03Simulator

Triple Product Simulator

One number decides whether fusion becomes a power plant. Here, you chase that number with a machine you design — computed by the same physics used to design real reactors.

This century's energy decisions turn on a physical threshold

Seventy-five percent of global CO₂ emissions come from the energy sector, and no source available today combines energy density, abundant fuel, and the absence of long-lived waste the way fusion does. But fusion only becomes a power plant if it crosses a precise threshold: the product of three factors — fuel density, plasma temperature, and energy confinement time — must exceed a minimum value. That threshold has a name, a number, and sixty years of history: the Lawson criterion, extended to the triple product. For deuterium–tritium, the ignition minimum is 2.9×10²¹ m⁻³·keV·s at an optimal temperature of 14 keV. No machine in history has reached it in sustained operation. That is exactly what makes the problem interesting.

You don't control the plasma; you control the machine — and that difference is the whole physics

In a real reactor, no one turns a “temperature” or “confinement time” knob. The operator chooses the machine: radius, magnetic field, current, injected fuel, heating power. Temperature emerges from the energy balance; confinement time emerges from an empirical law distilled from thousands of real tokamak shots — IPB98(y,2). This simulator works the same way. You design the machine; the physics solves the rest, including the walls nature imposes: the Greenwald density limit, above which the plasma disrupts, and the 4.3 keV floor, below which radiation beats any effort. Push the controls and see a truth no text teaches as well as ten seconds of trying: the three factors multiply, but they do not trade freely.

What this simulator is not — stated before what it is

This is not a toy with made-up numbers, and it is not an ITER digital twin — no browser model is, and be suspicious of any that promises otherwise. It is a 0-D power balance with the IPB98(y,2) confinement scaling, Bosch–Hale reactivity, and the Greenwald limit: the same published methodology the community uses as a starting point to predict reactor performance, validated here against the ITER design point (Q=10: 500 MW of fusion from 50 MW of heating) within declared tolerance. Every constant in the code carries the link to the source it was transcribed from. The full references are at the bottom of the page — and they are worth the visit.

T₀20.6 keV

n̄3.10×10²⁰ m⁻³

Q4.17

Breakeven

This is your reactor, drawn to the proportions of the machine you selected — and everything you see is live. The legend, element by element: the glowing ring is the plasma, a torus, because a straight tube would leak plasma out both ends and a ring has no ends. The thin shell around it is the vacuum vessel, shown as a cutaway so you can see inside. The dotted rings are schematic guides tracing the torus shape — they are not magnetic coils. The small figure gives human scale: switch presets and watch the machine grow or shrink around it. The badges are the plasma's vital signs — central temperature T₀, average density n̄, and gain Q — and the corner chip names the regime you are in. Now the cause-and-effect map. Switch machine size: the whole torus is rebuilt with the real radius and elongation of that design — the compact machine is visibly smaller and more stretched. Push heating or fuel: the glow responds, because its brightness is driven directly by Q — a plasma near breakeven smolders, a burning one blazes — and its color shifts with temperature. Cross the Greenwald density limit and the story changes abruptly: the glow collapses into an unstable flicker. That is a disruption — the plasma letting go — and it is the one signal here you would never want to see in a control room. One honest note: this is a physics schematic, not an engineering render. Coils, divertor and heating systems are not drawn — what is drawn is what the numbers are doing, in real time.

Sources: geometry (R, a, κ) from the ITER-FEAT baseline and published SPARC design point; glow driven by computed Q; scale note, 1 keV ≈ 11.6 million °C (ITER Organization).

Machine size

Two machines, two philosophies. ITER-class is the giant path: a plasma ring 6.2 m in radius, confined by a 5.3 T field — performance bought with sheer size, because bigger plasmas hold their heat longer. High-field compact is the opposite bet: one-third the radius (1.85 m), but more than double the field (12.2 T), made possible by a new generation of high-temperature superconducting magnets — the approach behind designs like SPARC. Same physics engine runs both; switch between them and watch how size and magnetic field trade places in the fight for confinement. There is no JET preset here on purpose: JET's famous 1997 record was a transient shot, and this simulator models steady state — so JET appears where it belongs, as a measured milestone on the Q-meter, not as a modeled machine.

Sources: ITER Organization (iter.org); machine parameters from the ITER-FEAT baseline and published SPARC design point.

Fuel (density n̄)3.10×10²⁰ m⁻³

Greenwald: 8.52×10²⁰ m⁻³

Heating power25.0 MW

You control two things — and the physics fights you on both. Fuel (density n̄): fusion needs nuclei colliding, so fusion power grows with the square of density — double the fuel, quadruple the reactions. But pack in too much and you hit the Greenwald limit: the plasma disrupts, collapsing in milliseconds. And with fixed heating, more particles share the same energy — the plasma cools. Heating power: hotter nuclei move faster and fuse more, with the sweet spot around 10–20 keV. The catch: confinement quality is not a knob. It emerges from the machine (the IPB98 scaling law), and it degrades as you push more power in — the harder you heat, the faster energy leaks. Since Q = fusion out ÷ heating in, you can raise fusion and still lower your gain. That tension — between more fuel, more heat, and a plasma that pushes back — is the entire engineering problem of fusion, compressed into two sliders.

Sources: IPB98(y,2) confinement scaling and Greenwald limit, PROCESS documentation (UKAEA); Greenwald, PPCF 44 (2002).

Lawson chart

1 keV ≈ 11.6 million degrees

You've reached breakeven. Push harder to start burning.

This chart is the map of the entire quest. The horizontal axis is plasma temperature; the vertical axis is the triple product — density × temperature × confinement time, the single number that says how close a plasma is to paying for itself. It is a product because all three must happen at once: enough nuclei, hot enough, held long enough. If any one is zero, everything is zero. The two curves are thresholds. The dashed blue line is breakeven (Q = 1) — above it, the fusion produced exceeds the heating injected. The solid line above is ignition (Q = ∞) — the plasma feeds itself and external heating can be switched off. Both curves form a valley: too cold (below ~4.3 keV) radiation always wins; too hot, the requirement climbs again. The floor of the valley sits near 14 keV — which is why reactors aim for the 10–20 keV window. "You are here" is your machine, live. In 1955, John Lawson drew the original version of this boundary; every fusion device since has been climbing toward it. This chart shows you exactly how far there is left to go.

Sources: Lawson criterion and triple product treatment, Wurzel & Hsu, Physics of Plasmas 29 (2022), arXiv:2105.10954; minimum D-T ignition triple product 2.9×10²¹ m⁻³·keV·s at 14 keV.

Q meter

Q = 4.17

JET 1997

Breakeven

Burning

ITER

Ignition

Q is the scoreboard: fusion power out, divided by heating power in. Q = 4 means every megawatt you inject comes back as four megawatts of fusion. The scale is logarithmic — fusion spans from machines that return a fraction of what they consume to plasmas that sustain themselves, and a linear ruler would crush half that story into a corner. The fixed marks are real history: JET 1997 (Q = 0.67 — the best experimental result of its era, still short of paying for itself), Breakeven (Q = 1, fusion equals heating), Burning (Q = 5, where alpha self-heating supplies half the plasma's heat), ITER (Q = 10, the design goal), and Ignition at the far end. One honest note, because this is where fusion headlines usually go wrong: Q measures the plasma's gain, not the power plant's. Heating systems, magnets and cryogenics all draw electricity — and no fusion device in history has yet produced net electric power. That is precisely the point of the race.

Sources: JET 1997 record and ITER Q=10 goal, ITER Organization (iter.org); gain definitions, Wurzel & Hsu (2022).

Energy balance

P_aux

25.0 MW

α self-heating

20.7 MW

Out

Transport

41.0 MW

Bremsstrahlung

4.7 MW

A plasma in steady state obeys one rule: everything in equals everything out. On the in side, two heaters: the external power you inject (your slider), and the alpha particles — each D-T reaction traps 20% of its energy inside the plasma as a helium nucleus, heating it from within. On the out side, two leaks: bremsstrahlung, the X-ray glow emitted when electrons swerve past nuclei, and transport, the heat that escapes despite the magnetic cage. The simulator finds the one temperature where the two sides balance exactly — which is why temperature here is a result, not a setting. Watch the alpha bar as you push Q higher: the more the plasma heats itself, the less it needs you. If the alphas alone could ever cover both leaks, you could switch the external heating off. That moment has a name — ignition.

Sources: 0-D power balance methodology, Wurzel & Hsu, Physics of Plasmas 29 (2022); alpha fraction 3.5/17.6 MeV, D-T reaction energetics.

Regime: Break-even

T₀: 20.6 keV

Q: 4.17

R / a / κ: 1.9 / 0.6 / 1.97

Drag to rotate · scroll to zoom

SCALE

What this energy would mean

If this heat were converted to electricity in a thermal plant (declared 33% efficiency): P_elec = 34.4 MW i

Turn the heating power up and watch the city light up.

LED bulbs lit

≈ 3,437,059

assumption: typical 10 W LED

American homes powered

≈ 27,921

ASSUMPTION: AVERAGE U.S. HOME — 10,791 kWh/yr (EIA 2022) ↗

U.S. city of comparable size

Burlington, VT

≈ 1.6× Burlington

BASIS: ~17,000 households (U.S. Census, ACS 2022) ↗

To release the same heat by burning fossil fuels:

heating oil burned per day (same heat)

≈ 62,084 gallons/day

HEATING OIL, EIA UNITS PAGE ↗

COAL BURNED PER DAY (SAME HEAT)

≈ 445 SHORT TONS/DAY

ASSUMPTION: 19.167 MILLION BTU PER SHORT TON (EIA FAQ) ↗

NATURAL GAS BURNED PER DAY (SAME HEAT)

≈ 0.23 MILLION m³/DAY

ASSUMPTION: 1 CUBIC FOOT = 1,036 BTU (EIA UNITS PAGE) ↗

Numbers in megawatts mean little until they become things. This block converts your reactor's output into objects you can picture — with one honest rule made explicit: these are "what if" conversions. Your simulated fusion power is heat; we assume a standard power plant turns it into electricity at 33% efficiency, and then count what that electricity could do — LED bulbs lit, average American homes powered (10,791 kWh per year each, the official U.S. average), a city of comparable size. The bottom row flips the question: to release the same heat by burning things, you would need this much oil, coal and natural gas — every single day. That daily mountain of fuel, versus grams of hydrogen isotopes, is the entire argument for fusion in one comparison. No fusion device feeds the grid yet; this ladder shows what is being reached for.

Sources: all conversion constants from the U.S. Energy Information Administration — home consumption (EIA 2022), heating oil 138,500 Btu/gal, coal 19.167 MMBtu/short ton, natural gas 1,036 Btu/cf (EIA Units & Calculators and FAQ pages, linked on each card).

Greenwald fraction

36%

Every tokamak has a density ceiling — and it is surprisingly simple. The Greenwald limit depends on just two things: how much current flows through the plasma, and how thick the plasma column is. Cross it, and the plasma doesn't gently underperform — it disrupts, dumping its entire stored energy into the walls in milliseconds. This bar shows how close you are: 0.85 means you are running at 85% of the ceiling, which is roughly where ITER's own design point sits — deliberately close to the edge, because density is too valuable to leave on the table. The limit is empirical: observed across machines worldwide for four decades, its deep cause is still an open research question. Think of it as an elevator's load plate — and remember that exceeding it doesn't slow the elevator down. It snaps the cable.

Sources: Greenwald, Plasma Physics and Controlled Fusion 44, R27 (2002); density limit implementation, PROCESS documentation (UKAEA).

Triple product

3.91×10²¹ m⁻³·keV·s

τ_E

0.61 s

T₀ (plasma centre)

20.6 keV

⟨T⟩ average: 7.2 keV

4.17

Breakeven

How this simulator computes

It is a 0-D power balance with the empirical IPB98(y,2) confinement scaling — the same published methodology the field uses for reactor prediction.

τ_E is computed (not chosen), density has the Greenwald ceiling, and below ≈4.3 keV bremsstrahlung wins — ignition is physically impossible.

The engine includes temperature profiles (ν_T=1.85) and the nominal ITER-FEAT plasma composition (He, Be, Ar) — parameters and sources in the code comments.

This simulator uses the published 0-D methodology and reproduces the ITER design point within a declared ±30% tolerance. It does not replace the laboratories' integrated codes — no model does.

Model parameters

Constants actually used by the solver at every step. Each row links to the primary source the value was transcribed from.

Symbol	Value	Meaning	Source
ν_T	1.85	Temperature profile exponent, T(x)=T₀·(1−x²)^ν_T. ITER-FEAT nominal parabolic-power profile.	ITER-FEAT (physics/0410118)
ν_n	0.00	Density profile exponent (ν_n=0 ⇒ homogeneous density, n₀=n̄).	ITER-FEAT (physics/0410118)
λ_B	0.5195	Profile factor for bremsstrahlung — volume integral of n²√T normalised by central values (W&H Eq. 39).	Wurzel & Hsu (2022), Sec. IV
λ_κ	0.3509	Profile factor for stored thermal energy — volume integral of nT, normalised (W&H Eq. 40).	Wurzel & Hsu (2022), Sec. IV
λ_F(T₀)	0.2230 @ T₀=20.6 keV	Profile factor for fusion power — volume integral of ⟨σv⟩(T(x)) normalised by ⟨σv⟩(T₀). Depends on T₀; recomputed each iteration (W&H Eq. 38).	Wurzel & Hsu (2022), Sec. IV
f_DT	0.8096	D+T fuel fraction of the electron density, set by quasi-neutrality with the nominal ITER-FEAT He, Be and Ar fractions.	ITER-FEAT (physics/0410118)
Z_eff	1.656	Effective plasma charge, ⟨Z²⟩. Enters bremsstrahlung as a multiplier — impurities push it up.	ITER-FEAT (physics/0410118)

References

Wurzel, S. E. & Hsu, S. C. — Progress toward fusion energy breakeven and gain as measured against the Lawson criterion. Phys. Plasmas 29, 062103 (2022). https://arxiv.org/abs/2105.10954
Slendebroek, T. et al. — Elevating zero dimensional global scaling predictions to self-consistent theory-based simulations. https://arxiv.org/abs/2305.07285
PROCESS systems code — plasma confinement scalings. https://ukaea.github.io/PROCESS/physics-models/plasma_confinement/
PROCESS — Bosch–Hale methods and constants. https://ukaea.github.io/PROCESS/physics-models/fusion_reactions/plasma_bosch_hale/
PROCESS — Density limits. https://ukaea.github.io/PROCESS/source/reference/process/models/physics/density_limit/
Bosch, H.-S. & Hale, G. M. — Nucl. Fusion 32, 611 (1992); ITER Physics Basis — Nucl. Fusion 39 (1999), ch. 2; Greenwald, M. — PPCF 44, R27 (2002).
ITER Organization — Q=10 goal and JET 1997 record. https://www.iter.org/fusion-energy/iter-goals

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