Exponential Overshooting in Stellar Models

• Exponential overshooting is a stellar mixing prescription where the diffusion coefficient decays exponentially outside the convective boundary instead of remaining uniform.
• It is implemented via models like the Herwig-type diffusive law and emerges naturally in non-local turbulence approaches such as the k–ω and TCM formalisms.
• The effective overshooting length is stage-dependent, influencing convective-core growth, μ-gradient smoothing, and the interpretation of seismic diagnostics.

Exponential overshooting is a stellar-mixing prescription in which the transport coefficient outside a formally convective boundary decreases exponentially with distance, rather than remaining spatially uniform over a fixed extension. In the literature summarized here, it appears both as the standard Herwig-type diffusive law imposed beyond a Schwarzschild-defined core boundary and as an emergent property of non-local turbulence models such as the $k$–$\omega$ model and related TCM formalisms. Its main astrophysical role is to regulate the extent of near-core mixing, the shape of the $\mu$-gradient, convective-core growth, and the seismic signatures of gravity modes; the same studies also show that the effective overshooting distance is stage-dependent rather than universal (Li et al., 2018, Pedersen et al., 2018).

1. Formal definition and canonical parameterizations

In the Herwig-type prescription adopted by Li et al., the overshooting region is treated as a diffusive tail attached to the usual instantaneously mixed convection zone, with

$$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$

Here $D_{\rm OV}(z)$ is the diffusion coefficient at distance $z$ beyond the Schwarzschild boundary of the convective core, $D_0$ is the convective diffusion coefficient at the boundary, $f_{\rm OV}$ is the dimensionless overshoot-efficiency parameter, and $H_P$ is the local pressure scale height evaluated at the boundary. In the formulation used for the SPB-star grids, the same exponential law is written as

$$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$

with $\omega$0 and a small ramp-in parameter $\omega$1 to ensure $\omega$2 just inside the core (Li et al., 2018, Pedersen et al., 2018).

This prescription contrasts with step overshooting, for which the overshoot region is a fully mixed extension with

$\omega$3

for $\omega$4 and $\omega$5 elsewhere. The step model therefore represents a uniform extension of the core, whereas the exponential model produces a gradual tapering of mixing efficiency. In the SPB analysis, this gradual tapering is explicitly connected to the assumption that convective motions penetrate the stable layer with velocities, and hence mixing efficiency, that decay exponentially with distance from the boundary (Pedersen et al., 2018).

A distinct but closely related line of work derives exponential decay from turbulence dynamics rather than imposing it. In the $\omega$6–$\omega$7 formulation for sdB stars, both $\omega$8 and $\omega$9 fall off exponentially in the formally stable region, which yields an approximately exponential mixing coefficient,

$\mu$0

with the e-folding scale set by the turbulence macro-length $\mu$1. The same qualitative behavior is obtained in the TCM asymptotic analysis, where the turbulent kinetic energy obeys $\mu$2 outside the convective boundary [(Li et al., 2018); (Zhang et al., 2012)].

2. Numerical implementation in stellar-evolution calculations

In the $\mu$3 calculations of Li et al., the Herwig-2000 exponential law is used in exactly the textbook form, with $\mu$4 taken to be the local convective-zone diffusion coefficient evaluated at the last convective mesh point. The adopted calibration is stage-dependent: $\mu$5 during the main sequence and $\mu$6 during core-helium burning. The same study emphasizes that all main-sequence models, Herwig and $\mu$7–$\mu$8, were calibrated so that the resulting convective-core sizes were virtually identical (Li et al., 2018).

In the SPB-star grids, the implementation is fully specified within MESA $\mu$9 using the Ledoux convection criterion, $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$0, semiconvection $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$1, and modified Asplund (2009) opacities with $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$2 Fe/Ni. The grid spans $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$3–$$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$4 in steps of $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$5, $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$6–$$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$7 in steps of $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$8, and two evolutionary stages centered on $$D_{\rm OV}(z)=D_0 \exp\!\left[-\frac{2z}{f_{\rm OV}H_P}\right].$$9 and $D_{\rm OV}(z)$0, each sampled by $D_{\rm OV}(z)$1 points. For exponential overshooting, $D_{\rm OV}(z)$2 is varied from $D_{\rm OV}(z)$3 to $D_{\rm OV}(z)$4 in steps of $D_{\rm OV}(z)$5; for step overshooting, $D_{\rm OV}(z)$6 is varied from $D_{\rm OV}(z)$7 to $D_{\rm OV}(z)$8 in steps of $D_{\rm OV}(z)$9 (Pedersen et al., 2018).

These implementations encode different physical assumptions. In the imposed Herwig law, the e-folding length is prescribed by $z$0. In the non-local turbulence treatments, the decay scale is an output of the closure and background stratification. A plausible implication is that numerical agreement in core size does not by itself imply agreement in the detailed radial mixing profile.

3. Main-sequence behavior and seismic diagnostics

For the $z$1 main-sequence models, Li et al. find that the Herwig-overshoot model produces an effective overshoot region of $z$2 beyond the formal Schwarzschild core. Their Figure 1 shows that Herwig’s $z$3 falls by about $z$4 dex over about $z$5, and Figure 2 shows the corresponding hydrogen-profile step at roughly the same scale. In the same stage, the $z$6–$z$7 model is described as equivalent to an overshooting distance of about $z$8, although it decomposes the mixing differently: the fully mixed zone cuts off sharply at about $z$9, while a longer exponential tail reaches about $D_0$0 overall (Li et al., 2018).

The asteroseismic study of SPB stars shows why this profile shape matters. Dipole prograde g-modes probe the Brunt–Väisälä frequency near the core, and a sharp $D_0$1-gradient left by a receding convective core produces spikes in $D_0$2, causing dips in the period-spacing series $D_0$3. Exponential overshoot smooths the $D_0$4-gradient over about $D_0$5, which reduces the amplitude of the dips and shifts them to longer periods. At $D_0$6, increasing $D_0$7 from $D_0$8 to $D_0$9 produces visibly shallower and broader dips in $f_{\rm OV}$0 versus $f_{\rm OV}$1 (Pedersen et al., 2018).

The same analysis quantifies when these differences are observable. Using a merit function $f_{\rm OV}$2 based on benchmark frequencies in the range $f_{\rm OV}$3–$f_{\rm OV}$4 and normalized by the Rayleigh limit $f_{\rm OV}$5, the best-matching step-overshoot model to an exponential-overshoot benchmark at $f_{\rm OV}$6 gives $f_{\rm OV}$7, implying a $f_{\rm OV}$8 discrepancy. At $f_{\rm OV}$9, however, the corresponding value is $H_P$0, so step and exponential overshooting are indistinguishable within $H_P$1. This establishes that the seismic identifiability of exponential overshooting is itself evolutionary-stage dependent (Pedersen et al., 2018).

The SPB study also reports that the three radiative-envelope mixing shapes considered behave the same in g-mode diagnostics, but that a constant envelope mixing requires a diffusion coefficient near the convective core five times higher than chemical mixing from internal gravity waves to obtain a surface nitrogen excess of about $H_P$2 dex within the main-sequence lifetime. This motivates combining average period spacing with measured surface abundances, notably nitrogen, to constrain both core overshoot and envelope mixing (Pedersen et al., 2018).

4. Core-helium burning and stage dependence

The clearest evidence that exponential overshooting is not characterized by a single universal distance comes from core-helium-burning models. In the post-main-sequence $H_P$3 calculations, Li et al. use the same Herwig formula with $H_P$4 and infer from their diffusivity and helium profiles a complete-mixing zone of about $H_P$5, a partial-mixing tail of about $H_P$6, and hence a total diffusion-dominated overshoot region of about $H_P$7. The same paper explicitly states that the overshooting distance in the core-helium-burning stage may be significantly smaller than that in the main-sequence phase for massive stars (Li et al., 2018).

The comparison with $H_P$8–$H_P$9 is particularly informative. In the same core-helium-burning regime, the $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$0–$$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$1 model gives a total overshoot region of about $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$2, but only about $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$3 of that is fully mixed; the remainder is partial mixing. Li et al. therefore conclude that the $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$4–$$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$5 model produces a similar complete-mixing region but a much wider partial-mixing region than the Herwig-based model. They also report that overshooting below the bottom of the intermediate convection zone beyond the hydrogen-burning shell can significantly restrict the size of the hydrogen-depleted core and can penetrate effectively into the hydrogen-burning shell, and that these two effects are crucial for the evolution of the core-helium-burning stage (Li et al., 2018).

The sdB calculations using the $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$6–$$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$7 model sharpen this stage dependence into three regimes. In the initial stage, when $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$8 decreases monotonically from the center, the overshooting mixing has exponential-decay behavior similar to Herwig (2000), and the overshooting distance is such that $$D_{\rm ov}(r)=D_0 \exp\!\left[-\frac{2(r-r_0)}{f_{\rm ov}H_{p,cc}}\right],$$9 at the convective-core boundary. Numerically, the sdB runs give a fully mixed extension of $\omega$00 and a partial-mixing tail out to $\omega$01, where $\omega$02 has fallen to about $\omega$03. In the middle single-zone case, the overshoot width contracts to only about $\omega$04 and no extra completely mixed pocket appears beyond the Schwarzschild boundary. In the middle double-zone case, the core-side tail extends about $\omega$05, while the shell-side tail is essentially zero (Li et al., 2018).

These sdB results are explicitly linked to the self-driving mechanism of Castellani, Giannone, and Renzini: small mixing outside the core boundary brings higher-opacity C and O into the He-rich envelope, raising $\omega$06 there and forcing the Schwarzschild boundary to move outward until $\omega$07. The same work states that the $\omega$08–$\omega$09 scheme is similar to the maximal overshoot scheme of Constantino et al. (2015), but achieves the throttling dynamically by letting the effective decay scale shrink when a buoyancy barrier appears (Li et al., 2018).

The following representative values illustrate the magnitude and variability of the reported scales.

Context	Prescription/model	Reported extent
$\omega$10 main sequence	Herwig, $\omega$11	$\omega$12
$\omega$13 core-He burning	Herwig, $\omega$14	$\omega$15
$\omega$16 core-He burning	$\omega$17–$\omega$18	total $\omega$19; fully mixed $\omega$20
sdB initial He-burning stage	$\omega$21–$\omega$22	complete $\omega$23; tail to $\omega$24
sdB middle single-zone stage	$\omega$25–$\omega$26	partial width $\omega$27

Taken together, these results support a narrow interpretation of “exponential overshooting”: the exponential shape can remain intact while the physically relevant width varies strongly with evolutionary state and local stratification.

5. Turbulent-convection theory and emergent exponential decay

The TCM analysis provides a semi-analytic explanation for why exponential decay appears so naturally in overshooting regions. For large turbulent Péclet number, the overshooting zone is partitioned into three parts: a thin region just outside the convective boundary with high efficiency of turbulent heat transfer, a power-law dissipation region of turbulent kinetic energy in the middle, and a thermal dissipation area with rapidly decreasing turbulent kinetic energy. In the asymptotic region, the turbulent correlations are written as

$\omega$28

with

$\omega$29

The decaying indices of $\omega$30, $\omega$31, and $\omega$32 are determined by the TCM parameters, and the theory also predicts an equilibrium value of the anisotropic degree $\omega$33 (Zhang et al., 2012).

The same analysis states that the overshooting length of the turbulent heat flux $\omega$34 is about $\omega$35, and that the boundary value $\omega$36 can be estimated by the “maximum of diffusion” method. A natural diffusion coefficient for one-dimensional stellar codes is then

$\omega$37

or, equivalently,

$\omega$38

with $\omega$39 tuned to give the same boundary mixing. Typical TCM-calibrated values quoted in the summary are $\omega$40, $\omega$41–$\omega$42, $\omega$43–$\omega$44, $\omega$45, and $\omega$46–$\omega$47, which imply $\omega$48–$\omega$49, $\omega$50–$\omega$51, and $\omega$52–$\omega$53 (Zhang et al., 2012).

Direct non-local TCM calculations in RGB and AGB envelopes are consistent with this picture. Li and Yang report that in the overshooting regions the turbulent kinetic energy follows

$\omega$54

to very good approximation over a substantial fraction of the overshooting zone, and that nearly perfect straight lines are obtained by fitting $\omega$55 versus $\omega$56. Their quoted example for the $\omega$57 AGB model gives $\omega$58 in the top overshooting region, which corresponds to $\omega$59 (Lai et al., 2011).

The same RGB/AGB calculations show that the e-folding length of $\omega$60 is larger than the overshooting distance of the heat-flux correlation. At the base of the convective envelope, the reported values are $\omega$61 for the $\omega$62 AGB model, $\omega$63 for the $\omega$64 RGB model, and $\omega$65 for the $\omega$66 RGB model, while the full overshooting distance of $\omega$67 is about $\omega$68 in all three cases. The same study also finds that the top-region values are $\omega$69, $\omega$70, and $\omega$71, respectively, and that these lengths decrease slightly as the stellar model moves up the Hayashi line (Lai et al., 2011).

6. Calibration, equivalence, and recurrent misconceptions

A recurring misconception is that exponential overshooting is simply another way of specifying a fully mixed extension of size $\omega$72. The cited studies do not support that identification. In the imposed Herwig prescription, the convection zone remains instantaneously mixed but the exterior region is a diffusive tail; in the $\omega$73–$\omega$74 and TCM approaches, the same exponential behavior often coexists with a narrow fully mixed extension plus a wider partially mixed zone. The $\omega$75 comparison makes this explicit: on the main sequence, Herwig and $\omega$76–$\omega$77 can be tuned to give nearly identical core masses and hydrogen profiles, yet the $\omega$78–$\omega$79 model decomposes the mixing into a narrower fully mixed core plus a longer tail (Li et al., 2018).

A second misconception is that a single $\omega$80 can be transferred unchanged between phases. The massive-star calculations use $\omega$81 on the main sequence and $\omega$82 in core-helium burning, and the sdB results go further by showing that the effective decay scale can shrink from about $\omega$83 in the initial stage to about $\omega$84 in the single-zone middle stage. This suggests that “exponential overshooting” names a functional form rather than a universal physical distance (Li et al., 2018, Li et al., 2018).

A third misconception is that g-modes always distinguish exponential from step overshooting. The SPB calculations show the opposite: such discrimination is possible at $\omega$85 within Kepler precision, but it disappears toward the terminal-age main sequence at $\omega$86. The same work also states that g-modes cannot discriminate between different envelope-mixing shapes outside the near-core region, which is why it recommends combining seismic information with measured surface abundances and, as a future strategy, additional diagnostics such as mixed p/g modes, rotational splittings, and 2D/3D modelling (Pedersen et al., 2018).

Within stellar-structure theory, exponential overshooting therefore occupies an intermediate position between phenomenological prescription and turbulence-derived behavior. It is “commonly adopted” as an exponentially decaying diffusion law, but several non-local models recover comparable exponential tails from the dynamics of $\omega$87, $\omega$88, and related turbulent correlations. The main unresolved issue is not whether exponential decay can occur, but how its amplitude, e-folding length, and fully mixed component should be calibrated as functions of evolutionary stage and local stratification.