Relativistic Dynamical Evolution of Kilonova Spectral Emission

The gist: A relativistic dynamical model for thermal evolution and spectral emission of expanding kilonova ejecta reveals that light-travel-time delays and Doppler boosting cause significant spectral distortions at high ejecta velocities, necessitating a self-consistent treatment to constrain underlying ejecta properties.

Thermal Evolution of the Ejecta

The internal energy evolution of the expanding ejecta is governed by the first law of thermodynamics, expressed as dEint/dt = Q˙in − L − PdV/dt (2). The model considers a spherically symmetric, homogeneous sphere expanding at a constant velocity v at its surface, where the radius is Rej(t) = vt and the mass density decreases as ρ(t) ≈ 3Mej / (4πv3t−3). In the early phase when the ejecta is hot and optically thick, U = 3P = aT4 (6). The transition from optically thick to optically thin regime is characterized by a characteristic timescale tc, defined as tc = (3κMej / 4πv2)(1/2) (5). The effective temperature Teff is determined by T4eff ≈ T4τs + 1/2, where τs(t) = t/tc−2 (7). The heating rate Q˙in is approximated by Q˙thin = (1 − e−τs)Q˙in = (1 − e−t2/c2t2)ϵ(t)Mej, where ϵ(t) follows a power-law form ϵ(t) = f c2/tc t(1+α) (9). The solution to the energy density evolution equation yields the dynamical temperature evolution Temit(t), which exhibits a broken power-law behavior as it transitions from optically thick to optically thin regimes, following Temit(t) ∝ [t−(1+α)/4] for t ≪ tc and Temit(t) ∝ [t−5/4] for t >> tc (14).

Relativistic and Geometrical Corrections

The model incorporates relativistic effects by mapping the dynamical temperature solution onto the equal-arrival-time surface (EATS; B. Zhang (2018)). The emission time in the rest frame is related to a single observer time tobs as t(µ, tobs) = tobs / (1 − βµ) (12). This leads to an equal–arrival–time radius R(µ, tobs) = βctobs / (1 − βµ), which is normalized by the emitting line–of–sight radius R(µ = 1, tobs) as 1 - β / (1 − βµ) (13). The observed temperature of a surface element is Doppler-shifted by Tobs(µ, tobs) = Temit(t(µ, tobs)) γ(1 − βµ) (15). The angular average of the observed temperature Teff,obs is computed using the bolometric luminosity Lbol(tobs) = A σ T4eff,obs (16), where the effective emitting area becomes A = 8π / Z1β R2(µ, tobs)µdµ (17). The resulting spectrum is a superposition of Doppler-shifted blackbodies originating from different latitudes on the emitting surface and different emission times (18).

Impact of Ejecta Parameters on Observed Temperature

The time evolution of the observed temperature Tobs(tobs, µ) shows distinct behaviors based on velocity and geometry. For mildly relativistic ejecta (β0 ≲ 0.3), the delay increases the limb temperature, while Doppler boosting enhances the temperature near µ ≈ 1, leading to a nearly uniform Tobs(tobs, µ), which explains why AT2017gfo appears approximately as a single-temperature blackbody despite its expanding geometry (Fig. 4). For higher velocities (β > 0.5), the cancellation of Doppler and EATS effects does not occur, and stronger angular temperature gradients are produced between the emission coming from θ = 0 and θ = π/2 (Fig. 4). Neglecting relativistic transfer effects increases the inferred dynamical temperature by up to 7.6% and the velocity by ≃ 11.9% for these epochs, in agreement with results reported by G. Sadeh (2025) (Page 12).

Fitting AT2017gfo Spectra

The model was applied to the early-time X-shooter spectra of AT2017gfo using Markov Chain Monte Carlo (MCMC) fitting with a Student’s t likelihood, which is motivated by capturing the underlying spectral continuum rather than detailed absorption and emission features (19). The best-fit parameters for four epochs were determined, including ejecta mass Mej, opacity κMej, velocity v/c, heating index α, and luminosity distance DL (Table 1). The effective temperature Teff,obs decreases from 5428 ± 4 K at 1.43 d to 3061 ± 5 K at 4.40 d, consistent with the progressive reddening of the spectra in Fig. 7 (Page 9). The inferred velocity parameter β decreases as time passes, from ≈ 0.20 at 1.43 days to ≈ 0.12 at 4.40 days (Page 11).

Conclusion and Future Directions

The work bridges the gap between dynamical models that describe thermal evolution and relativistic frameworks that account for geometrical effects by using a self-consistent dynamical temperature evolution as a function of ejecta properties before applying relativistic corrections to generate the observed spectrum (Page 5). A key outcome is that the emitted effective temperature follows a broken power-law evolution, emerging naturally from the transition between optically thick and optically thin ejecta at the transparency timescale (Page 14). The study suggests that for faster expanding ejecta, stronger angular temperature gradients are produced, leading to more significant spectral distortions (Page 12). Future work is needed to include detailed radiative transfer through a multi-frequency opacity structure and non-LTE effects to connect the continuum evolution more directly to the ejecta composition (Page 12). This model captures key features of the relativistic dynamical evolution of kilonova emission, which may shed light on the nature of these transients (Page 12).

How it works

The thermal evolution is described by solving Eq. (1) dEint/dt = Q˙in − L − PdV/dt (2), considering adiabatic expansion and radioactive heating. The ejecta is treated as a spherically symmetric, homogeneous sphere expanding at a constant velocity v at its surface, leading to the density ρ(t) ≈ 3Mej / (4πv3t−3) (2). The transition from optically thick to optically thin is marked by the transparency timescale tc = (3κMej / 4πv2)(1/2) (5). The heating rate Q˙thin is approximated as Q˙thin = (1 − e−τs)Q˙in = (1 − e−t2/c2t2)ϵ(t)Mej, where τs(t) = t/tc−2 and ϵ(t) follows a power-law form ϵ(t) = f c2/tc t(1+α) (8).

Relativistic Spectral Calculation

The thermal history is incorporated into a relativistic spectral calculation based on the EATS geometry and Doppler boosting. The emission time in the rest frame is related to a single observer time tobs as t(µ, tobs) = tobs / (1 − βµ) (12). This relation leads to an equal–arrival–time radius R(µ, tobs) = βctobs / (1 − βµ), normalized by the emitting line–of–sight radius R(µ = 1, tobs) as 1 - β / (1 − βµ) (13). The observed temperature of a surface element is Doppler-shifted by Tobs(µ, tobs) = Temit(t(µ, tobs)) γ(1 − βµ) (15). The angular average of the observed temperature Teff,obs is computed using the bolometric luminosity Lbol(tobs) = A σ T4eff,obs (16), where the effective emitting area becomes A = 8π / Z1β R2(µ, tobs)µdµ (17).

Parameter Dependence and Constraints

The model parameters—ejecta mass Mej, velocity β=v/c, opacity κ, heating index α, and heating efficiency f—are varied to show their impact on the observed temperature. Figure 2 displays the dependence of the angle-averaged observed temperature at 1.43 days after merger on these parameters (Page 6). The analysis involves fitting four X-shooter epochs using a Student-t likelihood, marginalizing over luminosity distance DL = 40.7 ± 2.2 Mpc (Table 1). The results show that the velocity parameter β decreases as time passes, from ≈ 0.20 at 1.43 days to ≈ 0.12 at 4.40 days (Page 11). The degeneracy between κ and Mej is noted, and the effective combination κMej increases mildly from ≈ 3.8×10−2 to ≈ 6.5 × 10−2 cm2/g−1 M⊙ between epochs (Page 12).

1. Improved AI system can perform self-consistent dynamical modeling of kilonova thermal evolution by solving Eq (11) to determine energy density evolution from which luminosity follows via Eq (10). This allows for predicting the broken power-law behavior of emitted temperature naturally arising from the transition between optically thick and thin regimes.

2. Improved AI system can incorporate relativistic and geometrical corrections, such as light-travel-time delays and Doppler boosting, into spectral calculations based on the equal-arrival-time surface geometry (EATS). This enables accurate modeling of observed spectra by accounting for light-travel–time delays and Doppler shifts which can produce significant spectral distortions at high ejecta velocities.

3. Improved AI system can infer physical properties of ejecta material—specifically ejecta mass, opacity, and velocity—by fitting early-time X-shooter spectra using Markov Chain Monte Carlo (MCMC) with a Student's t likelihood. This capability allows the system to constrain the underlying ejecta properties by sampling from the posterior distributions shown in Figure 8.

4. Improved AI system can analyze the angular dependence of observed temperature, Tobs, which is defined by Eq (15), and distinguish between effects: the delay increases the limb temperature, while Doppler boosting enhances the temperature near µ = 1. This allows for a more nuanced understanding of spectral shapes across different viewing angles.

5. Improved AI system can perform parameter sensitivity analysis by examining how model parameters like ejecta velocity, mass, opacity, and heating efficiency affect the observed spectra at specific observation times (e.g., tobs = 1.43 days). This helps identify which physical properties are most influential on the spectral shape and luminosity peak position.

6. Improved AI system can perform uncertainty quantification by marginalizing over unknown parameters like luminosity distance (DL) using priors derived from host galaxy measurements, as described in the fitting procedure for AT2017gfo. This ensures that inferred continuum properties are robust against uncertainties in distance estimates.