From: potamian@hpl.hp.com (Spyros Potamianos)
Subject: Re: Direction of the sun
Date: Thu, 24 Dec 1992 08:41:01 GMT
Organization: Hewlett-Packard


hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes

># Heard from a Japanese colleague -- Sorry again if this is in the FAQL
>
>Two boys, A and B, are watching the sunset.
>A points to the sun and says:
>"This is the direction where the sun is right now."
>B replies:
>"But don't you know that the rays take over 8 minutes to reach the Earth?
>The sun is actually already below the horizon."
>Which is correct?  Why? (Make it understandable to a ten year-old.)
>
># note: ignore subsidiary points like refraction by the atmosphere.
>
>BONUS: What would the answer be under the Ptolemaic system?
>
>-Yuzuru Hiraga

Nice puzzle! I have to admit that I got confused and thought of
the "obvious" (but painfully wrong) answer at first:

	Boy A is right, however if the earth was still and the sun
	was moving around it (as my ancestor Ptolemeos suggested)
	then boy B would be right.

However a friend of mine pointed out that both systems must give the same
answer, as this is a "kinematic" problem (is this the correct term???) and no
forces are involved, so both a geocentric and a "solarcentric" point of view
must provide the same answers. He gave me enough hints to figure out the
answer, so here it is (all the credit should go to my friend though)

Well this answer is not understandable by a 10 year old (unless we are
talkign about a *very* clever child! :-) but here it goes...

Before we go further, lets make some (not very correct) assumptions.
	a) ignore theory of relativity
	b) the earth does not rotate around the sun, it only rotates
	   around its axis (this is *very* wrong)

Let's start with the geocentric point of view. The earth is still,
positioned at point (0,0) and the sun moves in a circle of radius 'd'
clockwise (d = distnce between earth-sun, around 1.5*10^11 meters).

When the sun is at point (d,0) it emits a photon, that hits the earth
after 'd/c' seconds (c = speed of light). It's velocity is parallel to the X
axis, so at that time it seems (to an observer standing on earth) that the
sun lies on the X axis.
Meanwhile, the sun has moved by an angle
    a = w*t = w*d/c
where 'w' is the angular velocity of the sun (w = 2*Pi = 6.28 rads/day)
So the "real" position of the sun is 'w*d/c' rads "below" (clockwise)
the X axis.

Now, let's go to a "solarcentric" system.
The sun is still, positioned again at (d,0) and the earth is at (0,0)
rotating counterclockwise (around itself, *not* around the sun!)
with an angular velocity 'w'.
An observer standing on the earth (assume a zero earth radius, after all
it's very small compared with 'd') uses a different system of coordinates
X'Y'. 
Both systems have the same origin, but X'Y' rotates counterclockwise
with an angular velocity 'w'. The following formulas translate coordinates
from XY to X'Y':
    x' =  x*cos(wt) + y*sin(wt)
    y' = -x*sin(wt) + y*cos(wt)

A photon leaves the sun at time 0, and hits the earth at time 'd/c'.
It's coordinates as a function of time are given by:
    x = d - c*t
    y = 0

The coordinates in the geocentric system are:
    x' =  (d-ct)*cos(wt) + 0*sin(wt) =  (d-ct)*cos(wt)
    y' = -(d-ct)*sin(wt) + 0*cos(wt) = -(d-ct)*sin(wt)

Its velocity (as a function of time) is:
    V'x = dx'/dt = -(d-ct)*w*sin(wt) - c*cos(wt)
    V'y = dy'/dt = -(d-ct)*w*cos(wt) - c*sin(wt)

At time = d/c (when the photon hits the earth) its velocity will be
    v'x = -c*cos(w*d/c)
    v'y = -c*sin(w*d/c)

As you can see, its velocity has a non zero Y component. The photon will
appear to be coming from an angle:
    a = arctan( v'y/v'x ) = arctan (sin(w*d/c)/cos(w*d/c)) 
      = arctan(tan(w*d/c))
      = w*d/c

I.e. the sun will appear being at an angle 'w*d/c' "above" its current
position, a result consistent with the one we got from the "solarcentric"
system (BTW, this angle is around 2 degrees)

So, the final answer is that boy B is right in both cases.
Of course, this analysis ignores the movement of the earth around the sun,
(at a speed of 30Km/sec - in comparison, the speed at the surface of the
earth caused by the revolution around its axis is less than 0.5Km/sec!!!)
However, it is already 12:40, I have to wake up early tomorrow, and I
don't really feel up to the task of attacking this problem too, so....
zzzzzZZZZZZZZZZZZZZZZZZZZZZZZZ!!!!

Spyros Potamianos
potamian@hpl.hp.com
Newsgroups: rec.puzzles