Stellar magnetic activity: Is Rossby number really the key?

Stellar magnetic activity: X-rays Pallavicini et al (1981): L X ~ (v rot sin i) 1.9

Rotation period or Rossby number? P P/ c 4 Noyes et al (1984): normalized chromospheric emission R HK = F HK / T eff is better represented by P/ c than by rot. period alone ( c : convective overturn time from mixing-length models)

Convective overturn time: c vs. B V Noyes et al. (1984) 4 What if T eff has a similar dependence on B V as c?

Why Rossby? Rossby number: dimensional ratio of inertial force to Coriolis force Ro = U / 2 L = 1 / 2 (observers tend to omit the factor 2 ) Amount of twist of the Parker loop ( alpha-effect) is related to Ro: Coriolis acceleration: a cor ~ 2 u hor L ~ 2 u c c 2 ~ 2 u c /L c 2 ~ 2 c = Ro 1 Note that for > 90 deg, the alpha-effect decreases again and becomes negative (destructive) for > 180 deg

Why Rossby? Durney & Latour (1978), Noyes et al. (1984): simple -dynamo model (Parker wave) dynamo number D = d 4 / 2 > D crit for ~ d 2 / c, ~ d, ~ /d we obtain D ~ 2 c2 ~ Ro 2 This is the basis of the Rossby scaling. Flux Transport Dynamo: alpha-effect based on the tilt of bipolar regions resulting from flows along buoyantly rising flux tubes Rossby number of convection not relevant for this process

The saturated regime & empirical Ro Wright et al. (2011) For rapidly rotating stars, L X saturates at ~ 10 3 L bol Trust in the Rossby formulation has grown so much that even an empirical c is derived that minimizes the scatter of L X /L bol vs. Ro

c or normalization? Pizzolato et al. (2003) Empirical c agree reasonably well with values derived from mixing-length models, but ½ also scaled L bol has the same dependence. Does c just compensate for the normalization L X /L bol w/o further physical significance?

Generalized analysis Assume a general power-law relationship between normalized X-ray luminosity, rotation period, P, and stellar radius, R: Use the stellar sample of Wright et al. (2011) Determine the exponents and through minimizing the scatter of fits to the data in the unsaturated and in the saturated regimes (break point included in the minimization process) Result: = 4.3, = 2.2 2 intervals: Scatter changes only in the third digit for = 4, = 2 4.7 < < 3.8 2.3 < < 2.

Comparison to Wright et al. (2011) Wright et al. (2011): empirical Rossby number L X /L bol (P/ c ) 2.7 P 2.7 R 2.9 Scatter = 0.371 dex Reiners et al. L X /L bol P 2 R 4 Scatter = 0.346 dex (6% less than Wright et al. have)

Comparison to Wright et al. (2011) Scatter as a function of / Wright et al. (2011) 2 confidence interval (assuming a normal distribution for the uncertainties of L X /L bol )

Getting rid of L bol We have and for MS stars: L bol R 4 so that cf. Pallavicini et al. (1981)! Pizzolato et al. (2003)

Saturation

Saturation With = 4, = 2, and saturation sets in for

Relation to the magnetic field Pevtsov et al. (2003): L X ~ mag,surf (Zeeman broadening) Vidotto et al. (2014): L X ~ V 1.80, L X /L bol ~ V 1.82 (Zeeman Doppler Imaging) Since L X P 2 : magnetic flux depends only on rotation, not on any other stellar parameter (e.g., radius) Physical mechanism? Sun related to bipolar regions, loops How is that connected to the general dynamo amplitude, i.e., the amount of flux generated in the convection zone?

Dynamo amplitude: nonlinearity matters! Field amplitude depends on the kind of nonlinearity considered: D = D 0 * (B/B eq ) ~ Dcrit B/B eq ~ (D 0 /D crit ) 1 / D 0 ~ ~ for ~ and = const. D 0 ~ 2 for ~ and ~ A scaling B ~ 2 then requires = 1 or = 0.5 for these cases (a rather soft nonlinearity) In fact, nonlinear dynamo models with -quenching and = 2 show, e.g., B ~ D 0 0.65 (FT dynamo, Karak et al. 2014) B ~ D 0 0.5 (1D dynamo, Schmitt & S., 1989)

Possible origin of saturation Limit of the fraction of the total energy flux that can be converted to magnetic energy relation between E mag and L X?? P sat (L bol ) 1/2 R 2 critical rotation rate surface area surface filled with bipolar regions? Transition from quenching to quenching

Rossby or not Rossby Formally, our result L X /L bol P 2 R 4 Rossby number, Ro=P/ c : can be written in terms of a If we obtain Reliable quantitative information about Rossby numbers in stars?? Our rotational scaling involves no further assumptions about stellar properties. Dependence of dynamo driving, the nonlinearity determining the amplitude, and amount of emerging flux on rotation is quite unclear. At present, there is neither an empirical nor a theoretical basis for relating stellar activity and surface magnetic flux to the Rossby number.

Conclusions The theoretical basis of the Rossby scaling is limited to the classical turbulent-convective -effect. It also requires a specific scaling of the differential rotation. Other models (such as the FTD) are not expected to directly involve the Rossby number. The scaling of the field amplitude (and thus activity indices) with rotation (and other parameters) depends on the nature of the field-limiting nonlinearity. In simple models, the scaling of activity indices with 2 suggest a rather soft nonlinearity.

Rossby or not Rossby Saturation: L X /L bol = 10 3 Two branches in L X /L bol two different nonlinearities (soft & hard)? L X /L bol ~ sat 2 /L bol = const. sat ~ (L bol ) 1/2 ~ R 2 Surface area, meridional cross section ~ R 2

Rossby or not Rossby Pevtsov et al. (2003): L X ~ mag,surf Physical mechanism? Sun related to bipolar regions, loops here: L X /L bol ~ 2 * R 4 and L X ~ 2 (L bol ~ R 4 ) often: L X /L bol ~ ( c ) k ~ Ro k (e.g. k=2.7, Wright et al.) c :convective turnover time Ro: Rossby number effect of rotation on convection