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Vol. Science · Astronomy · How stars shine

Fusion
and the mass defect

A star shines because the product nucleus weighs less than the pieces that made it. The missing mass is energy, E = Δm c². Pick a reaction. The desk multiplies AME2020 atomic masses by the 2022 CODATA energy of one atomic mass unit, 931.49410372 MeV. Four hydrogen atoms to helium-4 release 26.7310 MeV. Deuterium plus tritium releases 17.5893 MeV. The binding-energy curve climbs to the iron group and turns down. In that table ⁶²Ni is the highest point, 8794.5555 keV per nucleon, with ⁵⁶Fe at 8790.3563 keV and ⁵⁸Fe between them at 8792.2534 keV. The Sun’s nominal luminosity, IAU 2015 Resolution B3, is 3.828×10²⁶ W. That wattage is 4.259×10⁹ kg of mass each second, and the hydrogen that supplies it is 0.712% of its mass — 0.7% to one digit after the decimal. How long the fuel lasts is the stellar lifetime desk. Where the light sits on temperature and luminosity is the HR diagram. The spectrum of the hydrogen is the spectra desk. The QCD binding inside one proton, a different ledger, is the Standard Model desk.

Educational calculator. Not a stellar-evolution model, not a reaction network, and not a measured solar luminosity. The nominal luminosity is a conversion constant. Neutrinos are named and not subtracted. The hue of a star is not drawn here.

Interactive · Mass defect

pp net · atomic masses · E = Δm c²

Reaction

Opens on the pp net. 4 × 1.007825031898 u minus 4.00260325413 u is 0.02869687 u.

Masses in the pp net · AME2020 atomic masses
Side	Count	Nuclide	Atomic mass (u)	Mass excess (keV)
In	4	¹H	1.007825031898	7288.971064
Out	1	⁴He	4.00260325413	2424.91587

The buttons need JavaScript. The arithmetic is still Δm times 931.49410372 MeV per u. pp net, 4 ¹H → ⁴He: 0.02869687 u, 26.7310 MeV, 4.2828×10⁻¹² J, mass fraction 0.712% (0.7% to one digit). CNO net uses the same masses. D+T, ²H + ³H → ⁴He + n: 17.5893 MeV (17.6 MeV to one digit). D+D → ³He + n: 3.2689 MeV. D+D → ³H + ¹H: 4.0327 MeV. Triple-α, 3 ⁴He → ¹²C: 7.2747 MeV. Two ⁴He → ⁸Be is −0.0918 MeV. Hydrogen-1 is 1.007825031898 u. Helium-4 is 4.00260325413 u. Not a stellar-evolution model.

Binding energy per nucleon

Mass number across · MeV per nucleon up · ⁶²Ni is the top

Curve
⁵⁶Fe
⁶²Ni peak

Landmarks · or tap the chart

Each point is [Z m(¹H) + (A−Z) m(n) − m(A,Z)] c² / A, with the atomic masses on this page. The AME2020 column, in keV, is the check: ⁶²Ni 8794.5555, ⁵⁸Fe 8792.2534, ⁵⁶Fe 8790.3563.

The Sun’s power budget

Nominal luminosity · the slice is the mass defect

Hydrogen consumed
Mass converted

Luminosity · relative to the nominal Sun

L / L☉

The hydrogen bar is the full width. The converted-mass bar is 0.712% of that width, at every luminosity. Sliding changes the kilograms per second. It does not change the fraction. 0.712% is 0.7% to one digit after the decimal.

Nominal Sun · IAU 2015 B3

4.259×10⁹ kg/s

mass equivalent of 1 L☉

Power3.828×10²⁶ W
Mass equivalent4.259×10⁹ kg/s
Hydrogen5.983×10¹¹ kg/s
Helium5.941×10¹¹ kg/s
pp reactions8.938×10³⁷ /s
Mass fraction0.712%

Nominal luminosity 1 L☉ is 3.828×10²⁶ W. Divided by c², the mass equivalent is 4.259×10⁹ kg/s. If every joule of the pp-net 26.7310 MeV leaves as that light, the hydrogen consumed is 5.983×10¹¹ kg/s and the helium made is 5.941×10¹¹ kg/s. The mass fraction is 0.712%, which is 0.7% to one digit after the decimal. Neutrinos carry away part of the Q, so a real star fuses a little more hydrogen than this. No neutrino fraction is subtracted here, because this page does not cite one.

Two chains, then the heavy elements

The pp chain in the Sun. CNO in hotter cores. Neutrons after iron.

pp and CNO are the same net masses. Both turn four hydrogen nuclei into one helium-4. The CNO cycle uses carbon, nitrogen, and oxygen as catalysts and hands them back, so the mass defect on this page is one number, 26.7310 MeV. Adelberger et al. 2011 put the division in the standard solar model at 99% pp and about 1% from the CN cycle. Heavier main-sequence stars are powered mainly by CNO, because that energy generation rises much more steeply with temperature: about T¹⁸ against about T⁴ for the pp chain, at solar-core temperature. Their figure shows the Sun on the pp side. This page does not read a crossing temperature off that figure. NASA’s Sun facts put the core near 15 million °C, hot enough for hydrogen to fuse to helium.

The curve says which way pays. Binding energy per nucleon rises from deuterium to the iron group, so fusing light nuclei releases energy. It falls again toward lead and uranium, so fission of those heavy nuclei also releases energy. AME2020’s own column peaks at nickel-62, 8794.5555 keV per nucleon. Iron-58 is 8792.2534 keV. Iron-56 is 8790.3563 keV. The familiar iron peak is this neighborhood. Nickel-62 is the highest point in the column, not a rounding of iron-56.

Heavier than iron needs neutrons. Fusion stops being a profit past that peak. Burbidge, Burbidge, Fowler, and Hoyle (1957) named the two neutron-capture paths that build most of the heavier nuclei: the s-process, slow enough that a nucleus can decay before the next neutron arrives, and the r-process, rapid, in a flood of neutrons. They associated the slow path with giant stars and the rapid path with supernova conditions. A neutron-star merger has since been watched doing r-process work. Abbott et al. 2017 reported GW170817, and Pian et al. 2017 identified r-process material in its kilonova. This page does not split the heavy elements into percentages between mergers and supernovae.

This set · binding energy per nucleon

AME2020 atomic masses, binding energy recomputed with 931.49410372 MeV per u
Nuclide	A	Atomic mass (u)	BE / A
²H	2	2.014101777844	1.1123 MeV
³H	3	3.01604928132	2.8273 MeV
³He	3	3.01602932197	2.5727 MeV
⁴He	4	4.00260325413	7.0739 MeV
⁶Li	6	6.01512288742	5.3323 MeV
⁷Li	7	7.01600343426	5.6064 MeV
⁸Be	8	8.005305102	7.0624 MeV
⁹Be	9	9.012183062	6.4627 MeV
¹⁰B	10	10.012936862	6.4751 MeV
¹²C	12	12	7.6801 MeV
¹⁴N	14	14.00307400425	7.4756 MeV
¹⁶O	16	15.99491461926	7.9762 MeV
²⁰Ne	20	19.99244017525	8.0322 MeV
²⁴Mg	24	23.985041689	8.2607 MeV
²⁸Si	28	27.97692653442	8.4477 MeV
³²S	32	31.97207117354	8.4931 MeV
⁴⁰Ca	40	39.962590850	8.5513 MeV
⁴⁸Ti	48	47.947940677	8.7230 MeV
⁵²Cr	52	51.940504714	8.7760 MeV
⁵⁶Fe	56	55.934935537	8.7904 MeV
⁵⁸Fe	58	57.933273575	8.7923 MeV
⁵⁸Ni	58	57.935341650	8.7321 MeV
⁶²Ni	62	61.928344753	8.7946 MeV
⁸⁴Kr	84	83.91149772708	8.7174 MeV
¹²⁰Sn	120	119.902202557	8.5045 MeV
²⁰⁸Pb	208	207.976652005	7.8675 MeV
²³⁸U	238	238.050786936	7.5701 MeV

Literacy · the words on the readout

Mass defect: Δm is the atomic mass of the reactants minus the atomic mass of the products, in u. Energy is Δm times 931.49410372 MeV. Joules are that same Δm, in kilograms, times c². c is 299792458 m/s. One u is 1.66053906892×10⁻²⁷ kg.
Atomic masses: AME2020 tabulates neutral atoms. For these reactions the proton count matches on both sides, so the electrons match too, except the pp and CNO nets. There, four hydrogen atoms carry four electrons and helium carries two. The extra two electron masses are the annihilation partners of the two positrons.
0.7%: The pp-net Δm divided by four hydrogen masses is 0.71185…, which this page prints as 0.712%. To one digit after the decimal, that is 0.7%. It is not a separate constant. It is this ratio.
Binding energy: BE = [Z m(¹H) + (A−Z) m(n) − m(A,Z)] c². Divide by A for the curve. Hydrogen-1 is zero by that formula. The free neutron is not plotted. Carbon-12 is exactly 12 u, which is the definition of u, so its mass excess is 0.0 keV.
Iron-56 and nickel-62: AME2020 prints 8790.3563 keV per nucleon for ⁵⁶Fe, 8792.2534 keV for ⁵⁸Fe, and 8794.5555 keV for ⁶²Ni. This page’s recomputation, using the 2022 CODATA MeV per u, agrees with that column to well under 0.001 keV per nucleon. The chart’s 8.7904 MeV and 8.7946 MeV are those results rounded to four digits after the decimal. Nickel-62 is the highest value in the AME2020 binding-energy column. Iron-56 is the nucleus the classroom name points at. Both are marked.
What is not subtracted: Neutrinos leave the star. They take a share of the 26.7310 MeV that never becomes heat or light. Adelberger et al. discuss that bookkeeping for the solar model. This page does not cite a single average neutrino energy, so it does not subtract one. The hydrogen rate is the rate if the whole mass defect becomes the radiated power. A real Sun fuses a little more.

Sources · the masses, the constants, and the chains

Cited, not invented

Wang et al. 2021, Chinese Physics C 45, 030003 — the AME2020 atomic mass evaluation. Atomic masses, mass excesses in keV, and binding energy per nucleon. The machine-readable table is mass_1.mas20.txt at the IAEA Atomic Mass Data Center. Every mass and every keV figure on this page is a central value from that table. ¹²C is exactly 12 u. ⁶²Ni is 8794.5555 keV per nucleon. ⁵⁸Fe is 8792.2534 keV. ⁵⁶Fe is 8790.3563 keV. Uncertainties are in the table and are smaller than the digits this page prints for these Q-values.
NIST CODATA — atomic mass constant energy equivalent in MeV — 2022 adjustment, 931.49410372 MeV, standard uncertainty 0.00000029 MeV. This page multiplies Δm in u by that central value. The kilogram equivalent is 1.66053906892×10⁻²⁷ kg, standard uncertainty 0.00000000052×10⁻²⁷ kg. Joules are Δm in kilograms times c².
NIST — speed of light and elementary charge — exact defining constants, c = 299792458 m/s and e = 1.602176634×10⁻¹⁹ C. One electronvolt is e joules. The MeV path times e and the kilogram path times c² agree far inside the CODATA uncertainty.
IAU 2015 Resolution B3 (Prša et al. 2016, AJ 152, 41) — nominal solar luminosity 3.828×10²⁶ W, exact as a conversion constant, not a measurement of the Sun. The same constant is 1 L☉ on the stellar lifetime and HR desks. The resolution text is also at the IAU archive.
Adelberger et al. 2011, Reviews of Modern Physics 83, 195 — Solar Fusion II. In the standard solar model the pp chain produces 99% of the Sun’s energy. The CN cycle produces about 1%. Heavier main-sequence stars are powered mainly by the CNO cycles. At solar-core temperature, CNO energy generation scales about as T¹⁸ and the pp chain about as T⁴. The paper’s Figure 1 puts solar-core conditions on the pp side. This page does not invent the temperature where the curves cross.
NASA Science — Sun facts — the core is about 27 million °F (15 million °C), hot enough to sustain nuclear fusion of hydrogen into helium. The core density they print, about 150 g/cm³, is not used in the arithmetic here.
Burbidge, Burbidge, Fowler, and Hoyle 1957, Reviews of Modern Physics 29, 547 — the s-process and the r-process, slow and rapid neutron capture, as the paths that build nuclei beyond the iron group. The slow path is tied to giant stars. The rapid path, in that paper, to supernova conditions.
Abbott et al. 2017, Physical Review Letters 119, 161101 — GW170817, a neutron-star merger seen in gravitational waves. Pian et al. 2017, Nature 551, 67 — spectroscopic identification of r-process nucleosynthesis in the kilonova of that merger. This page does not assign a fraction of the heavy elements to mergers or to supernovae.

Classroom mass defect. AME2020 atomic masses, 2022 CODATA 931.49410372 MeV per u, and the IAU 2015 nominal luminosity 3.828×10²⁶ W. The pp net is 26.7310 MeV. D+T is 17.5893 MeV. The binding column peaks at ⁶²Ni. Not a stellar-evolution model. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The clock

Chart	Binding energy per nucleon against mass number. Tap a point, or use the landmark buttons. Left and Right step along the curve.
⁵⁶Fe and ⁶²Ni	Both stay marked. Nickel-62 is the highest point. Iron-56 is the classroom name for the neighborhood.
Luminosity	0.01 to 100 times the IAU nominal solar luminosity, on a log slider. The mass fraction does not change. The kilograms per second do.

Chart	Left and Right move to the next nuclide. The focus ring is the accent color.
Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Fusion
and the mass defect

The pp chain in the Sun. CNO in hotter cores. Neutrons after iron.

Cited, not invented

Stellar lifetime

HR diagram

Blackbody

Stars

Standard Model

Solar map

Fusion desk

Reactions

The curve and the Sun

Keyboard

pp net	4 ¹H → ⁴He. The full mass defect, 26.7310 MeV. Neutrinos are not subtracted.
CNO net	The same masses. The catalysts are not in the sum. The readout is why the rates differ with temperature.
D+T	²H + ³H → ⁴He + n. 17.5893 MeV.
D+D	Two branches, helium-3 plus a neutron, or tritium plus a proton. No branching ratio.
Triple-α	3 ⁴He → ¹²C, the net. The ⁸Be step is shown because it is negative.

Fusionand the mass defect

The pp chain in the Sun. CNO in hotter cores. Neutrons after iron.

Cited, not invented

Stellar lifetime

HR diagram

Blackbody

Stars

Standard Model

Solar map

Fusion
and the mass defect