The stars outside the Solar System have parallaxes that are impossible to observe without a telescope because

Question

BlackTom AI · Accepted Answer

Parallax angle decreases inversely with distance: p ≈ baseline / distance. For stellar (annual) parallax the relevant baseline is 1 AU (astronomical unit, the mean Earth–Sun distance), and the parallax p (in arcseconds) relates to distance d (in parsecs) by p(") = 1 / d(pc). Human vision can resolve angles of order 1 arcminute (≈60 arcseconds), while even the nearest stars have parallaxes much smaller than that.

Step-by-step reasoning and calculations:
1. Relevant constants: 1 AU = 1.496 × 10^11 m; 1 light year = 9.461 × 10^15 m; 1 radian = 206,265 arcseconds.
2. Take the nearest star Proxima Centauri at ≈4.24 light years. Convert distance to meters:
   d = 4.24 ly × 9.461×10^15 m/ly = 4.01×10^16 m.
3. Small-angle approximation for parallax using baseline = 1 AU:
   p (radians) ≈ 1 AU / d = (1.496×10^11 m) / (4.01×10^16 m) = 3.73×10^−6 rad.
4. Convert to arcseconds:
   p(") = 3.73×10^−6 rad × 206,265 arcsec/rad ≈ 0.77 arcseconds.
5. Compare to naked-eye resolution (~60 arcseconds): 0.77" << 60", so the shift is far too small to be detected without optical magnification and precision instruments. More distant stars have even smaller parallaxes (e.g., a star at 10 pc has p = 0.1").

Why the correct answer is correct:
- The tiny parallax angles arise because stellar distances are enormous compared to the Earth–Sun baseline; therefore the apparent positional shift is minute and requires telescopes and precise detectors to measure.

Why the other options are incorrect:
- "The semi-major axis of the Earth's orbit around the Sun is so large." If the baseline were large it would increase parallax; the baseline is actually small relative to stellar distances, not the reason given.
- "The Earth–Moon distance is much smaller than the Earth–Sun distance." True but irrelevant — using the Moon as baseline would produce even smaller parallax, and this statement does not explain why stellar parallaxes are unobservable by eye.
- "The size of the Sun is much larger than the size of the Earth." The physical sizes are unrelated to parallax measurements.

Conclusion: Because stars are extremely far away, their parallaxes are too small to observe without a telescope. This confirms the correct answer: "the stars are so far away."

The stars outside the Solar System have parallaxes that are impossible to observe without a telescope because单项选择题

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