Most stars are main-sequence stars, a group of stars for which size, mass, surface temperature, and radiated power are closely related. The sun, for instance, is a yellow main-sequence star with a surface temperature of 5800 K. For a main-sequence star whose mass M is more than twice that of the sun, the total radiated power, relative to the sun, is approximately P/Pₛᵤₙ=1.5(M/Mₛᵤₙ)^3.5 . The star Regulus A is a bluish main-sequence star with mass 3.8Mₛᵤₙ and radius 3.1Rₛᵤₙ. What is the surface temperature of Regulus A?

Verified step by step guidance

1. Step 1: Identify the given values from the problem. The mass of Regulus A is given as 3.8 times the mass of the sun (Mₛᵤₙ), and the radius is 3.1 times the radius of the sun (Rₛᵤₙ).
2. Step 2: Use the formula for the total radiated power of a main-sequence star relative to the sun, P/Pₛᵤₙ = 1.5(M/Mₛᵤₙ)^3.5. Substitute M/Mₛᵤₙ = 3.8 into the formula to find the power output of Regulus A relative to the sun.
3. Step 3: Recall the Stefan-Boltzmann law, which states that the total power radiated per unit area of a black body is proportional to the fourth power of the black body's temperature (P = σA T^4, where σ is the Stefan-Boltzmann constant and A is the surface area).
4. Step 4: Calculate the surface area of Regulus A using the formula for the surface area of a sphere, A = 4πR^2, where R is the radius of the star. Since the radius of Regulus A is 3.1Rₛᵤₙ, substitute this value to find the surface area in terms of the sun's surface area.
5. Step 5: Combine the expressions for power and surface area to solve for the surface temperature of Regulus A. Use the proportional relationship between the power outputs and surface areas of Regulus A and the sun to set up an equation involving the temperatures, and solve for the temperature of Regulus A.

Key Concepts

Main-Sequence Stars

Main-sequence stars are a category of stars that are in a stable phase of stellar evolution, where they fuse hydrogen into helium in their cores. Their properties, such as size, mass, surface temperature, and luminosity, are interrelated, following specific relationships defined by the Hertzsprung-Russell diagram. The Sun is a prime example of a main-sequence star, and understanding this classification is crucial for analyzing other stars' characteristics.

Stellar Mass-Luminosity Relation

The mass-luminosity relation describes how the luminosity (total radiated power) of a star is related to its mass. For main-sequence stars, this relationship can be expressed as P/Pₛᵤₙ = (M/Mₛᵤₙ)³.⁵, indicating that more massive stars emit significantly more energy. This concept is essential for calculating the luminosity of stars like Regulus A based on its mass.

Surface Temperature and Color

A star's surface temperature is directly related to its color and spectral classification. Hotter stars emit more blue light and have higher temperatures, while cooler stars appear redder. The temperature can be estimated using the Stefan-Boltzmann law, which relates temperature to luminosity and radius, allowing us to derive the surface temperature of stars like Regulus A based on its known properties.