One way to calculate the radius of a star is to use its...

One way to calculate the radius of a star is to use its luminosity and temperature and assume that the star radiates approximately like a blackbody. Astronomers have measured the characteristics of central stars of planetary nebulae and have found that a typical central star is 16 times as luminous and 20 times as hot (about $110,000 \mathrm{k})$ as the Sun. Find the radius in terms of the Sun's. How does this radius compare with that of a typical white dwarf?

Key Concepts

Blackbody Radiation

This concept refers to the idealized physical body that absorbs all incident electromagnetic radiation and re?emits energy in a spectrum that depends only on the body's temperature. It provides a foundational model for understanding how stars emit radiation based on their temperature, which is crucial for calculating stellar properties such as luminosity and radius.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law quantifies the power emitted per unit area of a blackbody in terms of its temperature. It states that the total energy radiated per unit surface area is proportional to the fourth power of the temperature (T^4). This relationship is essential in astrophysics for linking the luminosity of a star to its radius and surface temperature.

Luminosity-Temperature-Radius Relation

This relation uses the Stefan-Boltzmann Law to connect a star's luminosity, surface temperature, and radius. By treating a star as a blackbody, one can express its luminosity as L = 4?R^2?T^4, where ? is the Stefan-Boltzmann constant. This formula is key when determining the radius of stars using known values of luminosity and temperature.

White Dwarf Characteristics

White dwarfs are stellar remnants with small radii and high densities compared to main-sequence stars. They typically have radii similar to that of Earth despite having masses comparable to the Sun. Understanding the typical size scales of white dwarfs offers a context for comparing stellar radii, such as when assessing the size of central stars of planetary nebulae relative to white dwarfs.

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