Re: [Phys-l] Teaching elementary astronomy topics

[Leigh Palmer <palmer@sfu.ca>]

I woke up this morning to a different scene outside. The first thing I see in the morning through my bedroom window is our 37.5 meter tall Douglas fir, which usually holds its branches rigidly horizontal. This morning they were hanging down at 45 degrees. Last night we had our first snowfall of the season, and I realized that I should finish a note I'd started more than a week ago regarding the cause of the seasonal variation in mean surface temperature.

In response to my earlier comment on "Private Universe":

> I suggest that teaching a student about "direct" and 
> "indirect" insolation perhaps confuses the student 
> unnecessarily. Talking about the seasonal variation in 
> duration of insolation is probably easier to sell, and it is 
> more important at most latitudes than angle of incidence of 
> insolation.

On 13 Nov 2010, at 03:18, Jeffrey Schnick wrote:

> Since the seasonal variation has to do with how long the sun is up each
> day and how directly it shines on your part of the world, I think it is
> better to teach both.  I haven't taught astronomy in a decade but I used
> to hold a cardboard cutout of the state in which my school is located in
> front of a screen illuminated by an overhead projector.  I'd tilt the
> cutout at angles corresponding to the orientation of the surface of the
> state relative to a line from the sun to the state, and the size of the
> shadow on the screen was indicative of the solar power being received by
> the state at local noon.

As it turns out I was incorrect in my original statements regarding the relative importance of angle of incidence of insolation and duration of insolation, something I only realized after I did a calculation which I will share with you below. Jeffrey's emphasis on angle of incidence is correct at intermediate northern latitudes.

Of course it is ideal to teach both influences, but I think the idea in the Private Universe film is to get something useful across to elementary and high school students about this topic. I haven't taught astronomy in a decade either, but I thought that there was a greater consciousness among the students of the lengthening and shortening of daylight hours than there was in the change in altitude of the Sun at noon for those students. I would use the diminishing time 'til sunset as the most obvious effect for my elementary astronomy university students since I taught in the fall semester. Once that was noticed I used either the overhead projector or a gooseneck lamp to illuminate a world globe. I didn't do it when I taught, but a string could be wrapped around the globe at a constant latitude and marked with ink at the terminators (the shadow edges) to illustrate the duration difference graphically. This would, perhaps, be appropriate for younger students to do themselves.

I have been away from this for some time, but let me try to calculate the asymmetry between summer and winter solstices as a function of latitude for both effects, duration of insolation and angle of incidence of insolation. This is a difficult calculation to do because the Sun has a finite angular diameter, refraction in the atmosphere affects the geometry, and there is even a +/- 3.4% variation due to Earth's orbital eccentricity. I will chicken out and substitute another, easier calculation which should give a similar result. I will calculate the ratio of the cosines of local noon zenith angles of the Sun at the solstices and compare it to the ratio of the durations of insolation at those times. (The zenith angle of an astronomical object is the angle of incidence at Earth's surface of light coming from that object. The zenith angle is the complement of the altitude of the object.)

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)

Ratio of solar irradiances at local noon at the solstices is R1(L)

                 cos Zs     cos (L - Q)
        R1(L) = -------- = -------------
                 cos Zw     cos (L + Q)

Ratio of durations of diurnal insolation at the solstices is R2(L)

                 arccos [tan(L)tan(-Q)]
        R2(L) = ------------------------
                 arccos [tan(L)tan(+Q)]

At my latitude (49°15' N) these ratios have values:

           R1 = 3.03   and   R2 = 2.01

So my own "Private Universe" has been altered significantly by Jeffrey, for which I thank him. Change in the angle of incidence of solar radiation is clearly more significant than  the change in duration of daylight in producing a difference in mean diurnal surface temperatures at different seasons. The joint effect is greater than I had expected; sunlight varies by a factor of six between the solstices! It is clear also that mean diurnal temperature does not vary nearly so much as surface insolation. The reason, I think, must be thermal inertia. The heat capacity of the surface (including crust, atmosphere and water) must buffer the temperature variation on both diurnal and annual time scales.

I would appreciate any help the group can give me in finding a similar analysis in the literature. If someone can check my answers or even extend them to include the other nonidealities I have neglected that would be appreciated. (No spherical trigonometry was used in the second calculation.) If someone wants to take on the challenge of solving the transcendental equation to find the latitude for which the two ratios are equal that might be interesting. I know it's farther north than Burnaby.

I could do the things I've listed above, but I am retired, and I don't work fast. I had a lovely walk around the lake this afternoon, in the snow, and I'm going out to a party tonight. I'm too mellow to worry further about this, but my curiosity is unabated.

Leigh