The Keck telescope on Mauna Kea has an angular resolution on Earth of half an arcsecond.
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?
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To determine how far away you could read a book with 3mm square type using the Keck telescope on Earth, we need to use the concept of angular resolution.
Angular resolution is essentially the smallest angle at which two objects can be distinguished as separate entities. For a telescope, it refers to the ability to distinguish between two closely spaced objects at a specific distance.
Given that the angular resolution of the Keck telescope on Earth is half an arcsecond (0.5 arcseconds), we can use this value to calculate the distance at which the 3mm square type can be resolved.
First, let's convert the angular resolution from arcseconds to radians:
1 arcsecond = (1/3600) degrees
1 degree = (π/180) radians
Therefore, 0.5 arcseconds is equal to:
0.5 * (1/3600) * (π/180) radians
Next, we can set up a trigonometric relationship between the distance, the size of the typeface, and the angular resolution:
tan(0.5 arcseconds) = (size of the typeface) / (distance)
Since we want to find the distance, we rearrange the equation to solve for "distance":
distance = (size of the typeface) / tan(0.5 arcseconds)
Plugging in the given values:
distance = 3 mm / tan(0.5 arcseconds)
Now, let's calculate the distance:
distance = 3 mm / tan(0.5 * (1/3600) * (π/180)) meters
Using a calculator, we find that the distance is approximately:
distance ≈ 2,425,636 meters
Therefore, using the Keck telescope on Earth, you could read the letters of a book with 3mm square type at a distance of approximately 2,425,636 meters.
To calculate the distance at which you can resolve the letters of a book with the given square type size using the Keck telescope, we need to consider the concept of angular resolution.
Angular resolution is a measure of the smallest angle that can be discerned by an optical instrument, such as a telescope. In this case, the Keck telescope has an angular resolution of half an arcsecond.
To determine the distance at which you can resolve the letters, we can use the following formula:
Distance = Size / Tan(Angular resolution)
Where:
- Distance is the distance from the observer to the object (the book in this case)
- Size is the size of the object being observed (3 mm in this case)
- Tan denotes the tangent function
- Angular resolution is the angular resolution of the telescope (0.5 arcseconds in this case)
First, convert the angular resolution from arcseconds to radians:
Angular resolution (radians) = Angular resolution (arcseconds) * (π/180)
Since there are 3600 arcseconds in one degree and π radians in 180 degrees.
Angular resolution (radians) = 0.5 arcseconds * (π/180) ≈ 8.73 x 10^-7 radians
Next, calculate the distance using the formula:
Distance = Size / Tan(Angular resolution)
Distance = (3 mm) / Tan(8.73 x 10^-7 radians)
We can use the small-angle approximation, which states that for small angles, the tangent of the angle is approximately equal to the angle itself in radians.
So, Tan(8.73 x 10^-7 radians) ≈ 8.73 x 10^-7 radians
Distance = (3 mm) / (8.73 x 10^-7 radians)
Now, convert the size from millimeters to meters:
Size (meters) = 3 mm * (1 meter / 1000 mm) = 3 x 10^-3 meters
Finally, substitute the values into the formula:
Distance = (3 x 10^-3 meters) / (8.73 x 10^-7 radians)
Calculating this gives you the distance at which you could read the book with 3 mm square type using the Keck telescope on Earth.