Re: [Phys-l] Teaching elementary astronomy

[Leigh Palmer <palmer@sfu.ca>]

I was curious to see what the equatorial seasons might be in this analysis, and taking orbital eccentricity into account, June solstice diurnal insolation is about 6.8 % less than that at the December solstice. However diurnal insolation is at minima rather than maxima near those times. The insolation maxima are reached near the equinoxes, so Ecuadorians get two "summers". The effect is only a bit more than 10 %, however. For real seasons to occur the differences must be much greater, like a factor of six as I calculated for Burnaby. Again, thermal inertia (or the heat capacity buffer of crust, water, and atmosphere) accounts for the observation that there are no dramatic seasons at the equator. I do remember a visit I made to Costa Rica for an eclipse in 1991. It was there that I learned of El Niño for the first time. El Niño is an important determiner of Costa Rica's "seasons" with regard to weather, indicating that between the tropics the thermal reservoir heat capacity (the ocean in this case) is a relatively more important influence than variation in insolation.

On 21 Nov 2010, at 00:43, Brian Whatcott asks:

> On 11/20/2010 5:48 PM, Leigh Palmer wrote:
> > /snip/
> > Ignoring refraction, eccentricity and assuming a point Sun and spherical Earth, and using northern hemisphere nomenclature:
> >
> > Let L = latitude
> >
> > Let Q = obliquity of the ecliptic = 23°27'.
> >
> > The declination of the Sun at the summer solstice is Q
> >
> > The declination of the Sun at the winter solstice is -Q
> >
> > The zenith angle of the Sun at the summer solstice is Zs = (L - Q)
> >
> > The zenith angle of the Sun at the winter solstice is Zw = (L + Q)
> /snip/
> > Leigh
> > _______________________________________________
> > Forum for Physics Educators
> > Phys-l@carnot.physics.buffalo.edu
> > https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
> Did you intend to express the Solar Altitude
> at mid Summer as a zenith angle of 90 deg - (L-Q)  ??
>
> Brian W

The zenith angle of the Sun at noon on midsummer day is (L - Q) in the northern hemisphere. In altitude-azimuth coordinates altitude is the complement of the zenith angle. 

In astronomical photometry one always works as near to the zenith as is practical to minimize atmospheric absorption. Since I used to do that kind of astronomy, and azimuth would play no role in the problem, and since the conventional use of the term "angle of incidence" refers to the surface normal, I chose to refer to zenith angle rather than altitude. In the more complete treatment of the problem (which I spared you because typing is not one of my skills) I would also have referred to Sun's hour angle* rather than its azimuth. If I had figured in refraction I would have had to change coordinates. 

Most problems suggest a best set of coordinates. This one didn't, but one could use alt-az as you suggest. The insolation is then proportional to the sine of the altitude rather than the cosine of the zenith angle. For teaching purposes I prefer the latter.

Leigh

After I wrote this I looked at Larry Woolf's excellent seasons poster <http://www.sci-ed-ga.org/modules/materialscience/color/images/SeasonsPoster.jpg> and I note that he mentions this specific issue of altitude and zenith angle.

*I know, hour angle is in yet another coordinate system, equatorial coordinates. I didn't say this problem wasn't confusing. It does not astonish me that few members of the general public understand the mechanism behind the seasons. As you have seen, I was under a misapprehension myself until I did the calculation. I am disheartened, however, that so few among them are sufficiently aware that they do not see that solving the problem of modeling climate is beyond the capability of those presently engaged in trying to do so.