SPICE -- modelling synthetic spectra of stars with non-homogeneous surfaces

Abstract

In the era of large time-domain spectro-photometric surveys, surface variations such as starspots, chemical inhomogeneities, pulsations, rotational distortions, and binary interactions can now be directly detected and modelled. Accurately interpreting these phenomena requires stellar spectral synthesis frameworks that go beyond the assumption of homogeneous surface properties. Yet most existing tools remain limited by this simplification, hindering their applicability to stars with complex surface structures. To address this need, we present SPICE (SPectral Integration Compiled Engine), an open-source Python package for generating high-resolution spectra and photometry from non-homogeneous stellar surface models. SPICE integrates angle-dependent specific intensities from each surface element, enabling forward modelling of both photometric and spectroscopic variability. Case studies demonstrate applications to spotted stars, Cepheid pulsations, and eclipsing binaries, making SPICE well-suited for analysing current and upcoming survey data. In addition, SPICE can directly import meshes from PHOEBE, enabling the modelling of complex binary configurations beyond these case studies.

Figures (24)

• Figure 1: Synthetic observed-flux computation model for a triangulated stellar surface. The left panel illustrates the geometry of the synthetic mesh, where each triangular element $i$ (highlighted in pink) contributes an intensity $I_{\lambda,i}$ that is modulated by the cosine of the angle $\mu_i = \cos\theta_i$ between the surface normal vector $\vec{n}_i$ and the observer’s line of sight. The projection of the surface element with area $S$ onto the observer’s plane is also indicated. SPICE assumes parallel light rays (i.e. the star is treated as a point source at infinite distance), which is appropriate for all but the nearest resolved stellar surfaces. The right panel shows an example of the resulting synthetic flux $F_\lambda$, obtained by summing the contributions from all visible mesh elements.
• Figure 2: Schematic overview of the spectrum-synthesis workflow in SPICE. The diagram summarises the main steps from mesh construction and surface-parameter assignment, through spectral-intensity emulation, to the integration of synthetic spectra.
• Figure 3: Three-dimensional velocity field of a rotating stellar model. White arrows show velocity vectors for a representative subset of mesh elements for clarity. The colour scale indicates line-of-sight (LOS) velocities computed from the full 3D velocity field and a chosen viewing direction. In SPICE, receding velocities are defined as negative.
• Figure 4: Example of a stellar surface with spots of different sizes and smoothness parameters. The illustration highlights how varying the spot radius and smoothness parameter $\alpha$ leads to sharper or more gradual transitions between the spot and the surrounding surface.
• Figure 5: Example stellar surface with a temperature distribution defined by a spherical harmonic function with $m = 4$ and $\ell = 4$, tilted by $45^\circ$ with respect to the rotation axis. This configuration illustrates how spherical harmonics can capture structured, non-axisymmetric surface features.