Astronomy midterm question: brighter, larger star is hotter?

Astronomy 210 Midterm 2, spring semester 2014
Cuesta College, San Luis Obispo, CA

An astronomy question on an online discussion board[*] was asked and answered:
qu: If you have two stars that are at equal distance, and one is brighter than the other, then the brighter one must be hotter if it is the larger one.
Discuss why this answer would not necessarily always be correct, and how you know this. Explain using Wien's law, the Stefan-Boltzmann law and/or an H-R diagram.

[*] answers.yahoo.com/question/index?qid=20140327175847AARrJhx.

Solution and grading rubric:
  • p:
    Correct. Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to discuss how a brighter/larger star does not necessarily have to be hotter than a dimmer/smaller star, by comparing:
    • bright/larger/cooler vs. dim/smaller/hotter stars;
    • bright/larger/(same temperature) vs. dim/smaller/(same temperature) stars;
    or makes other comparisons between:
    • bright/smaller/hotter vs. dim/larger/cooler stars;
    • bright/(same size)/hotter vs. dim/(same size)/cooler stars.
  • r:
    Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. Or as (p), but may instead compare:
    • (same brightness)/smaller/hotter vs. (same brightness)/larger/cooler stars;
    despite being given "one star is brighter than the other."
  • t:
    Contains right ideas, but discussion is unclear/incomplete or contains major errors.
  • v:
    Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least attempts to use Wien's law, H-R diagram and/or the Stefan-Boltzmann law.
  • x:
    Implementation/application of ideas, but credit given for effort rather than merit. Discussion not based on Wien's law, H-R diagram and/or the Stefan-Boltzmann law.
  • y:
    Irrelevant discussion/effectively blank.
  • z:
Grading distribution:
Section 30676
Exam code: midterm02sh7F
p: 26 students
r: 4 students
t: 3 students
v: 8 students
x: 1 student
y: 0 students
z: 0 students

A sample "p" response (from student 0417) showing that the brighter, larger star could be cooler:

Another sample "p" response (from student 1000) showing that the brighter, hotter star could have the same size:

Another sample "p" response (from student 1133) showing how the brighter, larger star could have the same temperature:

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