Astronomy midterm question: star size comparison

Astronomy 10 Midterm 3, Fall Semester 2007
Cuesta College, San Luis Obispo, CA

Astronomy 10 learning goal Q8.3

[15 points.] Consider the blackbody (continuous) spectra of two different objects, on the same intensity versus wavelength graph shown below. Explain using Wien's law and/or the Stefan-Boltzmann law why object X is smaller than object Y.

Solution and grading rubric:

• p = 15/15:
  Correct. Wien's law says that Object X (peak intensity at a shorter wavelength) must be hotter than Object Y. The Stefan-Boltzmann law says that luminosity (related to the intensity of the peak) is proportional to size and temperature^4. Since Object X is hotter, in order for it to be less luminous than Object Y is for Object X to be much smaller than Object Y. May only implicitly utilize Wien's law in discussion.
  
• r = 12/15:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors.
  
• t = 9/15:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. At least understands 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.
• 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 distributions:
Section 1073
p: 14 students
r: 4 students
t: 14 students
v: 9 students
x: 0 students
y: 0 students
z: 1 student

A sample of a "p" response (from student 4888) is shown below:

Another "p" response appeals to analyzing the Stefan-Boltzmann equation (from student 7276):

A few students (7635 shown here), upon immediately receiving the midterm at the start of class, had written down from short-term memory on the cover page the six special comparison cases discussed in class using the Stefan-Boltzmann law: