Astronomy midterm question: relative distances of same absolute visual magnitude, different apparent magnitude stars

Astronomy 210 Midterm 2, spring semester 2013
Cuesta College, San Luis Obispo, CA

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

admin: If two stars have the same absolute visual magnitudes, but different apparent magnitudes, how do you tell which star is farther away?
tham158: The star with the greatest difference between apparent magnitude and absolute visual magnitude is farther away.

Discuss an example (two stars with the same absolute visual magnitudes, but different apparent magnitudes) that would show that the response given by "tham158" is not correct, and how you know this. Explain using the properties of apparent magnitude, absolute visual magnitude, and distance.

[*]Adapted from alliknowisthatiknownothing.com/how-to-tell-which-star-is-farther-away-with-magnitudes-given/.

Solution and grading rubric:

• p:
  Correct. Understands difference between apparent magnitude (m) values and absolute magnitude (M_V) values, and that for two stars that have the same absolute magnitude (same brightness if both located 10 parsecs away) it is possible for the star with the greatest difference between m and M_V to be closer (instead of farther away) if its m is brighter than M_V. May cite examples of one star being farther and one star being closer than 10 parsecs, or two stars both closer than 10 parsecs.
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. As (p), but example may use stars that do not have the same M_V.
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. Some serious attempt at a contrary argument or example where the star with the greater difference between m and M_V would be closer.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least attempts to use relations between d, m, and M_V somehow.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Discussion not based on apparent magnitudes, absolute magnitudes, and distances.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 30674
Exam code: midterm02nNm3
p: 6 students
r: 3 students
t: 6 students
v: 16 students
x: 0 students
y: 0 students
z: 0 students

Section 30676
Exam code: midterm02s5Sz
p: 9 students
r: 4 students
t: 6 students
v: 24 students
x: 0 students
y: 1 student
z: 0 students

A sample "p" response (from student 2981), discussing two stars with the same absolute visual magnitudes:
A sample "r" response (from student 1022), discussing two stars with different absolute visual magnitudes: