A protostar with the mass of the Sun starts out with a temperature of about \(3500 \mathrm{K}\) and a luminosity about 200 times larger than the Sun's current value. Estimate this protostar's size and compare it to the size of the Sun today.

Stellar luminosity is a measure of the total amount of energy a star emits per unit time. This energy is radiated across all wavelengths and directions. In the study of stars, luminosity is usually compared to the Sun's luminosity, denoted as \( L_\odot \). Luminosity helps astronomers understand a star's energy output, which is crucial for determining its size and life cycle. The formula for luminosity involves several factors:
\[ L = 4 \pi R^2 \sigma T^4 \]
Here, \( R \) is the radius of the star, \( T \) is the temperature, and \( \sigma \) is the Stefan-Boltzmann constant. This equation shows that a star's luminosity depends on both its size and its surface temperature.

The radius-temperature relation indicates how a star's size changes with temperature. For a given luminosity, a hotter star must have a smaller radius than a cooler star. This relationship is vital for estimating the size of stars, including protostars, which are in the early stages of stellar development. If we know a star's luminosity and temperature, we can rearrange the luminosity formula to solve for the radius:
\[ R = \sqrt{ \frac{L}{4 \pi \sigma T^4} } \]
This relation is helpful for comparing different types of stars. It also allows us to understand how a star's structure changes as it evolves.

The Stefan-Boltzmann law connects the power emitted per unit area of a black body to its absolute temperature. It is expressed as:
\[ E = \sigma T^4 \]
Here, \( E \) is the energy emitted per unit area, \( T \) is the absolute temperature, and \( \sigma \) is the Stefan-Boltzmann constant \( ( \sigma = 5.67 \times 10^{-8} \ \text{W} \ \text{m}^{-2} \ \text{K}^{-4} \ ) \). This law is fundamental in astrophysics and helps describe how energy radiates from stars. By applying this law, we can understand the relationship between a star's temperature and its emitted energy, which in turn helps us determine its luminosity and size.

Comparing other stars to our Sun provides important benchmarks, as the Sun serves as a standard reference. When estimating the size of a protostar, we often compare its properties to those of the Sun. In the provided problem, the protostar’s luminosity is 200 times that of the Sun. Given that the Sun’s temperature is \( 5778 \ \text{K} \) and its luminosity is \( L_\odot \), the size of the protostar can be calculated using the radius-temperature relation and Stefan-Boltzmann law.
After using these principles, we find that the protostar is approximately 88.2 times the Sun's current radius. Such comparisons help students understand the scale and diversity of stellar objects in the universe.