Question

question 2
can a cool star be more luminous than a hot star?
yes, if the cool star is larger than the hot star.
no.
yes, if the cool star is closer to earth than the hot star.
yes, if the cool star is more dense than the hot star.
yes, if the cool star is smaller than the hot star.
yes, if the cool star is farther from earth than the hot star.

Explanation:

Luminosity \( L=\sigma T^{4} 4 \pi R^{2} \) (Stefan - Boltzmann law). Here, \( \sigma \) is the Stefan - Boltzmann constant, \( T \) is the temperature, and \( R \) is the radius. If a cool star (\( T \) is low) has a large enough radius (\( R \)), it can have a higher luminosity than a hot star (with a smaller radius). Distance from Earth affects apparent brightness (\( b = \frac{L}{4\pi d^{2}} \), where \( d \) is the distance), not luminosity. Density and size (smaller) do not help a cool star in terms of luminosity as per the Stefan - Boltzmann formula.

Answer:

Yes, if the cool star is larger than the hot star.