The star α centauri c and the star groombridge 34 b have the same apparent magnitude, but α centauri c is 1.3 pc away from earth and groombridge 34 b is 3.5 pc away. what is the luminosity of groombridge 34 b compared with α centauri c?

It can be concluded that groombrige 34 b is more luminous than centauri c since they have the same apparent magnitude.

Explanation:

The apparent magnitude is the brightness that a star appears to have as seen from the earth.

This can variate as a consequence of the distance or radius of the star.

for example, if one of two stars has a bigger radius that a second one, the free mean path of the photons will be higher than in the smaller one, so the bigger one will be more luminous.

In the case of the distance, if for example, two motorbikes with its lights on are approaching an observer, but one of the motorbikes is at a greater distance, it will appear that is light is dimmer compared to the one that is closer. Even if both emit the same amount of light per unit of time.  

Therefore, it can be concluded that groombrige 34 b is more luminous than centauri c since they have the same apparent magnitude.

b.a device that inputs heat into a cool object