In the limit of small obliquity, this equation reduces to 5 = y cos K(t). For a circular orbit, K(t) = Qt, where ^ is the orbital angular velocity (2n divided by the orbital period). In this special case, the subsolar latitude varies cosinusoidally over the year, with amplitude given by the obliquity. This is actually not a bad approximation even for the roughly 23o current obliquity of Earth and Mars, agreeing with the true value to two decimal places. At the opposite extreme, when 7 = 90o, the subsolar latitude is given by S = n/2 — k, which is not at all sinusoidal.
Exercise 7.3.3 Compute the length of day as a function of the time of year for the latitude at which you are currently located. Compare with data for the current day, either observed yourself or presented in the newspaper weather report. Compute the length of a shadow that would be cast by a tall, thin skyscraper of height 100m, as a function of the time of day and time of year at your latitude.
Contour plots of the flux factor for various obliquities are shown in Figure 7.5. These plots assume the orbit to be perfectly circular, so that there is no variation in distance from the Sun in the course of the year. Over the course of the year, the hot spot moves from south of the Equator to North of the Equator, and back again, passing over the Equator at the equinoxes. The amplitude of the excursion increases with obliquity, and goes all the way from pole to pole for sufficiently large obliquity. Earth, Mars, Saturn,Titan, and Neptune with present-day obliquities of 23.5o, 24°,26.7°,26.7°,and 29.6o respectively, are qualitatively like the 20o case. The pattern of variation of incident solar radiation which forces the seasonal cycle is similar in all these cases. However, the nature of the seasonal cycle will differ amongst these planets because the differing nature of the atmospheres and planetary surfaces will lead to different thermal response times. In the case of gas giant planets, another variable is the proportion of energy received from solar energy vs. the that received by transport from the interior of the planet. Insofar as the latter becomes dominant, the role of solar heating, and hence the prominence of the seasonal cycle, becomes less. Jupiter has a low obliquity (3.1o), which, compounded by a high proportion of internal heating (** per cent) should lead to a minimal seasonal cycle. At the opposite extreme is Uranus, which has an obliquity of nearly 90o, and a small proportion (about ** percent) of internal heating. Venus is so slowly rotating that its obliquity is of little interest. Obliquity is not constant in time; it varies gradually over many thousands of years. We will see in Section 7.6.1 that relatively slight variations in the Earth's obliquity are believed to contribute to the coming and going of the ice ages. The obliquity of Mars varies more dramatically, and perhaps with greater consequence; at various times in the past it could have reached values as high as 50o and as low as 15o.
If the thermal response time of the planet is a year or more, then a considerable part of the seasonal cycle is averaged out and the annual mean insolation becomes an informative statistic. It will be seen in the next section that this is the case for watery planets like the Earth. The annual mean flux factor is shown in Figure 7.6. When obliquity is small, the poles receive hardly any radiation. As obliquity is increased, the polar regions receive more insolation, at the expense of the equatorial regions. For Earthlike obliquity, the maximum insolation occurs at the Equator, which is why this region of Earth's surface tends to be warmest. When the obliquity exceeds 53.9o, the annual mean polar insolation becomes greater than the annual mean equatorial insolation. For such a planet, the poles will be warmer than the tropics, provided that the thermal response time is long enough to average out most of the seasonal cycle. Consider a planet with 20o obliquity, zero albedo, and a very long thermal response time. If the planet were put in Earth's orbit about the Sun, the Solar constant would be 1370W/m2, yielding equatorial insolation of 422W/m2 and polar insolation of 149W/m2, based on the flux factors given in Figure 7.6. In the absence of any greenhouse effect or lateral energy transport by atmospheres or oceans, the equatorial temperature would be 294K and the polar temperature would be 226K. If one takes into account the clear-sky greenhouse effect of an Earthlike atmosphere with 300ppm CO2 and 50% relative
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