Astronomy midterm question: example of a cooler star smaller than a hotter star?

Astronomy 210 Midterm 2, spring semester 2017
Cuesta College, San Luis Obispo, CA

An astronomy question on an online discussion board[*] was asked and answered:

Pdg: What is an example of a cooler star being smaller than a hotter star?
nin: The sun and a red star like Barnard's star.

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=20170301055021AAZuNwb.

Solution and grading rubric:

• p:
  Correct. Uses Wien's law to determine that the sun would be the hotter (yellow) star, while Barnard's star would be the (red) cooler star. Then uses the Stefan-Boltzmann law and/or interprets H-R diagram to demonstrate how Barnard's star would need to be a red dwarf in order to be cooler and smaller than the sun, a medium-mass main-sequence star.
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. Compares two stars (hotter, larger vs. cooler, smaller), where the hotter star is more luminous than the cooler star, but does not explicitly compare the sun versus a red (dwarf) star.
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. At least discussion demonstrates understanding of Wien's law, but the Stefan-Boltzmann law and/or H-R diagram discussion is garbled, with a hotter, larger sun having the same luminosity as a cooler, smaller red (dwarf) star; or may have erroneously claimed that the two stars have the same temperature, but Stefan-Boltzmann law and/or H-R diagram discussion is consistent with this mistake in Wien's 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, the Stefan-Boltzmann law, and/or H-R diagram.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Discussion not clearly based on Wien's law, the Stefan-Boltzmann law, and/or H-R diagram.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 30676
Exam code: midterm02sL0w
p: 22 students
r: 4 students
t: 9 students
v: 5 students
x: 3 students
y: 0 students
z: 0 students

A sample "p" response (from student 3428):