SCIENCE · INVESTIGATE AND EXPLAIN

Build an evidence-limited cosmic argument

Separate historical observations, modern distance evidence and synthetic mechanisms.

Learning outcome: Defend a bounded expansion argument without circular distance evidence or unsupported age claims.

Starting recap

A Hubble time is not automatically a cosmic age; six synthetic points do not establish acceleration.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At H = 70 km/s/Mpc, the inverse-rate timescale is approximately 13.968 Gyr.

Open the experiment

Predict

Does moving the observer in a uniformly scaled grid select a unique center?

Your prediction and reason

Experiment

Construct a four-column evidence ledger and translate the observer in a paper scaling experiment.

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

Sort claims by the evidence supporting them

Create four paper columns: historical observations, modern distance evidence, controlled model and unmeasured claims. Place every result you use into one column before writing a conclusion. A model can explain how calibration and local motion affect an inference without itself becoming evidence that the real universe behaves that way.

The required historical panel contains all 24 Table 1 pairs from Hubble’s 1929 paper. It shows an overall positive relation with substantial scatter and five negative velocities. This is genuine historical evidence, but its distance scale is obsolete, its individual uncertainties are unavailable here, and some entries share a group distance. Do not turn these limitations into a claim that the observed pattern has no scientific value.

Preserve the historical analysis boundary

Our through-origin and free-intercept fits describe the uncorrected Table 1 values. They do not reproduce the paper’s solar-motion-corrected Figure 1 or its joint fitted solution. Nor do they provide a modern expansion-rate measurement. Table 2’s velocity-derived distances are excluded because using them to verify the same assumed distance–velocity law would be circular.

Write a supported sentence about the historical pattern and a separate limitation sentence. Then add what the fictional six-row experiment teaches: independent wavelength standards distinguish additive calibration error from a common fractional shift, and alternative distance assumptions change descriptive slopes. That exercise is original synthetic data, not another observed sample.

The 2019 NGC 4258 maser result supplies a modeled geometric distance with separate statistical and systematic uncertainties. It illustrates a method not obtained by inverting an assumed redshift law. Do not combine its modern 7.576 Mpc distance with the historical Table 1 velocity of 500 km/s to announce a modern Hubble constant. Shared identity does not erase differences in measurement dates, calibration or reference-frame context.

Test whether a recession pattern selects a center

Use an optional paper grid with points at coordinates 0, 1 and 3. Scale every coordinate by 1.1 to get 0, 1.1 and 3.3. Relative to the first point, separations grow by ten percent. Now put the observer at the original middle point and compare signed differences: −1 and 2 become −1.1 and 2.2.

The same fractional separation change appears from that observer too. Translating the coordinate origin does not select one of these observers as a privileged center. This is a schematic geometry demonstration, not a model of the universe’s edge, finite size or complete spacetime. It does not imply gravitationally bound galaxies, solar systems or local rulers simply stretch according to the plotted relation.

Calculate a timescale without declaring an age

The inverse of an expansion rate has time units. For H = 70 km/s/Mpc, convert one Mpc to approximately 3.085678 × 10¹⁹ km, then divide by 70 km/s. Dividing the resulting seconds by 31,557,600 seconds per year gives about 13.968 billion years.

This is a Hubble time. A present rate alone does not supply the past expansion history, so it is not automatically the universe’s age. Hogg’s cosmographic definitions distinguish these concepts. The lab cannot establish acceleration, dark energy, curvature or the whole universe’s size from these limited data.

Construct a report with a discriminating next step

Combine calibrated spectral evidence with independently estimated distance in your final argument. State which sample is observed, which is fictional and which assumptions control the result. Identify a systematic limitation and propose an additional observation that could distinguish competing explanations. More points derived from the same assumed relation do not remove circularity; replotting a dataset does not create independent evidence.

Transfer: reject two unsupported extensions

A report claims that the fictional six points uniquely prove dark energy. It then estimates distance by dividing redshift proxy by an assumed H and calls the resulting straight line independent confirmation of that H. Repair both arguments before opening the response.

Check the expansion-claims transfer

The synthetic sample demonstrates a stipulated local relation and cannot establish dark energy or expansion history. Distances constructed from the same assumed H make the straight-line check circular. Use independently constrained distances and actual observations, retain calibration and motion limitations, and seek additional evidence appropriate to acceleration. This public worked response is guided practice, not independent mastery.

Apply

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

Name an independent next constraint and preserve historical versus modern measurement contexts.

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: Separate common calibration from scatter Next: Explain melting without a global molten layer

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.