Stefan-Boltzmann and Wien laws for stars
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Question
What is a 'black body'?
Answer
An ideal object that absorbs all radiation hitting it and re-radiates a spectrum set **only by its temperature**. A star is a good approximation.
Question
What is the 'peak wavelength' λ_{max} of a star?
Answer
The wavelength at which the star radiates **most intensely** — the top of its black-body curve. A shorter peak means a hotter star.
Question
State Wien's displacement law.
Answer
$\lambda_{max}T = 2.9\times10^{-3}$ m K (given). The peak wavelength and the absolute temperature are **inversely** related.
Question
How do you get a star's temperature from its spectrum?
Answer
Read off the peak wavelength λ_{max} (in metres), then T = 2.9 × 10⁻³ ÷ λ_{max}.
Question
State the Stefan-Boltzmann law for a star.
Answer
$L = \sigma A T^{4}$ (given), with A = 4πR² for a sphere, so $L = \sigma(4\pi R^{2})T^{4}$ and L ∝ R²T⁴.
Question
What is 'luminosity' L?
Answer
The **total power** a star radiates, in watts (W). It is set by the star's surface area and the fourth power of its temperature.
Question
Why does temperature dominate the luminosity?
Answer
Because it appears as **T⁴**. Doubling the temperature multiplies the luminosity by 2⁴ = **16**, while doubling the radius gives only 4×.
Question
How do you find the ratio of two stars' radii?
Answer
R_B/R_A = √(L_B/L_A) ÷ (T_B/T_A)² — take the ratio of the two Stefan-Boltzmann equations so σ and 4π cancel.
Question
What is the Stefan-Boltzmann constant σ?
Answer
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (a given data-booklet constant).
Question
A star's peak is at 580 nm. Its temperature?
Answer
T = 2.9 × 10⁻³ ÷ (580 × 10⁻⁹) ≈ **5000 K** — a yellow star.
Question
To estimate a star's radius, which two laws and in what order?
Answer
Wien first (peak → temperature T), then Stefan-Boltzmann (L and T → radius R, via L = σ(4πR²)T⁴).
Question
Two stars share a temperature; one is 4× as luminous. Radius ratio?
Answer
At equal T, L ∝ R², so R ratio = √4 = **2**.
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