Compare redshift with independent distance
Analyze the complete historical observed panel before contrasting a controlled fictional sample.
Learning outcome: Describe an overall relation while retaining scatter, negative velocities and historical limits.
Starting recap
Table 1 is uncorrected measured velocity, not corrected Figure 1 or a modern Hubble-constant sample.
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Try
The historical sums give a descriptive through-origin slope about 423.937 per historical Mpc.
Open the experimentPredict
Can negative nearby velocities coexist with a positive overall relation?
Experiment
Inspect all 24 observed rows, compute a slope from sums, compare the two historical fits and then examine fictional A and F.
Compare galaxy spectra, distance and expansion evidence
Fictional controlled comparisons · separate historical observations · Six original fictional records with artificial vacuum references. Low-redshift cz is a proxy, not an exact cosmological speed. Distance and peculiar-motion bounds are stipulated feasible ranges, not confidence intervals. A separate historical observed panel preserves all 24 raw Hubble 1929 Table 1 records. Neither fit is a current H0 estimate; Hubble time is not cosmic age.
Reset keeps your prediction, captures and practice history.
A galaxy is a gravitationally bound system of stars and other matter. Disk galaxies can include a central bulge and an extended halo; gas, dust and stellar populations need not share the same shape. Smooth elliptical and irregular appearances also occur. A viewing angle can change the appearance of one object without changing its physical identity. These schematic regions do not establish a galaxy's distance, age, stellar count or dark-matter distribution.
Try sketching the same disk from two directions, then list what a picture alone cannot determine. Morphological categories are not a universal evolutionary sequence. The NASA galaxy overview supplies background; the drawings here are original teaching constructions.
Controlled experiment: six fictional records
Selected D: Irregular. Appearance is supplied, not classified by the browser. All six records are original teaching cases. Selecting one highlights its evidence; it does not change the sample. The separate historical panel below supplies actual published observations.
The independent-method model distance is 40.00 Mpc and angular diameter is 15 arcseconds. With the small-angle local approximation, projected diameter is 2.90888 kpc. Mpc means million parsecs; kpc means thousand parsecs. One arcsecond is 1/3600 of a degree. These deliberately equal-size examples do not establish a real standard ruler.
| Reference nm | Raw nm | Corrected nm | Used nm | Fractional shift z |
|---|---|---|---|---|
| 400 | 406.002769 | 404.002769 | 404.002769 | 0.010006923 |
| 500 | 507.003461 | 505.003461 | 505.003461 | 0.010006923 |
| 600 | 608.004154 | 606.004154 | 606.004154 | 0.010006923 |
Consistent virtual-line shifts under the selected calibration; fictional local approximation. Accepted common z: 0.010006923; accepted cz proxy: 3000.000 km/s. The 10⁻¹⁰ consistency tolerance suppresses arithmetic noise in exact constructed lines; it is not instrument precision or a confidence interval.
| Record | Distance Mpc | Distance bounds Mpc | Accepted cz km/s | cz/d (km/s/Mpc) | Residual km/s | Feasible H km/s/Mpc |
|---|---|---|---|---|---|---|
| A | 5.00 | 4.50–5.50 | 50.000 | 10.000 | -303.093 | 0.000–77.778 |
| B | 10.00 | 9.00–11.00 | 1000.000 | 100.000 | 293.814 | 63.636–144.444 |
| C | 20.00 | 18.00–22.00 | 1100.000 | 55.000 | -312.371 | 36.364–77.778 |
| D (selected) | 40.00 | 36.00–44.00 | 3000.000 | 75.000 | 175.258 | 61.364–91.667 |
| E | 60.00 | 54.00–66.00 | 4000.000 | 66.667 | -237.113 | 56.061–79.630 |
| F | 80.00 | 72.00–88.00 | 5800.000 | 72.500 | 150.515 | 62.500–84.722 |
Descriptive slope H = 70.61856 km/s/Mpc. The calculation minimizes unweighted squared proxy residuals while forcing zero intercept. It is not an errors-in-variables analysis or a modern H₀ estimate. Wider bounds leave the central fit unchanged; multiplying all distances by 1.2 lowers the fitted slope by 1/1.2. A calibration change is not a new independent sample.
Connect the local approximation and conditional bounds
For each identified virtual feature, z = λused/λreference − 1. Multiplying by c = 299792.458 km/s expresses redshift as a proxy cz. The approximation cz ≈ Hd + u includes a prescribed line-of-sight peculiar-motion term u. It omits relativistic cross terms and distinctions between distance definitions. Neither cz as a speed nor cz/H as a distance is generally valid at high redshift without more cosmological assumptions.
The fitted slope is Σ(d × cz)/Σd²; Σ means add over all six records. The stated distance bounds and |u| ≤ 300 km/s give, for these positive proxies, Hlow = max(0, (cz − 300)/dhigh) and Hhigh = (cz + 300)/dlow. This is compatibility with rectangular assumptions and H ≥ 0, not a probability statement. No observational error bars are inferred from decimal precision.
Projected diameter in kpc ≈ 1000 × distance in Mpc × angular diameter in arcseconds / 206264.806. This uses the small-angle approximation and local angular-diameter distance. The distance and angular-size calculation remains available even when spectral inference fails.
For purely cosmological redshift of comoving sources, 1 + zcos = aobserve/aemit, where a is cosmic scale factor. A measured redshift may contain peculiar motion and other contributions, so this exercise's mixed proxy does not uniquely measure that scale-factor ratio.
Inspect how the fictional sample was constructed
The six central proxies were prescribed as 70 × distance plus offsets −300, +300, −300, +200, −200 and +200 km/s. These offsets are not random draws or population statistics. The model distances were supplied independently of the resulting proxies. Artificial references at 400, 500 and 600 nm are multiplied by 1 + z, then a +2 nm instrument offset is added. All six angular sizes were chosen to give the same nominal projected diameter. These choices teach controlled comparisons; they do not measure real expansion.
Observed reference: Hubble's 1929 Table 1
These 24 published historical distance–velocity pairs are observations, separate from the fictional calibration experiment. Its controls do not alter this panel. Distances use the obsolete historical calibration; velocities are the table's measured values before the fitted solar-motion correction used in the paper's Figure 1. These descriptive fits are neither a reproduction of that corrected figure nor a modern expansion measurement.
| Object | Historical distance Mpc | Measured velocity km/s | Group note |
|---|---|---|---|
| Small Magellanic Cloud | 0.032 | 170 | — |
| Large Magellanic Cloud | 0.034 | 290 | — |
| NGC 6822 | 0.214 | -130 | — |
| NGC 598 | 0.263 | -70 | — |
| NGC 221 | 0.275 | -185 | — |
| NGC 224 | 0.275 | -220 | — |
| NGC 5457 | 0.45 | 200 | — |
| NGC 4736 | 0.5 | 290 | — |
| NGC 5194 | 0.5 | 270 | — |
| NGC 4449 | 0.63 | 200 | — |
| NGC 4214 | 0.8 | 300 | — |
| NGC 3031 | 0.9 | -30 | — |
| NGC 3627 | 0.9 | 650 | — |
| NGC 4826 | 0.9 | 150 | — |
| NGC 5236 | 0.9 | 500 | — |
| NGC 1068 | 1 | 920 | — |
| NGC 5055 | 1.1 | 450 | — |
| NGC 7331 | 1.1 | 500 | — |
| NGC 4258 | 1.4 | 500 | — |
| NGC 4151 | 1.7 | 960 | — |
| NGC 4382 | 2 | 500 | Common Virgo distance |
| NGC 4472 | 2 | 850 | Common Virgo distance |
| NGC 4486 | 2 | 800 | Common Virgo distance |
| NGC 4649 | 2 | 1090 | Common Virgo distance |
| Description | Slope | Intercept km/s |
|---|---|---|
| All 24, forced through origin | 423.93732 | 0 imposed |
| All 24, fitted intercept | 454.15844 | -40.78365 |
| Grouping sensitivity: omit four Virgo rows | 446.35201 | 0 imposed |
The full sample remains the default evidence. The four common-distance objects are not automatically four independent distance calibrations. The disclosed omission is a sensitivity comparison, not a way to choose a preferred result. Neither fit models distance error, sample selection or the author's solar-motion solution.
Reproduce the historical arithmetic and provenance
For distance r and velocity v, the through-origin descriptive slope is Σrv / Σr². Here n = 24, Σr = 21.873, Σv = 8955, Σr² = 29.517795 and Σrv = 12513.695. Each Σ means add over all rows. Dividing 12513.695 by 29.517795 gives about 423.9373. A free-intercept line also estimates where the line crosses the velocity axis; changing that assumption changes its slope.
Table 1 used stellar/luminosity distance criteria; Table 2 instead inferred distances from velocity and is excluded. Using the latter to verify the same velocity–distance relation would be circular. The table is not a random, volume-complete survey. Scatter and missing individual error estimates limit inference.
Primary scan, printed page 169 · Original paper DOI. Retrieved 2026-10-26. The numerical transcription preserves every row and sign; normalized identifiers replace historical typography. Local record SHA-256: c7d41a68838eec76e7bcb539c77f2db39a28aa304cd8d376b86abce84c746ff5.
Calculate Hubble time without claiming cosmic age
For a hypothetical present rate of 70 km/s/Mpc, 1/H corresponds to 13.96846 Gyr using 1 Mpc = 3.085677581491367 × 10¹⁹ km and a Julian year of 31557600 seconds. This is Hubble time, not the age of the universe. A present rate does not determine the past expansion history. Neither these six fictional cases nor the historical table establishes dark energy, a cosmic edge, a privileged center or the entire Big Bang model.
For scale-factor and distance definitions, see Hogg's redshift derivation. Bound galaxies and solar systems do not simply grow according to the background expansion relation.
Fictional six-record descriptive slope 70.6186 km/s/Mpc; selected galaxy D has z=0.0100069.
Consistent virtual-line shifts under the selected calibration; fictional local approximation. Independent-distance angular-size estimate 2.9089 kpc. Historical observations are a separate fixed dataset; uncertainty envelopes are conditional, not probabilities.
Captures preserve the fictional galaxy selection and analysis assumptions. Historical observations remain fixed. Replay supplies conditional comparisons, not new observations or a current expansion-rate measurement.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Explain the obsolete distance scale, group dependence and why velocity-derived Table 2 distances would be circular evidence.
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Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
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