Simulating the Environment Around Planet-Hosting Stars - I. Coronal Structure

7.8 MB·

by  J. D. Alvarado-Gómez

Astronomy&Astrophysicsmanuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2 (cid:13)cESO2016
January19,2016
Simulating the Environment Around Planet-Hosting Stars
I. Coronal Structure
J.D.Alvarado-Gómez1,2,G.A.J.Hussain1,3,O.Cohen4,J.J.Drake4,C.Garraffo4,J.Grunhut1 andT.I.Gombosi5
1EuropeanSouthernObservatory,Karl-Schwarzschild-Str.2,85748GarchingbeiMünchen,Germany
e-mail:[email protected]
2Universitäts-SternwarteMünchen,Ludwig-Maximilians-Universität,Scheinerstr.1,81679München,Germany
3InstitutdeRechercheenAstrophysiqueetPlanétologie,UniversitédeToulouse,UPS-OMP,F-31400Toulouse,France
4Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
5CenterforSpaceEnvironmentModeling,UniversityofMichigan,2455HaywardSt.,AnnArbor,MI48109,USA
6
1 Received—–;accepted—–
0
2
ABSTRACT
n
a Wepresenttheresultsofadetailednumericalsimulationofthecircumstellarenvironmentaroundthreeexoplanet-hostingstars.A
J
state-of-the-art global magnetohydrodynamic (MHD) model is considered, including Alfvén wave dissipation as a self-consistent

8 coronal heating mechanism. This paper contains the description of the numerical set-up, evaluation procedure, and the simulated
1 coronalstructureofeachsystem(HD1237,HD22049andHD147513).Thesimulationsaredrivenbysurfacemagneticfieldmaps,
recoveredwiththeobservationaltechniqueofZeemanDopplerImaging(ZDI).Adetailedcomparisonofthesimulationsisperformed,

] where two different implementations of this mapping routine are used to generate the surface field distributions. Quantitative and
R
qualitativedescriptionsofthecoronaeofthesesystemsarepresented,includingsynthetichigh-energyemissionmapsintheExtreme
S Ultra-Violet(EUV)andSoftX-rays(SXR)ranges.Usingthesimulationresults,weareabletorecoversimilartrendsasinprevious
. observationalstudies,includingtherelationbetweenthemagneticfluxandthecoronalX-rayemission.Furthermore,forHD1237
h
weestimatetherotationalmodulationofthehigh-energyemissionduetothevariouscoronalfeaturesdevelopedinthesimulation.
p
Weobtainvariations,duringasinglestellarrotationcycle,upto15%fortheEUVandSXRranges.Theresultspresentedherewill
-
o beused,inafollow-uppaper,toself-consistentlysimulatethestellarwindsandinnerastrospheresofthesesystems.
r
Key words. stars:coronae–stars:magneticfield–stars:late-type–stars:individual:HD1237–stars:individual:HD22049–
t
s stars:individual:HD147513
a
[

11. Introduction et al. 2008; Alvarado-Gómez et al. 2015; Hussain et al. 2016).
v Long-term ZDI monitoring of particular Sun-like targets have
3Analogous to the 11-year Solar activity cycle, a large fraction showndifferenttime-scalesofvariabilityinthelarge-scalemag-
4of late-type stars (∼ 60%) show chromospheric activity cycles, neticfield.Thisincludesfastandcomplexevolutionwithoutpo-
4
withperiodsrangingfrom2.5to25years(Baliunasetal.1995). larityreversals(e.g.HNPeg,BoroSaikiaetal.2015),erraticpo-
4
Foraverylimitednumberofthesesystems,includingbinaries, laritychanges(e.g.ξBoo,Morgenthaleretal.2012)andhintsof
0
thecoronalX-raycounterpartsoftheseactivitycycleshavealso magneticcycleswithsingle(e.g.HD190771,Petitetal.2009),
.
1been identified (e.g. Favata et al. 2008; Robrade et al. 2012). anddouble(e.g.τBoo,Faresetal.2009)polarityreversalsina
0Theseperiodicsignaturesappearasaresultofthemagneticcy- time-scaleof1-2years.
6
cleofthestar.InthecaseoftheSun,thisiscompletedevery22 Furthermore, ZDI maps have proven to be very useful in
1
years over which the polarity of the large-scale magnetic field otheraspectsofcoolstellarsystemsresearch.Applicationscover
:
vis reversed twice (Hathaway 2010). These elements, the cyclic magneticactivitymodellingforradialvelocityjittercorrections
Xiproperties of the activity and magnetic field, constitute a major (Donati et al. 2014), transit variability and bow-shocks (Llama
benchmark for any dynamo mechanism proposed for the mag- etal.2013),coronalX-rayemission(Johnstoneetal.2010;Ar-
arneticfieldgeneration(Charbonneau2014). zoumanianetal.2011;Langetal.2014)andmasslossratesin
Recent developments in instrumentation and observational connectionwithstellarwinds(Cohenetal.2010;Vidottoetal.
techniqueshaveopenedanewwindowforstellarmagneticfield 2011).
studiesacrosstheHRdiagram(seeDonati&Landstreet2009). In the case of planet-hosting systems, ZDI-based studies
In particular, the large-scale surface magnetic field topology in have tended to focus on close-in exoplanet environments by
starsdifferentfromtheSuncanberetrievedusingthetechnique applying detailed global three-dimensional magnetohydrody-
of Zeeman Doppler Imaging (ZDI, Semel 1989; Brown et al. namic(MHD)models,originallydevelopedforthesolarsystem
1991; Donati & Brown 1997; Piskunov & Kochukhov 2002; (BATS-R-US code, Powell et al. 1999). This numerical treat-
Hussain et al. 2009; Kochukhov & Wade 2010). Several stud- ment includes all the relevant physics for calculating a stellar
ieshaveshowntherobustnessofthisprocedure,successfullyre- corona/wind model, using the surface magnetic field maps as
covering the field distribution on the surfaces of Sun-like stars, driver of a steady-state solution for each system. Within the
overawiderangeofactivitylevels(e.g.Donatietal.2008;Petit MHD regime, two main approaches have been considered: an
Articlenumber,page1of15
A&Aproofs:manuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2
ad-hoc thermally-driven polytropic stellar wind (i.e., P ∝ ργ, spectro-polarimetricdata.However,asdescribedbyBrownetal.
withγasthepolytropicindex,Cohenetal.2011;Vidottoetal. (1991),ZDIisnotabletoproperlyrecoververysimplefieldge-
2012, 2015), and a more recent description, with Alfvén wave ometries(e.g.dipoles),andismoresuitablefor complex(spot-
turbulence dissipation as a self-consistent driver of the coronal ted)magneticdistributions.ThislimitationisremovedintheSH-
heating and the stellar wind acceleration in the model (Cohen ZDIimplementation.Bothproceduresarerestrictedbytheincli-
et al. 2014). This last scheme is grounded on strong observa- nation angle of the star and therefore, a fraction of the surface
tionalevidencethatAlfvénwaves,ofsufficientstrengthtodrive field that cannot be observed, is not recovered in the maps. To
the solar wind, permeate the solar chromosphere (De Pontieu correctforthiseffect,previousnumericalstudieshavecompleted
et al. 2007; McIntosh et al. 2011). Additionally, this numerical the field distribution by a reflection of the ZDI map across the
approachhasbeenextensivelyvalidatedagainstSTEREO/EUVI equatorialplane(e.g.Cohenetal.2010).Morerecently,Vidotto
andSDO/AIAmeasurements(seevanderHolstetal.2014).The et al. (2012) have included complete symmetric/antisymmetric
modelspresentedinthispaperarebasedonthislatesttreatment SH-ZDImapstoshowthatthemapincompletenesshasaminor
oftheheatingandenergytransferinthecorona. impact over their simulation results. However, for the simula-
In this work we present the results of a detailed numerical tions performed here, which include the latest implementation
simulation of the circumstellar environment around three late- of BATS-R-US, this may not be the case. A larger impact may
type exoplanet-hosts (HD 1237, HD 22049 and HD 147513), be expected on the overall coronal structure, as the mechanism
using a 3D MHD model. This first article contains the results forthecoronalheatingandthewindaccelerationisdirectlyre-
ofthesimulatedcoronalstructure,whilethewindandinneras- lated to the field strength and topology (e.g. Alfvén waves, see
trosphere domains will be presented in a follow-up paper. The vanderHolstetal.2014).
simulationsaredrivenbytheradialcomponentofthelarge-scale
surfacemagneticfieldinthesestars,whichhavebeenrecovered
using two different implementations of ZDI (Sect. 2). All three
systemshavesimilarcoronal(X-ray)activitylevels(seetable1).
WhilethesearemoreactivethantheSuntheywouldbeclassified
asmoderatelyactivestarsandwellbelowtheX-ray/activitysat-
uration level. A description of the numerical set-up is provided
in section 3, and the results are presented in section 4. Section
5 contains a discussion in the context of other studies and the
conclusionsofourworkaresummarizedinsection6.
2. Large-ScaleMagneticFieldMaps
HD 1237, HD 147513 and HD 22049 are cool main sequence
stars(G8,G5andK2respectively)withrelativelyslowrotation
rates(P ∼7−12days).EachofthesesystemshostaJupiter-
rot
massplanet(M sini > M ),withorbitalseparationscompara-
p
ble to the solar system planets (Hatzes et al. 2000; Naef et al.
(cid:88)
2001;Mayoretal.2004;Benedictetal.2006).Table1contains
asummaryoftherelevantastrophysicalparametersforeachsys-
tem,takenfromvariousobservationalstudies.
Previousworkshaverecoveredthelarge-scalemagneticfield Fig.1.SurfaceradialmagneticfieldmapsofHD1237.Acomparison
onthesurfacesofthesestars,byapplyingZDItotime-seriesof betweenthestandardZDI(top)andtheSH-ZDI(bottom)ispresented.
circularlypolarisedspectra(Jeffersetal.2014;Alvarado-Gómez ThecolourscaleindicatesthepolarityandthefieldstrengthinGauss
et al. 2015; Hussain et al. 2016). For the stars included in this (G).Notethedifferenceinthemagneticfieldrangeforeachcase.The
work,thishasbeendonewiththespectropolarimeterNARVAL stellarinclinationangle(i=50◦)isusedforthevisualizations.
at the Telescope Bernard Lyot (Aurière 2003), and the polari-
Figures1and2showacomparisonbetweenthereconstruction
metricmode(Piskunovetal.2011)oftheHARPSechellespec-
proceduresappliedtoHD1237andHD22049,respectively.In
trograph (Mayor et al. 2003) on the ESO 3.6m telescope at La
general,themapsobtainedusingZDIshowamorecomplexand
SillaObservatory.Forconsistency,theZDImapsincludedinthe
weakerfielddistributionincomparisontotheSH-ZDI,wherea
simulations have been reconstructed using data from the same
instrument/telescope†(i.e.HARPSpol). smootherfieldtopologyisobtained.Whiletherearesimilarities
in the large-scale structure, discrepancies are obtained in terms
For the magnetic field mapping procedure, we considered
two different approaches; the classic ZDI reconstruction, in oftheamountofdetailrecoveredineachcase.Thesedifferences
arise as a consequence of the constraints imposed for complet-
which each component of the magnetic field vector is decom-
ing the SH-ZDI maps, which are all pushed to symmetric field
posedinaseriesofindependentmagnetic-imagepixels(Brown
distributions. In general, the spatial resolution of the SH-ZDI
etal.1991;Donati&Brown1997),andthesphericalharmonics
maps depend on the maximum order of the spherical harmon-
decomposition(SH-ZDI)wherethefieldisdescribedbythesum
ics expansion (l ). For each case this is selected in such a
of a potential and a toroidal component, and each component max
way that the lowest possible l value is used, while achiev-
is expanded in a spherical-harmonics basis (see Hussain et al. max
ing a similar goodness-of-fit level (reduced χ2) as the classic
2001;Donatietal.2006).Bothproceduresareequivalent,lead-
ZDI reconstruction (HD 1237: l = 5, HD 22049: l = 6,
ing to very similar field distributions and associated fits to the max max
HD147513:l = 4).Highervaluesofl wouldnotalterthe
max max
†Therefore,forHD22049((cid:15)Eridani)weonlyconsidertheJanuary large-scale distribution, but introduces further small-scale field
2010dataset(seePiskunovetal.2011;Jeffersetal.2014). withoutsignificantlyimprovingthegoodness-of-fit.Thisstepis
Articlenumber,page2of15
Alvarado-Gómezetal.:EnvironmentaroundPlanet-HostingStars-I.CoronalStructure
Table1.Planet-hostingsystemsandtheirobservationalproperties.
StarID S.Type Teff R∗ M∗ Prot i Age Activity Mpsini a (cid:104)εBr(cid:105)
[K] [R ] [M ] [days] [◦] [Gyr] log(R(cid:48) ) log(L ) [M ] [AU] ZDI SH-ZDI
(cid:12) (cid:12) HK X
HD1237a G8V 5572 0.86 1.00 7.00 ∼50 ∼0.88 −4.38 29.02 3.37 0.49 4.65 30.77
(cid:88)
HD22049b K2V 5146 0.74 0.86 11.68 ∼45 ∼0.44 −4.47 28.22 1.55 3.39 2.32 30.66
HD147513c G5V 5930 0.98 1.07 10.00 ∼20 ∼0.45 −4.64 28.92 1.21 1.32 −† 6.21
Notes.Thevalueslistedincolumns1−12aretakenfrompreviousstudiesofeachsystemandreferencestherein:(a)Naefetal.(2001);Alvarado-
Gómezetal.(2015)(b)Drake&Smith(1993);Hatzesetal.(2000);Benedictetal.(2006);Jeffersetal.(2014)and(c)Mayoretal.(2004);Hussain
etal.(2016).Thelasttwocolumnscontainthe(radial)magneticenergydensity,ε = B2/8π,averagedoverthevisiblesurfaceofthestar,and
Br r
estimatedfromthestandardZDIandtheSphericalHarmonicsimplementation(SH-ZDI).
(†)Duetothelowinclinationandsimplefieldgeometry,thestandardZDIreconstructionwasnotpossibleinthiscase(seeBrownetal.1991).
Fig.2.SurfaceradialmagneticfieldmapsofHD22049.Seecaptionof Fig. 4. Surface radial magnetic field maps of the Sun during activity
Fig.1.Thestellarinclinationangle(i = 45◦)isusedforthevisualiza- minimum(CR1922,top)andmaximum(CR1962,bottom)takenby
tions. SOHO/MDI. Note the difference in the magnetic field range for each
case.Aninclinationanglei=90◦isusedforthevisualizations.
To evaluate our numerical results, we have performed two
additionalsimulationstakingtheSunasreference.Themagnetic
field distributions during solar minimum (Carrington rotation
1922,endofcycle22),andsolarmaximum(Carringtonrotation
1962, during cycle 23) have been considered for this purpose.
The large-scale magnetic field is taken from synoptic magne-
tograms,generatedbytheMichelsonDopplerImagerinstrument
(MDI,Scherreretal.1995)onboardtheSolarandHeliospheric
Observatoryspacecraft(SOHO,Domingoetal.1995).Figure4
Fig. 3. Surface radial magnetic field maps of HD 147513 using SH- showsthecomparisonbetweentheglobalmagneticfielddistri-
ZDI. Two rotational phases (Φ) are presented. The stellar inclination butionfortheseactivityepochs.Duringactivityminimum,weak
angle(i=20◦)isusedforthevisualizations. magnetic regions (a few Gauss) tend to be sparsely distributed
acrosstheentiresolarsurface(nopreferentiallocationforthese
regionsisobserved).Strongersmall-scalemagneticfields,upto
particularly important for a consistent comparison, as the final twoordersofmagnitude,canbefoundduringactivitymaximum.
recovered field strengths depend on this. All these differences Inthiscasethedominantfieldsarehighlyconcentratedinbipo-
have a significant impact in the coronal and wind structure, as larsectors(activeregions)andlocatedmainlyintwolatitudinal
they depend on the field coverage and the amount of magnetic beltsat∼±30◦.Still,weakermagneticfieldscanbefoundalong
energy available in each case (see Table 1). In the case of HD theentiresolarsurface.
147513thestandardZDIreconstructionwasnotpossible,given Finally,asisshowninFigs.1to4,thenumericalgridforall
its low inclination angle (i ∼ 20◦) and fairly simple large-scale theinputsurfacemagneticfielddistributionsisthesame.There-
topology. Therefore, for this system we only consider the SH- fore, the resolution of the solar coronal models was adapted to
ZDI map presented in Fig. 3, previously published by Hussain match the optimal resolution of the stellar simulations. In this
etal.(2016). way, a more consistent comparison of the results can be per-
Articlenumber,page3of15
A&Aproofs:manuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2
formed. The surface grid resolution (∼10−2 R ) is sufficient to andmagneticfield,B,canbeextracted.Wepresentthesimula-
∗
resolve entirely the magnetic structures on the stellar ZDI/SH- tionresultsinthefollowingsection.
ZDI maps. However, in the solar case the internal structure of
theactiveregionsandthesmall-scalestructuresarenotresolved.
Theimpactofthislimitedresolutioninmagneticfieldmapsfor 4. Results
solar simulations has been investigated previously by Garraffo
Weperformadetailedevaluationofthesolutionsetsfortheso-
etal.(2013).Theyfoundthatthestructureofthestellarwindis
lar minimum and maximum cases in Sect. 4.1. Sections 4.2 to
less sensitive to this factor than the coronal structure and asso-
4.4containthesimulationresultsofthecoronalstructureforthe
ciatedemission(e.g.EUVandX-rays).Thiswillbeexploredin
starsconsidered.Ineachcasewepresentthedistributionofthe
moredetailintheevaluationprocedure,presentedinSect.4.1.
thermodynamicconditions(n,T),aswellasthemagneticenergy
density(ε ),associatedwiththeradialfield.Acommoncolour
Br
scaleisadoptedforallstarstofacilitatecomparison‡.
3. 3DMHDNumericalSimulation
In addition, synthetic coronal emission maps are generated
The numerical simulations presented here are performed using at SXR and EUV wavelengths. This is done by integrating the
the three-dimensional MHD code BATS-R-US (Powell et al. squareoftheplasmadensitytimestheemissivityresponsefunc-
1999) as part of the Space Weather Modeling Framework tion of a particular instrument, along the line-of-sight towards
(SWMF, Tóth et al. 2012). As discussed previously by Cohen theobserver.IntheSXRrangeweconsiderthespecificresponse
et al. (2014), the SWMF encompasses a collection of physics- function of the AlMg filter of the SXT/Yohkoh instrument, to
based models for different regimes in solar and space physics. synthesise images in the 2 to 30Å range (0.25 – 4.0 keV, red
Thesecanbeconsideredindividuallyorcanbecoupledtogether images).FortheEUVrangesensitivitytablesoftheEIT/SOHO
to provide a more realistic description of the phenomenon or instrument are used, leading to narrow-band images centred at
domain of interest. For the systems considered here, we have the Fe IX/X 171Å (blue), Fe XII 195Å (green), and Fe XV
included and coupled two overlapping domains to obtain a ro-
284Å (yellow)lines.Thecoronalemissionatthesewavelengths
bustcombinedsolution.Theresultspresentedinthispapercor-
hasbeenextensivelystudiedinthesolarcontext,servingalsoto
respondtothestellarcoronadomain(SCmodule).Afollow-up
calibratetheresultsfromtheSWMFinvariousworks(seeGar-
studywillcontainthewindandtheinnerastrospheredescription raffoetal.2013;vanderHolstetal.2014).Thisprocedurealso
(IHmodule).Thesolutionforeachdomainisobtainedusingthe
allowsthedirectcomparisonofthesyntheticimages,generated
mostup-to-dateversionoftheSWMFmodules†. fordifferentstars.ForHD1237andHD22049weadditionally
Thestellarcoronadomainextendsfromthebaseofthechro- compare the results driven by the different maps of the large-
mosphere (∼1R ) up to 30R . A three-dimensional potential
∗ ∗ scalemagneticfield(Sect.2).
fieldextrapolation,abovethestellarsurface,isusedastheinitial
condition. This initial extrapolation is performed based on the
photospheric radial magnetic field of the star (e.g. ZDI maps, 4.1. EvaluationoftheSolarCase
Sect. 2). In addition to the surface magnetic field distribution,
ThesimulationresultsfortheSunarepresentedinAppendixA.
thismodulerequiresinformationaboutthechromosphericbase
The synthetic images provide a fairly good match to the solar
density, n , and temperature, T , as well as the stellar mass,
0 0 observations obtained during 1977-May-07 (activity minimum,
M ,radius,R androtationperiod,P .Thisdiffersfromprevi-
∗ ∗ rot Fig.A.1)and2000-May-10(activitymaximum,Fig.A.2)§.The
ous ZDI-driven numerical studies, where these thermodynamic steady-statesolutionproperlyrecoversthestructuraldifferences
boundaryconditionsaresettocoronalvaluesandtherefore,not
forbothactivitystates.Anopen-fielddominatedcoronaappears
self-consistentlyobtainedinthesimulations(Cohenetal.2011;
in the solar minimum case, displaying coronal holes near the
Vidottoetal.2012,2015).
polarregionsoftheSun.Inturn,thesolarmaximumcaseshows
Forthestarsconsideredhere,weassumedsolarvaluesforthe
mainlyclose-fieldregionsacrossthesolardisk,withalmostno
chromosphericbasedensity(n =2.0 × 1016m−3),andtemper-
0 openfield-linelocations.Thiswillhavevariousimplicationfor
ature(T =5.0×104K).Thisisjustifiedfromthefactthatthese
0 the associated solar wind structure, which will be discussed in
systems,whilemoreactivethantheSun,arestillwithintheX-
thesecondpaperofthisstudy.
rayun-saturatedregimeandtherefore,thephysicalassumptions In general, the differences in the magnetic activ-
behind the coronal structure and the solar wind acceleration in ity/complexityareclearlyvisibleinthesteady-statesolution.As
the model are more likely to hold. This assumption permits a
expected, the thermodynamic structure of the corona, and the
consistentcomparisonwiththesolarcaseandbetweenthesys-
associated high-energy emission, show large variation in both
tems considered. The remaining initial required parameters for
activity states. To evaluate the simulation results, we need to
eachstararelistedinTable1.Forthesolarrunsweusetheside-
quantitatively compare the numerical solutions for the Sun to
realrotationrateof25.38days(Carringtonrotation). therealobservations(i.e.basedontheSXR/EUVdata).Aswas
Weuseanon-uniformsphericalgrid,dynamicallyrefinedat
discussed in Sect. 2, this is particularly important as the solar
thelocationsofmagneticfieldinversion,whichprovidesamax-
simulations presented here have been performed with limited
imumresolutionof∼10−3R∗.Thenumericalsimulationevolves spatial resolution (see also Garraffo et al. 2013). To do this,
until a steady-state solution is achieved. Coronal heating and we contrast the simulation results to archival Yohkoh/SXT and
stellarwindaccelerationduetoAlfvénwaveturbulencedissipa- SOHO/EIT data∗ covering both activity epochs (Carrington
tionareself-consistentlycalculated,takingintoaccountelectron
rotations1922and1962).
heatconductionandradiativecoolingeffects.Forfurtherdetails
thereaderisreferredtoSokolovetal.(2013)andvanderHolst ‡Exceptinthemagneticenergydensitydistributionforsolarmini-
etal.(2014).Fromthisfinalsolution,allthephysicalproperties, mumcase(Fig.A.1),wheretherangeisdecreasedbyafactorof10.
such as number density, n, plasma temperature, T, velocity, u §For a quick-look comparison with the observations from various
instrumentsduringthesedates,visithttp://helioviewer.org/.
†Codeversion2.4 ∗AvailableattheVirtualSolarObservatory(VSO)
Articlenumber,page4of15
Alvarado-Gómezetal.:EnvironmentaroundPlanet-HostingStars-I.CoronalStructure
Table2.EvaluationresultsfortheEUVrange.Thelistedvaluescorrespondtoaveragesoveranentirerotation,obtainedfromtheobservations
(Obs)andthesimulations(Sim).Thetwofilterwavelength-ratio(inÅ)usedfortheparametersestimationareindicatedineachcase.
Parameter Min(Obs) Min(Sim) Max(Obs) Max(Sim)
195/171 284/195 195/171 284/195 195/171 284/195 195/171 284/195
(cid:104)T(cid:105)[×106K] 1.06 1.77 1.14 1.63 1.13 1.79 1.16 1.66
(cid:104)EM(cid:105)[×1026cm−5] 4.51 5.08 1.19 0.98 8.42 14.9 5.52 5.06
FortheSXRrange,weusethedailyaveragesforthesolarirra- the observations (Obs). A similar level of agreement (with re-
dianceat1AU,describedinActonetal.(1999),andcomputea versedsign)isachievedforthehottercomponentofthecorona
meanvalueforeachCarringtonrotation.Thisleadsto1.02×10−5 (284/195ratio).Furthermore,thesimulatedSXRemissionprop-
Wm−2forsolarminimum,and1.21×10−4Wm−2forsolarmax- erly recovers the nominal estimates for both activity periods,
imum,inthe2–30Å range.IntermsofSXRluminosities,these withresultingvalueslyingbetweentheobservationalestimates
values correspond to 2.86×1025 ergs s−1 and 3.42×1026 ergs ofActonetal.(1999)andJudgeetal.(2003).Inasimilarman-
s−1,respectively.However,morerecentestimates,presentedby ner, the fiducial EUV luminosities during minimum and maxi-
Judgeetal.(2003),leadtolargervaluesintheSXRluminosities mum of activity are well recovered. However, we should note
during the solar activity cycle (i.e. 1026.8 ergs s−1 during activ- here that He II 304Å line tends to dominate the GOES-13 B
ity minimum, and 1027.9 ergs s−1 for activity maximum). From bandpass.Thislineisoverlystrongcomparedwithexpectations
the steady-state solutions, we simulate the coronal emission in based on collisional excitation (e.g. Jordan 1975; Pietarila &
the SXR band with the aid of the Emission Measure distribu- Judge 2004), and therefore our model spectrum is expected to
tion EM(T) (Sect. 5.1), and following the procedure described significantlyunder-predicttheobservedflux.Thatweobtainrea-
inSect.5.2.Thisyieldssimulatedvaluesof2.79×1026ergss−1 sonablygoodagreementislikelyaresultofouremissionmea-
and2.49×1027ergss−1duringactivityminimumandmaximum, suredistributionbeingtoohighattransitionregiontemperatures
respectively. (seeSec.5.1,Fig.11).Incontrast,largerdiscrepanciesarefound
A similar procedure is applied for the EUV range. Images for the EM distribution (over the sensitivity range of the EIT
acquiredbytheEITinstrumentduringbothactivityperiods,are filters)forbothcoronalcomponents.Duringactivityminimum,
used for this purpose. We consider 3 full-disk images per day differences up to factors of −3.8 and −5.2 appear for the low-
(oneforeachEUVchannel,excludingthe304Åbandpass),for andhigh-temperaturecorona,respectively.Slightlysmallerdif-
atotalof87imagesperrotation.Aftertheimageprocessing,we ference factors prevail during activity maximum for both com-
performedtemperatureand EM diagnostics,usingthestandard ponents,reaching−1.5and−3.0respectively.
SolarSoftWare(SSW)routinesforthisspecificinstrument†.This Someofthesediscrepanciescanbeattributedtoassumptions
leadstoaroughestimateofbothparameters,givenapair(ratios) ofthemodeloritsintrinsiclimitations(seevanderHolstetal.
ofEUVimages.Weusethetemperature-sensitivelineratiosof 2014).Inthiscase,asdiscussedpreviouslyinSect.2,theyarise
Fe XII 195Å/ Fe IX/X 171Å and Fe XV 284Å/ Fe XII 195Å mostlyduetothespatialresolutionofthesurfacefielddistribu-
(for a combined sensitivity range of 0.9 MK < T < 2.2 MK). tions. The overall lower densities of the corona and the imbal-
ThereaderisreferredtoMosesetal.(1997)forfurtherinforma- ance of emission at different coronal temperatures, are directly
tion.AswiththeSXRrange,wecomputemeanobservedvalues related with the amount of confining loops and therefore, with
of these parameters for both rotations, and compare them with the missing (un-resolved) surface magnetic field and its com-
simulated quantities, derived from the synthetic EUV emission plexity. In addition, as will be presented in the Sect. 5.2, the
maps.TheobtainedvaluesarepresentedintheTable2. simulated stellar X-ray and EUV luminosities appear underes-
We also compared the synthetic EUV emission to archival timated. This may indicate that some adjustments are required
data from the GOES-13/EUVS instrument‡. These measure- in the coronal heating mechanism, when applying this particu-
mentsspandifferentsolaractivityperiodsincomparisontothe lar model to resolution-limited surface field distributions (e.g.
epochs considered in the simulations (CR 1922 and CR 1962). ZDI data). Further systematic work will be performed in this
Therefore, we interpret these quantities as nominal values for direction, analogous to the numerical grid presented in Cohen
the EUV variation during minimum and maximum of activity. & Drake (2014), including also other coronal emission ranges
WeconsiderGOES-13datafromchannelsA(50−150Å)andB covered by current solar instrumentation (e.g. Solar Dynamics
(250−340Å),leadingtoaverageEUVluminosities,foractivity Observatory,Pesnelletal.2012).
minimumandmaximum,of∼2−5×1027ergs−1and∼1×1028
erg s−1, respectively. The simulated coronal emission, synthe-
4.2. HD1237(GJ3021)
sisedinthesamewavelengthranges,providesverygoodagree-
ment to the observations, leading to ∼1.4×1027 erg s−1 during The coronal structure obtained for HD 1237 shows a relatively
solarminimum,and∼1.3×1028ergs−1atsolarmaximum. simpletopology.Twomainmagneticenergyconcentrations,as-
Theresultsfromtheevaluationprocedureareconsistentbe- sociated with the field distributions shown in Fig. 1, dominate
tween the EUV and SXR ranges, showing a reasonable match thephysicalpropertiesandthespatialconfigurationinthefinal
between the simulations and the overall structure of the solar steady-state solution. The outer parts of these regions serve as
coronaforbothactivityperiods.Goodagreementisobtainedfor foot-pointsforcoronalloopsofdifferentlength-scales.Closeto
thelow-temperatureregion(195/171ratio),withdifferencesbe- thenorthpoleanarcadeisformed,whichcoversoneofthemain
low+8%inthemeantemperatureforbothepochs.Thesignin- polarityinversionlinesofthelarge-scalemagneticfield.Ascan
dicatestherelativedifferencebetweenthesimulation(Sim)and beseeninFigs.5and6,denserandcoldermaterialappearsnear
theselinesonthesurface,resemblingsolarprominencesorfila-
†MoreinformationcanbefoundintheEITuserguide ments.Largerloopsextendinghigherinthecorona,connectthe
‡Seehttp://www.ngdc.noaa.gov/stp/satellite/goes/. oppositeendsofbothmagneticregions.
Articlenumber,page5of15
A&Aproofs:manuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2
Fig.5.SimulationresultsforthecoronalstructureofHD1237drivenbytheZDIlarge-scalemagneticfieldmap.Theupperpanelscontainthe
distributionofthemagneticenergydensity(ε ,left),thenumberdensity(n,middle)andtemperature(T,right).Forthelasttwoquantitiesthe
Br
distributionovertheequatorialplane(z=0)ispresented.Thesphererepresentsthestellarsurfaceandselectedthree-dimensionalmagneticfield
linesareshowninwhite.ThelowerimagescorrespondtosyntheticcoronalemissionmapsinEUV(blue/171Å,green/195Å andyellow/
284Å)andSXR(red/2–30Å).Theperspectiveandcolourscalesarepreservedinallpanels,withaninclinationangleofi=50◦.
Fig. 6. Simulation results for the coronal structure of HD 1237 driven by the SH-ZDI large-scale field map. See caption of Fig. 5. The three-
dimensionalmagneticfieldlinesarecalculatedinthesamespatiallocationsasinthesolutionpresentedinFig.5.
Articlenumber,page6of15
Alvarado-Gómezetal.:EnvironmentaroundPlanet-HostingStars-I.CoronalStructure
Fig.7.SimulationresultsforthecoronalstructureofHD22049drivenbytheZDIlarge-scalemagneticfieldmap.Theupperpanelscontainthe
distributionofthemagneticenergydensity(ε ,left),thenumberdensity(n,middle)andtemperature(T,right).Forthelasttwoquantitiesthe
Br
distributionovertheequatorialplane(z=0)ispresented.Thesphererepresentsthestellarsurfaceandselectedthree-dimensionalmagneticfield
linesareshowninwhite.ThelowerimagescorrespondtosyntheticcoronalemissionmapsinEUV(blue/171Å,green/195Å andyellow/
284Å)andSXR(red/2–30Å).Theperspectiveandcolourscalesarepreservedinallpanels,withaninclinationangleofi=45◦.
Fig.8.SimulationresultsforthecoronalstructureofHD22049drivenbytheSH-ZDIlarge-scalefieldmap.SeecaptionofFig.7.Thethree-
dimensionalmagneticfieldlinesarecalculatedinthesamespatiallocationsasinthesolutionpresentedinFig.7
Articlenumber,page7of15
A&Aproofs:manuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2
Theseloopsconfinecoronalmaterialviamagneticmirroring,in- Alargefilamentcrossingtheentirediskisvisibleinbothsolu-
creasingthelocaldensityandtemperatureoftheplasma.Some tions,beingmoresmoothintheSH-ZDIasisexpectedfromthe
ofthisheatedplasmaisvisibleinthesyntheticemissionimages underlyingfielddistribution.
ofthelowercorona(lowerpanelsofFigs.5and6). ThecomparisonbetweentheZDIandSH-ZDIsolutionleads
Inside the two large magnetic energy regions, the coronal to similar results as for HD 1237. The variation in the average
fieldlinesaremainlyopen.Thisleadstothegenerationofcoro- coronal density, temperature and magnetic field strength reach
nal holes, where the material follows the field lines and leaves factors of 2.5, 1.6 and 4.3, respectively (see Table 3). The dif-
thestar.Inturn,thisdecreasesthelocalplasmadensityandtem- ferencesinthesyntheticemissionmapsarealsosomewhatpre-
peratureinbothregions,makingthemappeardarkinthecoronal served with respect to the HD 1237 simulations; Less coronal
emissionmaps.Thesecoronalholeswillhaveastronginfluence featuresarevisibleinEUVchannelsoftheSH-ZDIsolution,and
inthestructureofthestellarwindandtheinnerastrosphere.This theSXRemissionisdominatedbytheclosedfieldregions,dis-
willbediscussedindetailinthesecondpaperofthisstudy. tributedinthiscaseinvariouslocationsofthethree-dimensional
Intermsofthefielddistribution(i.e.ZDI/SH-ZDI,Sect.2), structure.
theglobalstructureofthecoronaofHD1237issimilarinboth Finally,itisinterestingtonoteherethesimilaritiesbetween
cases. This was expected since the largest features, in the sur- the quantitative average properties of the ZDI solution of HD
face field distributions, are common in both procedures. How- 22049andthesolarmaximumcase.Theresultingmeantemper-
ever,ascanbeseendirectlyinFigs.5and6,severalqualitative atures and field strengths are commensurate among these sim-
andquantitativedifferencesappearinvariousaspectsofthere- ulations. However, large differences are evident in the qualita-
sultingcoronalstructure.First,despitehavingthesamethermo- tiveaspectsofbothsolutions(seeFigs.7andA.2).Nocoronal
dynamicbaseconditions,theSH-ZDIsolutionleadstoalarger holes are obtained for the solar maximum case, and the high-
coronawithanenhancedhigh-energyemission.Thisisaconse- energy emission is highly concentrated from small portions of
quence of the available magnetic energy to heat the plasma, in thecorona(associatedwithactiveregions).Thisagaincanbeun-
combination with the size of the coronal loops (and therefore, derstoodintermsoftheamountofmagneticstructuresresolved
theamountofmaterialtrappedbythefield). inthesurfacefielddistribution.Despitethedegradedresolution
Toquantifythesedifferences,weestimatedtheaverageden- for the solar case, the number of bipolar regions on the surface
(sustainingdensecoronalloops)ismuchlargerthaninthelarge-
sity, temperature and magnitude of the coronal magnetic field,
scale field maps recovered with ZDI. Instead, the ZDI coronal
inside a spherical shell enclosing the region between 1.05 and
solution for HD 22049 is much more similar to the solar min-
1.50 R . This range captures the bulk of the inner corona, with
∗
imum case (Fig. A.1). This clearly exemplifies the importance
thelowerlimitselectedtoavoidpossiblenumericalerrorsinthe
ofcombiningquantitativedescriptions,togetherwithqualitative
average integration (due to the proximity with the boundary of
spatially-resolvedinformationforarobustcomparison.
thesimulationdomain).Theintegratedvaluesobtainedforeach
parameter,andfortheotherstars,arelistedintheTable3.
For HD 1237 we obtain differences by a factor of ∼1.4 in 4.4. HD147513(HR6094)
temperature,∼2.1indensityand∼3.5inmagneticfieldstrength,
We present the steady-state coronal solution for HD 147513 in
amongbothcases.AsthecoronaishotteranddenserintheSH-
Fig.9.Aswasmentionedearlier,weonlyconsidertheSH-ZDI
ZDI case, the resulting high-energy emission is almost feature-
lessintheEUVchannels(T ∼ 1−2MK).Inaddition,theim- fielddistributioninthiscase(seeSect.2).Thecoronalstructure
isdominatedbyarathersimpleconfigurationofpoloidalloops,
pactofthesurfacefieldcompletenessisclearintheSXRimage,
drivenbythesurfacefielddistribution(mainlyfromthedipolar
where the coronal holes are shifted to lower latitudes and the
and quadrupolar components). This generates bands of trapped
emissioncomesfrombothhemispheresofthestar(incontrastto
material,separatedbythemagneticpolarityinversionlinesand
thesimulatedemissioninthisrangefortheZDIcase).
distributedatdifferentlatitudes.Fewopenfieldregionsarevis-
As expected, HD 1237 shows enhanced coronal conditions
ible in the coronal structure, which are again located inside the
compared to the Sun, especially for the SH-ZDI case (see Ta-
largestmagneticenergyconcentrations.Oneoftheseregionsap-
ble 3). For the ZDI case the mean coronal density appears to
pearsinthenorthpoleofthestar,whichsuffersasmalldistortion
be lower than the Solar maximum value (by ∼25%). This may
in the EUV images due to a numerical artifact of the spherical
be connected with the incompleteness of the ZDI maps (Sect.
grid.Theline-of-sightSXRemissiondisplaysaring-likestruc-
2), since a similar situation occurs for the ZDI solution of HD
ture close to the limb, corresponding to the hottest material of
22049byroughlythesameamount.
thesteady-statecorona.Somefaintemissioncanbealsoseenin-
side the stellar disk. As the estimated inclination angle for this
4.3. HD22049((cid:15) Eridani) star is small (i ∼20◦), the coronal features are visible at almost
allrotationalphases.
ThesolutionsforHD22049arepresentedinFigs.7and8.The The coronal properties listed in Table 3, show an average
coronalstructureinthiscaseishighlycomplex,withseveralhot densitycomparabletothesolarcaseinactivitymaximum.How-
anddenseloopsconnectingthedifferentpolarityregionsofthe ever, as was presented in Sect. 4.1, the limited resolution of
surfacefielddistribution.Insomelocations,thematerialisable the surface field distribution can strongly affect this parameter.
toescapenearthecuspoftheloops,resemblinghelmetstreamers GiventherelativelylowresolutionfortheSH-ZDImapforthis
in the Sun. For the SH-ZDI simulation, some of this escaping star, we expect larger discrepancies than the ones obtained for
materialisevenvisibleintheEUVsyntheticmaps(inparticular the solar case. In this sense, the average values obtained from
inthe195Åchannel–GreenimageinFig.8). thesimulationcorrespondonlytoroughestimatesoftheactual
SimilartoHD1237,twolargecoronalholesarevisibleinthe conditions of the corona. This is considered in more detail in
synthetichigh-energyemissionmaps(especiallyintheZDIsim- Sect.5.1.Still,thegeometricalconfigurationofthissystempro-
ulation).Howeverinthiscase,thecorrelationwiththestronger videsaninterestingviewofthecoronalfeatures,thatcannotbe
magnetic features in the surface is less clear as for HD 1237. easilyobtainedevenforthesolarcase.
Articlenumber,page8of15
Alvarado-Gómezetal.:EnvironmentaroundPlanet-HostingStars-I.CoronalStructure
Table3.Averagephysicalpropertiesoftheinnercorona(IC)region(from1.05to1.5R ).
∗
Parameter HD1237 HD22049 HD147513 Sun
ZDI SH-ZDI ZDI SH-ZDI SH-ZDI CR1922(Min) CR1962(Max)
(cid:104)n(cid:105) [×107cm−3] 3.66 7.74 3.38 8.30 4.80 1.80 4.78
IC
(cid:104)T(cid:105) [×106K] 2.49 3.42 2.06 3.20 2.79 1.48 2.07
IC
(cid:104)B(cid:105) [G] 4.58 16.43 3.34 14.39 5.37 0.94 2.31
IC
Fig.9.SimulationresultsforthecoronalstructureofHD147513drivenbytheSH-ZDIlarge-scalemagneticfieldmap.Theupperpanelscontain
thedistributionofthemagneticenergydensity(ε ,left),thenumberdensity(n,middle)andtemperature(T,right).Forthelasttwoquantities
Br
thedistributionovertheequatorialplane(z = 0)ispresented.Thesphererepresentsthestellarsurfaceandselectedthree-dimensionalmagnetic
fieldlinesareshowninwhite.ThelowerimagescorrespondtosyntheticcoronalemissionmapsinEUV(blue/171Å,green/195Å andyellow
/284Å)andSXR(red/2–30Å).Theperspectiveispreservedinallpanels,withaninclinationangleofi=20◦.
5. AnalysisandDiscussion Figure 10 contains the computed EM(T) for all the considered
cases. As expected, the peak values are located at logT > 6.0,
Usingthesimulationresultswecanrelatethecharacteristicsof
andmovetowardslargeremissionmeasuresandhighertemper-
thesurfacefielddistributions,withtheobtainedcoronalproper-
atures,withincreasingaverage(radial)magneticenergydensity
tiesandtheenvironmentaroundthesesystems.Wewillfocusour
(cid:104)ε (cid:105)(seeTable1).
discussiononthreemainaspects,includingthethermodynamic Br
In a similar manner to the solar case (Sect. 4.1), we com-
structure,thecoronalhigh-energyemission,andthestellarrota-
pare the simulated quantities to observational values. The ZDI
tionalmodulationofthecoronalemission.
and SH-ZDI simulations of HD 22049 yield maximum EM
values of logEM(cid:39)49.1 (at logT(cid:39)6.4) and logEM(cid:39)50.0 (at
5.1. ThermodynamicCoronalProperties logT(cid:39)6.6), respectively. The peak temperature and emission
measure of the ZDI model are significantly lower than those
From the simulated 3D structure in each star, we calculate the
derived from both EUV and X-ray spectra (logEM(cid:39)50.7 at
emissionmeasuredistribution,EM(T),definedby
logT(cid:39)6.6±0.05, Drake et al. 2000; Sanz-Forcada et al. 2004;
Ness & Jordan 2008). The SH-ZDI emission measure fares
(cid:90)
somewhatbetter,withgoodagreementintermsofthepeaktem-
EM(T)= n2(T)dV(T), (1)
perature.However,thismodelstillpredictsanemissionmeasure
V(T) significantlylowerthanobserved,byroughlyafactorof5.
wheren(T)istheplasmadensityatthetemperatureT,theinte- ForHD147513availableobservations,fromthebroad-band
grationonlyincludesthevolumeofthegridcellsatthatpartic- filters of the Extreme Ultraviolet Explorer (EUVE) Deep Sur-
ulartemperature,andthevolumecoversalltheclosedfieldline veytelescope,onlyprovideroughestimatesofthecoronalcon-
regionsinthesteady-statesolutions.Weusetemperaturebinsof ditions, suggesting a probable emission measure in the range
0.1inlogT startingfromthebasetemperature(i.e.logT (cid:39)4.9), logEM ∼ 51–52 (Vedder et al. 1993) but with no discrimina-
up to the maximum temperature achieved in each simulation. tion on the temperature. In turn, the peak of the simulated dis-
Articlenumber,page9of15
A&Aproofs:manuscriptno.Alvarado-Gomez_et_al_2016A_low-res_V2
Fig.10.EmissionmeasuredistributionsEM(T)calculatedfromthe3D
steady-state solutions. Each colour corresponds to one of the simula-
tionspresentedinSect.4,includingthesolarruns(AppendixA).
tribution is located at logT(cid:39)6.5, with an associated value of
logEM(cid:39)49.7. The discrepancy in emission measure might be
expected given the relatively low spatial resolution of the SH-
Fig. 11. Simulated high-energy coronal emission vs. unsigned radial
ZDImapdrivingthesimulation(Sections2and4.4).Thepeak magneticflux(cid:104)Φ (cid:105).Eachpointcorrespondstooneofthesimulations
Br s
temperature is also slightly lower than what might be expected describedinSect.4,includingthesolarcasesasindicated.Thesevalues
based on the emission measure distribution and the observed arecalculatedfromsyntheticspectra,basedontheEM(T)distributions
peaktemperatureofHD22049. (Sect.5.1),andintegratedintheSXR(2−30Å,green),X-ray(5−100Å,
In the case of HD 1237, there are no observational con- red)andEUV(100−920Å,blue)bands.Thesolidlinescorrespondto
straintsintheliteratureregardingtheEMdistribution.Fromthe fitstothesimulateddatapoints.Thedashedlinesarebasedonobserva-
numericalsimulations,weobtainpeakvaluesoflogEM(cid:39)49.3 tionalstudiesusingX-ray,againstmagneticfieldmeasurementsusing
atlogT∼6.5fortheZDIcase,andlogEM(cid:39)50.2atlogT∼6.7 ZB(Pevtsovetal.2003)andZDI(Vidottoetal.2014).
fortheSH-ZDIcase.
Inallthestellarcases,thesimulatedEM distributionsshow
respect to the large-scale magnetic field flux (recovered with
maximaclosetotheexpectedvaluesforstarswithintheconsid-
ZDI).Theyalsofoundapower-lawrelationforbothparameters
eredlevelsofactivity(seeTable1).However,theemissionmea- (L ∝ Φ1.80±0.20). These observational results have been inter-
suresaresystematicallylowerthanindicatedbyobservation. X B
preted as an indication of a similar coronal heating mechanism
ThebehaviourofthesimulatedEMdistributionforthesolar
amongthesetypesofstars.
maximum case (red line in Fig. 10) is particularly interesting,
compared with the remaining simulations. Both the peak emis- In this context, we have considered this relation from a nu-
sion measure and temperature are in good agreement with as- merical point of view, by simulating the coronal high-energy
sessments from full solar disk observations (e.g. Laming et al. emission (based on the EM(T) distributions presented in the
1995; Drake et al. 2000). However, the observations indicate previous section) and comparing the predicted fluxes with the
a slope in the EM vs. temperature of order unity or greater, underlying surface magnetic field flux distributions (ΦB =
whereas the model prediction is much flatter. This results in a 4π|Br|R2∗). In this analysis, we have included the results from
substantial over-prediction of the cooler emission measure at all the considered cases (e.g. solar and ZDI/SH-ZDI), treating
temperatures logT ≤ 6 compared with observations. The so- the solutions independently. This allows us to explore a broad
lar minimum EM distribution (yellow line in Fig. 10) is more rangeforbothparameters,whilemaintainingtheconsiderations
similartothestellarcasesinthisregard.Thesedifferenceshave and limitations of the data-driven numerical approach. In prin-
aconsiderableimpactinthepredictedcoronalemission,asdis- ciple, this can be also studied from a more generic numerical
cussedinthenextsection. point of view (i.e. including different simulated field distribu-
tions). However, this would require implicit assumptions about
thefieldstrengthandspatialconfiguration(mostlyinfluencedby
5.2. High-EnergyEmissionandMagneticFlux thesolarcase),introducingstrongbiasesintheanalysis.There-
foreconsideringthedifferentrecoveredfieldmaps(e.g.ZDI/SH-
An observational study performed by Pevtsov et al. (2003)
ZDI) as independent observations, represents a reasonable ap-
showedarelationbetweentheunsignedmagneticfieldflux,Φ ,
B proximation.
and the X-ray emission, L , covering several orders of mag-
X
nitude in both quantities (L ∝ Φ1.13±0.05). The analysis in- Spectraweresimulatedforeachoftheemissionmeasuredis-
X B
cludedvariousmagneticfeaturesoftheSun,togetherwithZee- tributions,EM(T),overtheX-rayandEUVwavelengthregimes,
man Broadening (ZB) measurements of active dwarfs (spec- from1to1100Åona0.1Ågrid,coveringallthebandpassesof
tral types F, G and K), and pre-main sequence stars (see Saar interest to this work. Emissivities were computed using atomic
1996). More recently, Vidotto et al. (2014) investigated the be- datafromtheCHIANTIdatabaseversion7.1.4(Dereetal.1997;
haviour of various astrophysical quantities, including L , with Landi et al. 2013) as implemented in the Package for INTer-
X
Articlenumber,page10of15