Astronomy Principles and Practice Fourth Edition.pdf

The telescope and the collected energy 243
Figure 16.3. The relationship between the brightness of an extended object and the brightness of its image.
solid angle of the source per unit time. A convenient measure of the brightness might be expressed in
power units as J m −2 sr −1 s −1 or W m −2 sr −1 . Suppose that an extended source of brightness, B 1 ,at
a distance, d 1 , from an optical system of collecting area, A, presents an elemental area, σ 1 ,andthat
an image of this is formed (see figure 16.3). The amount of energy collected by the system per unit of
time, or the flux collected by the aperture is given by
( )
A
1 = B 1 σ 1
d1
2 (16.7)
where the bracketed term corresponds to the solid angle associated with the flux accepted by the
aperture. The ratio of d 2 to d 1 sets the magnification of the system and the corresponding area of
the image is given by
( )
d
2
σ 2 = σ 2 1
d1
2 . (16.8)
Now for this same optical system, we can also consider the case with the radiation originating at
the position of the first image from a source of brightness B 2 , providing an elemental area σ 2 , with the
drawn ray paths following the opposite direction to the original situation. The flux now entering the
aperture is now
( )
A
2 = B 2 σ 2
d2
2 . (16.9)
If the fluxes for the two scenarios are set equal, then by equations (16.7) and (16.9),
)
)
B 2 σ 2
(
A
d 2 2
= B 1 σ 1
(
A
d 2 1
so that
( ) ( )
σ1 d2
2 B 2 = B 1
σ 2 d1
2 .
By using equation (16.8),
( )( )
σ1 σ2
B 2 = B 1
σ 2 σ 1
with the result that
B 2 = B 1 .
The brightness of an image of an extended object, therefore, matches that of the object. A
telescope does not increase the brightness of an extended object—a fact not always appreciated. In
fact, because of transmission losses in the optics, the brightness of any image of an extended source