A common question asked at Star Parties, workshops, or other gatherings where Astronomy is discussed is:

### Why can't we see the flag that the Apollo Astronauts left on the moon?

The answer is some basic Wakefield High School mathematics* as follows:

A.    Lets assume that: • The height of the flag (a) pole in ft. = 6
• The average distance to the Moon (approx) in miles = 221,000
• Which means that the distance to the Moon (approx) in feet (b) = 1,166,880,000

B.    Given the above:

• The tangent (t) of the above in radians [tan(a/b)] = 0.0000000051, in radians
• Converted to Degrees [180*(t/pi)] = 0.000000295 degrees
• This means that a telescope would have to be able to resolve an angle of 2.95 x 10-07 degrees to see the flag left on the moon.

C.    That means:

 Since one Arc Degree (1/360 of a circle) = 1.000000000 Deg. And one Arc Minute (1/60 of a degree) = 0.016666667 Deg. And one Arc Second (1/60 of an arc-minute) = 0.000277778 Deg. We can calculate that: The Angle to be resolved is: 0.000000295 Deg. And the Angle to be resolved in Arc Seconds is: 0.001060596

So, to see the flag, a telescope must be able to resolve an angle of about 1/1000th of an arc second.

*The formulas in bold above can be used in most spreadsheets

#### "So how big of a telescope would that require?", you ask.

The Dawes limit calculates how close two objects can be and still be resolved by a telescope.

The formula is:  Resolution (in arc seconds) = 4.56/Diameter.

So, to determine the required diameter, the formula is:  D = 4.56/Resolution(arc seconds)

Doing the math, that is 4.56/.001060596 arc seconds, to see the flag a telescope would require a mirror 4,299.47 inches wide! That is 358.29 Feet!

A super-telescope THAT big would have to be in orbit, to eliminate the turbulence of the earth's atmosphere.

#### So, what distances can we see on the moon?

Actual distances to the moon based on Rukl's Atlas of the Moon:
KilometersMiles
Closest (perogee)356,400221,457
Average384,401238,856
Farthest (apogee)>406,700252,712
Lunar Diameter34762159.89

 The Moon's Angular Diameter at perogee (when it is closest to the earth) = 33' 28.8" (33 min. 28.8 sec.) Lunar Diameter in feet = 11,404,199.48 Total Angular diameter in Arc Seconds = 2008.80 Lunar Surface per Arc Second = 5677.12 (ft.) For a 10" telescope, Dawes Limit is 4.56/10 or 0.456000 Arc-Seconds So, the resolvable Distance with a 10" telescope is (5677.12)/(.456), or: 2588.77 ft. Our super-telescope distance resolved at .00106 Arc Seconds is: 6.02 ft.

Note: The Hubble Space Telescope can resolve .005 Arc Seconds, or about 280 Feet
(See Sky & Telescope 10/2004 pg 130)

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