Physics quiz question: surface temperature of Bellatrix

Physics 205A Quiz 7, fall semester 2015
Cuesta College, San Luis Obispo, CA

Stars can be modeled as spherical blackbodies that radiate heat out to an environment assumed to have a temperature of 0 K. Bellatrix is a star that radiates heat per time at a rate of 2.5×10³⁰ watts, and has a surface area of 2.2×10²⁰ m². The surface temperature of Bellatrix is:
(A) 4.2×10³ K.
(B) 2.1×10⁴ K.
(C) 1.1×10⁵ K.
(D) 6.1×10⁵ K.

Correct answer: (B)

The net power radiated by Bellatrix (–2.5×10³⁰ watts, where the negative sign indicates that heat is continuously leaving the surface of the star) given by:

Power = –e·σ·A·((T)⁴ – (T_env)⁴) = –e·σ·A·((T)⁴ – (0 K)⁴),

where the emissitivity e = 1 (for an ideal blackbody), the Stefan-Boltzmann constant σ = 5.670×10^–8 watts/(m²·K⁴), A is the surface area, and the surface temperature T is unknown, and can be solved for:

(T)⁴ = –Power/(e·σ·A),

T = (–Power/(e·σ·A))^(1/4),

T = (–(–2.5×10³⁰ watts)/((1)·(5.670×10^–8 watts/(m²·K⁴))·(2.2×10²⁰ m²)))^(1/4),

T = 2.11584362479×10⁴ K,

or to two significant figures, the surface temperature of Bellatrix is 2.2×10⁴ K.

(Response (A) is σ^(–1/2); response (C) is (Power/A)^(–1/2); response (D) is (Power·σ)^(1/4).)

Sections 70854, 70855
Exam code: quiz07zSOL
(A) : 12 students
(B) : 31 students
(C) : 15 students
(D) : 10 students

Success level: 45%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.35