For the questions below, refer to the accompanying Figure 2-5.
- In Figure 2-5,
how does the sun angle (the angle between the sun's rays and a horizontal
surface) at solar noon generally vary with increasing latitude? What does this imply about the intensity
of sunlight at high vs. low latitudes?
First, because the sun is far away and the earth is small compared to that distance, the earth receives only a tiny sliver of all the solar radiation that the sun emits, and so the solar radiation reaching the earth arrives from essentially the same direction everywhere on the planet. That is, "rays" of solar radiation striking the earth are parallel to each other. (We showed this in class by drawing the earth and sun to scale on the chalkboard. When they are drawn 15 feet apart, the sun is about 1.5" across and the earth is 0.015" across, which is a tiny dot on the board. The earth is such a small target on that scale that solar radiation can reach the earth basically from only one direction.) In contrast, because the earth is a sphere, a horizontal surface at one place will be oriented differently from horizontal surfaces at all other places on the earth. As a result, the sun angle, which is the angle between incoming rays of solar radiation and the earth's (horizontal) surface, will generally differ from place to place at any moment in time.
At any moment in time, there is only one place on earth that directly faces the sun—that is, one place where sunlight strikes a horizontal surface perpendicularly, so that the sun appears to be directly overhead. As the earth rotates over the course of a day, other places will move into that spot, but they will all be on the same latitude circle. Hence, on any particular day, there is only one latitude that experiences the sun directly overhead sometime during the day.
However, the axis of the earth's rotation tilts, by 23.5° all the time. Moreover, the axis is oriented toward the very distant North Star (Polaris) all the time, so on one side of the earth's orbit around the sun (from the late March equinox to the late September equinox) the Northern Hemisphere is oriented toward the sun, while on the other side of its orbit (from the late September equinox to the late March equinox) the Northern Hemisphere is oriented away from the sun. (The Southern Hemisphere is, of course, oriented away from and toward the sun opposite from the Northern Hemisphere.)
As a result, the particular latitude where the sun strikes the earth directly (and the sun appears directly overhead at some point during the day) varies with time of year. However, because the axis of rotation maintains its constant 23.5° tilt, no latitude farther north than 23.5°N (the Tropic of Cancer) and farther south than 23.5°S (the Tropic of Capricorn) can ever experience the sun directly overhead. Moreover, the farther north one is from the Tropic of Cancer and the farther south one is from the Tropic of Capricorn, the more the earth's horizontal surface is oriented away from the sun and the lower the sun angle is.
Hence, the farther you are north and south of the tropics, the lower the sun angle is.
When the sun angle is lower, solar radiation must travel a greater distance through the atmosphere before reaching the surface, and more sunlight will be scattered and reflected back to space and absorbed before reaching the surface, which reduces the intensity of solar radiation striking the surface. In addition, the lower the sun angle is, the more spread out the solar radiation will be on the earth's surface when the solar radiation does strike it. This also reduces the intensity of solar radiation striking the surface. Hence, we conclude that solar intensity on a horizontal surface at the earth's surface (that is, insolation at the earth's surface) decreases with increasing latitude (at least, outside of the tropics).
We observe warmer temperatures
at low latitudes and colder temperatures at higher latitudes. The decrease in sun angle with increasing latitude and the resulting decrease in insolation with increasing latitude might help explain the observation.
- Several globes are positioned around the room, simulating the position of the earth as it orbits the sun (represented by an overhead or slide projector at the center of the orbit). Notice that globes (should be) positioned so that the axis of
rotation of each globe points in the same direction, just as it does (pointing
toward Polaris, the North Star) when the real earth obits the sun.
- What time of year is it at the position of each globe? How can you tell?
The globe with the Northern Hemisphere tilted away from the "sun" and the Southern Hemisphere tilted toward it, must be at the December solstice (around December 21 or 22). The globe on the opposite side of the sun, where the Northern Hemisphere tilts toward the sun and the Southern Hemisphere tilts away from it (although the axis of rotation points toward the same distant star in both cases), must be at the June solstice (around June 21 or 22). The globe half way in between, where the axis tilts neither toward nor away from the sun but rather toward one side (still toward the same distant star), must be at the September equinox (around September 21 or 22). (We know this because the earth revolves (orbits) around the sun in the same sense that is rotates around it's own axis of rotation; the "right hand rule" helps us remember that direction or rotation.) Those three positions of the earth in its orbit are, respectively, the first day of winter in the Northern Hemisphere (first day of summer in the Southern Hemisphere), the first day of summer in the Northern Hemisphere (first day of winter in the Southern Hemisphere), and the first day of autumn in the Northern Hemisphere (first day of spring in the Southern Hemisphere)
- In Figure 2-5,
which figure (a, b, or c) corresponds to each globe? What perspective
(that is, from where in space) does the figure show your globe?
Figure (a) (the right-hand figure) shows the Northern Hemisphere tilted toward the sun (Southern Hemisphere tilted away from the sun), so that must correspond to the second of the globes mentioned above.
Figure (b) (the center figure) is harder to judge because, of the two middle figures of the earth shown in Figure 2-5, the topmost one is not tilted, whereas the bottom one shows the earth tilted but neither
toward nor away from the sun—instead, to one side. That must correspond to the middle of the three globes (the third one mentioned above).
Figure (c) (the left-hand figure) shows the Northern Hemisphere tilted away from the sun (Southern Hemisphere tilted toward the sun), so that must correspond to the first of the three globes mentioned above.
The perspective shown in the topmost views shown in Figures (a) and (c) is the same, looking directly at the equator in a direction perpendicular to a line from the sun to the earth (perpendicular to the sun's rays) and parallel to the plane of the earth's orbit around the sun. The perspective shown in the topmost view of Figure (b) is also looking directly at the equator, but that is necessarily at an angle to the orbit of the earth around the sun. It's in a direction perpendicular to the sun's rays, but it's also perpendicular to the direction of the other two views, which is necessary in order to see half of the earth lit by the sun, like the other two.
- At each time of year shown, how does the length of daylight seem to vary with latitude? How can
Over the course of one 24-hour day, the earth rotates once, and any particular spot moves in a circle around the axis of rotation. This circular path through space happens to correspond exactly to the latitude circle on which the spot lies.
We can tell something about the length of daylight at any particular spot simply by looking at what proportion of that spot's latitude circle is illuminated by the sun, compared to the proportion that is in the dark. If more than half of the latitude
circle is illuminated, then there must be more than 12 hours of daylight (and less than 12 hours of darkness). If exactly half of the latitude circle is illuminated, then there must be 12 hours of daylight and 12 hours of darkness. And so on.
At the December solstice, there is clearly 24 hours of daylight everywhere south of the Antarctic Circle (66.5°S, which = 90°−23.5°). The proportion of each latitude circle illuminated by the sun then decreases northward from the Antarctic Circle, o the equator, which gets exactly 12 hours of daylight at that time (half the latitude circle is illuminated). The length of daylight continues to decrease going northward in the Northern Hemisphere (where the length of daylight is less than 12 hours) all the way to the Arctic Circle (66.5°N, which = 90°−23.5°), where there is no daylight at all (and the same is true all the way to the North Pole).
At the June solstice, the opposite pattern is true: from the Arctic Circle to the North Pole there are 24 hours of daylight; from the Arctic Circle
to the equator the length of daylight decreases (to 12 hours at the equator) and continues decreasing in the Southern Hemisphere (where there is less than 12 hours of daylight) to the Antarctic Circle, which gets no daylight at all (and the same is true all the way to the South Pole).
At the September and March equinoxes (which are the autumn
equinoxes in the Northern Hemisphere and the Southern Hemisphere, respectively), every latitude circle is exactly half illuminated, so there are 12 hours of daylight everywhere on the planet. ("Equinox" means "equal nights", which means that the length of night, and hence the length of day, is the same everywhere on the earth.)
- Note the length of daylight at San Francisco's latitude at each time of year shown. Does it seem to vary with time of year? If so, describe the variation.
At the December solstice, San Francisco (at about 37.5°N)
gets less than 12 hours of daylight, while at the June solstice it gets more than 12 hours of daylight and at the autumn equinox it gets exactly 12 hours of daylight (and the same would be true at the other equinox, the vernal equinox).
- Note the sun angle at solar noon at San Francisco's latitude at each time of year shown. Does it seem to vary with time of year? If so, describe the variation.
At the June solstice, the Northern Hemisphere is oriented toward the sun while at the December solstice it is oriented away from the sun. As a result, a horizontal surface at San Francisco at solar noon (say) would be oriented more toward the sun at the June solstice than at the December solstice, and so the angle between the sun's rays and the horizontal surface at solar noon would be greater at the June solstice than at the December solstice. That is, the sun angle at solar noon would be greater at the June solstice than at the December solstice. Since those are the times when the Northern Hemisphere is most directly oriented
away from and toward the sun, respectively, and the difference in orientation is 23.5° toward the sun at the June solstice vs. 23.5° away from the sun at the December solstice, the difference in sun angle is a total of 23.5° (tilted in one direction) + 23.5° (tilted in the other direction) = 47°. This is a large difference! (At solar noon in San Francisco at the June solstice, when the sun is higher in the sky than at any other time of year, the sun reaches 76° above the horizon, while at solar noon at the December solstice, when it is lower than at any other day of the year at solar noon, the sun reaches only 29° above the horizon, fully 47° lower than at solar noon on the June solstice.)
- The seasons are defined in terms of key points in the annual cycle in the length of daylight and in sun angle at solar noon. Why do these two quantities vary over the course of the year? At what latitudes does this
variation seem to be the greatest? What are the seasons like at those latitudes, compared
to San Francisco?
The earth's axis of rotation is tilted (by 23.5°), and as the earth orbits the sun, the orientation of the axis remains constant, pointing toward the North Star (Polaris). As a result, on one side of the earth's orbit, the Northern Hemisphere is oriented toward the sun to varying degrees (and the Southern Hemisphere is correspondingly oriented away from the sun), while the opposite is true on the other side of the orbit (the other half of the year). Orienting a hemisphere toward the sun has two consequences: (1) it exposes more of each latitude circle in that hemisphere to the sun, thereby increasing the length of daylight at each location; and (2) it orients horizontal surfaces in that hemisphere more toward the sun, which increases the sun angle and therefore increases the intensity of solar radiation received there. Orienting a hemisphere away from the sun (on the other side of the orbit around the sun) has the opposite consequences.
North of the Arctic Circle and south of the Antarctic Circle, the length of daylight
varies from 24 hours to 0 hours over the course of the year. (In fact, at the North and South Pole, there is 24 hours of daylight for half the year and 24 hours of darkness the other half!). This extreme difference decreases the closer you are to the equator, which receives exactly 12 hours of daylight all year long. The variation in sun angle is a bit different. The highest the sun ever gets at the North and South Pole is 23.5°, so the sun angle at solar noon varies from 0° to 23.5°, a range of 23.5° over the course of the year. At the Arctic and Antarctic Circles, the highest the sun ever gets is 47°, and the lowest it gets is 0°, so the range is 47°. The range is 47° all the way from there to the Tropics of Cancer and Capricorn, but within the tropics it varies less and less with decreasing latitude. At the equator, the lowest the sun gets at solar noon is 66.5° above the horizon (90° - 23.5°). That happens twice a year, at the time of each solstice. At the June solstice the sun is in the northern sky, while at the December solstice it is in the southern sky. It is directly overheat at the equator only twice year, at the two equinoxes.
Although the sun angle varies less in the polar regions than it does between the Tropics and the Arctic and Antarctic Circles, it turns out that the impact of those variations in sun angle on solar intensity is greatest when the angle is lower. That, together
with the greater variation in length of daylight over the course of the year, means that the seasons tend to be the most extreme at the highest latitudes and least noticeable at the lowest latitudes. (In fact, the tropics don't experience seasons at all like ours.)
We tend to think of the seasons in terms of annual variations in temperature and perhaps in other aspects of the weather, more than we do in terms of day length and sun angle. Is there a connection? Sun angle helps determine the intensity of sunlight striking the earth's surface (in two ways, as described in the response to Question #1), and the length of daylight determines how long the sun shines each day. Both help determine the total amount of solar radiative energy received at a location over the course of a day. The amount of solar radiative energy that a location receives during a full day should have something to do with the temperature at that location. Hence, seasonal variations in temperature should be related in some way to seasonal variations in sun angle and length of daylight. How closely related the two are, though, requires further investigation.
- Bonus Questions: In Figure
2-5, in each of the figures (a), (b), and (c), when does the
sun rise (before 6 am, at 6 am, or after 6 am)? When does it set (before 6
pm, at 6 pm, or after 6 pm)? How can you tell?
Over the course of a day, any location moves in a circle around the axis of rotation, following the latitude circle on which it lies. We conclude that each point on a latitude circle corresponds to a different time of day. (Since each point on a latitude circle also has a different longitude, it follows that, at each latitude, different longitudes correspond to different times. There is, in fact, a very close relation between longitude and time of day. Time zones are drawn roughly one hour wide, and one hour with respect to the sun corresponds to 15° of longitude.)
Solar noon is the time of day when the sun reaches its highest point in the sky. Of course, that occurs exactly half way between sunrise and sunset, at which times the sun is on the horizon. Solar midnight occurs 12 hours before (or after) solar noon, and must therefore occur on the opposite side of the latitude circle form the point where solar noon occurs. Since 6 am is half way between midnight and noon, and 6 pm is half way between noon and midnight, it follows that 6 am and 6 pm must occur half way on the latitude circle between midnight and noon (on opposite sides of the latitude circle from each other). It should be straightforward to see on each of the three topmost figures in Figure 2-5, that it must be solar noon at places along the edge of each diagram that is oriented most directly toward the sun, from the North Pole all the way to the South Pole (since every latitude experiences solar noon some time during the day). Midnight is on the opposite side of each latitude circle, on each diagram along the opposite edge from solar noon (again from the North Pole all the way to the South Pole). And 6 am and 6 pm must lie half way in between, which lies along each latitude circle where the axis of rotation appears to intersect it (since from the perspectives shown the axis of rotation splits each latitude circle in half, with solar noon on one end and solar midnight on the other end of each latitude circle.)
At any one moment, the sun illuminates exactly half of the earth. The Circle of Illumination is the boundary between the sunlit and dark halves of the earth.
The point where a latitude circle crosses the Circle of Illumination marks the transition from dark to light or vice-versa, and corresponds to sunrise or sunset, respectively, for any particular location passing through that point as the earth rotates. (As noted above, at those moments, the sun would appear to be on the horizon as viewed from that location.)
Using the right-hand rule to determine the direction in which the earth rotates. (With your right hand, stick our your thumb, curl your fingers as if you were issuing the universal hitchhiker's request for a ride, orient your thumb parallel to the axis of rotation so that it points in from the South Pole to the North Pole, and your fingers will curl in the same sense as the earth rotates.) It should now be possible to compare the time when a point on any latitude circle crosses the Circle of Illumination (sunrise or sunset, depending on the sense of the earth's rotation and the particular side of the earth you're looking at) with the point where it is 6 am or 6 pm and determine whether the sun rises before, after, or exactly at 6 am and sets after, before, or exactly at 6 pm. (The results should be consistent with your conclusions about the length of daylight at different latitudes in Question #2c.)
In each figure, in what direction
(that is, where on the horizon) does the sun appear at sunrise and sunset?
Does this vary with latitude in each figure? [Hint: east is always in a direction
along a latitude circle; north is along a longitude circle toward the North
(There are many more questions of this sort that these diagrams can help you
answer—the topic is very rich.)
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