FR07-C · Which collisions burn deuterium?

Stellar Reaction Rates

Which collision energies set a star’s nuclear reaction rate? Most reactions come from an energy window where enough fast nuclei are present and tunneling is still probable. Heating moves that window upward. A rate inferred outside the measured energies depends strongly on the assumed cross-section shape.

Heat a proton–deuteron ensemble and watch the reacting energies shift. Fit published LUNA measurements, then see how much of the thermal rate comes from energies outside those measurements.

Molar thermal rate $\begin{gathered}5.38\\\mathrm{cm^3\,mol^{-1}\,s^{-1}}\end{gathered}$

Gamow peak estimate $23.0\,\mathrm{keV}$

Rate inside measured span $47.0\%$

Conditional rate uncertainty $4.23\%$

From cross section to a thermal rate

Move the burning window

Drag the ring to heat the nuclei.

Temperature

T

100\,\mathrm{MK}

Highest fitted energy

E_{\max}

263\,\mathrm{keV}

Calibration uncertainty and model

Shared scale uncertainty

\delta_N

2.40\%

Coherent energy uncertainty

\delta_E

0.300\,\mathrm{keV}

The calculation fits a quadratic to selected LUNA 2020 S-factor measurements and integrates a nonrelativistic Maxwell distribution for bare proton and deuteron nuclei. The shared 2.4% calibration component and energy-shift uncertainty are teaching assumptions; the resulting conditional interval omits extrapolation-model uncertainty. Plasma screening, resonances, stellar abundances and a full reaction network are omitted. Encounters show quantiles of the reaction-weighted energy distribution with enlarged nuclei and arbitrary animation time.

The covariance split is illustrative: the fixed independent errors retain the reported statistical errors and the remaining systematic variance after a 2.4% common component is removed. The scale slider changes the assumed common component. Left and right arrow keys change temperature when the scene has focus.

Model assumptions

Published LUNA data · calculated thermal average

Which energies did the experiment reach?

Points: LUNA measurements. Solid: selected-data quadratic. Dashed: published cubic.

Gold: reaction contribution. Green: selected measured span.

Independent point errors

3.13\%

Shared scale

2.40\%

Energy calibration

1.54\%

Sources, units and published comparison

The points are the thirteen published LUNA S-factor measurements in Extended Data Table 1. Short error bars show statistical errors; long bars combine statistical and systematic errors. The solid quadratic uses selected LUNA points, and the dashed reference is the published multi-dataset cubic in Eq. 2.

The gold density is normalized by the full thermal integral. The adaptive plot retains more than 99.99% of its contribution. Green marks the span from the lowest to highest selected measurement; the tails outside this span use polynomial extrapolation.

The quoted interval conditions on the quadratic model and chosen covariance. It does not estimate uncertainty in the extrapolation shape. The published rate used additional datasets and a different fitted function.

R_{\mathrm{pub}}=5.75\,\mathrm{cm^3\,mol^{-1}\,s^{-1}}

Mossa et al., Nature 587 (2020), Extended Data Tables 1–2 · Frozen numerical tables and provenance · Independent thermal-rate comparison

Study nuclear structure, scattering and stellar rates

Start with the essentials

Focus question: Which collision energies set a star’s nuclear reaction rate?
One-sentence intuition: The reacting collisions lie between the plentiful slow nuclei and the scarce fast nuclei.

Core mathematical model

A cross section with the Coulomb suppression separated

\sigma(E)=\frac{S(E)}{E}e^{-\sqrt{E_G/E}}

Energy is measured in the centre-of-mass frame. The S factor carries the remaining nuclear energy dependence. Both nuclear charges are one; their reduced mass fixes the Gamow energy.

The molar Maxwellian rate

R=N_A\sqrt{\frac{8}{\pi\mu}}\frac{\int_0^\infty S(E)e^{-E/(k_BT)-\sqrt{E_G/E}}\,dE}{(k_BT)^{3/2}}

The integral combines tunneling and the thermal population. Consistent SI units are used inside the calculation, then the result is converted to cubic centimetres per mole per second.

Where the exponential weight peaks

E_0=\left[\frac{E_G(k_BT)^2}{4}\right]^{1/3}

This estimate assumes that the S factor changes slowly across the window. The plotted reaction contribution includes the fitted S factor, so its maximum can shift from this estimate.

Common difficulties

The rate needs the whole window

Typical misconceptionThe rate is the cross section evaluated at the average thermal energy.

Better mental modelReaction probability rises rapidly with energy, so a small high-energy population can matter greatly. Integrating over the distribution gives each collision its proper weight.

Shared errors survive more points

Typical misconceptionThirteen points reduce every uncertainty by the square root of thirteen.

Better mental modelIndependent counting errors can average down. One calibration shared by all thirteen measurements moves them together, so adding points does not remove that scale uncertainty.

Run the experiment

01
Heat the ensemble
Predict the shift, then drag the bright temperature ring or choose A hotter early universe.
What to observe: The reaction contribution moves to higher energy. The encounter energies follow its quantiles, and the measured fraction changes with the overlap.
02
Keep the same measurements, shorten the fit
Choose Stop measuring too soon, then raise Highest fitted energy gradually.
What to observe: The saved measured points stay fixed. Added points constrain the fitted curvature and reduce the amount of the rate inferred beyond their span.
03
Move every point together
Double Shared scale uncertainty without changing temperature. Compare the coloured uncertainty bars.
What to observe: The central rate stays fixed. Its shared-scale uncertainty doubles because the calibration shifts all points together.

Stellar Reaction Rates

Which energies did the experiment reach?