Astronomy midterm question: same or different distance stars?

Astronomy 210 Midterm 2, spring semester 2015
Cuesta College, San Luis Obispo, CA

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

SM: If Star A has a dimmer apparent magnitude and a brighter absolute magnitude, and Star B has a brighter apparent magnitude and a dimmer absolute magnitude, they could not be the same distance from us.

Discuss why this reasoning is correct, and how you know this. Explain using the relationships between apparent magnitude, absolute magnitude, and distance.

[*] answers.yahoo.com/question/index?qid=20150326221032AADoWdf .

Solution and grading rubric:

• p:
  Correct. Understands difference between apparent magnitude m (brightness as seen from Earth, when placed at their actual distance from Earth) and absolute magnitude M (brightness as seen from Earth, when placed 10 parsecs away), and discusses how star A and star B cannot be located the same distance away from Earth by either arguing:
  1. how star A (which seems dim (m) but is actually bright (M) when placed at 10 parsecs away) must be located farther than 10 parsecs away, while star B (which seems bright (m) but is actually dim (M) when placed at 10 parsecs away) must be located nearer than 10 parsecs, such that it is not possible for these two stars to located the same distance away from Earth; or instead
  2. in order for star A and star B to both be located closer than 10 parsecs (or farther away than 10 parsecs), then both their apparent magnitudes (m) must be brighter (dimmer) than their respective absolute magnitudes (M). Since this is not true, then it is not possible for these two stars to located the same distance away from Earth, as they cannot be both be closer or farther away than 10 parsecs.
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors.
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. At least discussion demonstrates understanding of relationships between apparent magnitudes, absolute magnitudes, and distances.
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. At least attempts to use relationships between apparent magnitudes, absolute magnitudes, and distances.
• x:
  Implementation/application of ideas, but credit given for effort rather than merit. Discussion based on garbled definitions of, or not based on proper relationships between apparent magnitudes, absolute magnitudes, and distances.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Section 30674
Exam code: midterm02N4ra
p: 27 students
r: 4 students
t: 3 students
v: 4 students
x: 0 students
y: 0 students
z: 0 students

Section 30676
Exam code: midterm02s0Ju
p: 34 students
r: 9 students
t: 6 students
v: 0 students
x: 0 students
y: 0 students
z: 0 students

A sample "p" response (from student 4354):