Final answer:
At 34º N latitude, Rigil Kentaurus with a declination of -60º 52' will never be above the horizon because it is located more than 56º south of the celestial equator, beyond the visible limit for this latitude.
Step-by-step explanation:
To determine which of the stars Sirius, Canopus, or Rigil Kentaurus will never be above the horizon at 34º N latitude, we can use the concept of the celestial pole and declination. The altitude of the celestial pole above the horizon is equal to the latitude of the observer. This means that at 34º N, the celestial pole is 34º above the southern horizon, and stars within 34º of the south celestial pole will never rise. Therefore, the declination of a star tells us how far it is from the celestial equator, and if a star's declination is more than 56º (90º - 34º) south, it will never be above the observer's horizon.
Sirius, with a declination of -16º 44', is well above the threshold and can be seen above the horizon. Canopus, at -52º 42', is close to the limit but can still make an appearance above the horizon since it is less than 56º from the celestial equator. Rigil Kentaurus, however, with a declination of -60º 52', is too far south and will never be visible above the horizon for someone at 34º N latitude. Therefore, the correct answer to the question is c. Rigil Kentaurus.