Fill in the blank.

The light from a star can be reduced by the presence of dust between the observer and the star being viewed. The presence of this dimming would cause astronomers to calculate a distance that is __________ its true distance.

Question

Fill in the blank.

The light from a star can be reduced by the presence of dust between the observer and the star being viewed. The presence of this dimming would cause astronomers to calculate a distance that is __________ its true distance.

BlackTom AI · Accepted Answer

Interstellar extinction makes a star appear fainter, so an uncorrected distance estimate will be larger than the true distance.

1. Relevant knowledge points and concepts
- Flux follows the inverse-square law: observed flux F ∝ 1/d^2 for a given luminosity L.
- Astronomers use magnitudes and the distance modulus: m − M = 5 log10(d) − 5, where m is apparent magnitude, M is absolute magnitude, and d is distance in parsecs.
- Interstellar dust causes extinction A (in magnitudes), increasing the apparent magnitude: m_observed = m_without_dust + A. If extinction is not accounted for, the extra dimming is interpreted as extra distance.

2. Complete reasoning process step by step
- Start with the true relation including extinction:
  m_obs = M + 5 log10(d_true) − 5 + A.
- If extinction A is ignored when calculating distance, one sets
  m_obs = M + 5 log10(d_calc) − 5.
- Equate the two expressions and solve for d_calc in terms of d_true and A.

3. Calculations with intermediate steps
- From equality: M + 5 log10(d_calc) − 5 = M + 5 log10(d_true) − 5 + A.
- Cancel M and −5: 5 log10(d_calc) = 5 log10(d_true) + A.
- Divide by 5: log10(d_calc) = log10(d_true) + A/5.
- Exponentiate: d_calc = d_true × 10^(A/5).
- Since extinction A > 0 for dust, 10^(A/5) > 1, so d_calc > d_true.
- Example: if A = 1.0 mag, d_calc = d_true × 10^(0.2) ≈ 1.585 × d_true (about 59% farther).

4. Why the correct answer is correct
Because dust adds positive extinction A to the apparent magnitude, ignoring A leads to interpreting that extra faintness as coming from a larger distance via the distance modulus. The algebra shows d_calc is multiplicatively larger than d_true by factor 10^(A/5).

5. Why the other options are incorrect
- "Less than": would require the star to appear brighter than it really is (negative extinction), which is not caused by dust.  
- "The same as": only true if there is no dust (A = 0) or if extinction is correctly accounted for; not the general case described.

6. Conclusion
The calculated distance will be greater than the star’s true distance, confirming the provided answer: greater than.

Fill in the blank. The light from a star can be reduced by the presence of dust between the observer and the star being viewed. The presence of this dimming would cause astronomers to calculate a distance that is __________ its true distance. 单项选择题

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