The Rotation-Activity Correlations in K and M dwarfs. II. New constraints on the dynamo mechanisms in late-K and M dwarfs before and at the transition to complete convection

1.2 MB·

by  E. R. Houdebine

The Rotation-Activity Correlations in K and M dwarfs. II. New constraints on the dynamo
mechanisms in late-K and M dwarfs before and at the transition to complete convection1
E.R. Houdebine
7 ArmaghObservatory,CollegeHill, BT619DG Armagh,Northern Ireland
1 Universit´e de Toulouse,UPS-Observatoire Midi-Pyr´en´ees,IRAP, Toulouse,France
0 CNRS,Institut deRechercheenAstrophysique etPlan´etologie,14av. E.Belin, F–31400Toulouse,France
2 [e-mail]

n
a D.J. Mullan
J DepartmentofPhysics and Astronomy, University ofDelaware,Newark,DE 19716,USA

6 [e-mail]
2

] B. Bercu
R Universit´e deBordeaux,Institut de Math´ematiques,UMR5251,351coursdelalib´eration, 33405Talencecedex,France.
S [e-mail]
.
h
p F. Paletou
- Universit´e de Toulouse,UPS-Observatoire Midi-Pyr´en´ees,IRAP, Toulouse,France
o CNRS,InstitutdeRechercheenAstrophysiqueetPlan´etologie,14av.E.Belin,F–31400Toulouse,France
r
t
s
a M. Gebran
[ DepartmentofPhysics &Astronomy, Notre DameUniversity-Louaize, POBox72, ZoukMika¨el,Lebanon

1
v
8
0 ABSTRACT
6
7 We study the rotation-activity correlations (RACs) in a sample stars from spectral type dK4 to
0 dM4. We study RACs using chromospheric data and coronal data. We study the Caii line surface
.
1 fluxes-P/siniRACs. WefittheRACswithlinearhomoscedastic andheteroscedasticregression models.
0 We find that these RACs differ substantially from one spectral sub-typeto another. For dM3 and dM4
7 stars, we find that the RACs cannot be described by a simple model, but instead that there may exist
1 two distinct RAC behaviors for the low activity and the high activity stellar sub-samples respectively.
:
v Although these results are preliminary and will need confirmation, the data suggest that these distinct
i RACsmay be associated with different dynamoregimes.
X
We also study R’HK as a function of the Rossby number R0. We find that: (i) For dK4 stars,
r we confirm R’HK as a function of R0 agrees well with previous results for F-G-K stars. (ii) In dK6,
a
dM2, dM3 and dM4 stars, we find that, at a given R0, the values of R’HK lie a factor of 3, 10, 20
and90respectively below theF-G-KRAC.Ourresults suggest asignificant decrease in theefficiency of
the dynamo mechanism(s) as regards chromospheric heating before and at dM3, i.e. before and at the
TTCC.
We also show that the ratio of coronal heating to chromospheric heating LX/LHK increases by a
factor of 100 between dK4 and dM4 stars.
Subject headings: Stars: late-typedwarfs - Stars: late-typesubdwarfs - Stars: rotation - Stars: Activity
- Stars: Dynamo Mechanisms
1. Introduction
1Based on observations available at Observatoire de
Haute Provence and the European Southern Observatory In the present paper, we have two principal
databases andonHipparcosparallaxmeasurements. goals. First, we present data which have a bear-
1
ing on the existence of correlations between rota- E = α < B > is called the ”α-effect”. Parker
i ij j
tionandactivity(“RACs”)inoursampleofstars. (1955)firstsuggestedthattheα-effectmightarise
Second,weaskiftheempiricalcorrelationscanbe because cyclonic convective turbulence can sys-
interpreted in the context of stellar dynamo the- tematically twist a large-scale magnetic field, and
ories (which attempt to predict the properties of in the process, regenerate a large-scale poloidal
the magnetic fields which are generated in a star field. All of the dynamos we consider here rely
with a prescribed structure and rotation). The on the α-effect, as well as on (at least) one more
second goal is admittedly a challenging one: the factor.
basic mechanisms underlying stellar dynamo the- In order to construct a mean-field dynamo
ory are being continuously improved as computer model, ¡B¿ is written as the sum of poloidal and
resources permit inclusion of more realistic phys- toroidalcomponents, and ¡u¿ is assumedto be di-
ical effects. Rather than attempting to provide a rected purely in the azimuthal directionφ, and to
comprehensivediscussionofthesecomplexities,we be axisymmetric, i.e. the angular velocity Ω can
simplify our discussion by restricting attention to be a functionof r andθ, but notofφ. With these
certain broad classes of dynamos. We briefly out- assumptions, the induction equation for the time-
line the distinction between these classes in what varying magnetic field can be separated into two
follows. Whateverthesourceofthemagneticfield, equations, one for the poloidal field, the other for
the observational consequences of such a field in the toroidal field. Source terms (S , S ) appear
p t
a star is expected to be the same: dissipation of inbothequations,andthe dominantsourceterms
mechanical energy associated with magnetic pro- determine the dynamo class. For the poloidal
cessesleadstoenhancedemission(relativetowhat equation, S is a single term, namely, the φ com-
p
isgeneratedbythephotosphere)fromthechromo- ponent of E, i.e. the α-effect. For the toroidal
sphere (in spectral lines such as Caii H and K or equation, S contains two terms: (i) includes a
t
Hα) and from the corona (in X-ray continuum). spatial gradient of Ω; (ii) includes a spatial gradi-
These enhanced emissions arising from magnetic ent of E.
effectsinthestellaratmosphereontheareconsid-
Three classes of mean-field dynamos are de-
eredinthis paperto be genericindicatorsofwhat
fined as follows: (a) α-Ω dynamo, in which only
werefertoas“magneticactivity”,ormorebriefly,
term (i) is retained in the toroidal equation; (b)
“activity”. In this paper,we seek to quantify how α2 dynamo, in which only term (ii) is retained in
(anobservationalquantitythatisassociatedwith) the toroidal equation; (c) α2-Ω dynamo, in which
“activity” in low-mass dwarfs is correlated with
terms(i) and(ii)arebothretainedinthe toroidal
(anobservationalquantitythatisassociatedwith)
equation.
“rotation”. In order to setthe stage for interpret-
ingourresult,wefirstneedtodefinecertainterms
1.2. Stellar internal structure as it relates
which are related to dynamo models. to dynamo activity
1.1. Three classes of dynamo models Models of dwarfstars of spectral type F, G, K,
and early M possess a radiative core and a con-
Models of dynamos in the Sun and stars are
vective envelope. In such stars, magnetic fields
typically based on “mean-field electrodynamics”
can be generated by an α-Ω dynamo (also re-
(MFE): the fluid flow and the magnetic field are
ferredtoasaninterfacedynamo,orshelldynamo,
separated into mean (< u >, < B >) and turbu- Parker 1975)2. The strongest toroidal magnetic
lent(u,B )components(e.g. Racineetal. 2011).
′ ′ fields in these stars expected to be produced in
Althoughthemeanvaluesofu andB arebydef-
′ ′ thevicinityoftheinterface(“tachocline”)between
initionzero,the meanvalueoftheircross-product
the radiative core and the convective envelope
does not reduce to zero, but instead produces a
where the differential rotationis the strongest. In
non-zero turbulent MFE, E. The essence of MFE
stars which are massive enough to contain such
is to express the (vector) E in terms of the large-
scale (vector) magnetic field < B > by means of 2For a quantitative evaluation of such a dynamo in mid-K
a tensor expansion in the mean field plus its gra- to early-M stars with known rotation periods, see Mullan
dients. The leading order term in this expansion etal. 2015
2
an interface, and where therefore the possibility 0.3 M model arrives on the main sequence with
⊙
of an “interface dynamo” (ID) exists, data per- a radiative zone sandwiched between two convec-
taining to enhanced emission from such stars in tive zones: the models suggest that such a “sand-
chromospheric spectral lines and/or in coronalX- wich” structure exists for several gigayearsbefore
rays reveal clearly that there is a strong correla- themodelbecomescompletelyconvective(seealso
tionbetween rotationandactivity indicators(e.g. Rodriguez-Lopez et al. 2014). However, at lower
Pallavicini et al. 1981; Wright et al. 2011), i.e. masses, 0.25 M , stellar models are completely
⊙
the RAC has a slope with a numerical value that convective at all ages (see Rodriguez-Lopez et al.
is definitely non-zero. 2014). Thus, when we use the label “completely
However, since Limber (1958) derived models convective” for stars of masses in the intermedi-
of cool main sequence stars with lower and lower ate range (0.25-0.3) M , the label involves not
⊙
masses,it hasbeen widely believedby stellarevo- merely a question of the mass of the model, but
lution modelers that interfaces do not exist in all also of the age of the model. This complication
low-massstars. Ananonymousrefereehaspointed may cause some ambiguity when we attempt to
out that “there has been no definitive evidence interpret RACs in the vicinity of the TTCC.
validating this theoretical prediction”. In princi- Forsimplicity,itwouldbeconvenientifagiven
ple, asteroseismology of M dwarfs might eventu- star could be assignedto having either an ID or a
allyprovidep-modefrequencieswithenoughpreci- DD. But we recognize that such a simplified ap-
sionthat(byanalogywithhelioseismologicaldata) proach cannot be the whole story: in cool stars
signaturesoftheinterfacecouldbeidentified. But where an interface exists, the inevitable presence
although theoretical models have been calculated ofadeepconvectiveenvelopegivesrisetothepos-
for such stars (see Rodriguez-Lopez et al. 2014), sibility that ID and DD may both be operating
noempiricaldataonreliablep-modeperiodsinM (e.g. Brandenburg & Subramanian 2005; Brown
dwarfsareyetavailable. Despitethelackofdefini- et al. 2010).
tive evidence at the present time, we shall adopt In order to distinguish between the observa-
the widely held belief that stars on the main se- tionalpropertiesofIDandDD,Durney,DeYoung
quenceundergoaTransitionToCompleteConvec- & Roxburgh (1993) pointed out some differences
tion(TTCC)atacertainspectraltype. According betweenID andDD. First, the magneticfield cre-
to models, main sequence stars with masses that ated by an ID should depend strongly on rota-
arelessthanacriticalvalueofabout0.30-0.35M tion whereas the magnetic field created by a DD
⊙
arecompletely convective. Suchstarsshouldhave shouldnot. In terms ofthe notationto be usedin
no tachocline whatsoever on the main sequence. the present paper, this would lead one to expect
And yet the empirical evidence shows that stars that in an ID star, the RAC should have a slope
with masses less than 0.30-0.35M also show en- whichisdefinitelynon-zero,whereasinaDDstar,
⊙
hanced emission in chromospheric lines and in X- the RACslopeshouldbe essentiallyzero. Second,
raycontinuum. Since we have attributed suchen- although IDs can produce activity cycles in the
hanced emission in warmer stars to magnetic ac- large-scalefielditwasnotclear(in1993)howDD
tivity, then it seems natural also to ascribe the could ever give rise to a cycle.
enhancedemissionsincompletelyconvectivestars
Since 1993, the second feature has been called
to magnetic activity. Such activity must rely ona
into question. E.g. Stefani & Gerberth (2003)
2
non-interface type of dynamo, possibly an α dy-
have demonstrated that cyclic behavior is possi-
namo (distributive dynamo: DD), maintained by ble in α2 dynamo models provided that the fol-
convective turbulence alone (e.g. Roberts & Stix
lowing condition is satisfied: the α-effect needs to
1972, Rosner 1980, Dobler et al. 2006, Chabrier
have radial gradients which are sufficiently steep,
& Kuker 2006).
including changes in the algebraic sign.
It is important to note that the existence of
Moreover, recent modeling suggests that the
complete convection in a star does not mean that third class of dynamo models (α2-Ω) might be
the star is necessarily completely convective at
a better description to the solar dynamo (where
all stages of its main sequence phase of evolu-
there is certainly an interface) than an α-Ω dy-
tion. E.g.,FeidenandDotter(2013)reportthata
namo(e.g. Lawsonetal2015). Fromtheperspec-
3
2
tive of a change from an α-Ω dynamo to an α -Ω ditions are used for the stellar models (Mullan et
dynamo, perhaps in completely convective stars, al. 2015). (Even larger shifts of the TTCC to-
we might also encounter a change from an α2 dy- wards lower masses were at one time proposed by
namo to an α2-Ω dynamo. If this happens, then Mullan and MacDonald 2001 if interior magnetic
we mightfind thatratherthan seeinga zeroslope fields were to be as large as 107 8 G. If fields as
−
for the RAC in a completely convective star, the large as that were to exist inside stars, the defi-
inclusion of Ω in the dynamo model could lead to nition of a “fully convective star” could become
a non-zero RAC slope. moreambiguous: theonsetofcompleteconvection
would then depend not only on the mass and age
1.3. ObservationalevidencefortheTTCC? of a star, but would also depend on how strong
its magnetic field is. However,we may not in fact
A long standing problem has been the search
need to worry about this ambiguity: Browning et
foranobservationalsignatureoftheputativetran-
al. (2016) have recently argued that such strong
sition between different types of dynamos at the
fields would be unstable.
TTCC. There has been several attempts to de-
In standard main-sequence models, the radius
tect a change in magnetic activity diagnostics
ofamodelscalesalmostlinearlywithmass: there-
(e.g. Mullan&MacDonald2001)ormagneticfield
fore, the TTCC is expected to lie in the ra-
topologies (e.g. Donati et al. 2008, Morin et al.
dius range of roughly 0.25-0.4 R . According
2008, 2010, Phan-Bao et al. 2009, Stassun et al.
⊙
to the radius-T calibration of Houdebine et al.
2011) in the vicinity of the TTCC. None of these eff
(2016b) this radius range yields an effective tem-
studieshaveidentifiedunambiguousanddefinitive
perature range of 3200-3500K for the TTCC.
signatures of significant changes in magnetic ac-
tivity at the TTCC. However, in a recent paper, In this paper, we examine the topic of a possi-
West et al. (2015) have reported on a study of ble dynamo transition at the TTCC using a more
RACs in two groups of dwarfs: M1-M4, and M5- extensive and more fine-graineddata set than has
M8. West et al. (2015) found that the RAC in previously been available for study.
M1-M4 stars has a somewhat different slope than
the RAC in M5-M8 stars (see Sect. 4 below for 1.4. Dynamos: unsaturated and saturated
amoredetaileddiscussion). Oneinterpretationof
Empirically, evidence for rotationally driven
thisbehavioristhatsomethingchangesasregards
dynamos in cool stars first emerged when re-
the dynamo between M4 and M5. We will return
searchers plotted the strength of chromospheric
toadiscussionofthisinterpretation(aswellasan
emission versus stellar rotation. E.g. See Kraft
alternative interpretation) in Section 4 below.
(1967), Vaughan et al. (1981), Soderblom (1982),
InordertoaddresstheTTCC-dynamo-transition Vogt et al. (1983), Noyes et al. (1984), Marcy
topic meaningfully, we first need to identify at & Chen (1992), Patten & Simon (1996), Fekel
what spectral type the TTCC occurs. Limber (1997), Delfosse et al. (1998), Jeffries et al.
(1958) states that “in that part of the main se- (2000), Pizzolato et al. (2003), Mohanty & Basri
quence where the inner radiative region ... is (2003), Browning et al. (2010), Wright et al.
becoming vanishingly small”, the corresponding (2011), Rebassa-Mansergaset al. (2013), West et
spectraltypeisM3V-M4V.Subsequently,Dorman al. (2015).
et al. (1989) placed the TTCC at M 0.25M ,
∼ ⊙ The key signature is the following: emission
which corresponds to the spectral subtype dM4.
in chromospheric spectral lines is observed to be
Even more recently, Chabrier & Baraffe (1997)
stronger in stars with faster rotation speed v (or
r
predict that the TTCC occurs at M 0.35M ,
∼ ⊙ shorter rotation period P) (e.g. Vaughan et al
which corresponds to the spectral subtype dM2.
1981). This is defined as a “Rotation-Activity
Therefore, it appears that the TTCC may lie
Correlation” (RAC). The existence of an RAC is
somewhere in the range between subtypes dM2
consistentwiththeexpectationsthat(i)thefaster
anddM4,correspondingtomassesof0.25-0.4M
therotationis,thestrongerarethemagneticfields
⊙
(Stassun et al. 2011). The theoretical mass limit
which can be generated (e.g. Mullan et al. 2015),
at the TTCC has been found to shift towards
and(ii)strongerfieldsareassociatedwithstronger
slightly smaller masses if different boundary con-
4
chromospheric heating (Skumanich et al. 1975) reveals itself in two regimes: saturated and un-
and with stronger coronal heating (e.g. Mullan saturated. The properties of the two different dy-
2009). Thus, if chromospheric emission intensity namoregimeswefindbelow(seeSection3.8.1)for
is plotted as a function of P, it is found that over thelowandhighactivitysub-samplesrespectively
a certain range of periods, the RAC has a clearly cannot be due to effects of the chromospheric re-
negative slope. sponse to non-thermal heating mechanisms. In
However, as more data are accumulated, it fact, there does not seem to be any definitive ev-
emerges that the negative slope of the RAC does idence that the influence of a magnetic field in a
not extend indefinitely to shorter and shorter P. stellar atmosphere ever attains a saturated level.
Instead, when P becomes shorter than a certain Insupportofthisclaim,wenotethattheresponse
value P , no further increase in emission occurs: of stellar chromospheres to non-thermal heating
c
for P P , the RAC becomes flat (e.g. Vilhu mechanisms is continuous and monotonically in-
c
1984),≤with a slope of zero. By definition, in the creasingfrombasalchromospherestoflaringchro-
flattened portion of the RAC, increasing rotation mospheres (e.g. Houdebine 1992, Houdebine &
doesnotresultinincreasedchromospheric/coronal Doyle 1994a, Houdebine & Doyle 1994b, Houde-
emission. For solar type stars, the transition to a bine et al. 1995, Houdebine & Stempels 1997,
flat curve occurs for P 3 days. Vilhu suggested Houdebine2009,Houdebine2010). Therefore,the
the term “saturated” to≈refer to such conditions. two different types of RACs we find below for the
Similarly,Wrightetal. (2011)foundthatthe sat- low and high activity stars respectively highlight
uration regime occurs at about R0 0.8. Reiners thedifferentpropertiesofthedynamomechanisms
etal. (2009)foundthatthesturation≃occursatthe in these two types of stars.
critical Rossby number R0 0.1. We shall com- It seems to us that in order to study the prop-
≃
parethesevaluestothoseweobtainforourstellar erties of stellar dynamos mostprofitably,it would
samples below. Pizzolato et al. (2003)found that be preferable to concentrate as much as possible
saturationoccursfromaperiodofabout 2days on stars in the unsaturated regime. The reason
≈
insolartypestarstoabout 10daysinMdwarfs. for this claim is that, when conditions are sat-
≈
Rebassa-Mansergaset al. (2013) found that stars urated, there are extra factors which come into
with vsini 5 hm s 1 are all in the saturated play which may obscure some physical properties
−
≥
regime. This yields a rotation period of 3 days that are directly associated with dynamo action.
≈
for stars at the spectral type dM3 (see Paper I). InviewofVilhu’sidentificationofacriticalperiod
Therefore,weshallconsiderthatsaturationoccurs P ,itseemsthattheslowestrotatorshavethebest
c
forperiodsofabout 3daysorR0 0.5. Sincein chance of being in the unsaturated regime. For
≈ ≃
our stellar samples we have stars with shorter ro- thatreason,weconsideritworthwhiletopushthe
tationperiods orR0, we shallconsider below that spectroscopicmeasurements of stellar rotationto-
these stars likely lie in the saturated regime. wards the smallest possible values of vsini which
Vilhusuggestedthat“saturation”mightbedue can be reliably measured.
to a complete coverage of the star’s surface by In general, since slow rotationmeans less chro-
magnetic fields. Another explanation of “satura- mospheric heating, we expect that the slowest
tion”wasofferedbyMullan(1984)intermsofthe rotators in our dataset will be stars which are
maximum possible flux of mechanical energy that low-activity stars classified as dK and dM (i.e.
can be generated by convection. In the present those which by definition show no emission in the
paper, we shall not attempt to identify the phys- Balmer lines), while the fastest rotators will be
ical process which leads to saturation. Instead, stars which are highly active stars classified as
we shall adopt a purely empirical approach, and dKe and dMe (i.e. those where by definition the
we shall refer to the flat portion of an RAC as Balmer lines have an emission core). In what fol-
“saturated”. In the same vein, we shall refer to lows, we shall be especially interested in deter-
an RAC with a statistically significant negative mining the (negative) slope of the RAC for low-
slope as “unsaturated”. To the extent that an activitystars(e.g. seeFig.13below,lowerpanel).
RAC owes its existence to the operation of a dy-
namo of some kind, we can say that the dynamo
5
1.5. Extending our previous work magnetic field intensities are correlated with the
stellarrotationrate. This relationshipis tobe ex-
One reason for the present study has emerged
pected on the basis of standard dynamo theory
from recent work by Houdebine & Mullan (2015:
(e.g. Parker 1979, Krause & Radler 1980, Mullan
hereafterHM):theyfoundthatanotherdiagnostic
et al. 2015). However, most of the observational
of magnetic fields, namely, the efficiency of mag-
investigations cited above suffer from two inade-
netic braking (which manifests itself in the rota-
quacies: (i) they included only a few stars which
tional velocity), undergoes a detectable change at
are rotating slowly enough to be in the unsatu-
spectralsub-type dM3. Basingtheir analysisona
rated regime, and (ii) the targets included stars
new data set of precise rotational velocities, HM
whichwerespreadoutoverabroadrangeofspec-
found that the mean rotation period of M3 stars
tral types. In this paper, we attempt to remedy
is abnormally large compared to those of the ad-
both of these inadequacies.
joining spectral types dM2 and dM4. This indi-
cates that the dM3 stars have been slowed down
1.6. Aspects of the data used in the
more than the stars in the immediately adjacent
present study
sub-types dM2 and dM4. This excess slowing at
dM3 may be associated with a change in mag- In an effort to extend the RACs to slow rota-
netic properties of the stars at dM3. Specifically, torsamonglate-typedwarfs,wehavebeenreport-
HM suggested that the change might be associ- ing, over the past several years, improved spec-
ated with an earlier report (Mullan et al 2006) troscopic measurements of rotational broadening
thatthelengthsofflaringmagneticloopsundergo vsini in stars of spectral sub-types dK4 (Houde-
a significant increase at spectral type dM3. bine 2011a, Paper XVI thereafter), dK6 (Houde-
bine et al. 2016, Paper I), dM2 (Houdebine 2008,
In the present paper, we extend the work of
Paper VIII, Houdebine 2010a, Paper XIV), dM3
HM in two distinct ways. First, we expand the
(HM) and dM4 (Houdebine 2012a, Paper XVII,
database of precise rotationalproperties of K and
Houdebine et al. 2016, Paper I).
Mdwarfsofvarioussub-types. Second,weexpand
and analyze a separate database which deals with Combining our (previous) rotational measures
the second physical parameter which enters into with our measures of the Caii line equivalent
the RAC: the radiative properties which are asso- widths (EW), we have already reported RACs
ciated with magnetic “activity” in our sample of for dK4 (Houdebine 2011a, Paper XVI), dM2
rotating stars. Our goal here is to use the activ- ( Houdebine 2011b, Paper XV) and dM4 stars
ity data to construct RACs. Moreover, we quan- (Houdebine 2012b, Paper XVIII), for slow and
tify the RACs in two different parts of the stel- rapid rotators alike. In those earlier papers, we
laratmosphere: thechromosphere(usingtheCaii proposed empirical RACs which included large
lines) and the corona (using L ). The present samples of slow rotators. We found that for later
X
study is based on a larger sample of stars, and a spectral types it is crucial to examine a fine grid
finer grid of spectral sub-types, than have been of spectral sub-types. Specifically, we found that
used previously in constructing RACs for K and the RAC’s varysignificantlybetweendK4todM4
M stars. Our goal is to explore whether the un- stars: theRAC’swerefoundtohavedifferentgra-
usualrotationalsignaturereportedbyHMatdM3 dients and different saturation levels (Papers XV,
is accompanied by unusual behavior in the activ- XVIII andthe presentstudy). The previousdata,
ity indicators of either chromosphereor corona or combined with the present study, now provide us
both, either as regards the intensity of the radia- with large enough data-sets in each spectral sub-
tion, or as regards the slopes of the RACs. type that we can investigate with improved confi-
dencethedifferences(ifany)betweentheRAC’sin
Akeyphysicalfactorwhichisknowntobewell
fivedifferentspectralsub-types. Inviewofthere-
correlated with chromospheric emission and coro-
sultsreportedinHM,itisnotablethatthepresent
nalemissionhas todo with magneticfields onthe
studyenablesustostudytheRAC’sinthevicinity
stellar surface (e.g. Skumanich et al 1975; Schri-
of the TTCC.
jver et al. 1989). The existence of RAC’s may
be interpreted as an indication that the surface When we consider slow rotators, our data con-
firm that the mean rotation periods of stars in
6
the range dK4-dM4 in general decrease with de- 2. Selection of spectroscopic data
creasingeffectivetemperature(seeHMandPaper
Stepien´ (1989, 1993, 1994) has reported that
I). However, we also find that something unusual
RACs exhibit certaindifferences atdifferentspec-
happens in the rotation rates between dM2 and
traltypes. Asanextensionofthisfinding,wehave
dM4. The overall trend towards decreasing ro-
found,inpreviousstudies(PaperXVIII,HM)and
tation period as we go from dK4 to dM4 is in-
also in the present study, that it is important in
terrupted at spectral sub-type dM3: at that sub-
constructing RACs to select samples of stars with
type, the mean rotation period increases to a lo-
T values which are confined within a narrow
cal peak, such that the mean rotational period at eff
range. There are two principal reasons for this,
dM3 is longer than the overalltrendbetweendK4
one relatedto the choiceof anoptimalset of pho-
and dM4 would have predicted (HM). But when
tospheric absorption lines, and the second related
weextendourinvestigationtoincludefastrotators
toouranalysisofchromosphericemissionlines. As
amongthedK4-dM4stars,thereisfoundtobethe
regards the choice of photospheric lines, we have
following overall trend: the mean rotation period
already discussed the first of the above reasons at
tends to increase slightly from dK4 to dM4. But
length in Paper I in the context of optimizing the
once again, at dM3, an exception is found: the
measurementofrotationalvelocities at eachspec-
mean rotation period of fast rotators at dM3 is
tralsubtypebetweendK4anddM4. Inthepresent
locally significantly longer than the overall trend
paper, we turn now to the second reason.
would have predicted (HM). HM interpret these
abnormallylongrotationperiodsatsub-typedM3 In the context of chromospheric analysis, it is
as possibly being associated with the occurrence important to deal with stars with closely sim-
of increasing coronal loop lengths (previously re- ilar Teff when we are attempting to quantify
ported by Mullan et al. 2006 in a study of flare with as much precision as possible the EW of the
stars). The mean rotation period of the slow ro- chromospheric lines (Caii resonance doublet and
tators is an important constraint on the temporal Hα). These lines inevitably include some contri-
history of the dynamo mechanisms and magnetic butionsfromthebackgroundphotosphericcontin-
braking mechanisms (HM, Paper I). uum and from the temperature minimum region
(e.g. Houdebine & Doyle 1994, Cram & Mullan
1.7. Studying RACs in various formats 1979, Houdebine & Stempels 1997, Paper XV).
Initially, our samples of stars had been selected
An important aspect of the present paper is
for the purpose of chromospheric modelling stud-
that we wish to investigate the RACs in various
ies (e.g. Houdebine & Stempels 1997, Houdebine
formats. In the first place, we construct RACs
2009bPaperXII,Houdebine2010bPaperIX),and
separately for the chromosphere and the corona.
in these studies the selection of stars with closely
Moreover, we explore correlations between vari-
similarspectraltypes wasessentialinorderto de-
ous observations of the “activity” and various as-
velop reliable grids of semi-empirical model chro-
pectsof“rotation”. Asanexample,wewillexam-
mospheres,eachofwhichwouldbe superposedon
ine, for the H and K lines of Caii, a plot of the
a particular photospheric model.
quantityR asafunctionoftheRossbynumber
R0. We alsH′oKplot the Caii surface flux as a func- Based on our previous papers, we have found
that the most suitable initial selection parameter
tion of P/sini. We use these different formats in
when we wish to identify a homogeneous sample
an attempt to improve our chances of identifying
of K or M dwarfs belonging to a specific sub-
changes(ifany)inthedynamoregimeinthevicin-
type is the (R-I) color: this color is sensitive to
ity of the TTCC. We shall find that changes near
T , but less so to metallicity (e.g. Leggett1992,
the TTCC are more readily detectable in some eff
Ramirez & Melendez 2005, Mann et al. 2015).
RACs than in others. This may explain why the
Moreover,broad-band colors of high precision are
empirical detection of the TTCC has been elusive
widely available in the literature for many of the
in the past.
cool dwarfs which are of interest to us.
In the present paper, as an example of how we
selecteddataforourRACstudies,wenowdescribe
7
howwe gatheredthe relevantdatafor asample of obtained in the High Efficiency (HE) mode. The
dK6 stars. We selected a sample of 419 late K modes in which the spectra were obtained are in-
dwarfs on the basis of (R-I) measurements avail- dicatedinTable1togetherwiththecorresponding
able in the literature. For example, our sample Signal to Noise (S/N) ratio. For further details of
of dK6 stars (see Table 1 below) contains stars the spectrographs, see HM.
with (R-I)C (i.e. (R-I) color in the Cousins sys- As a second example of how we selected per-
tem) in the range [0.684;0.816] which also corre- tinent data for our spectral sub-samples, we con-
sponds to (R-I)K ((R-I) in the Kron system) in sider here briefly our sample of dM3 stars. As
the range [0.503;0.613] according to the transfor- described in HM, westarted off with a list of 381
mation formulae of Leggett (1992) (see Leggett dM3 objectsbasedon(R-I)datain the literature.
1992 for more information on the Cousin’s and Searching through the same databases as above,
Kronphotometricsystems). AccordingtoKenyon we found suitable observations which allowed us
and Hartmann (1995), this range of colors is cen- to determine reliable vsini values for 86 different
tered on (R-I)C=0.75, i.e. the spectral type dK7. dM3 stars. A subsequent search of the available
However, when we compiled and derived effec- spectra in the above databases allowed us to ob-
tive temperatures (see Paper I) for this sample tainreliablemeasurementsoftheEWsofCaiiand
of late-K dwarfs, we found in average higher tem- H for a sub-sample of only 59 M3 dwarfs for our
α
peratures than what would be expected from the chromospheric RAC (excluding probable spectro-
(R-I)C-Teff tabulation of Kenyon and Hartmann scopic binaries and low sini stars).
(1995)(see Paper I). Our dK6 stellar sample con-
tains stars that have similar (R-I)C colours and 2.1. Biases in our stellar samples
the same effective temperaturesto within 110K
± The stars in our samples include all stars from
(see Paper I). We refer to HM for a discussion of
all observing programs which have been carried
corresponding data for our sample of dM3 stars.
out with HARPS and SOPHIE for stars belong-
The literature provided us with a starting list
ingtothefollowingspectralsub-types: dK4,dK6,
of a large number (419) of late-K dwarfs. Search-
dM2, and dM3. For dM4 stars we compiled all
ing through databases at the European Southern
measurements of vsini available in the literature
Observatory (ESO) and Observatoire de Haute
(see Paper I). For the dK6 and dM3 samples, we
Provence (OHP), we identified spectra of 112 dif-
also supplemented our own measurements with
ferent stars which are suitable for our purposes.
measurements available in the literature, notably
The final list of 105 stars for which the available
for active stars (see Paper I).
spectra would allow us to determine reliable ro-
In the HARPS and SOPHIE databases, many
tational data (i.e. vsini values) in our sample of
of the spectra were obtained in connection with
late-K dwarfs has already been provided in Paper
planet-search programs. In such programs, ob-
I. For the present paper, we found spectra which
serverstendtoavoidstarswithhighlevelsofmag-
would allow us to make reliable measurements of
the EWs of Caii and H for a sub-sample of only netic activity. Therefore our spectral samples are
α
likely to be biased in general towards low activ-
89 late-K dwarfs. It is the combination of reliable
ity stars, i.e. stars which are in the unsaturated
rotations (Paper I) and reliable EWs of chromo-
portions of the RACs. For reasons outlined above
spheric lines which enable us to undertake, in the
(see Section 1.1), we consider this bias to be an
present paper, the study of the RAC in our sub-
advantage in the present study. The biases in our
sample of dK6 dwarfs.
samples may contribute somewhat to the density
The spectra which we use for determining the
ofthe samplingatdifferentpartsofthe RAC,but
Caii and H equivalent widths in the present
α thisisnotexpectedtocausesignificantdiscrepan-
study of dK4-dM4 stars came from three differ-
ciesasregardstheoverallRACs. Notethatforthe
ent ´echelle spectrographs; HARPS (High Accu-
dM4 and dK6 samples, the sampling of the RACs
racy Radial velocity Planet Search, ESO), SO-
should be more complete as regards the measures
PHIE (OHP) and FEROS (The Fiber-fed Ex-
ofP/sini (because we includedother measuresof
tended Range Optical Spectrograph). We in-
vsini from the literature, see Paper I).
cluded in our sample the SOPHIE observations
8
3. Therotation-activitycorrelations(RACs) the RACs (Sub-Section 3.7), and then as regards
in late-K and M dwarfs the absolute values of the chromosphericemission
levels (Sub-Section 3.8).
In this section, we first (Sub-section 3.1) eval-
In Sub-Section 3.9, we switch to a different ap-
uate the surface fluxes in the continuum in the
proachtoconstructingRACs. Asregardsrotation,
vicinity of the Caii resonance doublet by using
weswitchtotheRossbynumber. Asregardschro-
our estimates of T and also by using the syn-
eff mospheric activity, we switch to a quantity which
thetic spectra of de Laverny et al. (2012).
expresses the output power in the H and K lines
Then, we construct the RACs of M and K of Caii as a fraction of the star’s output power
dwarfs using a variety of approaches: some ap-
(L ) (Fig. 15).
bol
proaches may facilitate the extraction of informa-
Switching our attention to the corona (Sub-
tion that is more difficult to extract by means of
Section3.10),weconstructadifferenttypeofRAC
other approaches. In all of the approaches, we
forourtargets,thistimereferringtoconditionsin
plot a quantity related to “activity” as the ordi-
the corona, rather than the chromosphere. The
nate, and a quantity related to “rotation” as the
coronalRACisobtainedbyplottingL /L asa
abscissa. X bol
functionoftheRossbynumber(Fig.16). Compar-
Our first approach (Sub-Sections 3.2, 3.3, 3.4,
ing “activity” in chromosphere and corona (Sub-
3.5, and 3.6) specifies the “activity” of a star in
Section 3.11; Figs. 17, 18) can provide informa-
terms of the surface fluxes of chromosphericlines.
tion as to how any given M dwarf star partitions
Combining these surface flux results with our re-
its deposition of mechanical energy between chro-
sults for P/sini (Paper I), we construct a set of
mosphere and corona. To quantify this partition,
RACs for M and K dwarfs belonging to various
in Sub-Section 3.11 we examine the correlations
spectral sub-types (see Figs. 2, 3, 5, 8, and 12).
between L and L (see Figs. 17, 18).
X HK
We first analyze the RAC for K4 and K6 dwarfs.
In the final Sub-Section (3.12), we summarize
Then we re-visit the RAC in M2 dwarfs that was
thepropertiesofthevariousRACswhichourdata
first investigated in Paper XV. We then analyse
have enabled us to construct.
the RAC in M3 dwarfs for which vsini measures
werereportedinHMandforwhichwepresentnew
3.1. The mean fluxes in the vicinity of the
measurements of the chromospheric line equiva- Caii resonance doublet.
lent widths here. We also re-visit the RAC in M4
dwarfs that was studied in Paper XVIII with the In order to evaluate the efficiency of the dy-
new values of the stellar parameters determined namo mechanism(s) as a function of spectraltype
in Paper I. Finally, we compare the RACs for five in M and K dwarfs, it seems preferable to inter-
spectralKandMsubtypes,dK4,dK6,dM2,dM3 comparetheRAC’scalibratedintermsofabsolute
and dM4, and draw conclusions about the differ- energy fluxes, rather than confining our attention
ential variations of the RAC from mid-K dwarfs to the values ofthe equivalentwidth (EW). (Nev-
to M4 dwarfs. ertheless, the EW have proven useful in terms of
surface magnetic fields: see Mullan et al. 2015.)
We emphasize that our RACs are plotted as a
In order to convert from EW to energy fluxes, we
function of the projected rotation period P/sini
calculated the surface fluxes in the continuum in
and not the rotation period P. The average sini
the vicinity of the Caii lines from the theoretical
is 0.6. Then in order to recover the RACs as a
model atmospheres of de Laverny et al. (2012)
function of P, one must multiply our RACs by a
and Palacios et al. (2010) 3 for log(g) = 5.0,
factor of 1.67. The scatter due to variable sini
[M/H]=0.0 and α=0.0. We found that in M2, M3
is included in our correlations. Nevertheless, we
and M4 dwarfs, for our observations, the contin-
show that one can obtain reasonably good empir-
uum at about 3950 ˚A represents a good evalua-
ical RACs in spite of these uncertainties and the
tionofthebackgroundcontinuumfluxfortheCaii
uncertainties on our measures.
lines. But there are significant discrepancies be-
In Sub-Sections 3.7 and 3.8, we present an
overviewofthesystematicpropertiesoftheRACs
3 http://npollux.lupm.univ-montp2.fr/ or
inallofoursub-types,firstasregardstheslopesof
ftp://ftp.oca.eu/pub/laverny/DEPOT/AMBRE GridFlux/
9
Fig. 1.— Upper panel: the observed spectrum of Gl 205 (dM2, solid line) together with the theoretical
spectrum of de Laverny et al. (2012) for an effective temperature of 3500 K (dotted line). Note substantial
disagreements between the observations and the model. Lower panel: the observed spectrum of Gl 570A
(dK4, solid line) together with the model for an effective temperature of 4500 K (dotted line). Here again
there are significant differences between the model an1d0 observations. All the spectra are normalised for the
continuum flux at about 3950 ˚A, i.e. between the H & K line centers.