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Key hs di Thus, . α ff e-mail: 07 5, Sternwarte Tautenburg, Th¨uringer Landessternwarte e-mail: ntttfu srpyi,Georg-August-Universit¨at G¨ f¨ur Astrophysik, Institut a lnkIsiuefrSlrSse eerh Max-Planck Research, System Solar for Institute Planck Max ff rnilrtto ae r dnie.Teln hpso 43 of shapes line The identified. are rates rotation erential fdi of rnilrtto stogtt mrefo h inter- the from emerge to thought is rotation erential epeetpoetdrttoa eoiisadnwmeasure new and velocities rotational projected present We ff rjce oainlvelocity rotational Projected h u ipasdi displays Sun The [email protected] [email protected] rnilrotation! erential h vrl raeigpol sdrvdaayigspectra analysing derived is profile broadening overall The ff ff α tr:rtto tr:fnaetlprmtr Hertzspr – parameters fundamental Stars: – rotation Stars: rnilrtto scoeycnetdt mag- to connected closely is rotation erential so o,3 di 33 now, of As rnilrtto nohrstars. other in rotation erential ftednm utntb ofsdwt h pa- the with confused be not must dynamo the of ff rnilrtto a eosre di- observed be can rotation erential aucitn.di no. manuscript ff rnilrtto hti niaeycnetdt h solar the to connected intimately is that rotation erential ff v rnilrttr eetdb iepol nlsshv bee have analysis profile line by detected rotators erential ff sin cietmeaueadrpdrtto.Teei vdnefo evidence is There rotation. rapid and temperature ective i > v sin 2kms 12 mlrvnEi Ammler-von i smaue o oeta 1 ftesml tr.Rgdaddi and Rigid stars. sample the of 110 than more for measured is ff − rot 1 ff ff edtc di detect We . rnilrota- erential rnilrota- erential tign rerc-udPaz1 77 G¨ottingen, Germ 37077 1, Friedrich-Hund-Platz ottingen, α iiu.Tee The minimum. t n aiaiiyo lnt rudohrsasi simpo is it stars other around planets of habitability and ity - hl hr em ob iceac npro eednefo dependence period in discrepancy a be to seems there while Ω ff ABSTRACT 7 atnug Germany Tautenburg, 778 lope 5.D- teryFseta ye.Terneo oiotlserob shear horizontal of range The types. spectral early-F at e dy- oainb iepol nlssaecmie n aeavail made and compiled are analysis profile line by rotation a nyprl eepandb nuprbudfudfrteho the for found bound upper an by explained be partly only can rnetaon frltv di relative of amount trongest M. , tr r ossetwt ii oain vntog di though even rotation, rigid with consistent are stars ta- ial ff to al t ore transform. Fourier its t toghrzna ha n n fmd olt- yesta type late-F to mid- of one and shear horizontal strong ith iaino ai tla aafo htmtyadteident the and photometry from data stellar basic of pilation a Srß ,311Ktebr-idu Germany Katlenburg-Lindau, 37191 2, -Straße - rnilrtto ae of rates rotation erential et fterttoa rfieo oe10nab tr ihs with stars nearby 180 some of profile rotational the of ments 1 , h ufc fsascnuulyntb mgdwt su with imaged be not usually can stars of surface the dwarfs fr white split oscillations. rotationally stellar using 200 for of 2007) Solanki cies al. & (Su´arez et e.g. (Gizon stars stars B solar-type and studied 1999), al. been et (Kawaler has asteroseismology eetal suigta u-iednmsas oko oth d on rotating work be also dynamos to Sun-like stars that active assuming expect ferentially may one Yet, resolution. Ω=Ω = ∆Ω physics: stellar of description the to measurements rsn ok ti h oiotlshear horizontal the is it work, present eaiedi relative ufc di surface Ω law rotation surface the by described sfnto flatitude of function as ihteparameter the with 2 , 3 ( iesaefo udeso pcrllnsb h ehdof method the by lines spectral of hundreds from shape line l l n-usl n - diagrams C-M and ung-Russell ) n enr,A. Reiners, and tla di Stellar di stellar assess to potential The ff Ω = c fdi of ect Equ Equator ff ff rnilrtto sepesdb nua velocity angular by expressed is rotation erential − rnilrotation erential ff ff Ω rnilrtto antb bevddrcl since directly observed be cannot rotation erential rnilrtto ntebodnn rfiei best is profile broadening the on rotation erential (1 Pol yaoadhnerltdt oa ciiy the activity, solar to related hence and dynamo − δ Ω Ω = α α ofimd h rqec fdi of frequency The confirmed. n 2 ff = = α sin w ouain fdi of populations two r rnilrtto 5 )dtce yline by detected %) (54 rotation erential Ω n oe e tr ihsignificant with stars Ten more. and % 5 0 l 2 h oa di solar The . Equ . l frltv di relative of 2 (1) ) α steqatt esrdi the in measured quantity the is ff rnilrtto a be can rotation erential ff ff rnilrtto swell is rotation erential ∆Ω any rnilrtto.While rotation. erential ff tn ounderstand to rtant ff ff rnilrtto by rotation erential rnilrotators, erential rnilrotation erential hc oncsthe connects which aeFstars. late-F r ? Fstars: -F able. fiainof ification evdfor served

ff c ⋆⋆⋆ rizontal erential swith rs pectral S 2018 ESO ffi equen- cient (2) 4), if- er Ω 1 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle stars. On the other hand, a magnetic field may also inhibit differ- projected rotational velocity v sin i, low amounts of differential ential rotation through Lorentz forces so that it is not clear what rotation below the detection limit, shallow or strongly blended degree of differential rotation to expect in active stars. line profiles, asymmetric line profiles, or multiplicity. In the case of the solar rotation pattern, the angular momen- The amount of differential rotation depends on spectral type tum transport by the gas flow in the convective envelope offers an and v sin i. Reiners (2006) note that the frequency of differential explanation. Kitchatinov & R¨udiger (1999), K¨uker & R¨udiger rotators as well as the amount of differential rotation tends to (2005), and K¨uker & R¨udiger (2007) studied Sun-like differen- decrease towards earlier spectral types. Convective shear seems tial rotation in main-sequence and giant stars based on mean- strongest close to the convective boundary. While many slow field hydrodynamics. According to these models, horizontal Sun-like rotators (v sin i 10km/s) with substantial differen- shear varies with spectral type and rotational period and is ex- tial rotation are found, only≈ few rapid rotators and hot stars are pected to increase strongly towards earlier spectral type while known to display differential rotation. there is a weaker variation with rotational period. Horizontal The main goal of the present work is to better understand the shear vanishes for very short and very long periods and becomes frequencyand strength of differential rotation throughoutthe HR highest at a rotational period of 10days for spectral type F8 and diagram, in particular the rapid rotators and the hotter stars close 25daysin the case of a modelof theSun. A strong perioddepen- to the convective boundary. New measurements are combined dence is also seen in models of stars with earlier spectral type. with previous assessments of line profile analysis and presented K¨uker & R¨udiger (2007) reproduce observed values of horizon- in a catalogue containing the measurements of rotational veloc- 1 tal shear of mid-F type stars of more than 1radd− . ities v sin i, characteristics of the broadening profile, differential Although stars are unresolved, differential rotation can be rotation parameters α, and pertinent photometric data (Sect. 5). assessed by indirect techniques through photometry and spec- Furthermore, stellar rotational velocities and differential rotation troscopy. Photometric techniques use the imprints of stellar spots is studied in the HR diagram (Sect. 6) and compared to theoreti- in the stellar light curve. If the star rotates differentially, spots cal predictions (Sect. 7). at different latitudes will cause different photometric periods which are detected in the light curve and allow one to derive 2. Observations the amount of differential rotation. Stellar spots further leave marks in rotationally-broadened absorption lines of stellar spec- The techniques applied in the present work are based on tra. These features change their wavelength position when the high-resolution spectroscopy with high signal-to-noise ratio. spot is movingtowards or receding from the observer due to stel- Therefore, this work is restricted to the brightest stars. In order lar rotation. Then, Doppler imaging recovers the distribution of to improve the number statistics of differential rotators through- surface inhomogeneities by observing the star at different rota- out the HR diagram, the sample of the present work com- tional phases. Using results from Doppler imaging, Barnes et al. pletes the work of previous years (Reiners & Schmitt 2003b,a; (2005) studied differential rotation of rapidly rotating late-type Reiners & Royer 2004; Reiners 2006) and consists of 182 stars stars and confirmed that horizontal shear strongly increases with of the southern sky known to rotate faster than 10km/s, to be effective temperature. brighter than V = 6, and with colours 0.3 < ≈(B V) < 0.9. Both photometry and Doppler imaging need to follow stellar Emphasis was given to suitable targets rather than− achieving a spots with time and thus are time-consuming. Another method, complete or unbiased sample. line profile analysis, takes advantage of the rotational line profile The spectra were obtained with the FEROS and CES spec- which has a different shape in the presence of differential rota- trographs at ESO, La Silla (Chile). Five stars were observed with tion. While rigid rotation produces line profiles with an almost both FEROS and CES. The data was reduced in the same way as elliptical shape, solar-like differential rotation results in cuspy by Reiners & Schmitt (2003b,a). line shapes. Spectra of 158 stars studied in the present work were taken In contrast to the methods mentionedabove, line profile anal- with FEROS at the ESO/MPG-2.2m telescope between October ysis does not need to follow the rotational modulation of spots. 2004 and March 2005 (ESO programme 074.D-0008). The re- Instead, only one spectrum is sufficient. Therefore, hundreds of solving power of FEROS (Kaufer et al. 1999) is fixed to 48,000 stars can be efficiently checked for the presence of differential covering the visual wavelength range (3600Å-9200Å). Spectra rotation. Line profile analysis is complementary to other meth- of 24 objects with v sin i . 45km/s were obtained with the CES ods in the sense that it works only for non-spotted stars with spectrograph at the ESO-3.6m telescope, May 6-7, 2005 (ESO symmetric line profiles. programme 075.D-0340).A spectral resolving power of 220,000 The line profile analysis used in the present work was in- was used. troduced by Reiners & Schmitt (2002b). The noise of the stel- lar line profile virtually vanishes by least squares deconvolution of a whole spectral range averaging the profiles of hundreds of 3. Methods absorption lines. The Fourier transform of the rotational profile 3.1. Least-squares deconvolution displays characteristic zeros whose positions are characteristic of the rotational law. Thus, the amount of differential rotation The reduced spectra are deconvolved by least squares decon- can be easily measured in Fourier space. In principle, the in- volution of a wide spectral region following the procedure of fluence of all broadeners like turbulent broadening, instrumental Reiners & Schmitt (2002b). profile and rotational profile simply is a multiplication in Fourier The wavelength region 5440-5880Å has already been used space. Therefore, turbulent broadening and instrumental profile successfully by Reiners & Schmitt (2003a) for FEROS spectra leave the zeros of the rotational profile unaffected. of FGK starsand is thusused in thepresentwork,too. Inthe case Among hundreds of stars with spectral type A to early K, of A stars, the number of useful lines in this range is insufficient 31 differential rotators have so far been identified by line profile and one has to move to the blue spectral range. Reiners & Royer analysis (Reiners 2006). Actually, many more differential rota- (2004) analyzed A stars in the range 4210 4500Å using spectra tors might be present which escaped detection because of low taken with ECHELEC at ESO, La Silla. This− wavelength range

2 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle inconveniently includes the Hγ line right in the centre. In the 3.3. Derivation of the projected rotational velocity v sin i present work, we take advantage of the wide wavelength cover- age of FEROS and find that the range 4400 4800Åbetween Hγ The projected rotational velocity is taken as the average of the and Hβ contains a sufficient number of useful− spectral lines. The values derived from each one of the two zero positions in the CES spectra span a clearly narrower wavelength range of about Fourier transform. If the second zero is no longer resolved re- liably (according to the v sin i limits given above), v sin i is ob- 40Å. Reiners & Schmitt (2003b) succeeded with CES spectra in tained from the first zero only. the ranges 5770-5810Å and 6225 6270Å. Within the present The formal error bar does not reflect systematic uncertainties work the full range coveredby our CES− spectra of 6140 6175Å − which correspond to a relative error of 5% (Reiners & Schmitt is used. 2003a). In order to obtain a conservative≈ error estimate, the error The deconvolution process derives the average line profile of 5% is adopted. The formal error is taken if it is larger than from a multitude of observed line profiles. The spectral regions 5 %. used with the FEROS and CES spectra display a sufficient num- ber of useful absorption lines and are unaffected by strong fea- 3.4. Derivation of the parameter of differential rotation α tures and telluric bands. Although the deconvolution algorithm successfully disentangles lines blended by rotation, the deblend- The parameter α of differential rotation is calculated from the ing degenerates if lines are too close to each other. However, the ratio of the zeros of the Fourier transform q2 using the modeled q1 deconvolution was found to be robust in the wavelength ranges q relation of 2 and α given in Reiners & Schmitt (2002b, fig. 2). used. The regions can be used homogeneously for all the stars q1 analyzed (Reiners & Schmitt 2003a). Unknown inclination represents the largest uncertainty when in- terpreting measurements of rotational broadening from line pro- A template is generated as input to the processing, contain- file analysis. In order to reflect this uncertainty, the parameter ing information about the approximate strength and wavelength α of differential rotation was calculated in the present work for positions of spectral lines. The line list is drawn from the Vienna inclination angles 10◦ and 90◦. Period is assessed accordingly, Astrophysical Line Data-Base (VALD; Kupka et al. 2000, 1999; assuming inclinations of 10◦ and 90◦. Additional uncertainties Ryabchikova et al. 1997; Piskunov et al. 1995) based on a tem- originate from the unknown limb darkening. Values of the differ- perature estimate derived from spectral type. VALD is queried ential rotation parameter are also derived from the calibrations with the extract stellar feature and a detection limit of 0.02. given in Reiners & Schmitt (2003b) for comparison and consis- tency with previous work. There, the function α is obtained The optimization of the line profile (by regularised least √sin i square fitting of each pixel of the line profile) alternates with the which minimizes the scatter due to unknown inclination. optimization of the equivalent widths (by Levenberg-Marquardt A detection limit of α = 5% originates from the unknown minimization – see Press et al. 1992 – of the difference of ob- limb darkening and the consequent inability to pin down dif- served and template equivalent widths). Only fast rotators with ferential rotation rates lower than 5 %. In the following, all v sin i & 40km/s are accessible with the FEROS spectra. Then, stars with α & 5% are considered≈ Sun-like differential rotators the rotational profile dominates all other broadening agents. In in agreement with Reiners (2006). Adopting a Sun-like rotation the case of the CES spectra, much narrower profiles are traced law (Eq. 1), the maximum possible value of Sun-like differential at v sin i as low as 10km/s. Therefore, these spectra are de- rotation is α = 100% implying that the difference of the rotation convolved using Physical≈ Least Squares Deconvolution (PLSD, rates of the poles and the equator equals the angular velocity Reiners & Schmitt 2003b), accounting for the different thermal at the equator. Negative values of α would imply anti-solar dif- broadening of the spectral lines of the involved atomic species. ferential rotation with angular velocities being higher at higher latitudes. However, negative values of α are not interpreted in terms of true differential rotation. Rather it may be understood 3.2. Analysis of the overall broadening profile in Fourier as rigid rotation in the presence of a cool polar spot mimick- space ing differential rotation (Reiners & Schmitt 2002a). In principle, the cuspy line profile of solar-like differential rotation might also The shape of the deconvolved rotational profile is characteris- be mimicked by spots, namely in the configuration of a belt of tic of the type of rotation. Although differential rotation may be cool spots around the equator. However, this is a hypothetic case distinguished in wavelength space, the analysis works best in which has never been observed. the frequency domain (Reiners & Schmitt 2002b). The Fourier There are further effects causing spurious signatures of dif- transform of the rotational profile displays characteristic zeros. ferential rotation. In the case of very rapid rotation (v sin i > 1 The ratio of the first two zeros depends on the amount of differ- 200kms− ), the spherical shape of the stellar surface is distorted ential rotation. The convolution with other broadening agents, by centrifugal forces and the resulting temperature variations e.g. the instrumental profile, corresponds to a multiplication in modify the surface flux distribution. This so-called gravitational Fourier space and thus reproduces the zeros. The projected rota- darkening occurs in particular in the case of rapidly-rotating A tional speed has to be sufficiently high to resolve the frequency and F stars (Reiners 2003; Reiners & Royer 2004). In the case of the second zero. The Nyquist frequency of FEROS spec- of close binaries, the spherical shape of the star can be modified tra with a resolving power of 48,000 is 0.08s/km and for CES by tidal elongation and thus change the surface flux distribution. it is 0.37 s/km. In order to properly trace the broadening pro- It is actually the absolute horizontal shear ∆Ω which is re- file to the maximum of the second side lobe, v sin i has to be lated to the stellar interior physics while it is only the relative 1 1 ∆Ω ∆ΩP at least 45 kms− (FEROS) and 12 kms− (CES), respectively value α = Ω = 2π that can be measured by line profile analy- (Reiners & Schmitt 2003b,a). Actually, zero positions can be sis. Rotational periods are derived from photometric radius and measured for even lower v sin i but then they are beyond the measured v sin i, assuming inclinations of 10◦ and 90◦ to be con- actual sampling limit and affected by noise features. Then, an sistent with the values of α used. Thus, values of horizontal shear interpretation is not possible. are used which were obtained for both inclination angles to ac-

3 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle

count for uncertainty about inclination. Because of the relation 3.0 to α, very low values of horizontal shear may thus be detected in the case of very slow rotators which however is limited by the minimum v sin i required for line profile analysis. In the case of very rapid rotators, however, differential rotators might be unde- tectable even at strong horizontal shear. 2.5

4. Peculiar line shapes and multiplicity 1 q /

2 2.0

The deconvolved profiles of many stars show peculiarities like q asymmetries, spots, or multiple components. We widely treat these profiles according to Reiners & Schmitt (2003b). Instead of skipping all stars with peculiar profiles, we tried to get as much information as possible from the line profile, tagging the 1.5 results with a flag which indicates the type of the peculiarity. Many profiles are asymmetric in that the shape of the blue wing of the overall profile is different from the red wing. In these cases, the analysis is repeated for each of the wings sep- 1.0 arately. The average of the results is adopted with conservative 1 10 100 1000 error bars. vsini [km/s] In the case of distortions caused by multiples, three cases Fig. 1. Measurements of v sin i and q2 of the present work. Open are distinguished. In the first case, the components are separated q1 well and can be analyzed each. In the second case, the profile symbols indicate measurement from FEROS spectra and filled is blended but dominated by one component. If the secondary symbols data obtained from CES spectra. Squares show mea- component can be fully identified it is removed or a line wing surements of fully resolved rotational profiles. Circles denote unaffected by the secondary is taken. Third, if the components the cases where only v sin i was measured reliably. The trian- cannot be disentangled, then only an upper limit on v sin i is de- gles show measurements of upper limits to rotational velocities. rived from the overall broadening profile. The vertical lines indicate the sampling limits of CES (solid) and In some cases, the deconvolved profile is symmetric, even FEROS (dashed). though the star is a known or suspected multiple. This may affect both the assessment of global stellar properties and the interpre- tation of line profiles: spectroscopic multiple, results from apparent single-component line profiles might be spurious and are thus flagged by ’y’. The – In the case of two very similar but spatially unresolved com- analysis also gets the ’y’ flag if there are known visual compo- ponents, the colours and effective temperature will be the nents closer than 3′′. 0 and at a brightness difference of less than same as for a single object. However, luminosity will be 5mag in the V band. These values are based on experience with multiplied by the number of components. This results in a slit and fibre spectrographs under typical seeing conditions with- considerable shift in the HR diagram and thus to a wrong out adaptive optics. assessment of luminosity class (and logg). – In the case of very different, unresolved components, the de- rived parameters will be dominated by the primary, i.e. the 5. Discussion of measurements brightest component, but a considerable shift in the HR dia- gram is still possible. Results for all stars of the sample are presented in Table 1. For – In favourable cases, spatially unresolved components show clarity, only one peculiarity flag is given with the results. Our up as separate spectral components and can be separated choice is that flags indicating multiplicity override flags denot- for the rotational analysis – as was discussed above. In the ing asymmetries or bumps. In particular, spectral components of worst-case scenario, there is no relative shift between the multiples might also be affected by asymmetries. Therefore, the spectral components forming an apparently single symmet- reader is advised to check the other flag, too, which indicates the ric profile. This might be the case if components of a spec- type of the measurement. troscopic multiple are almost on the line of sight at the time Figure 1 displays the measurements of q2 and v sin i obtained q1 of observation or if the componentsform a wide but spatially 1 in the present work. v sin i measurements range from 4 kms− to unresolved multiple involving slow orbital motion. 1 q2 300 kms− while assessments of can be as low as 1 and ≈ q1 ≈ Therefore, in order to detect analyses which might be af- reach values of up to 2.7. As is discussed in Sect. 3.2, the mea- surements of q2 are indicative of the mode of rotation only at fected by multiplicity, the catalogue of Eggleton & Tokovinin q1 1 1 (2008) was searched. Suspicious analyses are identified by one v sin i & 12 kms− (CES) and v sin i & 45 kms− (FEROS), in of two flags. There is one ’photometric’ flag ’x’ and one ’spec- total 56 objects. If one zero of the Fourier transform is beyond troscopic’ flag ’y’. The photometric data is considered spurious the sampling limit, v sin i can still be measured. In total, v sin i and flagged by ’x’ if the star has spatially unresolved compo- was measured for 114 objects. If both zeros of the Fourier trans- nents. In some cases, the spectral types of the components are form are beyond the sampling limits of CES and FEROS, resp., similar but no individual brightness measurements are given in the v sin i derived must be interpreted as an upper limit. Indeed, the literature. In such a case, we derive an estimate of the pri- these values represent the lowest measured, are not much be- m 1 mary magnitude by applying an offset of +0. 75 to the given to- low 4 kms− , and cluster at the lowest measured velocities. The tal magnitude of the system. If the star is a known or suspected few upper limits at q2 2.5 are all due to peculiar line profiles q1 ≈

4 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle caused by binarity or spots. Upper limits on v sin i were derived We also mention HD80671 which does not display resolved in 68 cases. spectroscopic components but has been studied before. It is a blended multiple suspected by Reiners & Schmitt (2003a). In fact, it is a spectroscopic binary with a close visual com- 5.1. Measurements of multiple systems panion at a separation of 0′′. 066 and a period of 3 years (Hartkopf et al. 2000; Eggleton & Tokovinin 2008). Based on Among the sample, there are 21 stars with indications of multi- CES spectroscopy, the present work detects two blended spec- plicity detected in the overall broadening profile. Although mul- troscopic components. Only an upper limit on v sin i of the pri- tiplicity complicates the line profile analysis severely, a number mary is derived. of interesting objects could be measured. It is equally important to also discuss known multiples with One major achievement of the present work is the analysis an apparent single line profile. Signatures of differential rotation of resolved profiles of spectroscopic binaries. Among the 21 have to be regarded with care if the star does not show resolved stars in the sample with line profiles indicative of multiplicity, spectroscopic components although it is a known multiple. The rigid rotation could be assessed for HD4150, HD17168, and catalogue of Eggleton & Tokovinin(2008) was searched system- HD 19319. atically for multiples among the stars of the sample. Such ev- In addition, there are nine spectroscopic binaries with idence indicates the need of further studies but is incapable of both spectroscopic components analysed: HD 16754, HD 26612, firmly rejecting candidates of differential rotation by itself. The HD41547, HD54118, HD64185, HD105151, HD105211, findings in the work of Eggleton & Tokovinin (2008) are dis- HD147787, and HD155555. The line wings could be analysed cussed in the following in what concerns stars with signatures of separately and the results are shown in Table 1. differential rotation: One of these objects, HD64185, displays signatures of dif- ferential rotation. It is a binary system, possibly even triple. Two – HD493 is a visual binarywith similar componentsand a sep- spectroscopic components are detected in the present work and aration of 1′′. 433. It cannot be excluded that the light of both analysed separately. The stronger broad component (considered componentsentered the slit of the spectrograph which would HD64185A in the present work) indicates differential rotation produce a composite spectrum. The spectrum does not indi- with the strongest absolute horizontal shear encountered in the cate any spectroscopic multiplicity but the line profile could present work’s sample while the weaker and narrow compo- still be composed of two components. Additional studies of nent (HD64185B) is consistent with rigid rotation. However, this object are encouraged since it is located right in the lack ff as HD64185 is possibly a triple, it cannot be excluded that the of di erential rotators in the HR diagram at early-F spectral line profile of HD64185A is a blend mimicking the signatures types (Sect. 6.1). ff of differential rotation. Individual stellar parameters of the com- – Signatures of di erential rotation were detected for ponents are not available so that the position in the HR diagram HD105452 by Reiners & Schmitt (2003b). This cannot be might be erroneous. Therefore, the measured relative differential reproduced in the present work, the spectrum displays an rotation and horizontal shear are excluded from the studies in the asymmetric profile instead. HD105452 is a known spectro- present work. Nevertheless, additional studies of this star are en- scopic binary. Possibly, the blended line profile mimicked ff couraged since HD64185A shows the strongest horizontalshear di erential rotation in older spectra at an orbital phase when among the differentially rotating F-type stars. there is no displacement of spectral lines. Another explana- tion could be that the emitted flux is modified by tidal elon- Among the nine spectroscopic binaries with both compo- gation depending on orbital phase (see Sect. 3.4). nents measured, HD147787 and HD155555 are particularly – HD114642 is single according to Eggleton & Tokovinin interesting since they have been studied by line profile anal- (2008) but displays an asymmetric line profile in the present ysis before and are analysed once more in the present work. work. The line wings are analysed separately and both wings This means that line profiles are compared obtained at dif- indicate differential rotation. ferent orbital phase. HD147787 was found to be asymmetric – HD124425 is a spectroscopic binary with a short period of by Reiners & Schmitt (2003a). The present work resolved the 2.696 d. Although, it appears single in the spectrum available profile based on a CES spectrum and resolved two spectro- to this work, one cannot exclude that the broadening profile scopic components for which v sin i is measured. HD147787 is the sum of two blended profiles or affected by tidal elon- is a spectroscopic binary with a period of 40 days accord- gation. The star deserves particular attention since the line ing to Eggleton & Tokovinin (2008). HD155555 did not stand profile indicates the second largest amount of differential ro- out particularly in the work of Reiners & Schmitt (2003b) but tation (α = 44%) among the F stars of the sample. the present work detects two spectroscopic components. It is a short-period pre-main sequence spectroscopic binary accord- We also mention measurements of rigid rotators which might be ing to Strassmeier & Rice (2000, V824 Arae). They also derived affected by multiplicity. There is one group of stars, HD15318, v sin i of both components which agree with the values assessed HD42301, and HD57167, which appear single in the spectrum in the present work within the error bars. Dunstone et al. (2008) but are listed as binaries with unknown component parame- measured significant rates of differential rotation for both com- ters by Eggleton & Tokovinin (2008). This is also the case for ponents from Doppler imaging. As these translate to amounts HD215789 which shows signatures expected for anti-solar dif- of relative differential rotation of a few percent only, they are ferential rotation or a polar cap. below the detection limit of line profile analysis. In the present The multiples are excluded from the studies of relative dif- work, the quality of the deconvolved profiles is insufficient and ferential rotation and horizontal shear presented in Sect. 6 and 7 affected by asymmetries, so that the rotational profile could not since their position in the HR diagram and the line profile mea- be measured anyway, even though the rotational velocities are surements might be erroneous. This also concerns differential sufficiently large. The asymmetries probably are due to the spot- rotators detected in previous work. These stars are: HD17094, tedness which on the other side enabled the Doppler imaging HD17206, HD18256, HD90089, HD121370, and HD182640. analyses by Dunstone et al. (2008). Nevertheless, the data might still be correct but this needs to be

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20 obtained by Reiners et al. for stars in common is helpful as these results are based on data taken at another epoch with different

HD105452 HD198001 instrumentation. HD124425 In the present work, new CES and FEROS spectra have been taken of stars which have been analysed before by 0 Reiners & Schmitt (2003b, CES) and Reiners & Schmitt (2003a, HD105452 FOCES and FEROS). The more recent work of Reiners (2006) is mostly based on the same spectra. Reiners & Royer (2004) anal-

HD80671 ysed ECHELEC spectra with a resolution even lower than those −20 Reiners & Schmitt (2003b, CES) of FEROS and FOCES. Reiners & Schmitt (2003a, FEROS/FOCES) Reiners & Royer (2004, ECHELEC) Figure 2(a) displays differences between v sin i measure- Reiners (2006) (this work − prev. or CES FEROS) [km/s] ments of the present work and previous work and of measure- i this work (CES)

sin this work (FEROS)

v ments from different instruments. While most new measure- ∆ −40 1 1 10 100 1000 ments agree with previous assessments within a few kms− , 1 vsini [km/s] discrepancies of more than 5kms− appear in the cases of (a) HD80671, HD105452, HD124425, and HD198001 when comparing to Reiners & Schmitt (2003a) and Reiners & Royer

0.6 (2004), measurements which are based on FEROS, FOCES, and Reiners & Schmitt (2003b, CES) ECHELEC spectra. HD80671 and HD105452 are particular Reiners & Schmitt (2003a, FEROS/FOCES) Reiners & Royer (2004, ECHELEC) in that they are spectroscopic binaries. The shape of the rota- Reiners (2006) 0.4 this work (CES) tional profile varies because of the orbital motion of the com- this work (FEROS) ponents. Consequently, the v sin i assessed from different spectra HD105452 will be different. In particular, the spectra of HD105452 anal- HD89569 ysed in the present work display an asymmetric profile. While 0.2 HD128898 HD160032HD136359 HD64379 Reiners & Schmitt (2003a) flagged HD80671 as an object with a blended profile, the profile is resolved in the present work but HD197692HD4247 HD160915HD199260 HD200163 only an upper limit can be determined to the apparently very 0.0 HD210302 low intrinsic v sin i. HD124425 and HD198001 were measured

(this work − prev. or CES FEROS) HD4089 1 q /

2 at lower spectral resolution in previous work. HD210302 q HD22001 q ∆ Figure 2(b) shows discrepancies of 2 measurements to- HD23754 HD105452 q −0.2 1 0 10 20 30 40 50 gether with the corresponding v sin i measurements since these vsini [km/s] tell us whether a rotational profile can actually be resolved. (b) There are nine residuals larger than 0.10. Only three of them cannot be explained by peculiar line profiles or insufficient spec- Fig. 2. a) v sin i is compared for stars in common to previous and tral resolution. One of these cases is HD105452 as can be the present work or with both CES and FEROS measurements expected from the comparison of v sin i measurements above. available (see legend for details). The results mostly agree with Measurements for HD64379 differ by 0.16 which might be re- 1 lated to the fact that the CES spectrum shows a slight asym- differences below 5 kms− . The few outliers are labelled and dis- cussed in the text. Residuals are calculated following the scheme metry. The star is a candidate for spottedness according to this work - previous work and CES - FEROS. Arrows to the bot- Reiners & Schmitt (2003a). The profile of HD86569 suffers tom mark upper limits to measurements obtained in the present from bad quality in the present work so that the discrepancy of work or CES measurements of the present work. Arrows to the 0.24with respect to Reiners (2006) shouldnot be given too much left denote upper limits to values from previous work or FEROS weight and instead the older measurement be preferred. The re- measurements of the present work. The grey area encompasses maining residuals larger than 0.10 are probably due to the use of 1 different spectrographs. HD128898 and HD160032 were anal- all residuals with amounts of less than 5kms− . The vertical lines indicate the resolution limits of CES (solid) and FEROS ysed in the present work based on spectra with different resolu- (dashed). – b) Similar to (a), q2 is compared for stars in com- tion (CES and FEROS) and v sin i is below the sampling limit of q1 mon with previous studies or with both CES and FEROS mea- FEROS. HD22001, HD23754, and HD136359 were analyzed surements available. The grey area harbours all residuals with at higher resolving power in previous work. Generally, in case of profiles without pecularities, discrep- amounts of less than 0.10. Objects with larger deviations are dis- 1 cussed in the text. ancies between different measurements are less than 5kms− in v sin i and 0.10 in q2 . q1 proven by further studies. The measurements obtained for these 5.3. Distribution of measurements of differential rotation stars are tabulated in Table 1. These data flagged with ’m’, ’x’, or ’y’ offer a starting point for future studies. Figures 3(a) and 3(b) show the distribution of projected rota- tional velocities v sin i and of the indicator q2 of differential ro- q1 5.2. Comparison with previous work and discussion of tation for all stars of the sample. In the present work, the rota- systematic uncertainties tional profile of 56 stars is studied. In total, there are 300 objects with rotational profile studied by line profile analysis, also yield- Systematic errors may occur which can not be assessed a pri- ing precise projected rotational velocities v sin i. Many of the ori. This particularly concerns instrumental effects introduced stars studied in the present work display line profiles which can- by the spectrographs used. A comparison to the previous results not be studied by line profile analysis because of asymmetries

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ff 100 The amount of di erential rotation of six stars is below 6 % which is below our detection limit and consistent with rigid ro- tation. Yet, as is the case for many other objects with unde- 80 tected differential rotation, small rates of (relative) differential rotation – and even strong amounts of horizontal shear as is dis-

60 cussed later on – are still possible. Two F stars (HD104731 and 1 HD124425) with Teff 6500K and v sin i . 30kms− display the lowest values of q≈2 measured so far by line profile analy- number q1 40 sis corresponding to differential rotation rates of 54% and 44%, respectively. Five stars show signatures which can be explained

20 by anti-solar differential rotation or more plausibly by rigid rota- tion with cool polar caps. In total, there are 43 stars with spectra consistent with rigid rotation. 0 1 10 100 projected rotational velocity vsini [km s−1] (a) 6. Rotation and differential rotation in the HR diagram 300 The distribution of differential rotators in the HR diagram is of particular interest since it relates the frequency of differential rotation to their mass and evolutionary status. To construct the HR diagram, homogeneousphotometric and 200 stellar data are used for the sample of the present work. (B V) colours, V band magnitudes and parallaxes were drawn from− the Hipparcos catalogue when available and from Tycho-2 oth- number erwise. Bolometric corrections and effective temperatures Teff 100 were calculated from (B V) using the calibration of Flower − (1996). Absolute V band magnitudes MV were inferred from the apparent magnitudes and the parallaxes. Radii are calculated from effective temperature and bolometric magnitude which is 0 derived from absolute V band magnitude. 0.0 0.5 1.0 1.5 2.0 2.5 ff q2/q1 The study of di erential rotation throughout the HR dia- gram done in the present work includes previous analyses by (b) Reiners et al. (2003-2006). Not all photometric and basic stel- lar data used in the present work are equally available from Fig.3. (a) The distribution of projected rotational velocities there. Therefore, data needs to be complemented as consistently v sin i is shown for those stars for which measurements could as possible and without changing results previously obtained. be obtained from the analysis of the zeros of the Fourier trans- Therefore, photometry was adopted from Reiners et al. (2003- form of the overall broadening profile. Mesurements with upper 2006) when available and otherwise completed according to the limits on v sin i are omitted. The grey-shaded area represents the present work. Stellar parameters of previously analyzed stars are distribution of the sample of the previous work (Reiners et al.), adopted from Reiners et al. when available. Otherwise, they are the hashed area the sample of the present work. The white area derived in the same way as described above for the data of the shows the distribution of the combined sample. – (b) The distri- present sample. Table 2 presents rotational data and basic stellar bution of measurements of q2 is shown for the stars with resolved q1 data for all stars studied by line profile analysis. rotational profiles. The layout follows Fig. 3(a). Sometimes the stars have been studied more than once by line profile analysis. It is not always the most recent result that has been adopted in Table 2. If possible, the most plausible and caused by spots or multiplicity. However, information on v sin i conclusive result is used, for example in cases of peculiar pro- can still be obtained. v sin i is inferred for 68 stars and upper lim- files when the line profile could be resolved in separate compo- its for further 68 objects. These stars are no longer considered in nents later on. Generally, data are preferentially adopted which the present work but the v sin i measurements are tabulated in are based on spectra with higher resolution. Also, data with more Table 1. supplemental information given by flags are preferred. In the present work, ten stars show clear signatures of dif- Approximate surface gravities for all stars are inferred from ferential rotation between α = 10% and 54%. While three of photometry and from mass estimates which in turn are derived these are new discoveries, seven objects belong to the samples of from effective temperatures using the calibration for dwarf stars Reiners & Schmitt (2003b) and Reiners & Schmitt (2003a) and of Gray (2005, appendix B). Thus, the surface gravities will be are now identified or confirmed based on high-resolution CES upper limits only for giants and lower limits in the case of un- spectra. Including previous work, there are 33 differential rota- resolved binaries. The distribution of effective temperature and tors detected by line profile analysis (excluding those possibly surface gravity is shown in Fig. 4. affected by multiplicity). Not all parts of the HR diagram are equally accessible to line Three further stars with extremely fast rotation show line profile analysis. Figure 5 shows the distribution of the analyzed profiles indicative of differential rotation. However, in these stars in the HR diagramin termsof rotationalspeed. The location cases, the feature is plausibly caused by gravitational darkening of the stars is compared to evolutionary models of Siess et al. in the regime of rigid rotation (see Sect. 3.4). (2000) and the granulation boundary according to Gray & Nagel

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300 (1989). The granulation boundary indicates the location where deep convective envelopes form and the approximate onset of magnetic braking. Accordingly, the figure illustrates that Sun- like and cooler main-sequence stars are efficiently braked. Also the late-type giant stars display slow rotation. Consequently, 200 these and most cool dwarf stars are not accessible to the study of the rotational profile. Therefore, the present study is generally restricted to main-sequence and slightly evolved stars of spectral number types A and F. 100

6.1. Two populations of differential rotators The HR diagram in Fig. 6 shows that the present work adds 0 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 several rigid rotators in the hot star region between 8000 and effective temperature Teff [K] 10000K. Three stars are detected – the A stars HD30739, HD43940, and HD129422 at the transition to spectral type F 300 – with line shapes indicative of differential rotation. However, the low values of q2 might be caused to a certain extent by grav- q1 itational darkening in the regime of rigid body rotation since all 1 three objects rotate very rapidly with v sin i > 200 kms− . This 200 scenario is suggested by Reiners & Royer (2004) for the rapid rotator HD44892 which is the hottest and most luminous dif- ferential rotator in Fig. 6. In the cases of HD6869, HD60555,

number and HD109238 they argue, however, that this cannot be the sole explanation for the low values of q2 since rotational speed 100 q1 would exceed breakup velocity. These stars are all located at the granulation boundary and display the strongest absolute hori- zontal shear (Reiners & Royer 2004). There are two more dif-

0 ferential rotators close to the granulation boundary at its cool 1 2 3 4 5 side, Cl*IC4665V102 and HD72943 (Reiners 2006). V102 −2 logg [cms ] displays the strongest horizontal shear ∆Ω of the whole sam- ple. Reiners (2006) argue that a mechanism may be responsible Fig. 4. The distributions of effective temperature and surface for the strong shear at the granulation boundary which is differ- gravity are shown. The layout follows Fig. 3. ent from the mechanism at work in stars with deeper convective envelopes at cooler effective temperatures.

−5 At early-F spectral types, there is evidence for a lack of dif- vsini < 10 kms−1 ferential rotators (shaded area in Fig. 6). At later F and early-G 10 < vsini < 40 kms−1 40 < vsini < 80 kms−1 spectral types, however, a dense population of differential rota- 80 < vsini < 160 kms−1 vsini > 160kms−1 tors exists on the main sequence and a scattered population to- wards higher luminosity. At these later spectral types, there are 0 HD104731 and HD124425, the stars with the highest values of relative differential rotation α measured by line profile analysis.

V The object with strongest horizontal shear in this region, how- M ever, is HD64185A and it is the only object there that displaysa shear strength comparable to the stars at the granulation bound- 5 ary. This result has to be regarded with care since HD64185A is a component of a spectroscopic binary, or possibly a triple (Sect. 5.1). Reiners et al. (2003−2006) this work (CES) this work (FEROS) 10 6.2. Dependence of relative differential rotation on stellar 1.2•104 1.0•104 8.0•103 6.0•103 4.0•103 effective temperature T [K] eff parameters Fig. 5. HR diagram with the stars from previous (circles) and the Fig. 7(a) displays the basic quantity measured, i.e. the ratio ofthe q2 present (squares and triangles) work. Symbol size scales with zeros of the Fourier transform , vs. effective temperature Teff, q1 projected rotational velocity measured by the FT method. The including measurements of Reiners et al. (2003-2006). Values granulation boundary according to Gray & Nagel (1989) is in- of q2 between 1.72 and 1.85 are consistent with rigid body ro- q1 dicated by the hashed region. Evolutionary tracks for 1.0, 1.5, tation when accounting for unknown limb darkening. The value and 2.0 M and the early-main sequence of Siess et al. (2000, ⊙ of 1.76 represents rigid body rotation when assuming a linear solid, Z = 0.02 without overshooting) are added, using bolomet- limb darkening law with a Sun-like limb darkening parameter ric corrections by Kenyon & Hartmann (1995, table A5). Stars ε = 0.6. In the regime of rigid rotation and marginal solar/anti- with photometry possibly affected by multiplicity (flags ’x’ and solar differential rotation, the present work does not change the ’m’ in Tables 1 and 2) are not included. overall distribution found by Reiners et al. but the lack of early- F type differential rotators between 6700 and 7000K becomes

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−2 vsini < 10 kms−1 10 < vsini < 40 kms−1 40 < vsini < 80 kms−1 80 < vsini < 160 kms−1 vsini > 160kms−1

0 V M

2

4 10000 9000 8000 7000 6000 effective temperature Teff [K]

Fig. 6. The HR diagram with the data of the present work and the results from Reiners et al. (2003-2006). Differential rotators are indicated by filled circles while rigid rotators are shown by open circles. Evolutionary models and the granulation boundary are indicated as in Fig. 5. The shade highlights the region where no differential rotators are detected. Triangles denote HD30739, HD43940, HD129422 which are considered rigid rotators with gravitational darkening (Sect. 6). The fast differential rotators at the granulation boundary are HD6869, HD44892, HD60555, HD109238, and Cl*IC4665V102 which display strongest horizontal shear (Reiners & Royer 2004; Reiners 2006). Stars possibly affected by multiplicity (flags ’m’, ’x’, and ’y’ in Tables 1 and 2) are not included. more pronounced. Nevertheless, the strongest differential rota- ilar to Fig. 7(a). The dependence of the frequency of differential tors are located close to this gap. rotation on Teff and v sin i cannot be studied independently since The figure corroborates evidence of two different popula- v sin i and Teff are strongly correlated and thus degenerate in tions of differential rotators, one with moderate to rapid rotation what concernstheir effectondifferential rotation (Reiners 2006). at the cool side of this gap and one with extremely rapid rotation Therefore, it is not a surprise, that Fig. 8(a) looks very similar to at the granulation boundaryat the hot side. At the cool side, stars Fig. 7(a). Differential rotation is common among slow rotators are generally rotating more slowly than stars on the hot side of and becomes rare at rapid rotation (see Fig. 8(b)). From Fig. 7(a) the gap. and 8(a) one notices that almost all differential rotators at the 1 On average, the fraction of differential rotators among stars granulation boundary are fast rotators with v sin i & 100 kms− . with known rotational profiles significantly decreases with in- creasing effective temperature (Fig. 7(b)). Those fractions are The strength of differential rotation, in contrast to the fre- estimated from the number counts in bins of effective tempera- quency of differential rotation, does not vary for different val- ture. It is assumed that the number counts originate from a bi- ues of effective temperature or projected rotational velocity. The lowest values of q2 measured tend to be lower at cool effective nomial probability distribution with an underlying parameter p q1 which is the probability that a randomly chosen star among the temperature and slow rotation but these trends are not signifi- stars with measured rotational broadening is a differential rota- cant. tor. From this distribution and the actual number counts of differ- ential rotators in each temperature bin, 2σ confidence intervals In summary, the fraction of differential rotators decreases on the true value of p are derived (following Hengst 1967). significantly with increasing Teff and v sin i. In other words, the Fig. 8(a) displays the ratio of the zeros of the Fourier trans- fraction of hot and rapidly rotating stars among differential rota- form q2 vs. the projected rotational velocity v sin i in a way sim- tors is significantly less than among rigid rotators (see Table 3). q1

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Table 3. Fraction of hot, giant, and rapidly rotating stars among differential and rigid rotators. These fractions are compared to the fractions within the combined sample.

differential rotators rigid rotators combined

+18 +4 +5 fraction of hot stars (Teff > 7000 K) 45 17 % 88 6 % 82 6 % +−17 +−6 +−5 fraction of giant stars (log g < 3.5) 12 8 % 11 4 % 12 4 % 1 +−19 +−4 +−5 fraction of rapid rotators (v sin i > 50 km s− ) 42 17 % 88 6 % 81 6 % − − −

In terms of surfacegravity,the fraction of differential rotators The accessible range in horizontal shear is not fully covered increases towards high gravity but with the data at hand, differ- with measurements. Certainly, at long periods, the correspond- ences are not significant (Figs. 9(a), 9(b)). There are roughly as ing v sin i will be too slow for the measurement by line profile many giants among the differential rotators as among the rigid analysis as is indicated by the vertical lines in Fig. 10 and 11 rotators and the whole sample. We finally recall the reader that for a dwarf and a giant star. At short periods of the order of the sample is comprehensive but incomplete and biased towards 1d, some structure becomes apparent which is related to the fre- the selection of suitable targets. quency distribution in the HR diagram discussed above. In de- tail, two different groups of differential rotators stand out. The first group comprises the rapid differential rotators at the granu- 7. Discussion of horizontal shear lation boundary. The second group consists of the cooler differ- 7.1. Measured shear and rotational period ential rotators. Both groups cover values of relative differential rotation of similar magnitude (Figure 10) but the second group In order to study the strength of differential rotation, the mea- is located at longer periods. Figure 11, however, shows that the sured quantity q2 is converted to relative differential rotation α q1 first group of stars displays stronger shear than the second group. as described in Sect. 3.4 and via rotationalperiod to absolute hor- Reiners (2006) already noticed that the objects at the granulation 1 izontal shear ∆Ω (Eq. 2). Rotation periods are given in Table 2 boundary have high v sin i around 100 kms− and that all of these and are estimated from the v sin i derived and the photometric cluster at short periods at high horizontal shear. Reiners (2006) radius. suggest that an alternative process might be at work that is re- When interpreting the measured relative differential rotation sponsible for the strong shear observed in the first group. α, unknown inclination certainly is one of the most important An upper envelope can be identified to the horizontal shear uncertainties. Unknown inclination effectively turns the mea- of the second group while the four established differential rota- surements of α into upper limits because q2 is an approximate q1 tors at the granulation boundary remain clearly above. Reiners function of α . However, assuming a uniform distribution of √sin i (2006) identifies an upper envelope to the cooler F-type stars inclination angles, and according to the statistics of a sin i distri- which rises between periods of 0.5 and 3 days and then declines bution, much less stars will have low values of sin i while most towards larger rotational periods at constant α. Reiners (2006) will have sin i . 1. Although the uncertainties introduced by un- point out that the rising part of the upper bound to the horizon- known limb darkening and the measurement error of q2 might be tal shear of the second group between periods of 0.5 and 3 days q1 1 substantial, only the error bar due to inclination is indicated here could also be described by a plateau at ∆Ω 0.7radd− . Fig. 11 strengthens this interpretation and shows an≈ updated version of (represented by the values α90◦ and α10◦ introduced in Sect. 3.4). Figs. 10 and 11 present the relation of differential rotation the upper bound of Reiners (2006, fig. 4) suggesting a bit higher plateau close to values of ∆Ω 1radd 1. The envelope is trans- with rotational period. The measurements from line profile anal- ≈ − ysis cover rather short periods of 1 10d. This is not surprising formed to the α scale (Fig. 10). The shear of all the F stars in the 1 since the study only accesses rapid≈ rotators.− Relative differential second group is below 1radd− at periods of up to 3days while rotation ranges from the detection limit at 5% to values of more for longer periods, the shear is below the curve of maximum α = than 60% when assuming an incliation angle of 90 ◦. This corre- 1. 1 spondsto absolute horizontalshear of the order of 0.1 1radd− . It should be noted here that values of inclination very differ- − At each rotational period, a different range of horizontal ent from 90◦ may allow for shear measurements greater than 1. shear is accessible as given by the parallel dotted and dash- However, the measured shear scales with √sin i and the proba- dotted lines in Fig. 11. The detection limit of α corresponds to a bility to find an object with sin i very different from 1 is small. minimum detectable shear which decreases with increasing pe- 1 There are three particular objects, HD17094, HD44892, riod, e.g. a shear of 0.01radd− can be detected at a period of about 30d and above.≈ The line indicating α = 100% corre- and HD182640 that would be located at the upper envelope in sponds to the extreme case that the difference of angular veloc- Fig. 11. HD17094 and HD182640, however, are possibly af- ity between the poles and the equator is as large as the equato- fected by multiplicity while gravitational darkening mightbeim- rial rate of rotation. If this is considered as maximum attainable portant in the case of HD44892. Therefore, these objects are not 1 considered. relative differential rotation, no shear higher than 0.2radd− will be seen at periods larger than about 30d, for≈ example. In contrast to the F stars, the shear of the stars at the granu- 1 Substantial amounts of shear of the order of 1radd− and more lation boundaryat the transition from spectral type A to F shows can only be seen at periods shorter than 10d. It is worth to a clear period dependence – though based on a few data points note that line profile analysis allows one to≈ detect weakest hori- only – and seems to line up roughly parallel to the curve α = 1. zontal shear at long rotation periods. Then, the measurement is It cannot be excluded, however, that the line profiles of the hot only limited by v sin i. stars with short rotational periods of less than a day are rather

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2.0 2.0

1.8 1.8

1.6 1.6 1 1 q q / / 2 2 q q

1.4 1.4

vsini < 10 kms−1 vsini < 10 kms−1 1.2 1.2 10 < vsini < 40 kms−1 10 < vsini < 40 kms−1 40 < vsini < 80 kms−1 40 < vsini < 80 kms−1 80 < vsini < 160 kms−1 80 < vsini < 160 kms−1 vsini > 160kms−1 vsini > 160kms−1 1.0 1.0 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 10 100 1000 effective temperature Teff [K] projected rotational velocity vsini [km/s] (a) (a)

1.0 1.0 20 9 27 101 77 11 stars in each bin 0.8 30 42 38 38 22 18 17stars in each bin 0.8 12 3 6 7 5 0 diff. rotators 14 9 1 3 2 0 0 diff. rotators 0.6 0.6

0.4 0.4

0.2 0.2 fraction of differential rotators fraction of differential rotators 0.0 0.0 6000 7000 8000 9000 10 50 100 500 3.0 3.5 −1 effective temperature Teff [K] projected rotational velocity vsini [km s ] (b) (b)

q2 Fig. 7. a) The ratio indicates the amount of differential ro- Fig. 8. a) In a way similar to Fig. 7(a), q2/q1 is plotted vs. the q1 tation and is plotted here vs. effective temperature Teff. Dotted projected rotational velocity v sin i. Symbols are the same as in lines indicate the range of q2 for rigid body rotation and un- Fig. 7(a). Although v sin i are given by the bottom axis the v sin i q1 known limb darkening. The solid line denotes q2 for the case scaling of symbol size is kept to facilitate the identification of q1 data points when comparing to Fig. 7(a). – b) 2σ confidence = of rigid rotation and Sun-like limb darkening (ε 0.6) Symbol limit on the fraction of differential rotation with respect to pro- size scales with projected rotational velocity v sin i as in Fig. 5. jected rotational velocity. Open circles denote rigid rotators and filled circles differential rotators. The triangles show the cases where the low fraction of q2 is probably caused by gravitational darkening. The data q1 2007). They describe a strong dependence on rotational period. point with lowest q2 represents HD104731 which displays the q1 K¨uker & R¨udiger (2005) notice maxima in the period depen- strongest relative differential rotation discovered in the present denceof shear in the F8 and G2 models. Themaximumof the F8 work. The lack of differential rotators at early-F types can be star is higher and at shorter rotational periods. They further point spotted easily. – b) 2σ confidence limit on the fraction of differ- out that the increase of maximum horizontal shear with increas- ential rotation with respect to effective temperature. Only bins ing effective temperature is much stronger than the variation due with a number of at least 15 objects are displayed. to rotational period. The comparison to the predictions of K¨uker & R¨udiger (2005) is complicated since these cover longer periods of 4d due to gravitational darkening, in parts at least, than due to dif- and more. There is only a small overlap with the present work ferential rotation. between 4 and 10 days. In this range, the measurements of late- F type stars scatter widely and are larger than the prediction for 7.2. Comparison to theoretical predictions the F8 star. Only HD114642 agrees nicely with the prediction for an F8 star although the rotational period of this particular Figures 10 and 11 compare the measurements to models of object is not much different from the other late-F differential ro- K¨uker & R¨udiger (2005) who modeled differential rotation for tators. None of the stars studied displays rotational shear below an F8 and a G2 star and later for hotter F stars (K¨uker & R¨udiger the shear predicted for an F8 star which is in agreement with

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2.0 1.0

Teff < 6350 [K]

6350 [K] < Teff < 6700 [K]

6700 [K] < Teff < 7050 [K] T > 7050 [K] 1.8 0.8 eff α

1.6 0.6 1 q / 2 q

1.4 0.4 relative differential rotation

vsini < 10 kms−1 1.2 0.2 10 < vsini < 40 kms−1 40 < vsini < 80 kms−1 80 < vsini < 160 kms−1 vsini > 160kms−1 1.0 0.0 2 3 4 5 0.1 1.0 10.0 100.0 logg [cms−2] rotational period P [d] (a) ff Fig. 10. The figure displays relative di erential rotation α90◦ vs. estimated period for all differential rotators detected by pro- 1.0 file analysis. The dotted and dashed-dotted lines indicate the dependence of α on period for constant absolute shear ∆Ω = 0.8 1 1 5 20 29 51 81 54 2 stars in each bin 1.0radd− and 0.1radd− , resp. Open symbols indicate dwarfs 0 4 2 3 10 13 1 differential rotators and filled symbols evolved stars with logg 3.5. Symbol size 0.6 scales with effective temperature as indicated≤ in the legend. The straight solid lines relate the data points to the values derived 0.4 for 10◦ thus illustrating the error due to inclination. Errors in-

trinsic to α90◦ and α10◦ are omitted for clarity but might be 0.2 substantial. The scaling of symbol size with effective temper-

fraction of differential rotators ature allows one to compare with the theoretical predictions 0.0 3 4 5 6 for an F8 star (vertically hashed) and a G2 star (horizontally logg [cms−2] hashed) (K¨uker & R¨udiger 2005, width of relations given by cases cv = 0.15 and4/15). The vertical lines indicate period lim- (b) its accessible to measurement. The vertical solid line gives the shortest rotational period of the stars with a measurement of the Fig. 9. a) In a way similar to Fig. 7(a), q2/q1 is plotted vs. the rotational profile while the other vertical lines denote the periods surface gravity log g. Symbols are the same as in Fig. 7(a). – b) corresponding to the v sin i limit of CES for a dwarf star (short- 2σ confidence limit on the fraction of differential rotation with dashed) of 1 solar radius and an evolved star (long-dashed) of8 respect to surface gravity. solar radii. The FEROS limit will be at even shorter periods. The location of the Sun is also shown ( ). The grey solid line gives the upper envelope to mid and late-F⊙ type differential rotators the predictions as none of the differential rotators identified has discussed in the text. spectral type G or later. A cool, slowly-rotating G-type star like the Sun (which is also highlighted in Figs. 10 and 11) is beyond the detection limit. at least, partly has a shape similar to that of the curves predicted Although the observed α is larger than predicted at spec- by theory. It can be noticed from Fig. 10, that the rising parts tral type late-F, the total range of relative differential rotation of these curves roughly follow the lines of constant horizontal α observed roughly agrees with the total range predicted by shear. K¨uker & R¨udiger (2005). A close inspection of Figs. 10 and 11 The situation looks different for hotter F stars. Among later- shows that it is not necessarily due to underestimated shear that type stars, K¨uker & R¨udiger (2007) also modeled F stars up to the observed shear seems higher. Instead, the apparent under- 1.4 M corresponding to spectral type F5 on the main sequence. estimation might also be due to disagreements in period. The A total⊙ range in horizontal shear is predicted (not shown in observed α occur at much shorter rotational periods than the the figures) which roughly agrees with the horizontal shear ob- predicted data which translates to higher values of shear. The served. Again, a strong variation will be involved due to the de- consequence is that the measured shear is higher than predicted pendence on rotational period. The rotational periods assumed shear. In principle, there is a large uncertainty in observed peri- in the calculations are not available to us so that we cannot study ods because of unknown inclination. However, inclinations very the behaviour of relative differential rotation α. The theoretical different from 90◦ will result in even shorter periods as is indi- calculations agree with observations in that horizontal shear of 1 cated by the straight solid lines in Figs. 10 and 11. 1.0radd− is not exceeded. So far, there are no predictions from While a clear trend with rotational period is not discernible models of more massive, rapidly rotating which would be very among the observations of the F stars, the upper envelope to α, interesting however.

12 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle

10.00 10.0000

Teff < 6350 [K]

6350 [K] < Teff < 6700 [K]

6700 [K] < Teff < 7050 [K] T > 7050 [K] 1.0000 eff

] 1.00 ] −1 −1

0.1000 [rad d [rad d ∆Ω ∆Ω

0.0100

horizontal shear 0.10 horizontal shear

0.0010 Teff < 6350 [K]

6350 [K] < Teff < 6700 [K]

6700 [K] < Teff < 7050 [K]

Teff > 7050 [K] 0.01 0.0001 0.1 1.0 10.0 100.0 3000 4000 5000 6000 7000 8000 rotational period P [d] effective temperature Teff [K]

∆Ω ∆Ω ff Fig. 11. The figure displays the absolute shear 90◦ vs. es- Fig. 12. The figure displays the absolute shear vs. e ec- timated projected rotation period. The dash-dotted and dotted tive temperature for all measurements of differential rotation in- lines indicate absolute shear for constant α = 0.05 (detection cluding previous measurements by other methods (diamonds, limit) and 1.0, resp. The other curves and the symbols have have Barnes et al. 2005), compared to the location of the Sun ( ). the same meaning as in Fig 10. The dashed line represents the fit by Barnes et al. (2005). Circles⊙ identify results from line profile analysis as given in the cap- tion to Fig. 10. Values derived from the α calibration instead √sin i

There are basic principles, both physical and concep- of α90◦ or α10◦ are presented in order to be consistent with pre- tual, which strongly constrain observable shear. According to viously published versions of this figure. Error bars due to un- K¨uker & R¨udiger (2007), as early F-type stars have thin convec- known inclination are omitted for clarity. 1 tive envelopes, strong horizontal shear of the order of 1rad d− can only be sustained at rotational periods of the order of 1 day. This means that rotation necessarily has to be fast in order to more easily detected at slow rotation while they might be un- sustain strong shear. On the other hand shear that strong cannot detected at fast rotation (K¨uker & R¨udiger 2005; Reiners 2006; appear at slow rotation. Fig. 11 shows that only at short periods, K¨uker & R¨udiger 2007). As temperature and rotation are corre- relative differential rotation will not be above 100%. In other lated, this might explain the general trend of less detections of ff ff words, the relative differential rotation of a slower rotator with di erential rotation at higher e ective temperature and faster ro- similarly strong shear would be larger than 100 %. tation.

Table 4. Number of stars with measured rotational profile, dis- 7.3. Horizontal shear and effective temperature tinguished by effective temperature and by rotational period. Figure 12 updates fig. 5 of Reiners (2006) and displays the rela- tion of horizontal shear and effective temperature. The measure- early-F mid-F ments of the present work are added. 7050 6700 K 6700 6350 K The trend detected by Barnes et al. (2005) is generally fol- − − P< 1 d total 26 9 lowed by the results from line profile analysis which fill the plot diff.rot. 0 2 at spectral types F and earlier although the hotter differential ro- tators clearly stand out and display stronger shear. At spectral P> 1 d total 48 55 type F8 corresponding to an effective temperature of 6200 K, diff. rot 1 15 ≈ 1 horizontal shear covers a wide range of 0.1 1.0d− while the stars at the granulation boundary are≈ above− that and reach 1 ∆Ω 3radd− . ≈ Together with the upper bound to horizontal shear of F stars, this might also explain the lack of differential rotators among 7.4. Implications for the frequency of differential rotators early-F type stars. Adopting the upper bound to horizontal shear 1 An interesting question is how far the attainable values of hor- of F stars of ∆Ω 1radd− in Fig. 11 and a detection limit of α = 0.05, we can≈ only detect differential rotation in stars ro- izontal shear and the rotational periods are related to the fre- 1 ∆Ω 1 rad d− quency of differential rotation seen across the HR diagram. The tating slower than Ω = α 0.05 , i.e. periods longer than 2π ≈ detectable values of α play a crucial role. Assuming the same P = Ω 0.31 days. This means that no differentially rotating shear ∆Ω at different rotational periods, α will be lower at F star is≈ expected at periods much less than a day. This limit shorter periods and no longer detectable below a certain period cannot be accessed by the measurements since the shortest de- that depends on shear. Therefore, stars with less shear will be tected period among the stars with rotational profile measured

13 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle

is 0.4 d. It is instructive to look more closely at stars with 100.00 ≈ < 6700 Teff 7050K corresponding to early-F spectral types T < 6350 [K] ≤ eff and compare them to stars with 6350 < Teff 6700K in the 6350 [K] < Teff < 6700 [K] ≤ 6700 [K] < T < 7050 [K] mid-F range. Furthermore, we distinguish in both groups stars eff T > 7050 [K] with rotational periods shorter than a day and stars with longer eff period (Table 4). First of all, the numbers in the table agree with 10.00

the finding that hotter stars tend to rotate faster. In other words, ] the fraction of early-F type stars with rotational periods less than −1

1d is larger than the fraction of mid-F type stars at the same pe- [rad d ∆Ω riods. 1.00 Indeed there is no single rapid differential rotator with a rotational period of less than a day among the early-F stars.

However, there are two differential rotators among the mid-F horizontal shear stars, approximately one fourth. Similarly, there are 15 differ- entially rotating mid-F stars at periods larger than 1 day, again 0.10 one fourth, but there is only one early-F star. While this agrees with the previous finding that the frequency of differential ro- tators decreases towards hotter stars, it is somewhat unexpected that the fractions are the same for both period bins. This indi- 0.01 cates that the rotational periods and the implied detection limit 5500 6000 6500 7000 7500 8000 effective temperature Teff [K] of α do not fully explain the gap of differential rotation among early-F stars. Fig. 13. The figure enlarges the hot portion of Fig. 12 and dis- In a certain respect, the lack of differential rotation among plays the absolute shear ∆Ω vs. effective temperature for mea- early-F stars contrasts the results from modeling. According to surements by line profile analysis. Symbols identify measure- models by K¨uker & R¨udiger (2007), horizontal shear will be ments of differential rotation from line profile analysis as given higher at hotter effective temperature that would facilitate the in the caption to Fig. 10, but here assuming an inclination of 90◦. detection of differential rotation. Therefore, a substantial frac- In addition, the ranges of shear accessible to observation are in- tion of differential rotators must be expected for early-F type dicated by short horizontal lines. The cases of α = 0.05 and stars even though they rotate faster than mid-F stars and even α = 1 are distinguished by black and grey, respectively. These though, consequently, the detection limit on shear is higher. This ranges are determined by rotational period alone, in a way that is not the case however. A possible explanation might be found is given in Fig. 11. Therefore, the detectable ranges can not only in the details of the period dependence and in particular the ex- be given for those objects which display differential rotation. act rotational periods at the predicted maximum of horizontal Instead they can be assessed for all objects with the rotational shear. For example, there will be less detections of differential profile measured, including the stars that are considered rigid rotation among early-F stars if the corresponding maximum is rotators. Among these limits indicated by short horizontal lines, 1 below 1radd− and located at periods shorter than 1 day and if stars with periods longer than a day are distinguished from stars at the same time the maximum for mid-F stars is at longer pe- with shorter periods by dotted lines. The two solid (black and riods. More modeling of early-F stars at very short periods is grey) flexed lines represent envelopes to the bulk of the limits. clearly needed. These are based on the maximum v sin i observed which is most Figure 13 enlarges the hot portion of Fig. 12 and shows the probably due to sin i.1. Error bars are omitted for clarity. The temperature effect more clearly. The range of shear accessible dotted line denotes the upper bound on shear of F stars identi- to line profile analysis is shown for all stars with rotational pro- fied in Fig 11. The dashed line reproducesthe fit of Barnes et al. files measured. The lower limits to measurable shear are derived (2005). from the rotational period determined for each star assuming α = 0.05. The upper limits are inferred the same way assum- ing α = 1.0. Therefore, these two limits enclose the whole range bound results from the detection limit on α(= 0.05) while the of measurable shear for each star. In accordanceto Fig. 11, high- upper bound is given by α = 1. The correlation of faster rotation est values of shear are accessed at very short rotational periods with higher Teff explains at least in parts the trend of increasing P < 1d. shear with hotter effective temperature. Error bars due to inclination are omitted. Lower values of The case of high inclination deserves closer inspection since inclination will essentially increase the shear measurements but Fig. 13 shows values of shear assuming an inclination of 90◦ to 1 also raise the limits and the upper bound ∆Ω = 1radd− identi- facilitate comparison with Fig. 11. It is a reasonable assumption fied in Fig. 11. that the stars with the highest v sin i measured will be those with The distribution of the limits with respect to effective temper- sin i.1. Consistently, these stars will have the shortest rotational ature actually reflects the correlation with v sin i as periods were periods so that the upper bound to v sin i corresponds to a lower calculated from v sin i and photometric radii. The detection lim- bound to periods. Shear is derived from periods and from α mea- its generally decline towards cooler effective temperatures since sured by line profile analysis such that a lower bound to periods rotational periods increase and thus smaller amounts of shear are corresponds to an upper envelope to the limits on shear. The up- detectable. At the same time, extremely large values like for the per bounds to the bulk of the limits are indicated by the blue and transition objects at the granulation boundary cannot be detected red flexed lines in Fig. 13. any more. These upper bounds consist of a rising part at temperatures A majorconclusionfromFig.13is thatthe fit of Barnes et al. of mid-F spectral types and by a flat part at hotter temperatures. (2005) in Fig. 12 cannot be tested with our data. The detectable The bound to the lower limit of detectable shear (black solid 1 range of shear is constrained by selection effects. The lower line) and the upper bound to the shear of F stars of 1radd−

14 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle identified in Sect. 7.1 (dotted line) define a detectable range of netic braking on spectral type. This stops us from identifying the shear which becomes very narrow towards hotter temperatures quantity responsible - temperature or rotation. 1 ( 0.5 1radd− at Teff 6600 K). Therefore, differentially ro- On the other hand, the fraction of differential rotators does ≈ − ≈ 1 tating F stars with horizontal shear below 0.5radd− will not be not vary notably with surface gravity. In the present work, sur- detected if their projected rotational period is shorter than a day. face gravity was estimated from photometry to be used as in- 1 However, F stars with shear higher than 0.5radd− should still dicator of the luminosity class. The study of the rotational be- ≈ be detectable even at rotational periods that short. Furthermore, havior of evolved stars will certainly benefit from a more accu- the number of lower and upper limits in Fig. 13 at effective tem- rate assessment of surface gravity and allow for a comparison peratures above 6600K is large so that there are many suitable with theoretical predictions (e.g. Kitchatinov & R¨udiger 1999). objects at early-F spectral types. However, all these seem to be The analysis of giants will be particularly promising since at the rigid rotators or, at least, display undetectable rates of differential same v sin i longer rotation periods can be studied. Of course, rotation. Therefore, the detection limits to differential rotation do the challenge is to find giants with sufficiently large v sin i to be not fully explain the lack of differential rotators at early-F spec- accessible to line profile analysis. tral types. The lack of differential rotators around Teff = 6750K (spec- In summary, three issues have been identified which play tral type F2-F5) detected in previous work was consolidated. At an important role in explaining the lack of differential rotators the cool side of this gap, there is a large population of differ- among early-F type stars: an apparent upper bound to horizontal ential rotators. The differential rotator with the highest differ- shear of F-type stars, the detection limit on shear which depends ential rotation parameter measured so far by line profile analy- on rotational period, and the correlation of rotational periods and sis (HD104731, α = 54%) is right at the cool edge of this √sin i effective temperature. However, these cannot fully explain the gap. This object and also other strong differential rotators in this lack of differential rotators among the early-F type stars. A di- region exhibit moderate rotational velocity. At the hot side of rect dependence on temperature is apparent. We speculate that the gap at the location of the granulation boundary, all differ- the physical conditions and/or mechanisms of differential rota- 1 ential rotators are rapid rotators (v sin i & 100 kms− ) and show tion change at early-F spectral types. This change might also be strongest horizontal shear (∆Ω & 1radd 1). related to the detection of extremely large horizontal shear at the − Theoretical calculations are at hand for F-stars and were transition to A spectral types. compared to the observations. While the observed range of hori- zontal shear in mid-F type stars agrees with the predicted range, the observed values in late-F type stars were found to be much 8. Summary and outlook larger than predicted. This was explained rather by a discrepancy in period dependence than in absolute shear strength. Further In the present work, stellar rotation of some 180 stars has been observations with other techniques are needed to measure differ- studied through the analysis of the Fourier transform of the over- ential rotation at periods of 20d and more. Strong shear seems all spectral line broadening profile, extending previous work of possible for very rapid rotators only. Models of more massive Reiners et al. 2003-2006. In total, the rotation of 300 stars has and rapidly rotating stars are clearly needed. been studied by line profile analysis. Among these, 33 differen- Even more, it was found that the observations of F stars do tial rotators have been identified. not follow a clear period dependence. Still, the upper envelope In order to study frequency and strength of differential rota- which traces the upper bound to horizontal shear and the curve tion throughout the HR diagram, particular emphasis was given of relative differential rotation α = 1 exhibit a shape similar to to the compilation of astrometric and photometric data which the predicted period dependence of an F8 and a G2 star. has been drawn homogeneously from the Hipparcos and Tycho- Regarding the comparison with theoretical predictions, it is 2 catalogues. Complications introduced by multiplicity were dis- worth discussing the role of inclination. In the present work, the ff cussed in detail and a ected measurements flagged. Multiplicity uncertainties of inclination were properly accounted for by de- ff might explain exceptionalamounts of di erential rotation, e.g. in riving relative differential rotation and horizontal shear assum- the case of HD64185A. The flagged objects were not included ing different, extreme values of inclination. A remarkable fact in the discussion. It will be worthwhile to further study these ob- is that a better assessment of inclination will not allow for a jects since they are are located in interesting regions of the HR better agreement with the predicted period dependence. Instead, diagram. limb darkening and the assumed surface rotation law might be The results were found to be in good agreement with previ- more critical sources of uncertainties in this respect. This does ous observations for stars in common. Substantial deviations can not mean that there is no need to better assess inclination, as it be explained by instrumental limitations or peculiar line profiles 1 causes large part of the uncertainty about measurements of dif- due to binarity or spots. Otherwise discrepancies exceed 5 kms− ferential rotation. in v sin i and 0.10 in q2 only when v sin i is below the limits for q1 The horizontal shear of F stars remains below a plateau of 1 1 1 line profile analysis (e.g. 12kms− for CES and 45kms− for ∆Ω 1radd− . This might explain the low frequency of dif- FEROS). A compilation of all stars with new or previous rota- ferential≈ rotation among hotter and rapidly rotating F stars con- tion measurements by line profile analysis was presented. sidering the detection limit on relative differential rotation α but The present work detected several rigid rotators among A cannot fully account for the gap of differential rotation at early-F stars. Some rapidly rotating A stars show line profiles probably spectral types. We assume, that conditionschange in early-Ftype affected by gravitationaldarkening.As of now, no differential ro- stars so that differential rotation is inhibited while at somewhat tators are known with effective temperatures higher than 7400K. hotter temperatures at the transition to spectral type A, differen- The frequency of detections of differential rotators gener- tial rotation reappears with extreme rates of differential rotation. ally decreases with increasing effective temperature and increas- The shear of rapidly rotating A stars was found to be higher 1 ing projected rotational velocity. Effective temperature and rota- than 1radd− and possibly increases with decreasing rotational tional velocity are correlated because of the dependence of mag- period. So far, there are no predictions from models which could

15 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle explain the shear of A-type stars. Although K¨uker & R¨udiger Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593 (2007) set out to understandthe strongshear detected at the gran- Snodgrass, H. B. 1984, Sol. Phys., 94, 13 ulation boundary by Reiners & Royer (2004), the stellar models Strassmeier, K. G. & Rice, J. B. 2000, A&A, 360, 1019 Su´arez, J. C., Garrido, R., & Goupil, M. J. 2007, Communications in discussed are not beyond 1.4 M , i.e. mid-F type stars. Asteroseismology, 150, 211 The strength of differential⊙ rotation generally follows the increasing trend with effective temperature identified by Barnes et al. (2005) but the trend with effective temperature can actually not be tested with our data since the spread is large and the detection limits on shear also follow the trend with tempera- ture as v sin i and Teff are correlated. As the surface rotation law is identified as a possible source of uncertainty, it is important to emphasize that the present work is based on the assumption of a very simple surface rotation law (Eq. 1). This assumption is reasonable since it is motivated by the solar case and has been consistently used by similar stud- ies. However, K¨uker & R¨udiger (2008) point out that in the case of the Sun there are other measurements favouring a rotation law dominated by a cos4 term of co-latitude instead of the cos2 term (Snodgrass 1984). The adoption of such a rotation law will introduce a second parameter of differential rotation and might change the shear measurements from line profile analysis.

Acknowledgements. We would like to thank our referee, Dr. Pascal Petit, for a very clear and constructive report. M.A. thanks Dr. Theo Pribulla for dis- cussion which helped to improve the analysis of the A stars among the sam- ple. M.A. and A.R. acknowledge research funding granted by the Deutsche Forschungsgemeinschaft (DFG) under the project RE 1664/4-1. M.A. further ac- knowledges support by DLR under the projects 50OO1007 and 50OW0204. This research has made use of the extract stellar request type of the Vienna Atomic Line Database (VALD), the SIMBAD database, operated at CDS, Strasbourg, France, and NASA’s Astrophysics Data System Bibliographic Services.

References Barnes, J. R., Cameron, A. C., Donati, J., et al. 2005, MNRAS, 357, L1 Dunstone, N. J., Hussain, G. A. J., Collier Cameron, A., et al. 2008, MNRAS, 387, 1525 Eggleton, P. P. & Tokovinin, A. A. 2008, MNRAS, 389, 869 Flower, P. J. 1996, ApJ, 469, 355 Gizon, L. & Solanki, S. K. 2004, Sol. Phys., 220, 169 Gray, D. 2005, The observation and analysis of stellar photospheres, 3rd edn. (Cambridge: Cambridge University Press) Gray, D. F. & Nagel, T. 1989, ApJ, 341, 421 Hartkopf, W. I., Mason, B. D., McAlister, H. A., et al. 2000, AJ, 119, 3084 Hengst, M. 1967, Einf¨uhrung in die mathematische Statistik und ihre Anwendung (Mannheim: Bibliographisches Institut) Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8 Kawaler, S. D., Sekii, T., & Gough, D. 1999, ApJ, 516, 349 Kenyon, S. J. & Hartmann, L. 1995, ApJS, 101, 117 Kitchatinov, L. L. & R¨udiger, G. 1999, A&A, 344, 911 K¨uker, M. & R¨udiger, G. 2005, A&A, 433, 1023 K¨uker, M. & R¨udiger, G. 2007, Astronomische Nachrichten, 328, 1050 K¨uker, M. & R¨udiger, G. 2008, Journal of Physics Conference Series, 118, 012029 Kupka, F., Piskunov, N., Ryabchikova, T. A., Stempels, H. C., & Weiss, W. W. 1999, A&AS, 138, 119 Kupka, F. G., Ryabchikova, T. A., Piskunov, N. E., Stempels, H. C., & Weiss, W. W. 2000, Baltic Astronomy, 9, 590 Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Jeffery, C. S. 1995, A&AS, 112, 525 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical recipes in C. The art of scientific computing, ed. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. Reiners, A. 2003, A&A, 408, 707 Reiners, A. 2006, A&A, 446, 267 Reiners, A. & Royer, F. 2004, A&A, 415, 325 Reiners, A. & Schmitt, J. H. M. M. 2002a, A&A, 388, 1120 Reiners, A. & Schmitt, J. H. M. M. 2002b, A&A, 384, 155 Reiners, A. & Schmitt, J. H. M. M. 2003a, A&A, 412, 813 Reiners, A. & Schmitt, J. H. M. M. 2003b, A&A, 398, 647 Ryabchikova, T. A., Piskunov, N. E., Kupka, F., & Weiss, W. W. 1997, Baltic Astronomy, 6, 244 Schou, J., Antia, H. M., Basu, S., et al. 1998, ApJ, 505, 390

16 Please give a shorter version with: \authorrunning and/or \titilerunning prior to \maketitle, Online Material p 1

Table 1. Results obtained in the present work from the analysis of the broadening profile and pertinent stellar data. The first 25 data rows provide measurements from CES spectra while the remainder lists the results from FEROS spectra. The last column provides flags which characterize the broadening profile and the type of measurement done for a particular object: d differential rotator, r consistent with ∆Ω = 0, i.e. rigid rotation, p anti-solar differential rotation (probably cool polar cap), v only measurement of v sin i, < only upper limit on v sin i, g symmetric profile, a asymmetric profile (e.g. spots and multiplicity), f noisy, distorted profile, failed deconvolution, m2 spectroscopic binary, m3 spectroscopic triple, m? multiple with unknown number of components. For clarity, only one flag is given to indicate peculiarity. See the text for more details. Two additional flags refer to indications of multiplicity found in the literature: x photometry affected by unresolved components, and y spectroscopic analysis affected by unresolved components.

q2 α P name (B V) Teff MV v sin i ∆Ω √sin i flags q1 √sin i sin i − 1 1 [K] [kms− ] d[radd− ] HD32743 0.42 6632 3.28 22.7 1.1 1.62 0.02 0.19 0.09 3.3 0.36 0.17 dg HD33262 0.53 6158 4.38 15.4 ± 0.8 1.68 ± 0.01 0.10 ± 0.07 3.5 0.18 ± 0.13 dg HD64379 0.47 6409 3.88 43.1 ± 2.2 1.72 ± 0.01 0.03 ± 0.06 1.4 0.13 ± 0.27 ra HD68456 0.44 6541 3.19 8.8 ± 1.1 2.25 ± 0.01± 9.2± vfx HD80671 0.41 6678 3.13 7.4 ± 1.0 1.34 ± 0.00 10.7

q2 α P name (B V) Teff MV v sin i ∆Ω √sin i flags q1 √sin i sin i − 1 1 [K] [kms− ] d[radd− ] HD19319 0.35 6964 1.99 79.2 4.0 1.74 0.00 0.00 0.07 1.5 rm2 HD20121 0.44 6541 2.71 64.4 ± 3.2 1.72 ± 0.00 0.00 ± 0.11 1.6 rf HD20606 0.34 7014 1.97 126.6 ± 6.3 1.84 ± 0.00 0.00 ±-0.18 1.0 rg HD20610 0.90 5047 0.85 4.5 ± 0.2 1.15 ± 0.00± 101.1

q2 α P name (B V) Teff MV v sin i ∆Ω √sin i flags q1 √sin i sin i − 1 1 [K] [kms− ] d[radd− ] HD59380 0.49 6324 3.64 7.7 0.4 1.57 0.00 9.2 vg HD59984 0.54 6117 3.52 4.1 ± 0.3 1.11 ± 0.00 19.8

q2 α P name (B V) Teff MV v sin i ∆Ω √sin i flags q1 √sin i sin i − 1 1 [K] [kms− ] d[radd− ] HD135153 0.37 6867 3.9 0.2 1.09 0.01

Table 2. Compilation of all stars measured to date by line profile analysis and pertinent stellar data. Line profile measurements (v sin i, q2 , α , q1 √sin i Ω √sin i, and flags) are adopted from the source indicated in the last column: 1 this work (CES measurements), 2 this work (FEROS measurements), 3 Reiners & Schmitt (2003b), 4 Reiners & Schmitt (2003a), 5 Reiners & Royer (2004), 6 Reiners (2006, table B1), 7 Reiners (2006, table B2). P The basic stellar data (Teff, MV, and sin i ) are only partly taken from the given sources as is described in more detail in the main body of this paper. Flags are given as in Table 1 for the present work which widely follows the use of flags in previous work. In Reiners & Schmitt (2003a), there are additional flags b for suspected blended multiples, s for suspected spotted stars, and c for stars with symmetric profile but which are candidates for spottedness.

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD432 6763 0.96 71.0 3.6 1.78 0.03 0.00 0.04 3.0 6 HD493 6678 2.05 116.8 ± 5.8 1.68 ± 0.00 0.10 ± 0.06 1.1 0.56 0.34 dgy 2 HD693 6366 3.51 3.8 ± 0.8± ± 19.4± 3 HD739 6496 3.55 2.4 ± 1.2 28.9 < 3 HD1581 6002 4.56 2.3 ± 0.4 22.7 < 3 HD2151 5855 3.45 3.3 ± 0.3 28.1 3 HD2726 6916 2.48 13.5 ± 1.5 7.3 x3 HD3302 6453 2.72 17.8 ± 0.8 5.8 3 HD4089 6161 2.75 23.5 ± 1.2 1.82 0.02 0.00 -0.08 4.8 x 6 HD4125 7914 1.68 15.1 ± 4.7± ± 7.8 5 HD4150 9261 0.03 133.1 ± 6.7 1.79 0.00 0.00 -0.06 1.4 rm2x 2 HD4247 6825 3.08 42.7 ± 2.1 1.77 ± 0.04 0.00± 0.09 1.8 6 HD4338 7123 1.70 88.4 ± 2.7 1.80 ± 0.15 0.00 ± 0.15 1.6 5 HD4757 6706 1.34 93.8 ± 4.7 1.73 ± 0.03 0.00 ± 0.14 1.9 6 HD4813 6240 4.22 2.6 ± 0.8± ± 21.4 < 3 HD6178 8612 1.28 78.7 ± 3.6 1.73 0.16 0.00 0.26 1.6 5 HD6245 5047 1.10 5.1 ± 1.8 1.05 ± 0.00± 79.8

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD16825 6587 2.52 9.0 0.5 1.29 0.03 12.0 vg 2 HD16920 6678 2.62 4.0 ± 0.6± 25.1 xy3 HD16978 10612 0.76 99.5 ±18.6 1.1 5 HD17051 6040 4.22 4.2± 0.6 14.3 3 HD17094 7141 2.17 45.1 ± 2.3 1.49 0.02 0.34 0.10 2.4 0.90 0.26 y 6 HD17168 9026 1.06 50.5 ± 2.5 1.81 ± 0.01 0.00 ±-0.07 2.3± rm2 2 HD17206 6371 3.68 25.6 ± 1.3 1.70 ± 0.02 0.07± 0.07 2.7 0.16 0.16 x 6 HD17793 5047 -0.01 4.3 ± 0.2 1.30 ± 0.00± 157.5 ±

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD30020 5751 0.51 7.6 0.4 1.06 0.00 49.4

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD49095 6324 3.99 5.8 0.3 1.30 0.00 10.4

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD66358 7909 0.63 75.6 15.7 2.5 5 HD66540 7249 0.60 105.6± 3.8 0.87 0.06 2.1 5 HD67228 5751 3.46 3.7 ± 0.3± 26.0 3 HD67483 6366 1.59 52.3 ± 4.1 1.50 0.07 0.33 0.12 3.4 0.61 0.22 c 4 HD67523 6453 1.41 8.0 ± 0.4 1.20 ± 0.01± 23.6 ±

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD83962 6507 1.53 140.3 7.0 1.72 0.08 0.00 0.26 1.2 6 HD84607 7000 1.78 93.1 ± 4.7 1.79 ± 0.01 0.00 ±-0.03 1.4 6 HD84999 7267 1.11 124.2 ± 6.2± ± 1.4 s4 HD85396 5047 1.59 4.3 ± 0.2 1.16 0.00 75.4

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD106022 6651 2.28 77.2 3.9 1.86 0.06 0.00 -0.08 1.5 6 HD106086 7553 1.33 84.2 ± 2.7 1.69 ± 0.12 0.08± 0.22 1.8 0.28 0.78 5 HD107192 7014 2.93 68.1 ± 3.4± ± 1.2± a4 HD107326 7107 2.08 132.2 ± 6.6 1.80 0.05 0.00 0.04 0.9 6 HD108722 6490 1.71 97.0 ± 4.8 1.78 ± 0.04 0.00 ± 0.07 1.7 6 HD108968 5819 22.0 ± 1.8 1.49 ± 0.00 ± vax 2 HD109074 8011 0.76 90.8 ± 3.8± 1.9 5 HD109085 6813 3.13 60.0 ± 3.0 1.75 0.05 0.00 0.14 1.3 6 HD109141 6881 2.84 135.7 ± 6.8 1.79 ± 0.03 0.00 ± 0.02 0.6 6 HD109238 7184 1.31 103.5 ± 3.9 1.51 ± 0.11 0.32 ± 0.16 1.6 1.26 0.63 d 5 HD109272 5136 2.15 4.0 ± 0.2 1.23 ± 0.02± 59.6 ±

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD121107 5159 15.7 2.2 1.32 0.00 vg 2 HD121370 6024 2.36 13.5 ± 1.3 1.46 ± 0.03 0.37 0.10 10.6 0.22 0.06 x 6 HD121416 4587 1.12 2.5 ± 0.3± ± 219.5 ± < 3 HD121932 6964 1.53 155.4 ± 7.8 1.75 0.03 0.00 0.11 1.0 4 HD122066 6395 2.18 40.6 ± 2.0 1.81 ± 0.01 0.00 ±-0.08 3.3 6 HD122797 6725 2.25 33.6 ± 5.0± ± 3.5 a4 HD123255 7014 1.76 157.8 ± 7.9 1.81 0.05 0.00 0.02 0.8 c 4 HD123445 10396 59.9 ±22.0 ± ± 5 HD123492 8105 2.08 105.8± 3.2 1.76 0.12 0.00 0.16 0.9 5 HD123999 6117 2.86 15.0 ± 1.0± ± 7.3 xy3 HD124425 6324 2.20 31.0 ± 3.9 1.37 0.01 0.44 0.11 4.4 0.62 0.16 dgx 1 HD124689 6725 2.39 146.9 ± 8.4± ± 0.7± 5 HD124780 7204 2.21 70.7 ± 3.5 1.80 0.04 0.00 0.02 1.5 6 HD124850 6075 2.85 15.0 ± 0.8 1.91 ± 0.04 0.00 ±-0.26 7.4 6 HD124915 7293 2.42 66.9 ± 2.0± ± 1.4 5 HD125276 6240 4.72 2.8 ± 0.4 15.9 3 HD125283 8625 1.75 248.4 ±12.4 1.85 0.01 0.00 -0.18 0.4 rf 2 HD125442 7164 1.49 148.0 ± 10.0± ± 1.0 b4 HD125451 6820 3.33 40.5± 2.0 1.7 4 HD125473 10082 -0.44 122.8 ±18.1 1.8 x5 HD126722 8538 1.88 94.4± 4.7 1.71 0.01 0.04 0.06 0.9 0.27 0.41 rg 2 HD126868 5589 1.88 14.8 ± 0.7 1.64 ± 0.01± 14.4± vg 2 HD127486 6366 1.15 22.4 ± 1.1 1.91 ± 0.03 0.00 -0.27 9.8 pa 1 HD127739 6787 2.20 55.8 ± 2.8 1.81 ± 0.01 0.00 ± -0.08 2.1 6 HD127785 7334 2.57 15.0 ± 4.8± ± 5.9 5 HD127821 6601 3.76 55.6 ± 2.8 1.75 0.04 0.00 0.13 1.1 6 HD127964 8187 0.07 125.9 ± 4.9 0.59 ± 0.05± 1.9 5 HD128898 7426 2.18 13.2 ± 0.7 1.81 ± 0.04 0.00 -0.02 7.5 ra 1 HD129153 7592 2.30 105.7 ± 5.3 1.78 ± 0.06 0.00± 0.11 0.8 c 4 HD129422 7164 2.05 201.7 ±10.1 1.67 ± 0.00 0.12 ± 0.07 0.6 1.35 0.79 df 2 HD129502 6695 2.94 47.0± 2.4 1.80 ± 0.03 0.00 ±-0.01 1.8± 6 HD129926 7164 2.76 112.5 ± 5.6 1.74 ± 0.03 0.00± 0.13 0.7 c 4 HD129956 9840 -0.69 82.9 ± 4.1 1.75 ± 0.00 0.00 ± 0.06 2.9 rg 2 HD130701 5684 7.4 ± 0.4 1.15 ± 0.00 ±

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD142703 9274 0.28 105.1 7.7 1.7 5 HD142860 6409 3.62 10.0 ± 1.3 6.9 3 HD142908 6964 2.34 75.7 ± 3.8 1.80 0.05 0.00 0.04 1.4 c 4 HD143333 6240 2.99 8.2 ± 0.2± ± 12.0 x3 HD143466 7426 2.31 141.3 ± 7.1 1.77 0.05 0.00 0.11 0.7 c 4 HD143790 6409 2.31 5.5 ± 3.0± ± 23.0 4 HD143928 6678 2.91 23.5 ± 1.2 1.82 0.01 0.00 -0.09 3.7 rg 1 HD144069 6453 4.86 16.9 ± 1.5± ± 2.3 3 HD144415 7267 2.14 7.8 ± 1.7 13.4 b4 HD145005 7014 3.16 57.2 ± 1.9 1.2 5 HD145191 7427 2.16 102.1 ± 2.9 1.76 0.12 0.00 0.16 1.0 5 HD145876 7297 2.18 113.7 ± 3.3 1.76 ± 0.12 0.00 ± 0.15 0.9 5 HD146836 6409 2.54 18.3 ± 1.2 2.01 ± 0.01 0.00 ±-0.59 6.2 pa 1 HD147084 5386 4.7 ± 0.2 1.12 ± 0.01 ±

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD165499 5927 4.34 2.5 0.1 23.9

q2 α P name Teff MV v sin i ∆Ω √sin i flags source q1 √sin i sin i 1 1 [K] [kms− ] [d] [radd− ] HD201636 6700 2.20 58.8 2.9 1.86 0.03 0.00 -0.15 2.0 6 HD203608 6324 4.39 3.7 ± 0.7 1.18 ± 0.04± 13.5