What Thin Flux Tube Models Can Tell Us About Star Spots Thomas Granzer, Vienna,

Observation of large-scale magnetic phenomena has a long-standing history:

What is a sunspot?

How does the flux tube get there? 1)They are generated close to the photosphere. 2)They are generated somewhere else, but are than transported to the photosphere. Observations tend to possibility 2

The Model, outline:  Magnetic flux tubes are stored and amplified in the overshoot layer (tachocline).  The star must have an outer convection zone and a radiative core underneath.  Parker instability sets in as the magnetic field grows.  As the crests penetrate into the convection zone, the superadiabatic stratification accelerates the growth of the instability, a loop forms.  The summit of the loop emerges in the photosphere as an active region.

Details of the simulation:  Initial field strength of flux tube from linear stability analysis. For the solar case yields field strength of 10 5 G.  Total flux set to Mx.  Semi-implicit numerical scheme for eq. of motion of a thin, adiabatic flux tube.  Study evolution until the thin-flux-tube approximation breaks down (0.98 R * ).  Project the end-points to photosphere, defines emergence latitude.

Buoyant rise of a flux tube: Sun Giant

Main forces: The ratio between F B and F C determines the emerging latitude.

Results for our Sun: 1)Butterfly diagram 2)Proper motion of sunspots 3)Joy’s law 4)Hale-Nicholson’s law

Butterfly diagram: No explanation, but at least not contradictory

Proper motion of sunspots:  At all latitudes, leading spots rotate 0.6% faster than followers.  Young groups expand rapidly, while old groups contract slowly.

Joy’s law Varies with latitude: -) 6-7º at 30º, -) 1-2º at the equator.

Hale-Nicholson rules  Preceding and following spots are of different polarity.  The leader polarity is opposite in opposite hemispheres.  The magnetic axes are inclined.  The leader polarity changes at the end of the cycle.

Why not extend this model to other stars?  Outer convection zone.  Radiative core/zone.

HRD

Stellar case:  Stellar model from spherical evolution.  Spin up to different  (0.25, 1, 4, 10, 25, 63  , rigid rot. profile).  Mainly PMS-stars, a few (sub)giants for comparison.  Initial field strength of flux tube from linear stability analysis.  Total flux set to Mx.

Which stellar models have been chosen?

Forces, extremely small cores Axisymmetric modes rise to the pole while m >0 modes are driven to the equator.

Spot probabilities 1)We consider twelve flux tubes at latitudes 5 º to 60 º. 2)The end-points are converted into a probability of spot formation.

T-Tauri stars  No significant change in spot pattern for  < 10  .  At   10   an almost immediate transition to polar spots occurs.  For  higher than   25   : bimodal spot patterns: polar spots and medium-latitude spots.

Young PMS stars  Spots on the 0.6 M  star are independent of .  Spots on the 1.0 M  star are higher than on the 1.7 M  star.  Spot latitude increases with .

PMS stars:  Spot latitude increases with  and decreases with mass.  Spot latitudes are higher than for our Sun, even for the 1 M , 1   model.  At  > 25  , a saturation effect for the 0.6 M  model is visible.

ZAMS stars:  Spot latitude increases with  and decreases with mass.  Spot latitudes are lower than for the PMS models.  The 0.4 M  -ZAMS model shows an almost identical spot pattern as the 0.6 M  -PMS model.

Spot latitude vs. core size Spot probability vs. core size for fixed  = 10  .  For anchored flux tubes the core size determines the emerging latitude for fixed rotational rates.  For fully detached flux tubes the instability mode (m = 0 or m > 0) determines the emerging latitude.

PMS-stars vs. giants  Decrease of spot latitude with mass.  Broader region of spot latitudes at giants. PMS-stars, R c /R * =0.4,  = 1  .Giants, R c /R * =0.4,  = 1  .

Comparison to stars. Four well-observed PMS stars:  V410 Tau  Par 1724  HDE  AB Dor

V410 Tau Doppler Image of V410 Tau, flux tube model a 1 M , 1 Myr,  = 14.1    The high-latitude feature and the low- latitude spot are well reproduced

Par 1724 Doppler Image of Par 1724, flux tube model for a 3 M , 0.2 Myr,  = 5.68    The spot centered at 30° latitude is reproduced.

HDE Doppler Image of HDE , flux tube model for a 1.8 M , 2.5 Myr,  = 17.0    High latitude features found, but no polar spot.

AB Dor Doppler Image of AB Dor, flux tube model for a 1 M , 15 Myr,  = 51.2  .  The polar spot is not reproduced.

Questions not addressed: 1)Flux storage for amplification. 2)High field strength necessary. 3)Flux tube coherence on rise. 4)Post-emergence evolution. Thanks To P. Caligari, M. Schüssler

Orion Nebular Cluster:  Spot latitude increases with  and decreases with mass for higher mass stars.  Spot latitudes for a 1.3 M  independent of .  The fast-rotating 1.0 M  model shows activity at all latitudes.

Pleiades Cluster:  General increases of spot latitude with  and decreases with mass.  Spot latitudes are lower than expected for the 4 M  model.  The 4 M  - model shows no spots at  > 4  .