On the Possibility of an Upper Limit on Magnetically Induced Radius Inflation in Low-mass Stars

The radii of low-mass stars are observed to be inflated above standard model predictions, especially in magnetically active stars. Typically, the empirical relative radius inflations ΔR/R are ≤10% but in (rare) cases may be ≥20%. Our magneto-convective stellar models have already replicated many empirical ΔR/R values. Here, we ask: is there any theoretical upper limit on the amount of such inflation? We use our magneto-convective model to compute ΔR/R using empirically plausible values of the surface field strength parameter δ. Inside each model, the maximum internal field is set to a particular value: B ceil = 10, or 100 kG, or 1 MG. When B ceil = 10 kG, peak inflation with ΔR/R ≈ 90% occurs in stars with masses of 0.7 M ⊙. With B ceil = 100 kG, peak inflation with ΔR/R ≈ 140% occurs in stars with M ≈ 0.5 M ⊙. But with B ceil = 1 MG, we find no peak in ΔR/R as a function of δ; instead, the larger δ is, the larger ΔR/R becomes, reaching 300%–350% in the case of the largest δ considered. Thus, magneto-convective modeling can accommodate ΔR/R values which are considerably larger than any reported empirical inflations. We find that a maximum occurs in ΔR/R as a function of δ only in model stars where the field reaches its maximum strength B ceil inside the convective envelope. Moreover, our models of completely convective stars undergo smaller amounts of relative radius inflation than models with radiative cores, a result consistent with some previous reports.


Introduction
Empirical evidence indicates that certain late-type dwarfs (especially those which are magnetically active) have radii R which exceed the predictions of standard stellar models (Leggett et al. 2000;Mullan & MacDonald 2001, hereafter MM01).The excesses in radii are observed to be typically several percent of the standard radii, although some are reported to be as large as 20% (or more).Models of such "inflated radii" were addressed by MM01 by means of a magneto-convective model based on a particular criterion due to Gough & Tayler (1966): We refer to this criterion for the onset of magneto-convection as the GT66 criterion.According to GT66, when a medium with high electrical conductivity and gas pressure p g is permeated by a vertical magnetic field B v , magneto-convection sets in only when ∇ > ∇ ad + δ.Here,  = d T d P ln ln , while ∇ ad = ( ) d T d P ln ln ad .The magnetic inhibition parameter δ, introduced by GT66, is defined by where γ is the local ratio of specific heats.The parameter δ depends on B v , scaling roughly as B v 2 /p g for small values of δ, but saturates at a maximum value of δ = 1 when the field is infinitely strong.In the absence of a (vertical) magnetic field, the GT66 criterion for convective onset reduces to the standard Schwarzschild criterion ∇ > ∇ ad .
Of course, no star contains a field which is infinitely strong.Therefore, although in this paper we will explore a range of δ values, and determine quantitatively how the structure of the star is altered in each case, we will certainly not need to explore the case δ = 1.Even the case δ = 0.9 may be too extreme to be relevant to real stars: In such a case, the magnetic field would be so strong that the field would exert a pressure of order 10 times the local gas pressure.From a physics perspective, it is not clear how such a field would ever be generated in the turbulent medium which exists inside a convective envelope or core of a low-mass star.The quantitative question is: What range of δ values should we explore and still retain a hope of having a plausible connection with "real stars"?Once we have calculated some models of the internal structure of low-mass magnetic stars, we shall return to this quantitative question: We shall find that several "real stars" exhibit at their surface a value of δ which may be as large as 0.8 (see Section 5.1).As a result, we recognize that some of the models we have calculated (i.e., the "high-δ cases" with δ > 0.8) are probably less relevant in the real world than our "low-δ cases." The GT66 criterion provides a quantitative statement of the fact that large B v makes it more difficult for convection to set in in highly (electrically) conductive material.In fact, when B v is large enough, convective motions in such a medium can in principle be completely suppressed.Observational confirmation that magnetic fields interfere with convection in the Sun is provided by observations of reduction in convective velocities in sunspot umbrae by more than an order of magnitude (Löhner-Böttcher et al. 2018).Moreover, in support of the GT66 criterion, Jurcak et al. (2018) have reported that a preferred value of B v (≈1800 G) occurs in sunspots at the boundary between umbra and penumbra.
In our initial paper on modeling magneto-convection (MM01), we found that our models could replicate the empirical parameters for certain inflated low-mass stars, although the data which were available at that time had relatively large error bars.The major uncertainty in our models was that we were forced to assume a particular radial profile of the magnetic field strength inside the star; in some cases, our assumed profile led to very strong fields at the center of the star.In an attempt to avoid such strong fields, and arguing on the basis of a limit on the available rotational energy in a star, in 2012 we imposed an ad hoc upper limit (a "ceiling field" B ceil ) of 1 MG on the (vertical) field strength inside a model of the (fast-rotating) stars in CM Draconis (MacDonald & Mullan 2012).Subsequently, Browning et al. (2016) demonstrated that the central field strengths in low-mass stars could not be stable if they exceeded a value of order 1 MG: Our choice of B ceil in CM Dra in 2012 turned out to be serendipitously consistent with this limit.In subsequent papers, we computed magnetoconvective models of stars in which various numerical values were assigned for B ceil 1 MG; the smallest value we assigned to B ceil in these models was 10 kG.These models indicated that, even though the steadily improving empirical precision of stellar parameters made it increasingly challenging to fit the data, our models were successful in replicating the empirical R and effective temperatures T eff in a sample of about 20 stars (e.g., MacDonald & Mullan 2017b; Mullan & MacDonald 2020;MacDonald & Mullan 2021).In most cases, we obtained satisfactory fits to the empirical data even when we set B ceil at its lowest value (10 kG).
In this paper, our principal goal is to explore the following question: Is there a maximum magnetically induced inflation in the radius of a low-mass star as the surface magnetic field increases in strength?However, before reporting on the numerical results on which that exploration is based, we consider it worthwhile to report in Section 2 on independent theoretical confirmation of the inflation results we have already published using our magneto-convective model.Then, in Section 3, we also revisit the question of how to choose the ceiling field in our model.In Section 4, we summarize a recent literature search which yields up-to-date information on radius inflations in a sample of several dozen cool dwarf stars.In Section 5, we report on new results from our magnetoconvection code which suggest that, in certain conditions, an upper limit exists on magnetically induced radius inflation in cool dwarfs as the surface field increases.Discussion and conclusions are provided in Section 6. Ireland & Browning (2018) The stellar models which were reported in MM01 and in subsequent papers have been computed by means of a stellar evolution code which has been developed over the course of some years by J. MacDonald.That code started off in a nonmagnetic version, and the magnetic effects started to be included from the time of MM01 onward.The essential aspect of the magnetic version of this code is that it includes a quantitative description (the GT66 criterion) of how (vertical) magnetic fields interfere with the onset of convection.

The Work of
As was recognized from the beginning of our studies (MM01), the principal source of uncertainty in our models concerns the question: How are we to specify the radial profile of the (vertical) magnetic field strength?The approach we have adopted starts with assigning a numerical value to the inhibition parameter δ at the photosphere of a model with given stellar mass.The principal assumption in our models in the nearsurface layers is to hold the value of δ constant: This has the effect that as we move deeper into the stellar interior, i.e., into regions of increasing gas pressure, the local value of the (vertical) field strength must increase with increasing depth.This depth-related increase in field strength is allowed to continue until the local value of (vertical) field strength becomes equal to B ceil .At greater depths, the model assumes that the field strength no longer increases with increasing depth, but instead remains constant at B ceil all the way to the center of the star (see, e.g., MacDonald & Mullan 2017b, their Figure 1).The enforced constancy of B v = B ceil has the effect that the local numerical value of the parameter δ decreases monotonically between the depth z ceil at which the ceiling value is first reached and the center of the star (see MacDonald & Mullan 2017b, their Figure 2).
Can the results we have obtained using our magnetoconvective code for magnetically induced radius inflation be corroborated by other investigators using an independent evolution code?

Numerical Modeling of Magneto-convection
Ireland & Browning (2018, hereafter IB18) have reported on this issue using 1D stellar models constructed using Modules for Experiments in Stellar Astrophysics (MESA; Paxton et al. 2015).The MESA code is in widespread use for "standard" stellar models; we use the word "standard" to refer to models in which no magnetic effects were originally included.In order to extend the MESA code to a magnetic case, IB18 incorporated into the code a slightly modified GT66 criterion which includes the departure from ideal gas behavior (MacDonald & Mullan 2014, hereafter MM14).Here, we shall refer to the IB18 modified version of the MESA code as the "magneto-MESA" code.In their first approach to the inflation problem, IB18 repeated the procedure of MM01 (and later papers, especially MM14) to compute a numerical model of a particular magnetic star: This model provides them with a "numerical" value of the radius R num (MM) of that stellar model.Comparing R num (MM) with the radius R s of the nonmagnetic stellar model having the same mass, IB18 determined a "numerical" estimate of the fractional inflation ΔR/R = (R num (MM)-R s )/R s .By varying the GT66 parameter δ over a range of values from 0.01 to 0.03, and by varying the magnitude of the "ceiling field" (see Section 1 above) from 1 to 100 kG, IB18 computed more than a dozen models for young (10 Myr) magnetic stars, all with a mass of 0.3 M e .They obtained a "numerical" estimate of ΔR/R for each model.In Figure 10(a) of the IB18 paper, they plotted, along the x-axis, this estimate of the fractional inflation, labeling the axis as "Inflation [%] (MM14)": The numerical inflations were found to range from 1% to 2% up to 13%.Moving to older stars (1 Gyr), IB18 repeated the process for a sample of about 20 models, using a range of δ values from 0.04 to 0.06.In this case (see their Figure 10(b)), the numerical inflations were found to range from essentially zero up to 6%.

Entropy and Its Connection with Radius Inflation
In order to check the "numerical" inflations that they obtained using the above approach, IB18 then switched to an independent approach which had played no role in the modeling of MM01 or MM14.In developing this independent approach to the inflation problem, IB18 replied explicitly on the physical parameter entropy of each stellar model.To do this, they first calculated the specific entropy (i.e., entropy per unit mass) at each mass coordinate inside the star and then integrated over the stellar mass to determine the total entropy S tot in the star.In particular, they stressed that, although in principle deep convection in a star is isentropic, "the structure models calculated by MESA (or any other stellar structure code) are not isentropic."The level of departure from isentropy depends on the details of the models, and in particular on the convective mixing length.In practice, the models indicate that "most of the entropy resides in the deep interior with nearly constant specific entropy s ad ."As a result, in a fully convective star (such as IB18 consider, with mass 0.3 M e ), it is acceptable to write S tot ≈ s ad M, where M is the mass of the star.However, in the case of a star which is not fully convective, i.e., those on the main sequence with masses in excess of (roughly) 0.35 M e , it needs to be recognized that the entropy associated with nonconvective zones inside the star (in the deep radiative core) is lower than in the convective envelope.As a result, the total entropy in such a star has a numerical value which is less than s ad M.
For present purposes, the essential point to note is that the specific entropy s ad in a stellar convection zone turns out to have a larger numerical value if the efficiency of convective transport is (for some reason, for example due to the presence of vertical magnetic field) reduced.The reason for the larger s ad is that in the presence of reduced convective efficiency, the magnitude of the superadiabatic gradient ∇ s = ∇ − ∇ ad is forced to take on a larger numerical value. in order to transport a given flux F c of convective energy.The magnitude of the flux F c is determined by nuclear processes occurring near the center of the star, where local conditions are controlled mainly by the (total) mass of the star.According to the mixing length theory (MLT) of convection, the numerical value of F c is proportional to the product of local gas density ρ times convective speed v times ∇ s .The magnitude of v depends on ∇ s ; in MLT, v ∼  s 0.5 .Thus, F c ∼ ρ s 1.5 .Therefore, in a medium where ∇ s is enhanced (e.g., where convective efficiency is smaller), the requirement that a certain flux of energy F c is to be transported has the effect that, in a region where the local temperature is T, the local gas density ρ must be reduced (relative to conditions where convective efficiency is larger).Given that the specific entropy in a gas varies as ln(T a /ρ) (where a is positive definite; see IB18), it follows that in a gas with a certain T, smaller convective efficiency, with its accompanying reduction in ρ, leads to larger specific entropy.
To make this conclusion quantitative, IB18 consider the case of convection in a star of given mass (0.3 M e ) as described by a "mixing length model": The mixing length is set equal to α MLT times the local pressure scale height.IB18 show explicitly (see their Figure 1) that when α MLT is assigned a small value ( = 0.5, representing convection with low efficiency), the numerical value of ∇ s is indeed larger at all depths compared to a model with α MLT = 2.0 (representing convection of high efficiency).IB18 find that the value of ∇ s in their model with α MLT = 0.5 exceeds by a factor of almost 10, at all depths, the value of ∇ s in their model with α MLT = 2.0.IB18 demonstrate (see their Figure 3) that this larger value of ∇ s in the model with α MLT = 0.5 translates to a systematically larger value of specific entropy throughout the deep interior relative to the more efficient model with α MLT = 2.0.

Magneto-convection Based on the GT66 Criterion
In this context, we note that according to the GT66 criterion, ∇ > ∇ ad + δ, models in which the GT66 criterion is used (e.g., MM01; MM14) are in essence forcing ∇ s to take on the numerical value of δ.As a result, when we consider models of active stars in which δ is assigned to take on increasingly large values, for example increasing from (say) 0.01 to (say) 0.1, the superadiabatic gradient inevitably is forced to become increasingly large.This translates into larger values of entropy in stars where (in some cases which we have already reported; MacDonald & Mullan 2017b) the observed inflated radii required δ values which were no larger than 0.1 (or so).In the present paper, we shall present models in which the value of δ is chosen to be considerably larger than in any of our previous models.As a result, some of the results in the present paper will deal with stars in which the efficiency of convection is significantly smaller than we have previously reported.In that sense, we shall report here on models that can be regarded as "more extreme" in their magnetic effects than any models we have reported hitherto.As a measure of how much "more extreme" the models in the current paper can be, we shall report (in Section 5.3) on significant changes in global structure.

Enhanced Entropy: Relationship with Inflated Stellar Radius
Once IB18 have a numerical estimate of S ad for any particular magnetic model, they use theoretical relations in the textbook of Hansen et al. (2004) to show that the stellar radius is exponentially sensitive to the value of S tot .Using their value of S tot in any particular magnetic model, IB18 compute an entropy-based estimate of the radius R ent for each magnetic model.In combination with the nonmagnetic radius R s , the amount of "entropy" inflation is calculated using ΔR/R = (R ent -R s )/R s .They then compare this "entropy" inflation with the "numerical" inflation which emerged from the corresponding magneto-MESA model.The results (which appear in their Figures 10(a) and (b)) are plotted along the y-axis, labeled as "Inflation [%](ΔS ad )," in their Figures 10(a) and (b).The plots show remarkably good agreement between the "entropy" and "numerical" inflations, at least for the results that are plotted (which extend up to a maximum fractional inflation of about 13%).
In summarizing their analysis of our magneto-convective model, IB18 note that "in accord with MacDonald & Mullan (2017b), we find that if magnetic fields indeed influence convective transport in the manner assumed here, fields of a plausible strength (10 kG or less) could noticeably 'inflate' the stellar radius."

Summary of the Comparison between IB18 Results and Our Magneto-convective Models
The work of IB18 provides a significant corroboration of our magneto-convection modeling approach because the IB18 computer code differs in detail from our code, including differences in the choice of mixing length and in computational techniques.As IB18 stress, these coding differences inevitably give rise to differences in the numerical values of S tot between their models and ours in the deep interior.Moreover, since the "entropic radius" depends exponentially on the value of S ad (Hansen et al. 2004), it is in principle possible that even rather small differences in S ad might have led to significantly different inflations.In view of this, it is gratifying to see that no such differences are seen between the IB18 "entropy" inflations and our results for magneto-convective inflations.
In what follows, we will use the results of our magnetoconvective calculations to report for the first time on the increase in entropy which occurs in each of our stellar models.Moreover, we examine how this increase in entropy is correlated with radius inflation.

The Strength of the Ceiling Field in Our Magnetoconvection Models
As mentioned in Section 1, one of the major uncertainties of our modeling approach concerns the radial dependence of field strength inside the star.In particular, how should a plausible limiting value for the "ceiling field" be chosen?If dynamo mechanisms of particular kinds are at work inside the stars we are modeling, it may be possible to estimate upper limits on what the field strengths could be.
Several possible regimes have been discussed by Bugnet et al. (2021).( 1) The turbulent equipartition regime: the convective kinetic energy (KE) density of the gas is fully converted into magnetic energy (ME) density, giving rise to a field strength B eq (KE).If the ME becomes greater than the KE, the star is in a superequipartition regime.(2) The magnetostrophic regime: Coriolis acceleration balances the Lorentz force in the momentum equation.(This regime generates a field with maximum strength, and may be applicable to stellar dynamos; Augustson et al. 2019).(3) The buoyancy dynamo regime: Lorentz, Coriolis, and buoyancy forces are all comparable in magnitude.In this case, the limiting field strength may be intermediate between those of mechanisms (1) and (2).(4) A possible fourth regime (not discussed by Bugnet et al. 2021) involves an equipartition that is not based on convective KE, but is based instead on the local gas pressure p; in this case, since convective flows are typically subsonic (so as to avoid rapid dissipation), the strength B eq (p) of mechanism (4) certainly exceeds the field strength B eq (KE) obtained in regime (1).As an example of a model of differential rotation in which the magnetic pressure is large enough to balance the gas pressure, see Hotta et al. (2022).
Using the physical conditions which exist at the bottom of the convective envelope inside a star with mass <1.1 M e , Bugnet et al. (2021) estimate that the typical values of field strength in regimes (1), (2), and (3) may be of order 10 kG, 1 MG, and 100 kG, respectively.In regime (4), with field strengths in excess of those in regime (1), the typical values of field strength are expected to exceed 10 kG.
In our calculations of various magneto-convective models during the past ∼15 yr, we have assumed that the ceiling field ranges from as weak as 10 kG (e.g., MacDonald & Mullan 2017b) to as much as 1 MG (e.g., MacDonald & Mullan 2012).In fact, the limit of 10 kG was based on models of 3D convection inside a solar-like model by Yadav et al. (2015), where the field is generated by the KE of convective flows; this corresponds to Bugnet et al.ʼs (2021) regime (1).Our choice of 1 MG for the ceiling field was based on an estimate of how strong a field would be in a low-mass star (CM Dra) which rotates some 20 times faster than the Sun; in other words, we were relying on rotational speeds to estimate how strong a field might be generated.This corresponds (roughly) to Bugnet et al.ʼs (2021) regime (2).
We note that IB18 consider a range of ceiling fields from 1 kG to 100 kG in their magneto-MESA models.This range of ceiling fields overlaps significantly with the range that we have used over the years in our magneto-convective models, although the mean fields in the three regimes cited by IB18 are roughly 1 order of magnitude weaker than the ceiling fields we have used in various models.
A result which has emerged from our earlier models, where we considered only relatively low values of δ, is that the strength of the surface magnetic field B surf is very insensitive to the choice of the ceiling field B ceil .MacDonald & Mullan (2017a) have shown that, when B ceil is varied over the range 10 kG to 1 MG (i.e., precisely the range we have considered here for the values of B ceil ), the value of B surf scales as B ceil −0.07 .This is a very slowly varying function, showing that our results for low δ values do not depend sensitively on the choice of value for B ceil .In work to be reported below (Section 5.6), we shall revisit the relationship between B surf and B ceil in the regime of large values of the GT66 parameter δ.

Empirical Data on Radius Inflation
Plotted in Figure 1 are empirical values of the relative radius inflation (expressed as a percentage), which can be found in the literature as a function of the effective temperature T eff of the star.The data, which were assembled from a variety of references, are listed in Table 1.Error bars have been estimated from data given in the relevant paper.We make no claim for an exhaustive list of targets: We merely scanned the literature over a random time interval until we amassed information on a few dozen stars.But, given the randomness of our selection process, we can think of no reason why our choice of targets should not be considered as more or less representative of coolstar data.
Almost all of the inflation data listed in Table 1 have been determined from eclipsing-binary data.Such analysis can yield radii which (in the best cases) are precise to 1% or so.Only one of the "systems" listed in Table 1 involves a single star, for which interferometry or spectrophotometry can (in the best cases) also yield fairly reliable radii.
The empirical values of the relative radius inflation in the data, which we used in plotting Figure 1, spans a range that has a minimum value as small as −0.03.In such a case, the empirical radius of the star fits the standard nonmagnetic evolutionary model within the error bars.At the upper end of the range of empirical relative inflations in Figure 1, the maximum value of inflation in our sample is ∼30%.Most of the empirical inflations in Figure 1 seem to be clustered around 15% or below.A straightforward arithmetic average of all of the inflations plotted in Figure 1 turns out to have a value of 8%.
In an extensive (Bayesian) analysis of inflation data for a different (and larger) sample of stars than we have used here, the mean relative radius inflation with maximum probability (Kesseli et al. 2018) has been reported to be 7% ± 1%.Thus, our simple arithmetic average of the empirical data in our independently selected sample is not far from the results of the Bayesian analysis.We note that in the work of Kesseli et al. (2018), inflations in particular cases can be as large as 17.5%, depending on which evolutionary codes are used for the "standard" (nonmagnetic) radii.Moreover, the authors reported larger inflations (on average) in M dwarfs with radiative cores (stellar mass >0.35 M e ) than in the fully convective M dwarfs (stellar mass 0.35 M e ).Since we have plotted relative inflations as a function of T eff in Figure 1, we do not (at this point in the paper) make any attempt to search for a distinction between fully convective stars and partially convective stars; as a result, we refrain in this section from commenting on whether or not the data in Figure 1 are consistent with the mass-based division of Kesseli et al. (2018).However, we shall make sure to return to this mass-based division in our discussion below (Section 5.1) of the results of magneto-convective modeling.
In the early years of the exploration of radius inflations, the values derived for the inflation in certain systems were in certain cases found to be subject to much larger errors than those which typically emerge nowadays from eclipsing-binary analysis.Prominent examples can be found in the important paper by Lopez- Morales (2007), who helpfully separated their sample of stars into groups of single stars and groups of stars in binaries.Since binary components are subject to tidal forces which may contribute distortion(s) to one or both of the components, we consider it prudent to concentrate on their sample of single stars here.Thus, in Figure 2 of Lopez-Morales (2007), values of inflations are plotted which, in certain single stars, have mean values of as much as 32%.To be sure, the error bars on some of the data points reported by Lopez-Morales (2007) are quite large.On the one hand, the inflation values which lie at the lower limit of the error bars is about 16%, consistent with the upper limits plotted in Figure 1.It is also consistent with the upper limits which are cited by Kesseli et al. (2018).On the other hand, the upper limit error bars reported by Lopez- Morales (2007) indicate that the inflations might in some stars be as large as about 48%: This certainly exceeds the maximum value in the sample we plot in our Figure 1.
If the results of Lopez- Morales (2007) are taken at face value, it appears that radius inflations in certain low-mass stars might lie, in cases of the extreme upper limit, in the range 45%-50%.As far as the current authors know, no empirical  values of radius inflation of this magnitude have so far been reported in the literature.It will be interesting to see if reliable eclipsing-binary data will, in the future, corroborate extreme upper limits in the range 45%-50%.
For future reference, we note that among the single stars reported by Lopez- Morales (2007), the five maximally inflated stars, with mean fractional inflations between 16% and 32%, have masses between 0.401 (for GJ 687) and 0.631 M e (for GJ 205).None of these five stars has a mass that is small enough for the star to be completely convective.This result is consistent with the report by Kesseli et al. (2018) that stars with radiative cores in their sample were found to exhibit larger inflations.

Magneto-convective Models Based on the GT66 Criterion
Our magneto-convective modeling is similar to that described in MacDonald & Mullan (2017b) and in the Appendix of Mullan & MacDonald (2020).In these models, the major magnetic effects are incorporated in two ways: (i) magnetic inhibition of convection as a result of the presence of a vertical component of the magnetic field, and (ii) addition of a corresponding magnetic pressure.We also add to the internal energy a contribution from the magnetic energy corresponding to the magnetic pressure.The conversion of magnetic energy to thermal energy and vice versa is allowed to occur by means of the energy equation.In all of the models to be reported here, we have evolved stars from the pre-main-sequence phase to an age of 5 Gyr, keeping the value of the magnetic inhibition parameter δ (at the surface of the star) fixed.We have considered three values for the magnetic ceiling, B ceil .For models with B ceil = 10 kG, the ceiling is kept constant throughout the evolution up to an age of 5 Gyr.However, we found that this limitation to a constant ceiling field led to difficulties in attaining a magneto-convective model when B ceil was assigned a time-independent value as large as 100 kG or 1 MG; the difficulties arose because of lack of convergence during the contraction to the main-sequence phase.In order to avert these computational difficulties, we allowed B ceil to linearly ramp up with age from 0 to a value at the age of the zero-age main-sequence nonmagnetic model of either 100 kG or 1 MG.

Radius Inflation in Models with B ceil = 10 kG
In Figure 2, we show (on the vertical axis) the relative radius inflations (in fractional units) which emerge (at age 5 Gyr) from our models with masses in the range from 0.1 to 1.0 M e , in increments of 0.1 M e .Along the horizontal axis, we show the magnetic inhibition parameter δ (at the stellar surface) for models in which the ceiling field strength was set to have the value 10 kG.The δ values that we used in the plotted models range from δ = 0 (no magnetic field present) to δ = 0.9.We note that in all of our magneto-convective models which we have hitherto published (e.g., MacDonald & Mullan 2017b), the values of δ which were needed to account for the observed inflations were in almost all cases smaller than 0.1; therefore, the models we are exploring in the present paper are a lot more "magnetic" than any models we have previously reported.
Since the magnetic inhibition parameter δ plays an essential role in our treatment of stellar structure, it is important to make an effort to estimate the maximum value d max which plausibly exists in a "real star."In the context of empirical measures of magnetism in low-mass stars, the surface field strength, averaged over the surface of the star, has so far been found Figure 2. Values of fractional radius inflation which are obtained in our magneto-convective models at age 5 Gyr, with values of the magnetic inhibition parameter δ ranging from 0 to 0.9, and with masses ranging from 0.1 M e (red line) to 1.0 M e (blue line) in increments of 0.1 M e .In all cases, the ceiling field is time independent with a constant strength of 10 kG.The results suggest that, in the context of the GT66 criterion, the amount of fractional radius inflation is not a monotonic function of stellar mass, nor is the inflation a monotonic function of the surface magnetic field strength (as quantified by the GT66 inhibition parameter δ).
to extend up to values of order 〈B〉 = 5-8 kG in several stars (Reiners et al. 2022).However, in order to determine the corresponding upper limit on the empirical value of d max , we need to know specifically the vertical component of the field strength B v (see Equation (1) in Section 1 above).How can we estimate a plausible value of B v given an empirical value of 〈B〉?In the extreme case that the field at all points of the surface happens to be disordered and randomly distributed in three dimensions, then the Pythagorean theorem might lead us to guess that B v could be of order á ñ B 3 .If such a conclusion is applicable to a star with an empirical field of 〈B〉 = 8 kG, this leads to d max ≈ 0.4 (using results for the relationship between surface vertical field strength and δ presented in Figure 8).An independent argument for estimating B v can be obtained by considering stars where a particular observational technique (Zeeman-Doppler imaging) has allowed observers of certain low-mass stars to separate the surface field into its radial, azimuthal, and meridional components; in such stars, it is found that the maximum field strength on the surface may exceed 〈B〉 by factors of 2-5 (see Tables 3-8 in Morin et al. 2010).In such stars, an appropriate estimate of B v (which certainly cannot exceed the maximum field strength) might be comparable to 〈B〉.In such a case, we find (by referring again to Figure 8) that d max ≈ 0.8.Thus, although we will nominally extend the range of δ in the present study to as large as 0.9 for the sake of completeness (see, e.g., Figure 2), the range of δ values in "real stars" likely spans a somewhat narrower range from 0 up to d max (the "low-δ" models).Models in the "high-δ" category with δ > d max are of lesser physical relevance.It appears that the cutoff between low-δ and high-δ models can be considered to occur in the vicinity of d max ≈ 0.4-0.8.Referring to Figure 2, we see that the most interesting aspect of our results (i.e., the existence of maxima in inflation) are found to occur around δ = 0.6-0.7.In view of the uncertainty in our estimate of the numerical value of d max , there is a finite probability that the phenomenon of maximum inflation occurs in our low-δ models, i.e., in models which are pertinent to "real stars." Several conclusions are suggested by the results presented in Figure 2. First, the fractional radius inflation does not increase monotonically with increasing δ, i.e., with increasing strength of the (vertical) magnetic field.For example, for M = 0.7 M e , maximum inflation occurs for δ ≈ 0.6.This means that in any particular star with mass M = 0.7 M e , the same amount of relative radius inflation (e.g., ΔR/R = 80%) can arise from two different field strengths, one relatively weak B w (δ = 0.4), the other relatively strong B s (δ = 0.8).(Of course, the terms "weak" and "strong" are relative terms.However, for observational evidence that δ values in the range 0.4-0.8 can plausibly be expected to be present in "real stars," see the preceding paragraph.)Second, models that are fully convective in the absence of magnetic fields (M < 0.35 M e ) experience relatively little inflation, no more than ∼10%.Third, models of stars which are not fully convective in the absence of magnetic fields (M > 0.35 M e ) experience relatively larger inflations, leading to magnetic models which can have radii that are almost twice as large as the corresponding nonmagnetic model.Fourth, in models where the ceiling field is 10 kG, a maximum fractional inflation of ∼90% occurs for stellar models with mass 0.7 M e .There are at least two competing effects that are responsible for determining the mass at which maximum inflation occurs.(i) Higher-mass models have smaller convective envelopes, and so as the stellar mass increases, magnetic inhibition of convection has the ability to affect a progressively smaller fraction of the star's mass.For example, in main-sequence stars with a mass of 1 M e , the convective envelope contains no more than roughly 1% of the stellar mass (Hansen et al. 2004).In such a star, some 99% of the mass is not directly subject to magnetic inhibition, and so, even if magnetic inhibition is "strong" (in some sense) in the convection zone, this does not mean that the global structure (e.g., the radius) would necessarily undergo significant inflation.(ii) Lower-mass stars, with their smaller radii (R ∼ M) have higher mean densities (∼1/M 2 ) and, as a result, the convective transfer of heat by denser gas remains efficient even in the presence of relatively strong magnetic fields.As IB18 have demonstrated, higher convective efficiency corresponds to lower specific entropy, and therefore inflation is reduced in such stars.
Despite the results in Figure 2, which point to a clearly nonmonotonic dependence of the relative radius inflation on the GT66 parameter δ, it is striking to find that, in the context of entropy, the dependence of radius on entropy is monotonic.In Figure 3, values of the specific entropy are plotted as a function of log R for our 0.7 M e models.In both panels in Figure 3, the range in δ is from 0.0 to 0.9.In the left panel, we plot the specific entropy at the base of the convective envelope, and in the right panel, we plot the mean specific entropy of the model.To show clearly that, although the dependence of the relative radius inflation on δ is nonmonotonic, the radius is essentially a function of the entropy alone; the parts of the curves for which radius increases as δ is increased are plotted in black, while the parts of the curves in which radius decreases as δ is increased are shown in red.For low-mass models that are fully convective in the absence of magnetic fields, we find that there is an essentially linear relationship between log R and entropy, in agreement with the analysis of IB18.This relationship is illustrated in Figure 4 for the case of M = 0.3 M e models.Remarkably, a linear relationship is found in these models for all values of δ, even though we find that models of an M = 0.3 M e star in which δ has a numerical value of 0.3 or larger are not fully convective but instead are found to have a small radiative layer (i.e., a layer where convection has been suppressed by magnetic effects) lying between a convective core and a convective envelope.
In Figure 5, we summarize the results of our models with B ceil = 10 kG by plotting the maximum value of the relative radius inflation which the models predict as a function of mass.The results in Figure 5 indicate that the most extreme empirical inflations that have been reported so far in the literature (i.e., relative inflations as large as 50%; e.g., Lopez-Morales 2007) can be readily replicated provided we allow the GT66 inhibition parameter δ to take on a value as large as 0.6.Moreover, the results in Figure 5 also support the empirical reports that lower-mass (fully convective) stars are observed to have relatively small inflations, while stars with radiative cores exhibit relatively larger inflations (e.g., Lopez-Morales 2007; Kesseli et al. 2018; see also Section 4).Independent of the empirical data, a theoretical conclusion that magnetic stars with radiative cores undergo larger radius expansion than completely convective stars do was reported by Chabrier et al. (2007).However, the authors admitted that their "approach is phenomenological" and they made "no claim to present a consistent description of the effect of a large magnetic field on the evolution of dominantly convective objects."In the present paper, we go beyond a phenomenological approach and seek a quantitative physical description (based on GT66, whose work was based on a fundamental and well-established energy principle in plasma physics) of the way in which magnetic fields affect the structure and evolution of low-mass stars.
As discussed in Section 5.1, it is not clear whether δ values as large as the largest value considered here (9) are actually Figure 3. Specific entropy plotted against log R for models of M = 0.7 M e and B ceil = 10 kG.In the left panel, the specific entropy is that which pertains to the bottom of the surface convection zone.In the right panel, the specific entropy is the average over the model.The red (black) portions of the curves correspond to models in which the stellar radius decreases (increases) as δ is increased.realized in stars.Hence, we also show in Figure 5 how the relative radius inflation varies with stellar mass for δ = 0.4, a value that is certainly realizable for M dwarfs (see Section 5.1). Figure 5 indicates that, even when δ is constrained to be no greater than 0.4, the maximum radius inflation (ΔR/R ≈ 85%; see red curve in Figure 5) is comparable to the maximum inflation which is found for δ = 0.9 (ΔR/R ≈ 93%; see black curve in Figure 5).

Radius Inflation in Models with B ceil = 100 kG and 1 MG
In Figure 6, we show the relative radius inflation at age 5 Gyr for models of masses 0.1-1.0M e , in increments of 0.1 M e , plotted against δ for ceiling values of 100 kG and 1 MG.
Due to limitations of the opacity tables, we were not able to evolve models to age 5 Gyr for all combinations of M, δ, and B ceil .In cases of very large magnetic field strengths, the structural alterations in stellar structure can be so severe (see Section 5.3) that they lead to local values of temperature/ density which are too low/high to be included in the available opacity tables.For B ceil = 100 kG and M = 0.1 M e , the largest successful value of δ is 0.59.For the mass range 0.2-1.0M e , the largest inflation of 139% occurs for M = 0.5 M e .Due to the larger ceiling value, interior magnetic fields are larger than in the B ceil = 10 kG case, which shifts the largest inflation to denser, lower-mass models.For B ceil = 1 MG, we could evolve models to age 5 Gyr only for masses 0.4 M e and higher, and δ values of 0.90 and lower.For lower values of mass, the highest values of δ for which we could evolve models are 0.13, 0.35, and 0.89 for masses 0.1, 0.2, and 0.3 M e , respectively.
For models with B ceil = 100 kG, a maximum in the inflation occurs for models with mass 0.5 M e ; but in models with higher masses, we found no maxima in the inflations.A fortiori, in the case of models with B ceil = 1 MG, we did not find any maxima in the inflations for models of any mass.
As shown by MacDonald & Mullan (2010), in their study of pre-main-sequence evolution, magneto-convection slows the contraction of a stellar model on its way to the main sequence: This behavior arises because magnetic effects, with their concomitant influence on global stellar structure, reduce the central temperature at any given age, thereby reducing the rate of consumption of protons via nuclear fusion.Indeed, for models with M = 0.2 M e and B ceil = 1 MG, with δ in the range 0.20-0.35(B surf = 388-415 G), the main sequence (defined as the onset of proton fusion in the core) is not reached even at an age as advanced as 5 Gyr.The delay in reaching the main sequence is a contributing factor to the radius inflation of these models.

Magnetically Induced Structural Changes in Low-mass Stars
In standard (nonmagnetic) stellar models, it is well known that a main-sequence star with mass 1 M e has a radius of ∼650 Mm, and the star contains a convective envelope which extends inwards to a depth z base of about 200 Mm below the surface.Between the base of the convective envelope and the center of the star, a radiative zone (core) exists.For standard models with masses lying below a certain limit (typically 0.3-0.35M e ), theory indicates that the convective envelope engulfs the entire star; in such a star, no radiative core should exist.
This standard scenario changes significantly in the presence of magnetic fields that are as strong as those which we are considering in the present paper.In this subsection, we review the structural changes which emerge in a small (but representative) sample of our models.
Figure 5. Maximum relative inflation plotted against stellar mass for models with B ceil = 10 kG.The black line corresponds to the case in which δ is limited to a maximum value of 0.9 and the red line corresponds to the case in which δ is set to a value of 0.4.

Ceiling Field = 10 kG
Let us consider first the models in which the maximum internal field is weakest, i.e., B ceil = 10 kG.And let us start by considering the star with which we are most familiar, a star with mass 1 M e .In the presence of a global (vertical) surface field of almost 700 G, corresponding to δ = 0.6, our model indicates that the radius of the star is inflated to 1016 Mm (i.e., relative inflation = 55%).Even more striking than the inflation of the star as a whole is the effect on the convective envelope: Our models indicate that the depth of the convective envelope is reduced by more than an order of magnitude, to z base = 14 Mm.In such a model, the physical conditions are much less "convective" than they are in a nonmagnetic model with the same mass.Of course, the "real" Sun never exhibits global fields with strengths as large as 700 G covering the surface: Such large fields are confined to sunspots, which never occupy more than about 1% of the surface area.Nevertheless, it is clear that if a global field of several hundred gauss were to exist on the solar surface, dramatic changes would occur in the internal structure of the Sun (according to our model).And the sense of the changes is clear: The field would cause a dramatic reduction of the thickness of the convective envelope.This reduction is directly tied to the presence of a strong magnetic field which interferes with convective flows, thereby reducing the effectiveness of convection as a heat-transport mechanism.
As a second example of the dramatic alteration which is produced in the global properties of a star by magnetic fields, let us consider a model with a mass of 0.3 M e .In the nonmagnetic case (i.e., δ = 0), such a model has a radius of 203 Mm, and the star is completely convective, i.e., the convection zone has a depth of z base = 203 Mm, extending from the surface to the center of the star.But in a magnetic model with δ = 0.6, the stellar radius is found to be 223 Mm, i.e., an inflation of the radius by almost 10%.The surface field strength in such a model is found to be 2.2 kG.The depth of the convection zone is found to be z base = 140 Mm, considerably thinner than in the nonmagnetic case.In the magnetic model, we find a dramatic difference from the nonmagnetic model: The convection zone no longer extends all the way to the center of the (magnetic) star.In the magnetic model, there is now a radiative core extending from the center of the star out to a radial distance of 83 Mm.The magnetic effects are such that a star with mass 0.3 M e containing a field no stronger than 10 kG is no longer completely convective.
However, in the case of a model with mass 0.1 M e , the condition of complete convection persists not only in the nonmagnetic model but also in the magnetic model with δ = 0.6, where the global surface field is 4 kG; the magnetic effects in a model where the maximum field is 10 kG are not strong enough to suppress convection in the 0.1 M e model.Our magnetic model for a star with mass 0.1 M e remains completely convective.
At what depth z ceil does the magnetic field strength reach the "ceiling" value of 10 kG?In our magnetic models for 1, 0.3, and 0.1 M e , this depth occurs close to the top of the convection zone.

Ceiling Field = 100 kG
In a 1 M e model, and again considering the case of a magnetic model with δ = 0.6, the surface field is again close to 700 G.In this model, the stellar radius is found to be 1074 Mm, i.e., inflated by 64% relative to the nonmagnetic solar model.The depth of the convection zone is z base = 15 Mm, i.e., much thinner than in the nonmagnetic Sun.
In a 0.7 M e model, and again considering the case of a magnetic model with δ = 0.6, the surface field is 740 G.In this model, the stellar radius is found to be 899 Mm, i.e., inflated by 100% relative to the nonmagnetic model.The depth of the convection zone is z base = 16 Mm; as a result, most of the stellar model is a radiative zone.
In a 0.3 M e model, also with δ = 0.6, the surface field is almost 220 G.This leads to a radius of 367 Mm, representing an inflation of 81% relative to the nonmagnetic model.This model is not completely convective: The convective envelope extends to a depth of only z base = 90 Mm.Most of the remainder of the star is radiative, except for a very small convective core within 10 Mm of the center.
In a 0.1 M e model, also with δ = 0.6, the surface field is 380 G.This leads to a radius of 122 Mm, representing an inflation of 50% relative to the nonmagnetic model.This model is completely convective, even in the presence of a field of 100 kG; this suggests that if convection were ever to be suppressed in such a star, fields of strength ?100 kG would be required.
The ceiling field of 100 kG is reached inside the convection zone for the models with masses 0.1 and 0.3 M e , but in the models with masses 0.7 and 1.0 M e , the ceiling field is not reached until we descend well below the base of the convection zone.

Ceiling Field = 1 MG
In a 1 M e magnetic model with δ = 0.6, the surface field is again close to 700 G.In this model, the stellar radius is found to be 1300 Mm, i.e., inflated by 98% relative to the nonmagnetic model.The depth of the convection zone is z base = 23 Mm, much shallower than the convection zone in the nonmagnetic solar model.
In a 0.7 M e magnetic model with δ = 0.6, the surface field is 750 G.In this model, the stellar radius is found to be 1067 Mm, i.e., inflated by 150% relative to the nonmagnetic model.The depth of the convection zone is z base = 27 Mm; as a result, most of the star is a radiative zone.
In a 0.3 M e model with δ = 0.6, the surface field is 140 G.This leads to a radius of 550 Mm, representing an inflation of 180% relative to the nonmagnetic model.This model is almost entirely radiative: The convective envelope extends to a depth of only z base = 30 Mm.The remainder of the star is radiative, i.e., very different structurally from the nonmagnetic star with the same mass.
The ceiling field of 1 MG is reached below the base of the convection zone in all three of these models.
No converged model could be obtained for a 0.1 M e model with δ = 0.6 because of limitations of the opacity model.

Why Does a Maximum Inflation (as a Function of δ) Occur
in Some Models but Not in Others?
We note that in computing a particular model of a magnetoconvective star, a value is pre-assigned for the ceiling field B ceil (e.g., 10 kG or 1 MG).In any particular magneto-convective model, we impose the constraint that the field strength increases at first as we penetrate deeper below the surface.This occurs by construction, since the model assumes that the magnetic inhibition parameter δ remains constant in the nearsurface layers of the star.But in all cases the field strength eventually reaches, for the first time, the value of B ceil at a certain depth z ceil below the surface.
In the context of whether or not a maximum occurs in the radius inflation (as a function of δ), we consider it worthwhile to emphasize the following conclusion which emerges from our models.Whether or not a maximum occurs in the relative radius inflation depends primarily on how the depth z ceil is related to the depth of the convective envelope z base .For a given value of B ceil , if the depth z ceil lies inside the surface convection zone for all values of δ, our results indicate that a maximum of relative radius inflation occurs at a certain value of δ.But if the depth z ceil lies below the convection zone, the material in the convection is never subjected to the full effects of the ceiling field; in this case, our results indicate that inflation increases monotonically with δ.It will be interesting to see if future modeling of magnetized low-mass stars, based (presumably) on a more complete physical description of magneto-convection than we have incorporated here, will confirm or refute our conclusion regarding the condition required in order to give rise to a maximum value of radius inflation.

Dependence of Radius Inflation on Vertical Surface Field Strength
For models with B ceil = 10 kG, the surface magnetic field strength increases monotonically with δ for the complete range in mass.Figure 7 is a color plot showing the dependence of the radius inflation on stellar mass and B surf , the strength of the vertical component of the surface magnetic field, for the case in which B ceil = 10 kG.The largest relative inflation, amounting to about 90% of the standard (nonmagnetic) stellar radius, is found for a star with mass ∼0.7 M e and B surf ∼ 2 kG.For models that are fully convective in the absence of magnetic field (M  0.35 M e ), the maximum radius inflation is about 20% and requires B surf ∼ 5 kG.
For the higher ceiling values, the surface magnetic field strength does not increase monotonically with δ for mass 0.4 M e .This behavior is illustrated in Figure 8 for M = 0.3 M e .
For the lower-mass models, which have relatively low effective temperature even in the absence of magnetoconvection, strong interior fields lead to large radius increases and the photospheric temperature becomes sufficiently low (∼10 3 K) that molecule formation significantly increases the opacity.In response, the total pressure at the photosphere decreases significantly, leading to a reduction in the magnetic pressure and surface magnetic field strength.
However, for models with B ceil = 1 MG, the relative inflation increases monotonically with δ, which allows us to show in Figure 9 a color plot of the predicted surface vertical field.
The largest surface fields of ∼4 kG are found for models of mass 0.6-0.7 M e , and give relative inflations of ∼2-3.

Dependence of B surf on B ceil for 20% Relative Inflation
A result which has emerged from our earlier models, where we considered only relatively low values of δ, is that the strength of the surface magnetic field B surf is very insensitive to the choice of the ceiling field B ceil .MacDonald & Mullan (2017a) showed that, when B ceil is varied over the range 10 kG to 1 MG, the value of B surf required to obtain a specific amount of inflation scales as - B a ceil , where a ∼ 0.07-0.08 for models with δ assigned a value less than ∼0.1.This is a very slowly varying function, showing that our results for low δ values do not depend sensitively on the choice of value for B ceil .The largest empirical inflations for main-sequence stars is ∼20% (see Section 4).From our models, we find that, for 20% inflation, a varies from 0.06 for models of mass 0.5 M e to 0.027 for models of mass 0.3 M e , with a mean value of 0.12 over the mass range 0.1-1.0M e .With the exception of models near the fully convective boundary, we again conclude that B surf depends only weakly on the adopted value for B ceil .

Discussion and Conclusion
The model of magneto-convection which we rely on in the present paper is based mainly on the work of Gough & Tayler (1966, referred to herein as GT66).This model has several  important features.(i) The model quantifies the effects of magnetic inhibition of convection by means of the GT66 parameter δ, which is the ratio of magnetic pressure of the vertical component of the field to the sum of magnetic pressure and the product of the first adiabatic exponent and the gas pressure.(ii) The numerical value of δ is assumed to remain constant in the outermost regions of the star; in these regions, the local field strength increases as one moves deeper into the star, and the local gas pressure increases monotonically with depth.(iii) The model incorporates the effects of magnetic pressure in the equation of hydrostatic equilibrium when calculating stellar structure.(iv) The model imposes an upper limit ("ceiling") on the field strength inside a star.Using a model with these features, we report here on a systematic study of the question: How large can the radius of a star with a mass in the range 0.1-1.0M e become due to magnetic effects?
Although in principle the GT66 parameter δ is allowed any numerical value between zero (absence of magnetic field) and unity (infinitely strong field), stars in the "real world" will have only finite field strengths.As a result, "real stars" will have maximum values of δ which are less than unity.The physical process which sets an upper limit on field strength in any given star is probably associated with some sort of equipartition with one of the dynamical variables involved in the dynamo process.Comparison with empirical data on field strengths in low-mass stars (Reiners et al. 2022) suggests that δ has a limiting value no larger than 0.8, or perhaps (depending on how one estimates the strength of the vertical component) no larger than 0.4.In the present paper, we have formally explored a range of δ values between zero and 0.9; we recognize that there exists inside this range a transition between models with "low δ" (realistically acceptable models) and models with "high δ" (where models become increasingly less realistic if δ exceeds a limiting value of 0.4-0.8).
In order to establish a connection between our models and observations, we have gathered empirical results from the literature of radius inflation in a sample of a few dozen lowmass stars.Although we make no claim to a complete sample of inflation measurements, the largest inflations which we came across in our literature search were 30%-50% (with large error bars).However, the mean inflations were considerably smaller, of order 10% or less, consistent with empirical results reported independently by other investigators using different samples of stars.
As regards a choice of the strength of the ceiling field B ceil , we have already suggested (MacDonald & Mullan 2017b) that a plausible lower limit would be 10 kG; it would be hard to justify reducing the ceiling value below this, since the field strengths observed at the surface of several low-mass stars are already observed to be approaching values which are almost as large as 10 kG (e.g., Shulyak et al. 2017).As regards an upper limit on the ceiling field, Browning et al. (2016) have demonstrated that fields in excess of 1 MG would not be stable inside low-mass stars.In view of these results, we have concentrated in this paper on computing stellar models in which the value of B ceil is assigned values in the range from 10 kG to 1 MG.
Our results indicate that, in the presence of a ceiling field of 10 kG, the relative radius inflations are expected to be smallest (<10%) among fully convective stars.However, in more massive stars, where a radiative core exists, our models predict that the larger empirical inflations (up to 50%) can readily be accommodated by the model results provided that we use δ values in the range 0.0-0.6.(Such a range is permissible if the limiting value of δ is in fact as large as 0.8.)According to our results, there exists a maximum in the relative radius inflation among stars with mass (0.6-0.7) M e .In such stars, in the presence of a ceiling field of 10 kG, the relative radius inflation may rise to a value as large as 90%.As far as we know, this is well in excess of any inflation which has been reported to date in the literature.For ceiling fields of 100 kG, even larger inflations are found in our models.A maximum relative inflation of almost 140% is found in stars with mass 0.5 M e .Relative inflations are predicted to become smaller in more massive stars, rising to no more than 75% in stars with mass 1.0 M e .
For ceiling fields of 1 MG, even larger inflations emerge in the models, reaching values as large as 250%-300% for stars of all masses in the range from 0.1 to 1.0 M e .In this case, the models do not predict that a maximum relative inflation should occur at any particular value of δ.
In the presence of internal fields as large as 100 kG and 1 MG, the results in Figure 8 indicate that the alterations to the internal structure of the star are so serious that the surface field strength is no longer a monotonic function of the magnetic inhibition parameter δ.In this regard, we note that surface fields as large as 8 kG are already known to exist in low-mass stars (Reiners et al. 2022).If the nonmonotonic behaviors in Figure 8 for B ceil = 100 kG and 1 MG can be confirmed by more realistic magneto-convective modeling, it will allow us to draw an important conclusion: Our magneto-convective models apply to "real stars" only if the internal fields are chosen to be no stronger than 10 kG.
In conclusion, we predict that, if our model results are applicable to "real stars," it would be of great interest if future observations were to identify certain low-mass stars where the relative radius inflation is significantly larger than the values which have hitherto been reported in the literature.This prediction is expected to be especially true if the internal fields in low-mass stars are in fact as large as the 1 MG upper limit reported by Browning et al. (2016).This leads us to offer the following speculation: If future surveys never detect any mainsequence stars with inflations as large as factors of 2-3 (values that are greatly in excess of any that have been reported so far), might this mean that internal fields as large as 1 MG are actually quite rare (or even nonexistent) in "real stars"?In this context, we note that the work of Browning et al. (2016) merely sets an upper limit on the strength of a stable field in a lowmass star; it is not obvious that any particular low-mass star must have an internal field as large as 1 MG (especially since the generation of such a field would probably not be physically trivial).If an upper limit on radius inflation in low-mass stars could be confirmed observationally, this might lead to robust estimates of the maximum field strengths inside such stars.On the other hand, an anonymous referee has pointed out that, although the ceiling field does indeed have an effect on stellar radii, the effects of varying δ (which depends on the surface field) are larger (see Section 5.6).In view of this, "an absence of large inflations might be telling us more about the nearsurface field."

Figure 1 .
Figure 1.Empirical values of the percentage relative radius inflation ΔR/R reported in the literature as a function of the effective temperature T eff of the star.

Figure 4 .
Figure 4. plotted against log R for models of M = 0.3 M e and B ceil = 10 kG.The red (black) portions of the curves correspond to models in which the stellar radius decreases (increases) as δ is increased.

Figure 6 .
Figure 6.As Figure 2, except for magnetic field ceiling values of 10 5 G (left panel) and 10 6 G (right panel).

Figure 7 .
Figure 7. Color map showing dependence of relative inflation on mass and surface vertical field strength for models with B ceil = 10 kG.

Figure 8 .
Figure 8. Vertical surface field B surf plotted against the magnetic inhibition parameter δ for models of mass M = 0.3 M e and three values of the magnetic field ceiling, B ceil = 10 4 , 10 5 , and 10 6 G.
9. map showing dependence of surface vertical field strength on mass and relative inflation for models with B ceil = 1 MG.

Table 1
Data on Relative Radius Inflation in Low-mass Stars, Mostly in Binaries