Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay
Rotation of the earth causes day and night. Before sunrise and after sunset some light is scattered over the horizon by the atmosphere, giving us twilight. Civil twilight is defined as the time when the sun is less than 6 degrees below the horizon, about half an hour before sunrise or after sunset in middle latitudes. During civil twilight it is light enough to conduct many activities without artificial light. Nautical twilight is the period when the sun is 6-12 degrees below the horizon, when the horizon is still distinct enough for navigational sightings. Astronomical twilight is the period when the sun is 12-18 degrees below the horizon. Once the sun is more than 18 degrees below the horizon, the sky is dark enough for astronomical observations. At high latitudes, the sun may not get far enough below the horizon for certain kinds of twilight during the summer. Saint Petersburg, Russia, at almost 60 degrees north, is famous for its "white nights," when civil twilight barely ends in summer.
At low latitudes, the sun rises and sets much more steeply. At the equator, since the earth rotates one degree in four minutes, six degrees below the horizon is only 24 minutes and 18 degrees is only 72 minutes. Thus, twilight is very short in the tropics.
We use the earth's rotation as the basis for keeping time. Noon is the moment when the sun is due south, or at its highest elevation in the sky. We define the average rotation of the earth (noon to noon) to be 24 hours.
Astronomical noon is different for every longitude on earth, a problem that became acute when travel became rapid enough for people to notice the difference. Nowadays we divide the earth into time zones, mostly separated by an hour (there are exceptions). As measured by a clock, astronomical events happen earlier on the east side of a time zone and later on the west side, by 4 minutes per degree of longitude.
However, the sun appears to move about a degree per day in the sky because of the earth's motion in its orbit. But the earth doesn't move at a uniform speed, which means the sun doesn't appear to move at a uniform rate in the sky. Sometimes in 24 hours the earth spins a bit too far to catch up to the Sun, and other times it doesn't spin quite enough. Over the course of a year, the sun, or a sundial, can be up to 18 minutes fast or slow compared to a uniform clock. This variation is called the equation of time. If you photograph the sun at the same clock time every day, over the course of a year it will move north and south with the seasons, but also a little east and west, tracing out a skinny figure 8 called an analemma. Globes often have an analemma, usually out in the Pacific, to show where the sun is overhead or how much sundials and clocks differ.
During the course of a day, the earth rotates with respect to the stars, then has to rotate about a degree more to make up for its orbital motion. It takes a bit less than 24 hours for a given star to reappear at the same place in the sky: 23 hours and 56 minutes to be exact. Time measured with respect to the stars is called sidereal time. Sidereal time is easy to determine with a star chart - it's equal to the right ascension that is overhead at the moment. The difference between solar and sidereal time means that the stars rise about four minutes earlier each night, or two hours earlier each month.
Before GPS systems, people used the stars for navigation.
The earth moves around the Sun every 365.25 days in an orbit that is slightly elliptical. Nowhere is it written that the earth's rotation has to mesh neatly with the earth's orbital motion. It doesn't. Every four years we have to add an extra day to make up for the discrepancy. We actually overcorrect a bit, and we adjust by dropping century years unless they are divisible by 400. Thus 1700, 1800 and 1900 were not leap years but 1600 and 2000 were. Leap Year Day, 2000 was scheduled before the Pilgrims landed, and the rarest scheduled event in human history slipped by most people completely unnoticed. There are even longer term discrepancies, which we will deal with some time in the distant future when they become significant.
Because the earth's axis is tilted relative to the plane of its orbit, the amount of sunlight any point on earth receives during the course of a year varies, resulting in the seasons. There is a zone around each pole where the sun never rises or sets at certain times, bounded by the Arctic or Antarctic Circles. The latitude of the Arctic or Antarctic Circles is 90-23.5 = 66.5 degrees north or south. In a zone either side of the equator, the sun can be overhead at some time of year. The limits of this zone are the Tropics of Cancer and Capricorn, 23.5 degrees north or south.
The diagram above addresses two of the biggest misconceptions about the seasons:
Just like the axis of a tilted top changes orientation, the earth's axis changes orientation, or precesses because of the gravity of the moon, sun, and other planets. In fact, all rotations of any kind experience precession if an oblique force is applied to the moving object.
The tilt of the earth's axis remains constant, but the orientation of the earth's axis sweeps out a cone over a period of about 26,000 years. As it does, the position of the north celestial pole shifts among the stars. In 14,000 A.D., the bright Star Vega will be a fairly useful pole star and the Southern Cross will be visible over most of the U.S.
Precession means that right ascension and declination of the stars change. Star atlases are drawn with coordinates that are correct for the time of publication, but a century or so later, the coordinates are noticeably different from the real sky. Fortunately it is easy nowadays to generate correct star charts for any point in time by computer. The Tropics of Cancer and Capricorn got their names because at one time the solstices were in those constellations. The summer solstice is now in Gemini and the winter solstice is in Sagittarius.
So far we've used the term orbit without really defining it. Now the time has come to understand how they work.
An orbit is the path an object takes in response to two laws. Inertia causes the object to move in a straight line. Gravity causes the object to fall toward some other object. The combined effect of the two laws is to cause the object to move in a fixed path, or orbit. The two objects can be the Space Shuttle or Moon around the Earth, the Apollo command module around the Moon, or a planet around the Sun. The more massive object is called the primary and the less massive one is called the secondary.
Note that in the example above, the orbit isn't a circle. The object moves fast close to the Sun, loses speed as it pulls away and gravity pulls it back, and then begins falling inward, gaining speed again. The astronomer Johannes Kepler first worked out the laws of planetary motion in the early 1600's.
Kepler's First Law is that planets move in elliptical orbits with the Sun at one focus.
There are many ways to draw ellipses. The top diagram shows one way. Stick two pins in a board, loop a string loosely around them, then pull the string tight with a pencil and keep it tight as you move the pencil around the pins. Each pin is a focus of the ellipse.
The lower figure shows why the focus is called that. If you put a light at one focus and line the ellipse with mirrors, all the light reflects to the other focus. Some telescope designs take advantage of this fact. It also works with sound. An elliptical room, called a whispering gallery, enables two people at the foci to converse in whispers while people not at the foci hear nothing.
What's at the other focus of an orbit? Nothing. It's called the empty focus and has no physical significance.
Kepler's Second Law is that a line from the planet to the Sun sweeps out equal areas in equal times. In the diagram above, the wedges are of different shapes but their areas are equal.
Kepler's Third Law is that the square of a planet's period and the cube of its distance are proportional. A good example in approximate whole numbers is Jupiter, which is 5 times as far from the Sun as Earth and takes 11 years to orbit the Sun. 5 x 5 x 5 = 125 and 11 x 11 = 121. There are two reasons for this law:
To describe a planet's orbit completely, we need to describe:
These six quantities are called the elements of the orbit. They can be determined from three observations of an object.
The size of a planet's orbit is given as half the length of the long axis, or major axis of the orbit, and is usually symbolized with the letter a. It is equal to the average distance of the object from the Sun.
The eccentricity, symbolized by the letter e, is the fraction of the way the focus lies from the center to the end of the ellipse. For an ellipse, e is always between 0 and 1. A circle is a special ellipse with e = 0. Apart from Mercury, none of the planets have eccentricities even as large as 0.1. Comets, however, can have eccentricities greater than 0.9. Perihelion distance of a planet equals a(1-e) and aphelion equals a(1+e).
Even if we know the shape and size of an orbit, that doesn't tell us how it's oriented. We do that by specifying the longitude of perihelion, usually measured from the vernal equinox.
Finally, we want to know how the angle is tilted with respect to the Earth's orbit (ecliptic). This angle, usually denoted i, is the inclination. The direction of the tilt is described by the intersection of the two orbital planes. Usually we specify the direction where the planet passes from below the ecliptic to above it, called the ascending node. The symbol „ is often used for this quantity. Finally, we need the position of the planet at some specified time, called the epoch. Usually we tabulate the longitude of the planet from perihelion.
A time-lapse movie of the Moon's motion shows several striking effects:
The illustration shows how libration works. The gray represents the side of the Moon that always faces earth and the purple represents the side that faces away. The Moon is rotating at a uniform rate. Unrealistic colors are deliberately used to avoid confusing this with the phases of the Moon. As the Moon moves away from perigee (counterclockwise) it is moving around the earth faster than it's rotating. We see around the right side of the Moon, but as the Moon approaches perigee, its orbital motion is slower than its rotation. The rotation catches up and outpaces the moon's orbital motion, and we begin to see around the left side of the Moon.
In the bottom diagram, when the Moon is north of the ecliptic plane we can see past its south pole, and when the Moon is south of the ecliptic we can see over its north pole.
One of the most entrenched myths about the Moon is that the Moon does not rotate. True, it always keeps the same face to the earth, but as seen by an observer on another planet, the Moon does rotate. It rotates in the same length of time it orbits the earth. Note in the diagram above that someone watching from outside sees all sides of the Moon.
When objects in the sky line up as seen from Earth, a variety of interesting sights may occur:
Because of the tilt of the Moon's orbit, most of the time the Earth's shadow misses the Moon and the Moon's shadow misses the Earth, and there is no eclipse (top).
Only when the Earth, Sun and Moon line up close to the nodes of the Moon's orbit are eclipses possible (bottom).
Shadows consist of two parts, a dark inner umbra where the light source is completely covered, and a light outer portion where the light source is partially covered. In the umbra of the Moon's shadow, the Sun is completely covered and we have a total eclipse. In the penumbra, the Sun is partially covered and we see a partial eclipse.
The umbra of the Moon is just barely long enough to reach the earth, so a total eclipse is visible over a narrow band no more than a couple of hundred kilometers wide. The duration of a total eclipse is at most 7-1/2 minutes, with 2-3 minutes being more typical.
Often the umbra of the Moon doesn't reach the earth at all, and the Moon is too small to cover the Sun completely. Then we see a thin bright ring of Sun around the Moon, an annular eclipse. Annular eclipses can be up to 12 minutes long.
Sometimes only the Moon's penumbra strikes the earth and then all we see is a partial eclipse. Rarely, the umbra just nicks the polar regions and there is a small patch where a total or annular eclipse is visible.
Occasionally, the Moon's umbra is just long enough to hit the earth and we have a total eclipse in the center of the eclipse paths but an annular eclipse at either end. The band of annularity or totality is very narrow and neither the annular nor the total eclipses are very long.
The map above shows the next total eclipse of the Sun in the United States on August 21, 2017. The penumbra first touches down in the North Pacific (gray) but the total eclipse is already over when it does. The umbra hits the earth a little later and viewers at the start of the totality track would see a total eclipse sunrise. The umbra sweeps eastward (purple band), touching land in Oregon and then slides across the U.S., with the longest eclipse in southern Illinois, finally entering the Atlantic off South Carolina. It sweeps across the Atlantic, leaving the surface before touching Africa about 3-1/2 hours after first touching down. Beyond that (gray) there is a band where the sun sets before maximum eclipse.
Around the umbra is an oval penumbra. The shaded bands represent zones where the maximum eclipse is 20, 40, 60, 80%, or greater. Most of the continental United States will see at least an 80% eclipse. A third of South America and the fringes of Europe, Africa and Asia will also see a partial eclipse.
The U.S. had a long drought without a total eclipse: from 1979 to 2017 is 38 years, but then we hit the jackpot. During the 21st century, there are seven total eclipses visible in the lower 48 states, two more visible in Alaska, and one in Mexico that just misses the border. These are all impressively long eclipses, too. One has over six minutes of totality, two others are over five minutes.
Mercury and Venus (and any other objects that orbit within the earth's orbit, like minor planets or spacecraft) never get far from the Sun as seen from Earth. When they are farthest from the sun as seen from Earth, they are at greatest elongation. When they are at greatest eastern elongation they are east of the Sun and visible in the evening sky, and at greatest western elongation they are west of the sun and visible in the morning sky.
They also can experience conjunction with the Sun. When they are beyond the Sun, they are at superior conjunction. When they are between Earth and the Sun, they are at inferior conjunction. Rarely, Mercury and Venus pass across the face of the Sun in a transit. Earth can transit the Sun as seen from the outer planets.
|Venus and Mercury show phases like the Moon. When they first reappear in the evening sky, they are on the far side of their orbit, fully illuminated, and tiny. As they move slowly away from the sun they become bigger but less illuminated. When farthest from the sun they look like quarter moons. Then as they drop back toward the sun they become crescents. They are brightest during the "fat crescent" stage because they are close and still well illuminated. They become thin crescents and plunge quickly in toward the Sun.|
|When Venus and Mercury first appear in the morning sky, they shoot quickly out of the solar glare. They emerge as crescents but become more illuminated and smaller. At greatest elongation they appear like quarter moons. Then as they round the far sides of their orbits they become smaller, more fully illuminated, and drop slowly back toward the Sun.|
Because Venus is the closest planet to Earth at inferior conjunction, it can pass as much as 7 degrees above or below the Sun. At such times, it can easily be seen with a telescope in daylight as a thin crescent. At high latitudes, where the ecliptic in summer makes a very low angle with the horizon, it is even possible to see Venus at inferior conjunction with the unaided eye just before sunrise or after sunset.
Outer planets can appear anywhere in the sky. When they are on the far side of the Sun, they are in conjunction. It is always a superior conjunction. When they are opposite the Sun in the sky, they are at opposition. Planets at opposition are at their closest to Earth and at their biggest and brightest in a telescope. They rise around sunset and are highest in the sky at midnight.
As a planet moves out of conjunction with the sun, it appears to move eastward among the stars. But when it is 90 degrees from the Sun, or at quadrature, the earth is moving directly toward the planet. As seen from the planet, the earth is at greatest eastern elongation. As seen from the Earth, the planet appears to stand still. Then, as we approach opposition, the earth passes the planet and it appears to move backward. Finally, when the planet is again 90 degrees from the Sun, the earth is moving directly away from the planet and it again appears to stand still, before resuming its normal eastward motion among the stars.
Above are the loops traced in the sky by Mars during two oppositions. The ancient astronomer Ptolemy accounted for the looping motion by assuming the planets moved on a small circle attached to its main orbit. The astronomer Copernicus finally proposed that the loop was due to Earth overtaking the planet.
The time it takes two planets to repeat some configuration, say oppositions of Mars or inferior conjunctions of Venus, is called a synodic period (a synod is a meeting). The formula for calculating synodic periods is not difficult. Let's say Planet 1 has period P1 and Planet 2 has period P2. In one day, Planet P1 completes 1/P1 of its orbit and Planet P2 completes 1/P2. If Planet 1 is moving faster, it gains on Planet 2 by 1/P2 - 1/P1. The time it takes to gain a complete orbit, then, is 1/(1/P2 - 1/P1). We can also write the formula:
1/Psyn = 1/P2 - 1/P1.
For example, Earth has a period of 365.25 days and Venus has a period of 225 days, so the synodic period of Venus with respect to Earth is 1/Psyn = 1/225 - 1/365.25 = 0.0017066, or Psyn = 586 days. It is rarely useful to be very precise because synodic periods vary due to the eccentricity of planets' orbits.
The earth need not be one of the planets. For example, how often are Jupiter and Saturn in conjunction, that is, the same right ascension? Jupiter has a period of 11.9 years and Saturn a period of 29.4 years, so 1/Psyn = 1/11.9 - 1/29.4 = 0.05, so Psyn = 20 years.
Any repeating motion has a synodic period. For example, Venus circles the Sun in 225 days and rotates in the opposite direction in 243 days, that is, has a period of minus 243 days. How often does the Sun rise on Venus? We have 1/Psyn = 1/225 - (-1/243) = 1/225 + 1/243 = 0.00856, or Psyn = 117 days.
Created 07 July 2008, Last Update 13 September 2018