Astronomy midterm question: cooler star more luminous than a hotter star?

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

An astronomy question on an online discussion board[*] was given the following answers:

CM: I keep thinking that a cooler star could be more luminous than a hotter star--is this true?
Okl: No, it cannot be true.
Do: There are situations in which this may be true.

Discuss why this answer is incorrect, and how you know this. Explain using Wien's law, the Stefan-Boltzmann law and/or an H-R diagram.

[*] Adapted from https://answers.yahoo.com/question/index?qid=20090316143453AAmTD1h.

Solution and grading rubric:

• p = 20/20:
  Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to show specifically how it is possible for a cooler star to be more luminous than a hotter star, provided that the cooler star is sufficiently larger than the hotter star.
• r = 16/20:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. As (p), but may instead demonstrate how a cooler star could be equally luminous as a hotter star.
• t = 12/20:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors.
• v = 8/20:
  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 = 4/20:
  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 = 2/20:
  Irrelevant discussion/effectively blank.
• z = 0/20:
  Blank.

Grading distribution:
Section 30674
Exam code: midterm02n4iL
p: 22 students
r: 4 students
t: 1 student
v: 0 students
x: 1 student
y: 0 students
z: 0 students
z: 0 students

A sample "p" response (from student 0507) using an H-R diagram:
Another sample "p" response (from student 0794) using the "box model" of the Stefan-Boltzmann law:
Yet another sample "p" response (from student 3946) showing all the possible entries in the Stefan-Boltzmann "box model":