A simplified survey gives (distance in 10²³ m, recession speed in km/s) pairs (1, 220), (2, 440), (3, 660), (4, 880). (a) Use the gradient to estimate H₀ in s⁻¹. (b) Estimate the expansion timescale 1/H₀ in years, taking 1 year = 3.16 × 10⁷ s. (c) Explain how redshift supports expansion and why 1/H₀ is a model-based age estimate.

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Accepted Answer

H₀ = 2.2e-18 s⁻¹; 1/H₀ = 14400000000 years; increasing wavelength indicates recession and the estimate assumes a simple expansion history. The line's gradient is 220 km/s for every 10²³ m. Convert km/s to m/s: H₀ = 220000/10²³ = 2.2e-18 s⁻¹. It is a rate of expansion, not a speed. The reciprocal is a time: 1/H₀ = 454500000000000000 s. Dividing by 3.16 × 10⁷ s/year gives approximately 14400000000 years for the expansion timescale. Compared with the same spectral lines measured in a laboratory, light from receding distant galaxies has longer wavelengths. The general tendency for more distant galaxies to recede faster supports a Universe whose large-scale separations are increasing. Working backwards with the present distance-to-speed ratio gives an approximate age scale. It is not an exact clock reading: expansion can change with time, and actual astronomical measurements have uncertainty. These deliberately simplified data do not replace a modern measured cosmological parameter.

A simplified galaxy survey 3

What a complete response should do

Galaxy distance and recession

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Galaxy distance and recession — supplied values
Distance / 10²³ m	Recession speed / km s⁻¹
1	220
2	440
3	660
4	880