A. A BRIEF HISTORY OF THEORIES AND DISCOVERIES.
The ancients recognized 7 objects in the sky that moved relative to the
"fixed" stars. These
were the Sun, Moon, and
5 relatively bright stellar-like objects. The latter were called wandering
stars or planets and were
assigned the names of ancient gods. The Earth was not considered a
planet but the center of
the universe. This remained the situation up through the 16th century
until
the telescope became an
instrument to study the sky.
The
Geocentric Theory or Model of the universe has its origins in prehistoric
time, that is, going
back more than 3,000 years.
The Geocentric Model assumes the Earth is the immobile center of
the universe and everything
else seen in the sky revolves around the Earth.
The Heliocentric
Model assumes that the Sun is the center of the universe, that the Earth
is a
planet, and that the planets
all revolve around the Sun. This is closer to what we recognize as
the
model of the Solar System
today. It must be remembered that the ancients had no awareness of
the universe other than
what they could see with their eyes,
Aristotle,
circa 350 BC
Presented arguments for the geocentric theory of the universe, such as,
no evidence for the
parallactic displacement
of nearby stars, which should result from the revolution of the Earth
around
the Sun. He also argued
that diurnal motion could not be explained by the Earth rotating, since
the
Earth would have to rotate
so fast that it would break apart.
Aristarchus,
c. 250 B
Rebutted Aristotle's arguments. He argued that parallax can not be
observed because all the
stars are so far away that
this angular displacement could not be detected, which is true. He
promulgated
and taught the validity
of the heliocentric theory of the universe, which was commonly not accepted.
Hipparchus,
c. 150 BC
One of the greatest of the ancient astronomers was the Greek philosopher
and mathematician,
Hipparchus.
He lived and worked in Alexandria, Egypt. Apparently he made measurements
of planetary positions without
a telescope and developed a mathematical model for computing future
positions of the planets.
Unfortunately, he adopted the geocentric theory. As a result, the
model did
not predict things very
well. Hipparchus also invented trigonometry, calculated the distance
of the
Moon, made the first known
star chart, and classified stars according to their apparent brightnesses,
by assigning them numbers
that we now call magnitudes.
Ptolemy, Claudius c. 120 AD
Attempted to improve the geocentric theory by introducing epicycles into a planet's motion.
Wrote a compendium of astronomy referred to by Arab scholars as the Almagest.
Copernicus,
Nicholas, 1543 AD
Nicholas Kopernik was a polish cleric of the Church. He resurrected
and updated the
heliocentric theory including
the development of a mathematical model that predicted the positions
of the planets more accurately
than any of the geocentric theory models The heliocentric model of
the solar system was also
able to explain the apparent retrograde motions of the planets, which the
geocentric model was not
able to do successfully. He published his results in 1543, shortly
before
his death.
Brahe,
Tycho, 1580 - 1600
Made the most precise measurements of the positions of the planets before
the use of
telescopes. He used
sighting instruments that he designed and had constructed by skilled craftsmen.
He was able to measure positions
with an uncertainty of only 4 arcminutes, a precision never
achieved before.
Kepler,
Johannes, 1600 - 1620
Mathematically analyzed the observations of Brahe and developed 3 laws
of planetary motion.
In 1609, Galileo Galilei began to observe the planets telescopically. He saw that Jupiter had
satellites. He watched Venus go through a cycle of phases that is different than the Moon's cycle.
Galileo described Saturn as a planet with ears. His telescope could not separate Saturn from its
rings. It was some years later, in 1659, that Huygens discovered that Saturn had rings.
In 1671,
Issac Newton developed 3 laws of motion and the universal Law of
Gravity. To accom-
plish the latter, he invented
calculus.
On March
13, 1781, W. Herschel came across and object
which he first believed was a new
comet. After observing this
object for some time, he noticed that it did not move like a comet nor
was it
changing like comets usually
do. He finally announced that he had discovered a new planet orbiting
the Sun! This news
astonished the entire world and made him instantly famous, for no one had
ever
thought there was another
planet in the Solar System. Herschel named his new planet "Georgium
Sidus" (George's Star) out
of gratitude to King George III of England, because Herschel had migrated
from Germany to England.
This was not accepted by many other astronomers, especially the French.
An international body of
astronomers decided that the naming of new planets should follow the
tradition of using names
of the ancient gods. Therefore, it was decided that the new planet should
be
called Uranus, the grandfather
of Saturn and the god of the universe.
In 1801, Piazzi accidentally discovered another
planet, which was called Ceres. In the following
few years several other
planets were discovered. However, Ceres was found to have a diameter of
only 1000 km and the other
newly discovered planets were even smaller. Furthermore, all of these
small planets moved in closely
spaced orbits at about 2.8 AU. They then became known as minor
planets.
In the meantime, all was not well with Uranus. It was not following the
predicted orbit that had
been calculated for it,
based on observations. It was concluded that there must be another distant,
large planet that was disturbing
the orbit of Uranus. Two young mathematicians, Urbain Jean Leverrier
and John Adams,
working independently of one another, set about to analyze mathematically
the
observed departures of Uranus
from its predicted path in order to calculate where this other planet
might be located in the
sky. Adams sent his results to George Airy at the Royal Greenwich
Observatory but Airy did
not bother to look for the planet. Leverrier was more successful in getting
an astronomer at the Berlin
Observatory to look for the planet. And so on the night of September 23,
1846,
the eighth planet was discovered very close to where Leverrier had calculated
it should be.
Later Airy realized that
Adams had also calculated the location of the new planet correctly. Adams
and Leverrier are now both
credited with having discovered the new planet which was named
Neptune. Of course,
all of this would not have been possible without Newton's Laws.
Astronomers
were convinced that there could still be other undiscovered planets and
so the
search for new planets continued
for a long time without success.
In 1930, Clyde Tombaugh, an assistant at the
Lowell Observatory in Arizona, succeeded in finding
another planet by photographic
means. It was named Pluto and was considered to be another major
planet because its orbit
was found to be larger than that of Neptune.
Pluto was later found to be a rather small planet both in size and mass.
It also had a strange orbit
that departed from the orbital
characteristics of the major planets. This has led to debates as
to whether it should be
classified as a major planet. In 2006, the International Astronomical Union's
committee on planetary designations
decided to reclassify Pluto as a dwarf planet.
In 1992
Jewitt and Liuu at the University of Hawaii discovered a relatively large
body moving in
orbit around the Sun at
a distance greater than the distance of Pluto. Since
1992, hundreds of other
bodies smaller than the
major planets have been discovered beyond the orbit of Neptune.
These bodies are called
Trans-Neptunian
Objects (TNOs) or Kuiper Belt Objects (KBOs). They
appear to made of a lot
of ice, like Pluto. The names given to some TNOs are Orcus, Sedna,
Quaoar,
Varuna, and Ixion.
In 1994,
the first solid evidence for the existence of a very massive planet orbiting
another star
(51 Pegasi) was obtained
by Swiss astronomers. Such planets are called exoplanets.
Exoplanets
are discovered by appying
the Doppler effect to an analysis of a star's spectrum, thereby measuring
the changing speed
of the star in its motion around the barycenter of the star and its planets.
Recently,
the satellite telescope Kepler has been placed in orbit to search for other
exoplanets. This
is done by monitoring the
brightnesses of hundreds of stars in the hope of detecting minute changes
in brightness that would
indicate the transit of a planet across the surface of the star around
which the
planet revolves. A
detailed analysis of how the planet dims its star reveals certain physical
properties
of the planet such as the
size of its orbit and the size of the planet. To date, Kepler has
detected more
than 1000 possiible exoplanet
candidates and 19 are confirmed exoplanet findings, including one that
orbits a binary star system.
In 2005,
a relatively large TNO was discovered moving in orbit around the Sun at
a distance of about
97 AU, which is its aphelion
distance. Its elliptical orbit brings it to 38 AU at perihelion.
It was given the
designation 2005 UB313 and
was nicknamed Xena by its discoverer, Mike Brown of Cal. Tech. However,
the IAU has now officially
named this body to be Eris. Eris has a small moon called Dysnomia.
Estimates
of the size of Eris indicate
it is about 1.5 times larger in size than Pluto.
In August
2006, the IAU voted to demote Pluto as a major planet and placed it in
a new category
called "dwarf planets" along
with the largest asteroid Ceres, and Eris. Some of the other dwarf
planets
were given in the previous
chapter.
The number of new TNOs being discovered is growing rapidly.
B. ORBITAL PROPERTIES OF THE PLANETS
The orbits of the major planets are nearly concentric circles, that is,
they do not intersect.
Actually the orbits are
ellipses of small eccentricity. This was discovered by Johannes Kepler
circa 1600 AD.
The eccentricity
of an ellipse is defined by the spacing of 2 points called the foci.
The foci are
located on the major axis
of the ellipse, symmetrically positioned on either side of the minor axis.
Kepler also discovered that the planets speed up and slow down as they
move in an elliptical
orbit.
When the planet is closest to the Sun, the perihelion point, the planet
has its maximum speed.
When
the planet is farthest from the Sun, the aphelion point, the planet's speed
is a minimum.
In the table below, Pluto and Eris are also listed, though they are not major planets.
Planet Sidereal
Orbital Period Mean Orbital Radius
Discovery
(in years)
(in AU)
Mercury
0.24
0.4
unknown
Venus
0.62
0.7
unknown
Earth
1.00
1.00
unknown
Mars
1.9
1.5
unknown
Jupiter
11.9
5.2
unknown
Saturn
29.5
9.5
unknown
Uranus
89.0
19.2
W. Herschel, 1781
Neptune
165.
30.
Adams & Leverrier. 1846
Pluto
248 .
39
C. Tombaugh, 1930
Eris
557.
50
Brown et al.
Mercury,
Venus, Earth, and Mars are classified as the inner planets because of their
location. The other
planets are called the outer planets.
The
orbits of the major planets lie nearly in the same plane, that is, the
orbits are nearly
coplanar.
Pluto and Eris deviate considerably
from this statement. Pluto's orbit is inclined 17 degrees to the
plane of the Earth's orbit
and Eris has an orbit that is inclined 44 degrees to the plane of the ecliptic.
The plane of the Earth's
orbit is also called the plane of the Ecliptic or Ecliptic plane.
All the major planets move in orbit around the Sun in the same direction,
viz., counter-
clockwise.
This is defined as the prograde direction and in the sky, this appears
as a motion
from
west to east, or eastward.
C. Kepler's Laws of Planetary Motion
1. Kepler's 1st Law of Planetary Motion: All
the planets move in elliptical orbits
around the Sun, which is located at one of the foci of the ellipse.
The point in an orbit where a planet is closest to the Sun is called the perihelion point.
The point where a planet is farthest from the Sun is called the aphelion point.
2. Kepler's 2nd Law of Planetary Motion:
The
radius vector of a planet sweeps out
equal areas in equal time intervals.
The radius vector is the line connecting the planet and the Sun.
So imagine this line sweeping
out the blue areas in the diagram below as the planet moves in its orbit.
Diagram courtesy of Addison-Wesley
Educational Publishers
Kepler's 2nd Law implies
that a planet speeds up and slows down as it moves in an elliptical
orbit.
At perihelion, a planet
has its maximum orbital velocity. When a planet is at the aphelion
point of
its orbit, it has
its minimum velocity.
Surprisingly, the Earth arrives
at its perihelion point on or about January 2nd, whereas it arrives at
aphelion around July 2 nd.
Obviously then, the seasons can not be explained in terms of the changing
distance of Earth from the
Sun. Instead, it is the obliquity of
the ecliptic, or tilt of the Earth's axis of
rotation that results in
the seasons. See section F below.
3.
Kepler's 3rd Law of Planetary Motion: For
every planet, the square of its sidereal orbital
period is proportional to
the cube of its mean distance from the Sun.
The sidereal orbital period
is the length of time it takes a planet to revolve exactly 360o
in
its orbit around
the Sun. For the Earth
this is 365d 6h 9m 10s.
Kepler's laws are
a consequence of the nature of gravity and Newton's laws of motion.
In
particular, Kepler's 3rd
Law follows from Newton's Law of Gravity. Kepler had in his grasp
the law
of gravity but did not realize
this.
Homework Assignment:
Do Exercise 15.1 in the Course Manual. It is to be
submitted on
a date to be announced.
Read sections
I
and II
in Ex. 7.0 in the Course Manual and memorize the table on
page 50 regarding
the equinoxes and solstices.
As the
Earth revolves around the Sun, in the sky the Sun appears to move eastward
relative to the fixed stars
by slightly less than 1
degree per day. The apparent path of the Sun amongst the fixed stars
is called the
ecliptic.
The
ecliptic is may also be thought of as the projection of the Earth's orbit
onto the celestial sphere.
See
the diagram below.
The Earth's
axis of rotation is tilted from a line perpendicular to the plane of the
Earth's orbit around the Sun
(plane of the ecliptic)
by an angle of about 23.42 degrees. This angle is often referred
to as the tilt of the
Earth's axis of rotation
or the obliquity of the ecliptic.
The imaginary line that is
perpendicular to the plane of the Earth's orbit, intersects the celestial
sphere at 2 points
directly opposite one another
( 1800 apart).
One point is called the north ecliptic pole (NEP) and the
other is
called
the south ecliptic pole (SEP).
See the diagrams above and below. A corollary of this is that
the plane of
the
Earth's geographic equator (the same as the plane of the celestial equator)
is tilted with respect to the plane
of
the ecliptic by about 23.42 degrees. This angle between the plane
of the ecliptic and the plane of the celestial
equator
is really what defines the term the obliquity of the ecliptic mentioned
above. This angle does not change
as
the Earth revolves in orbit around the Sun. This is shown in the
dagram below. As the Earth revolves in
orbit,
its axis of rotation remains parallel to itself and always points to the
NCP, which is always about 23.5
degrees
from the NEP.
The ancients established a band of 12 constellations, centered on the ecliptic,
and girdling the celestial sphere.
This
band of constellations is called the Zodiac and these constellations are
referred to as the zodiacal
constellations.
This is depicted in the diagram above and also shows the relation
between the ecliptic and the
celestial
equator. When the Earth is at the position in its orbit lableled
as Dec 22, the Sun is seen in the sky to
be
at the point W in the constellation Sagittarius. This point is the
winter soltice and is located at a declination
of
-23.5 degrees. The instant the Sun arrives at this point denotes the beginning
of winter in the northern
hemisphere.
As the Earth moves in its orbit from Dec. 22 to Mar. 21, the Sun appears
to migrate along the ecliptic from
Sagittarius
through Capricorn, Aquarius, to the point V in Pisces. This
is the vernal equinox and the Sun's
arrival
at this point marks the beginning of spring. Note that as the Sun
moves along the ecliptic from W to V,
the
declination of the Sun is increasing from -23.5 degrees to 0 degrees.
Subsequentlly. as the Earth revolves in orbit from Mar. 21 to June 21,
the Sun is observed to move from
Pisces
through Aries and Taurus to the point S in Gemini. This
point is the summer solstice and the arrival of
the
Sun at this point marks the beginning of summer. The declination
of the Sun is now +23.5 degrees. The
Earth
then continues to move from the June 21 position to the September 22 postion
and the Sun moves
from
Gemini through Cancer and Leo to the point A in Virgo. Note the declination
of the Sun is now
decreasing
to zero again. The point A is the autumnal equinox and this is now
the beginning of Autumn.
To complete the cycle of the seasons, the Earth rolves from Sep. 22 to
Dec. 22 and the Sun appears to
move
from Virgo, through Libra and Scorpio to the winter solstice.
A
table of the Sun's right ascension and delination when it at an equinox
or solstice is given in Ex. 7.0 in the
Course
Manual, page 48. This table should be memorized
These
changing positions of the Sun on the ecliptic may also be represented on
a rectanglular star chart using
equatorial
coordinates. The chart below is such a chart. The ecliptic
is the light blue curve connecting the
points
V, S, A and W. These points are the vernal eqinox, the summer solstice,
the autumnal equinox and the
winter
solstice. The position of the Sun on the ecliptic is shown for Feb.
21. You should be able to determine
where
the Sun is in such a chart for any day of the year.
E.
Cause of the Seasons
The seasons cannot be explaned
to be the result of the Earth moving in an elliptical orbit around the
Sun.
The surprising thing is
that the Earth is an perihelion on January 2nd and at aphelion on July
2nd. It is the
tilt of the Earth's axis
of rotation to the plane of its orbit, that is, the obliquity of the ecliptic,
that results in
the seasons. Because
of the obliquity of the ecliptic, as the Earth revolves around Sun, the
Sun changes its
declination. Therefore,
the diurnal circle of the Sun changes throughout
the year.
During summer for the northern
hemisphere, the Sun is north of the celestial equator.
Therefore, the Sun is higher
in the sky during the day and its rays are more intense,
thereby heating the northern
hemisphere more so than the southern hemisphere. Also, a
larger fraction of its diurnal
circle is above the horizon than below, so the length of time
the Sun is above the horizon
is more than 12 hours resulting in a longer time that the
Sun provides its heat.
In the
winter, the Sun is south of the celestial equator and therefore lower in
the sky
during the daytime. When
the Sun is lower in the sky its rays are not as intense and the
Sun heats the northern hemisphere
less than it does the southern one. Also, the length
of the daylight period is
shorter in winter because a smaller fraction of the Sun's diurnal
circle is above the horizon.
This means the Sun rises later in the morning and sets earlier
in the afternoon.
So, there is insufficient time each day for the Sun to heat the continents
or oceans to make up for
the heat energy lost by the Earth emitting infrared radiation into
space during the night.
Angle of Incidence.
This the angle between the zenith and the line of sight to an object,
such as the Sun.
The angle of incidence is also called the zenith distance of an object
and is the compliment of
an object's altitude.
These things may be summarized as follows:
1.
The angle of incidence of the Sun's rays is smaller in summer than in winter.
That is, the
Sun's rays are more perpendiuclar to the ground during the daytime.
Hence, the Sun's
rays are more efficient in heating the continents and oceans.
2.
The length of the daylight period is longer in summer than winter.
Hence, this permits the
Sun to provide more heat to the continents and oceans in one day.
All
bodies cool spontaneously by emitting electromagnetic radiation from their
surfaces into
space.
This
is true for both stars, planets, and anything else, like pumpkins and people.
The type and amount of raiation
emitted by a body depends on its surface temperature.
Because planets are relatively
cool, they emit mostly infrared radiation, whereas stars, which
are hot, emit much more
infraed and also visible light and ultraviolet.
In the summer, the
amount of thermal energy absorbed by the Earth during the day is greater
than what the Earth emits
into space at night. In the winter, the opposite is true.
Lag of the Seasons
After the Sun arrives at
the summer solstice on or about June 21, the northern
hemisphere continues to
receive more thermal energy from the Sun than it loses by radiation
during the night until about
July 23. Thus the hottest time of the year is near the end of July.
After that, the Earth begins
to lose more thermal energy during the night than it receives from
the Sun during the day and
the average temperature begins to decline. A similar phenomenon
occurs for the winter, so
that the coldest time of the year is around January 23 instead of at the
winter solstice.
F. Properties of Diurnal Circles
1. A diurnal circle
is the apparent path that an object traces on the celestial sphere as a
result of
the rotation
of the Earth.
2. They are actually parallels of declination.
3. All diurnal circles lie in planes parallel to one another and parallel to the celestial equator.
4.. All diurnal circles are centered on the axis of rotation.
5. Diurnal circles are slanted with respect to the celestial horizon by an angle that depends on
of the object. .
9. Circumpolar
stars have there diurnal circles entirely above the horizon and do not
rise or set.
They are circumpolar stars of perpetual apparition.
10. Circmpolar stars
of perpetual occultation never rise and have their diurnal circles entirely
below the horizon.
End of Chapter
3, SSII