Download OBSERVATIONS (1)

OBSERVATIONS – 1 – Days and Years
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Every day the Sun rises in the east and sets in the west.
At midsummer’s day (in the northern hemisphere) the sun is higher off the southern horizon
than at any other time. By midwinter’s day its elevation has gotten lower and lower until it
has reached a minimum. Then it begins moving higher again, cycling back and forth between
the two extremes annually.
As solar elevation increases or decreases, so does day length, varying directly. The longest
day corresponds to the highest elevation, the shortest to the lowest, and so on.
When the Sun is high in the sky by day the weather is hottest. When it is low, the weather is
coldest. When the Sun is between the extreme elevations the weather is similarly between
its extreme temperatures. We call this combination of things “seasons”.
Over the course of a year we see different stars in the sky between sundown and sunup.
That is, the stars do almost exactly the same thing, but it takes an extra day for us to see the
same ones in the same relative positions to the sun. This means that we cycle one day past
an entire zodiac in the course of a year, the stars seeming to outrun the sun very slightly.
Let’s think about these things one (or three) at a time.
• Every day the Sun rises in the east and sets in the west.
This is evidently what the Sun is doing, and there are still people who “believe”
that it does exactly that. They have elaborate explanations for why all the
observations that science has made are illusions and their “belief” trumps them.
The alternative, of course, is that the Sun is not moving around us, but rather that
we are rotating beneath it. You can get first-graders to understand this by having
them twirl on the playground and describe how the trees, classmates, swing sets,
etc. look as they do it.
zzzzzzzzzzz
NIGHT
wheeeee!
DAY
As we will see, a rotating Earth explains other things’ motions much better than a
bunch of revolving other things does.
S
U
N
• At midsummer’s day (in the northern hemisphere) the sun is higher off the
southern horizon than at any other time. By midwinter’s day its elevation has
gotten lower and lower until it has reached a minimum. Then it begins moving
higher again, cycling back and forth between the two extremes annually.
• As solar elevation increases or decreases, so does day length, varying directly.
The longest day corresponds to the highest elevation, the shortest to the lowest,
and so on.
• When the Sun is high in the sky by day the weather is hottest. When it is low, the
weather is coldest. When the Sun is between the extreme elevations the
weather is similarly between its extreme temperatures. We call this combination
of things “seasons”.
The imaginary plane in which the Earth revolves around the Sun is called the “ecliptic”. The axis of the Earth is not perpendicular
to the ecliptic, but is inclined from that position by about 23.4° presently. (This number changes gradually over time, but we
don’t have time to discuss that.)
This means that as the Earth revolves around the Sun its direction of tilt changes, day by day, with respect to the Sun. Note that
its actual tilt in space does not change. The northward extension of the North Pole always points toward Polaris, it’s only the
orientation with respect to the Sun that changes.
Two extreme ends of this system are shown – at midsummer’s (A) and midwinter’s (B) days.
A
The orange arrows point out the directions that the solar rays come to the Earth. Notice the angle between the raypaths and the
North Pole, which has been extended for clarity. The Sun is more nearly overhead at any given latitude at A than at B. Indeed, at
A the North Pole is lighted (though still at a low angle) even at midnight and at B it is dark, even at noon! This is like the
stick/shadow system that we have seen. Make sure you see why.
You should be able to infer that the angle of incidence at any latitude in the northern hemisphere will gradually decrease as the
Earth moves away from A toward B, reach a minimum at B, and then begin to increase again toward A, exactly as we see it do
across the seasons.
B
There is another thing going on here that zooming in makes clearer. Remember that the sun is to the right of A and the left of B in
the original picture as indicated by the parenthetical Sun in the diagram. When the axis is tilted toward the Sun (A) the length of
line segment A1 (representing the arc traveled at night at that latitude) is shorter than A2 (arc traveled by day). In reality this
means that night is shorter than day.
Note that the opposite is true when the axial tilt is away from the Sun – night is longer than Day.
Notice also that this inequality is not true at the Equator. Day and night are always equally long there!
A
B
( )
You should now see why day length correlates to axial tilt. When the Sun is more
directly overhead (in summer) the time we spend on the lighted side of the planet is
greater. In winter the opposite of both these things holds. Both things result directly
from the axial tilt.
You should be able to infer that from the equator (where days and nights are always
equally long) to the poles (where it remains dark for months in winter and light for
months in summer) the difference in day and night length varies directly with
latitude.
We will hold off explaining why, but the angle of incidence is also the cause of more
efficient warming of the atmosphere (at high angles of incidence – summer) and less
efficient heating at low angles (winter). Summer is not warmer becauuse we are
closer to the sun. How much closer we are ((a couple of thousand miles, at most) is
miniscule in proportion to the distance to the sun (93,000,000 miles, on average). It’s
all about the angle of incidence.
So it is the axial tilt that gives us seasons, with all that entails – differences in the
height of the sun, differences in day length, and differences in climate.
• Over the course of a year we see different stars in the sky between sundown and sunup.
That is, the stars do almost exactly the same thing, but it takes an extra day for us to see
the same ones in the same relative positions to the sun. This means that we cycle one
day past an entire zodiac in the course of a year, the stars seeming to outrun the sun
very slightly.
This picture from your book explains this as well as I’ve ever seen it explained. After each rotation
the Earth has moved a little farther along its orbit around the Sun. The next time the Sun “sets” we
can see a few stars that were behind the Sun the previous evening at sunset.
In January You’d
see these stars at
midnight. The
ones 180° away
would be
overhead when
the sun was
overhead and
you’d not be able
to see them.
In July You’d see
these stars at
midnight. The
ones 180° away
would be
overhead when
the sun was
overhead and
you’d not be able
to see them.
One last thing. The Earth rotates on its axis just about 365-1/4 times in the time it
makes one revolution around the Sun. That is, there are ~ 365-1/4 days in a year.
Because the calendar has no mechanism for dealing with a partial day we give three
out of four years 365 days, save the quarters, and add them as an extra day in
February in leap years.
This doesn’t quite fix the problem. In fact, it over-corrects, so in each century year
(like 1990) the leap day is omitted and February only has 28 days.
This also over-corrects, so in each millennial year (like 2000) the leap day is added
back in, even though it’s a century year. So there was a Feb 29th in 2000, but there
won’t be one in 2100.
Once again, this over-corrects and so …
OBSERVATIONS – 2 – Months
• Some nights the Moon rises in the east and sets in the west, but it moves more
slowly and is “lapped” buy the sun. As this happens we see more or less of it
and we call this the Moon’s phases. Consequently, sometimes the Moon rises in
the morning and sets in the evening. Indeed, its set/rise happens at a different
time each day, creeping later and later. It obviously moves differently than the
Sun.
• The timing of daily tide cycles corresponds to the timing of a lunar orbit and the
phases of the moon correlate to tidal cycles.
* Some nights the Moon rises in the east and sets in the west, but it moves more slowly and is “lapped” buy the sun.
As this happens we see more or less of it and we call this the Moon’s phases. Also, sometimes the Moon rises in the
morning and sets in the evening. Indeed, its set/rise happens at a different time each day, creeping later and later.
It obviously appears to move differently than the Sun.
The reason the sun seems to lap the Moon is that we rotate on our axis
ever so slightly faster than the Moon revolves around us. Thus each night
it is not as far along as it was the night before on its orbit. This diagram is
very schematic – we do not make that much progress on our orbit in “a
few days”.
On the other hand, the Moon rotates on its axis at exactly the same rate
that it revolves around us. That is why we always see the same face. This
is not magic, there is a simple physical basis for it called “tidal locking”.
Eventually that same basis will apply to Earth as well and we will rotate at
the same rate too. The Moon will no longer rise and set, it will always be
above the same spot. That orbital rate means that the Moon returns to
its same position with respect to a distant star in about 27.5 days, on
average. This is a “sidereal lunar month”.
If you study the diagram carefully you will also be able to see why the
Moon has phases. Remember that it has day/night too – one side is
toward and the other away from the Sun, but that side changes day-byday with respect to our line of sight. One cycle of the phases lasts a full
“synodic lunar month” (~29.5 Earth days on average.) The difference in
length of these two lunar months is because of the forward movement of
the Earth Moon system as the cycle proceeds. This “synodic month” is the
basis for all lunar calendars like the ones used for religious purposes in
Islamic countries.
The situation a
few days later.
(Imagine the
intermediates
day by day.)
The situation
one day
A one month cycle of the Moon
(At ~3.5 day intervals. Rotation and revolution directions indicated at start.)
1 – NEW MOON
*Moon is between Earth
and Sun.
*Far side is lighted and
near side dark.
*Rises at dawn and sets at
sundown.
3 – FIRST QUARTER
*Moon has lagged 1/4 of its
orbital path.
5 – FULL MOON
*Earth is between Moon and Sun.
*Right-hand 1/2 visible.
*Near side lighted and far side
dark.
*Rises at noon, obvious in
night sky before midnight.
*Rises at sunset and sets at dawn.
*Lunar eclipse possible.
7 – THIRD QUARTER
*Moon has lagged 3/4 of its
orbital path.
*Left-hand 1/2 visible.
*Rises at midnight and sets
at noon,
9 – NEW MOON
*Back to beginning of
cycle.
*Solar eclipse possible.
2 – WAXING CRESCENT
*Moon has lagged 1/8 of its
orbital path.
*Right-hand 1/4 visible.
*Rises late morning, obvious
in early evening sky.
4 – WAXING Gibbous
*Moon has lagged 3/8
of its orbital path.
6 – WANING Gibbous
*Moon has lagged 5/8
of its orbital path.
*Right-hand 3/4 visible.
*Left-hand 3/4 visible.
*Rises mid afternoon,
obvious in night sky until
about 3:00AM.
*Rises very early AM,
obvious in night sky
after about 3:00AM.
Sets mid-afternoon.
8 – WANING
CRESCENT
*Moon has lagged
7/8 of its orbital
path.
*Left-hand 1/4
visible.
*Rises late evening,
obvious until very
early AM.
* The timing of daily tide cycles corresponds to the timing of a lunar orbit and the phases of
the moon correlate to tidal cycles.
It is not quite accurate to say that the Moon revolves around the Earth. What actually happens is that the two
objects revolve around a common center of mass called the barycenter. (“Center of mass” is just a translation of the
Greek roots.)
It is also not quite accurate to say that the Moon’s gravity causes tides. It is the entire gravity system of the two that
does it. One high tide each day is a direct result of the Moon’s gravity on the water, the other is the inertial result of
the wobbly path taken by the Earth as the two of us revolve around the Sun. There is a smaller tidal effect from the
Earth/Sun system.
At new moon and full moon the lunar and solar bulges coincide on the Earth’s surface and we get particularly large
tide ranges (low lows and high highs) called spring tides. They have nothing to do with the spring season, occurring
once a (lunar) month all year.
At first and third quarters the two tides are 90° out of phase and tend to cancel each other. Because the lunar bulge
is the larger it is still seen as high tide, but the range is very small (high lows and low highs) because low tide occurs
where the solar high is. These we call neap tides.
Solid black line shows path of barycenter on common orbit around Sun. Dashed line shows path of center of Earth.
Dotted line shows path of center of Moon. Earth/Moon system wobbles along its orbit!
So, we see lunar phases because, to a first approximation, the Moon revolves around
us and so its lighted face becomes more or less visible depending on where it is in the
orbit.
This cycle takes about 29.5 days from start to finish, but because there is also motion
of the Earth with respect to the Sun this is actually longer than the time it takes the
Moon to complete an true orbit – about 27.5 days. As with days/year, the “about” is
important in both cases.
What a lunar month is depends on who you ask, that is, whether they are thinking of
synodic or sidereal months (or one of the other three potential definitions). At any
rate, though the synodic lunar phase is the original basis for calendar months it is
obvious that the length of all modern calendar months, except possibly February, are
longer than a true lunar month, however defined.
Early calendar makers wanted to cram 12 Moons into a year to match the 12
constellations, but that didn’t work because the lunar, daily, and annual periods are
simply mismatched and cannot be brought into phase. Ultimately what we got was
this: “Thirty days hath September …”.
OBSERVATIONS – 3 – Other Planets
• There are other star-like objects that move in yet a different way. Day by day
they rise in the east and set in the west like the other things, but sometimes they
outrun the stars and sun and sometimes they lag behind them.
• These objects are called “planets” from a root that means “wanderer”. We will
use Venus as an example.
The planets (including Earth (blue) and Venus (green)) orbit the Sun because of the interactions of two things. Their inertia
gives them a momentum in a direction tangent to their orbital direction (green arrows). Were there no other force acting
upon them they would continue in a straight line in that same direction (dotted arrows). What keeps them from doing so is
their gravitational attraction to the Sun (red arrows). Were it not for their momentum they would fall directly into the Sun.
These two things are in precise balance. There is no magic here, things in orbit automatically balance these two things.
Otherwise they either escape the orbit or fall into the thing they orbit.
You should realize that the effect of this is that we are perpetually falling into the sun and perpetually missing it!
Because Venus orbits closer to the sun than
Earth (it has an “inferior orbit”) the
gravitational attraction on it is greater, and it
falls faster. This means that its momentum
must be proportionally greater to ensure that it
keeps missing the Sun. That is, it moves faster.
Because it’s orbital path is shorter and its
orbital velocity faster it completes one
revolution around the sun in a distinctly shorter
time than Earth – about 223 Earth Days rather
than ~365. It outruns us, in other words. To be
specific (but not absolutely precise) Earth takes
about 1.64 times longer to make its journey
than Venus does. An Earth Year is different
from a Venutian year.
Let’s look at the consequences. Earth makes roughly 30° of arc in the course of a month. In the same time,
Venus makes about 49° of arc. The radial dotted lines mark out 30° segments of the orbits.
The numerals indicate the relative
positions of the two bodies at the
same time. For a hypothetical
example, at time 0 Earth and
Venus are both at a
corresponding point on their
orbits and are aligned with
respect to the Sun. (Earth returns
to this point in one year, Venus
does not.
At time 1 Earth has advanced 30°
and Venus 49°.
6
7
5
11
4
8
3
10
5
2
9
3
9
At time 2 Earth is 60° into its orbit
and Venus 98°.
6
1
10
And so on … Venus outruns us.
Examine the diagram and follow it
step by step. Note that Venus
completes its year in about 7-1/3
Earth months, between positions
7 and 8. Its position is shown by
lighter images beyond that.
4
12
8
7
0
1
11
0/12
2
First let’s examine how the Sun and Venus look to Earthlings through a year.
At “time 0” (the beginning of the year)
the Sun is immediately behind Venus
when viewed from Earth.
Like the Moon, Venus will be “new”
and would appear relatively dark if
you could see it.
It would also be up only in daylight
hours so spotting it would be a real
trick, particularly against the sun.
There are telescopic images of Venus
crossing the Sun, but it doesn’t
happen quickly, so don’t expect a
movie. You can probably find such
images on the web if you look.
Our convention will be to show the
line of sight to the Sun with a black
arrow and the line of sight to Venus
with a green one. Additionally, what
is illustrated will be what is seen from
Earth at either sunup or sundown as
appropriate. We start with sunup.
At time 1, because we have both moved by
different amounts from our original positions
Venus and the Sun will not align. Venus will rise
earlier than the Sun, and by the time it rises
Venus will be fairly high in the sky. In this phase
we colloquially call Venus the “Morning Star”.
At time 2 (after about two months) Venus will
come up even earlier than the Sun than it did at
time 1. The dashed green arrow shows how
high it was above the horizon at sunrise at time
1, the solid arrow at time 2.
At time 3 things change. Again, the dashed green
arrow shows how high it was above the horizon at
sunrise at time 2, the solid arrow at time 3. There is
virtually no change in its elevation. In fact, it is ever
so slightly lower.
You should be able to infer that for roughly half the
time (2 weeks) between times 2 and 3 its height
gradually increased, then decreased to this point.
The trend continues at time 4. Venus is now lower,
not much but noticeably, when the sun rises than it
has been.
And at time 6 …
The trend continues at time 5. Venus has become
about as much lower at sunrise than between times
3 and 4.
By time 7 the rate of Venus’s “approach” to the sun
speeds up. The angle is now distinctly smaller.
Remember that this means it rises nearer sunup
than it did at previous times.
And even more so at time 8.
By time 9 Venus is barely above the horizon when
the Sun rises, and has preceded the Sun by only a
few minutes. It will be pretty close to “full Venus” –
the entire face toward us will be lit by the Sun.
By time 10 something very interesting happens. Venus
is again at a very low angle to the Sun but notice that it
is now on the other side of the Sun from where it has
been – we see the sun first and Venus second as we
rotate. Until now it has been the other way round.
This means that at sunup Venus will still be below the
horizon and will not rise for a few minutes more. We
will only see it just after our rotation has taken the Sun
out of our line of sight, in the early evening. Venus has
become the “Evening Star” instead of the “Morning
Star”! You can probably predict what happen from
here.
You can see where this goes. By time 11 Venus is
pretty far above the horizon when the Sun rises,
and lags the Sun by a proportional amount of time.
And finally, after exactly one Earth year (time 11)
The angle and lag time have both increased again.
You should also be able to predict the future. The
angle/lag time increase will slow and then reverse.
Venus will again approach the Sun, eventually
“cross” it and be the “Morning Star” once again.
(OPTIONAL) The geometry of this is easy to visualize if you think about how Venus looks only from the perspective of Earth. That
is, we’ll pretend that Earth is still and Venus’s movement is not its actual orbital movement but just how much it outruns us.
Think first about the two tangents to Venus’s orbit as seen from Earth (A & B). At and near these places Venus will not seem to
get either farther or nearer the Sun, but stay at about the same distance from it. It is moving mostly toward (A) or away from (B)
us, not across our field of view
Now think about when Venus is on the same side of the sun as we are (C and D). At C it will appear to approach the Sun fairly
rapidly and at D to diverge from it at the same rate. It is moving mostly (or entirely) across our field of view, not toward or away
from us.
Something similar happens on the far side of the tangent points A and B. The approach phase (E) or divergence (F) phase toward
or from the sun lasts much longer because it takes a greater amount of the arc of the Venutian orbit. However, the rates when it
is crossing our line of sight (G and H) are the same as at C and D.
H G
F
E
A
B
C D
Because our orbit has a larger diameter than is Venus’s any change in relative position of
Venus and the stars behind is very subtle when Venus is on the far side of the Sun (where
there is less parallax) but it is very pronounced on the near side (there is much more parallax
when a moving objects is close to us). The following diagram illustrates the very obvious
differences in apparent motion of Venus and the background stars when it is “lapping” us –
catching up and passing us on the near side of the sun. Extending the model for Venus
beyond what this picture shows is not very useful. For most of its course the rate of
progression and retrogression are so small that it’s hard to illustrate at this scale. It takes
weeks or months to notice that it has drifted against the background stars so the angular
changes, even over months are pretty small.
On the other hand, you can see the difference in position almost daily when Venus laps us.
We’ll start this section similarly to how
we started the last, but this time instead
of comparing the direction to Venus and
the Sun we will compare the directions
to Venus and some distant star,
immediately aligned with both the Sun
and Venus. We want one that shows no
parallax with other stars because then it
will always be exactly the same direction
to that star no matter where we are.
At time 0 the star and Venus (and the
Sun) are aligned. Venus has caught up to
us and is about to pass us.
In this position it rises at the same time
as the Sun. It is a “new Venus”.
To distant
“marker” star
Time 1 – Venus is “ahead of” the marker. It rises
earlier and is above the horizon when the marker
comes onto the horizon. Venus is progressing (or
outrunning the stars).
Time 2 – Venus is “behind” the marker. It rises later
and is below the horizon when the marker comes
onto the horizon. Venus is retrogressing (or lagging
behind the stars).
Like Venus, Mercury has an inferior orbit and so appears sometimes in the morning and sometimes in the
evening. Because it is smaller, more distant, and inferior to Venus it is much harder to see, rising to a much lower
maximum height above the horizon. Because it is closer to the Sun than Venus its orbital velocity is even faster.
That, coupled with its shorter orbit, means that its cycle will be completely different from Venus’s. These two
inferior planets are always close to the sun from our perspective and we only see them in the morning or
evening, never near midnight. (Incidentally, if you know where to look it is fairly easy to find Venus on a very
clear day during the daylight hours – late afternoon or early morning.)
The other planets all have superior orbits, meaning that we can see them quite distant from the sun at any time
of night, even directly opposite Earth from the Sun at midnight. Because they are farther from the Sun and their
orbits longer they have different periods to their progression/regression cycles, and they are all different from
each other.
To explain the planetary orbits as orbiting Earth at their center is very cumbersome, and in the case of the
inferior ones, impossible. Each planet would have to be attached to a separate “celestial sphere”, and all rotating
at different rates. Furthermore, each one would have to be able to “back up” to explain how they sometimes
approach and sometimes diverge from the Sun. This is called “retrograde motion”.
In the case of the inferior planets, the situation is even worse for a geocentric model. Any object that orbits Earth
would have to be visible at any point in the sky along its orbit and be visible at any time of the day or night. That
is simply a consequence of the definition of “orbit”. We should be able to see Venus at midnight directly
overhead if it were orbiting us, but we never do. It (and Mercury) is a “Morning Star” or an “Evening Star”, never
a “Midnight Star”.
The model of all the planets orbiting the Sun makes all these issues go away. Retrogression, Morning/Evening
Stars are all simply consequences of simple solar orbits, not a series of special cases in Earth orbit.