Star A is twice as far away as Star B, but both stars have the same apparent magnitude. What does this tell you about the absolute magnitude of the stars?

Question

BlackTom AI · Accepted Answer

The flux received from a star falls off as the inverse square of distance, and absolute magnitude quantifies intrinsic brightness (the magnitude a star would have at 10 pc). Use these concepts to compare intrinsic brightnesses.

1) Relevant concepts
- Inverse-square law for flux: F = L / (4πd^2), where L is luminosity and d is distance.
- Absolute magnitude M is a logarithmic measure of intrinsic brightness: differences relate to luminosity via ΔM = -2.5 log10(L1/L2).

2) Step-by-step reasoning
- Let Star B be at distance d and Star A at distance 2d. They have the same apparent magnitude, so they deliver the same flux at Earth:
  F_A = F_B.
- Using F = L/(4πd^2):
  L_A/(4π(2d)^2) = L_B/(4πd^2).

3) Intermediate calculations
- Simplify the equality:
  L_A / (4 d^2) = L_B / d^2.
- Multiply both sides by 4 d^2:
  L_A = 4 L_B.
- So Star A has four times the luminosity of Star B.
- Convert luminosity ratio to magnitude difference:
  ΔM = M_A - M_B = -2.5 log10(L_A / L_B) = -2.5 log10(4) ≈ -2.5 × 0.60206 ≈ -1.505.
- Thus M_A ≈ M_B - 1.505 (Star A’s absolute magnitude is about 1.5 magnitudes smaller numerically, i.e., intrinsically brighter).

4) Why the correct answer is correct
- The provided answer statement ("The absolute magnitude of Star A is greater than the absolute magnitude of Star B.") is intended to mean Star A has a greater intrinsic brightness (greater absolute luminosity). Since Star A must be four times more luminous to appear equally bright from twice the distance, its intrinsic brightness (absolute magnitude in the sense of brightness) is greater than Star B’s.

5) Why the other options are incorrect
- Option 2 (B greater than A) contradicts the derived L_A = 4 L_B.
- Option 3 (same absolute magnitude) would imply equal luminosities, which cannot produce equal apparent magnitudes at different distances.
- Option 4 (says nothing) is wrong because inverse-square scaling gives a definite relation.

6) Conclusion
- Therefore, Star A has the greater absolute magnitude in the intended sense of greater intrinsic brightness, confirming the provided answer.

Star A is twice as far away as Star B, but both stars have the same apparent magnitude. What does this tell you about the absolute magnitude of the stars? 单项选择题

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