Astronomy midterm question: Acrux misprint?

Astronomy 210 Midterm 2, fall semester 2009
Cuesta College, San Luis Obispo, CA

Information regarding the star Acrux from an astronomy textbook[*] is shown below.

Apparent magnitude (m): +0.90
Distance: 80 parsecs (260 light years)
Absolute magnitude (M): +3.5

Discuss whether or not the absolute magnitude value of M = +3.5 for Acrux has been misprinted, and how you know this. Explain using the properties of apparent magnitude, absolute magnitude, and distance.

[*] Michael A. Seeds and Dana E. Backman, Perspectives in Astronomy, 1/e, Thomson Brooks/Cole (2008), p. 338 (Table A-6).

Solution and grading rubric:

• p:
  Correct. Apparent magnitude is how bright the star appears at its real distance of 80 parsecs away; absolute magnitude (its intrinsic brightness) is how the bright the star would be if brought to the "fair distance" of 10 parsecs. Bringing a star that is farther away than 10 parsecs to this fair distance should result in a absolute magnitude brighter, not dimmer than its apparent magnitude. Since Acrux's absolute magnitude M = +3.5 (at 10 parsecs) is dimmer than its apparent magnitude m = +0.90 (at 80 parsecs), the absolute magnitude must be incorrect (as it is expected to be brighter than +0.90). May instead argue that distance is misprinted, but at least recognizes discrepancy in how m, M, and d are related. (Cf. wikipedia.org/wiki/Alpha_Crucis, where Acrux has a distance of 99 ± 5 parsecs, an apparent magnitude of +0.77 and an absolute magnitude of –4.14.)
  
• r:
  Nearly correct (explanation weak, unclear or only nearly complete); includes extraneous/tangential information; or has minor errors. Does not clearly explain why d, m, M data is inconsistent, but at least indicates that m and M are "switched," the distance must be much smaller, etc.
  
• t:
  Contains right ideas, but discussion is unclear/incomplete or contains major errors. May switch definitions of m and M and/or state data is correct, but at least distinguishes between brightnesses that are seen and are intrinsic.
  
• v:
  Limited relevant discussion of supporting evidence of at least some merit, but in an inconsistent or unclear manner. Garbled definitions/relations between d, m, and M.
  
• x:
  Implementation/application of ideas, but credit given for effort rather than merit.
  
• y:
  Irrelevant discussion/effectively blank.
  
• z:
  Blank.
  

Grading distribution:
Section 70160
p: 27 students
r: 5 students
t: 6 students
v: 0 students
x: 3 students
y: 0 students
z: 1 student

A sample "p" response (from student 1543):
A sample "x" response (from student 6307):
Another sample "x" response (from student 6364):