Lab 07: Hubble's Law: Distance, Redshift, and the Age of the Universe

Apparatus and Setup

You will use the classic five-cluster redshift-distance data set, ../data/hubble_diagram_clusters.csv (representative of the type of data compiled in early Hubble-diagram studies; see ../reference-log.md for spot-check status), to fit Hubble's law and estimate the age of the universe.

Materials and Data

Procedure

1. For each cluster, compute v/d in km/s/Mpc, and note how consistent the five ratios are with each other.
2. Compute a best-fit Hubble constant using H₀ = Σ(v×d) / Σ(d×d) (a line through the origin, since Hubble's law predicts v = 0 at d = 0), and show your sum-of-products and sum-of-squares calculations explicitly.
3. Convert your fitted H₀ (in km/s/Mpc) into a Hubble time in years, using 1 Mpc = 3.0857e+19 km and 1 year = 3.156 × 10⁽ s. Show the full unit conversion.
4. Compare your Hubble-time estimate to the independently measured age of the universe from the cosmic microwave background (about 13.8 billion years). State whether your estimate is higher, lower, or consistent, and explain one physical reason an exactly-constant-expansion-rate assumption (implicit in using 1/H₀ as an age) might not exactly match the true age.
5. Using your fitted H₀, predict the recession velocity of a cluster at 700 Mpc, and state whether such a velocity is physically sensible (compare to the speed of light).

Analysis and Uncertainty

The five v/d ratios you computed in Step 1 are not identical to each other, reflecting both measurement uncertainty and each cluster's own "peculiar velocity" (motion relative to the smooth cosmic expansion, caused by local gravitational interactions). State which cluster's v/d ratio deviates most from your fitted H₀, and explain why a single galaxy's or cluster's peculiar velocity matters more for nearby objects (like Virgo) than for very distant ones (like Bootes).

Deliverables

• Completed v/d ratio table for all five clusters.
• Fitted H₀ with the sum-of-products/sum-of-squares calculation shown.
• Hubble-time calculation with full unit conversion shown.
• 700 Mpc prediction and physical-sensibility discussion.
• Peculiar-velocity discussion identifying the most-deviant cluster.

Assessment Criteria

• Correct v/d arithmetic for all five clusters.
• Correct H₀ fit using the specified through-the-origin method, with visible sums.
• Correct unit conversion from km/s/Mpc to a Hubble time in years.
• Physically sound comparison to the CMB-based age and to the 700 Mpc extrapolation.

References and Provenance

• OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 29.1-29.3.
• Cluster data: materials/ASTR230/data/hubble_diagram_clusters.csv; classic pedagogical data set, logged with spot-check status in materials/ASTR230/reference-log.md.