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A 103 Notes, Week 1B (C-M 0.1-0.2)

0. Latitude and longitude

I. The obvious view: appearance of the sky

A. Constellations B. Celestial sphere, NCP, SCP, celestial equator, zenith, horizon

II. Apparent motion of the stars and real rotation of the Earth. View from

A. Poles B. Equator C. Milwaukee

III. Apparent motion of the Sun through the Zodiac and real revolution of the Earth

A. Zodiac constellations, ecliptic, and equinox B. The solar day (ordinary day) and the sidereal (star) day:

Stars rise (and set) 4 minutes earlier each night. C. The seasons and the tilt of the Earth's axis of rotation

IV. Precession, the third motion of the Earth: Apparent precession of the equinoxes and real precession of the Earth's axis of rotation.

You, King Gelon, are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface. (Archimedes, The Sand Reckoner)

Aristarchus (310-230 BC)

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0. Latitude and longitude The longitude of a place, say Milwaukee, is the angle between the meridian through the place and the meridian through Greenwich. That is, longitude is the number of degrees along the equator from the Prime Meridian to the meridian through Milwaukee: 87° 57' W (of Greenwich). At nearly 90o W, Milwaukee is nearly ¼ the way around from Greenwich. The latitude of a place is the number of degrees measured along its meridian from the equator to the place. Milwaukee is 43° N (of the equator) almost halfway to the pole. Green Bay is halfway to the North Pole, at 45°N

latitude lines longitude lines (meridians)

I. The Obvious View

A. Constellations These are apparent groups of stars. The stars in a constellation are usually far from each other, but happen to be in the same direction as seen from Earth; they are not bound to each other by gravity. Many of the constellations – the patterns of stars we use – were picked out by the Greeks and still have Greek names. These days the sky is divided by astronomers into some 88 pieces, each given the name of an ancient constellation that it contains, so that every star—every point of the sky—officially belongs to some constellation. The Chaisson-McMillan picture of Orion shows the varied distances to the stars in Orion, the brightest constellation in the winter sky.

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B. The celestial sphere The sky has the appearance of a sphere, and at night it looks like the sphere rotates, with all of the stars moving as if the sphere were rotating. The name given to this imaginary sphere is the celestial sphere.

The celestial sphere appears to rotate once a day from east to west (counterclockwise looking up at the north celestial pole). This is because the Earth really rotates once a day from west to east: counterclockwise, looking down at the north pole. (See, for example, the animation at http://www.opencourse.info/astronomy/introduction/03.motion_earth/ )

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• Thus, for us, standing on the Earth in the northern hemisphere, each star, the Sun, the Moon, and the planets seem to move in a counterclockwise circle about Polaris, the North Star.

In a day, the celestial sphere appears to rotate once around. It rotates 360° in 24 hours, so each hour it rotates 360o / 24 = 15o . Thus stars appear to move l5° per hour east to west.

NCP, SCP, celestial equator, zenith, horizon

• NCP: the point on the celestial sphere directly over Earth's north pole • SCP: the point on the celestial sphere directly over Earth's south pole • celestial equator: the circle on the celestial sphere directly over Earth's equator

The meaning of the next two locations depends on where you are:

• zenith: the point on celestial sphere that at any moment is directly overhead • horizon: the stars that lie along your horizon are on a circle on the celestial sphere 90°

away from zenith

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II. Apparent motion of the stars and real rotation of the Earth What you can see at night depends on where on Earth you are standing. You can see only half the sky: The Earth block light from stars that are below the horizon. As the Earth rotates, stars rise along the east half of the horizon, move in circles centered about the NCP, and set along the west half of the horizon. • The star Polaris, the North Star, is close to the NCP, and stars appear to move in circles

about Polaris.

Stars close enough to Polaris on the celestial sphere never rise or set and are called circumpolar. Seafarers, Polynesians, Babylonians, and Libyans, knew from sailing that as you move north, the NCP appears higher in the sky. (The Greeks finally understood the reason—that the Earth is a ball). As seen from the equator, Polaris is at the horizon. If you travel 1° north from the equator, Polaris will be 1° above your horizon. By the time you reach the North Pole, 90° N, Polaris is directly overhead, 90° above the horizon, and in general • The altitude of Polaris = your latitude • Because Milwaukee is 43o north of the equator, Polaris is 43o above our horizon.

From a place south of equator, Polaris is below the horizon, never visible.

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A. Sky from N,S Poles

This diagram and two more below are from Nick Strobel's Astronomy Notes at www.astronomynotes.com. (This is a good set of notes, with excellent illustrations.)

• Stars move in horizontal circles, never rising or setting (all stars circumpolar)

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Here’s a photo taken at the north pole with a camera that’s pointed straight up.

B. Sky from equator

• The poles are at the horizon. Stars rise straight up along the east half of the horizon and set straight down along the west half of the horizon. Poles are at the N and S points of the horizon. Here’s the photo, followed by the diagram.

N W S

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C. Sky from Milwaukee: Milwaukee is 43° N of the equator so • Polaris is 43° above the horizon. Stars rise obliquely in the east, move across the sky and set

in the west. Here’s a diagram from Seattle, about 46o N

You can see that the stars move in circles as the celestial sphere appears to rotate about the north celestial pole. What is really happening is that the Earth is rotating, carrying us and the camera with it. In Milwaukee, we and the camera move along the circle of latitude 43o N, and as we move along the circle, we rotate once each time we move once in a circle. If this is not clear, stand up and turn in a circle, holding your hand out with your finger pointing outward.

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As you turn, your hand moves in a circle and it also rotates once, your finger changing the direction it points in as your hand moves once about the circle. Here’s long time exposure taken from a place about as far north as Milwaukee by a camera that’s sitting with its shutter open, facing Polaris

NW N NE Each circular arc is about 1/3 of a circle, implying the exposure time was about 1/3 of 24 hours, or about 8 hours. The picture dramatizes the fact that stars near Polaris (the bright circular arc near the NCP – north celestial pole do not rise or set; stars farther than about 43o from the NCP are below the horizon at some time in each 24-hour period. They rise and set, moving counterclockwise (NE to NW) in the photo. (I think the reddish line on the right is an airplane taking off.)

• Stars within 43° of Polaris are circumpolar in Milwaukee, never dipping below our horizon. Our circumpolar constellations include Ursa Major, Ursa Minor (the great and small bears, commonly called the Big and Little Dippers), and Cassiopeia.

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III. Apparent motion of Sun through Zodiac and real revolution of the Earth A. Zodiac constellations, plane of ecliptic, equinox

The Sun is always at some point on the celestial sphere. When it is above the horizon, blue light from it is scattered—it bounces off air molecules, water droplets, and dust to produce the blue sky. (We’ll talk later about why the longer-wavelength red light is scattered less). Because the sky's blue light is brighter than the light from the stars, you cannot see the stars during the day. The effect is like the glare of headlights that prevents you from seeing, for example, the face of a driver coming toward you at night. If he turned off his headlights, your own lights would be more than bright enough to reveal his face.

In a year the Earth moves 360 degrees about the Sun, so each day it moves about 1°.

The figure above shows that for us, standing on the Earth, the Sun appears to move against the background stars. As the Earth moves 1o from west to east, the Sun appears to move 1o from east to west. As the Earth moves in a circle (an ellipse, but it’s nearly a circle) around the Sun from west to

1o

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east each year – counterclockwise looking down on the north pole – the Sun appears to us to move in a circle west to east against the background stars. The stars that lie in the plane of the Earth’s orbit are the constellations of the Zodiac – If we could see the stars in the day when the Sun is out (and we can in fact see them during a total eclipse of the Sun) we would see the Sun move through these constellations during the course of the year.

The stars in the diagram that we can see from the Earth are the stars above the side of the Earth that faces away from the Sun. You can see from the diagram that these stars change during the course of the year. Gemini and Taurus are in our winter sky (on opposite sides of Orion). Scorpius, Capricornus and Sagittarius in our summer sky. Notice that Polaris is always 43o above the horizon in Milwaukee, so it and the circumpolar constellations are always visible at night. An animation of the Earth’s motion about the Sun is at http://esminfo.prenhall.com/science/geoanimations/animations/01_EarthSun_E2.html Here’s another equivalent but prettier diagram:

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On January 1 the Sun is in Sagittarius, implying that stars in or near Sagittarius in the sky can only be seen (if at all) just before sunrise or just after sunset.

• The Sun appears to move 1°/day east along the celestial sphere • The apparent path of the Sun on the celestial sphere is called the ecliptic. The constellations

along the path are the Zodiac constellations. • The stars that one sees at night change during the year. The daily motion of the earth in its

orbit is 1/365 of a circle or about 1o per day. As a result, stars appear to shift their position by about 1o per night. After a year the Earth is back in the same position. Equinoxes: The equinoxes are the days when the Sun is directly over the equator. The vernal equinox is the first day of spring, March 21. The autumnal equinox is September 23 (for convenience, fall starts on September 21, a couple of days before the equinox). The locations of the Sun on the celestial sphere on the vernal equinox and autumnal equinox are also named equinoxes. If you look at the diagram on p. 12, you’ll see that the vernal equinox is near the boundary between the constellations Pisces and Aquarius: That is, the line from the Earth through the Sun on March 21 points to stars near the Pisces-Aquarius boundary.

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B. Solar and sidereal day Here’s the Chaisson-McMillan diagram showing the change in the position of the Earth in one day. Compare the top and middle pictures. The middle picture shows the Earth after one rotation relative to the stars (one sidereal day or star day), bringing the same star to the same position in the sky. If the Earth had not moved around the Sun, the Sun would also be at the same position in the sky. But the has moved around the Sun by 1o, and it has to rotate by 1o for the Sun to be above the same point of the earth. [There is one confusing thing about this picture: the continents in the diagram don’t rotate with the Earth! - An artist’s mistake apparently not caught by the authors.]

~1o

~1o

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What this means is that a solar day is longer than a sidereal day by the time it takes the Earth to rotate through 1o, 1/360 of a circle. How long is that? Well the Earth rotates by 360o in one day, so it rotates by 1o in 1/360 of a day. This is 4 minutes: 1 day = 24 hours = 24x60 minutes = 1440 minutes. 1/360 (1440 minutes) = 4 minutes (Or: 24x60/360 = 24/6 = 4) The time from sunrise to sunrise (1 day) is 4 minutes longer than the time from one rising of the star Betelgeuse (the brightest star in Orion) to the next. • Stars rise 4 minutes earlier (and set four minutes earlier) each day.

If Betelgeuse rises at 4:00 pm this afternoon, it will rise tomorrow at 3:56. In a month it will rise about 30x4 = 120 minutes = 2 hours earlier. After 12 months it will rise 24 hours earlier: After a year, the Earth will be back to its original position: Betelgeuse will rise at 4:00 pm. In fact, an accurate ancient measurement of a year was the rising time of a star: On the same date of each year, the same star will rise at the same time.

C. The seasons and the tilt of the Earth's axis of rotation Why do globes of the Earth have their axes of rotation tilted? The reason is that the Earth’s axis of rotation is not exactly perpendicular to the plane of the Earth’s orbit about the Sun:

• The Earth's rotation axis is tilted by 23 ½ o away from perpendicular to its orbit— the plane of the equator is 23 ½ o from the plane of the orbit. If you look back at the figure on p. 12 or at Fig. 0.10 of Chaisson-McMillan, you will see the Earth’s axis pointing in the same direction (toward Polaris) as the Earth circles the Sun. In summer in the northern hemisphere the north part of the axis is tilted toward the Sun, and the Sun is directly over a point north of the equator. On the summer solstice, when the axis is most directly tilted toward the Sun, the Sun is directly over a point 23 ½ o north of the equator. On the winter solstice, with the north part of the axis tilted away from the Sun, the Sun is over a point 23 ½ o south of the equator.

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The plane of the Earth’s orbit about the Sun is called the plane of the ecliptic (we’ll see why in the discussion of eclipses next week). The last diagram had the plane of the Earth’s orbit horizontal and the equator tilted by 23 ½ o. In this one we’re looking at the celestial sphere with the equator horizontal, so the plane of the Earth’s orbit is tilted by 23 ½ o. The diagram makes it easy to see how the position of the Sun changes from 23 ½ o north to 23 ½ o south of the equator and back again each year.

• The seasons are caused by the tilt of the Earth's axis. The tilt has two effects: 1. Sunlight is more direct (the Sun higher in the sky) in summer than in winter 2. The Sun is up longer in summer than in winter. In our summer, the northern hemisphere is tilted toward the Sun, in winter it is tilted away from the Sun. The diagram below illustrates the fact that in winter, the same amount of Sunlight has to heat up a larger area. The effect is essentially the same as the lengthening of shadows in morning and evening when the Sun is low. The same rays that hit the paper are spread over larger area in winter than summer. The same amount of sunlight has to heat a larger area in winter.

SMALLER AREA

SUMMER

SUN’S RAYS

SUN’S RAYS

WINTER

LARGER AREA

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Here’s another version of the same picture:

Together, the fact that (1) the same amount of sunlight has to heat a larger area in winter and (2) the Sun is up for a shorter amount of time in winter mean that the amount of sunlight—of energy from the Sun—that reaches the northern hemisphere is less in our summer than in our winter.

• When the northern hemisphere is tilted away from the Sun, the southern hemisphere is tilted

toward the Sun. Consequently, during our summer, it is winter in the southern hemisphere, and vice-versa.

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One final diagram, this one showing why days are shorter in winter than in summer:

In the middle is someone standing in Milwaukee. Up in the diagram is up in Milwaukee. In a 24-hour period, the Sun appears to move once around a circle centered about the dotted axis through the poles –for example, the red, green and blue circles. On the summer solstice, the Sun as shown is on the red circle 23 1/2 o north of the equator (above the Tropic of Cancer). The Sun is above our horizon for more than 12 hours, because more than half the red circle is above the horizon. You can see from the diagram that the Sun rises in the northeast and sets in the northwest. On the vernal and autumnal equinoxes, the Sun is on the celestial equator, the green circle. Half the circle is above the horizon, half below, so the Sun is up for exactly 12 hours. It rises due east and sets due west. Finally, on the winter solstice, the Sun is on the blue circle, 23 1/2 o south of the equator, over the Tropic of Capricorn. Less than half the blue circle is above the horizon, so the Sun is up for less than 12 hours, rising in the southeast and setting in the southwest.

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IV. Precession, the third motion of the Earth

Animation on D2L under Content / Week 1 and at http://visibleearth.nasa.gov/view_rec.php?id=164

A top that is not spinning falls. When it is spinning, it stays upright. Instead of falling, it precesses: the direction of its spin slowly changes. The Earth’s spin also precesses, and this is related to the fact that the Earth bulges at the equator. Because the Sun’s gravity gets stronger as you get closer to the Sun, the part of the bulge that is closer to the Sun is pulled slightly more strongly. If the Earth did not spin, the Sun would pull the bulge so that the equator was aligned with the plane of the Earth’s orbit. Because the Earth spins, its axis precesses instead. The plane of the equator is tilted by 23 ½ o from the plane of the Earth’s orbit, and the precession of the axis keeps the 23 ½ o tilt. As shown in the figure, the gravitational tug of the Moon on the bulging equator has the same effect (again trying to align the Earth’s equator).

• The Earth’s axis of rotation precesses with a 26,000 year period: In 13,000 years, what are now winter stars will be summer stars, and vice-versa. In 13,000 years, Vega, not Polaris will serve as our north star. In 26,000 years, the Earth’s axis will again point in the direction it now points, and Polaris will again be the North Star.

•

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The effect of the Earth’s precession was first observed by the Greek astronomer Hipparchus (in about 150 BC). He noticed that the position of the stars on the same day of the year had changed slightly over the 50 years he had made observations. The effect has been called the

• precession of the equinoxes, because the position of the Sun on the celestial sphere at an equinox slowly changes. You’ve seen most of this picture before. Look now at the two

dotted lines (along the Earth-Sun direction) that mark the spring equinox at the boundary between Pisces and Aquarius. These show the difference in the Sun’s position at the spring equinox that is caused by the precession of the Earth’s axis of rotation. The spring equinox is the date when the Sun is directly over the equator, when the axis of rotation is perpendicular to the line from Earth to Sun. Because of the slow precession of that axis, the year ends slightly before the Earth-Sun line points to the same stars it pointed to one year earlier. The precession of the Earth’s rotation axis takes 26,000 years, so each year it precesses by the small fraction 1/26,000 of a circle, or 50'' arc. In astrology, the sign under which you are born is supposed to be the constellation in which the Sun is on your day of birth. Because of the precession of the equinoxes, however, the constellation corresponding to a given day gradually changes. For example, on the spring equinox some 6,000 years ago, as ancient priests were devising a religion tied to the stars and planets, the Sun was in Taurus, the bull. On the spring equinox 4,000 years ago, the Sun was in the constellation Aries, and Aries is still the sign in astrology associated with someone born on the spring equinox, March 21. For the last 2,000 years, however, the Sun has been in Pisces on the spring equinox. Its entrance into Pisces at about the time Christ was born is related to the use of the fish as a symbol for Christianity. Now at the spring equinox, the Sun is at the boundary between Pisces and Aquarius (hence a song about the dawning of the Age of Aquarius and the phrase “new age”, as in “new-age music”).