Astronomy midterm question: same luminosity giant and main-sequence stars

Astronomy 210 Midterm 2, Fall Semester 2009
Cuesta College, San Luis Obispo, CA

[20 points.] Discuss whether or not a giant star and a main sequence star could have the same luminosity, and why. Support your answer using Wien's law and/or the Stefan-Boltzmann law.

Solution and grading rubric:

• p = 20/20:
  Correct. Discusses how the Stefan-Boltzmann law (luminosity (brightness) proportional to size * (Temperature)^4) explains that two stars can have the same luminosity if the smaller star is hotter, and the larger star is cooler.
  
• 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. At least recognizes how the Stefan-Boltzmann law is applicable, but argument is garbled (e.g., has giant star smaller but hotter than main sequence star of the same luminosity).
  
• v = 8/20:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least recognizes how size and temperature are relevant, but Stefan-Boltzmann law itself is garbled.
  
• x = 4/20:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.
  

Grading distribution:
Section 70158
p: 32 students
r: 6 students
t: 3 students
v: 6 students
x: 4 students
y: 0 students
z: 0 students

A sample "p" response (from student 8260) using both details from an H-R diagram (provided with the midterm), and a table comparing relative luminosities, sizes, and temperatures in the Stefan-Boltzmann law:

A sample "r" response (from student 5678), with the giant correctly being larger and cooler than (but the same luminosity as) a main-sequence star, but with a garbled quantitative expression of the Stefan-Boltzmann law:

A sample "v" response (from student 0733), mangling the comparative size and temperatures:

Another sample "v" response (from student 1030), at least recognizing that size and temperature are relevant factors:

A sample "v" response (from student 3803), while recognizing that size and temperature are relevant factors, also explicitly points out that these two stars indeed can have the same luminosity, as demonstrated on an H-R diagram:

A sample "x" response (from student ), discussing factors other than size and temperature: