Astronomy midterm question: small hot stars always more luminous than cool big stars?

Astronomy 210 Midterm 2, fall semester 2012
Cuesta College, San Luis Obispo, CA

An astronomy question on an online discussion board(*) was asked and answered:

Pd: Is it possible that a small hot star and a cool big star [located the same distance from us] can be equally bright?
Star Dust: Small-hot is [always] more luminous than cool-big.

Discuss why this answer is not correct, and how you know this. Explain using Wien's law, the Stefan-Boltzmann law and/or an H-R diagram.

*Adapted from: http://answers.yahoo.com/question/index?qid=20120928201352AA8dDbB.

Solution and grading rubric:

• p = 20/20:
  Correct. Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to show that small, hot stars can be either (a) equally luminous, or (b) less luminous than cool, big stars.
• r = 16/20:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors.
• 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-Boltzman 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-Boltzman law.
• y = 2/20:
  Irrelevant discussion/effectively blank.
• z = 0/20:
  Blank.

Grading distribution:
Section 70160
Exam code: midterm02NuF7
p: 20 students
r: 5 students
t: 1 student
v: 1 student
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 5411) demonstrating how a small hot star could be as luminous as a large cool star:
Another sample "p" response (from student 0716) demonstrating how a small hot star could be less luminous than a large cool star: