If we have two objects \(\theta(")\) apart on the sky, how far apart, \(x,\) are their images on the film of a camera with a focal length \(f\). (Assume that we wish to express \(x\) and \(f\) in the same units.)

The focal length (denoted as \( f \)) of a camera is a key parameter that determines how light is focused to produce an image. It is the distance between the camera lens and the film or sensor when the subject is in focus. Understanding how to calculate the focal length is essential in photography and astronomy. When an image is captured, the light from the object converges at the focal point. From this point to the lens is the focal length.
For practical purposes:

• A short focal length (e.g., 18mm) provides a wider field of view.
• A long focal length (e.g., 200mm) provides a narrower field of view but greater magnification.

Focal length directly influences how much of the sky or the object will be captured in the frame and is crucial when calculating the apparent separation between two points on the film.

The small angle approximation is an essential concept in astronomy and optics. It simplifies the calculation by assuming that angles measured in radians are very small. An angle measured in degrees can be converted into a radian by using the formula \(1 \text{ radian} \approx 57.2958 \text{ degrees}\). For angular separations much smaller than one radian, the approximation holds that:
\(\sin(\theta) \approx \theta\) when \(\theta\) is in radians.
This simplification allows using linear calculations instead of trigonometric functions, thus making it easier to determine distances. Specifically:

• \(x = f \theta \)

In this exercise, this formula specifies that the distance \( x \) between the images on the camera's film is directly proportional to the focal length \( f \) and the angular separation \( \theta \). This approximation is valid because we assume that the angles are measured in arcseconds, which are usually small.

Often, angles are measured in degrees or arcseconds. To use them in calculations involving arcs or circular functions, they need to be converted to radians. One degree is equivalent to \(3600\) arcseconds, and one arcsecond is a very tiny angle:
\(1 \text{ arcsecond} = \frac{1}{206265} \text{ radians} = \frac{\text{\pi}}{648000} \text{ radians}\).
To convert an angle \(\theta_{arcsec}\) from arcseconds to radians, we use:
\(\theta_r = \theta_{arcsec} \times \frac{\text{\pi}}{648000}\).
This conversion is vital in the step-by-step solution. Correct conversion ensures accurate results when applying the small-angle approximation formula. For instance, if the angular separation is 1 arcsecond, it converts to:

• \(\theta_r = 1 \times \frac{\text{\pi}}{648000} \approx 4.85 \times 10^{-6} \text{ radians}\).

Thus, correctly converting and then applying this in the formula \(x = f \theta_r\) is straightforward and essential for determining the separation accurately.