Star A and star B are two stars nearly the same distance from Earth. Star A is blue, and star \(\mathrm{B}\) is red, but they have equal brightness. Which of the following statements is true? a. Star A is more luminous than star B. b. Star A is larger than star B. c. Star A is smaller than star B. d. Star A is less luminous than star B.

Stars come in a variety of colors, and these colors tell us a lot about their temperatures. Blue stars are the hottest, with temperatures exceeding 25,000 Kelvin, while red stars are cooler, often below 3,500 Kelvin.
The relationship between color and temperature can be summarized as follows:

• Blue stars: Very hot, high temperature
• White stars: Hot
• Yellow stars: Moderate temperature (like our Sun)
• Red stars: Cooler, low temperature

It's essential to understand that the color of a star is a direct indicator of its surface temperature. In our exercise, star A is blue, making it hotter than the red star B. This temperature difference plays a crucial role in determining other characteristics of these stars, such as their luminosity and size.

Luminosity is the total amount of energy a star emits per unit of time. It depends not just on the star's temperature but also on its size. Hotter stars, like our blue star A, generally emit more energy than cooler stars, like the red star B.
However, in our exercise, both stars A and B appear equally bright to us. Brightness as observed from Earth, also known as apparent magnitude, may be misleading because it doesn't account for distance or intrinsic luminosity. Since both stars are at nearly the same distance from Earth and exhibit equal brightness, we deduce that star A, which is hotter, must emit more energy per square meter (surface area) compared to the cooler star B.

• Higher temperature = More energy emitted
• Equal brightness but different temperatures = Impact on size

Thus, luminosity helps to infer the comparative sizes of the stars, considering they have the same apparent brightness.

The size of a star, referred to as its radius, significantly impacts its luminosity and brightness. For two stars with equal luminosity but different temperatures, the hotter star must be smaller in size to maintain the same brightness.
In our example, star A is hotter and yet they both shine with the same brightness. To balance this, star A must be smaller in size compared to star B. This is because a hotter star will emit more energy per unit area, so if it's smaller, it can still match the brightness of a cooler, larger star.

• Equal brightness but hotter star = Smaller size
• Cooler star = Larger size if equal luminosity

This makes sense as the increased energy output of the hotter star A compensates for its smaller surface area, making it appear as bright as the larger but cooler star B. Therefore, the correct statement about our stars is that star A is smaller than star B.