A star is modeled as a spherical blackbody with luminosity 3.000 × 10^27 W and maximum spectral intensity per unit wavelength at 5.000 × 10^-7 m. Use Wien's constant b = 2.90 × 10⁻³ m K, Stefan–Boltzmann constant σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ and solar radius R_sun = 6.96 × 10⁸ m. Determine its surface temperature and radius, including radius in solar units. Compare its radius with that of a second star of the same luminosity but twice the surface temperature, and explain whether temperature alone determines luminosity.

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Temperature 5800 K; radius 1.929 × 10^9 m = 2.771 solar radii. The twice-hotter equal-luminosity star has one-quarter the radius. Wien's displacement law for the stated wavelength-spectrum peak is λ_max T = b. Therefore T = b/λ_max = 5800 K; the wavelength must be in metres. A spherical blackbody radiates luminosity L = 4πR²σT⁴. Rearranging gives R = √[L/(4πσT⁴)], where the full emitting surface area is required. Substitution gives R = 1.929 × 10^9 m. Relative to the supplied solar radius, R/R_sun = 2.771; this is a dimensionless comparison. At fixed luminosity, R²T⁴ is constant, so R is proportional to T⁻². Doubling surface temperature therefore requires one-quarter the radius, not one-half, for the same total luminosity. Temperature sets radiated power per unit area, while radius sets emitting area. A cool large star can therefore be more luminous than a hot small star; temperature alone is insufficient to determine a star's luminosity.

Infer a star's radius from its spectrum · set 5

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