To Explain the World: The Discovery of Modern Science (53 page)

(This works only for thin lenses; otherwise
d
also depends on the angle
θ
introduced in Technical Note 22.) If the rays of light from
some distant object strike the lens in a range
Δγ
(delta gamma) of angles, they will be focused to a strip of height
Δd
, given by

(As usual, this formula is simpler if
Δγ
is measured in radians, equal to 360° /2
π
, rather than in degrees; in this case it reads simply
Δd
=
f Δγ
.) This strip of focused light is known as a “virtual image.” (See Figure 17a).

We cannot see the virtual image just by peering at it, because after reaching this image the rays of light diverge again. To be focused to a point on the retina of a relaxed human eye, rays of light must enter the lens of the eye in more or less parallel directions. Kepler’s telescope included a second convex lens, known as the eyepiece, to focus the diverging rays of light from the virtual image so that they left the telescope along parallel directions. By repeating the above analysis, but with the direction of the light rays reversed, we see that for the rays of light from a point on the light source to leave the telescope on parallel directions, the eyepiece must be placed at a distance
f’
from the virtual image, where
f’
is the focal length of the eyepiece. (See Figure 17b.) That is, the length
L
of the telescope must be the sum of the focal lengths

L
=
f
+
f
ʹ

The range
Δγ'
of directions of the light rays from different points on the source entering the eye is related to the size of the virtual image by

Figure 17. Telescopes.
(a) Formation of a virtual image. The two solid lines marked with arrows are rays of light that enter the lens on lines separated by a small angle Δ
γ
. These lines (and others parallel to them) are focused to points at a distance
f
from the lens, with a vertical separation Δ
d
proportional to Δ
γ
. (b) The lenses in Kepler’s telescope. The lines marked with arrows indicate the paths of light rays that enter a weak convex lens from a distant object, on essentially parallel directions; are focused by the lens to a point at a distance
f
from the lens; diverge from this point; and are then bent by a strong convex lens so that they enter the eye on parallel directions.

The apparent size of any object is proportional to the angle subtended by rays of light from the object, so the magnification
produced by the telescope is the ratio of this angle when the rays enter the eye to the angle they would have spanned if there were no telescope:

By taking the ratio of the two formulas we have derived for size
Δd
of the virtual image, we see that the magnification is

To get a significant degree of magnification, we need the lens at the front of the telescope to be much weaker than the eyepiece, with
f
>>
f’
.

This is not so easy. According to the formula for the focal length given in Technical Note 22, to have a strong glass eyepiece with short focal length
f'
, it is necessary for it to have a small radius of curvature, which means either that it must be very small, or that it must not be thin (that is, with a thickness much less than the radius of curvature), in which case it does not focus the light well. We can instead make the lens at the front weak, with large focal length
f
, but in this case the length
L
=
f
+
f’
f
of the telescope must be large, which is awkward. It took some time for Galileo to refine his telescope to give it a magnification sufficient for astronomical purposes.

Galileo used a somewhat different design in his telescope, with a concave eyepiece. As mentioned in Technical Note 22, if a concave lens is properly placed, converging rays of light that enter it will leave on parallel directions; the focal length is the distance behind the lens at which the rays would converge if not for the lens. In Galileo’s telescope there was a weak convex lens in front with focal length
f
, with a strong concave lens of focal length
f’
behind it, at a distance
f’
in
front
of the place where there would be a virtual image if not for the concave lens. The magnification of such a telescope is again the ratio
f
/
f’
, but its length is only
f
–
f’
instead of
f
+
f’
.

24. Mountains on the Moon

The bright and dark sides of the Moon are divided by a line known as the “terminator,” where the Sun’s rays are just tangent to the Moon’s surface. When Galileo turned his telescope on the Moon he noticed bright spots on the dark side of the Moon near the terminator, and interpreted them as reflections from mountains high enough to catch the Sun’s rays coming from the other side of the terminator. He could infer the height of these mountains by a geometrical construction similar to that used by al-Biruni to measure the size of the Earth. Draw a triangle whose vertices are the center
C
of the Moon, a mountaintop
M
on the dark side of the Moon that just catches a ray of sunlight, and the spot
T
on the terminator where this ray grazes the surface of the Moon. (See Figure 18.) This is a right triangle; line
TM
is tangent to the Moon’s surface at
T
, so this line must be perpendicular to line
CT.
The length of
CT
is just the radius
r
of the moon, while the length of
TM
is the distance
d
of the mountain from the terminator. If the mountain has height
h
, then the length of
CM
(the hypotenuse of the triangle) is
r
+
h.
According to the Pythagorean theorem, we then have

(
r
+
h
)
²
=
r
²
+
d
²

and therefore

d
²
= (
r
+
h
)
²
−
r
²
= 2
rh
+
h
²

Since the height of any mountain on the Moon is much less than the size of the Moon, we can neglect
h
²
compared with 2
rh.
Dividing both sides of the equation by 2
r
²
then gives

So by measuring the ratio of the apparent distance of a mountaintop from the terminator to the apparent radius of the Moon, Galileo could find the ratio of the mountain’s height to the Moon’s radius.