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Question
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what is the light-collecting area of the 50-mm objective?____ mm2?
what is the light-collecting area of the 203-mm objective?___mm2?
Rowan Introduction to Astronomy
Lab 5 / Properties of Telescopes: Light-Gathering
Power, Magnification, Resolution
Name:
________________________________________________
Score:
__________________________
Summary:
The student will learn about the relationship between objective size, resolution, focal length, and magnification.
.
Light-Gathering Power
[33 pts]
Light-gathering power of a telescope is directly proportional to the area of its primary lens or mirror. All lenses and mirrors have a
circular circumference. The area of a circle is given by the formula:
A
= π
r
2
. Because π is a constant, the radius,
r,
of the mirror or lens
is the most important factor in determining the light-gathering power of a telescope. Note that area of a circle varies by the square of
the radius. Thus, a lens or mirror that is twice the radius (or diameter) of another telescope objective has 2
2
or 4 times the light-
gathering power.
1.
A typical pair of binoculars has an objective lens of 50-mm diameter. A typical amateur telescope is an 8-inch reflector that has a
mirror diameter of 203 mm. (Give answers in
a
and
b
as a number; that is, when multiplying, use π as 3.14159.)
(a) What is the light-collecting area of the 50-mm objective? _________________________ mm
2
[Round to
1
decimal place]
(b) What is the light-collecting area of the 203-mm objective? _________________________ mm
2
[Round to
1
decimal place]
(c) The 203-mm objective collects _______________ times the light of a 50-mm objective.
[Round to
nearest
whole number]
(d) The brightness of celestial objects usually is expressed in terms of magnitude. A 1
st
magnitude star is defined as being 100
times brighter than a 6
th
magnitude star (5 magnitude steps). A single magnitude jump equals a brightness change of about
2.512 (given that 2.512
5
= 100). Using the factor of 2.512 for a single magnitude jump, about how many magnitudes
fainter can the 203-mm objective “see” than the smaller 50-mm objective?
[Round to nearest whole number]
__________ magnitudes
[Hint: 2.512
1
= 2.512; 2.512
2
= ?; 2.512
3
= ?; 2.512
4
= ?; 2.512
5
= 100]
2.
Compare an amateur telescope of 100 mm (a typical “4-inch” telescope, usually a refractor) with that of the Keck telescope,
which is 10 meters across. [Hint: Work in powers of ten; “2 decimals” means after the decimal point in powers of ten notation.]
(a)
Area of 100-mm objective in mm
2
: _______________ mm
2
[Write in scientific notation and round to
2
decimals; same for part
b
]
(b)
Area of 100-mm objective in m
2
: _______________ m
2
(Careful! Note the conversion from millimeters
2
to meters
2
. Working
with powers of ten can make this step easier. Hint: How many mm in 1 meter? How many mm
2
in 1 m
2
?)
[Round to
2
decimals]
(c) Area of 10-m Keck objective: ____________________ m
2
[
Write answer in scientific notation and round to
2
decimals]
(d) 10-m objective collects _______________ times the light of a 100-mm objective
[Round to
nearest
whole number]
(e) The answer to (d) represents how many magnitudes? _______________
(
Hint
: Look for the
x
y
function on a scientific calculator. If (2.512)
5
= 100 and represents 5 magnitude steps, then how
many steps does the answer to
d
represent? If 100 = 10
×
10 or 10
2
, then how many powers of ten is the answer to
d
?)
3.
A quicker way to calculate light-gathering power is to use the formula:
2
B
A
B
A
D
D
LGP
LGP
[Hint: Don’t forget that on the right side the values are squared after being divided.]
where
LGP
A
and
LGP
B
are the light-gathering powers of A and B, respectively, and
D
A
and
D
B
are the diameters of objectives A
and B, respectively. If the human eye has a diameter of 8 mm (actually the pupil of the eye), and we have both a 50-mm telescope
and a 203-mm telescope, how much more light will the telescopes gather than our eye?
[Round answers to
nearest
whole number]
(a)
LGP
50mm
/
LGP
eye
= _______________ times more light
(b)
LGP
203mm
/
LGP
eye
= _______________ times more light
3 pages
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