High-energy emission

All high-energy emission is computed from the instantaneous rotation state in physicalmodel.ExtendedQuantities. Throughout this section, surface fluxes are evaluated at the stellar surface (in erg s\(^{-1}\) cm\(^{-2}\)) and luminosities are in erg s\(^{-1}\).

1. X-ray emission

The ratio \(R_X = L_X / L_\mathrm{bol}\) is related to the Rossby number by a broken power law constrained using the sample of Wright et al. (2011) ² with convective turnover times from Spada et al. (2013) ³:

The constants \(C_1\) and \(C_2\) are derived from the requirement that the two power laws are equal at the saturation point (\(R_{X,\mathrm{sat}} = C_1 Ro_\mathrm{sat}^{\beta_1} = C_2 Ro_\mathrm{sat}^{\beta_2}\)). The fitted parameters are:

The \(R_X\) relation is shallower in the unsaturated regime than many previous estimates, consistent with the Sun being less X-ray active than other stars with similar parameters ⁴. The X-ray luminosity and surface flux then follow as:

Implementation: physicalmodel._Xray.

X-ray variability

Real stellar X-ray emission varies around the average relation. The observed scatter in the \(Ro\)–\(R_X\) distribution can be described as a log-normal centred on zero with standard deviation \(\sigma = 0.359\) dex (params['sigmaXray']), meaning stars spend approximately 90% of their time within one standard deviation of the average ¹. This scatter can be sampled with physicalmodel.XrayScatter or physicalmodel.XUVScatter, which apply correlated random offsets consistently across all XUV bands.

2. Coronal temperature

Stars with higher X-ray surface fluxes have hotter coronae. The emission-measure-weighted average coronal temperature is estimated from \(F_X\) following Johnstone & Güdel (2015) ⁵:

This relation is mass-independent when expressed in surface fluxes. Since coronae dominate emission at wavelengths below \(\sim\)40 nm, stars with higher \(F_X\) emit a larger fraction of their XUV at shorter wavelengths. Implementation: physicalmodel._Tcor.

3. EUV emission

EUV emission is empirically related to \(F_X\) rather than to \(L_X\) or \(R_X\), since \(F_X\) best captures the physical state of the emitting plasma ⁵. The relations are derived from EUVE observations of nearby F, G, K, and M stars ⁶ and solar spectra.

EUV band 1 (10–36 nm)

Constrained from the EUVE stellar sample using the OLS Bisector method (Johnstone et al. 2021 ¹ Eq. 19):

which gives

Implementation: physicalmodel._EUV1.

EUV band 2 (36–92 nm)

Constrained using solar spectra only, considering solar values with \(L_X > 10^{27}\) erg s\(^{-1}\) (Johnstone et al. 2021 ¹ Eq. 21):

which gives

This relation is less reliable than the \(F_X\)–\(F_\mathrm{EUV,1}\) relation since it is derived from the Sun alone, which samples only a small fraction of the parameter space. Implementation: physicalmodel._EUV2.

Total EUV

Implementation: physicalmodel._EUV.

4. Ly-\(\alpha\) emission

The Ly-\(\alpha\) line at 121.6 nm is formed in the transition region and upper chromosphere ⁷ and is often more luminous than the entire X-ray and EUV combined. Although most of the line is absorbed by the ISM, reconstructions of the intrinsic line flux are available for a large number of stars ⁸.

The Ly-\(\alpha\) surface flux is related to \(F_X\) following Wood et al. (2005) ⁸ and Linsky et al. (2013) ⁹. When expressed in surface fluxes (rather than luminosities or 1 AU fluxes), the relation becomes mass-independent (Johnstone et al. 2021 ¹ Eq. 23):

which gives

where both fluxes are in erg s\(^{-1}\) cm\(^{-2}\). Implementation: physicalmodel._Lymanalpha.

5. Habitable zone fluxes

\(F_X\), \(F_\mathrm{EUV,1}\), \(F_\mathrm{EUV,2}\), \(F_\mathrm{EUV}\), and \(F_\mathrm{Ly\alpha}\) are all also computed at the habitable zone distance (FxHZ, Feuv1HZ, Feuv2HZ, FeuvHZ, FlyHZ) and stored on every evolutionary track. The habitable zone distance is defined as half-way between the moist and maximum greenhouse limits, calculated at 5 Gyr stellar properties. See Habitable Zone for details.