Your telescope has a 1000 -mm-focal-length objective. You...

Your telescope has a 1000 -mm-focal-length objective. You use your telescope to take a photo of the full moon, which has an angular size of $0.52^{\circ}$ that night. You position a $\mathrm{CCD}$ detector so that it captures the image produced by the objective lens. What is the diameter of that image?

Key Concepts

Image Scale

The image scale or plate scale in optics defines the relationship between an angular measurement in the object space and the corresponding linear measurement in the image space, typically given by multiplying the focal length by the angular size (when the small angle approximation is valid). This relationship is fundamental in determining the size of the image formed by an optical system on a detector, linking geometric optics with practical imaging outcomes.

Small Angle Approximation

The small angle approximation is a mathematical simplification used when dealing with small angles (typically when the angle is measured in radians and is much less than one radian). It states that the sine or tangent of an angle is approximately equal to the angle itself (in radians). This concept is important in optics for converting an object's angular size into a linear image size on the detector, thereby simplifying calculations.

Focal Length

The focal length of an optical system is the distance from its primary lens or mirror to the point where it converges parallel rays of light into a focused image. It is a key parameter in determining the magnification and image scale of the system. A longer focal length results in a larger image for a given angular size, directly influencing the resolution and imaging capability of the instrument.

Angular Size

Angular size is the apparent size of an object as seen by an observer, measured as the angle it subtends at the observer's eye. It is crucial in determining how large an object will appear in an optical instrument, and it is typically measured in degrees or radians. In optical systems, understanding angular size helps in relating the actual image size to the object's angular extent.

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