ABSTRACT 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. INTRODUCTION 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: }{\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, η 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 Ω 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 . This review will be based on and should start by discussing the main ideas in this review. Other papers to discuss not included in : 1.6 year cycle and multiple cycles in ι Horologii The idea about a second dynamo: , , , , , Discussion of TSI: , , Oscillation amplitudes and activity 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, ) - 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 To our knowledge the first observations of emission in Ca II H & K lines in solar-like stars were done by 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 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 , 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 and a smaller update of 35 stars was presented in , 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 . Unfortunately, these two sets of observations have never been combined. In fact non of the Mount Wilson observations are public and 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 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. used the results from 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 $/\Omega$ as a function of the Rossby number Ro (which is defined as ...). In this way stars of intermediate age generally showed a $ \propto {\rm Ro}^{-0.7}$ relationship, whereas the ratio $/\Omega$ increased by a factor of 6 for stars of solar age or older. This led them to suggest that the dynamo α-parameter increases with magnetic field strength, contrary to the conventional idea of α-quenching. names the two groups of stars active and inactive branch stars. A naming that is still being used today. extended the analysis of to also include so-called RS Canum Venaticorum variables (binary stars with shows very high activity levels) and less certain cycles and evolved stars from . This study not only confirmed the result by it also showed that the most active stars with rotation period below 3 days occupied a third branch call the superactive branch and that the intermediate age and the solar age and older stars could in fact have cycles on both the active and the inactive branch, where the intermediate age stars tend to have their primarily cycle on the active branch and the solar age and older stars tend to have their primarily cycle on the inactive branch. The analysis was redone by , who compared the cycle periods to the rotation period in what is now known as the Böhm-Vitense diagram, which show the active and inactive branches very clearly. What is very peculiarly about the Böhm-Vitense diagram falls right between the active and the inactive branches. suggested that cycles on the different branches are driven by different kind of dynamos and thus that the discontinuous break between for stars around the age of the Sun is caused by a change in the dynamo action which she suggest is due to abruptly increased deep mixing. The idea behind the Böhm-Vitense diagram was taken up by and in their _sounding stellar cycles with Kepler_ project, who rely on the fact the asteroseismic observations from the _Kepler_ mission could be used to test the hypothesis that the discontinuous break between for stars around the age of the Sun is caused by a change in the dynamo action which she suggest is due to abruptly increased deep mixing. This is possible as asteroseismology can be used to measure both ages, differential rotation and ages of the stars. Unfortunally, _Kepler_ was only fully operational for four years, which appears to be to short to allow any firm detection of stellar cycles using _Kepler_ observations alone. The project does however continue using ground-based facilities to measure activity cycles. was the first to realize that the high precision space photometry could be used to search for stellar cycles. They reported the detection of a stellar activity cycle with a period of at least 120 days in the F type star HD 49933 using observations from the CoRoT satellite. This is however much shorted than any of the activity cycles detected in the observations from the Mount Wilson and Lowel observatories. Though it might be immature to call the signal in HD 49933 detected by CoRoT a cycle, the reason why the possible cycle period might be so sort could be that this star is an F type star with a very thin outer convection zone. This is supported by a number of other studies, including who identified a 1.6 year cycle in the F8 star ι Horologii using Ca HK measurements from the Small and Moderate Aperture Research Telescope System 1.5 m telescope at Cerro Tololo Interamerican Observatory and the who identified possible activity periods in the two F type stars KIC 3733735 and KIC 9226926 with periods of 800 and 500 days, respectively, using what they call a photometric S index, which is calculated as the mean value of the standard deviations of the subseries of length 5 times the rotation period. When analyzing observations on F type stars from CoRoT and _Kapler_ we think it is important to remember the conclusion from that stars younger than 1 Gyr are characterized by fast rotation and high average activity levels, they often show large variability, but rarly cycles. F type stars do not stay for much more than 1 Gyr on the main-sequence, so most of them fall in under this catagory. F type stars often show one or two cycles in the CoRoT and /it Kepler observations, but never more and before or after or in between you have more or less quite periods. used a three-dimensional magnetohydrodynamic anelastic spherical harmonic code to simulate differential rotation and a magnetic dynamo in a 1.2 M⊙ F-type star at two rotation rates. These simulations showed that when the rotation rate was increased above 20R⊙ dynamo action would start to rise in the models with polarity reversal every 1600 days. The magnetic energy could however rise and fall a few times within a polarity period. Another approached was used by to follow the idea by . used a mean field dynamo model to model the F-type star KIC 5955122 using stellar parameters obtained with asteroseismology and a differential rotation profiled obtained from spot modeling. This analysis revealed that KIC 5955122 had started to evolve off the main-sequence and that is has strong differential rotation with a rotation rate between 18-26 days. In other words in the domain where the models by could not produce dynamos. THe analysis by also indicated that this KIC 5955122 had evolved from an active to a quiet status and the mean field models predicted that the characteristic activity cycle is of the order of the solar one. In other words the dynamo generated by the very thin convection zone in these F-types stars may be much more unstable that the solar dynamo and jump rapidly in and out of quiet periods and periods characterized by stable cycles. This agrees with the picture by . There is another point where the observations does not agree with the interpretation by and the is the absolute level of the chromospheric emission in these stars, the S index. KIC 5955122 has a S index of only 0.14. This could suggest that the dynamos in these evolved F-type stars generally only produce weak chromospheres. This idea is partly supported by the S-indexes as a function of B − V color index measured by . Here it is clear that KIC 5955122 with a B − V value of 0.54 is located in a sweet spot where stars with similar color generally show very low chromospheric activity – another star with B − V of 0.54 is Procyon, for example. A number studies have also searched and found indication of stellar cycles in stars later than F. tried to measure rotation periods in 16 FGK main-sequence stars using observations from CoRoT. Periods between 33 and 650 days are found. Though simulations predict that only half of these cycle period should be true, it is remarkable to see the nice relation between rotation and cycle period, including the identification of a active and an inactive branch. analyzed to G-type stars, with rotation periods of 6.0 and 14.7 days and found indication of activity cycles of 1.3 and 2.5 years respectively. Again, this is shorter than the periods found by and here both studies are looking at Sun-like stars. By comparing the photometric variability observed in these two stars with the Sun, arrives at another very interesting conclusion, i.e. that the rotational modulations of the light curves observed by _Kepler_ for these two stars are caused by bright faculae and not dark spots. A number of studies have been published analyzing photometric variability in _Kepler_ light curves , without attributing this variability to stellar cycles. We will come back to these studies in the next sections. ROTATION The study of stellar rotation was pioneered Robert P. Kraft in a series of papers . These observations, together with the early observation from Odie Wilson led to the famous Skumanich spin-down law , which connects to rotation of a star to its evolutionary stage. The simple idea is that a star, as it goes through life, loses angular momentum to a stellar wind and thus spin down. The general idea behind the Skumanich spin-down law is still accepted today, but the theory no contains a lot more details and if fact also, a lot more unknowns. Stars like the Sun are formed from rotating molecular cloud. As these cloud contract, they will stars to spin-up, in order to conserve angular momentum. This means that Sun-like stars, as they arrive on the main-sequence, will be rotating relatively fast. Naively, we would assume that Sun-like stars would have close to homogenous rotation when they arrive on the main sequence, but even this assumption is like not corrects (ref). We would then expect that the star spin-down through out the convection zone and thus build up radial differential rotation in the convection zone. This is how ever also not correct. This was proven from the first helioseismic observations of progrande and retrograde sectroal oscillations modes by . These observations revealed that the solar convection zone rotated at more or less the same rotation profile as the solar surface leaving all the radial shear to the thin border between the radiation and the convection zone. A region that have later been names the _tachocline_ . used observations from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) spacecraft to calculate was should properly be names the standard to rotation profile. This profile shows that the decrease in angular velocity with increasing latitude seen on the surface continues all the way down to the tachocline, where a strong shear layer change the latitudinal rotation profile of the convection zone to a solid body rotation rotation in the radiation zone. The paradigm change to to a very small radial differential rotation in the convection zone had a huge impact on our understanding of the solar dynamo. Here the idea of a so-called αΩ dynamo _distributed_ of the convection zone had to be abandoned . Instead a so called _interface_ dynamo was proposed, where the α effect is seated at the tachocline was more accepted, but this subjected has attached renewed attention in recent years, partly do to new observations of _slower_ rotating cores in red giants. Mean-field dynamo models of the solar are most often so-called αΩ dynamos, where the the toroidal component of the magnetic field is driven by rotational shear (the Ω effect) and the poloidal component is driven by turbulence (the α effect). For fully convective M stars the dynamo responsible for generating the magnetic field is expected to be a so-called α² dynamo, where the α effect is driving both the toroidal and the polodial component of the magnetic field (ref). In young stars with week differential rotation the dynamo could be a so-called α²Ω dynamo, where the toroidal component of the magnetic field is driven by both the α and the Ω (as discussed for the F-type stars ). The existing of this zoo of dynamo models suggest that the nature of the solar dynamo might not has been as it is now, throughout the whole life of the Sun and that we should be carefully with assuming that we understanding the dynamo in the Sun. THe theoretical description of how stars loss angular momentum to stellar winds was formulated by and tested on stellar models by . This analysis suggested that though stars are form with a verity of different initial angular momentum, but the time the stars reach an age of 80 million years, then have also spun so much down that their rotation rate is independent of the initial angular momentum. This explains the strong relation between rotation rate and age seen by , but the general picture is still that the angular momentum is distributed evenly over the convection zones of the stars. This was changes with the model by who used a set of coupled diffusion equations to describe the internal transport of angular momentum throughout the convection zone. These calculation show that F and G type stars losses angular momentum more efficient than K and M type stars. As the F and G type stars have thiner convective zone and larger radiative zones than K and M stars, F and G type stars will build up a stronger tachocline. THis will lead to a coupling between the radiative interior and the outer convection zone. In the K and M type stars there will be no such coupling and the angular momentum can therefore only be lost from the (thick) convective zone, which will lead to a relative faster spin-down rate. combined these studies into the term he call _gyrochronology_ – i.e. how to relate stellar rotation periods to stellar ages. The idea is that FGKM stars separate into three different sequences: the I sequence, the C sequence and the R sequence. I stand for _interface_ and represents stars that have a coupling between the radiative interior and the convective envelope. C stands for _convective_ and represents stars with deep convective zones, where the radiative interior is decoupled from the convective envelope and where a αΩ and an α²Ω dynamo have not yet set in. R stands for _radiative_ and represents early F-type stars with very thin convective zones that are thus dominated by the radiative interior. A note of caution here. The I, C and R sequences by have nothing to do with the active and inactive branches by . Stars on both the active and the inactive branch are likely to have an αΩ or an α²Ω dynamo, where as stars on the C sequence are likely to have an α² dynamo. Though dynamos in the stars on the R sequence are likely to be α²Ω dynamos (ref), they are also likely to have a rather different nature than the dynamos on the I sequence. The stars on the I sequence, however, can belong both to the active and the inactive branch. The first study of rotation periods in a large ensemble of stars using _Kepler_ observations was done by who identified 3200 stars that show signs of rotational modulation and of activity in the light curve, using a supervised classifier on a number of Fourier parameters of the light curves. have done a series of studies of rotation modulation of _Kepler light_ curves with periods between 0.2 and 70 days. These studies claim a number of interesting findings, including: a) that fast rotating stars with large photometric variability tend to have lower proper motions, indicating that they are located closer to the galactic plane; b) that A and F type stars have short (less than 2 days) rotation period and show larger photometric variability, whereas G,K and M type stars have longer (larger than 5 days) rotation periods and show smaller photometric variability; c) that the size and life-time of the active regions responsible for the photometric variability increase towards later spectral type; d) that M-type dwarfs on average have a bimodal period distribution with peaks at ∼19 and ∼33 d (from here they unfortunately go on to speculate that this is caused by the Vaughan–Preston gap and two distinct waves of star formation ); e) that only slowly rotating stars with rotation periods longer than 5–10 days can host close-in planets with orbital periods shorter than 2–3 days. We will discuss these results in details below, but star by noting that simulations have shown that if _Kepler_ had observed the Sun, if would not had been possible to reliable estimate the solar rotation period from these observations [figure] results from CoRot and Kepler Recently, CONVECTION
