Yonetoku Relation in Gamma-Ray Bursts

• The Yonetoku relation is an empirical correlation between a GRB’s rest-frame spectral peak energy and its 1-second peak isotropic luminosity, providing a practical framework for examining burst energetics.
• Its calibration relies on precise sample selection, spectral modeling, and accounting for viewing-angle effects, which help reduce intrinsic scatter and refine energy estimates.
• The relation is interpreted through both photospheric and synchrotron emission models, offering insights into GRB population classification and potential usage as a cosmological standard candle.

The Yonetoku relation is an empirical power-law correlation between the intrinsic spectral peak energy ($E_{\rm p,rest}$) of a gamma-ray burst’s (GRB’s) $\nu F_\nu$ prompt emission spectrum and its 1-second peak isotropic luminosity ($L_p$ or $L_{\rm iso}$). Identified originally in long GRBs, the relation remains a key phenomenological tool in the paper of GRB energetics, population classification, and as a putative cosmological standardizable candle. The following sections review the quantitative foundation, data calibration, theoretical interpretations, population-dependent variants, and current constraints on the relation.

1. Formal Definition and Empirical Calibrations

The Yonetoku relation expresses a scaling law between the rest-frame spectral peak energy and the 1-second peak luminosity. One widely adopted form, as established using Fermi/GBM bursts with known redshift, is: $$E_{\rm p,rest} = 667^{+295}_{-310} \left(\frac{L_p}{4.97\times10^{53}\,\mathrm{erg}\,\mathrm{s}^{-1}}\right)^{0.48\pm0.01}\,\mathrm{keV}$$ Here, $E_{\rm p,rest}$ is measured in the source frame, and $L_p$ is the 1-second peak isotropic-equivalent luminosity over the rest-frame 1–10000 keV band (Gruber, 2012). The exponent of $\sim0.48$ indicates that a decade change in $L_p$ corresponds to roughly a factor of 3 change in $E_{\rm p,rest}$. The tightness of the correlation is quantified by a Spearman’s rank coefficient of $\rho\approx0.8$, $p\approx 9\times 10^{-8}$.

Several alternative functional forms have been employed: $$L_{\rm iso} = A \left(\frac{E_{\rm p,rest}}{\mathrm{keV}}\right)^p$$ with $L_{\rm iso}$ in $10^{52}$ erg/s, $A\sim 3.6\times 10^{-5}$, and $p\sim2.04$ for Swift LGRBs with high photon flux (Zitouni et al., 2016), and a linearized version: $$\log_{10}\left(\frac{E_{i,p}}{100\,\mathrm{keV}}\right) = m\,\log_{10}\left(\frac{L_{\rm iso}}{10^{52}\,\mathrm{erg/s}}\right) + k$$ where $m$ and $k$ are regression parameters determined for each sample (Aldowma et al., 18 Oct 2025). The intrinsic rest-frame energy is computed as $E_{i,p} = E_p(1+z)$, and luminosity as $L_{\rm iso}=4\pi d_L^2 P_{\rm bol}$, with $P_{\rm bol}$ the observed bolometric flux and $d_L$ the cosmological luminosity distance.

2. Methodological and Sample Selection Issues

The reliability and scatter of the Yonetoku relation are highly sensitive to sample selection, spectral modeling, and measurement techniques.

Selection criteria include:

• Well-determined redshift;
• Secure $E_p$ measured within the detector’s energy window and with $<70\%$ relative error;
• Adequate photon flux ($P_{\rm ph}\geq2.6\,\mathrm{ph\,cm^{-2}\,s^{-1}}$ for Swift bursts) (Zitouni et al., 2016);
• Band or cutoff power-law (CPL) spectral fits appropriate for the data (Li, 2022);
• Joint spectral analysis across multiple instruments to reduce uncertainties (Fermi GBM + LAT + LLE) (Aldowma et al., 18 Oct 2025).

Best-fit parameters and intrinsic dispersion are minimized when these criteria are strictly enforced (see Table).

Sample/Method	$E_p$–$L_{\rm iso}$ Index	Regression Approach	Notable Scatter/Remarks
Fermi/GBM LGRBs (Gruber, 2012)	$0.48\pm0.01$	Bisector OLS, outlier rejection	Outlier for short GRB 080905A
Swift/BAT LGRBs (Zitouni et al., 2016)	$2.04$ (power index)	Monte Carlo, flux thresholding	Tightening with restrictive sample
Fermi (GBM+LAT+LLE) (Aldowma et al., 18 Oct 2025)	Sample-dependent	Global joint fitting	Joint fits reduce error/dispersion
Model-wise (Band/CPL) (Li, 2022)	$0.34\pm0.04$ (Band), $0.40\pm0.08$ (CPL)	Linear in log–log	Model-dependent separation between populations

Short or otherwise atypical bursts may be strong outliers (e.g., GRB 080905A (Gruber, 2012); GRB 170817A (Zhang et al., 2020)), indicating that the relation is not strictly universal.

3. Theoretical Interpretations and Physical Origin

The observed correlation is broadly interpreted within two emission paradigms: (a) photospheric quasi-thermal models, and (b) non-thermal (internal shock, synchrotron) models.

Photospheric Models

3D relativistic hydrodynamical + Monte Carlo radiative transfer simulations (Ito et al., 2018) show that the $E_p$–$L_p$ scaling arises naturally as a function of observer viewing angle in a structured jet. The simulated relation,

$$L_p = 10^{52} \left(\frac{E_p}{355\,\mathrm{keV}}\right)\ \mathrm{erg/s}$$

is robust over jet kinetic powers and durations. The key driver is the observer’s angle relative to the jet axis; on-axis views sample hotter, higher-$\Gamma$ regions, yielding higher $E_p$ and $L_p$, whereas off-axis observers view cooler, less beamed emission. The viewing angle thus produces a tight locus in the observed $E_p$–$L_p$ plane.

Synchrotron and Internal Shock Models

Within the standard fireball/internal shock scenario (Xu et al., 2022), analytic scaling yields:

• On-axis: $L_p\propto E_p^{0.5}$
• Off-axis: $L_p\propto E_p^{0.21-0.24}$

This bifurcation emerges as the effective Doppler factor, emitting region, and thus observed $E_p$ and $L_p$, depend sensitively on both $\Gamma$ and viewing angle. Off-axis (“low-luminosity”) GRBs systematically deviate towards shallower slopes.

Recent model-wise studies (Mei et al., 12 Sep 2024) show that when prompt spectra are fit with a physical synchrotron model (allowing for two spectral breaks), the classical $E_{p,z}$–$L_{iso}$ Yonetoku relation is only recovered in intermediate-cooling regimes ($1<\nu_m/\nu_c<3$). In fast-cooling ($\nu_m/\nu_c\gg1$), $E_p$ is not tightly correlated with $L_{iso}$. The tight physical correlation instead links the rest-frame cooling frequency $\nu_{c,z}$ to $L_{iso}:\ \nu_{c,z}\propto L_{iso}^{0.53\pm0.06}$.

4. Spectral Modeling, Width Correlations, and Population Segregation

The spectral model adopted in fitting the prompt emission has direct impact on the derived Yonetoku relation:

• Band-like spectra typically conform to the canonical relation, while CPL-like spectra may populate a parallel but displaced locus in $E_{p,z}$–$L_{p,iso}$ space (Li, 2022).
• The relation can be subsumed within a broader family of spectral-width–luminosity correlations (“width–$L_{iso}$”), where the absolute spectral width (at 90% of $EF_E$ maximum) nearly overlaps the traditional $E_{p,i}$–$L_{iso}$ regression (Peng et al., 2020).
• Both E-I and E-II subgroups (classified by how well they follow energy correlations rather than prompt duration) among GRBs with extended emission obey distinct Yonetoku relations with differing normalizations and slopes (Zhang et al., 2020). This suggests divergent progenitor or jet properties for these subclasses.

Anomalous events such as GRB 170817A may be outliers, even after off-axis corrections are applied (Zhang et al., 2020), providing diagnostic leverage on population diversity.

5. Cosmological Applications and Limitations

Due to the tight, albeit scattered, $E_p$–$L_p$ correlation, the Yonetoku relation has been proposed as a tool for estimating GRB pseudo-redshifts and as a cosmological standard candle:

• Pseudo-redshift estimation proceeds by expressing both $E_{p,z}$ and $L_{iso}$ as functions of $z$ given observed $E_{p,obs}$ and peak flux, then inverting the Yonetoku relation numerically (Yorgancioglu et al., 18 Feb 2025). The relevant equations are:

$$E_{p,z} = E_{p,obs}(1+z),\quad L_{iso} = 4\pi D_L(z)^2 f_\gamma k$$

and

$$\log[E_{p,obs}(1+z)] = a_Y \cdot \log[4\pi D_L(z)^2 f_{\gamma}] + b_Y$$

• In practice, the Yonetoku-based redshift is mathematically well-behaved (single intersection), but the intrinsic scatter (e.g., $\sigma_{\log E_p} \sim 0.25$ dex) produces large systematic uncertainties. The mean Pearson correlation with true redshift is weak (e.g., $r \approx 0.40\pm0.04$ for simulated samples) (Yorgancioglu et al., 18 Feb 2025).
• Using the relation for Hubble diagram construction or $H_0 / \Omega_\Lambda$ constraints requires minimizing both the statistical and extrinsic scatter, for which joint spectral analysis (with broader spectral coverage and multi-instrument synergy) is critical (Aldowma et al., 18 Oct 2025).

6. Implications for GRB Physics and Open Questions

• The Yonetoku relation is robust for long GRBs fitted with appropriate models, but may break down or acquire significant outliers in the case of short, nearby, or unusual bursts, and in populations where the underlying emission regime differs (e.g., photospheric vs. shock-dominated, viewing angle variations).
• The physical origin remains debated, with current evidence favoring viewing-angle–mediated photospheric emission as the main driver of the tightest correlations at high luminosities (Ito et al., 2018), while the standard internal shock model yields the correct power-law indices in idealized on-axis configurations but introduces diversity off-axis or with varying microphysical assumptions (Xu et al., 2022).
• Model-wise analyses and extension to more general width–luminosity correlations highlight the importance of comprehensive spectral characterization and caution against over-interpretation of the classical $E_{p,i}$–$L_{iso}$ scaling (Peng et al., 2020, Li, 2022, Mei et al., 12 Sep 2024).
• The relation’s utility as a cosmological probe is fundamentally limited by its intrinsic scatter and sensitivity to sample selection and spectral modeling (Yorgancioglu et al., 18 Feb 2025, Aldowma et al., 18 Oct 2025).

7. Summary Table of Representative Calibrations

Reference	Regression Formula	Sample/Comments
(Gruber, 2012)	$$E_{\rm p,rest} = 667^{+295}_{-310}(L_p/4.97\cdot10^{53})^{0.48\pm0.01}$$ keV	Fermi/GBM, 39 LGRBs, outlier short GRB 080905A
(Zitouni et al., 2016)	$$L_{iso}/10^{52} = 3.6\cdot10^{-5}[E_{p,i}/\rm keV]^{2.04}$$	Swift "good" LGRBs, reduced dispersion
(Ito et al., 2018)	$L_p = 10^{52}(E_p/355\ \rm{keV})$ erg/s	3D hydro, photosphere model, viewing-angle origin
(Li, 2022)	$E_{p,z} = 307 (L_{p,iso}/2\cdot10^{52})^{0.34}$ keV	Fermi/GBM, Band-like LGRBs, model dependence
(Mei et al., 12 Sep 2024)	$\nu_{c,z}\propto L_{iso}^{0.53\pm0.06}$	Synchrotron fit: fundamental relation with $\nu_c$
(Aldowma et al., 18 Oct 2025)	$\log(E_{i,p}/100) = m\log(L_{iso}/10^{52})+k$	Fermi (GBM+LAT+LLE), joint spectral fits

The Yonetoku relation persists as both a tool and a probe—its calibration, functional form, and physical basis now understood to depend sensitively on spectral fitting methodology, population substructure, jet geometry, and emission mechanisms. Future progress will likely require simultaneous advances in broad-band spectroscopy, time-resolved model selection, and theoretical modeling of jet dynamics and emission physics.