7. Earth-sun relations on or about 21 June. a. On or about this date (and it can vary from about the 20th to the 23rd, depending on the leap year cycle), the North Pole points towards the sun, which allows the direct ray to strike well north of the equator, at 23½° N, which corresponds to the axial tilt. This concentrates the lion's share of the solar radiation in the Northern Hemisphere, making for warm or hot weather. The Southern Hemisphere is cheated of the solar radiation, giving it cold weather. b. This latitude is called the Tropic of Cancer (geotrivia: Why not, say, Scorpio or Aries? Because the astrological sign of Cancer is said to begin on this date -- this should make you popular at parties). c. This means the northernmost tangent ray is displaced past the North Pole, 23½° past 90°: 23½° closer to the equator on the other side of the pole. Subtract 23½° from 90° and you get a latitude of 66½° N. d. This latitude is called the Arctic Circle. On that date, the sun never completely sets there. This is the midnight sun experience! Areas within the Arctic Circle are called the "Land of the Midnight Sun." e. Places north of there go longer and longer periods without sunset. At the Arctic Circle, it's only one day, but the period of midnight sun increases until, at the North Pole, you go six whole months without sunset! Now you know what's behind the story of the North Pole having a six month day: June 21st could be thought of as the "noon" of the six month day! f. Places along the equator, however, get 12 hours of day and 12 hours of night. North of the equator, any place spinning around the earth's axis spends more of its time facing the sun than the night sky. Now you know why the days are long in summer and the nights are short. The difference between night and day gets greater and greater as you move north: At high latitudes, you can have something like 20 hours of daytime and only 4 hours of nighttime! g. In the Southern Hemisphere, everything is just the opposite: i. The equator has 12 hours of day and 12 hours of night. ii. The farther south you go, the less time a location spends in the daylight and the more time it spends facing the night sky: Nights are progressively longer and days shorter. iii. Finally, at 66½° S, you reach the Antarctic Circle, but now it's 24 hours of nighttime and 0 hours of daytime. iv. As you move farther south from the Antarctic Circle, the time spent in darkness increases from one day to one week to one month until finally, at the South Pole, you go for 6 months of nighttime (this would be the midnight of the six month night down there). h. In our Northern Hemisphere, this date is called the summer solstice and marks the beginning of our summer; in the Southern Hemisphere, this date is the winter solstice and marks the start of their winter. i. So, Earth-Sun relations account for the greater heat in the Northern Hemisphere (direct ray and other high angle rays are concentrated north of the equator), the longer daylength in the Northern Hemisphere, and the midnight sun experience in the Arctic regions. j. Now you can understand why it is that the northern and southern hemispheres have opposite seasons. k. A bit of geotrivia. This date is called a solstice, which, in Latin, means "the sun stands still." What this means is, if you were to stand at one particular spot each evening and systematically recorded where exactly the sun set each day all year round, you would find that the sun set in a noticeably different place each night, particularly in September and March. In June and July, however, you would hardly notice any difference in the location of sunset (north of due west). So, the ancients said "the sun stood still" and this time is still called the solstice. 8. Earth-Sun relations on or about 21 September (again, this date changes, depending on the leap-year cycle) a. As the earth swings along its orbit around the sun, the constant tilt of its axis means that the North Pole begins to shift away from pointing into the sun. b. By September 21st, the North Pole is only tangentially visible from the sun, pointing to the upper left -- and the South Pole has just swung around into being barely tangentially visible to the lower right. The equator would now appear as a straight line trending from the upper right to the lower left. c. Swinging around the planet to a point above the equator, where we can see half of the tangent rays on the planet, we see that both poles are touched by the tangent rays and the circle of illumination forms a straight line crossing the equator at right angles: d. Looking at the situation, we see that the direct ray is now directly (sorry) on the equator on this date. e. Looking at any location, we can see that it will spend half of one rotation facing the sun and half facing away from the sun. f. This means that all places on Earth experience 12 hours of daylight and 12 hours of night. i. The equator always has 12 hours each, all year round. ii. But so does Long Beach at 34°N iii. And Buenos Aires, Argentina, roughly 35° S iv. And at the North Pole, the first 12 hours of this date are the last 12 hours (technically, given the twilight effect) of the six month day and the last 12 hours of the date are the first 12 hours of the six month night: This is sunset at the North Pole. v. Similarly, the first 12 hours of the date at the South Pole are the last 12 hours of the six month night and the last 12 hours of the day are the first 12 hours of the six month day (with caveats for the twilight quality). vi. So, everywhere on Earth, night is (technically) equal to day on this date: Everywhere on Earth has the same amount of nighttime today. This is, therefore, an equinox, which, in Latin, means "equal night." g. We call this date the fall equinox in the Northern Hemisphere or the autumnal equinox, and it marks the beginning of fall (or autumn) for our hemiphere; in the Southern Hemisphere, this is the spring equinox or the vernal (green) equinox, and it marks the start of spring there, when greenery pops out all over. 9. Earth-Sun relations around 21 December (again, remembering that the date changes from one year to the next) a. The earth continues along its orbit, all the while maintaining the constant tilt of its rotational axis. This means that the North Pole begins to point away from the sun, while the South Pole now points toward the sun. The equator would look like a curve bent upwards. b. This produces exactly the opposite situation we analyzed for the 21st of June. c. The direct ray is now positioned well into the Southern Hemisphere. How well? 23½° S. Want to guess the name of this latitude? Yup -- it's the Tropic of Capricorn, because astrology buffs tell you it's the beginning of the "sun sign" Capricorn (even though, because of the precession of the equinoces, the sun no longer rises in the constellation Capricorn on this date). This concentrates the most direct rays of the sun in the Southern Hemisphere, together with the heat they produce. The Southern Hemisphere is now hotter than the Northern. d. The northernmost tangent ray again strikes at the Arctic Circle, but this time in such a way as to shade the North Pole and the Arctic regions. The Arctic Circle now gets 24 hours in which the sun technically fails to rise. The farther north you go, the longer the time the sun fails to rise, until, at the North Pole, you are now in the middle of the six month night. This date is the "midnight" of the six month night. e. The southernmost tangent ray again strikes at 66½° S, along the Antarctic Circle, but this time so that the South Pole and the Antarctic regions are now bathed in constant sunshine: 24 hours at the Antarctic Circle and progressively longer and longer periods of constant sunshine until, at the South Pole, we experience the "noon" of the six month day there. It's now the Antarctic regions (those bounded by the Antarctic Circle) that are the "Lands of the Midnight Sun." f. The equator still spends half a rotation in the daytime and half in the nighttime (hence, the constant equality of night at day at the equator. g. All places north of the equator spend a greater and greater proportion of their rotational periods in nighttime and a smaller and smaller period facing the sun: It's now the Northern Hemisphere that has short days and long nights. This gets progressively more extreme, until at the Arctic Circle, we're talking 24 hours of nighttime and 0 hours of daytime. h. All places south of the equator not only enjoy the warmer temperatures that the concentrated sunlight in their hemisphere brings, but the longer and longer days of summer and shorter nights of winter. At the Antarctic Circle, this disparity between day and night finally hits the 24:0 ratio. i. This date, then, is the winter solstice in the Northern Hemisphere, which marks the start of our winter, and the summer solstice in the Southern Hemisphere, marking the beginning of the Austral summer. 10. Earth-Sun relations on or about 21 March (don't forget the changeability of this date). a. The earth continues along its orbit, maintaining its constant axial tilt, and this means that the South Pole no longer points toward the sun. b. By the time the March equinox rolls around, the North Pole and the South Pole are again just tangentially visible from the sun, the North Pole now pointing to the upper right and the South Pole to the lower left. c. Again, the equator would form a straight line, this time trending from the upper left to the lower right. d. Other than that, the geometry of direct rays and of tangent rays is exactly as it was for the 21st of September or thereabouts. i. The equator again gets a boring 12:12 ratio of day to night. ii. So does every other place on Earth, including the North Pole (coming out of its six month night: sunrise) and the South Pole (coming out of its six month day: sunset). e. This date, too, is an equinox ("equal night"): the spring or vernal equinox here in the Northern Hemisphere and the fall or autumnal equinox for those in the Southern Hemisphere. 11. To view an animation illustrating Earth-Sun relations at equinoces and solstices, click here. 12. Quick review of the cartographic terms associated with revolution, the tilt of the earth's axis, and seasonality. a. The tropics are the outer limits of the noon overhead sun, of the direct ray of the sun. i. The Tropic of Cancer is the one in the Northern Hemisphere, at 23½° N, and the direct ray of the sun is experienced there on one day of the year, on or about the 21st of June. ii. The Tropic of Capricorn is the one in the Southern Hemisphere, at 23½° S, and it experiences the noon overhead sun one day each year, too, but on or around the 21st of December. iii. All places in between the tropics experience the direct ray of the sun two days a year, once when the direct ray is migrating north to Cancer and again when the direct ray is migrating south to Capricorn (for instance, the equator, halfway between the two tropics, receives the direct ray of the sun on or about the 21st of September and again on or about the 21st of March). b. Declination of the sun refers to that latitude experiencing the noon overhead sun on a given day. i. The declination of the sun is 23½° N on or about the 21st of June. ii. It's 23½° S on or around the 21st of December. iii. It's 0° around the 21st of September and again around the 21st of March. iv. You can learn the precise declination of the sun for any date by plugging in the following formula: d = 23.44 * sin [360/365 * (284 + N)] where: d = declination N = the number of a day in the year (so, 1 January would be 1, 25 February would be 56, you get the idea). You would add N to 284 and divide 360 by 365. Then, you would multiply those two answers and take the sine (you can try this convenient online scientific calculator, which has a sine button). After doing that, you'd multiply that answer by 22.44 (which is 23°26'28" expressed as a decimal to make the math easy). Voilà! -- declination for a given day. v. You can also learn the declination for a given day by simply consulting a declination chart, such as this one: vi. Alternatively, you can consult an analemma, which provides day by day information on the sun's declination and information on whether the sun is fast or slow (that is, whether the apparent motion of the sun takes less than 24 hours or more than 24 hours, because of the difference in tropical time and sidereal time and where we are in our orbit -- more on that in a bit). c. The Arctic and Antarctic circles are the outer limits of the midnight sun phenomenon. i. They are the farthest latitudes from each pole that ever experience the midnight sun or the sunless noon. ii. This happens only one day at the Arctic or Antarctic circles and for progressively more and more days as you move closer to the poles, until at the poles we're talking six months of constant sunshine and six months of constant nighttime. iii. Their latitudes, 66½° N or S, reflect the axial tilt of 23½° from the vertical of the plane of ecliptic: You get them by subtracting the axial tilt from 90° N or S, the latitude of each pole. 13. Mentioning the analemma reminds me about the sun-fast/sun-slow thing. a. Because the earth's orbit is elliptical, that means its speed varies over the course of the year: It moves fastest around perihelion and slowest around aphelion. b. This is predicted by Johannes Kepler's laws. Kepler, who lived from 1571 to 1630, is one of the folk heroes of many sciences. He formulated three laws for the understanding of planets' motions: i. Each planet travels in an elliptical orbit, with the sun at one of the two foci of that orbit. ii. Most relevant here: The imaginary line connecting a planet with the sun sweeps out equal areas in equal times. That is, let's say we drew a line from Earth to the sun at one point in time (oh, July 1st) and then did it again some time later (oh, how about ten days later, July 11th?). We'd figure out the area in the wedge between the sun and the earth at those two points in its orbit. Then, we'd repeat the experiment half a year later, drawing a line from the earth to the sun on, let's say, January 1st and then again ten days after that, January 11th. Again, we'd calculate the area. What we'd find is the longer wedge at aphelion (July) would have to be skinnier so that it included the same area as the shorter wedge at perihelion (January). This means the earth wouldn't have travelled along its orbit as far in July as it would in January. Thus, the earth speeds up in January and slows down in July to preserve Kepler's Second Law. iii. For the sake of completeness, Kepler's Third Law states that the square of the period of a planet (its year) is proportional to the cube of its mean distance from the sun. The farther it is out there, the longer its year. That's why, when astronomers figure out how long an extrasolar planet's revolution takes, they can infer how far it is from its star. c. This variation in speed means that, at the end of a 24 hour rotation of 360°, the planet has moved farther along its orbit than it would have in 24 hours on a perfectly circular orbit -- or not so far along (depending on the time of year). So, rotating 360° means you return to the 0° point, which would be obvious if you were looking at a far distant star outside the solar system (assuming you could in broad daylight!). But the sun will be relatively little or somewhat more to your left 24 hours later, because you've moved along the orbit while you were rotating. It won't be in the same relative position with respect to some other distant star. To see the sun in the same relative position in a circular orbit, you need to rotate a shade under 361°. If the orbit is elliptical, though, 361° will overcompensate at some times (aphelion) and undercompensate at others (perihelion) because of the different rates of speed and distance covered in the orbit at different times of year. So the sun would be seen as farther east or west than you expected it to be when your watch reads noon. This is also why sunrise and sunset are not perfectly symmetrical, the same time before and ahead of noon on your clock. d. So, what does this have to do with the analemma? The analemma gives you the declination of the sun (its N/S latitude) for a given day, and it also gives you the "equation of time," the amount of time by which sun time is off in comparison with a perfectly 24 hour clock. e. You can create a really cool analemma of your own, if you have a lot of time and self-discipline (only about a half dozen people on Earth have had the time and obsessiveness!). What you do is set up a camera to take really fast exposures of the sun on the same exact frame at exactly noon (according to your clock) on a regular basis throughout the year (every ten days or every month). When, after a year's labor, you develop that one frame, you'll see the analemma in your own sky. It'll come out looking kind of like here. G. Well, this lengthy section (F) takes care of revolution as one of the kinds of motions carried out by Earth (the other major one being rotation, discussed in section E). There are some other motions, though, in which the earth is involved. 1. The axis wobbles. a. Right now, the axis is about 23½° from the perpendicular of the plane of ecliptic, but, because of a wobble in the axis, that tilt has varied through time from 21°39' to 24°36'. b. The periodicity of this wobble is 40,600 years. c. Because of this wobble, the North Pole points to different "pole stars" through time. i. Right now, it's Polaris (the North Star) ii. In about 12,000 years, the North Star will be pointing to Vega, the brightest star in the constellation Lyra (the Harp), which is in our summer skies. d. The wobble means that our Northern Hemisphere summer, which now takes place around aphelion, will in 10,000-11,000 years coïncide with perihelion, making our summers a bit hotter (the more so since the Northern Hemisphere heats up more than the Southern Hemisphere in summer anyhow, because of the greater preponderance of landmasses in the Northern Hemisphere). 2. The eccentricity of the orbit also varies. a. Right now, the semimajor axis is 149,597,870.7 km from the sun and the semiminor axis is 149,576,880.8 km, which is not really very different: The earth's orbital eccentricity, then, is only 0.017 (I got that by squaring both the semimajor and semiminor axes, subtracting the smaller from the larger and raising the answer to the 0.5 power and then dividing that answer by the semimajor axis) b. The earth's orbital eccentricity varies over time from a minimum of 0.01 to a maximum of 0.07, kind of bouncing from a more perfectly circular shape to a more oval one over complex cycles of 95,000 years and 413,000 years. We're pretty close to a minimum now (orbit almost circular). When we attain maximum eccentricity, perihelion will be about 5 percent closer, which will exaggerate our seasons: Summers will be hotter and winters will be colder. These changes in seasonality may have something to do with the timing of the great ice ages, so it's not a trivial difference in the long run. 3. The earth is part of the solar system, which itself revolves about the core of the Milky Way galaxy, to the suburbs of which we belong, moseying along at some 225 km/sec (around 140 mps)!!! A complete revolution of the galactic center takes about 250 million years. We're out about two thirds of the way from the core, about 30,000 light years out in the disk part of the galaxy, in one of its spiral arms (the Orion-Cygnus arm). There are some 400 billion stars in this galaxy alone! 4. The Milky Way is itself moving within the Local Group of galaxies (which includes Andromeda and the Magellanic Clouds) some 300 km/sec (185 mps). 5. With respect to the microwave background radiation, the echo of the Big Bang, we're moving about 380 km/sec (around 235 mps) towards the constellation Leo -- this is our absolute velocity with respect to the structure of space itself. Well, that's it for the size and shape of the planet and its complicated motions in space. In the next lecture, I'll introduce the geographic grid (latitude and longitude) and how to navigate with the stars, sun, a peculiar watch, and an analemma.
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First placed on web: 09/04/00 Last revised: 06/04/07