... of an optical telescope to resolve detail is governed b

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Telescope Resolving Ability

• Joe Roberts
• joe_at_rocketroberts.com

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Telescope Resolving Ability

• We will cover the basics of an optical
  telescope's ability to resolve detail on a
  subject
• An often asked question Can Hubble see the Flag
  on the Moon?
• Answer not a chance, and we'll show why not!

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Dawes Limit

• The ability of an optical telescope to resolve
  detail is governed by the Dawes Limit
• Resolution (in arc seconds) 4.56/diameter of
  the telescope mirror/lens (in inches)
• 1 degree 60 arc minutes 1 arc minute 60 arc
  seconds, therefore 1 degree 3600 arc seconds

Example 1 Telescope Mirror Diameter 6
inches Resolving ability 4.56/6 .76 arc
seconds
Example 2 Hubble Telescope Mirror Diameter
94.5 inches Resolving ability 4.56/94.5
0.048254 arc seconds
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Can we see the Flag on the Moon?

• First we need to make some basic assumptions
  about how big the flag is
• We then do some basic geometry to determine how
  big (in arc seconds) the flag would look at the
  distance of the Moon
• We will assume that we want to just barely be
  able to make out the stripes on the flag
• Once we figure this out we apply the Dawes Limit
  equation to find out what size telescope is
  required!

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American Flag Dimensions
B
A1.0 B1.9 A Stripes are 1/13 of A
A
Flag on the Moon Assume it is 3 feet wide
therefore stripes are 36 inches x (1/1.9) x
(1/13) 1.46 inches tall We'll round this up
to 1.5 inches for our calculations...
So, we need a telescope that can just resolve
something about 1.5 inches wide on the Moon. So
how do we figure this out? Basic geometry is all
that is required!
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Earth - Moon Geometry(Sketch not to scale!!)
Moon
Angle A
b
Earth
a diameter of the Moon 2160 miles b mean
distance from Earth to Moon 239,000 miles
For our diagram tan(A) (a/2)/b tan(A)
(2160/2)/239,000 0.004519 Therefore angle A
0.259 degrees The TOTAL diameter of the Moon (as
seen from Earth) is 2 x 0.259 .518 degrees
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Earth Flag on Moon Geometry(Sketch not to
scale!!)
A
Part of One Flag Stripe
b
Earth
a height of a flag stripe (1.5 inches for our
assumptions) a in miles 1.5 inches /
(5280x12) inches/mile 2.3674e-5 miles b mean
distance from Earth to Moon 239,000 miles
For our diagram tan(A) (a/2)/b tan(A)
(2.3674e-5/2)/239,000 4.9528e-11 Therefore
angle A 2.838e-9 degrees The TOTAL width of
the flag stripe (as seen from Earth) is 2 x
2.838e-9 5.675e-9 degrees
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Apply Dawes Limit Equation
From previous slide The TOTAL width of the flag
stripe (as seen from Earth) is 2 x 2.838e-9
5.675e-9 degrees

• There are 3600 arc seconds per degree therefore
  5.675e-9 degrees 2.043e-5 arc seconds
• Dawes Limit Resolution (in arc seconds)
  4.56/diameter of the telescope mirror/lens (in
  inches)

2.043e-5 4.56/x therefore x 223183 inches
3.52 miles
Therefore it is not even close to possible to see
the Flag on the Moon from Earth with telescopes
available today!
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So what can Hubble see on the Moon?

• Hubble is in orbit 375 miles above Earth, so
  basically it is no closer to the Moon than we are
  right now
• Hubble's ability to resolve was determined to be
  0.048254 arc seconds
• Using geometrical methods previously used it can
  be shown that 0.048254 arc seconds is equal to an
  object about 295 feet wide at the distance of
  the Moon
• This is the smallest object that could just
  barely be discerned!

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What could Hubble see if it was pointed at the
Earth?

• Using geometrical methods previously shown it can
  be calculated that 0.048254 arc seconds is equal
  to an object about 5.56 inches wide at the
  distance of 375 miles
• Basically Hubble could just make out the size and
  shape of a car license plate (assuming it was
  laying flat on the ground)
• Reading the plate would be out of the question!