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
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 00h
to 24h.
(We will clearly define the line of right ascension at 00h in a moment.) Each right-ascension hour-angle is actually
15° of right ascension, since 360° divided by 24h
equals 15°/1h. 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 00h
right ascension by the line of right ascension that passes through the vernal
equinox (the spring equinox). The
autumnal equinox is 12h 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 06h right ascension since it is halfway from the
vernal equinox at 00h to the autumnal
equinox at 12h. The winter solstice is 18h
right ascension since it is halfway from the autumnal equinox at 12h to the vernal equinox at 24h
(which is the same as 00h since 24h is all the way around the Celestial Sphere
back to 00h). 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 00h
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 06h 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 12h 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 18h 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 Ephoton = 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: 11Na3–. (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 rperihelion;
the distance the Sun to a planet’s aphelion is called the aphelion distance and
is denoted raphelion. 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, rperihelion + raphelion
= 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 rperihelion
= 6 au and raphelion
= 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.
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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