Telescope Resolving Ability Joe Roberts

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!

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 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 = arc seconds Example 2: Hubble Telescope Mirror Diameter = 94.5 inches Resolving ability = 4.56/94.5 = arc seconds

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!

B A American Flag Dimensions A=1.0 B=1.9 A Stripes are 1/13 of 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!

Earth - Moon Geometry (Sketch not to scale!!) b 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 = Therefore angle A = degrees The TOTAL diameter of the Moon (as seen from Earth) is: 2 x = ~.518 degrees Angle A Earth Moon

Earth – Flag on Moon Geometry (Sketch not to scale!!) b a = height of a flag stripe (1.5 inches for our assumptions) a in miles = 1.5 inches / (5280x12) inches/mile = e-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 = e-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 A Earth Part of One Flag Stripe

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 = 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!

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 arc seconds Using geometrical methods previously used it can be shown that 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!

What could Hubble see if it was pointed at the Earth? Using geometrical methods previously shown it can be calculated that 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!