Using Angular Size Relationship to Find the Diameter
Using Optical Observations to find the Diameter of HT CasAbove is an optical image of the sky including HT Cas. Only the central star in the HT Cas binary system is visible at optical wavelengths. Optical images can be used to find the angular size of an object, such as a galaxy or a supernova remnant. Then, if you have the distance to the star from the Earth, you can calculate the actual diameter of the central star from its angular size using the relationship:
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Because of this relationship, you can solve for one unknown (the actual size, or diameter, for example) if you know the other two (the angular size and the distance, in this case). In fact, that is exactly what you are going to do to find the size of the central star in HT Cas and answer the challenge question. Once you find or measure HT Cas's angular size and its distance from Earth, you can plug the values into the equation above and solve for its actual size.
The 1.5 x 108 is a conversion factor to account for the units used in this problem, which need to be consistent. Parsecs are based on parallax, as measured from the Earths orbit which has a radius of 1.5 x 108 km.