Astronomy midterm question: different temperature, same luminosity stars

Astronomy 210 Midterm 2, Fall Semester 2008
Cuesta College, San Luis Obispo, CA

[20 points.] Two stars have the same luminosity, but different temperatures. Explain whether the hotter star or the cooler star is larger in size, using the Stefan-Boltzmann law and/or an H-R diagram.

Solution and grading rubric:

• p = 20/20:
  Correct. According to the Stefan-Boltzmann law, the luminosity of a star is proportional to its size (surface area) and its temperature, raised to the fourth power. Thus for the cooler star to have the same luminosity as a hotter star, the cooler star must be much larger in size.
  
• 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. Discussion of concepts other than the Stefan-Boltzmann law, such as stellar evolution rates, pressure-temperature thermostat, etc.
  
• 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 70160
p: 21 students
r: 3 students
t: 0 students
v: 6 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response, graphically demonstrating the dependence of luminosity on size and temperature (from student 0228):

A sample "p" response, using an H-R diagram (from student 1186):

Another "p" response, using two specific stars as examples (from student 2020):

A "p" response organizing the Stefan-Boltzmann law in tabular form (from student 8187):