Solar System Astronomy

The Perspective From Earth

OBSERVATIONS: 1 – Days and Years

1 – The Sun frequently “rises” in the east and “sets” in the west. We call this a “day” from one rise to the next.

2 – In the northern hemisphere on the summer solstice (also, erroneously, called “midsummer’s day”) the Sun is higher off the southern horizon than at any other time. Every day its elevation gets lower and farther south, until by the winter solstice (“midwinter’s day”) it is low in the southern sky. Then it begins moving higher again, cycling back and forth between the two extremes annually. The southern hemisphere is exactly out of phase with the northern. Summer here is winter there, and vice versa. “Year” is what we call this cycle.

3 – As solar elevation increases or decreases, so does day length, varying directly. The longest day corresponds to the highest elevation (summer solstice), the shortest to the lowest elevation (winter solstice).

4 – 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. “Seasons” is what we call this combination of things – the Sun’s position and the resulting weather.

5 – Over the course of a year we see different stars in the sky at night. That is, the stars appear to move almost like the Sun, but slightly faster. 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.

As we proceed we will compare the heliocentric and geocentric explanations for these two things as an example of hypothesis testing. Treat each as a hypothesis and see which one makes more sense at the end.

This is a long slide show, but is easily broken up at obvious points into a series of study sessions rather than just one.

Let’s think about these things one at a time.

1 – 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. A merry-go-round is perfect, if you can find one.

As we will eventually see, a rotating Earth explains other things’ motions (planets and stars) much better than a bunch of revolving other things does.

NIGHT DAY(Sun not visible) (Sun visible)

SUN

wheeeee!zzzzzzzzzzz

Sun orbits Earth in 24 hours.

Here is the necessary setup in the heliocentric model. The Sun orbits us once in a single day.

Newton’s laws of motion indicate that this cannot be true. Why can’t it?

2 – 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.

The small arrows in the diagram remind you that we rotate continually as we revolve around the Sun. The bigger ones indicate the direction of revolution. Notice that the combined

motion is like a ball rolling along curved path – both types of motion are in the same direction. The plane in which we move, tilted slightly here, is called the ecliptic.

We will have to examine Earth’s orientation in space and its revolution around the Sun in order to understand observation 2 and the remaining observations.

A B

ECLIPTIC PLANE

This line isperpendicularto the ecliptic.

~23.4°

Earth’s axis is not perpendicular to the ecliptic, but rather tilted at about 23.4°. This means that one or the other hemisphere leans toward the Sun most of the time. The only exception is on the equinoxes, and we’ll see why later. In this diagram the Northern Hemisphere is tilted toward the Sun and the Southern Hemisphere away from it. This changes over the course of a year.

However, it doesn’t change because Earth “wobbles” in space, at least on a timescale that interests us. The axis always points in the same direction: toward the North Star. That star is called Polaris because it stays directly above the pole. (There is a very slow wobble – once in about 26,000 years – but we do not have to care about that in this class.)

SUN

As Earth revolves around the Sun its direction of tilt with respect to the Sun changes slowly day by day, but only with respect to the Sun. Note that in this picture the axis is everywhere tilted to the right by the same degree. It does not change orientation in space! The extreme ends of this system are shown by the bigger Earths below, whose axes are drawn tilting directly toward the right in each case. The Northern Hemisphere is tilted toward the Sun at the summer solstice (A) and away from it at the winter solstice (B). Make sure you see that the opposite is true in the Southern Hemisphere.

At points Y and Z that same tilt will be neither away nor toward the Sun. These are the transition points from one to the other state. As Earth approaches Z from A the tilt angle with respect to the Sun decreases to zero. Between Z and B it increases to a maximum (away from the Sun). On the far side, going back to A, the pattern reverses. Z is the fall equinox and Y is the spring equinox – the days the Sun was overhead at Albemarle Island at noon in our first experiment. Remember that the tilt of the axis with respect to the rest of space – all the other stars, for example, doesn’t change. It still tilts in the same direction as always, but that direction is neither toward nor away from the Sun.

This revolutionary or orbital motion is the basis for the year. To get all the way around from A back to A takes one year: ~365-¼ days. The seasons within that year are dictated by the axial tilt.

A B

PolarisPolaris

Y

Z

A

SUN

N

S

BN

S

Remember that the aspect of seasons that we are currently considering is #2 – the height of the Sun at different times of the year. In summer it is highest at noon, in winter noticeably lower. Another way to say this is that the angle of incidence (i) is higher in summer and lower in winter. Make sure you see that.

“A” shows the situation in Northern Hemisphere summer. If you were at the point labeled “N” the angle of incidence (i) would be high. That is, you would have to look up to see the Sun at Noon and the ground would be out of your view. At the same latitude in the Southern Hemisphere (S) the angle would be low – the Sun would be low on the (northern) horizon and you could see it and much of the ground in front of you.

In “B” the situation is exactly reversed. At point “N” the angle of incidence is low and the Sun would be low on the (southern) horizon. At the same latitude in the Southern Hemisphere the Sun would be high in the sky.

Summer

Winter

“Down Under”Summer

Winter

i

i

i

i

The angle of incidence being higher in local summer and lower in local winter is a major part of the reason why summer is warmer than winter.

We will look at how the Sun heats the Earth in some detail later in the course. For now take it for granted that the solar radiation hitting the ground at a higher angle (summer) allows more of it to be retained, and hitting at a lower angle (winter) means that more of it is reflected away. Greater absorbance means warmer weather, and the converse.

It is a widely held belief that summer is warmer because the Sun is closer, but this is not true! The Sun is not appreciably closer to us in any season. At the latitude of Americus the difference that results from the axial tilt is only a few hundred km. If we call it 1000km and compare that to the roughly 150,000,000km to the Sun, you can see that the difference (0.0000667% -- about 7/100000ths of a percent) doesn’t really matter. The difference in summer and winter weather is almost entirely about angle of incidence. We will see what controls the rest next.

3 – As solar elevation increases or decreases, so does day length, the two things varying directly. The longest day corresponds to the highest elevation of the Sun and the shortest day to its lowest elevation. Day length is directly proportional to position in the orbit, just as angle of incidence is.

A B

PolarisPolaris

Y

ZAHighest angle of incidence

andlongest day of the year in

northern hemisphere.

BLowest angle of incidence and

shortest day of the year in northern hemisphere.Y & Z

Average angle of incidence;

day and night the same length in

both hemispheres.

SUN

A B

At the solstices the plane of the terminator – the line between the sunlit part of Earth and the dark part – is at an angle to that of the axial tilt. The latitude lines drawn below show the consequence. Visualize each of the three as two segments – one in the night sector (A1) and one in the day sector (A2) of the globe. The length of the line you see is directly proportional to the arc degrees around the globe at that latitude. Remember that this is summer.

At the equator the two are equally long. A place spends the same amount of time in both sectors and day and night lengths are the same.

In northern latitudes A2 is longer than A1 so day is longer than night. Make sure you see that the opposite is true in the southern hemisphere.

At very high latitude, above the Arctic Circle the entire line is in the day sector. A day lasts 24 hours here, on this one day. Even farther north 24 hours of daylight becomes more and more common. At the pole a day is all summer long, without any dark.

In winter the situation is reversed.

At the equator the two segments are still equally long. Indeed, day and night length are equally long every day of the year at the equator.

In northern latitudes B2 is longer than B1 so night is longer than day. Again, make sure you see that the opposite is true in the southern hemisphere.

Above the Arctic Circle the entire line is in the night sector. A night lasts all day here, on this one day. Even farther north 24 hours of dark becomes more and more common.

At every latitude day/night lengths are the same on the two equinoxes. Refer back to slide 6 and figure out why before moving on.

SUN

Position “A” onSlide 6

A1 A2B2B1

Position “B” onSlide 6

Viewed from the perspective in Slide 6 the axis would be tilted directly away from you at “A” and the sun would be directly to the right. The terminator cuts the latitude lines exactly in half everywhere.

At “B” the axis would be tilted directly toward you and the sun would be directly to the left. The terminator again cuts all latitude lines exactly in half. “Equinox” translates from Latin “equal night” – equal to the day length. How does this change from this point to the next solstice? Figure it out.

You should now see why both solar angle of incidence and day length correlate to axial tilt. When the Sun is more directly overhead (in summer) the Sun hits us from higher overhead and the time we spend on the lighted side of the planet is greater. In winter the opposite of both these things holds.

You should be able to infer that, year-round, 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 angle of incidence and in day and night length varies directly with latitude. The weather stays colder poleward all year because the angle of incidence is the dominant factor in the weather. There is a rough equator to pole gradient in temperature, modified by other considerations that we’ll get to later in the course.

Summer not only give us a higher angle of incidence of solar energy, it lets us spend more time each day soaking up that energy. Both things make summer warm. Winter forces a lower angle of incidence and shorter duration of daylight on us, so the weather is colder. Things don’t change much at the equator because day length never changes and angle of incidence only changes slightly (~13.25°), but they change dramatically toward the poles, where day length varies from 0-24 hours and angle of incidence stays at, effectively, zero for months. The temperature gradient is therefore ameliorated in the summer and exacerbated in the winter.

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 the resulting differences in weather. The revolution around the Sun means that that tilt direction changes with respect to the Sun throughout the year.

4 – 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”.

Sun orbits Earth in 24 hours.

Sun’

s orb

ital p

ath

wob

bles

up

and

dow

n an

d ba

ck d

own

one

full

cycl

e in

~36

51/4

days

Here is the necessary setup in the heliocentric model. The Sun orbits us once in a single day and over the course of a year its path swings up, down and back up in a year.

Newton’s laws of motion prove that this cannot be true. Why not? This is the second case where the heliocentric model is overly complicated and the geocentric model very simple. What was the other one?

5 – Over the course of a year we see different stars in the sky at night. That is, the stars appear to move almost like the Sun, but slightly faster. This means that we see the stars 366 time for every 365 times we see the Sun – the stars seem to outrun the Sun very slightly.

Every morning my wife and I walk about a mile up and down the road we live on. During the academic year this usually happens before sunrise so we get to watch the morning sky day by day for most of the year.

Every year when the fall semester starts, Aquarius, a constellation of very faint stars that is nearly impossible to visualize, is low in the west, Pisces is above that, and Ares is near the zenith. From there to the eastern horizon we see Taurus, Gemini, and the little fuzzy cluster that is the only part of Cancer you can really see very well.

As the year progresses we lose constellations to the west and gain new ones from the east. Aquarius goes and Leo arrives. Pisces goes and Virgo arrives, then out with Ares and in with Libra. By Christmas break Taurus is very low in the west and the claws of Scorpio start to show, and by the start of Spring semester the one is gone and the other mostly up.

By the end of the semester all the constellations we saw at the start of the year have gone and new ones have come in. Aquarius has gone and come back, now approaching the zenith. They come and they go, east to west, one by one, slowly and surely, forever and ever, regular as clockwork. In fact, farmers have always used them as a clock. The Sun comes up just behind the appropriate one in the queue, but that one changes throughout the year. It’s like we have one hand on the clock that runs too slowly compared to the others. It does so in a very predictable way, but the Sun and the stars appear to move on slightly different schedules viewed from here.

How does this work?

I can’t see the stars because the Sun is too

bright.

What gorgeous stars. Good

thing the Sun is down.

There are two things to remember while you think of this. The first is that you can only see the stars when you can’t see the Sun. In other words, you have to be on the side of Earth presently turned away from the Sun. Don’t worry, if you can’t see them now you’ll roll around where you can in a little while (small dashed arrows).

The other thing is that we are on different “sides” of the Sun as we revolve around it (big solid arrow). The view here is toward the North Pole of Earth from above and the people are at noon and midnight, local times. Other hours would work as well. The next three slides bring the two together.

DAY 2

DAY 1

On day 2, because the Earth has revolved a bit, this is the dividing line between visible and not visible stars.

Make sure you see that the observer would be able to see stars now that had formerly been below the horizon at dawn.

DAY 2

DAY 1

At the nightfall end of the same two days:

This picture from your book explains the whole system 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.

As you study it remember that the Earth is continually rotating. Twelve little Earths are drawn to show their position relative to the Sun in each month. The side toward the Sun is in daylight, so no stars are visible. The darkened outside is away from the Sun, so it’s nighttime there. Midnight comes when the world has rotated right to the middle of that darkness – when the Sun is exactly 180° away.

The next slide shows what constellations my wife and I see toward the south at about 5:30 AM each day as we take our morning walk.

In July You’d see the stars in the “July” box at midnight.

Those 180° away –the ones in the “January” box –would be overhead with the Sun and you’d be unable to see them.

In January you’d see the stars in the “January” box at midnight. Zenith would be between Cancer and Gemini.

The stars 180°away (the “July” box) would be overhead when the Sun was also overhead and you’d be unable to see them.

1 - Due South, Jan 1, 5:30 AM

2 - Due South, Feb 1, 5:30 AM

3 - Due South, Mar 1, 5:30 AM

4 - Due South, Apr 1, 5:30 AM

5 - Due South, May 1, 5:30 AM

6 - Due South, Jun 1, 5:30 AM

7 - Due South, Jul 1, 5:30 AM

8 - Due South, Aug 1, 5:30 AM

9 - Due South, Sep 1, 5:30 AM

10 - Due South, Oct 1, 5:30 AM

11 - Due South, Nov 1, 5:30 AM

12 - Due South, Dec 1, 5:30 AM

Leo

Virgo

LibraCapricornus

Sagittarius

Aquarius

Pisces

Aries

Cancer

GeminiGemini

TaurusScorpius

Back to

Leo

STARS

STARS

STARS

STARS

Here is the heliocentric setup, viewed from above.

The Sun goes around us at a certain speed.

The stars goaround us just enough faster that we see them oneextra time every year.

Which explanation is simpler?

In the course of a year we see the Sun rise roughly 365 times. If it’s going around us it’s doing it ~365 times from one solstice back to the same solstice (or any other specific point on the orbit).

Because different stars come into view each day, if we count the number of times we see them in the course of a year it’s 366 times. This is just another way of saying that the Sun and stars apparently go around us on different schedules. Thus we recognize both a “solar year” (which we use on calendars) and a “sidereal year”. One has ~365 days, the other has ~366 days, but both have exactly the same number of seconds or minutes or hours. The definition of a “day” is what differs.

But, of course, this is a result of our progress along the orbital path, not because the stars and the Sun are moving differently. They aren’t moving at all. They just look like they are because we’re spinning, and they look like they are doing it differently because we are forevermore spinning in a slightly different part of space – a little farther along on our path around the Sun.

Here we find another case of the geocentric model requiring a complex set of motions that the heliocentric model does not need. What were the other cases we saw?

One last thing about years. 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 1900) – which is divisible by 4 and would be a leap year – the leap day is omitted and February only has 28 days.

This also over-corrects, in the opposite direction, 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 slightly over-corrects and so …

1 – Some nights the Moon rises in the east and sets in the west, but it moves more slowly than, and is “lapped” by, the Sun. This means that sometimes it rises or sets in the daytime. As this happens we see more or less of its face, and we call this pattern the Moon’s “phases”. Its set/rise happens at a different time each day, creeping later and later. It appears to move differently than the Sun.

2 – The timing of daily tide cycles corresponds to the timing of a lunar orbit and therefore the phases of the Moon correlate to tidal cycles.

3 – Beyond the idea of dividing the year into 12 roughly equal intervals there is no real relationship between lunar months and the months on western calendar. In fact, there are different ways of defining what a lunar month is. Some societies, particularly in the Middle East do maintain a separate lunar calendar for religious and cultural customs, like Ramadan, even though their financial and political relations are managed on a western calendar.

OBSERVATIONS – 2 – Months

The reason the Sun seems to lap the Moon is that we rotate on our axis a little faster than the Moon revolves around us. Thus each night it is not as far along in its orbit around us as it was the night before. (This diagram is very schematic – we do not make that much progress on our orbit in “a few days”.)

The Moon rotates on its axis at exactly the same rate that it revolves around us, so we always see its same side. This is not magic, there is a physical basis for it called “tidal locking”. Eventually Earth will rotate at the same rate as well. The Moon will not rise and set, but will always be above the same spot, and there will be no tides.

The moon’s orbital rate means that it returns to the same position with respect to a given star in about 27.5 days. This is a “sidereal lunar month”.

If you study the diagram carefully you will also be able to see why the Moon has phases. It has day/night too – one side is toward and the other away from the Sun, but that side changes day-by-day 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 the sidereal and synodic months months is because of the forward movement of the Earth Moon system as the cycle proceeds. The synodic month is the basis for all lunar calendars like the ones used for religious purposes in Islamic countries.

1 – 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.

1) The situation one day

2) The situation a few days later.(Imagine the intermediates day by day.)

NEW MOON

The lunar cycle begins with new Moon. This is also called “dark of the Moon” because none of the visible surface is illuminated. Schematically the Moon looks like the image on the left. It is visible, but throws no light toward Earth.

Nor does it need to because, as the right diagram shows, new Moon always occurs during the day. The Moon rises and sets (almost) simultaneously with the Sun and crosses the sky all day. Line of sight from the Earth at noon is shown by the dotted line. All the system views are from above the northern hemisphere.

This side of theMoon is in shadow This side

is lit bysunlight.

What you’d see.Arrangement

of Objects.

FIRST QUARTER MOON

A little over 7-1/3 days later half of the side of the Moon toward Earth is seen lit. The other half is still dark. Increasing illumination of the surface we see is called “waxing”. In the northern hemisphere we see the Moon wax starting on the right side and progressing leftward, so by first quarter the right half is bright. Line of sight at about sunset is shown.

This is also called “waxing half” Moon, but first quarter is better because: 1) we actually see only a quarter of the Moon illuminated – half of the half toward us, not half of the whole, and 2) this is one quarter the way through the lunar cycle.

This sideThis side is lit byof the sunlightMoon is in shadow

What you’d see.Arrangement

of Objects.

FULL MOON

A little over 7- 1/3 days later we get to full Moon. The Moon is directly opposite the Earth from the Sun so the entire illuminated half is toward us. Line of sight is at about midnight.

Beyond this point in the cycle we will see less of the Moon each night. The Moon is “waning” from this point.

What you’d see.Arrangement

of Objects.

THIRD QUARTER MOON (aka “waning half”)

Third quarter comes, predictably, seven days and change after full Moon. The left side is illuminated as seen from the northern hemisphere because the Moon has waned from the right. The line of sight is at about sunrise.

The light dashed arrow shows what happens during the rest of the synodic month – the Moon returns to its original position with respect to Earth and to new Moon.

The diagrams have not been perfectly accurate because they do not show the progression of the Earth (and Moon) in the orbit around the Sun. The solid arrow shows which way it should have gone. In this time it would have made nearly 1/12 of the orbit.

What you’d see.Arrangement

of Objects.

The next step will be to return to new moon here.

A one month cycle of the Moon(At ~3.5 day intervals. Rotation and revolution directions indicated at start. View to

Moon directly overhead – approximate times given at bottom.)

1 – NEW MOON*Moon is between Earth and Sun.

*Far side is lighted and near side dark.

*Rises at dawn and sets at sundown.

*Solar eclipse possible.

5 – FULL MOON*Earth is between Moon and Sun.

*Near side lighted and far side dark.

*Rises at sunset and sets at dawn.

*Lunar eclipse possible.

9 – NEW MOON*Back to beginning of cycle.

4 – WAXING Gibbous*Moon has lagged 3/8 of its orbital path.

*Right-hand 3/4 visible.

*Rises mid afternoon, obvious in night sky until about 3:00AM.

3 – FIRST QUARTER*Moon has lagged 1/4 of its orbital path.

*Right-hand 1/2 visible.

*Rises at noon, obvious in night sky before midnight.

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.

6 – WANING Gibbous*Moon has lagged 5/8 of its orbital path.

*Left-hand 3/4 visible.

*Rises very early AM, obvious in night sky after about 3:00AM. Sets mid-afternoon.

7 – THIRD QUARTER*Moon has lagged 3/4 of its orbital path.

*Left-hand 1/2 visible.

*Rises at midnight and sets at noon,

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.

noon 3 PM sunset 9 PM midnight 3 AM dawn 9 AM noon

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, although the synodic lunar phase is the original basis for calendar months it is obvious that the length of all modern calendar months, except usually 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 …”.

2 – 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.) The picture below, reproduced from the previous figure, shows this. The barycenter is within Earth because we have the greater mass by far.

If we imagine only the moon’s path as we orbit the Sun it weaves back and forth (thin red dotted line) across the common orbit of both bodies (solid black line) Earth also wobbles like this, considered alone, but to a much lesser extent (dashed red line).

It is usually claimed that the Moon’s gravity causes tides, but this is not accurate. It is the entire gravity system of the two bodies rotating around the barycenter 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 slinging the water away from the Moon. The next slide shows the effect on 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!

GRAVITYof

MOON

INERTIAof

WATER

LPath of barycenter (labeled “B”)

H H

L

B

The view is from above the north pole.

Gravity and inertia displace water from the places labeled “L” (for low tide) into the places labeled “H” (for high tide) as the blue arrows indicate. This creates two bulges of water (much exaggerated here): one toward (gravitational) and one away from (inertial) the Moon. As the system rotates the bulges rotate with it, following the Moon*.

*This description of how tides work is a substantial oversimplification. In order to understand how high and low tides actually work within any ocean, a much more complex model is needed, but the complexity is simply added onto this basic model, it doesn’t replace or negate it.

Rotation of system

B

Earth’s rotation on axis is a little faster than system rotation around barycenter

1) If you were on an island here the water would almost cover the island. This is high tide.

B

2) In a bit over 6 hours you would have turned to here. The water would have fallen continuously, exposing more of the island. This is low tide.

B

3) Another 6+ hours would bring you here. The water would have risen continuously, covering more of the island at high tide.

B

4) Back to low tide.

5) High tide.

Make sure you realize that the appearance of tide behavior at a beach is similar in one important respect to the appearance of the Sun “rising” and “setting”.

Just as the Sun is not really doing the moving, the high and low water levels created by the bulges are not moving. They are stationary (in this simple model) and Earth is turning beneath them, just as it is turning beneath a stationary Sun. The erroneous perception that we are immobile throws us off in both cases. We are moving, but everything around us (except the Sun and tide bulges) is moving just as we are. We don’t perceive our motion any more than we feel ourselves move in a closed car on a straight road.

The bulges do, of course move with the Moon, but that doesn’t cause us to see low tides and high tides. Our motion does.

Full New

1st

3rd

The tide cycle just described happens every day, so it cannot correspond in any way to the phases of the Moon. What does correspond is a consequence of the interaction of the lunar and the solar tides.

The Sun pulls/slings a bulge about 1/3 the size of the lunar bulges. At new and full moons the lunar and solar bulges are in the same place and the tide range is at a maximum: higher than average highs and lower than average lows. This is spring tide. At 1st and 3rd quarter moons the bulges are at right angles and literally work at cross purposes to produce the lowest tide ranges: lower than average highs and higher than average lows. This is neap tide. At every other phase of the Moon the tide range is increasing toward spring tide or decreasing toward neap tide.

Axis of spring tides

Axis of neap tides

The apparent motion of the Moon around Earth is different from the apparent motion of the Sun around us. It seems to go a fair bit slower and so loses ground to the Sun, as it were.

In the old geocentric (Earth-centered) model of the universe this difference was accounted for by having the two do literally that: move at different rates around us.

In the heliocentric (sun-centered) system the apparent motions result from different actual paths. The Sun does not move, we spin. The Moon, on the other hand, does revolve around us.

This two part system explains all of the apparent motions and the related phenomena (like tides) without any additional hypotheses.

And as Newton pointed out, an object with the size of Earth could never hold something the size of the Sun in orbit. It would inevitably hold us – the barycenter of the system would be very near the center of the Sun. The Moon is a different story because it is so much smaller.

There are other star-like objects that appear to move in yet a different way than the Moon and Sun. 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. Mercury’s behavior would be very similar because it too is an inferior planet. The superior planets’ apparent motions can be explained with the same model, but because their orbits are much larger they are hard to use as examples.

OBSERVATIONS – 3 – Other Planets

Scorpius

VirgoLibra

12/1/18

12/16/18

ScorpiusLibra

Virgo

ScorpiusLibra

Virgo

ScorpiusLibra

Virgo

1/1/19

1/16/19

These rose after sunrise and so were not visible.

Direction of apparent daily motion(in all cases)

VENUS

Constellations have “risen” higher in the morning sky,But Venus has “lagged” behind them.

There is one interesting point I want you to try to follow. This year (2018-18 academic year) the observed apparent motions of Venus have taken place during the early morning hours from late in the Fall, 2018 semester to early in the Spring, 2019 semester. Venus will do this same sort of thing again in a few months, but not in the same part of the year, and not against a backdrop of the same constellations! Next time will be late summer and maybe into early fall of 2019 against whatever constellations are in the morning sky at that time.

If the geocentric model of the universe were correct and everything revolved around Earth, then the revolutions should be smooth and unidirectional and they should have some consistency to their schedules. Something that goes one way and then another is not moving smoothly enough to fit the bill. When considered as a group, each planet does this completely differently. We have to assign completely different motions to each planet, which doesn’t really sound much like “consistency of scheduling”.

When we see Venus, or any other planet, we see it going forward faster than the nearby stars sometimes and lagging behind them at others, as has happened since early December. We see them doing that against a backdrop of some stars this time and against other stars next time, and still others the time after that. The planets are “wanderers” because they cannot seem to stay in one relative position like all the other things in the sky seem to do.

Aristotle explained this within the geocentric model by having the planets orbit something on their orbital path around us – sometimes literally reversing their path and going backwards. Newton explained why such orbits cannot exist. We are about to see how the heliocentric model makes this far easier to understand. One set of smooth, regularly scheduled motions explains everything we see.

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°.

At time 2 Earth is 60° into its orbit and Venus 98°.

And so on … Venus outruns us.

Examine the diagram and follow it step by step. Note that Venus completes its year in about 71/3 Earth months, between positions 7 and 8. Its position is shown by lighter images beyond that. 0/12

0

1

1

2

2

3

3

4

5

6

4

5

7

8

9

10

11

6

7

8

9

10

11

12

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. Venus is even higher in the sky here when the Sun comes up.

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.

The trend continues at time 5. Venus has become about as much lower at sunrise than between times 3 and 4.

And at time 6 …

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, and we will not be able to see it in the morning for the same reason we can’t see stars in the daytime. We will be able to see it, barely, just after our rotation has taken the Sun out of our line of sight, in the early evening. Venus has therefore 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 toget 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.

C D

H G

A B

F E

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 (geocentric model) 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 and other stars in “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 heliocentric 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. Occam’s Razor lops off the geocentric model because it is far too complex.

SUMMARY

1. Earth rotates on its axis. The most obvious consequence is to make everything else appear to revolve around us. Another effect we saw was to make the tide rise and fall everyday.

2. Earth revolves around the Sun with the other planets. The time required is one year. the other planets require different amounts of time, in general longer the farther from the Sun they are. Their years are longer than ours.

3. Our axis is tilted from the vertical in the ecliptic plane. this means that the hemispheres “lean” toward or away from the Sun at varying angles as the revolution proceeds. A hemisphere leaning toward the Sun is in summer, with all that entails. One leaning away is in winter. Season-related concepts are:

A. Angle of incidence of sunlight/position of Sun;B. Day length; andC. Weather

4. Revolution around the Sun also causes us to see different constellations in the night sky throughout the year because the dark side of Earth, where those stars are visible, is facing away from the Sun throughout the entire orbit, bringing new stars into view every night.

5. The Moon appears to orbit Earth slightly slower than the Sun. It does orbit Earth, and is the only thing that does. Actually it is more correct to say that Earth and the Moon revolve together around a common center of mass within Earth. This revolutionary motion of Earth is distinct from its rotation by a small amount. The rotation on the axis is a little faster than the barycenter rotation.

6. The lunar orbit means that we see it in different places relative to the Sun day by day, with different amounts illuminated and dark. These are the moon’s phases. A complete cycle of phases takes roughly a month, and is the basis for the calendar month.

7. The barycenter rotation causes tidal bulges in the oceans – one on the side toward the Moon and one on the side away from the Moon. earth’s rotation

8. The planets all seem to move at different rates around Earth, sometimes faster and sometimes slower than the Sun and stars. The apparent complexity of their motion is a consequence of their revolution around the Sun.

9. All the apparent motions of bodies in space are explained by the simple hypothesis of a heliocentric system. The old geocentric system required a separate hypothetical motion for every object.