Suppose we were observing the globular cluster in problem 5...

Suppose we were observing the globular cluster in problem 5 through a telescope. At the distance of the globular cluster, its radius appears to be 100 arcsec, so it covers an area on the night sky of $\pi \times(100 \text { arcsec })^{2}$ a. Supposing each star has the radius of the Sun, what is its radius in arc seconds? b. What fraction of the area of the globular cluster is covered by the summed area of the million stars?

Suppose we were observing the globular cluster in problem 5 through a telescope. At the distance of the globular cluster, its radius appears to be 100 arcsec, so it covers an area on the night sky of $\pi \times(100 \text { arcsec })^{2}$
a. Supposing each star has the radius of the Sun, what is its radius in arc seconds?
b. What fraction of the area of the globular cluster is covered by the summed area of the million stars?

Key Concepts

Small-Angle Approximation

This is a fundamental principle used in astronomy to relate the physical size of an object to its angular size when the angle is small. It asserts that for small angles (measured in radians), the angle is approximately equal to the linear size divided by the distance, allowing astronomers to easily convert between physical dimensions and observed angular measurements.

Angular Diameter

Angular diameter refers to the apparent size of an object as seen by an observer. It is defined as the angle the object subtends at the point of observation. In astronomy, angular diameters are crucial for deducing the physical dimensions of celestial bodies from their observed sizes when the distance to the object is known.

Conversion between Physical and Angular Size

This concept involves using geometrical relationships, notably the small?angle formula, to convert the actual physical size of an astronomical object (e.g., radius of a star) to its angular size (e.g., in arc seconds) as seen from Earth. It is essential for understanding how distant objects appear much smaller in the sky compared to their actual sizes.

Fractional Area Coverage

Fractional area coverage is a concept used to determine the proportion of a larger area that is occupied by one or more smaller areas. In an astronomical context, it can be applied to calculate what fraction of the sky, or a defined region such as a globular cluster, is covered by the summed areas of many stars, thus providing insights into their spatial distribution and density.

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