This is one of the webpages of Libarid A. Maljian at the Department of Physics at CSLA at NJIT.
New Jersey Institute of Technology
College of Science and Liberal Arts
Department of Physics
Introductory Astronomy and Cosmology
Summer 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 approaching the age of our planet Earth, 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 approaching 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 cosmic 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. Three other
stars in the Big Dipper asterism can be used to find
Arcturus, the brightest star in Boötes (the
shepherd). Cassiopeia (the queen of Aethiopia) has five bright stars shaped like the letter
W. 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. (We cannot have 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 directly
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, electrons contribute a
miniscule amount to the mass of an atom.
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, this absolute zero of 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 a couple million 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. Caution: 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 space 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 at 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 at 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 from 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 visible 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 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 for 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 everyday
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 a 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 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 letter 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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