Arnett's Rule: Why a Supernova Peaks When Radioactive Heating Equals Radiated Light

What Arnett's Rule Actually States

A supernova's optical light does not come from the explosion shock itself — that energy is largely spent unbinding and accelerating the ejecta within seconds. What we see for weeks afterward is a delayed glow powered by radioactivity. The explosion synthesizes unstable nickel-56, which decays to cobalt-56 and then to stable iron-56, releasing γ-rays and positrons that thermalize deep inside the opaque, expanding gas.

Arnett's Rule (Arnett 1979, 1982) says that at the moment of peak bolometric luminosity, the surface luminosity equals the instantaneous rate of radioactive energy deposition:

• L_peak ≈ ε_decay(t_peak) × M_Ni

In words: while the ejecta are still trapping most of the heat, the interior is charging up with radioactive energy faster than it can leak out. The light curve keeps rising. Once escape catches up to input, the two balance — that instant is the peak. It is a statement about a crossing point, not about total energy.

The Mechanism: A Race Between Heating and Diffusion

The physics is a competition between two clocks. Radioactive heating declines on the ⁵⁶Ni e-folding time (τ_Ni ≈ 8.8 days). Photons, meanwhile, random-walk out of the ejecta on the diffusion timescale t_diff, which itself shrinks as the ejecta expand and thin. Arnett showed that for a homologously expanding sphere with roughly constant opacity, the bolometric light curve L(t) obeys an integral relation:

• L(t) = e^(-t²/t_d²) × ∫₀ᵗ 2(t'/t_d²) e^(t'²/t_d²) L_Ni(t') dt'

where t_d is a characteristic diffusion time. Differentiating, the peak (dL/dt = 0) falls exactly where the running integral of past heating balances present losses — which reduces to L_peak = L_heating(t_peak). This is sometimes called the Arnett–Katz or 'rise-time equals diffusion-time' condition. Crucially, the peak brightness depends almost entirely on how much ⁵⁶Ni was made, while the timing of the peak depends on the ejecta mass, opacity, and kinetic energy through t_d ∝ (κ M_ej / v)^½.

Key Numbers and a Worked Example

The decay chain sets the energy budget precisely. One gram of ⁵⁶Ni releases about 3×10¹⁶ erg over the full chain; the specific heating rate is often written:

• ε(t) ≈ 3.9×10¹⁰ e^(−t/8.8 d) + 6.8×10⁹ (e^(−t/111 d) − e^(−t/8.8 d)) erg s⁻¹ g⁻¹

The first term is ⁵⁶Ni, the second ⁵⁶Co. Plugging in the ~18-day rise typical of a Type Ia yields the famous calibration:

• L_max ≈ 2.0×10⁴³ (M_Ni / M_sun) erg s⁻¹

Example: a normal Type Ia peaking at L ≈ 1.2×10⁴³ erg/s implies M_Ni ≈ 0.6 M_sun — a huge fraction of the Chandrasekhar mass (1.4 M_sun) burned to iron-group elements. A subluminous 1991bg-like event might peak near 3×10⁴² erg/s, giving M_Ni ≈ 0.15 M_sun, while SN 1987A's tiny 0.07 M_sun of nickel made it faint for a core-collapse event. This linear L∝M_Ni scaling is what makes the rule so practical.

How Astronomers Use and Test It

Applying Arnett's Rule requires a bolometric light curve — integrated flux across UV, optical, and infrared — not just a single filter. Observers assemble this from multi-band photometry, correct for host-galaxy reddening and distance (via Cepheids, redshift, or the SN itself as a standard candle), and read off the peak.

• The peak luminosity gives M_Ni directly.
• The rise time to peak constrains ejecta mass and kinetic energy through t_d.
• The late-time tail (after ~60 days) should decline at the ⁵⁶Co rate of 0.98 mag per 100 days — a direct fingerprint of the decay chain.

Two landmark confirmations stand out. In SN 1987A, astronomers detected the ⁵⁶Co γ-ray lines directly with balloon- and satellite-borne detectors, proving radioactivity powers the tail. Decades later, NuSTAR imaged ⁴⁴Ti and ⁵⁶Co γ-rays from SN 1987A and Cassiopeia A, and INTEGRAL caught the 158 keV and 812 keV ⁵⁶Ni/⁵⁶Co lines from the Type Ia SN 2014J — closing the loop between the observed light and the isotope inventory Arnett assumed.

Where the Rule Breaks — and Its Cousins

Arnett's Rule is an approximation, and its assumptions matter. It presumes constant opacity, spherical symmetry, homologous expansion, and — critically — that the energy-density profile stays self-similar as it evolves. Real supernovae violate these to varying degrees:

• Type Ib/Ic (stripped-envelope): longer diffusion times and more centrally concentrated ⁵⁶Ni make the rule overestimate nickel masses, sometimes by 50%.
• Superluminous supernovae: require implausible ⁵⁶Ni masses (several M_sun) under the rule, hinting at other engines — magnetar spin-down or circumstellar interaction.

Khatami & Kasen (2019) rederived the peak condition and showed L_peak/L_heating depends on where the nickel sits and on t_peak/t_diff; they replaced the strict equality with a calibrated relation L_peak = (2/β²)∫... that corrects the bias. The rule's close relatives are the radioactive diffusion model for the full curve, and the photospheric-phase versus nebular-phase analyses that cross-check M_Ni from emission lines.

Significance and Open Questions

Arnett's Rule is foundational to modern time-domain astrophysics. Because a Type Ia's peak brightness maps to its ⁵⁶Ni mass, and because that mass correlates with light-curve width (the Phillips relation, brighter = broader), the rule underpins the standardization that made Type Ia supernovae the standard candles behind the 1998 discovery of dark energy by Perlmutter, Schmidt, and Riess.

• Open question: what causes the ⁵⁶Ni spread — different white-dwarf masses, sub-Chandrasekhar double-detonations, or white-dwarf mergers?
• Debate: how much does asymmetry and clumping in the ejecta bias Arnett-derived masses?
• Frontier: applying (and correcting) the rule for exotic transients — kilonovae, fast blue optical transients, and pair-instability candidates — where the power source may not be ⁵⁶Ni at all.

The enduring lesson is conceptual: a supernova's brightest moment is not the explosion — it is the quiet instant when a decaying atomic nucleus finally loses its race with escaping light.

The ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe decay chain that powers the light curve, plus how Arnett's Rule performs across supernova classes.

Isotope / class	Half-life or rise	Energy / behavior
⁵⁶Ni → ⁵⁶Co	6.08 days	~1.75 MeV per decay, mostly γ-rays; dominates the rise
⁵⁶Co → ⁵⁶Fe	77.2 days	~3.73 MeV per decay (γ + positrons); sets the late tail
Type Ia SN	rise ~17–19 d	Arnett's Rule accurate to ~10%; M_Ni ~ 0.3–0.9 M_sun
Type Ib/Ic SN	rise ~15–25 d	Rule overestimates M_Ni; longer diffusion, central heating
SN 1987A (II-pec)	rise ~85 d	M_Ni ≈ 0.07 M_sun; γ-ray leakage seen directly