Unified line profiles for hydrogen perturbed by collisions with protons: satellites and asymmetries

2.2 MB·English

by  Ingrid Pelisoli

Mon.Not.R.Astron.Soc.000,1–13(2015) Printed23January2015 (MNLATEXstylefilev2.2)
Unified line profiles for hydrogen perturbed by collisions
with protons: satellites and asymmetries
5
1 Ingrid Pelisoli1⋆, M. G. Santos2 and S.O. Kepler1
0
1Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, 91501-900 Porto Alegre, RS, Brazil
2
2Campus Litoral Norte, Universidade Federal do Rio Grande do Sul, 90040-060 Porto Alegre, RS, Brazil

n
a
J
Accepted 2015January22.Received2015January21;inoriginalform2014November19

2
2
ABSTRACT
]
R
We present new calculations of unified line profiles for hydrogen perturbed by
S
collisions with protons. We report on new calculations of the potential energies and
.
h dipolemomentswhichallowtheevaluationofprofilesforthelinesoftheLymanseries
p up to Lyman δ and the Balmer series up to Balmer 10. Unified calculations only
- existed for the lines Lyman α to Lyman γ and Balmer α including the H+ quasi-
o 2
molecule. These data are available as online material accompanying this paper and
r
t shouldbe includedinatmospheremodels,inplaceofthe Starkeffectofprotons,since
s
a thequasi-molecularcontributionscausenotonlysatellites,butlargeasymmetriesthat
[ areunaccountedfor inmodels thatassume Starkbroadeningofelectronsandprotons

are equal.
1
v Key words: stars: white dwarfs – line: profiles – atomic data.
9
0
6
5
0 1 INTRODUCTION been included in atmosphere models, offering a good diag-
1. nostics of stellar parameters (e.g. Allard & Koester 1992;
Collisional broadening occurs whenever a radiating parti-
0 Allard, Kielkopf & Loeillet 2004). With the improvement
cle interacts with a perturber during the process of emis-
5 of observational techniques and increase in data qual-
sion or absorption of radiation. This interaction is known
1 ity, satellites in other spectral lines have also been iden-
to cause not only a broadening of the line profile, but also
: tified. Koester et al. (1996) reported two strong absorp-
v thepossibility of appearance of satellite-lines. Satellites are
tions in the red wing of Lyman β, at 1060 and 1078 ˚A,
i explained by the formation, during the collision, of a tran-
X in the spectra of the DA white dwarf Wolf 1346 ob-
sient bond state which has energy levels other than those
r tained with the Hopkins Ultraviolet Telescope (HUT). In
a ofthenon-interactingparticle,producingaspectrumunlike the same paper they already presented a model for the
theobtained for the isolated particle.
absorptions assuming a constant dipole moment for the
Hydrogen satellite-lines were already observed in some
transitions.Allard, Kielkopf & Feautrier(1998)approached
linesoftheLymanseriesinstars,andintheLymanandthe
the problem again, now considering the true variation of
Balmer series in laboratory plasmas. In stars, these detec-
the dipole moment. In Lyman γ, Wolff et al. (2001) iden-
tionsaremainlyinspectraofwhitedwarfstars,butarealso
tified a satellite with wavelength around 990 ˚A in the
reported in A-type Horizontal Branch stars (which may be
Far Ultraviolet Spectroscopic Explorer (FUSE) spectrum
extremelylowmasswhitedwarfs,ELMs)andλBootisstars of the star CD –38◦ 10980. Models were evaluated by
(Allard et al. 2000). The first detections were two satellites
Allard et al.(2004),assumingconstantdipolemoments,and
intheredwingofLymanα,closeto1400(Greenstein1980;
Allard, Noselidze & Kruk (2009), considering their depen-
Wegner 1982) and 1600 ˚A (Holm et al. 1985), of DA-type
dencewith distance.
white dwarfs with effective temperatures around 15000 K.
Astheycould notbeexplained bytransitions ofthehydro- In laboratory plasmas, experiments focused on quasi-
gen atom, which are theonly lines expected to be observed molecular contributions were done for Lyman α and the
in DA white dwarfs, they were explained by Koester et al. Balmer series. Lyman α was studied by Kielkopf & Allard
(1985) and Nelan & Wegner(1985) asquasi-molecular con- (1995)usinglaser-producedplasmainpurehydrogen.They
tributions dueto H+ and H , respectively. detected five satellites: the already known features at 1400
2 2
Since then, the effect of said two contributions has and 1600 ˚A and three others closer to the line centre, at
1234, 1172 and 1268 ˚A, explained by their models as con-
tributions due to the quasi-molecules H+ (1234 ˚A) and H
2 2
⋆ E-mail:[email protected] (1172 and 1268 ˚A), but not detected in white dwarfs spec-
2 Ingrid Pelisoli, M. G. Santos and S. O. Kepler
tra because the large pressure broadening of the spectral Thislackofaccuracyresults,forexample,inadiscrep-
lines hides them. Kielkopf, Allard & Decrette (2002) stud- ancy in the calculated mean masses of DA white dwarfs
ied Balmer α and identified six satellites in the theoretical when distinct methods are applied. Falcon et al. (2010b),
wing, of which only a stronger absorption at 8650 ˚A is ob- analysing 449 DA white dwarfs from the European South-
servedintheplasmaspectrum.ForBalmerβ andBalmerγ, ernObservatorySupernovaIaProgenitorSurvey (SPY),via
ontheotherhand,nosatelliteswereidentified,becauseonly thegravitational redshift method,obtained amean massof
the line centre experimental profiles were published so far hMDAi = 0.647+−00..001134 M⊙. A consistent value of hMDAi =
(Falcon et al. 2012, 2014) and usually there are no satel- 0.630 ± 0.028 M⊙ was obtained by Romero et al. (2012)
litesinthisregion,whichcorrespondstoradiationwhenthe with seismology of 44 ZZ Ceti white dwarfs. On the other
particles are far apart. In these papers and in Falcon et al. hand, when analysing 2217 DA stars from the Sloan Dig-
(2010a), they use x-rays generated from a z-pinch dynamic ital Sky Survey (SDSS) data release 7 with signal-to-noise
hohlraum to drive plasma formation in a macroscopic gas ratio larger than 15 and T > 13000 K [to avoid the sys-
eff
cell and generate absorption spectra for the Balmer lines, tematic super-estimative reported by Kepleret al. (2006)],
with the intention of benchmarking white dwarf stars at- Kleinman et al. (2013) found a mean mass of hMDAi =
mosphere models, though the published data are scarce to 0.593±0.002 M⊙, which is significantly smaller than the
date. values obtained by Falcon et al. (2010b) and Romero et al.
Accuracyinmodelatmospheresforwhitedwarfstarsis (2012). Applying corrections to 3D convection, as calcu-
essentialforagood determinationoftheirphysicalparame- lated by Tremblay et al. (2013), Kepler et al. (2015) ob-
ters,especially effectivetemperatureandsurfacegravity,as tained a mean mass of 0.662 ± 0.003 M⊙ in the analy-
thefitofsyntheticspectraisthemost successfulmethodto sis of 1659 DA white dwarfs from the SDSS data release
obtain those parameters (e.g. Tremblay & Bergeron 2009). 10 with signal-to-noise ratio larger than 15 and Teff >
White dwarf stars are the final evolutionary state of all 10000K,whichisavalueclosertotheresultsofFalcon et al.
starswithmasses below8–10.5 M⊙,dependingonmetallic- (2010b) and Romero et al. (2012). Hence the inconsisten-
ity (e.g. Iben,Ritossa, & Garc´ıa-Berro 1997; Smartt et al. cies between the spectroscopic method and other meth-
2008; Doherty et al. 2014), corresponding to over 97 % of ods are reduced when corrections to 3D convection are ap-
the total number of stars. They generally do not present plied, but are still without a complete solution at present
ongoing nuclear fusion and only emit their residual ther- (Genest-Beaulieu & Bergeron 2014).
mal energy, resulting in a slow cooling of the star. Because Falcon et al. (2014) noted that the inferred electronic
they have radii of the same order of the Earth’s radius, density of a plasma when fitting H γ was ∼ 40 % smaller
theyhaveasmall surfacearea, sotheircoolingrateissmall than the density obtained when fitting H β, regardless of
and therefore their cooling times are large; it takes approx- the set of models used. They suggested that this inconsis-
imately 1010 years for the effective temperature of a nor- tency would lead to lower determinations of surface grav-
mal mass white dwarf to decrease from 100000 K to near ities and, by consequence, a lower mean mass would be
5000K.Asaresult,thecoolonesareamongtheoldest ob- obtained. The probable cause for the different values ob-
jects in the Galaxy, being extremely useful in the study of tained with each line is some missing physics in current
stellar formation and evolution history of the Milky Way atmosphere models. In this work we calculate line profiles
(e.g. Winget et al. 1987; Bergeron, Saffer & Liebert 1992; for the quasi-molecular contributions of collisions between
Liebert, Bergeron & Holberg 2005; Moehler & Bono 2008; hydrogen and free protons, which is one of the effects not
Tremblay et al. 2014). yet consistently included in themodels. The potentials and
Currently the atmosphere models in use to fit white dipole moments for H+2 used here are extensions of those
dwarf stars and estimate their physical parameters only publishedbySantos & Kepler(2012).Ourresultsshowthat
take into account quasi-molecular contributions for a few the quasi-molecular effects should be included in the atmo-
lines of the Lyman series, when they consider such con- spheric models, regardless of the detection or not of satel-
tributions at all: Lyman α, Lyman β and Lyman γ (e.g. lite lines, since they cause also large asymmetries in every
Allard, Noselidze & Kruk 2009). For the other lines of the line.Astheyarenotcontemplatedinthecurrentcalculated
LymanseriesandtheBalmerseries,theinteractionbetween spectra, they are probably affecting the determinations of
hydrogen atom and free protons is approximated simply as temperatureand gravity for theanalysed data.
electron Stark effect, which causes only a broadening that
adds up to the effect of the electrons. The quasi-molecular
effectsarethusneglected,disregardingthattheyarerespon-
2 THEORETICAL BASIS
sible not only for a number of known satellites but also for
strong asymmetries between the red and blue wings of all There are basically four possible approaches for the calcu-
lines.ThefarredwingofLymanα(λ>2500˚A),forexam- lation of collisional profiles: the impact theory, the quasi-
ple,turnedouttobethemissingopacityinthebluepartof static approximation (statistical theory), the unified the-
the optical spectrum of cool white dwarfs (T < 6000 K) ory and the quantum method. The impact theory (e.g.
eff
of pure hydrogen composition. The inclusion of the broad- Lindholm 1945) assumes that the duration ∆t of a colli-
eningbycollisionswithH wasthekeyelementinobtaining sion is much smaller than the interval between successive
2
such result, as shown by Kowalski & Saumon (2006). How- collisions (∼ 2π/∆ω, where ∆ω is the angular frequency
ever, contributions to other lines remain unaccounted for, measured from the line centre), being useful for determin-
despite the fact that their inclusion may be important to ing the behaviour of the line centre, but not being able to
obtainaccuratevaluesforeffectivetemperatureandsurface accountforthesatellites.Ontheotherextreme,thereisthe
gravity. quasi-static approximation, which assumes ∆t ≫ 2π/∆ω,
Unified line profiles for H perturbed by protons 3
beingthenagood methodforobtainingthelinewings(e.g.
RAonhdremrsaonn, A(1l9t5h2a)ussh&owKeedpletrha2t01b1o;thSaanptopsro&acKheesplecrou2l0d12b)e. −18 Lyd Lyg Lyb Lya
obtained from the same formulation, taken the appropri-
ate limits, proposing the unified theory, which can give −20
the shape of the whole line. Allard et al. (1999) developed
such theory in a quantum mechanical approach, obtain- 2m]
c
ing from it better results for the shape and amplitude of [l−22
the satellites. The pure quantum mechanical method can sog
l
also give the whole line, but, for its implementation, one
−24
must know all of the states involved in a transition, in-
cludingrovibrational,demandinglargercomputationaltime
(Zygelman & Dalgarno 1990). We chose the unified theory −26
asproposedbyAllard et al.(1999)totheevaluationsinthis
work. 1000 1100 1200 1300 1400
l (Å)
In the unified theory, the normalized line profile mea-
suredfrom thelinecentreisgivenbytheFouriertransform Figure1.OpacityprofilesforeverycomputedlineintheLyman
of an autocorrelation function Φ(s): series for a temperature of 10000 K and perturber density of
1017 cm−3.Strongsatellitescanbeidentifiedatthewavelengths
I(∆ω)= 1Re ∞Φ(s)e−i∆ωsds. (1) of 1400, 1080, 1060, 990 and 965 ˚A. The last one has not been
π Z identifiedinobservations yet.
0
The autocorrelation function measures the influence of the
presenceof perturberparticles –in thiscase, free protons– atmospheremodelsbutespecially toallow comparison with
in the radiation of the particle – here, hydrogen. As shown newexperimentaldatalikethosepublishedbyFalcon et al.
by Allard et al. (1999), it can bewritten as: (2010a,2012, 2014).That is thescope of thepresent work.
Φ(s)=eng(s), (2)
where n is the perturber density and g(s), for a transition 3 LINE PROFILES FOR THE LYMAN SERIES
α=(i,f) between the initial state i and the final one f, is
TheLymanseriescorrespondstotransitionswhichhaven=
given by
1asthefinallevel,forabsorptionlines,orastheinitiallevel,
1 ∞ ∞
gα(s) = αe,e′|dee′|2 Xe,e′αZ−∞dx0Z0 2πbdbd˜ee′[R(0)] irnegcioanse, boeftwemeeisnsi9o1n2. I˚Ats(wLayve∞le)ngatnhds 1a2re16in˚Ath(Leyulαtr)a.vHioeleret
×P e~i R0sVee′[R(t)]dtd˜∗ee′[R(s)]−d˜ee′[R(0)] , (3) pwreofielveasluaraetesdhoLwynαin,FLiyg.β1,foLryaγtemanpderaLtyurδe,owf1h0os0e00opKacaintdy
n o
perturber density of 1017 cm−3. The behaviour is linearly
where e and e′ denote the energy subspaces containing the
extrapolatedforlog(σ ).−26.Manysatellitescanalready
states which approach, respectively, the initial and final λ
beidentified,aslistedinTable1,buteachprofiledisshown
atomic states of the transition as R → ∞, with R the dis-
in more detail in the next subsections, for clarity.
tance between radiator and perturber, dee′ the dipole mo-
ments for the transition, d˜ee′ the dipoles modulated by the
the Boltzmann factor e−βVe, x0 the perturber initial posi- 3.1 Lyman α
tion, b its impact parameter and Vee′(R)=Ve(R)−Ve′(R)
Lyman α has contributions due to six transitions of the
thepotential energy difference between states i and f.
H+ quasi-molecule. The total profile accounting for each
We adopted this approach to evaluate the line pro- 2
contribution is shown in Fig. 2 for temperatures of 10000,
files I(∆ω) for the Lyman series up to Lyman δ and
15000 and 20000 K and perturber densities of 1016, 1017
to the Balmer series up to Balmer 10. The input data
and 1018 cm−3. Four satellites can be identified, with ap-
for potentials and dipole moments were calculated by
proximate wavelengths of 980, 1100, 1240 and 1400 ˚A. The
Santos & Kepler (2012) and are available at their web
page1, except for the potential energy data used for the firsttwoareprobablytoofainttobedetectedinstellarspec-
tra,atleast atthesetemperatures.Theoneat980˚Awould
evaluation of the line centres, which were obtained with
also be blended with the Lyman γ line in stellar spectra.
an improvement of the code, as will be described in
The other two were already observed, as cited in the intro-
Section 6. Calculations via unified theory existed in the
duction,inlaboratoryplasmasandinwhitedwarfs.Wecan
literature only for Lyman α (Allard, Kielkopf & Loeillet
alsonotethat,inlargescale,thetemperaturedoesnotseem
2004), Lyman β (Allard, Kielkopf & Feautrier 1998), Ly-
toaffecttheprofiles,but,inthedetailedbox,itcanbeseen
man γ (Allard, Noselidze & Kruk 2009) and Balmer α
thatthesatelliteat∼1400 ˚Ahasaconsistentbehaviour:it
(Kielkopf, Allard & Decrette2002).Fortheotherlines,cal-
broadenswiththeincreaseintemperature,whileitspeakis
culations were done only by Santos & Kepler (2012), but
displaced to smaller wavelengths. Such behaviour might be
viathequasi-staticapproach. Evaluationscovering also the
useful as a good temperature diagnostic for plasmas. The
line centre were in order, not only to improve white dwarf
effect of increasing density is simply a linear shift (in log
scale) and a larger broadening.
1 http://astro.if.ufrgs.br/marcios/index.htm InFigs.3and4,weshowcomparisonbetweenobserved
4 Ingrid Pelisoli, M. G. Santos and S. O. Kepler
Table1.WavelengthsforthemaximumofthestrongersatellitesineachcalculatedlineintheLymanseries
(forT =10000Kandn=1017 cm−3),thetransition(s)thatoriginatesthemandnotesonobservationand
theoretical models.
Line Satellites (˚A) Transition(s) Observed? Previousmodel
Lyα 979.55 2pσu-3dπg no Allardetal.(1994)a
1118.42 2sσg-2sσg no Allardetal.(1994)a
1230.88 1sσg-4fσu yes(Kielkopf&Allard1995) Kielkopf&Allard(1995)a
1sσg-2pπu
1396.40 2pσu-3dσg yes(Koester etal.1985,e.g.) e.g.Allardetal.(1999)a
Lyβ 1035.77 1sσg-6hσu no Allard,Kielkopf&Feautrier(1998)a
1060.99 1sσg-4fπu yes(Koesteretal.1996) Allard,Kielkopf&Feautrier(1998)a
1078.34 2pσu-5gπg yes(Koesteretal.1996) Allard,Kielkopf&Feautrier(1998)a
Lyγ 978.24 1sσg-8kσub no Allard,Noselidze&Kruk(2009)a
992.05 1sσg-8kσu yes(Wolffetal.2001) Allard,Noselidze&Kruk(2009)a
996.03 2pσu-7iπg yes(Wolffetal.2001) Allard,Noselidze&Kruk(2009)a
Lyδ 953.03 1sσg-10mσu no Santos &Kepler(2012)c
2pσu-9lπg
961.61 1sσg-8kπu no Santos &Kepler(2012)c
2pσu-9lσg
a
viaunifiedtheory
b The level 8kσu is often miswritten inthe literature as 8jσu. The letter j, by convention, must not used
fororbitalrepresentation.
c
viaquasi-staticapproach
10 the dotted line uses tables provided by N. F. Allard2. The
T = 10 000K
T = 15 000K models are not fitted to the data, but simply over-plotted
T = 20 000K
considering normalization at an arbitrary wavelength and
assumingeffectivetemperatureandsurfacegravityfoundin
5
theliterature.Inbothcases,therearetwomajordifferences
betweenthemodels:thebluewingofthecentrallineandthe
) 1018cm−3 position of the satellite. Allard’s models seem to fit better
log I ( 0 11001167ccmm−−33 the central line, but the central wavelength of the satellite
l isunderestimated,beingbetterdeterminedbyourmodel.A
1.8
1.7 possible explanation for such difference is that Allard’s cal-
1.6
−5 1.5 culations use potential energy data only to small distances,
1.4 so the real asymptotic value of each level is not reached.
1.3
1.2 Asasubtractionofthecalculatedasymptoticvalue[sothat
1.1
1370 1380 1390 1400 1410 V(R ) = 0] is necessary to the numerical evaluation of
−10 max
900 1000 1100 1200 1300 1400 1500 theline,alargervaluethantherealoneissubtracted,over-
l (Å)
estimating the shift in the difference potential and making
Figure 2. Normalized profiles for the Lyman α line. In black the satellite appear closer to the line centre. Such problem
for T = 20000 K, dark grey for T = 15000 K and light grey doesnot happenin ourcalculations, sincewe evaluated the
for T = 10000 K. The dotted line shows calculations for per- potential energy datato large distances.
turber density of 1016 cm−3, the continuous for 1017 cm−3 and
the dashed for 1018 cm−3. The same labels are adopted for the
otherLymanlines.Thesmallerboxshowsindetailthebehaviour
ofthesatelliteat∼1400˚Awithchangesintemperatureforden- 3.2 Lyman β
sityof1017 cm−3.
Ten transitions contribute to the Lyman β profile. Their
combined effect gives the total profile shown in Fig. 5. Our
calculations are shown for temperatures of 10000, 15000
dataforthestarsHS0507+0434A andWD0231-054 inthe and 20000 K and perturber densities of 1016, 1017 and
region of Lyman α and calculated models using the codes 1018 cm−3. Apart from the strong satellites at 1060 and
Tlusty (Hubeny 1988) and Synspec (Hubeny& Lanz
2011).The dashed line shows themodel obtained using the
quasi-molecular contributions calculated in this work, and 2 http://mygepi.obspm.fr/∼allard/
Unified line profiles for H perturbed by protons 5
3.5 10
HS 0507+0434A T = 10 000K
This work T = 15 000K
Allard, Kielkopf and Loeillet, 2004 T = 20 000K
3 8
Å] 6
0 2.5
0
ormalized at 13 1 .25 1390 1400 1410 llog I () 24 111000111678cccmmm−−−333
) [n 0 33..68
log I ( 1 −2 33..24
l 3
0.5 2.8
−4 2.6
2.4
0 1060 1070 1080
−6
1150 1200 1250 1300 1350 1400 980 1000 1020 1040 1060 1080 1100
l (Å) l (Å)
Figure 3. HST spectrum of the white dwarf HS 0507+0434A Figure5.NormalizedprofilesfortheLymanβ line.Thesmaller
(grey) and Tlusty/Synspec models using the quasi-molecular box shows in detail the behaviour of the two stronger satellites,
contribution tables evaluated by us (dashed line) and by N. at 1060 and 1078 ˚A, with changes in temperature for density
F. Allard (dotted line), assuming temperature of 20000 K and 1017 cm−3.
log(g) = 7.75 [close to the values determined by Andrewsetal.
(2012)].Inthesmallbox,thespectrumwassmoothedtoenhance
thepositionofthesatellite. 10
T = 10 000K
T = 15 000K
T = 20 000K
8
WD 0231−054 6
5 Allard, Kielkopf and LoeiTllheits, 2w0o0r4k 1018cm−3
1017cm−3
300 Å] 4 llog I () 24 1016cm−3
d at 1 3 1390 1400 1410 44..56
malize 0 44..34
llog I () [nor 12 −−42 9 445 ..0412 990 960 995 970 980 990 1000
l (Å)
0 Figure6.NormalizedprofilesfortheLymanγ line.Thesmaller
box shows in detail the behaviour of the satellites at 990 and
1150 1200 1250 1300 1350 1400
l (Å) 995˚Awithchanges intemperaturefordensity1017 cm−3.
Figure4.HSTspectrumofthewhitedwarfWD0231-054(grey)
and Tlusty/Synspec models using the quasi-molecular contri-
the window function in the Fourier transform [as reported
bution tables evaluated by us (dashed line) and by N. F. Allard
byAllard, Sahal-Brechot & Biraud (1974)].
(dottedline),assumingtemperatureof15000Kandlog(g)=8.80
[similartothevaluesdeterminedbyGianninas,Bergeron&Ruiz
(2011)]. Again we show the smoothed spectrum in the satellite
3.3 Lyman γ
regiontoenhance thesatelliteposition,whichisbetter fitted by
ourdata. FourteentransitionsoftheH+ quasi-moleculecontributeto
2
Lyman γ. The calculated total profiles, for temperatures of
10000, 15000and20000 Kandperturberdensitiesof1016,
1078 ˚A already studied in white dwarf spectra, there is an- 1017 and 1018 cm−3, are shown in Fig. 6. The noticeable
othersatellitearound1035˚A,probablytooclosetotheline satellite features areat 979, 990 and 995 ˚A.Thefirst oneis
centre to be identified in white dwarfs, since the broaden- again to close to the line centre for detection in stars, but
ing of the line due to other mechanisms hides it. Temper- thecombinedeffectoftheothertwowasalreadydetected,as
ature is again not an important factor, result that is cur- detailedintheintroduction.Sincethissatellitehastwocom-
rent in the literature. However, again in a more detailed ponents, as already reported by Allard, Noselidze & Kruk
scale, the strong satellites do show some variation. They (2009), its behaviour with temperature shall not be very
also show small oscillations. Possible reasons for that are simple, because each component has its independent varia-
the fact that the calculations presented here assume that tion. The rest of theline shows no significant change.
the perturbers have a single constant velocity and no aver- In Figs. 7 and 8, we show FUSE observed spectra in
aging was done, causing an effect similar to diffraction, as the region of the lines Lyman β and Lyman γ compared to
mentionedbyAllard(1978)and,mainly,thecontributionof calculated models using the codes Tlusty (Hubeny 1988)
6 Ingrid Pelisoli, M. G. Santos and S. O. Kepler
70 10
Wolf 1346 T = 10 000K
This work T = 15 000K
T = 20 000K
60 Allard, Noselidze and Kruk, 1998
8
50
61Å] 6 1018cm−3
at 9 40 1060 1070 1080 1017cm−3
d )
malize 30 llog I ( 4 1016cm−3
nor 20 5.4
) [ 2 5.3
lI( 5.2
10
5.1
0 5
0 4.9
4.8
958 960 962 964
−10 −2
960 980 1000 1020 1040 1060 1080 935 940 945 950 955 960 965 970
l (Å) l (Å)
Figure 7. FUSE spectrum of the white dwarf Wolf 1346 (grey) Figure9.NormalizedprofilesfortheLymanδline.Thesmaller
and Tlusty/Synspec models using the quasi-molecular contri- box shows in detail the behaviour of the satellite at 962 ˚A with
bution tables evaluated by us (dashed line) and by N. F. Allard changes intemperaturefordensity1017 cm−3.
(dottedline),assumingtemperatureof20700Kandlog(g)=8.00
[asdeterminedbyGiammichele,Bergeron,&Dufour(2012)].
3.4 Lyman δ
Lymanδ has contributionsof 18 transitions. It isevaluated
1.4
CD 38°−10980 with theunified theory for thefirst time in this paper. The
This work
1.2 Allard, Noselidze and Kruk, 1998 normalized profiles are shown in Fig. 9 for temperatures of
10000, 15000and20000 Kandperturberdensitiesof1016,
Å] 1 1017 and 1018 cm−3. We findstronger satellites around 953
1100 0.8 1060 1070 1080 and962˚A.Neitherhasreporteddetection intheliterature.
at The former probably is to close to the line centre for that,
d
malize 0.6 bpuretsetnhteilnattthere,FaUlthSoEugsphenctortummeonftiConDed–3in8◦t1h0e98li0tepruatbulirseh,eids
) [nor 0.4 in fig. 4 of Wolff et al. (2001). Similar to what was noted
lI( 0.2 for the1400 ˚A satellite in Ly α, we see that its peak moves
to smaller densities and that it broadens as temperature
0 increases,soitsbehaviourisprobablyalsoagooddiagnostic
for temperature determinations.
−0.2
960 980 1000 1020 1040 1060 1080
l (Å)
Figure 8. FUSE spectrum of the white dwarf CD-
38◦10980 (grey) and Tlusty/Synspec models using the
quasi-molecular contribution tables evaluated by us (dashed
line) and by N. F. Allard (dotted line), assuming temper- 4 LINE PROFILES FOR THE BALMER
ature of 26000 K and log(g) = 8.00 [as determined by SERIES
Giammichele,Bergeron,&Dufour(2012)].
The Balmer series is composed by transitions that have
n = 2 as the final level, in case of absorption, or the ini-
tial level,for emission. It islocated in thevisible portion of
thespectrum,between3646˚A(H∞)and6563˚A(Hα).We
andSynspec(Hubeny& Lanz2011).Asbefore,thedashed evaluatehereeverylinefromHαtoH10,ascanbeseenin
line shows the model which uses the quasi-molecular con- the opacity profiles in Fig. 10, for temperature of 10000 K
tributions calculated in this work and the dotted line uses and perturberdensity of 1017 cm−3. The stronger satellites
tables by N. F. Allard. In this case, there seems to be no arelisted in Table2.Onlyoneof themwas observedsofar,
difference between the central line of each model, but the whichis probablyduetolack of laboratory dataaway from
position of thesatellite is again underestimated byAllard’s the line centre for the other lines. Such satellites would be
models,probablyduetothesamereasonasfortheLymanα stronger for temperatures around 10000 K, but the ioniza-
satellite.Theunderestimationbecomesmoreevidentasone tionrateofhydrogenislowatsuchtemperature,sotheyare
moves to high-order lines because higher levels converge at effectively stronger around 15000 K, when the H+ popula-
larger distances, so that the higher the line, the larger is tionismoresignificant.Inwhitedwarfs,linesaboveH11are
the error in the calculated asymptotic value and, by con- hardlyeverseen,sincethehighpressurecausesthentoover-
sequence, the larger the shift of the satellite towards the lapanddisappear.Theprofilesevaluatedhereareprobably
central line. sufficient to study any spectra in thevisible region.
Unified line profiles for H perturbed by protons 7
Table 2.Wavelengths forthemaximumofthestrongersatellites(ifexistent)ineachcalculatedlineinthe
Balmerseries (for T = 10000 K and n = 1017 cm−3), the transition(s) that originates them and notes on
observationandprevioustheoretical models.
Line Satellite(˚A) Transition(s) Observed? Previousmodel
Hα 8506.60 3dπg-4fπu yes(Kielkopf,Allard&Decrette2002) Kielkopf,Allard&Decrette(2002)a
4fσu-5gπg
Hβ 5413.33 3dπg-6hπu no Santos&Kepler(2012)b
Hγ 4392.21 3dπg-8kδu no Santos&Kepler(2012)b
3dπg-9kσu
3pσu-7gσg
4571.88 2pσu-7iδg no Santos&Kepler(2012)b
3dσg-8kπu
3dπg-8kπu
Hδ 4203.87 2pσu-10lσg no Santos&Kepler(2012)b
3dπg-9kπu
Hǫ 4005.35 2sσg-10kπu no Santos&Kepler(2012)b
2pσu-10iδg
2pσu-11lδg
2pσu-12nσg
3pσu-12nσg
3dπg-10kπu
4fσu-12nσg
a
viaunifiedtheory
b
viaquasi-staticapproach
The strongest satellite is around 8500 ˚A and has been ob-
−17 H10 Ha −19 servedinplasmaspectrabyKielkopf & Allard(1995).How-
H9 −20
−18 HH8eHd Hg Hb −−2221 esovners,:ifinrwsth,iitteidswtaorofs,cliotsheastonotthebereengiiodnenwtiefireedtfhoertPwaoscrheean-
−23
−19 series is and, second, because, for temperatures at which
−24
2m]−20 3800 3900 4000 4100 tehmeitHs+2prpedoopmuliantaionntlyisinreltehveanbtlu(e1,0h0a0v0i–n2g50li0t0tleKfl),uxthienstthaer
c
[l−21 region of this satellite. The satellite also contributes to the
sog line asymmetry.
l
−22
−23 4.2 Balmer β
−24 Forty-sixtransitionscontributetoBalmerβ.Itstotalprofile
is shown in Fig. 12 for a temperature of 10000 K and den-
−25
4000 5000 6000l (Å) 7000 8000 sities of 1016, 1017 and 1018 cm−3. We can see many faint
satellitesintheredwing,causingasymmetry.Thestrongest
Figure 10. Opacity profiles for every computed line in the satellite is around 5400 ˚A,butit isprobably too faint com-
Balmerseriesforatemperatureof10000Kandperturberdensity
pared to the line intensity to be observed in white dwarfs.
of1017 cm−3.Thesatellitesarenotveryintenseinthiscase,but
It is important to notice, though, that it contributesto the
itisworthnoticingthatalllinesareveryasymmetrical,especially
asymmetry,which,isworthremembering,doesnotappearif
the high order ones (inthe detailed box), inwhich the redwing
theinteractionbetweenhydrogenandprotonistreatedsim-
fallsofveryquickly,whilethebluewinghasasmotherbehaviour.
plyasStarkeffect,asisdoneincurrentatmospheremodels.
4.1 Balmer α 4.3 Balmer γ
FortheBalmerαline,therearecontributionsdueto32tran- To Balmer γ there are contributions due to 60 transi-
sitions. The total profile is shown in Fig. 11 for a tempera- tions. The total profile is shown in Fig. 13 for a temper-
ture of 10000 K and densities of 1016, 1017 and 1018 cm−3. ature of 10000 K and perturber densities of 1016, 1017 and
8 Ingrid Pelisoli, M. G. Santos and S. O. Kepler
8 9
8
6
7
6
4
) ) 5
lg I( 1018cm−3 lg I( 1018cm−3
lo 2 1017cm−3 lo 4 1017cm−3
1016cm−3 3
0 1016cm−3
2
−2 1
0
5000 6000 7000 8000 9000 10000 3900 4000 4100 4200 4300 4400 4500 4600 4700
l (Å) l (Å)
Figure 11.NormalizedprofilesfortheBalmerαline.Thetem- Figure 13.NormalizedprofilesfortheBalmerγ line.Thereare
peratureis10000Kanddensitiesare1016(lightgrey),1017(dark two strong satellite features, at 4400 and 4600 ˚A, none of them
grey) and 1018 cm−3 (black). The same labels are adopted for reportedinobservedspectrauntilnow.
theotherBalmerlines.Varioussatellitesappearintheredwing,
makingthelineveryasymmetric.
8
8 6
1018cm−3
)
6 lg I( 4 1017cm−3
o
l 1016cm−3
)
lg I( 4 1018cm−3 2
o
l
1017cm−3
2 1016cm−3 0
3800 3900 4000 4100 4200 4300
0 l (Å)
Figure 14.Normalizedprofiles forthe Balmerδ line.Themost
4200 4400 4600 4800 5000 5200 5400 5600
l (Å) noticeable satellite is at 4200 ˚A, but again it seems too faint to
beobservedinwhitedwarfspectra.
Figure 12. Normalized profiles for the Balmer β line. The
strongersatelliteisat5400˚A,butitisstilltoofaintforobserva-
tion,itsmainimportancebeingthustheasymmetryitcauses.
4.5 Balmer ǫ
There are 88 transitions that contribute to Balmer ǫ. We
showthetotalprofileinFig.15foratemperatureof10000K
1018 cm−3.Therearetwonoticeablesatellites,closeto4400 and perturber densities of 1016, 1017 and 1018 cm−3. We
and 4600 ˚A. The former is too close to the line centre to can see a very faint satellite around 4005 ˚A, with intensity
be observed in white dwarfs, the latter, despite being away of the same order of the continuum in the blue wing of the
from the centre, has very small intensity over the contin- line,meaningitisprobablynotobservable.However,asthe
uum, what would difficult or even prevent its observation. intensityoftheredwingrapidlydecreasesafterthesatellite,
However, both of them cause asymmetry. while the intensity in the blue line changes slowly, knowing
thesatellite is important to model theline asymmetry.
4.4 Balmer δ 4.6 Balmer 8, Balmer 9 and Balmer 10
We have to account for contributions of 74 transitions in The lines of Balmer 8, Balmer 9 and Balmer 10 have con-
Balmer δ.Itsprofileforatemperatureof 10000 Kandper- tributionsdueto 102, 116 and130 transitions, respectively.
turberdensities of 1016, 1017 and 1018 cm−3 can be seen in Their profiles are shown in Fig. 16, Fig. 17 and Fig. 18,
Fig.14.Thestrongestsatelliteiscloseto4200˚A,but,asfor again foratemperatureof10000Kandperturberdensities
the4600˚AsatelliteinBalmerγ,itsintensitydoesnotseem of 1016, 1017 and 1018 cm−3. It is worth noticing that, for
to be strong enough, when compared to the continuum, to n=1018cm−3,thesehighorderlinesstarttofadeduetothe
allow its detection in white dwarf spectra. high pressure. There are no satellites visible in those lines.
Unified line profiles for H perturbed by protons 9
9
8
8
6 1018cm−3 7 1018cm−3
1017cm−3 6 1017cm−3
) )
lg I( 4 1016cm−3 lg I(
o o
l l 5
1016cm−3
2 4
3
0
2
3850 3900 3950 4000 4050 3790 3800 3810 3820 3830 3840 3850 3860
l (Å) l (Å)
Figure 15.NormalizedprofilesfortheBalmerǫline.Asatellite Figure17.NormalizedprofilesfortheBalmer9line.Inspiteof
featureappears at4005 ˚A,and theintensity of the profiledrops theabsenceofsatellites,wecannotealineasymmetrythatdoes
afterit. notexistinpure-Starkprofiles.
9
9
8 8
7 7
1018cm−3 1018cm−3
6 1017cm−3 6 1017cm−3
)
llog I( 5 1016cm−3 llog I() 5 1016cm−3
4
4
3
3
2
2
3850 3860 3870 3880 3890 3900 3910 1
l (Å) 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840
l (Å)
Figure16.NormalizedprofilesfortheBalmer8line.Nosatellites
Figure18.NormalizedprofilesfortheBalmer10line.Againno
areseem,butthelineislargelyasymmetric.
satellites,butalargeasymmetrybetweenredandbluewingscan
beseen.
One might wonder in that case if the quasi-molecular con-
tributions really matter. However, it is very clear that they
do affect the line behaviour, again causing a strong asym- determination of temperature and surface gravity. We plot
here the calculated lines for T = 10000 K and log(n) = 17
metry between the red and blue wings, which is until now
together with the correspondent electronic Stark effect to
completelyignoredinwhitedwarfstarsatmospheremodels.
show that. Profiles for the Lyman series are shown in Fig.
19 and for the Balmer series in Fig. 20. It is easy to no-
tice two large differences: theexistence of satellites and the
5 COMPARISON TO ELECTRONIC STARK
large asymmetries between red and blue wings. The former
In current synthetic spectra for DA white dwarf stars, the is more important for low order lines (nf 6 5), when the
effect of the interaction between free protons and radiating redwingusuallyshowsatleastonestrongsatellite.Thelat-
hydrogen is assumed to be exactly the same of the inter- ter is evident for high order lines (n > 5), in which the
f
action of free electrons with hydrogen. So the profiles are red wing falls quickly for the quasi-molecular profile, while
calculated with density ne of electrons, and such density theredwingintheelectronicStarkbehavessimilarlytothe
is simply doubled to account for protons as well, as the bluewinganddecreasesslowly.Thesedifferencesmaybethe
number of free protons must be the same as the number cause of the inconsistent densities reported by Falcon et al.
of free electrons, because both originate, in this case, when (2014) and Genest-Beaulieu & Bergeron (2014), which are
hydrogenisionized.Asalreadysaidintheintroduction,such obtained when different lines are fitted.
approach disregards the quasi-molecular contributions that It is worth mentioning that the validity domain of the
appearwhen temperature andpressure are adequateto the VCS theory, in which the Stark models shown here are
formation of a transient bond state between hydrogen and based, is ∆ω/ωp 61 (Vidal, Cooper & Smith 1971), where
proton. ∆ω is the angular frequency measured from the line cen-
This is not a good practice and must be affecting the tre and ωp = 4πnee2/me is the plasma frequency (cgs
p
10 Ingrid Pelisoli, M. G. Santos and S. O. Kepler
10 10 is only appropriate for the frequencies in the line wings.
8 Lya 8 Lyb Therefore,thecentralbehaviouroftheprofilesevaluatedby
6 6
Santos& Kepler (2012) should not be relied upon. Besides
4 4
2 2 that,suchprofileswerecalculatedsummingupcontributions
0 0 due to different transitions, when, actually, they should be
−2 −2 convoluted,sincelifetimesforhydrogenexcitedstatesareof
) −−64 −−64 the order of 10−8 s, while the typical duration of collisions
lg I( 800 900 1000 1100 1200 1300 1400 1500 950 975 1000 1025 1050 1075 1100 in thephysical conditions of white dwarfs is around 10−13–
lo 1 80 Lyg 1 80 Lyd 10−12 s,implyingthatdifferentstatesmayformthroughout
6 6 theelectronic transition and contributeto theline.
4 4 The profiles calculated here are based on the unified
2 2
theory,in which no assumption is made concerning the du-
0 0
−2 −2 ration of collisions, and the partial profiles are convoluted,
−4 −4 sotheformofthelinecanberelieduponforthewholewave-
−6 −6
950 975 1000 930 940 950 960 970 length range. Nevertheless, we recall that both the unified
l (Å)
theory and thequasi-static approach assume adiabatic per-
Figure 19.Comparisonbetween each calculated lineinthe Ly- turbations, disregarding interference between different lev-
man series (black continuous line) and the correspondent pure els, which can result in underestimation of the line shift
electronic Stark profile (grey dashed line) for T = 10000 K and
(Allard et al. 1999). However, as the behaviour of the line
log(ne)=log(np)=17.Thereisnogreatdifferenceifoneconsid- centreinwhitedwarfstarsisdominatedbyelectronicStark
ers only the blue wing, but the red wingis very different, show-
andDopplereffects,beingthequasi-molecularcontributions
ingsatellite features when the quasi-molecular contributions are
moreimportant in thewings, theprofiles shown in thepre-
takenintoaccount.
vious sessions should be satisfactory for model atmosphere
applications.
10 10 However,withtheadventofexperimentssuchastheone
8 Ha 8 Hb donebyFalcon et al. (2012) at theZ Pulsed Power Facility
6 6
4 4 at Sandia National Laboratories, more accurate profiles in
2 2 the central region of the line are necessary. When trying to
0 5000 6000 7000 8000 9000 04000 4250 4500 4750 5000 5250 5500 evaluate such detailed profiles using the published data of
10 10 Santos& Kepler (2012), we noticed that the data was not
llog I() 468 Hg 468 Hd oafcctuhreatpeoetnenotuigahl dfoartath,adtu.eTthoerceownveerergdeinscceonitsisnuueistiiensitnhesoimme-
2 2 plemented computational methods, as can be seen on Fig.
0 3900 4050 4200 4350 4500 4650 4800 0 3750 3900 4050 4200 4350 21. That lead to abnormal behaviour in the line centre, as
10 10 shown for Lyman α in Fig. 22. To avoid that behaviour, it
8 He 8 H8 was necessary to truncatethe potentials at ∼180 a , where
6 6 0
4 4 a0 ≈0.5292 ˚A is the Bohr radius, but that way the asymp-
2 2 toticbehaviourisnotreachedandthelinecentreisstillnot
0 0
3800 3900 4000 4100 l (Å) 3800 3850 3900 3950 accurately determined (Fig. 23), since it is formed mainly
when the particles are far apart. Besides, as the mean dis-
Figure 20. Comparison between calculated lines in the Balmer tancebetween theparticles for density of1017cm−3,for ex-
series (black continuous line) and the correspondent pure elec- ample,isaround 250a ,going upto180 a isnotsufficient
0 0
tronic Stark profile (grey dashed line) for T = 10000 K and
if oneis concerned with theline centre.Asat that distance
log(ne)=log(np)=17.Forthelowerorderlines,weseethatthe the asymptotic behaviour of the potential energy, which is
satelliteinthe bluewingmusthave aconsiderablecontribution.
∝r−2,isstillnotreached,itisalsonotpossibletoextrapo-
Ontheotherhand,inthehighorderlines,themajordifferenceis
latethedatawith thenecessary precision.Forlargervalues
therapiddecreasingoftheredwingcomparedtothebluewing,
effectthatdoesnotappearintheelectronicStarkprofiles. ofn,theasymptoticbehaviourisreachedevenfartheraway,
sothetruncationmaylargelyaffectthedeterminationofthe
line centreprofile.
units).Accordingtothat,theprofileswouldbevalidonlyin Thus, we recalculated the potential data for the H+
2
the central 1–10 ˚A of the line. However, the validity of the quasi-molecule.Inordertochecktheconvergenceofthepo-
theory has been extrapolated in the current models, since tentials to their asymptotic values, we had to make use of
observed lines of white dwarf stars are usually more than a larger numberof terms in the recurrence relations3, what
100 ˚A wide. leads to a larger set of homogeneous linear equations and
demands, in consequence, larger matrices to solve. We cal-
6 THE LINE CENTRE
Quasi-staticprofilesareinherentlyapoorapproximationto 3 InSantos &Kepler(2012),therecurrencerelationsweretrun-
the line centre, since its main assumption of zero velocity catedatindexesequalto25.Now,forn=1andn=2forexample,
implies that only collisions whose duration is larger than wehadtotruncate the indexes at 300toget the correctasymp-
theinterval between them are taken into account, and that toticvaluetoaprecisionatthe21st decimal.