Astronomy midterm question: cooler, brighter star versus dimmer, hotter star

Astronomy 10 Midterm 3, Spring Semester 2008
Cuesta College, San Luis Obispo, CA

Astronomy 10 learning goal Q8.5
[15 points.] How is it possible that a more luminous star can be cooler than a less luminous star? Using an H-R diagram, explain your reasoning with Wien's law and/or the Stefan-Boltzmann law.

Solution and grading rubric:

• p = 15/15:
  Correct. The Stefan-Boltzmann law states that luminosity depends on both the size (surface area) of the star, and its temperature. Thus a cooler star can still be more luminous than a hotter star if the cooler star is much bigger in size than the hotter star.
  
• r = 12/15:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. Stefan-Boltzmann law is garbled, but at least understands how size can be independent from temperature in determining the luminosity of a star. May confound mass with size in discussion.
  
• t = 9/15:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. Discussion based somehow on Wien's law and/or the Stefan-Boltzmann law.
  
• v = 6/15:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Typically confuses luminosity with apparent magnitude by saying that the cooler star could be closer to the Earth than the hotter star, when in fact luminosity (and absolute magnitude) are unaffected by distance.
  
• x = 3/15:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y = 1.5/15:
  Irrelevant discussion/effectively blank.
  
• z = 0/15:
  Blank.
  

Grading distribution:
Section 5166
p: 21 students
r: 10 students
t: 9 students
v: 11 students
x: 2 students
y: 1 student
z: 1 student

A sample "p" response (from student 1315):
This same student, like many others, made a thumbnail note of Wien's law and the Stefan-Boltzmann law on their exam paper at the start of the midterm. Note the explicit use of the word "proportional" instead of the symbol, which some students this semester called "The Pliers."
Another student 1652, after writing a "p" response, decried the abstractness and irrelevance of astronomy at this point in the semester.