Astronomy midterm question: bigger, more luminous stars always cooler or hotter?

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

An astronomy question on an online discussion board[*] was asked:

Pd: Are the bigger, more luminous stars always the cooler stars?
Sa: You wouldn't be right because "more luminous" means more energy output, which means bigger and hotter.

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.

[*] answers.yahoo.com/question/index?qid=20141031235350AAd46qs.

Solution and grading rubric:

• p:
  Correct. Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to discuss how a more luminous star does not necessarily have to be both bigger and hotter, by comparing:

  bright/larger/cooler vs. dim/smaller/hotter stars;
  bright/larger/(same temperature) vs. dim/smaller/(same temperature) stars;
  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;
  brighter/larger/hotter vs. dim/smaller/cooler star;

  thus not sufficiently discussing why the response would not always be correct.
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. At last discussion demonstrates understanding of Wien's law, H-R diagram and/or the Stefan-Boltzmann law.
• 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 clearly based on Wien's law, H-R diagram and/or the Stefan-Boltzmann law.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 70158
Exam code: midterm02s0vA
p: 20 students
r: 14 students
t: 6 students
v: 3 students
x: 1 student
y: 0 students
z: 0 students

Section 70160
Exam code: midterm02n4Rs
p: 20 students
r: 7 students
t: 2 students
v: 2 students
x: 2 students
y: 0 students
z: 0 students

A sample "p" response (from student 1327) comparing a bright/larger/cooler star versus a dim/smaller/hotter star:
A sample "p" response (from student 5656) comparing a bright/larger star that has the same temperature as a dim/smaller star:
A sample "p" response (from student 5309) comparing a bright/smaller/hotter star versus a dim/larger/cooler star:
A sample "p" response (from student 1795) comparing a bright/hotter star that has the same temperature as a dim/cooler star: