The Universe in the Classroom

To Every Season There is a Reason

Springtime in the Solar System

In our case, we can use the tilt theory of the seasons to make a prediction: that any tilted planet will have seasons, and that the strength of these seasons will depend on the amount of the tilt. Astronomers have tested this prediction by looking at the other planets (see table below).

Inclination of equator to solar orbit
Mercury 2 degrees
Venus 2.7 degrees
Earth 23.5 degrees
Moon 1.5 degrees
Mars 25.2 degrees
Jupiter 3.1 degrees
Saturn 26.7 degrees
Uranus 82.1 degrees
Neptune 29.6 degrees
Pluto 57.5 degrees

Venus and Jupiter are tilted by only 3 degrees, so the seasonal variation is quite mild. Every day is almost an equinox. Some might think it boring; personally, I wouldn't mind an eternal spring.

Uranus has extreme seasons, because its axis is tilted 82 degrees. It lies flat on its side as it orbits the Sun. At summer solstice, the north pole points straight at the Sun; the north polar regions have tropical weather, but the Southern Hemisphere is in total darkness. At winter solstice, the north pole points straight away from the Sun, plunging the Northern Hemisphere into total darkness.

Of all the planets, Mars has the seasons that most closely resemble Earth. Mars is tilted by 25 degrees. Its seasons were first seen by the English astronomer Frederick William Herschel in the late 18th century. He noticed changes in the polar caps: They enlarged in winter and shrank in summer. Other areas on Mars seem to get brighter and darker depending on the season, a phenomenon that many scientists used to attribute to vegetation growing on a seasonal basis. On closer observation, Marsologists realized that the seasonal changes were not plants, but dust storms.

Currently it's spring in the martian Northern Hemisphere. Northern summer begins in April. The next vernal equinox occurs on Aug. 26, 1996. On Mars, summers are hotter in the Southern Hemisphere than in the Northern Hemisphere, because the planet's orbit is so noncircular.

There's one missing link in the tilt theory of the seasons. The solstice occurs in December, but December isn't the coldest month in the Northern Hemisphere. A delay occurs for the same reason that water doesn't start boiling just as soon as you turn on the stove. The ground, air, and oceans need time to warm up and cool down. The Earth's atmosphere is so thick that it takes about a month to adjust to the new season. For the same reason, the daily maximum temperature occurs not at noon, when the Sun is highest, but at 2 p.m. On Mars, the atmosphere is so thin that it adjusts almost instantaneously to seasonal changes.

It's pretty incredible that the tilt of the Earth can cause so many effects. Scientists have found that the simplest explanation for a phenomena, as long as it explains all the facts, is probably the best one. It's equally incredible that by looking at the sky, we can understand processes and worlds so much mightier than we.

Geometry of the Seasons

You don't need math to understand why the seasons occur, but it helps. With a little trigonometry, you can calculate the angle of the Sun and the intensity of solar heating.

In Figure A, I've drawn a cross section of the Earth. Because the Earth is a sphere, its cross section is a circle. The equator is the line EOW, perpendicular to the rotation axis NOS. To find the latitude of any point P on the surface of the Earth, draw a line from P to the center of the circle, O. The acute angle between the line OP and the equator EOW is the latitude, L. To distinguish between north and south, we use positive angles for north latitudes and negative angles for south latitudes. At the equator, L = 0 degrees; at the north pole, L = 90 degrees; at the south pole, L = -90 degrees. Each degree of latitude corresponds to a fixed distance on the circumference, 110 kilometers (69 miles) for the Earth. That is, if you drive due north 110 kilometers, you will have gone 1 degree in latitude, or 1/360 around the Earth.
figure a
Figure A

The Sun is directly overhead at a latitude S. Overhead means that the Sun's rays are perpendicular to the tangent to the circle. The latitude S changes depending on the seasons. At the vernal and autumnal equinox, S = 0 degrees; at the summer solstice, S = 23.5 degrees; at the winter solstice, S = 23.5 degrees. In fact, S is a simple trigonometric function of time. (Can you figure out what this function is?)

Once you know S, it's easy to calculate the angle of the Sun, theta, at any latitude L. Using the principle of similar triangles, the solar angle theta = L - S. Therefore, the solar angle increases in direct proportion to latitude: If you go north 10 degrees, the solar angle increases by 10 degrees. That's why the Sun is lower in the sky the farther north you go. You can also work backwards. For example, on an equinox or solstice, you can figure out your latitude. How would you go about doing this?

When the solar angle is bigger, sunlight spreads over a larger area of ground. In Figure B, I've shown a beam of sunlight. The beam has the same width, b, at all times. The beam illuminates a patch of ground of width g. When the Sun is directly overhead, g = b. When g > b, the beam shines on a wider area. This dilutes the energy in the beam, causing each spot on the ground to be cooler. Therefore, we can estimate the intensity of the beam as i = b/g. When i = 1, the intensity is greatest; when i = 0, the intensity is weakest.
figure b
Figure B

As figure B shows, b is the adjacent side of the triangle and g is the hypotenuse. Therefore, i = cos theta. To check whether this equation is correct, test it in simple cases that we already know the answer for. When the Sun is directly overhead, theta = 0 degrees, so that i = 1. When the Sun is on the horizon, theta = 90 degrees, so that i = 0. Our little equation seems to work.

Now we can combine the two equations into one: i = cos(L - S). This is the key formula for the seasons. It tells us how strong the Sun is at different times of year and at different latitudes. Suppose it is the summer solstice. At what latitude is the solar intensity strongest? At what latitude is the intensity weakest? Compare the seasons at two different latitudes: Your own city and some other city in the same hemisphere. What is the solar intensity (i) at the summer solstice? What is the solar intensity at the winter solstice? By what percentage does the intensity change? Is the percentage change greatest at the higher latitude or the lower latitude?

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