20160503

Astronomy midterm question: same luminosity, but different temperature and size stars

Astronomy 210 Midterm 2, spring semester 2016
Cuesta College, San Luis Obispo, CA

An astronomy question on an online discussion board[*] was asked and answered:
??: If two stars have the same luminosity, is it always true that the star with the lower temperature must be bigger?
qcp: Yes, that's true. A cooler star means that it must be bigger to be of equal 
luminosity.
Discuss why this answer is correct, 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=20160402221522AAZkrDm.

Solution and grading rubric:
  • p:
    Correct. Uses Wien's law, the Stefan-Boltzmann law and/or interprets H-R diagram to demonstrate how a cooler star must be bigger than a hotter star to have the same luminosity.
  • r:
    Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors.
  • 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 30674
Exam code: midterm02n4s5
p: 12 students
r: 3 students
t: 3 students
v: 1 student
x: 2 students
y: 1 student
z: 0 students

A sample "p" response (from student 1503), using both the "box method" to fill in relative Stefan-Boltzmann law parameters, along with using the diagonal lines on an H-R diagram to compare size:

A sample "x" response (from student 1096), appealing to recent tragic events in pop culture:

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