Which of the following shows evidence of the existence of dark matter?

Question

BlackTom AI · Accepted Answer

Galaxy rotation curves reveal the key evidence: stars in the outer parts of spiral galaxies orbit at nearly constant speeds rather than slowing down.

1. Relevant concepts
- Newtonian circular motion: centripetal force = gravitational force.
- For a star orbiting at radius r with speed v, v^2/r = G M(r)/r^2, so v(r) = sqrt[G M(r)/r].
- If all mass were concentrated in the visible luminous disk, mass M(r) would approach a constant beyond the visible edge, predicting v ∝ 1/sqrt(r).
- Observations show flat rotation curves: v(r) ≈ constant at large r, implying M(r) keeps increasing with r — indicating additional, unseen mass (dark matter halo).

2. Step-by-step reasoning
- Write the orbital speed relation: v(r) = sqrt[G M(r)/r].
- Assume visible mass M_vis becomes approximately constant for r beyond the luminous disk. Then predicted v_pred(r) = sqrt[G M_vis / r] ∝ r^(-1/2) — speeds should decrease with radius.
- Observed rotation curve: v_obs(r) ≈ v0 (constant). Set v_obs^2 = G M(r)/r ⇒ M(r) = v_obs^2 r / G, so M(r) grows linearly with r at large radii.
- This increasing enclosed mass cannot be accounted for by observed stars and gas, so there must be a diffuse, non-luminous mass component (dark matter) extending beyond the visible galaxy.

3. Intermediate calculations (illustrative)
- If v_obs = 200 km/s at r = 20 kpc, convert units if desired and compute M(20 kpc) = v^2 r / G. Using G in appropriate units gives a much larger mass than luminous matter accounts for, demonstrating the discrepancy.

4. Why the correct answer is correct
- "Stars within galaxies move faster than expected" directly describes flat rotation curves and the mass discrepancy; this is classic, direct observational evidence requiring extra mass in galaxy halos — dark matter.

5. Why the other options are incorrect
- "The night sky appears dark" refers to Olbers’ paradox and cosmic expansion, not dark matter.
- "Far away galaxies appear dimmer than expected" is due to cosmological redshift and distance, not evidence for dark matter.
- "There are fewer galaxies than expected" involves galaxy formation and observational limits, not the dynamical evidence for unseen mass.

6. Conclusion
- The observed faster-than-expected stellar speeds in galaxies indicate extra unseen mass; this matches the provided correct answer: Stars within galaxies move faster than expected.

Which of the following shows evidence of the existence of dark matter?Single choice

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