The Radius and Entropy of a Magnetized, Rotating, Fully Convective Star: Analysis with Depth-dependent Mixing Length Theories

Abstract

Some low-mass stars appear to have larger radii than predicted by standard 1D structure models; prior work has suggested that inefficient convective heat transport, due to rotation and/or magnetism, may ultimately be responsible. We examine this issue using 1D stellar models constructed using Modules for Experiments in Stellar Astrophysics (MESA). First, we consider standard models that do not explicitly include rotational/magnetic effects, with convective inhibition modeled by decreasing a depth-independent mixing length theory (MLT) parameter α MLT (following Cox et al. 1981;Chabrier et al. 2007). We provide formulae linking changes in α MLT to changes in the interior specific entropy, and hence to the stellar radius. Next, we modify the MLT formulation in MESA to mimic explicitly the influence of rotation and magnetism, using formulations suggested by Stevenson (1979) and MacDonald & Mullan (2014) respectively. We find rapid rotation in these models has a negligible impact on stellar structure, primarily because a star's adiabat, and hence its radius, is predominantly affected by layers near the surface; convection is rapid and largely uninfluenced by rotation there. Magnetic fields, if they influenced convective transport in the manner described by MacDonald & Mullan (2014), could lead to more noticeable radius inflation. Finally, we show that these non-standard effects on stellar structure can be fabricated using a depth-dependent α MLT : a non-magnetic, non-rotating model can be produced that is virtually indistinguishable from one that explicitly parameterizes rotation and/or magnetism using the two formulations above. We provide formulae linking the radially-variable α MLT to these putative MLT reformulations.

Exploring the M-dwarf Luminosity–Temperature–Radius relationships using Gaia DR2

“…Hence there has been much theoretical work over the last decade to develop consistent stellar evolution models that correctly account for dynamo effects and stellar magnetism, including the resulting starspots (Mullan & MacDonald 2001;Feiden & Chaboyer 2013;Somers & Pinsonneault 2016;MacDonald & Mullan 2017). Recently these effects have been modelled in 1D stellar structure models by Ireland & Browning (2018), who adopt a depth dependent mixing length theory parameter α MLT ; emulating the effect of convective inhibition. In their work they observe radii inflated by 10 -15% when compared to models that do not treat convection in this way, though they caution that such treatments of magnetic inhibition are highly uncertain and may be difficult to calibrate.…”

Rotation suppresses giant-scale solar convection

“…The approach used in the present manuscript is a force-balance approach intimately dependent on the processes that dictate the magnitude of the Bernoulli pressure. Detailed notions of rotational anisotropy have been slow to gain traction in the stratified stellar modeling community and astrophysics in general, although that has been changing in recent years (82)(83)(84)(85)(86)(87). Overall, it seems that a consensus is forming surrounding the proper mechanical and thermal balances in rapidly rotating convection.…”

Existence of Rotating Stars with Variable Entropy

“…Our problem is thus reduced to finding solutions of ( 11) and ( 12) subject to the boundary data on the floor ( 13), (17) and the mass constraint (18) on the large ball B R . The existence result for our reformulated div-curl system is stated as follows.…”

“…Let s 0 ∈ C 1,β (B R ∩ {z = 0}) and ω 2 ∈ C 2,β (B R ) be given. There exist ǫ > 0, ǫ 1 > 0 such that if |κ|+|µ|< ǫ 1 , then there exists a unique solution (V * , α * ) ∈ C 2,β (B R ) × R, and S * ∈ C 1,β (B R ) to (11), (12), (13), (17), (18), with (V * , α * ) − (V 0 , α 0 ) C 2,β (BR)×R ≤ ǫ. Furthermore, the solution has the properties that V * + is supported on B R 0 +R 2 , and S * = 1 outside B R1 for some fixed R 1 ∈ (R 0 , R).…”

“…When it comes to rotating star solutions with variable entropy, however, very few results are available, despite its physical importance [18,31]. The goal of this paper is to establish the existence of rotating star solutions for self-gravitating perfect fluids with variable entropy.…”