Final answer:
Given that the two stars are equally far from Earth and one appears brighter, the valid conclusion is that the first star has a higher luminosity than the second star. Size and proximity are not factors since the distance is constant and size isn't mentioned in relation to brightness.
Step-by-step explanation:
When an astronomer observes two stars in the sky that are equally far from Earth, with one star appearing brighter than the other, it can lead to a few possible conclusions. Since the first star is brighter but at the same distance, the first star might be more luminous than the second one. However, the first star being brighter does not necessarily mean it is closer to Earth, as distance has already been established as equal for both stars.
Now, a star's brightness as seen from Earth (its apparent brightness) is directly related to its luminosity, which is the total energy a star emits per second. If two stars are equally distant, the one appearing brighter has higher luminosity. This is evidenced by the brightness calculations provided for stars in varying distances and luminosities in the text references.
Therefore, options 1) and 2) can be eliminated since size is not given as a reason for brightness, and the equal distance has been specified. Option 3) is a possible conclusion since stars with higher temperatures tend to be more luminous. However, this isn't confirmed by the given evidence, as we can only speculate about the stars' temperatures. Option 4) is the most valid conclusion based on the given information. A brighter star at the same distance would indeed have a higher luminosity, reflecting a greater output of energy.