A rotation age is a distribution on a chosen scale

Start with the distribution of stars at a given age

Let $t$ denote age in gigayears, $T$ effective temperature and $P$ an observed surface rotation period in days. An empirical calibration predicts a distribution $p(P\mid t,T)$. Stars of the same adopted age and temperature need not all have one identical period. Initial histories, surface patterns and unknown measurement errors can widen a sequence.

All logarithms in this chapter are natural logarithms. The lognormal width is the standard deviation of natural log period.

Our teaching calibration fits separate robust cubic curves in log period against temperature for Praesepe, NGC 6811 and Ruprecht 147. Their fixed ages are 0.67,1.0 and 2.7 Gyr. The temperature-limited fits use 64,103 and 24 stars respectively. We interpolate mean log period and an effective log width linearly in log age. We do not force the curves to have a universal age exponent or erase crossings.

The predictive distribution is lognormal over the declared 0.5–60 day period support. Its effective width is the larger of the residual MAD scale and an explicitly imposed one-day young-sequence or two-day old-sequence width converted into log period. This mixes astrophysical and unknown training-error dispersion. It is a declared predictive prescription, not a measurement of intrinsic scatter.

Only the actual common temperature interval 4245.37–5832.12 K is supported by all three anchors. NGC 6819 supplies a warm-star whole-cluster holdout. Extending its warm coverage to cool stars would require an unvalidated extrapolation. Its warm-star data remain a whole-cluster holdout.

Reading: Curtis et al. (2020), When Do Stalled Stars Resume Spinning Down? Advancing Gyrochronology with Ruprecht147, arXiv2010.02272v1 · Bouma, Palumbo & Hillenbrand (2023), The Empirical Limits of Gyrochronology, arXiv2303.08830v1

Use the light likelihood once

The saved light curve supplies the marginal likelihood $Z(g)$ from the previous chapter. For clock hypothesis $m$, the trial frequency is $g=m/P$. A temperature measurement supplies a Gaussian likelihood $L_T(T)$ with its stated standard error. We condition its temperature integral on the actual common support; if the measured central temperature lies outside that support, this implementation declines an age.

The period plot has already multiplied the likelihood by a period prior. Feeding that posterior straight into the age integral would count that prior again. The implementation instead keeps the original frequency likelihood measure and combines it directly with the calibrated period distribution.

For numerical integration, put $u=\log g$. Since $P=m e^{-u}$, the magnitude of $dP$ is $Pdu$. That factor cancels the inverse-period term in the lognormal. Positive weights integrate the remaining Gaussian in log period against $Z(g)du$. An independent integral and a doubled grid check the same age curve.

The simulation settings sit outside this calculation. The injected equatorial period and latitude may be useful for validating a synthetic test, but an age inferred from saved light cannot read them. The empirical cluster sequences retain observed rotation periods, including their surface-pattern and unmeasured-latitude effects.

Choose the prior and normalize the result

An age likelihood is not yet an age probability density. Multiply by a proper age prior and normalize over the supported 0.67–2.7 Gyr interval. A uniform-age prior and a uniform-log-age prior are both explicit choices, with different weights inside that finite interval.

In the default synthetic record, the ten-day and twenty-day clock alternatives favour different parts of the empirical sequences. Under a uniform-age prior, the central 95% interval is roughly 0.725–2.67 Gyr. It spans a broad, structured density. A single median would be a poor description of that shape.

A central 95% interval leaves 2.5% of probability below it and 2.5% above it. When the density has two separated peaks, that interval also crosses the low-probability space between them; read its shape as well as its endpoints.

Switching to the log-age prior gives younger ages more weight. The same saved light now leads to a different age density. This is a visible sensitivity to a stated population assumption, especially useful when the data leave separated period alternatives. The period likelihood itself stays unchanged.

A stalled clock gives little leverage on age

Compare a cool slice of the cluster diagram. If the typical period hardly changes between two ages, a precise period can still match both. The age likelihood becomes broad along that nearly stalled segment. Reducing measurement noise does not restore information that the mean relation barely contains.

An intentionally simpler test makes the problem concrete. Fit a Praesepe temperature–period curve and impose a separable period growth law with exponent 0.5 or 0.62. For 27 NGC 6811 stars at 4200–4600 K, median inferred ages are about 0.709 or 0.702 Gyr, against the fixed 1 Gyr reference. At 5500–6000 K,27 stars give about 1.16 or 1.04 Gyr. One global exponent does not remove the temperature-dependent mismatch.

The nonseparable calibration preserves that changing leverage. Its model-generated validation draws 400 ages and periods from the declared prior and predictive model. The central 95% and 68% intervals include their generated ages in 94.75% and 67.0% of draws. That checks the stated conditional calculation. Real field stars can violate the model, selection or external age scale.

Reading: Curtis et al. (2019), A Temporary Epoch of Stalled Spin-Down for Low-Mass Stars: Insights from NGC6811 with Gaia and Kepler arXiv1905.06869v3 · Curtis et al. (2020), When Do Stalled Stars Resume Spinning Down? Advancing Gyrochronology with Ruprecht147, arXiv2010.02272v1

Separate uncertainty sources before quoting a number

Measurement errors belong to saved fluxes and temperature. The predictive width describes effective period dispersion at given temperature and age. Resampling training stars changes the fitted sequence. External age-scale uncertainty moves the cluster anchors themselves. These are different experiments and do not collapse into one training-bootstrap interval.

The interface includes a shared ten-percent older or younger rescaling of all anchors as a sensitivity scenario. It rescales the age axis without pretending that each cluster provided an independent random error of exactly that size. Absolute age precision would require a separately justified model of the external scale and its correlations.

For NGC 6811, a published rotation-based age cannot serve as an independent check of another rotation clock. Eclipsing-binary component ages also share stellar-model assumptions; discrepant components should not be multiplied as independent truths. Even the adopted NGC 6811 extinction in the source work was refined using a rotation model, so our imported temperature scale carries that dependence.

Reading: Curtis et al. (2019), A Temporary Epoch of Stalled Spin-Down for Low-Mass Stars: Insights from NGC6811 with Gaia and Kepler arXiv1905.06869v3 · Sandquist et al. (2016), eclipsing-binary constraints on NGC6811, arXiv1608.03917v1

Read a boundary as a request for more information

If more than 15% of probability lies in either outer 5% of the supported age interval, the Lab displays a calibration-edge notice. A rising curve at 2.7 Gyr can mean that older ages outside the calibration would still fit. Reporting 2.7 Gyr as a physical upper bound would answer a question the supported model never tested.

Temperature uncertainty near an edge is similarly conditioned on the available support. The retained likelihood fraction is recorded. For a weak signal, a changing pattern or too few rotations, the analysis withholds an age before entering this stage. Better observations, broader validated calibrators or a different surface model can then address the particular missing information.