Tilt of earth's axis and the four seasons

Task

The geometry of the earth-sun interaction plays a very prominent role in many aspects of our lives that we take for granted, like the variable length of days throughout the year and the four seasons. This problem will explore the role geometry plays in these experiences. The picture below shows the sun and the earth in four different positions of its annual orbit:

The small blue arrows show the direction of the earth's orbit (counterclockwise) around the sun. The red line is perpendicular to the plane of the earth's orbit and the (blue) axis of rotation is tilted approximately $23.5^\circ$. The tilt is directly toward the sun when the earth is in the position marked (A) and directly away from the sun when the earth is in the position marked (C). Here is a close up of how the sun's rays hit the earth in position (C):

In positions (B) and (D), the tilt of the earth's axis is neither toward the sun nor away from the sun.

In addition to orbiting around the sun once each year, the earth spins on its axis, making one complete revolution each day: it makes a little more than $365$ of these revolution in each year.

1. At which point in the earth's orbit are the days in the United States shortest? At which point in the earth's orbit are the days in the United States the longest? Explain.
2. Indicate in the picture which sections of the orbit correspond to the four seasons (winter, spring, summer, fall) in the Unitied States. Justify your choices.
3. The tropics are the region of the earth where the sun's rays meet the earth perpendicularly at some point in the year. Explain why the tropics are the area between the lines of latitude of about $23.5^\circ$ in the northern and southern hemispheres.

IM Commentary

The four seasons are a familiar part of the lives of people who have grown up in North America and the differences in the seasons becomes accentuated the further one goes north. The source of the four seasons, as well as their characteristics at different latitudes, is the fact that the axis on which the earth rotates is tilted relative to the plane of its orbit. Because of this tilt, in the middle of the (North American) summer, the north pole is pointed toward the sun and so, in fact, the sun does not set at night at the north pole during this period. Similarly, in the middle of the (North American) winter, the north pole is pointed away from the sun and the sun never shines at the north pole in these months.

This task gives students a chance to relate their weather experiences with a simple geometric model which explains why the seasons occur. It would be good for the students to have (or make) a physical model to aid in this visualization. The orbit of the earth around the sun is not circular as depicted in the picture above but it is closely approximated by a circle. If students use a flashlight for the sun, they can experiment to estimate the length of days, at the latitude where they live, at different points in the earth's orbit around the sun. Perhaps unsurprisingly given its strongly "real world" nature, the task is very high in cognitive complexity, and would represent a rather significant time investment. The benefits of the time are an increased understanding of mathematical modeling, but also an illustration of nearly every Standard for Mathematical Practice (in particular, those dealing with modeling and problem-solving).

The solution to the problem mentions the tropics (the parts of the earth which the sun's rays meet perpendicularly at some time during the year). Also important are the arctic and antarctic circles, the places on the earth where, at some time in the year, there is at least one day when the sun does not rise. More information is available at the following website:

Arctic Circle

The task can be taken much further. Students might measure, at noon perhaps once a week, the length of the shadow cast by some fixed object like a flagpole. They could plot this information and eventually think about how to determine what time of the year it is by studying the shadow.

Teachers may wish to refer students to the following video after they have worked on this activity:

"> Cause of the four seasons

This misconception that the seasons are created by the relative distance between the earth and sun during the earth's orbit is a very widespread.

Solution

1. In the picture, when the earth is in position (A) the northern hemisphere is tilted toward the sun and consequently the days will be longest and the rays of the sun will shine on the United States most directly. So this is the summer solstice (Summer Solstice). This position also corresponds to the longest day of the year in the northern hemisphere because the tilt of the earth toward the sun means that as little of the earth as possible is able to obstruct the sun's rays.

   So when the earth is in the opposite position marked (C) this should be the shortest day in the northern hemisphere or the winter solstice (Winter Solstice). This is the point in the earth's orbit when tilt of the earth's axis leaves the northern hemisphere pointed directly away from the sun. This means both that the sun's rays reach the northern hemisphere more obliquely and that the days are the shortest because the largest portion of the earth is able to obstruct the sunshine.

2. First, using the information from part (a) we know which points in the orbit correspond to the summer and winter solstices in the northern hemisphere. The summer solstice is the beginning of summer while the winter solstice marks the beginning of winter. Since the full orbit represents one calendar year this means that summer in the northern hemisphere corresponds to the arc from position (A) to position (B), fall is the arc from (B) to (C), winter is the arc from (C) to (D) and spring is the arc from (C) to (D).

   Thinking about this geometrically, there will be a large difference in North America between the weather at position (A) and at position (C) since these represent the points when the northern hemisphere receives the most direct and least direct sunlight. On the other hand, thinking of the positions (B) and (D) in the picture, here the tilt of the earth plays no role and these two positions are identical as far as the directness and amount of sunlight received. If we imagine for a moment that the earth's axis pointed directly out of the plane of its orbit about the sun, then at any point (other than the north and south poles) there will be equal daylight and night. Indeed, in this situation, the sun would illuminate, at any given moment, all points on the surface of the earth between $x^\circ$ longitude and $(180 + x)^\circ$ degrees longitude for some $x$ between $0$ and $180$ degrees. Other than the north and south poles each point on the earth has a fixed longitude and so would be illuminated during exactly half of the earth's rotation so for $12$ hours.

   So the points in the diagram marked (B) and (D) are the points where we have very close to equal amounts of daylight and darkness. These are the autumnal and vernal equinoxes (Equinox). Once all of this information has been gathered, we can again reconfirm that the position of (A) in the diagram is the beginning of summer (in the northern hemisphere), (B) the beginning of fall, (C) the beginning of winter, and (D) the beginning of spring.

3. If we imagine the earth spinning for one full revolution in position (A) in the diagram, the places on the surface of the earth hit directly (that is, at a right angle) by the sun's rays map out a circumference of the earth. If the axis of the earth were not tilted, that is if it were perpendicular to the plane of the earth's orbit, then this circumference would be the earth's equator. Because of the earth's tilt, instead of sweeping out a regular equator, the points met perpendicularly by the sun map out a ''tilted'' equator which reaches $23.5^\circ$ north at its northerly most point and $23.5^\circ$ south at its southerly most point. Similarly, if the earth makes a full revolution at position (C) in the diagram, the places on the surface of the earth hit directly by the sun's rays map out a ''tilted'' equator which is the reflection of the previous ''tilted'' equator about the plane of orbit of the earth around the sun. Since these ''tilted'' equators move smoothly as the position of the earth changes, for each point on the earth between $23.5^\circ$ south and $23.5^\circ$ north latitude, there will be exactly one time in each half year (corresponding to the motion from (A) to (C) and then from (C) to (A)) when the sun is directly overhead.

   The previous paragraph represents an idealized model. In practice, the spots on the earth which are actually hit perpendicularly by the sun's rays would represent a fraction more than $365$ ''tilted'' equators (since there are 365 and a fraction days, or full rotations of the earth about its axis, during a year represented by one completion of the earth's trajectory around the sun). This is because we assumed that the earth makes a full orbit at each individual point whereas in fact the earth is moving in its orbit as the days pass. Nonetheless, for any point on the earth in the tropics the sun will be very close to being directly overhead twice in the year.