Final answer:
The luminosity of a contracting protostar that is five times the radius of the sun and has a temperature of 2000 K would be approximately 35 times more luminous than the Sun when applying the Stefan-Boltzmann law.
Step-by-step explanation:
To determine the luminosity of a contracting protostar compared to the sun, we apply the Stefan-Boltzmann law which states that luminosity (L) is proportional to the fourth power of the star's surface temperature (T) and directly proportional to the square of the star's radius (R). The formula is given by L = 4πR²σT´, where σ represents the Stefan-Boltzmann constant. According to this law, the luminosity of a protostar five times the radius of the sun at a temperature of 2000 K is calculated by comparing it to the sun's luminosity and temperature (approximated at 5800 K).
Using the provided formula and assuming all other factors are equal, we can say:
L (protostar) / L (sun) = (R²protostar * T´protostar) / (R²sun * T´sun)
L (protostar) / L (sun) = (25 * R²sun * 16 * 10⁶ K⁴) / (R²sun * 11.34 * 10¹ K⁴)
L (protostar) / L (sun) = 400 / 11.34L (protostar) / L (sun) = 35.27
Thus, the protostar would be approximately 35 times more luminous than the Sun.