"I think the moments
leading up to it were surreal. The lighting changes, and it was unlike any
lighting I've ever experienced in my life, like something you can only see on
set or manufactured," she said. "And then when it actually happened,
it was... I had like a genuine giggle that was only brought on prior by, like,
a roller coaster." - Burlington VT eclipse viewer
It was perhaps the country’s most viewed celestial event ever, spreading the darkness of the moon’s shadow across the homes of some 32 million people, including in major cities from Dallas to Cleveland and smaller communities such as Russellville, Ark., and Littleton, Maine. Millions more flocked into the 115-mile-wide path of totality, while the moon blocked at least half of the sun’s face across most of the rest of the United States.
Millions, including excited school children, put down cell phones yesterday to pay full attention to one of the most sensational space spectaculars: the total eclipse of the Sun. In televised specials from ABC, CBS, in places as far removed as Dallas, Indianapolis and Cleveland the temporary night brought a range of emotions all generated by awe. In Indy kids told ABC's Gio Benitez how they now wanted to study astronomy and maybe even learn how such awesome celestial events occur.
But do they really? Or is this merely part of the media hype? Let's actually introduce some basic spherical astronomy to see how these solar eclipses are predicted. The basics of eclipse prediction begin with the spherical reference frame (Fig. 1) used in most spherical astronomy texts. It then follows on by calculating what we call the "Besselian elements" (which are also used to predict occultations).
Thus a line EC follows the z-axis from the Earth's center directed outwards as shown. The plane DBA is called the "fundamental plane" to which EC is normal (i.e. at 90o ). Similarly, EA and EB mark the x and y axes, respectively.
We proceed by taking (a, d) to be the apparent right ascension and declination of the Sun and (a’, d’) to be the corresponding coordinates for the Moon. We then let (a,d) be the right ascension and declination of point C on the reference sphere. We then let (x,y,z) be the rectangular coordinates of the Sun. Then if X is the Sun's position on the sphere:
x = r cos AX, y = r cos BX, z = r cos CX
with r the Sun's geocentric distance. Then if A is the pole for the plane CPB (and on the equator) we have: PA = FA = 90o . Hence, the right ascension of A is (90o + a) and therefore the angle <XPA = ( 90o + a - a ) .
Since PX = (90o - d) we have for the first Besselian element:
x = r cos d sin (a - a)
We then refer to the spherical triangle PBX, for which BP = d.
Also since angle <APB = 90o then <XPB = 180o + a - a.
Hence, the next Besselian element:
y = r [sin d cos d - cos d sin d cos (a - a)]
Going to the spherical triangle PCX we have: PC = 90o - d, PX = 90o - d
And: <XPC = (a - a)
Hence: z = r [sin d sin d - cos d cos d cos (a - a)]
Analogous to the preceding we can derive the corresponding equations for the Moon with (x',y',z') the corresponding rectangular coordinates.
x' = r1 cos d’ sin (a’ - a)
y' = r1 [sin d’ cos d - cos d’ sin d cos (a’ - a)]
z' = r1 [sin d’ sin d - cos d’ cos d cos (a’ - a)]
Since the x-axis is parallel to the line joining the centers of the Sun and the Moon we must have:
x = x' and y = y'
Thus the coordinates (x,y) or (x', y') are the coordinates of the center of the Moon's shadow on the fundamental plane, so that:
r cos d sin (a - a) = r1 cos d’ sin (a’ - a)
r [sin d cos d - cos d sin d cos (a - a)] = r1 [sin d’ cos d - cos d’ sin d cos (a’ - a)]
r [sin d sin d - cos d cos d cos (a - a)] =r1 [sin d’ sin d - cos d’ cos d cos (a’ - a)]
So at any instant the respective values, r, r1, a, d, a’, d’
may be presumed known.
As a further - and final note - the formulae can be put in a simpler form since at or near the time of eclipse a, d are little different from a’ and d’. We can write:
r1/ r = b
Which can also be expressed: b = sin P/ sin P1
Since by definition: 1/r = sin P and 1/ r1 = sin P1
Thus b can be computed at any time, and is a small quantity on the order of:
1/400
Writing: [(a’ - a) + (a - a)] for [(a’ - a) ]
We obtain:
sin (a - a) [1 - b sec d cos d’ cos (a’ - a)] =
[b sec d cos d’ cos (a’ - a)]sin (a’ - a)
Or, with sufficient accuracy:
a = a - b sec d cos d’ (a’ - a) / (1 - b )
Similarly, for Besselian element d:
d = d - b (d’ - d )/ 1 - b
And the quantities a and d can be calculated at intervals of one hour, even with this basic (i.e. trig) approach.
To be sure, awe and wonder in response to a total solar eclipse are a natural accompaniment of the event- especially when shared with so many others - who may be of a different mind politically, culturally. But to fully appreciate what one beholds one needs to grasp it at least at a basic level. That understanding, often via mathematics, does not destroy wonder and awe, but enhances it. Also, showing our science is real and can actually predict these events, many thousands of years in advance. Does it take away the "magic"? Nope, it shows that our human minds can develop tools to probe and uncover many aspects of the cosmos.
See Also:
Post Eclipse - Can The Awe and Wonder Be Translated Into Learning, Scientific Endeavor?
And:
Selected Questions -Answers From All Experts Astronomy Forum (Astronomical Coordinates)
And:
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