarter Moon), the lunar tidal force and the solar tidal force cancel each other, causing modest flood tides and modest ebb tides. These are called the neap tides. The Moon’s orbital period around the Earth is roughly one month. (In fact, the word month is derived from the word moon. Take the word month, remove the last two letters and insert one extra o, and we get the word moon!) One month is roughly four weeks. Therefore, if today is New Moon, we will have spring tides (severely high flood tides and severely low ebb tides). Roughly one week later will be First Quarter Moon, and we will have neap tides (modest flood tides and modest ebb tides). Roughly one week later will be Full Moon, and we will have spring tides again (severely high flood tides and severely low ebb tides). Roughly one week later will be Third Quarter Moon, and we will have neap tides again (modest flood tides and modest ebb tides). Roughly one week later, we have returned to New Moon, roughly four weeks since the previous New Moon. For thousands of years, humans already noticed that there is a correlation between the changing appearance of the Moon in the sky and the changing tides in the ocean, but it was Isaac Newton who explained mathematically why this happens. The lunar tidal force and the solar tidal force not only cause the Earth’s oceans to bulge, but they also cause the shape of the solid Earth itself to bulge. The shape of the solid Earth itself suffers two flood tides and two ebb tides every day. When the solid Earth itself is suffering a flood tide, then we are slightly further from the center of the Earth. Later when the solid Earth itself is suffering an ebb tide, then we are slightly closer to the center of the Earth. Each and every day of our lives, we move up and down roughly one meter twice a day, even while we believe ourselves to be remaining still!

Although many Europeans were convinced that the Newtonian Model of the universe is correct, many other Europeans were not convinced. When Halley’s comet passed the Earth during Newton’s lifetime, its orbital parameters were measured, and Newton’s equations were used to calculate that it had an orbital period around the Sun of roughly seventy-four years. After seventy-four years, Halley’s comet did not return as scheduled, and the enemies of Newton rejoiced since they believed that this disproved the Newtonian Model of the universe. However, the original calculation only included the gravitational attraction of the Sun. What about the gravitational attractions of the planets? Physicists and mathematicians recalculated the orbit including the gravitational attractions of the planets in addition to the gravitational attraction of the Sun and realized that the orbital period of Halley’s comet was not seventy-four years; it was seventy-six years. In other words, the original calculation was two years in error. Europeans waited two more years, and Halley’s comet returned! Isaac Newton, who had died a few decades earlier, was already considered a genius by many, but the return of Halley’s comet as predicted convinced not only his admirers but his enemies as well that he may have been the most brilliant person who ever lived. The great British poet Alexander Pope wrote the following poem in honor of Isaac Newton. “Nature and Nature's laws lay hid in night. God said, ‘Let Newton be!’ and all was light.” Isaac Newton’s achievements defined the Age of Reason of seventeenth-century Europe which led directly to the Age of Enlightenment of eighteenth-century Europe. During the Age of Enlightenment, scholars in many different disciplines began to approach their subjects with mathematical logic and scientific reasoning. For example, mathematicians began to insist on rigorous proofs before any mathematical statement would be regarded as a true theorem. As another example, political philosophers debated different systems of government in a reasoned and logical fashion. As yet another example, religious scholars began to study and teach the Bible in a reasoned and logical fashion. Isaac Newton is not only one of the greatest figures of scientific and mathematical history for his model of the universe, but he is also one of the greatest figures of world history for his singularly-important role in the Age of Reason and for inspiring the Age of Enlightenment. The most authoritative biography of Isaac Newton is Richard S. Westfall’s Never at Rest: A Biography of Isaac Newton. No one would dare question the Newtonian Model of the universe until two hundred years after Isaac Newton died. In the early twentieth century, Albert Einstein dared to question the Newtonian Model of the universe. The great British poet J. C. Squire wrote the following poem in honor of Albert Einstein as a sequel to Alexander Pope’s poem. “It did not last: the devil howling ‘Ho! Let Einstein be,’ restored the status quo.” We will study the Einsteinian Model of the universe later in this course.

Links

New Jersey Institute of Technology

College of Science and Liberal Arts at NJIT

Department of Physics at CSLA at NJIT

Libarid A. Maljian at web.njit.edu

Libarid A. Maljian at the Department of Physics at CSLA at NJIT

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