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sun's distance as lab exercise?



I really enjoyed the recent discussion of how to measure the distance to
the sun, especially because it is germane to what was happening in my
class room last week. There was a problem I couldn't figure out at that
time - here's the question (with long-winded intro...)

I let myself get talked into teaching a one-semester Astronomy class this
year (high school) and since I am not an astronomer I'm kind of feeling my
way along. (BTW, thanks again for much excellent advice last spring from
PHYS-L ers while I was in the planning process)

I started the semester with a quasi historical approach -- sort of a "what
did the ancients know and how did they know it?" approach.

Recently I asked my students to determine the diameters of the moon and
sun and their average distances from the earth "as experimentally as
possible." (I'd describe the assignment in more detail but this is
already a lengthy posting; I'll elaborate if anyone is really interested.)

In the process, a student had what I thought was a terrific idea which I
couldn't figure out how to implement -- so I'll ask the list if anyone
knows how to do this:

The hardest part of the process is finding the ratio of our distance to
the two bodies: d_moon/d_sun. The ratio can be approximated by carefully
observing the angle separation between the sun and moon at precisely 1st
or 3rd quarter. (When exactly half of the moon's face is lit, the
earth-moon-sun angle is 90). I believe this process is called something
like "dichotomized moon." The S-E-M angle at this moment is between 89
and 90 degrees and the cosine of the angle is the ratio of the distances.
If you can measure the angle, you can find the ratio.

I have a colleague at another school who built a little brass instrument
for making this measurement, but with homemade equipment it's pretty touch
and go making a measurement to better than a tenth of a degree.

I asked my students to suggest methods for finding the angle and one came
up with the following idea: at the moment when the moon is at first
quarter find out what two places on earth have either the sun or the moon
at the zenith. The angle separating those two points on the globe is the
same as the M-E-S angle. Other kids liked this a lot because it made
really good conceptual sense to them.

The ancients certainly could not have done this, but my students certainly
should be able to do it using the web:

1. look up the time and date for the quarter moon
2. Find (using on-line tables of ephemerides) what two locations have the
sun and moon respectively at zenith at that moment
3. Find angle difference between those two points
4. cosine of that angle is the ratio dist_moon/dist_sun

The problem is that all the sites that calculate tables of ephemeris take
a lat-long as inputs and output azimuth and altitude over time. What I
want is to input altitude = 90 at a given time and find the lat-long.

If anyone knows a quick way to do this I'd love to hear about it.

David Strasburger
Noble and Greenough School
Dedham MA