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Suppose one of the 10 m-diameter Keck Telescopes in Hawaii is trained on San Francisco,...

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Suppose one of the 10 m-diameter Keck Telescopes in Hawaii is trained on San Francisco, 3400 km away. 

Assuming 550 nm light, and ignoring atmospheric distortion, would it be possible to read newspaper headlines?

Would it be possible to read a billboard sign at this distance?

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To solve this problem one could use the Raleigh Criterion.  This is a mathematical relationship which defines the minimum distance between two objects that will allow them to still be distinguished as two separate objects.  The further the observer is from two objects the harder it is to distinguish them as being separated in space.  Eventually, at a great enough distance between the observer and the objects, the observer will not be able to distinguish the distance of separation and they will appear as one object.

There are two general forms of the Raleigh Criterion.  One deals with using a slit diffraction grating to separate objects and the other uses the diameter of the opening of a light collector like a telescope.  Obviously, we will use the second.

Rayleigh's formula is   Sin(theta) = 1.22 (wavelength/d)

Where theta is the angle created by the two rays of light leaving the observing apparatus and going through the adjacent edges of the objects to be resolved creating an isosceles triangle, and d is the diameter of the light collecting apparatus.

Finding sin(theta) at great distances can be simplified because when measure in radians, the sin(theta) and theta are virtually equal.  Using this simplification gives us an angle of separation of 

theta = 1.22(wavelenghth/d) =1.22x550x10^-9m/10m 

theta = 6.71x10^-8 radians.

We can can find the approximate distance of separation by using right angle trig

tan(theta) = x/z  where x is the x is the midpoint distance between the objects and z is the distance the objects are from the observer.  So the distance of separation is

2x = 2zTan(theta)  =2x3400x10^3mxTan(6.17x10^-8) = 0.456m

Thus, the objects to be observed have to have their edges at least 45.6 cm apart to be distinguishable.  So, it would be impossible to read the type on a newspaper, but it may be possible to read the information on a billboard depending on its size and clarity of print. 


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