1/2 " = 2.424*10^-6 radians
So, find r using
s = rθ
How far away (in meters) could you read ("resolve the letters of") a book with 3 mm square type, using the Keck telescope on Earth?
So, find r using
s = rθ
First of all, we need to convert the 3 mm square type into an angular size. To do that, we can use the small angle formula:
Angular size = Size of object / Distance to object
So, let's substitute the values we have:
Angular size = 3 mm / Distance
Now, we know that the angular resolution is half an arcsecond, which we can also write as 1/7200th of a degree. So, we can set up the following equation:
1/7200th of a degree = 3 mm / Distance
Now, let's solve for Distance by cross-multiplying:
Distance = (3 mm / 1/7200th of a degree)
Calculating this will give us the distance in meters that you can read the book from using the Keck telescope.
But let me tell you, if you're reading a book with 3 mm square type from that far away, you better have some pretty fantastic eyesight!
θ = 1.22 * (λ / D)
Where:
θ is the angular resolution in radians,
λ is the wavelength of the light used in meters, and
D is the diameter of the telescope's aperture in meters.
Since you are using visible light to read the book, you can assume a wavelength of around 550 nm (550 * 10^-9 meters).
The angular resolution is given as half an arcsecond, which is 0.5/3600 degrees. Converting this to radians:
θ = (0.5/3600) * (2π/360)
Now, we can solve for D to find the diameter of the telescope's aperture:
D = λ / (1.22 * θ)
Substituting the values:
D = (550 * 10^-9) / (1.22 * (0.5/3600) * (2Ï€/360))
Calculating this value, we find:
D ≈ 5.364 meters
Therefore, using the Keck telescope with an angular resolution of half an arcsecond, you could read the letters of a book with 3 mm square type at a distance of approximately 5.364 meters.
The angular resolution of the Keck telescope is given as half an arcsecond. An arcsecond (arcsec) is a unit of angular measurement, dividing a circle into 360*60*60 = 1,296,000 equal parts.
To calculate the distance at which a book's 3 mm square type can be resolved using the Keck telescope, we need to convert the angular resolution to radians and use the tangent function.
First, convert the angular resolution from arcseconds to radians:
1 arcsecond = (1/1,296,000) radians
Next, convert the 3 mm square type to radians:
1 mm = (1/1000) meters
So, 3 mm = (3/1000) meters
Now, we can calculate the distance at which the 3 mm square type can be resolved using the following formula:
distance = size / angular resolution
Plug in the values:
distance = (3/1000) meters / ((1/1,296,000) radians)
Now, simplify the formula:
distance = (3/1000) * (1,296,000/1) meters
Calculate the result:
distance ≈ 3,888 meters
Therefore, using the Keck telescope on Earth, you could resolve the letters of a book with 3 mm square type at a distance of approximately 3,888 meters.