The solar-stellar connection after CoRoT and Kepler


The solar-stellar experiment started with the HK Project at Mount Wilson Observatory initiated by Olin Wilson in 1966, continued with the hunt for solar analogs up through the eighties and nighties and have culminated so-far with the remarkable results from CoRoT and Kepler.


With the term the solar-stellar connection we understand primary the field where studies of Sun-like stars are used to increase our understanding of the Sun. In this review we will focus on how Sun-like stars can be used to increased our understanding of the solar dynamo. The solar dynamo can theoretical be described by the magnetohydrodynamical (MHD) induction equation(Charbonneau 2005):

\[\frac{\partial {\rm \bf B}}{\partial t} = \nabla \times \left({\rm \bf u} \times {\rm \bf B} - \eta \nabla \times {\rm \bf B} \right)\]

where \({\bf B}\) is the magnetic field, \(\eta\) is the magnetic diffusivity and \({\bf u}\) is the large-scale flow field. This flow filed can be expressed as a sum of an axisymmetric azimuthal component originating from the differential rotation ans a axisymmetric poloidal component originating from the meridional circulation:

\[{\rm \bf u} = {\rm \bf u_p}\left( r,\theta \right) + r ~{\rm sin}~\theta ~ \Omega \left( r, \theta \right) \hat {\rm e}_{\phi}\]

where \(r\) is the distance from the center of the Sun and \(\Omega\) is the angular velocity.

It is thus clear that in order for the Sun or other Sun-like stars to host a dynamo they need to have magnetic fields, differential rotation and meridional circulation or convection. These ingredience have thus been chosen as the subject to be covered by this review:

  • Magnetic fields

  • Rotation

  • Convection

One chapter will thus be dedicated to each of these subjects in this review. We have also dedicated a chapter to the discussion of the ’Solar Analogs’ and ’The Sun in Time’ projects (Strobel, 1996; Güdel, 2007).

This review will be based on (Hall 2008) and should start by discussing the main ideas in this review.

Other papers to discuss not included in (Hall 2008):

1.6 year cycle (Metcalfe 2010) and multiple cycles (Metcalfe 2013) in \(\iota\) Horologii

The idea about a second dynamo: (Fletcher 2010), (Broomhall 2011), (Simoniello 2012), (Vecchio 2012), (Laurenza 2012), (Simoniello 2013)

Discussion of TSI: (Shapiro 2011), (Shapiro 2013), (Shapiro 2014)

Oscillation amplitudes and activity (Chaplin 2011)

Magnetic fields

It is not possible to measure stellar magnetic fields directly with e.g. a Hall sensor. Instead the effect of magnetic fields can be observed through mainly:

  • Observations of chromospheric non-thermal emission generated by magnetic fields (This is mainly done in the Ca II H & K lines, Hall (2008))

  • Observations of coronal non-thermal emission generated by magnetic fields (This is mainly done in X-ray, ref)

  • Observation of degeneration of spectral lines due to the Zeeman effect (Zeeman 1897)

To our knowledge the first observations of emission in Ca II H & K lines in solar-like stars were done by (Eberhard 1913) and they noted that this emission was similar to what was seen in sun-spots. In fact they also speculated It remains to be shown whether the smission lines of the star have a possible variation in intensity analogous to the sun-spot period. This glove was taken up by (Wilson 1968, Wilson 1978) who measured variability in the chromospheric emission from 91 main-sequence stars (F2 to M2) over the length of a solar cycle. These observation revealed that cyclical variations occur with periods ranging from about 7 years to probably at least twice as long. These observations were updated by (Duncan 1991), who also introduced the canonical S-index at a measurement of the chromospheric emission. The last big update from what is know known as the Mount Wilson project was (Baliunas 1995) and a smaller update of 35 stars was presented in (Radick 1998), which also made the first connection between the observations at Mount Wilson and Lowell observatories. The observations at the Mount Wilson Observatory were terminated in 2003, but complementary observations of the 97 stars began in 1996 at the Lowell Observatory and are still ongoing (Hall 2007, Hall 2009). Unfortunately, these two sets of observations have never been combined. In fact non of the Mount Wilson observations are public and (Baliunas 1995) only include observations up to 1992, so in somewhere a decade of unpublished observations is laying around just waiting to me published.

The results from (Baliunas 1995) can be summerise as follows: out of 112 stars with spectral class between F2 and M2 including the Sun 52 showed cycles with periods between 2.5 to 25 years, 29 showed variability, but no cycles and 31 showed no variability or only a linear trend. For stars with spectra class between G0 to K5 V a pattern of changes in the rotation and chromospheric activity on an evolutionary timescale was indetified. This pattern suggested that these stars could be separated into three distinct groups: 1) stars younger than 1 Gyr that were characterized by fast rotation and high average activity levels. These stars often show large variability, but rarly cycles; 2) stars of intermediate age that were characterized by moderate rotation rates and activity levels. These stars often had shooth cycles and 3) stars as old as the Sun or older that were characterized by slow rotation and low activity levels. Some of these stars showed smooth cycles and some showed flat activity levels.

(Brandenburg 1998) used the results from (Baliunas 1995) to show that the difference between stars of intermediate age and stars of solar age or older give rise to a discontinuous dependence of the ratio of the cycle to rotation frequency \(\omega_{\rm cyc}/\Omega\) as a function of the Rossby number Ro (which is defined as ...). In this way stars of intermediate age generally showed a \(\omega_{\rm cyc} \propto {\rm Ro}^{-0.7}\) relationship, whereas the ratio \(\omega_{\rm cyc}/\Omega\) increased by a factor of 6 for stars of solar age or older. This led them to suggest that the dynamo \(\alpha\)-parameter increases with magnetic field strength, contrary to the conventional idea of \(\alpha\)-quenching. (Brandenburg 1998) names the two groups of stars active and inactive branch stars. A naming that is still being used today.

(Saar 1999) extended the analysis of