NavList:

George, you wrote:

" As I see it, if it's made exactly equal to the
(fixed) tilt angle of the normal to the horizon mirror, with respect
to the telescope axis, then the light will be reflected from the added
mirror, not toward the horizon mirror, but directly toward the target.
In that way, it will fulfil the requirement of sending a flash of
sunlight to the second observer at the distant target, and show him
the position of the first observer, with his Gauss instrument."

 

That sounds about right to me. It strikes me as the sort of solution that would occur to a well-funded mathematician first. But of course, it's over-kill. When you see the final design of the heliotrope which Gauss came up with later, it's a fairly trivial device. It can be trivial because the Sun's angular diameter is half a degree. And if you reflect the Sun's light off a small mirror, you only need to aim it accurately to the nearest half a degree (actually, the angle of the light cone is half a degree, so the aiming accuracy is +/-0.25 degrees).

 

And you wrote:

"It seems to me, then, that it's intended to solve the same problem
that faces distressed mariners, who are advised to use a mirror to
flash reflected sunlight at a possible rescuer, in an attempt to
attract attention. But how does he decide to angle the mirror, to
ensure that the light reaches its target? Gauss's instrument solves
that same problem, scientifically."

 

Which brings up an interesting point. If you want to signal someone several miles away with a mirror, how do you get the angle right if you don't happen to have a Gaussian heliotrope in your survival kit? Here's a simple method: Drive a stake in the ground with a nice point at the top to serve as a sight. Step back ten feet from it and hold the mirror in front of you so that you can look just over the top of it at your target lined up directly behind the point at the top of the stake --your eye, the mirror, the point of the stake, and the distant target are now all lying on one line. Roll the mirror around until the beam of sunlight reflected from it just strikes the top of the stake. If you can maintain the pointing within about an inch at the distance of the stake, then you can be assured that your beam is reaching its target, since an inch at ten feet is about half a degree. Flick the mirror up and down so the projected beam shifts six inches along the stake, and you're all set to send code:

-... .   ... ..- .-. .   - ---   -.. .-. . -. -.-   -.-- --- ..- .-.   --- ...- .- .-.. - .. -. .

 

 

How bright is the spot of light from a heliotrope or a hand-held mirror if you've aimed it correctly? Surprisingly quite bright. If the angular diameter of the mirror as seen by the observer at the target is half a degree, then you can see the whole image of the Sun in it. Assuming it's a very good mirror, it will only be slightly less bright than the Sun itself (a fraction of a magnitude). That's our starting point. If we're twice as far away from the mirror, the light from the mirror would appear four times fainter, ten times farther away, a hundred times fainter, and so on. Suppose the mirror is two inches across and the targeted observer is five miles away. The angular size of the mirror at this distance is only 1.3 seconds of arc. That makes the point of light from the mirror about one-half of one-millionth of the brightness of the Sun (5*10^-7). But in magnitude terms since a factor of 100 is five magnitudes (and a factor of 10,000 is ten magnitudes, and a factor of 1,000,000 is fifteen magnitudes), that works out to just about 15.7 magnitudes fainter than the Sun itself. You can throw in an extra couple of tenths of a magnitude for the loss due to the fact that no mirror is perfectly reflective, so call it 16 magnitudes fainter than the Sun. The Sun's magnitude under clear skies when it's well up is just about -26.5. That means that a little two inch mirror reflecting the Sun seen from a distance of five miles will have an apparent magnitude of -10.5, about as bright as the gibbous Moon, but concentrated into a tiny dot! In other words, it would be plainly visible as a brilliant point of light as long as there is not too much intervening haze.