Forward-Mapping Flux¶

For on-axis viewing angles (\(\theta_v \approx 0\)), blastwave supports a forward-mapping flux computation that replaces the default EATS adaptive quadrature. This is selected via flux_method="forward".

Background: EATS vs forward-mapping¶

The standard approach to computing observer-frame flux from a relativistic blast wave is inverse mapping via the Equal Arrival Time Surface (EATS). For each observation time \(t_\mathrm{obs}\) and each angular cell, one must root-find the source-frame time \(t_\mathrm{src}\) satisfying:

This root-finding, combined with adaptive quadrature over \((\theta, \phi)\), dominates the cost of computing multi-frequency light curves.

Forward-mapping inverts this: instead of finding \(t_\mathrm{src}\) for a given \(t_\mathrm{obs}\), it pre-computes \(t_\mathrm{obs}\) and the radiation at every hydro grid point, then queries via binary search.

Algorithm¶

Pre-computation¶

After the hydro solve, for each theta cell \(j\) and source time step \(k\):

1. Compute observer-frame time:

   \[ t_\mathrm{obs}[j][k] = t_\mathrm{src}[k] - \frac{R[j][k] \cos\theta_j}{c} \]

2. Compute spectral luminosity per steradian:

   \[ \frac{dL}{d\Omega}[j][k] = I(\nu_\mathrm{src}) \cdot R^2 \cdot \mathcal{D}^3 \]

where \(\mathcal{D}\) is the Doppler factor and \(I(\nu_\mathrm{src})\) is the specific intensity from the chosen radiation model.

Query¶

For a desired observation time \(t_\mathrm{obs}\):

1. In each theta cell, binary search the pre-computed \(t_\mathrm{obs}\) array
2. Log-linear interpolate \(dL/d\Omega\)
3. Sum over solid angle:

Optimizations¶

• Cell skipping: Cells with negligible energy (peak \(\beta\Gamma^2 < 10^{-6} \times \max\)) are excluded
• Time-grid subsampling: When the hydro output exceeds 300 time steps (PDE mode produces ~1000), the time grid is subsampled to limit pre-computation cost
• Off-axis fallback: For \(\theta_v > 0\), flux_method="forward" silently falls back to EATS (forward-mapping assumes azimuthal symmetry)

Usage¶

# On-axis: uses forward-mapping
fd = blastwave.FluxDensity_tophat(t, nu, P, flux_method="forward")

# Or via the Jet class
fd = jet.FluxDensity(t, nu, P, flux_method="forward")

# Off-axis: silently falls back to EATS
P_offaxis = {**P, "theta_v": 0.3}
fd = blastwave.FluxDensity_tophat(t, nu, P_offaxis, flux_method="forward")

Accuracy¶

Forward-mapping agrees with EATS to within:

Verified across top-hat (0.002 dex), Gaussian (0.002 dex), and power-law (0.004 dex) jet profiles.

Performance¶

Forward-mapping reduces flux computation from milliseconds to ~0.1 ms, making the remaining performance gap vs VegasAfterglow (~1.5 ms) entirely hydro solve overhead.

17 cells, top-hat, on-axis, X-ray, 100 time points.

References¶

• Wang, Y., Zhang, B., & Huang, B. (2025). "VegasAfterglow." ApJ.