Astronomy final exam question: comparing temperatures and sizes of main-sequence stars

Astronomy 210 Final Exam, spring semester 2018
Cuesta College, San Luis Obispo, CA

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

Srr: Is it possible for two main-sequence stars to have different temperatures and different sizes but the same luminosity?
Rgr: No, because more massive main-sequence stars are hotter and larger than the lesser massive main-sequence stars, they cannot have the same luminosity.

Discuss why this answer is correct for main-sequence stars, 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=20110511035317AAeHphE.

Solution and grading rubric:

• p:
  Correct. Discusses how the Stefan-Boltzmann law and/or H-R diagram shows that along the main-sequence, the more-massive stars will be located in the hotter, larger, brighter upper left-hand corner, while the less-massive stars will be located in the cooler, smaller, dimmer lower right-hand corner, so they cannot have the same luminosity.
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. May have compared two stars in general, rather than explicitly comparing two stars on the main-sequence line.
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. At least 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 based on Wien's law, H-R diagram and/or the Stefan-Boltzmann law.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 30676
Exam code: finalSuB1
p: 24 students
r: 7 students
t: 4 students
v: 1 student
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 0707):
Another sample "p" response (from student 0219):