[image: 240px-Fontaine_Médicis_Luxembourg.jpg]
*Polyphemus:   The Inhospitable Cyclops*
At some point during their journey from Troy to Ithaca following the Trojan
War, Odysseus and his remaining crew landed on Sicily, then a sparsely
populated island.    Their wanderings took them to an immense cave that was
food laden and -for the moment- uninhabited.   Odysseus and his
hunger-weakened crew entered the cave and dined cheerfully from its full
larder.  The meal was all the more delectable for the bags of sweet wine
that they brought with them.     Just after dusk, the cave's owner, the
giant cyclops Polyphemus, returned with the flock of sheep to which he had
been attending since releasing them that morning.   Polyphemus was
Poseidon's most fearsome son.  He stood as high as ten men, but was as
strong as a hundred.  Most notably, he had only one large eye, located at
his forehead's center.     Odysseus and his crew bowed to Polyphemus and
thanked him for providing  them with much needed nourishment.  Laughing,
Polyphemus grabbed two of the men in one hand, dashed their brains against
the cave floor and ate them: an act that traduced most customs of
hospitality.   Before the horrified survivors could flee, Polyphemus
blocked the cave entrance with a giant rock and promptly fell asleep.
 Throughout the night Odysseus and the others desperately sought another
escape route, but in vain.   Despite frantic searching they couldn't even
find a suitable hiding place. Even the stacks of logs next to the
perpetually burning hearth fire offered no concealment.  Though immense,
the cave also lacked any tunnels.   The next morning Polyphemus awoke,
quickly grabbed two other crew members and devoured them.  He then pushed
the blocking rock away to permit his sheep out to graze.   He followed his
flock outside and set the rock back in place, trapping Odysseus and his
men.   That day the clever Odysseus tried to formulate a plan.   Yes, he
could kill the Cyclops in his sleep, but then he and those that remained
wouldn't be able to move the rock. They would be imprisoned forever.
 Then, he thought of something just around dusk when the Cyclops returned
with his sheep.    Polyphemus took two other men, ate them and sat down on
his cave floor.  This time, Odysseus greeted him warmly despite the terror
the monster had inspired in him.   He spoke lovingly of the Cyclops'
beautiful seaside home which seemed to appease his host.  He offered
Polyphemus the remaining sweet wine he had brought with him.  The Cyclops
accepted it at once and quickly consumed it all.    Having never tasted
wine before, Polyphemus was soon in good humor.  He chatted jovially for a
few moments about life and the other Cyclopes who resided nearby.   He then
asked Odysseus his name.   Odysseus replied, "Outis," the word for
"nobody."      "I like you, Nobody," Polyphemus murmured as he was drifting
off into a drunken sleep. "I will eat you last."       While the Cyclops
slept, Odysseus and his men carved one end of a log into a sharp point.
They then held it in the fire until it nearly burst into flame.   Working
together they plunged their makeshift spike into Polyphemus' eye.  The
Cyclops' anguished screams were so loud as to summon his nearest
neighbors.     "What's wrong?!" they shouted toward his closed cave.
 "Nobody is hurting me!  Nobody is hurting me!!"  he cried.   Thinking that
Polyphemus was just in one of his rages, they all left.      Now blinded
and furious, Polyphemus resolved to capture and eat the remaining men.   He
moved the rock from the entrance to release the sheep. He felt the back of
each sheep as it passed for he believed that Odysseus and the others would
try to escape on top of his flock.    Having felt the backs of all the
sheep without touching a single man, he was satisfied that his captives had
remained in the cave.  They didn't. The cunning Odysseus had tied his men
and then himself to the underside of the sheep.   Once out in the open,
Odysseus and the others detached themselves from the sheep and returned to
their boat, taking the flock with them for food.    Once offshore, Odysseus
recklessly shouted taunts the Cyclops. He told him his true name and
thanked him for the the sheep.  Polyphemus, now in consuming fury, prayed
to his father Poseidon to avenge him.  From that moment Poseidon hated
Odysseus and through his influence with the ocean waves delayed the
warrior's return home by many years.

THE SOUTHWORTH PLANETARIUM
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Altitude:  10 feet below sea level
Founded January 1970
Julian Date: 245948.16
2019-2020:  CXXII

THE DAILY ASTRONOMER
Wednesday, April 8, 2020
Remote Planetarium 8:  Tides and Tropics

We're dividing today's class into two separate topics.
One is about the tides and the other about the Tropics.
--end of award-worthy poetry---

___________________________________
Helpful terms for today:

*Perigee: * the point of least distance between Earth and the moon

*Perihelion:*  the point of least distance between the Earth and Sun.
____________________________________

*TIDES*
And, yes, we understand that tide talk is the astronomical equivalent of
visiting a Medieval dentist who harbors irrational grudges. Nevertheless,
despite the torment, we proceed.  Earth generally experiences two high
tides and
two low tides a day,  The tides result from a differential gravitational
force.   To understand this idea we now turn to Newton:

[image: download.jpg]

*"The magnitude of the gravitational force exerted between two massive
objects is proportional to the masses of both and inversely proportional to
the square of their separation distance."*

If you double the distance between two objects, the force is reduced to a
quarter; triple the distance, the force is one ninth of the original
value.     Now, if all massive objects were point masses, as described in
elementary physics texts, then the matter would be simple: the forces would
be equal along each mass and the math would treat us sweetly. Of course,
the physical world doesn't have point masses. Instead all massive objects
occupy a specific volume, whether they're volleyballs or planets.
Differential gravity arises from this inconvenient breadth.


[image: earth-and-moon-space-view-johan-swanepoel.jpg]
Regard Earth and the Moon.    Both bodies exert a force on one another.
However, the magnitude of the force varies within each body because
different regions experience a different gravitational force.  The portion
of Earth closest to the Moon feels a greater tug than the planet's center
because gravity falls  off with the square of the distance, which is 6,371
km (3,959 miles).


Regard Earth and the Sun.   Yet, again, both bodies exert a force on each
other.  And, likewise, the gravitational force the Sun exerts on Earth
varies along its volume.   The part of Earth closest to the Sun (noon)
experiences less gravity than the part farthest away (midnight.)  However,
the Sun is much farther away from us than the Moon. The Sun's  mean
distance is 93 million miles; the Moon's is 240,000 miles, so the 'tidal
force' the Sun causes will be less than the Moon (44% as great, actually.)
 This is because the force difference between the near and far sides of
Earth relative to the Sun is not as profound as the difference the Moon
induces.


Both the Sun and Moon contribute to the high tides, although, as mentioned
previously, the latter is more influential than the former.   Each day, as
Earth rotates beneath the Moon, the planet 'bulges' along the Earth-moon
line.  This bulge region has a high tide. Yet, so, too does the point on
Earth that is diametrically opposite of the first bulge.  Here, our
intuition fails us.  Though it might be easier to understand why the area
just under the Moon has a high tide because of the differential gravity,
one is at a loss to know why the most distant region also experiences one.

Think of this: the pull of the Moon, and to a lesser extent, the Sun,
exerts a "bulging effect" on Earth, similar, in principle, to the effect of
squeezing the planet into an oblong shape.    When we regard this issue
physically, we can think of Earth's center as being stationary, with the
two opposite points along the Moon-Earth line pulled away.  The water is
drawn toward both these bulges, while the two areas perpendicular to it
experience low tides.   The water amount is constant, so an excess in one
region must create a deficiency in another.


*Spring and neap tides  Both the moon and the Sun exert tidal forces on
Earth. The Sun's tidal force influence is 44% that of the moon's.   When
the moon and Sun are aligned,as they are during a full or new moon,
the tides will be at their highest. We refer to these higher tides as
"spring tides."   When the moon is at the quarter phase, the Sun and moon
will be working in opposite directions and the tides will be lower.   We
call these lower tides "neap tides."     When the moon is new or full
around the same time it is near or at perigee, we'll have perigean
spring tides,which will be higher than other spring tides.   They are known
as "King Tides."  Finally, when Earth is at or near perihelion around the
same time as the full or new moon it at or near perigee,we'll experience a
perihelic perigean spring tide.  We call these "Poseidon's Tides."*



To add a little more complexity, we remind the reader (yes, we know you
haven't forgotten) that the angle between the Sun, Moon and Earth is always
changing.   During New Moon, the Moon is between the Sun and Earth, so both
Moon and Sun are pulling in the same direction and their gravitational
influence is combined.   At full moon, Earth is between the Moon and Sun.
  They're pulling in different directions, of course, but they're both
operating along the bulges.      The tides occurring around the  new and
full moon are known as "spring tides."     The lowest high tides happen
during the first and last quarter (quadrature), when the Moon and Sun are
operating perpendicularly and therefore diminishing the tidal effect.

[image: download.jpg]
Now, to make life even more interesting, the Moon's distance from Earth and
Earth's distance from the Sun are both constantly changing.    This
continuous change happens because both Earth and the Moon travel along
elliptical orbits, which we can envision as 'ovals.'    If the orbits were
perfect circles, the distance would be constant.  Of course, it isn't.
 During every orbit the Moon reaches a point of least distance, called
"perigee."  Just as Earth, itself, also has a closest point, called
"perihelion."     When the Moon reaches perigee, the tidal forces are
enhanced because the differential gravitational force varies with the cube
of the distance.   The same effect occurs, to a lesser extent, when Earth
reaches perihelion because at that time the Sun is closest to us,



Here is where we watch the gears turning:   Every so often, the Moon will
reach perigee around the time it is either at full moon or new moon.
This coincidence is called 'astronomical high tide,' a term also applied
merely to the tides corresponding to the full and new moon.  (Although the
term 'spring tide' is more apt.)    Perigee doesn't generally correspond to
the full or new moon because the period between successive perigees, called
an 'anomalistic month,' is about 27.5 days, whereas the phase cycle, called
a 'synodic month,' is 29.5 days.  However, when they correspond, the tides will
be quite high.  We call these "King Tides."

We're almost done.
Now, every so often, Earth will be at or near perihelion when the moon is
new or full when at or near perigee.

[image: unnamed.png]
Having Earth close to the Sun while the moon is close to Earth at new or
full moon produces the highest tides, known appropriately as 'Poseidon's
Tides."
Differential gravity is never distance sensitive so that these close
approaches of Sun, Earth and moon can produce a dramatic effect.

We quickly review the four main tide types

*SPRING TIDE*: The high tide that occurs when the moon is new or full.
These tides are higher than any other tides during the lunar cycle.

*NEAP TIDE*:  The high tide that occurs when the moon is at first or last
quarter,   These high tides are lower than any other tides during the lunar
cycle.

*KING TIDE:* The high tide that occurs when the moon is at or near perigee
when the moon is new or full.

*POSEIDON'S TIDE: * The high tide that occurs when the moon is at or near
perigee when full or new at the time when Earth is at or near perihelion.


*TROPICS*

*Lamentable fact:* The Sun doesn't pass directly overhead everywhere on
Earth.

Recall from a previous lesson that the ecliptic, the Sun's annual path
through the sky, passes north and then south of the celestial equator.   It
reaches its northernmost point on the June solstice, when it is 23.5
degrees north of the celestial equator.  It attains its southernmost point
on the December solstice, when it is 23.5 degrees south of the celestial
equator.

[image: sunspath.jpg]

The Sun's position relative to the Celestial Equator determines the
latitude where the Sun can reach the zenith, the point directly overhead.

On the March and September equinox the Sun is on the Celestial Equator and
will pass directly overhead at Earth's equator.    *More accurately, an
equatorial observer will see the Sun pass directly overhead when the Sun
crosses the meridian.  *

Today (April 8), the Sun is about 7.5 degrees north of the Celestial
Equator.  One will therefore find the Sun over head at the latitude 7.5
degrees North.

On the June solstice, the Sun will be 23.5 degrees north of the Celestial
Equator.    The Sun will pass through the zenith at the latitude 23.5
degrees North on this date.    This line defines the "Tropic of Cancer."

On the December solstice, the Sun will be 23.5 degrees south of the
Celestial Equator. The Sun will pass through the zenith at the latitude
23.5 degrees South on this date.    This line defines the "Tropic of
Capricorn."

[image: SciOFPPR-ArcticCircle.jpg]

Only in the tropics will one ever see the Sun overhead.  And, yes you are
correct.  Earth's 23.5 degree tilt is responsible for the 23.5 degree north
and 23.5 degree south latitude boundaries.  if Earth were tilted by 43
degrees, we here in Portland, ME, would be living in the tropics!
Oh,well.

One added note.  Only in two regions would one ever be able to see the
midnight Sun effect, when the Sun is above the horizon for twenty four
hours for at least part of the year.    In the Northern Hemisphere, the
midnight Sun effect is visible at or north of the Arctic Circle.  In the
Southern Hemisphere, the midnight Sun effect is visible at or south of the
Antarctic Circle.

The Arctic Circle's latitude is 66.5 degrees north
The Antarctic Circle's latitude is 66.5 degrees south

To understand why these boundaries are located at this latitude, just
subtract 23.5 from 90 degrees:    90 - 23.5  =  66.5

The midnight Sun effect is only visible at the Arctic Circle around the
June Solstice.
The midnight Sun effect is visible at the North Pole for a little more than
six months.  The closer one is to the north pole, the longer the period of
the midnight Sun effect.

*Today, April 8th, the Sun passes directly overhead at 7.5 degrees north.
Observers at the latitude 82.5 degrees north will now start to see the
Midnight Sun effect.  (90 - 7.5).    *

Tomorrow we'll move back off Earth to observe the planets.




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