Re: Apparent Distance

In article <1114779645.671559.5180@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Ante Perkovic <anteperkovic@xxxxxx> wrote:
>> A couple of observations:
>>
>> From the geometrical picture, clearly:
>>
>> 0 < a < M*a <= Pi/2. (4)
>>
>> (The object cannot be at an "apparent distance" smaller than 0 to the
>> observer under any circumstances and it is, say of finite size, so a
>= 0
>> only if OB = +oo).
>>
>> The above though constraints (3) badly. For example:
>>
>> For the Moon, a = 1/2 degree, and OB ~ 380,000 km.
>>
>> After doing the appropriate conversions and plugging in the numbers,
>(4)
>> gives a maximum magnification of 180x. However, it is well known that
>one
>> can use magnifications in excess of x180 for the Moon for example.
>>
>> Why the apparent discrepancy?
>
>Because the telescopes do not really "decreased distance". Instead, the
>magnify.
>
>What is the difference?
>
>Well, the sky seen throught the telescope has circumference not just
>360 degrees but M*360 degrees, so objects _can_ have M*a higher then
>Pi/2.
>
>Basicaly, equation (4) is wrong.

Yes, equation (4) is naive, but it has the correct idea. For a scope
centered on an object, what is multiplied via magnification is not the
angle, but the tangent of half the angle. That is, if we have a 2000mm
focal length telescope viewing a 30' diameter moon through a 25mm
eyepiece, we get the tangent of half the apparent angle to be

2000/25*tan(1/4 degree) = .349 = tan(19.24 degrees)

thus, the apparent size of the moon is 38.48 degrees, not 40
degrees as 2000/25*1/2 would compute. Taking this to an extreme,
try this with a 5mm eyepiece. The standard formula would give
an apparent diameter of 200 degrees; obviously impossible. The
formula with tangents gives

2000/5*tan(1/4 degree) = 1.745 = tan(60.18 degrees)

thus, the apparent diameter would be 120.36 degrees, not 200.

The reason that the naive formula, which multiplies angles, is still in
use is that it is a good approximation, better when the angles are
small.

Now, since the Moon is spherical, when you move closer, you see less of
it, but the part you can see is closer than the center. The result is
that what is magnified is the sine of half the angle. Thus, if we were
to move 80x closer to the center of the Moon (80x is the magnification
of the 25mm eyepiece) the sine of half the angle is multiplied by 80:

2000/25*sin(1/4 degree) = .349 = sin(20.43 degrees)

Thus, the moon would appear 40.86 degrees, not 38.48 as with the 25mm
eyepiece.

We are closer to the center of the Moon than 400 lunar radii, so moving
400x as close to the center of the Moon would put us inside the Moon.
Therefore, it is senseless to do this comparison with the 5mm eyepiece.

However, when the angles are small, multiplying angles is still a good
approximation (40.86 is not too different from 38.48).

Rob Johnson <rob@xxxxxxxxxxxxxx>
take out the trash before replying
.

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