SCIENCE · INVESTIGATE AND EXPLAIN

Recover a fractional shift after calibration

Compare virtual reference features with independently corrected wavelengths.

Learning outcome: Calculate consistent fractional shifts and distinguish them from additive instrument error.

Starting recap

The 400, 500 and 600 nm vacuum features are artificial, not real atomic transitions.

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Try

Corrected 404, 505 and 606 nm features each give z = 0.01.

Open the experiment

Predict

Does adding two nanometers to every line add the same fractional shift?

Your prediction and reason

Experiment

Compare D in uncorrected and calibrated cases; use both independent lamps and calculate each line ratio.

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.

Fictional galaxy: Galaxy D: Irregular Analysis assumptions: Calibrated wavelengths; baseline distances

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.

Selected virtual features; z = used wavelength / reference − 1
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.

Full six-record sample; intervals are conditional feasibility bounds, not probabilities
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.

All 24 original Table 1 pairs; uncertainty unavailable, not zero
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

Fits describe this historical table; units km/s per historical Mpc
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.

Observation to attach to a capture

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

Compare a shift with a fraction

Three line centers move toward longer wavelengths. Is that enough to accept one common redshift? Begin with fictional galaxy D and Uncorrected wavelengths; baseline distances. The three reference features are exactly 400, 500 and 600 nm in vacuum. They are an artificial reference pattern, not hydrogen, helium or another real element’s transitions.

A redshift z compares a wavelength difference with its reference wavelength: z = observed/reference − 1. It is dimensionless. The same fractional change stretches different reference wavelengths by different numbers of nanometers. An additive instrument offset, in contrast, moves every center by the same number of nanometers.

Calibrate with separate standards

The lab supplies two independent calibration standards: known 450 and 650 nm features measured at 452 and 652 nm. Subtract known from measured in each case. Both indicate a +2 nm additive offset. The correction is therefore raw wavelength minus 2 nm.

Make a paper table with columns reference, raw, correction and corrected. Standards must be independent of the unknown galaxy pattern: choosing a correction solely because it makes a desired galaxy answer appear would not be an independent calibration check. The standards here are exact teaching inputs, not a model of real instrument uncertainty.

For D, raw line centers are approximately 406.002769, 507.003461 and 608.004154 nm. After subtracting 2 nm they become approximately 404.002769, 505.003461 and 606.004154 nm. Retain enough digits while calculating, but do not call those displayed decimals observational precision.

Calculate three redshifts before accepting one

The corrected 400 nm feature gives 404.002769/400 − 1, approximately 0.010006923. The corrected 500 and 600 nm features give the same result within floating-point rounding. The common fraction is about one percent, not a common shift of one nanometer.

Before correction, the additive error contributes 2/400 = 0.005 to the first fractional shift, 2/500 = 0.004 to the second and 2/600 ≈ 0.003333 to the third. These fractions disagree. Do not average them into an accepted redshift simply because their mean looks plausible.

Switch to Calibrated wavelengths; baseline distances and compare your table with the lab. The uncorrected mode withholds accepted common redshift and downstream slope inference. Its missing result is an evidence failure, not zero expansion. The numerical consistency tolerance tests exact constructed values; it is not a confidence interval for real spectra.

Capture the raw and calibrated cases with a prediction explaining why subtraction repairs the common fractional pattern. Selection of a galaxy highlights an existing record; it does not generate an independent new observation.

Interpret the velocity-like number carefully

Multiplying z by c = 299792.458 km/s expresses redshift in velocity units. For D the result is 3000 km/s. Here cz is a low-redshift velocity proxy, not an exact general conversion from cosmological redshift to physical recession speed. All fictional cases are below z = 0.02, and the experiment uses a first-order local approximation.

Hogg’s redshift discussion relates genuine cosmological redshift to the ratio of cosmic scale factors at observation and emission. Actual observed shifts can also contain peculiar motions and other contributions. Do not insert a mixed observed shift into that relation as though every contribution had been separated. Real measurements also require identified physical transitions, wavelength conventions and reference-frame corrections absent from this virtual pattern.

The historical Hubble panel supplies published velocities, not wavelengths. We cannot pretend to apply these artificial three-line calculations to those observed records.

Transfer: a new corrected pattern

A paper-only pattern has corrected centers 404, 505 and 606 nm for references 400, 500 and 600 nm. Calculate each z and contrast the result with adding exactly 4 nm to every reference feature.

Check the cosmic-redshift transfer

Each corrected ratio gives z = 0.01. A common additive 4 nm shift instead gives 0.01, 0.008 and approximately 0.006667, which disagree. Multiplicative stretching and additive calibration error are different mechanisms. The exercise assumes the stated virtual references are correctly matched.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why inconsistent raw shifts withhold accepted inference rather than produce zero.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Identify a galaxy from complementary evidence Next: Compare redshift with independent distance

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