ERTH 535: Planetary Climate Change (Spring 2018) Problem #2 Dr. Dave Dempsey Dept. of Earth & Climate Sci., SFSU

(10 pts total; due in class Monday, Feb. 26)

Introduction. In Lab Activity #2: The Seasons, using satellite observations, we calculated the global- and monthly-average insolation on horizontal surfaces at the top of the atmosphere for December and for June, 1987. These averages differed from each other by a modest but noticeable amount. (In contrast, year to year variations for any particular month are much, much smaller.)

We proposed several possible explanations, some more plausible than others. One very plausible explanation (but not necessarily the only one, or even the correct one) is that the earth is closer to the sun during one of the two months than the other. This explanation is consistent with the observations.

However, although the proposed explanation is qualitatively consistent with the observations (which means that the observations don't rule it out as a possible explanation), by itself this is not an adequate test. For example, if the effect of distance from the sun on insolation were much smaller than the observed difference, then qualitative consistency is almost irrelevant because the proposed explanation is not nearly enough to explain the observations quantitatively. One or more other mechanisms with the same qualitative effect but larger quantitative effect(s) would have to be operating.

Similarly, if the effect of distance from the sun on insolation were much larger than the observed difference, then the proposed explanation is quantitatively more than enough, but one or more other mechanisms, nearly as large but with the opposite effect, would also have to be happening to compensate partly, so the proposed explanation by itself isn't adequate, either, though it is at least quantitatively relevant (that is, it has to be taken into account).

In contrast, if the effect of distance from the sun on insolation is quantitatively similar to the observed difference in insolation, then this provides stronger support that the explanation is by itself adequate to account for the observations, and we can have more confidence in it.

(Even this isn't enough to "prove" that the explanation is correct, though. There could be errors in the observations, or the theory underlying the explanation could be faulty, or there could be multiple other mechanisms that are quantitatively significant but tend to compensate for each other quantitatively, etc. In science, an explanation can never really be considered "proven" even though it is qualitatively plausible and quantitatively adequate to account for the observations.)

The Assignment. Test the explanation that variations in distance from the sun can quantitatively account for the observed difference in global, monthly average insolation between December and June. Do this by estimating the global, monthly average insolation at the top of the atmosphere in, or as close as you can to, December and June, and compare your estimates to the observed global, monthly average values that you calculated in Lab Activity #2. This should be based on theoretical considerations. Use only information presented in class, in class handouts, or in assigned reading.

When writing up your solution, follow the guidelines summarized in the handout, "Good Problem Solving Strategy". (See summary of class meetings for Monday & Friday, Feb. 19 & 23 for an example of how to organize and format the solution for a problem of this general type.)

[Hint #1: In addition to providing an example of the format for organizing and presenting your solution, the summary of class meetings for Monday & Friday, Feb. 19 & 23 also includes some background information relevant to solving this problem.

Hint #2: The intensity of solar radiation on a surface directly facing the sun is not the same as the global average insolation because at any moment in time, only one spot on the earth directly faces the sun (and fully half of the planet is dark!). You can calculate the global average insolation (a flux) by dividing the total rate at which solar radiation strikes the earth by the surface area of the earth. See the derivation of the effective radiating temperature of a planet [PDF file], posted on the same date, for relevant information about this.

Another, possibly helpful reference: Some Facts about the Earth and Its Orbit around the Sun.]