Start with the measured rows
Extended Data Table 1 has four columns: centre-of-mass energy, S factor, statistical standard error and systematic standard error. There are thirteen rows. Preserve those columns separately when you copy them. The frozen numerical file also records the manuscript address and SHA256, so a future calculation can identify the exact source that was checked.
For example, the row at gives , with statistical and systematic errors and . Adding their variances gives a marginal standard error of about . It does not specify the correlation with the next row.
The table contains processed cross-section information, not individual detected photons. It supports a fit of the published S factors. Reconstructing detector efficiency or background subtraction would require the acquisition and calibration records that are absent from this table.
Source: V. Mossa et al.
Keep the two fitted curves separate
In Figure 1, first find the filled LUNA points, then the published smooth fit. Methods Eq. 2 describes a cubic fit that also used other datasets. The teaching workbench makes a new quadratic fit to selected LUNA rows. Its solid curve and the published dashed reference answer related questions with different inputs.
A new fit should save the selected point indices, basis, weights and coefficient covariance. Here the diagonal weights retain the statistical errors and the part of each systematic variance left after an assumed 2.4% common component is removed. The common scale is propagated separately. This explicit decomposition is a teaching choice because the table does not supply a full covariance matrix.
Read the residuals against the original central values. A small residual in the measured interval tests how the chosen curve describes those points. It cannot tell you how the same polynomial behaves far below the lowest point. The lowest collision energies are especially relevant when the temperature is reduced.
Source: V. Mossa et al.
Check the integral in physical units
The reproducible benchmark integrates the published cubic, using the nuclear masses and SI conversions stated in the preceding chapter. Composite Simpson quadrature in logarithmic energy resolves the low-energy tail. A separate test instead integrates cross section times speed times Maxwell density on a uniform energy grid. Their different variables and integrands make the agreement useful as an implementation check.
At , the first integral gives . Extended Data Table 2 prints , a relative difference of about 0.206%. The seven checked temperatures range from to ; the largest relative difference is about 0.594%.
The mismatch is retained in the downloadable comparison. It has not been assigned to a particular cause. The published all-dataset covariance and its low and high rate columns would require a fuller analysis. These numbers check the central-rate scale, and the two independent numerical integrals agree much more closely with each other.
Source: R. Longland et al.
Ask which energies carried the answer
Return to the fitted thermal contribution, which is the S factor multiplied by both exponential weights. Shade the interval from the first selected energy to the last. Integrate inside the shading and divide by the full fitted rate. This fraction measures how much of that model calculation lies inside the sampled energy span.
At 100 MK, the new LUNA-only quadratic places about 47% inside that span. Its rate is about 5.38, lower than the separately integrated published cubic. At lower temperature the contribution moves below the first point. Different smooth low-energy continuations can then change the integral even when they describe the measured interval comparably well.
A pointwise error band is also insufficient to propagate a rate. Two curves can have identical uncertainty at each energy and different correlations across energies. A common normalization moves the entire integral in one direction; independent fluctuations partly cancel. Carry coefficient covariance or sample coherent whole curves before integrating.
State the result at the scale actually checked
The measurement is a published experimental result. The quadratic and its uncertainty decomposition are an instructional reanalysis of published numerical values. The independent quadratures are numerical evidence for the implemented integral. The exact reaction model outside the data, plasma effects and a complete nucleosynthesis calculation remain separate research questions.
For another reaction, begin by writing down the isotope pair, energy frame and data version. Identify every shared calibration before fitting. Save the selected measurements, parameter covariance and the integrand. Quote the energy coverage alongside the rate. These steps make it possible for another reader to repeat your calculation and see where new measurements would help.