Chroma+ model stellar surface intensities: Spherical formal solution

TL;DR

This work extends the Chroma+ package by implementing a spherical formal solution for the radiative transfer equation (RTE) and its coupling to hydrostatic balance (HSE) in a finite spherical atmosphere, using Chapman’s integrating-factor approach. A discretized 1D spherical framework (64 layers, 32 Gauss-Legendre directions) computes the emergent surface intensity and updates the pressure structure, with a new ifSphere option to compare spherical and plane-parallel geometries. Results show that sphericity induces measurable differences in spectral energy distributions, center-to-limb intensity variations, and exoplanet transit light curves, especially for low surface gravity models, while remaining fast and accessible across languages. The paper demonstrates a practical, platform-independent tool for teaching and rapid diagnostic experiments in stellar atmospheres and transit photometry, complementing existing spherical codes by enabling interactive exploration of sphericity effects.

Abstract

We announce V. 2025-08-08 of the Chroma+ suite of stellar atmosphere and spectrum modelling codes for fast, approximate, effectively platform-independent stellar spectrum synthesis, written in a number of free well-supported programming languages. The Chroma+ suite now computes the emergent surface intensity and flux distributions and the hydrostatic pressure structure assuming a spherical atmosphere rather than local flatness by implementing the analytic formal solution of the 1D spherical radiative transfer equation of Chapman (1966} based on an integration factor. We present our adaptation and discretization of the solution and demonstrate the resulting impact of our sphericity treatment on a number of computed observables, including exo-planet transit light-curves. All codes are available from the OpenStars www site: www.ap.smu.ca/OpenStars.

Figures (3)

• Figure 2: Spectral energy distribution (SED) for the test models of flr90 of $T_{\rm eff}$ of $10\,000$ K, $M$ of 2.0 M$_{\rm Sun}$ and $\log g$ 4.0 (upper panel) and 1.5 (lower panel) computed with the spherical solution (blue solid line) and the PP solution (red dashed line), comparable to Fig. 2 of flr90.
• Figure 3: The continuum limb darkening curve for the test model of neilson13a of $T_{\rm eff}$ of 5000 K, $\log g$ of 2.0, and $M$ of 5.0 M$_{\rm Sun}$ (lower panel) computed with the spherical solution (blue solid line) and the PP solution (red dashed line). For comparison we also include the limb darkening for a model of the same parameters except for a $\log g$ value of 4.5 (upper panel). The limb darkening is shown for a $\lambda$ value from our background continuum $\lambda$ grid of 653.1 nm, near the centre of the Kepler bandpass, and is comparable to Fig. 1 of neilson13a.
• Figure 4: Upper panel: The transit light-curve at ingress for the model of Fig. \ref{['CompareNeilsonLDC']} and a planet of radius equal to $1 R_{\rm Jup}$ and orbital radius equal to 1 AU computed with the spherical solution (blue solid line) and the PP solution (red dashed line). Middle and lower panels: The relative difference between the light-curve computed with the spherical and the PP model.