MAGNETIC INHIBITION OF CONVECTION AND THE FUNDAMENTAL PROPERTIES OF LOW-MASS STARS. II. FULLY CONVECTIVE MAIN-SEQUENCE STARS

Our treatment of magneto-convection depends on how we assume the magnetic field is generated. Assuming that rotation drives the dynamo, as in a standard shell dynamo (Parker 1979), leads to perturbations consistent with the idea that magnetic fields can stabilize a fluid against thermal convection (e.g., Thompson 1951; Chandrasekhar 1961; Gough & Tayler 1966). This assumption forms the basis of our magneto-convection formulation (Feiden & Chaboyer 2012b, 2013). However, permissible magnetic field strengths can reach upward of 6 kG at the model photosphere with exact upper limits determined by equipartition with the thermal gas pressure. This also results in interior magnetic field strengths that can grow nearly without limit owing to large internal gas pressures. Models that rely on stabilizing convection with a magnetic field are hereafter referred to as having "rotational dynamos."

The form of the perturbation used for a star with an assumed rotational dynamo is not necessarily valid in all stellar mass regimes. This is particularly true in fully convective stars, where the interface region between the radiative core and convective envelope is thought to disappear. To address this, and the problem that magnetic fields in the rotational dynamo can grow without limit, we introduced a "turbulent dynamo" mechanism (Feiden & Chaboyer 2013). This formulation assumes that the magnetic field strength at a given grid point within the model receives its energy from the kinetic energy of convecting material. Therefore, the local Alfvén velocity cannot exceed the local convective velocity. It is a simple approach developed to address zeroth-order effects within the already simplified convection framework of mixing length theory. For low-mass stars, this methods places an upper limit of around 3 kG for surface magnetic field strengths, consistent with observed upper limits of average surface magnetic field strengths (Reiners & Basri 2007; Shulyak et al. 2011). Although rotation is still required for turbulent dynamo action (Durney et al. 1993; Dobler et al. 2006; Chabrier & Küker 2006; Browning 2008), magnetic field strengths are more sensitive to properties of convection.

We note, again, that the term "dynamo" is used loosely. Our formulations of magneto-convection do not rigorously solve the equations of magnetohydrodynamics. Instead, we seek to capture physically relevant effects on stellar structure in a phenomenological manner consistent with actual dynamo processes. Each of the above descriptions rely equally on a prescribed magnetic field strength profile within the star.

Models of the "dipole radial field" variety are the standard sort introduced in Feiden & Chaboyer (2012b). This profile is characteristic of a magnetic field generated by a single current loop centered on the stellar tachocline. Given a surface magnetic field strength, the radial profile of the magnetic field is determined by calculating the peak magnetic field strength at the tachocline. Lacking a tachocline, we define fully convective stars to have a peak magnetic field strength at 15% of the stellar radius (0.15R_⋆). This is loosely based on results from three-dimensional (3D) magnetohydrodynamic (MHD) models of fully convective stars (Browning 2008). MHD models indicate that the magnetic field reaches a maximum around ∼0.15 R_⋆ (Browning 2008). Note that this maximum is not a sharply defined peak in the magnetic field strength profile since the profile is based largely on equipartition of the magnetic field with convective flows. Still, we adopt 0.15 R_⋆ knowing this caveat, which we address in a moment. The rest of the interior magnetic field is then calculated by assuming the magnetic field strength falls off steeply toward the core and surface of the star. Explicitly,

$$\begin{equation} B(R) = B_{\rm surf}\cdot \left\lbrace \begin{array}{@{} ll}R^3/R_{\rm tach}^6 & R < R_{\rm tach} \\ R^{-3} & R > R_{\rm tach} \end{array}, \right. \end{equation} \tag{ 1 }$$

where B_surf is the prescribed surface magnetic field strength, R_tach is the radius of the tachocline normalized to the total stellar radius, and R is the radius within the star normalized to the total stellar radius.

To increase the peak magnetic field strength for a given surface magnetic field strength, we implemented a Gaussian profile (Section 4.4.1 in Feiden & Chaboyer 2013). The peak magnetic field strength is still defined at the tachocline in partially convective stars and at R = 0.15 R_⋆ in fully convective stars. However, instead of a power-law decline of the interior magnetic field strength from the peak, the peak was set as the center of a Gaussian distribution. Thus,

$$\begin{equation} B(R) = B(R_{\rm tach}) \exp \left[ -\frac{1}{2}\left(\frac{R_{\rm tach} - R}{\sigma _{g}}\right)^2 \right], \end{equation} \tag{ 2 }$$

where σ_g controls the width of the Gaussian and B(R_tach) is defined with respect to the surface magnetic field strength. The width of the Gaussian depends on the depth of the convection zone. Deeper convection zones have a wider Gaussian profile compared to stars with thin convective envelopes (Feiden & Chaboyer 2013). Hereafter, we will refer to these as "Gaussian radial profile" models. Note that this prescription has no physical motivation beyond providing stronger interior magnetic fields.

We define a third radial profile for this study: a "constant Λ radial profile." When a turbulent dynamo is invoked, the magnetic field radial profiles described above can cause the Alfvén velocity to exceed the convective velocity within the model interior. This happens quite easily in models of fully convective stars where the peak magnetic field strength is defined deep within the model. As a result, perturbed convective velocities become imaginary leading convergence problems when solving the equations of mixing length theory.

If we assume that kinetic energy in convective flows generates the local magnetic field, an Alfvén velocity exceeding the convective velocity will lead to a decaying magnetic field strength. Eventually, equipartition will be reached. We avoid iterating to a solution by using a profile that assumes a constant ratio of magnetic field to the equipartition value, B(r) = ΛB_eq, where $B_{\rm eq} = (4\pi \rho u_{\rm conv}^2)^{1/2}$. This factor, Λ, was introduced in Feiden & Chaboyer (2013) as a means of comparing reduced mixing length models (i.e., Chabrier et al. 2007) to our turbulent dynamo models.

Perturbations to the equations of mixing length theory are now expressed as functions of Λ. Removing energy from convection slows convective flows such that

$$\begin{equation} u_{\rm conv}= u_{\rm conv,\,0} (1 - \Lambda ^2)^{1/2}, \end{equation} \tag{ 3 }$$

where u_conv, 0 is the convective velocity prior to losing energy to the magnetic field. We restrict 0 ⩽ Λ ⩽ 1 to avoid imaginary convective velocities. Reduction of convective velocity causes a significant reduction in convective energy flux since $\mathcal {F} \propto u_{\rm conv}^3$. Steeping of the background temperature gradient, ∇_s, occurs as radiation attempts to transport additional energy. This increase, Δ∇_s, over the non-perturbed temperature gradient is

$$\begin{equation} \Delta \nabla _{\rm s}= \frac{(\Lambda u_{\rm conv\, 0})^2}{C}, \end{equation} \tag{ 4 }$$

where $C = g\alpha _{\rm MLT}^2H_P\delta /8$ is the characteristic squared velocity of an unimpeded convecting bubble over a pressure scale height. This is a sort of terminal convective velocity, where g is the local acceleration due to gravity, α_MLT is the convective mixing length parameter, H_P is the pressure scale height, and δ = (∂ln ρ/∂ln T)_P, χ is the coefficient of thermal expansion. Resulting effects on convection likely represent an upper limit on the effects of such a dynamo mechanism in inhibiting thermal convection. Note that when using this radial profile and dynamo mechanism, modifications to the Schwarzschild criterion (i.e., the formalism outlined in Feiden & Chaboyer 2012b) are neglected.

We first focus our attention on Kepler-16 A, as the analysis will directly influence our analysis of the secondary. Plotted in Figure 1 are non-magnetic Dartmouth models computed at the measured masses of the primary provided by D11 and B12. Three separate tracks are illustrated for each mass estimate, corresponding to [Fe/H] =−0.12, −0.04, and +0.04 (Winn et al. 2011). In this figure, the horizontal shaded region highlights the observed radius with associated 1σ uncertainties. Ages are determined by noting where the mass tracks are located within the bounds of the empirical radius constraints. Similarly, we confirm that when the mass track has the required radius it also has an appropriate effective temperature in Figure 1(b).

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Given a primary mass from D11, we find that standard stellar evolution models match the stellar radius and temperature at an age of 2.1 ± 0.5 Gyr. When the model radius equals the precise empirical radius (0.6489  R_☉), the associated model effective temperature is 4354 K, which is within 17 K of the spectroscopic effective temperature (Winn et al. 2011). We note that this age is also consistent with the estimated age from Winn et al. (2011).

Agreement between models and observations does not guarantee the validity of the D11 masses over the B12 masses. In fact, this agreement is rather expected. The effective temperature and metallicity for the primary were determined using Spectroscopy Made Easy (hereafter SME; Valenti & Piskunov 1996), which relies on theoretical stellar atmospheres. Our models also rely on theoretical atmospheres, although phoenix model atmospheres, used by our models, adopt a different line list database (see Hauschildt et al. 1999 and references therein) than the theoretical atmospheres used by SME (VALD: Vienna Atomic Line Database; Piskunov et al. 1995). In a sense, the fact that we find such good agreement with the effective temperature for a given log g and metallicity (i.e., those of Kepler-16 A) may be a better test of the agreement between different stellar atmosphere models rather than a test of the interior evolution models. What is encouraging, is that we derive appropriate stellar properties at an age consistent with gyrochronology and age-activity relations Winn et al. (2011). Age consistency is not ensured by agreement between stellar atmosphere structures.

Our previous discussion may be erroneous if we adopt an incorrect mass. B12 suggest this is the case. The effect of adopting the lower B12 masses is displayed in Figure 1. Assuming the radius measurement remains constant, we derive an age of 10.5 ± 0.8 Gyr for the primary star. The effective temperature associated with the model also appears too cool compared to observations at the measured metallicity (−0.04 dex). Relief is found by lowering the metallicity by 0.1 dex, which increases the temperature by 30 K. This is enough to bring the model temperature to within 1σ of the spectroscopic value (Winn et al. 2011).

There is a caveat: the spectroscopic analysis by Winn et al. (2011) relied on fixing the stellar log g as input into SME. Thus, the temperature and metallicity are intimately tied to the adopted log g. Reducing the mass of the primary by 5%, as is done by B12, but leaving the radius fixed to the D11 value leads to a decrease in log g of 0.02 dex. Such a change in the fixed value of log g may decrease the derived effective temperature and bring the model and empirical temperatures into agreement. All things considered, we believe it is safe to assume that a shift in mass does not introduce any significant effective temperature disagreements at a given radius. This is predicated on the fact that the temperature is safely above ∼4000 K. Below this temperature, theoretical atmosphere predictions start to degrade with the appearance of molecular bands. More simply stated, we have no reason to doubt the Dartmouth model predicted temperature for Kepler-16 A, regardless of the adopted mass.

A model age of 11 Gyr for the B12 primary mass appears old given the multiple age estimates provided by Winn et al. (2011). Is it possible that the system is actually 11 Gyr old, but appears from rotation and activity to be considerably younger? Consider that a 35.1 ± 1.0 day rotation period of the primary, as measured by Winn et al. (2011), is nearly equal to the pseudo-synchronization rotation period, predicted to be 35.6 days (Hut 1981; Winn et al. 2011). One might think that tidal effects are unimportant in a binary with a 41 day orbital period, however, subsequent tidal interactions briefly endured when the components are near periastron can drive the components toward pseudo-synchronous rotation. Pseudo-synchronous rotation will keep the stars rotating at a faster rate than if they were completely isolated from one another. The timescale for this to occur is approximately 3 Gyr (Winn et al. 2011), meaning that the rotation period is not necessarily indicative of the system's age. The primary will have approximately the same rotation period at 11 Gyr as it will at 3 Gyr. Furthermore, the timescale for orbital circularization is safely estimated to be between ∼10⁴ and 10⁵ Gyr (Winn et al. 2011). Tidal evolution calculations are subject to large uncertainties and should approached as an order of magnitude estimate. However, we are unable to immediately rule out the possibility that Kepler-16 has an age of 11 Gyr.

We have so far neglected any remark on the agreement between standard models and Kepler-16 B. This comparison is carried out in Figure 2. No effective temperature estimates have been published, explaining our neglect of the T_eff–radius plane. As with Figure 1, mass tracks are shown for multiple metallicities. Standard model mass tracks for Kepler-16 B are unable to correctly predict the observed radius at an age consistent with estimates from the primary. The disagreement is independent of the adopted metallicity, which introduces ∼0.5% variations in the stellar radius at a given age.

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No evidence is available to support the idea that Kepler-16 B is magnetically active. Still, we look to magnetic fields to reconcile the model predictions with the observations. All possible scenarios relating to the various stellar mass estimates are considered. Explicitly, we compute models for both the D11 and B12 masses and then attempt to fit the observations using the magnetic Dartmouth stellar evolution models.

The D11 primary mass implies that Kepler-16 is approximately 2 Gyr old, as shown in Figure 1(a). Due to the consistency between the model derived age and the empirically inferred age, we see no reason to introduce a magnetic perturbation into models of Kepler-16 A. Winn et al. (2011) observe only moderately weak chromospheric activity coming from the primary, further supporting our decision. Thus, we seek to reconcile models of Kepler-16 B with D11 masses at 2 Gyr.

Magnetic models of Kepler-16 B were computed for a range of surface magnetic field strengths. A dipole radial profile was used and the perturbation was applied over a single time step at an age of 1 Gyr. Mass tracks with a 4.0 kG and 6.0 kG surface magnetic field strength are shown in Figure 3 alongside a standard mass track for comparison. Note, that even though the perturbation is applied at an age close to the age we are trying to fit, the models adjust to the perturbation rapidly. We are unable to produce a radius inflation larger than 1%, even with a strong surface magnetic field strength of 6.0 kG. The peak field strength in the 6.0 kG model (located at R = 0.15 R_⋆) is approximately 1.8 MG (ν ≈ P_mag/P_gas = 10⁻⁶). Discussion about how real such a magnetic field might be is deferred until Section 4.3.2. For the moment, we are interested in knowing what magnetic field strength is required to reconcile models with observations.

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We next constructed magnetic models using a Gaussian radial profile, which are shown in Figure 4. The magnetic perturbation was again introduced as a single perturbation at 1 Gyr. Two surface magnetic field strengths were used, 4.0 kG and 5.0 kG. The model with a 5.0 kG surface magnetic field strength causes the model to become over inflated compared to the observed radius at 2 Gyr. From Figure 4, we see that a magnetic field intermediate between 4 kG and 5 kG is required to produce agreement between the models and observations. In contrast to results for partially convective stars (Feiden & Chaboyer 2013), dipole and Gaussian radial profiles produce different results for a given surface magnetic field strength in fully convective stars. This is caused by a difference in peak magnetic field strengths (dipole: 1.8 MG, ν = 10⁻⁶; Gaussian: 30 MG, ν = 10⁻⁴). We will return to this issue in Section 4.1.

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Finally, Figure 5 shows the influence of a constant Λ profile on a model of Kepler-16 B. Recall, this radial profile invokes a turbulent dynamo mechanism. We started with a rather high value of Λ = 0.9999 to gauge the model's reaction to this formulation. Impact on the radius evolution of a M = 0.203 M_☉ star is effectively negligible. Further increasing Λ has no effect on the resulting radius evolution.

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Adopting B12 masses mainly alters the age derived from stellar models. Instead of 2 Gyr, we infer an age of 11 Gyr from models of Kepler-16 A, as was shown in Figure 1(a). The relative radius discrepancy noted between models and Kepler-16 B is increased by approximately 2% over the D11 case.

Magnetic models were computed for Kepler-16 B with the B12 mass estimate using a Gaussian radial profile introduced at an age of 1 Gyr. These models are shown in Figure 6. We did not generate models with a dipole radial profile or with a constant Λ profile given the lack of radius inflation observed for these magnetic field profiles for the D11 masses. We find that a surface magnetic field strength slightly weaker than 6.0 kG is required to fit the observations. This translates to a nearly 40 MG (ν = 10⁻⁴) peak magnetic field strength. A stronger field strength was required when using the B12 mass instead of the D11 mass because of the 1% increase in the radius discrepancy mentioned above.

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Kepler-16, although it has components with fundamental properties measured with better than 3% precision, must be approached cautiously when comparing to stellar models. Mass differences quoted in the literature obscure how well models perform against the observations. Though the masses are determined with high precision by each group, the 3%–5% uncertainty introduced by their disagreement overwhelms the measurement precision. This produces an age difference of 9 Gyr for the primary star, calling into question inferences drawn about this system from stellar models. It is also unclear how revising the masses would affect estimates of the component radii, the system's metallicity, and the primary star's effective temperature. Furthermore, since the primary may be rotating pseudo-synchronously, it is not possible to rule out either of the age estimates.

Until the mass differences are resolved, care must be taken when comparing Kepler-16 to stellar models. However, the disparity between the observed radius of Kepler-16 B and model predictions is apparent regardless of the adopted mass estimate. Changing the mass estimate simply changes the level of inferred radius inflation. This result appears robust and can be used to test the magnetic hypothesis for low-mass star radius inflation. Our magnetic models require magnetic field strengths of similar magnitudes. Surface magnetic field strengths are on the order of 4 kG–6 kG with interior field strengths of a few tens of MG.

Methods used in previous studies were, in some respects, similar. Each used a method for treating magneto-convection. Four techniques have been employed thus far: a reduced-α_MLT approach (Chabrier et al. 2007; Morales et al. 2010), stabilization of convection by a vertical magnetic field (MacDonald & Mullan 2012), stabilization by a more general magnetic field (not specifically vertical; this work), and then a turbulent dynamo approach that is similar to a reduced-α_MLT (this work).

Including effects of star spots is inherently difficult in a 1D stellar evolution code. Spots are blemishes scattered across the stellar surface that extend a non-fixed distance into the surface convection zone. Spots have largely been treated in the same fashion in previous investigations and rely on reducing the total stellar surface flux by a fractional amount,

$$\begin{equation} \beta = \frac{S_{\rm spot}}{S}\left[1 - \left(\frac{T_{\rm spot}}{T_{\rm phot}}\right)^4\right], \end{equation} \tag{ 5 }$$

where S_spot/S is the surface areal coverage of spots and T_spot/T_phot is the spot temperature contrast. This approach is based on the "thermal plug" spot model advanced by Spruit (1982a, 1982b) and Spruit & Weiss (1986). The modified surface flux is then $\mathcal {F} = (1 - \beta)\mathcal {F}_\star$, where $\mathcal {F}_\star$ is the flux of the star if the photosphere is spot free.

Morales et al. (2010) performed a detailed analysis to establish how specific star spot properties effect results from both theoretical modeling and light curve analyses. For a given star spot β and assumed distribution of spots over the stellar surface (uniform, clustered at mid-latitudes, and clustered at the poles), they evaluated the reliability of light curve analyses in determining stellar radii. If spots are preferentially located at the poles, they showed that stellar radii may be over-estimated by as much as 6%. On the other hand, if spots are more evenly distributed across the surface, or clustered at mid-latitudes, radius determinations proved reliable for β < 0.3.

Combining the aforementioned results with the influence of spots on stellar evolution models, via Equation (5), Morales et al. (2010) found that β = 0.17 was required to fit the stars of CM Dra. Magneto-convection using a reduced-α_MLT was found to be ineffective and was not required. Thus, they predict that the stars in CM Dra are 35% covered by spots that are 15% cooler than the surface (this latter value was fixed in their analysis). This was deemed sufficient to correct model radii with observations.

MacDonald & Mullan (2012), on the other hand, were able to produce agreement between their model and observations using a combination of magneto-convection and star spots. They identified regions of δ_MM − β (β ≡ f_s in their paper) parameter space that reconciles their models with CM Dra (see their Figures 13–15). Their magnetic inhibition parameter is defined as

$$\begin{equation} \delta _{\rm MM}= \frac{B^2}{B^2 + 4\pi \gamma P_{\rm gas}}, \end{equation} \tag{ 6 }$$

where B is the magnetic field strength, γ is the ratio of specific heats, and P_gas is the total gas pressure. This quantity is added directly to the adiabatic gradient in the Schwarzschild stability criterion. What strongly distinguishes the MacDonald & Mullan (2012) study from the Morales et al. (2010) investigation is that MacDonald & Mullan (2012) do not include model inflation due to spots as described by Equation (5). Instead, they only adopt the adjustment made to the observed stellar radii due to the influence of polar spots on the light curve analysis. Model radius inflation is then caused by stabilization of convection using their inhibition parameter described above.

The full range of values for which they find agreement is 0.15 ≲ β ≲ 0.28 with 0.0 < δ_MM < 0.025. To further constrain the parameter space, they assume the best-fit star spot β = 0.17 from Morales et al. (2010). This narrows the acceptable range for the magnetic inhibition parameter to 0.020 < δ_MM < 0.025. These values of δ_MM correspond to vertical magnetic field strengths of approximately 500 G at the stellar photosphere and a (capped) interior vertical magnetic field strength of 1 MG.

Our modeling results are largely consistent with those of Morales et al. (2010) and MacDonald & Mullan (2012). Comparing first with MacDonald & Mullan (2012), we focus on the dipole and Gaussian radial profile models with a rotational dynamo (Figures 8(a) and (b)). The magnetic perturbation applied in these models is qualitatively similar to magneto-convection method favored by MacDonald & Mullan (2012). We have estimated that our magnetic field strengths should be up to an order of magnitude larger than those of Mullan & MacDonald (2001) as a consequence of our formulation (Feiden & Chaboyer 2013). Our "inhibition parameter" is primarily controlled by the quantity,

$$\begin{equation} \nu \nabla _{\rm \chi }= \frac{P_{\rm mag}}{P_{\rm gas} + P_{\rm mag}}\left(\frac{d\ln \chi }{d\ln P}\right), \end{equation} \tag{ 7 }$$

where ν is a magnetic compression coefficient and ∇_χ is the gradient of the magnetic energy per unit mass with respect to the total pressure. We showed that ν ∼ δ_MM (Feiden & Chaboyer 2013), but our formulation has the additional term ∇_χ ∼ 0.1. Therefore, to achieve the same results as MacDonald & Mullan (2012), we should need ν to be about 10 times larger than δ_MM. Thus, our requirement of a 6.0 kG surface magnetic field strength with a roughly 40 MG peak magnetic field strength for the Gaussian radial profile is consistent with the values we would expect given the values from MacDonald & Mullan (2012). Although the dipole radial profile model has a 5.0 kG surface magnetic field strength, the peak magnetic field strength is only 1.5 MG, a factor of 10 or so too small to impart the necessary structural changes.

To compare with Morales et al. (2010), we refer to our constant Λ models, which closely match the reduced-α_MLT formulation. We found that even if the magnetic field were a considerable fraction of the equipartition magnetic field strength (Λ = 0.9999), there is little impact on the stellar radii. This is consistent with Morales et al. (2010), who found that reducing α_MLT has a negligible impact on the stellar model predictions for fully convective stars, especially after star spots were invoked. In that sense, we confirm that reducing convective efficiency does not appear to be a viable method to fully account for inflation among fully convective stars.

Note that we have not adopted the reduced stellar radius presented by Morales et al. (2010) nor have we explicitly addressed reductions in surface flux due to star spots. Models presented in Feiden & Chaboyer (2013) did not require star spots to provide an adequate fit to the data. This reinforces a known degeneracy between magneto-convection implementations and introducing star spots (MacDonald & Mullan 2010). One may not be too surprised that these two issues are so closely connected, as spots are the physical manifestation of suppressed convection near the stellar surface. Therefore, it is reasonable to assume that reduction in convective flux from either magneto-convection or star spots should produce similar results. The degeneracy, in a sense, can be broken by the idea that star spots could bias light curve analyses toward larger radii. Magneto-convection, strictly speaking, has no influence on the light curve analysis, if we assume global changes to stellar properties. In that sense, particular configurations of spots (clustered near the pole) could have a significantly larger impact on the mass–radius problem than magneto-convection.

Despite the aforementioned results that support the magnetic field hypothesis, we are skeptical of the interpretation that magnetic fields are driving the observed radius inflation. We will now assess the results primarily by considering available observational data.

Reductions in convective efficiency appear to be inadequate for producing the observed stellar radii (Section 3 of this work; Chabrier et al. 2007; Morales et al. 2010). We showed that for Kepler-16 and CM Dra, radius inflation induced by the inhibition of convective efficiency was effectively negligible (∼0.1%). This is in agreement with Chabrier et al. (2007) and Morales et al. (2010), who find reducing the convective mixing length from α_MLT = 2.0 to α_MLT = 0.1 has little effect on the radii of fully convective stars. Reducing α_MLT further could begin to provide reasonable effects on fully convective stellar radii. The question then becomes, "is it possible for α_MLT → 0, and if so, by what mechanism?"

Results from the stabilization of convection are encouraging. A proper amount of radius inflation can be achieved with somewhat reasonable surface magnetic field strengths (Section 3 of this work; MacDonald & Mullan 2012). Instead of surface field strengths posing a problem as they did with partially convective stars (Feiden & Chaboyer 2013), it is the interior magnetic field strengths, which range from 1 MG to 50 MG, that require attention.

There is presently no observational evidence to suggest a 1 MG (or greater) magnetic field could exist within a fully convective star. To be fair, there is also no direct evidence to rule out the possibility. We must be honest about the lack of observational data concerning interior magnetic field strengths. However, there is some indirect evidence that casts doubt on the existence of super-MG interior magnetic fields. We will now carefully examine the possibility that such fields do exist deep in fully convective stars.

From a theoretical perspective, there is concern about how a super-MG magnetic field is generated. Simulations suggest that turbulent dynamos reach saturation at field strengths of ∼50 kG (Chabrier & Küker 2006; Dobler et al. 2006; Chabrier et al. 2007; Browning 2008), two orders of magnitude below the 1 MG magnetic field required by stellar models. We will return to this a bit later. For now, let us assume that super-MG magnetic fields cannot be generated by dynamo action. Instead, it could be assumed that super-MG magnetic fields are the result of the amplification of a primordial μG seed magnetic field during the proto-stellar collapse.

Assuming magnetic flux is conserved during collapse of the proto-star, one finds that super-MG magnetic fields could plausibly exist. However, the super-MG magnetic fields must then survive several Gyr within the stellar interior without decaying. There are three primary timescales of interest: that given by macroscopic diffusion in a non-convecting medium, the rise time for a buoyantly unstable flux tube, and the advection timescale in a convecting medium with relatively high conductivity. We'll consider the latter, first.

If the stellar plasma is highly conducting, which is a valid assumption throughout the interior of stars, then the field lines will be frozen into the plasma. We might then expect that the flux tubes will be carried by convection from deep within the star to the stellar surface, where the they will then dissipate their energy. The timescale for this to occur is roughly τ_conv = R_⋆/u_conv, where R_⋆ ∼ 10¹⁰ cm and u_conv ∼ 10³ cm s⁻¹. We find that τ_conv ∼ 10⁷ s, or about 1 yr. Even if we assume u_conv ∼ 10¹–10² cm s⁻¹, τ_conv ∼ 10⁸–10⁹ s. Thus, the timescale for this type of advection is about 10 yr. Although convection can quickly carry a flux tube from the deep interior to the stellar surface, the process can also carry flux tubes from the surface to deep in the interior.

A more unidirectional process that can transport magnetic flux to the stellar surface is the magnetic buoyancy instability (e.g., Parker 1955, 1979). Magnetic flux tubes are assumed to be in pressure equilibrium with their surroundings deep in the interiors of stars,

$$\begin{equation} p_{{\rm gas},\, i} + \frac{B^2}{8\pi } = p_{{\rm gas},\,e}, \end{equation} \tag{ 8 }$$

where p_gas, i and p_gas, e are the interior and exterior gas pressures acting on the flux tube, respectively, and B²/8π is the magnetic pressure within the flux tube. Since flux tubes are supported by both the internal gas pressure and the magnetic pressure, the gas density within a flux tube is lower than the surrounding gas density, provided that the temperature inside the flux tube is the same as the temperature of the surroundings. This density perturbation creates a buoyancy force toward the surface of the star, but will be counteracted by radially inward hydrodynamic forces resulting from convective down-flows.

If we assume a polytropic EOS, a reasonable approximation for deep stellar interiors, and that p_gas can be expressed as a power-series expansion about ρ_gas, e, the gas density exterior to the flux tube, then truncating the series to first order gives

$$\begin{equation} p_{{\rm gas},\,e} \left[1 + \gamma \rho _{{\rm gas,\, e}}^{-1}\Delta \right] + \frac{B^2}{8\pi } = p_{{\rm gas},\,e}, \end{equation} \tag{ 9 }$$

where γ is the ratio of specific heats and Δ = ρ_gas, i − ρ_gas, e is the density perturbation. The specific buoyancy force resulting from this density perturbation is

$$\begin{equation} f_b = \frac{g}{\gamma }\left(\frac{B^2}{8\pi p_{{\rm gas},\,e}}\right). \end{equation} \tag{ 10 }$$

Balancing the buoyancy force with the hydrodynamic force (Fan 2009) yields the condition whereby a magnetic flux tube will be unstable to buoyant rise,

$$\begin{equation} B \gtrsim \left(\frac{H_P}{a}\right)^{1/2} B_{\rm eq}, \end{equation} \tag{ 11 }$$

where B is the magnetic field, H_P is the local pressure scale height, a is the flux tube radius, and B_eq is equipartition magnetic field given in Section 2.2.4. This is a simplified picture that ignores effects due to curvature and the response of the magnetic tension that may act to prevent the onset of buoyant instability. However, it provides a reasonable order of magnitude approximation. Given the propensity for magnetic stellar models to invoke magnetic fields with B ⩾ 10⁶ G, we can estimate a typical flux tube radius needed to maintain this magnetic field strength stable against buoyancy.

For M-dwarfs, H_P ∼ 10⁹ cm and B_eq ∼ 10⁴ G. Thus, to maintain strong 10⁶ G magnetic fields deep in the interior (as in MacDonald & Mullan 2012), flux tubes must be no larger than 10⁵ cm, or 1 km. This means the ratio of the pressure scale height to the flux tube radius must be ∼10⁴. By comparison, the values estimated for this ratio near the solar tachocline are of the order of unity (Fan 2009). This ratio rises to 10⁶ (a ∼ 10 m) if the deep interior magnetic field is to be of the order of 10⁷ G (this work). Once the flux tube becomes unstable to buoyant rise, an estimate of the rise time can be approximated assuming that the tube travels at the Alfvén velocity (Parker 1975). A 10⁶ G magnetic field will traverse a stellar radius R ∼ 10¹⁰ cm in approximately 10^5.5 s, or about 10 days, assuming a constant density of 10² g cm⁻³. This rise time will be increased by non-adiabatic heating effects, but remains small compared to evolutionary timescales (Parker 1974).

Figure 10 indicates that a radiative shell develops deep within the star in the presence of a super-MG magnetic field. This means that the dissipation time of the magnetic field does not necessarily obey the advection timescale discussed above. Instead, we may look at the timescale for diffusion given by the induction equation. In the absence of any favorable current networks, we have that

$$\begin{equation} \frac{\partial \bf B}{\partial t} = \eta {\bf \nabla }^2{\bf B}, \end{equation} \tag{ 12 }$$

where η is the magnetic diffusivity. The approximate timescale for diffusion, assuming η ∼ 10²–10³ cm² s⁻¹ (Chabrier et al. 2007), is then τ_diff ∼ L²/η, where L is the size of the radiative shell. The field diffuses through the radiative shell until it reaches the convective boundary, which then efficiently transports magnetic flux to the stellar surface. Based on our models of CM Dra, we find the size of radiative shell is L ∼ 10¹⁰ cm. Therefore, τ_diff ∼ 10¹⁰–10¹¹ yr, assuming the magnetic field fully diffuses out of the radiative zone. However, even within a radiation zone, magnetic buoyancy must be considered and can lead to a rapid rise time for magnetic flux tubes (e.g., MacGregor & Cassinelli 2003).

These are only order of magnitude approximations, so it is possible that the field could survive for a shorter or longer time in any scenario. Instead, this exercise suggests that an amplified seed field, or any super-MG magnetic field, present in a fully convective star will likely decay away rapidly owing to the magnetic buoyancy instability. We note that this is still possible, even if a sizeable radiative zone develops as a result of the strong magnetic field. Furthermore, the radiative zone is a product of the magnetic field and therefore exists in an unstable equilibrium. As the magnetic field decays away, so will the radiative zone.

MacDonald & Mullan (2012) recognized that primordial magnetic fields would likely decay and thus were not a suitable solution for the existence of super-MG magnetic fields within CM Dra. Instead, they proposed that the dynamo mechanism may be able to generate the necessary magnetic fields. Their proposal contradicts what we mentioned earlier, that simulations are unable to generate magnetic fields greater than ∼50 kG in fully convective models. MacDonald & Mullan (2012) reason that this is a result of those simulations using rotational velocities similar to those in the Sun. If one nominally assumes that magnetic field strengths increase with rotation rate, then a star like CM Dra will have considerably stronger magnetic fields than does the Sun.

Using a scaling relation between magnetic field strength and rotational angular velocity, MacDonald & Mullan (2012) find that CM Dra should have a surface magnetic field strength of about 500 G with an internal magnetic field strength anywhere between 0.3 MG and 1.3 MG. This could then explain the existence of super-MG magnetic field strengths. Note that the magnetic field strength is predicted to be the large-scale magnetic field strength, not necessarily the total magnetic field strength.

However, the peak interior magnetic field strength seen in simulations is moderated by the kinetic energy available in convective flows. This was discussed in Section 2.2.4. Simulations find that a dynamo driven by turbulent convection can create both small-scale magnetic fields and large-scale magnetic fields (Chabrier & Küker 2006; Browning 2008). It is not immediately clear that rotation would not drive stronger magnetic field strengths, so we will proceed in our discussion assuming that it might to see if we encounter any inherent contradictions.

The scaling relation adopted by MacDonald & Mullan (2012) would then also explain why KOI-126 can be fit by stellar evolution models. KOI-126 is a triply eclipsing system found earlier in the Kepler data (Carter et al. 2011) that contains two fully convective stars that are nearly identical to CM Dra. Yet, the properties of KOI-126 (B, C) are well reproduced by standard stellar evolution models (Feiden et al. 2011; Spada & Demarque 2012). Assuming a scaling relation between magnetic field strength and rotational angular velocity may support this finding. KOI-126 (B, C) have an orbital period of 1.77 days, to be compared to the 1.27 day orbit of CM Dra. This 0.5 day difference in orbital period means KOI-126 (B, C) would have weaker magnetic fields than CM Dra and would experience considerably less radius inflation (MacDonald & Mullan 2012).

MacDonald & Mullan (2012) predict that KOI-126 should be inflated by approximately 2%–3%. We find this difficult to understand in context of the agreement between standard stellar evolution models and the observed radii. Inflating model radii by 2%–3% would mean that models would over-predict the radii of KOI-126 (B, C). If one assumes that spots are biasing the observed radius measurements, it would decrease the measured radii, again breaking the agreement between the observations and the models. There seems to be problems with simultaneously increasing model radii and decreasing the measured radii of KOI-126 (B, C) while still maintaining model-observation agreement.

Furthermore, the scaling relation suggested to validate super-MG magnetic fields may not be compatible with observations. Results seem to show that at a given mass, the large-scale magnetic field strength of fully convective stars do not appear to depend on rotation below a critical threshold (see, e.g., Figure 3 in Donati & Landstreet 2009). For low-mass stars, this threshold corresponds to a rotation period of about 3 or 4 days. Magnetic field saturation is also observed in data regarding the total magnetic field strength of fully convective stars. Magnetic fields saturate around 3 kG for vsin i > 3 km s⁻¹ (Reiners et al. 2009; Shulyak et al. 2011). Saturation inherently implies that scaling relations are invalid. Since both CM Dra and KOI-126 (B, C) have components rotating faster than the threshold for saturation, there is no reason to assume that their magnetic fields are that different. There is the possibility that their large-scale components (and thus vertical) magnetic fields are sufficiently different, but this is not immediately obvious.

Finally, we note that magneto-convection techniques alone have not been sufficient to rectify the model-observations disagreements. Effects of star spots must be included and must therefore be assessed on their own basis.

The most efficient means of inflating low-mass stellar models of CM Dra is by including star spots (Morales et al. 2010; MacDonald & Mullan 2012). Although we did not explicitly include spots, we will take a careful look at the results of these two previous studies. We feel this analysis is required in light of recent observational data, including confirmation that CM Dra has a sub-solar metallicity of [Fe/H] ≈−0.3  ±  0.1 (Rojas-Ayala et al. 2012; Terrien et al. 2012).

We use approximate fits to the Morales et al. (2010) spot-modeling results to estimate how a metallicity reduction alters their conclusions. Assuming polar spots, they find that spots tend to bias observations toward larger radii. From the few data points in their Figure 7, we calculate that spots produce larger radii according to the function

$$\begin{equation} R_{\rm spot}/R_{\rm real} = 0.187\beta + 1.0, \end{equation} \tag{ 13 }$$

where R_spot/R_real is the ratio of the radius derived when polar spots are present to the actual radius. This function can be used to correct for the radius bias produced by spots. In stellar models, spots can also increase radius predictions. Given the data in Figure 8 of Morales et al. (2010), model radii can be approximated as,

$$\begin{equation} R/\,R_{\odot }= R_0 (0.172\beta ^2 + 0.0834\beta + 1.0) \end{equation} \tag{ 14 }$$

where R₀ is the radius of a non-magnetic model in units of solar radii. We note that this formula approximates the effects of spots only for model masses of ∼0.23 M_☉. The intersection of these two relations in the β–R plane provides an estimate of the star spot parameter required to correct models.

Figure 11 illustrates that a model of CM Dra A with [Fe/H] =−0.3 (see also Figure 7) needs a star spot β ∼ 0.19 to match observations. Small differences in our results from those of Morales et al. (2010) are due to the adopted base model (Dartmouth versus Lyon) and our method of fitting polynomials to the spot radius bias and inflation data in Morales et al. (2010). Changing β has a proportionally larger effect on correction for radius measurement biases than it does on inflation caused in stellar models. If this correction is not applied, β = 0.38 is needed to produce agreement using model radius inflation at sub-solar metallicity.

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Whether the required star spot parameter is realistic is a more difficult question. Spot properties for M-dwarfs are relatively unconstrained, as identifying spot sensitive features in spectra is complicated by uncertainties in modeling molecular features. Nonetheless, there are three components to the spot parameter that are of interest: spot coverage, spot temperature contrast, and spatial distribution. We will address each separately and then present a unified picture afterward.

Temperature contrast. Spot temperature contrast is defined, here, to be the ratio of the spot temperature to the temperature of the unblemished stellar photosphere. As seen from Equation (5), constraining the spot temperature contrast provides insight into the required filling factor predicted by the spot model. Although no direct empirical constraints exist for M-dwarfs, there have been studies performed to measure spot temperature contrasts of K stars using TiO bandhead features (O'Neal et al. 1998, 2004). We will take a closer look at those results, here.

Properties of spots were determined by fitting the shape of TiO bandheads with a spectrum created using a two temperature spot model. This technique involves convolving two template spectra for which the effective temperature is assumed to be known and finding the best combination of template spectra to produce the molecular bandhead features in spectra of a given star (see, e.g., O'Neal et al. 1998). The result is an estimate of the temperatures for the unblemished photosphere and the spotted regions, as well as the fractional surface coverage of each feature (see below). Using this technique, it was found that active K stars—specifically II Peg and EQ Vir—have spots with a characteristic temperature contrast of 75% (25% cooler than the unblemished photosphere).

This estimate, however, relies on the temperature for the two template spectra being correct. Results from interferometric studies (e.g., Boyajian et al. 2012) have revised the temperature estimates for the stars whose spectra were used as templates in the O'Neal et al. investigations. These revisions only affect the temperatures for template spectra used as the spot (cool) component in the fit, namely, the temperatures associated with cooler M-dwarf spectra, and increase the associated effective temperature by 300 K–400 K. Taking into account the revised temperature scale for the cooler template spectra, one finds the temperature contrast for K stars inferred from molecular bandhead features is between 82% (II Peg) and 90% (EK Vir). The warmer spot temperatures are supported by Doppler Imaging (DI) studies of II Peg, which indicate temperature contrasts between 80% and 85% (Hackman et al. 2012).

Assuming that spot temperature contrasts become weaker toward later spectra types (Berdyugina 2005), then one might expect M-dwarfs to have spot temperature contrasts around 90% (10% cooler than the unspotted surface). Using this value, one finds that a spot surface coverage of 55% is required to satisfy the condition that β = 0.19 for CM Dra. Of course, slightly warmer or cooler spots may be permitted, depending on the efficiency at which spots temporarily suppress surface convection. For the sake of argument, we take 90% to be a typical spot contrast. We now look whether a spot areal coverage of 55% is consistent with empirical evidence.

Surface coverage. To assess what constitutes typical spot surface coverages, we look to results from DI and molecular bandhead fitting. Using DI, spot coverages for three M-dwarfs have been estimated. DI maps of HK Aqr, RE 1816+541, and V374 Peg (Barnes & Collier Cameron 2001; Barnes et al. 2004; Morin et al. 2008) reveal varying levels of surface coverage from 2% (V374 Peg) up to 40% (HK Aqr). Similarly, surface coverages derived by modeling the shape of TiO bandheads of active K stars—again II Peg and EK Vir—are typically around 30%–40%, consistent with values found in DI investigations (O'Neal et al. 1998, 2004). There are no measurements of spot coverages on M-dwarfs using this technique, which limits the applicability of these results, but we may cautiously extrapolate and assume that M-dwarfs are able to possess at least similar surface coverages.

For one of the K stars studied using the molecular bandhead technique, II Peg, information on spot coverage from DI is also available. Hackman et al. (2012) mapped the spot coverage of II Peg and find seasonal variation in spot coverage between 5% and 20%, a factor of eight and two lower than the coverage inferred from TiO bandheads, respectively. Seasonal variation is expected if magnetic activity cycles are present in other stars, so it's not surprising that one observes this phenomenon. However, the difference between the maximum surface coverage revealed by DI (20%) and that obtained using molecular bandhead features (40%) highlights a known limitation with DI studies: they are only sensitive to spot features larger than the grid resolution in the DI analysis. In other words, small-scale spot features will be missed by the DI reconstruction. This is especially relevant since comparison of spectro-polarization measurements of M-dwarf magnetic fields using Stokes I and V polarization indicate a majority of the magnetic energy is contained in small-scale features (Reiners & Basri 2009). Therefore, spot coverages inferred from DI maps must be taken as lower limits to the true spot coverage. One may then plausibly expect spot surface coverages at least as high as 40% for M-dwarfs.

Evidence from both DI and molecular bandhead fitting support the existence of spot coverages around 40% for active late K-dwarfs and M-dwarfs. This is below the 55% needed to satisfy the use of a β = 0.19 spot parameter. Whether 55% is unrealistically high is unclear. If, for instance, the spot temperature contrast were around 85%, as opposed to 90%, then the required spot surface coverage would be 40%, and thus consistent with observations. One could just as well argue the opposite, that a larger temperature contrast of 94% would require a surface coverage of 87% and push the requirement further from the current empirical evidence.

One further scenario is that stars are able to achieve spot coverages near 100%. Observations suggest that the filling factor of magnetic fields at the surface of fully convective stars (both main-sequence and pre-main-sequence) approaches unity (Johns-Krull et al. 2004; Shulyak et al. 2011; Johnstone et al. 2014). We must then try to understand the connection between the presence of magnetic fields and star spots. This includes determining how strong of a magnetic field is necessary at the surface of an M-dwarf to generate spots of appreciable temperature contrast. It may be that pervasive 1 kG magnetic fields covering the whole surface of fully convective stars may be too weak to cause any noticeable effects on convection. Such an even distribution of small spots would also likely not cause any noticeable light curve modulation, if filling factors approach unity. However, pockets of strong 5–8 kG magnetic fields (Shulyak et al. 2011) may produce the noticeable light curve modulation, bias stellar radii measurements, and suppress surface convection.

Spatial distribution. Arguably, whether spots cluster near the poles of active M-dwarfs is of greatest consequence for the present assessment. Evidence for spots clustered at the poles is less definitive than temperature contrasts and surface coverages. About half of all stars studied using DI show signs of polar caps (Berdyugina 2005). Exceptions are very active stars and M-dwarfs, which tend to have spots distributed across all latitudes. Of the three DI spot maps generated for M-dwarfs, none display evidence for polar caps (Barnes & Collier Cameron 2001; Barnes et al. 2004; Morin et al. 2008). A DI study of CM Dra to search for possible signs of significant polar coverage of spots is needed. Ideally, this would be performed with existing data that was used to derive the stellar properties, but a look at the signal to noise of the observations reveals that the data is likely unsuitable for such an investigation (see Metcalfe et al. 1996). In the meantime, we must be cautious as even polar spots would require large areal coverages (half or more of the stellar surface) to provide agreement between models and observations.

Additional evidence for non-polar-cap distributions comes from studies of the ∼150 Myr open cluster NGC 2516. The statistical distribution of light curve modulations assumed to be caused by star spots in the low-mass star population can only be reproduced if the stars have randomized spot latitudinal distributions and rotational inclination angles (Jackson et al. 2009; Jackson & Jeffries 2013). We do note that these latter studies also require spot temperature contrasts of 75%, which we have just discussed may be too dramatic for K- and M-dwarfs. How using a more realistic spot contrast would alter their results is unclear as there are strong degeneracies when modeling spot modulation in light curves. A first estimate would be that typical surface coverages would increase from the 40% ± 10% that was required in their study.

There is as yet no definitive empirical evidence for polar cap spots among fully convective main-sequence stars. However, there is a theoretical expectation that spots should be located near the poles in rapidly rotating stars (e.g., Schüssler & Solanki 1992; DeLuca et al. 1997). Whether this expectation applies to fully convective stars is questionable, as spots distributed more evenly at lower latitudes may be more reasonable for more distributed dynamos (DeLuca et al. 1997). Based on only three data points, the apparent lack of polar spots among fully convective main-sequence stars is not robust and requires further confirmation.

Is it then possible to abandon the idea of a largely polar cap distribution and still rely on spots to address the radius discrepancies? In Figure 11, we see that abandoning the idea of polar spots requires β = 0.38 if model inflation is to alone correct for the observed radii. In this case, a spot temperature contrast of 89% with areal coverage of 100% would be required. Temperature contrasts greater than 89% lead to areal coverages greater than 100% and are thus unphysical. Even as one approaches 100% surface coverage (for temperature contrasts below 89%), it becomes increasingly difficult to produce spot modulation in light curves as there are fewer regions of unblemished photosphere left to produce the asymmetries needed, as discussed in the previous section. Considering patches of varying spot temperature contrast may provide the necessary asymmetries, but has not yet been investigated for consistency with the empirical data.

The case for magnetic fields inflating the radii of fully convective stars appears to be tenuous. Stabilization of convection in stellar models can produce the necessary radius inflation with reasonable surface magnetic field strengths. However, the models require super-MG magnetic fields in the interior. We are hesitant to suggest that real stars contain such strong interior magnetic fields for several reasons.

First, turbulent dynamos cannot generate magnetic fields with strengths greater than about 50 kG (Dobler et al. 2006; Chabrier & Küker 2006; Browning 2008). How magnetic fields with strengths greater than 1 MG would accumulate in the interior of fully convective stars would be unknown. If super-MG magnetic fields are of primordial origin, they would probably not survive to the present day. Nor would magnetic field strengths of that magnitude be generated within the star and survive on evolutionary timescales. To avoid this, one can assume a scaling relation between magnetic field strengths and rotation, as suggested by MacDonald & Mullan (2012) to explain the super-MG magnetic fields. However, this breaks down for most stars in DEBs. Observations indicate a saturation of magnetic flux for fully convective stars that have rotation periods below ∼2.5 days (Donati & Landstreet 2009; Reiners et al. 2009; Shulyak et al. 2011). Also, KOI-126 (B, C) agree with stellar evolution models (Feiden et al. 2011). These stars exist in a regime where magnetic flux saturation should occur, but are seemingly uninfluenced by the presence of any magnetic phenomena. If super-MG magnetic fields are causing dramatic inflation in CM Dra, one would expect to observe it in KOI-126 (B, C) as well. Finally, suppression of convective efficiency is unable to provide a suitable solution on its own. Instead, star spots may be invoked for fully convective stars, which introduces additional concerns.

The presence of star spots on the stellar surfaces may provide an adequate solution, mostly through biasing the analysis of light curve data. From observations, star spots on fully convective stars can be characterized as: having temperatures about 10% cooler than the surrounding photosphere (O'Neal et al. 1998, 2004; Berdyugina 2005; Hackman et al. 2012), having surface coverages ranging from a couple percent (Morin et al. 2008; Hackman et al. 2012) up to around 40% (Barnes & Collier Cameron 2001; Barnes et al. 2004; O'Neal et al. 1998, 2004), and having spot distributions that appear to be more random and not clustered at the poles.

At this point, there is not sufficient empirical data regarding spot properties of M-dwarfs to draw firm conclusions. Still, the lack of noticeable spots at the poles is a concern. Results regarding how spots bias radius measurements rely on concentrated regions of spot coverage at the poles that are darker than their surroundings. At present, this feature is not observationally supported, but more work needs to be performed on this matter.