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Sandbox Physics
Star cluster colours and populations

AC32 · L3

An age comparison needs a model

Sandbox Physics ·

An age comparison begins with a model that predicts the observation. Its age nodes, composition, passbands and phase support must be explicit. A curve that looks close to a scatter plot has not yet supplied an age probability.

More fuel can still run out sooner

A star has finite central fuel and continuously radiates energy. A larger birth mass supplies more matter, but its energy-production rate can grow faster still. Massive stars can leave central hydrogen burning earlier. In a coeval population, the mass range remaining near the main-sequence endpoint therefore changes with time.

tfuel∼Eavailable/Lt_{\rm fuel}\sim E_{\rm available}/L
This is a dimensional fuel-over-power picture, not an M67 age calibration. Luminosity, mixed fuel and structure evolve. The Lab does not turn this proportionality into a replacement age scale.

Colour predictions require stellar structure and evolution, followed by atmosphere and instrument passband calculations. Composition, mixing and boundary choices all enter. An age result therefore depends on more than the position of a line on a plot.

Reading: Reyes et al. (2024), a precise M67 isochrone; native ISOA data · PARSEC/CMD3.8 official input help and phase labels · MIST official isochrone interface and user documentation

Identify the actual curve

The public ISOA slice has one age, 3.95 Gyr, and native Gaia DR2 photometry. It tests pairing and selection within its supported population; an age slider cannot manufacture a second native age. Importing the specified complete PARSEC or MIST model set enables twelve actual returned age nodes, approximately 0.398–5.012 Gyr.

Both local sources use their original Gaia EDR3 columns, but their composition definitions differ. PARSEC fixes solar-scaled metallicity; MIST fixes solar iron abundance, with a different native metal mass fraction. Attributing every discrepancy solely to one code’s quality would confound composition and model choices.

Local synthetic components are drawn on 0.1–8 solar masses. When an age table lacks the required mass or phase, the record withholds that light. It does not turn an out-of-support component into a dark remnant. Source phase labels are approximate. An equal label is a useful refusal rule, not a mathematical certificate of continuity. The model retains source order and uses only valid same-phase mass intervals.

Reading: Reyes et al. (2024), a precise M67 isochrone; native ISOA data · PARSEC/CMD3.8 official input help and phase labels · MIST official isochrone interface and user documentation

Interpolation must earn its domain

Nearby ages may look similar, but that is not a validation. Turnoff and later stages can move rapidly; averaging curves at fixed mass can bridge a phase that has ended at one age.

An actual withheld-age experiment removed log age 9.5 and attempted to recover it from 9.4 and 9.6. Even with equal phase labels at all three locations, the maximum discrepancy among 324 supported masses was approximately 0.107949 magnitudes; another 158 masses were refused for support or phase. Continuous age motion was therefore rejected. The interface selects only the twelve native nodes.

The public ISOA slice has a separate retained failure. Alternating 48 withheld nodes produced a maximum 0.003018 magnitude discrepancy, beyond its prespecified 0.002 exploratory gate. The default instead draws its 98 native mass nodes with integrated mass-cell weights and takes their original light. The dashed single-star reference helps locate the source nodes; connecting them does not establish an accepted continuous prediction.

An independent check is useful precisely because it can change which operations remain allowed. Smooth plots, finite numbers and small average errors cannot erase a failed region.

Reading: Reyes et al. (2024), a precise M67 isochrone; native ISOA data · PARSEC/CMD3.8 official input help and phase labels

Define a distance on the diagram

The reader-local real-source tutorial makes a descriptive comparison. Divide colour residuals by 0.05 magnitudes and G residuals by 0.2 magnitudes, then find the nearest equal-phase line segment. These are chosen display scales, not measurement standard deviations estimated for each source.

r(s)=a+s(b−a),0≤s≤1\mathbf r(s)=\mathbf a+s(\mathbf b-\mathbf a),\qquad0\le s\le1
The candidate point moves from one endpoint to the other as the coefficient runs from zero to one. Minimize its squared distance to the scaled observation.
y=(c0.05 mag,G0.2 mag),s∗=clip⁡[0,1](y−a)⋅(b−a)∣b−a∣2\mathbf y=\left({c\over0.05\,\mathrm{mag}},{G\over0.2\,\mathrm{mag}}\right),\qquad s_*=\operatorname{clip}_{[0,1]}{(\mathbf y-\mathbf a)\cdot(\mathbf b-\mathbf a)\over|\mathbf b-\mathbf a|^2}
Transform the observation and segment endpoints with the same dimensionless scales. Minimizing squared distance along the segment gives the projection coefficient; clip it to the endpoints. Never connect different source phases into a usable segment.

A colour residual of 0.1 magnitudes and a G residual of 0.2 magnitudes give distance approximately 2.236 under these scales. That does not mean “more than two sigma”, and a chi-square significance does not follow. Changing scales, selection or multiplicity can move the descriptive minimum.

The fixed real-source comparison reaches the oldest tested age and the lowest tested distance modulus. A search boundary marks where the calculation stopped, not a data-derived age limit. This calculation supplies neither an interior optimum nor an age posterior.

Reading: Gaia DR3 documentation and archive · PARSEC/CMD3.8 official input help and phase labels

Probability needs a generative observation model

An age probability requires a model for birth masses, binary pairing, field contamination, correlated measurement errors, extinction and sample inclusion. Exponentiating a nearest-curve distance does not supply those missing distributions.

p(y∣selected,ϑ)=S(y) p(y∣ϑ)Z(ϑ),Z(ϑ)=∫S(y) p(y∣ϑ) dyp(\mathbf y\mid{\rm selected},\vartheta)={S(\mathbf y)\,p(\mathbf y\mid\vartheta)\over Z(\vartheta)},\qquad Z(\vartheta)=\int S(\mathbf y)\,p(\mathbf y\mid\vartheta)\,d\mathbf y
This conditional normalization is a requirement for future inference, not a likelihood implemented by the current Lab. Selection depends on observations, and its normalization generally depends on model parameters. Omitting it changes the relative weight of models with different sample-inclusion probabilities.

Even a normalized conditional likelihood does not remove shared uncertainty in the external age scale or stellar physics. Those need separate scenarios or systematics. Ages tied to the same calibration must not be multiplied as independent truths. This tool stops at recorded forward populations and descriptive source comparisons, which already reveal several tempting but invalid assumptions.

Reading: Gaia DR3 documentation and archive · Reyes et al. (2024), a precise M67 isochrone; native ISOA data

Exercises and explanations

  1. Can a DR2 curve be overlaid on DR3 data after an arbitrary constant shift and called a same-band comparison?

    Read the explanation

    No. Passband response and zero-point conversions require evidence and can depend on spectral shape. The public DR2 diagram remains self-contained; the real DR3 tutorial uses explicitly native EDR3 columns.

  2. A descriptive loss is smallest at the oldest grid node. Does the grid maximum become an age upper bound?

    Read the explanation

    No. Only the tested nodes were searched, with no exterior comparison. Extinction, membership and binary assumptions have not become a complete likelihood. Report the boundary minimum and leave age inference open.

  3. Two models give equal unconditional density at an observed point but have different probabilities of entering the sample. Must their selected-sample densities agree?

    Read the explanation

    No. Each density is divided by its own selection normalization. Being observed carries information; ignoring that denominator does not compare the probability of the same selected sample.

Try the prediction

Check when age selection becomes available

  1. The public model should show fixed 3.95 Gyr without a movable age slider.
  2. Import a specified local model, choose exact native nodes and check the saved logAge and passband label.

Keep missing support explicit

  1. At an older imported age, export a record and find a system with supported false and its reason.
  2. Do not interpret its withheld light as physically zero. Check the supported count and the modelled-light denominator.

Sources and calculation scope

  1. Reyes et al. (2024), a precise M67 isochrone; native ISOA data

    The original ISOA file supplies one 3.95 Gyr age. Our CC BY 4.0slice uses98 monotonic rows with central hydrogen fraction above0.001; its native absolute magnitudes areGaia DR2, notEDR3.

  2. PARSEC/CMD3.8 official input help and phase labels

    PinnedPARSEC 1.2S, solar-scaled composition, OBC andGaia EDR3Vega, native returned ages andAvnodes. Source labels are approximate; our equal-phase rule prevents bridging labelled boundaries but does not certify continuous evolutionary branches.

  3. MIST official isochrone interface and user documentation

    The local comparison usesMIST 1.2, MESA 7503, nonrotating, solar iron abundance, Av 0, and the originalGaia EDR3columns. Full original numerical hashes identify the twelve-node file; code licensing does not establish table redistribution permission.

  4. Gaia DR3 documentation and archive

    Astrometric reference epoch2016.0 is not a universal epoch for mean photometry. The tutorial uses a fixed cone and explicit astrometric cuts; source rows and real point plots are made only on the reader computer.

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