While the Stefan-Boltzmann Law is primarily used for stars, it can also be applied to estimate the radii of planets, particularly those that emit significant thermal radiation. This is especially relevant for exoplanets or planets outside our solar system. By measuring the infrared radiation emitted by these planets and knowing their surface temperatures, the Stefan-Boltzmann Law can be used to estimate their radii. However, this application is more complex for planets than stars due to additional factors like reflected light from their host stars and the greenhouse effect in their atmospheres, which can affect the emitted radiation. Moreover, planets do not emit radiation as uniformly as stars, making the application of this law more challenging. Nonetheless, when used with other observational data and techniques, the Stefan-Boltzmann Law can contribute valuable information in the study of exoplanets, particularly those that are large and radiate significant heat.

The Stefan-Boltzmann Law plays a significant role in understanding the life cycle of stars by providing insights into various stages of a star's evolution. As a star ages, its radius and surface temperature change, directly affecting its luminosity as described by the Stefan-Boltzmann Law. For example, in the main sequence phase, a star burns hydrogen in its core, and its size and temperature are relatively stable, leading to a stable luminosity. As the star exhausts its hydrogen and moves to later stages like the red giant phase, its radius increases and surface temperature decreases, but overall luminosity increases due to the larger surface area. These changes in luminosity, radius, and temperature, understood through the Stefan-Boltzmann Law, provide vital information about the star's age, composition, and the nuclear reactions occurring within it. This law, therefore, is key in mapping the lifecycle of stars from their formation to their eventual demise, whether as white dwarfs, neutron stars, or black holes.

The application of the Stefan-Boltzmann Law to variable stars, which are stars that exhibit significant changes in luminosity over time, presents several limitations. Firstly, variable stars, such as Cepheid variables or RR Lyrae stars, often undergo changes in size and temperature, leading to fluctuations in luminosity. Since the Stefan-Boltzmann Law assumes a constant surface temperature and radius, it may not accurately reflect the star's luminosity at different stages of its variability cycle. Additionally, variable stars may have non-uniform surface temperatures and radiate energy differently at various wavelengths, which the law does not account for. These stars might also have complex atmospheric phenomena that affect their radiation, further complicating the use of this law. Therefore, while the Stefan-Boltzmann Law provides a basic framework, additional observational data and more sophisticated models are required to accurately estimate the radii and understand the behaviour of variable stars.

Knowing the radius of a star is crucial in astrophysics for several reasons. Firstly, it allows for the classification of stars into various types (like dwarfs, giants, supergiants) based on size, which is essential for understanding their evolutionary stages and life cycles. Secondly, the radius, combined with temperature, helps in determining the luminosity of a star, a key factor in understanding its energy output and distance from Earth. Thirdly, the radius is vital in studying the structure and dynamics of stars, including their internal processes like nuclear fusion. Additionally, knowing the radius is essential in determining the habitable zones around stars, which is critical for the search for extraterrestrial life. Finally, understanding the size of a star aids in the study of stellar phenomena such as flares, sunspots, and pulsations, which have broader implications in understanding the universe.

The Stefan-Boltzmann Law and Wien's Displacement Law are complementary in astrophysical observations. While the Stefan-Boltzmann Law relates a star's luminosity to its radius and temperature, Wien's Displacement Law provides a method to determine the star's surface temperature by identifying the peak wavelength of its emitted radiation. By using Wien's Law to find the temperature, and then applying the Stefan-Boltzmann Law, astronomers can estimate the radius of the star. This synergy allows for a more comprehensive understanding of a star's physical characteristics. For instance, by observing the spectrum of a star, its peak wavelength can be determined, which then gives its surface temperature. With this temperature and the star's known luminosity, the Stefan-Boltzmann Law can be used to accurately estimate the star's radius. This process is essential for classifying stars and understanding their evolutionary stages.