Abundance determinations in photoionized nebulae

Grazyna StasinskaObservatoire de Meudon

Mexico october 2006

Why care ?

1- abundances in HII regions probe the present-day abundances in galaxies

2- abundances of some elements in planetary nebulae probe nucleosynthesis in intermediate-mass stars

Good news1- observations of emission lines are easy, even for distant objects

2 - the methods for abundance derivations are easy

Contents

1- theory of photoionized nebulae in a nutshell

2- abundance determination through photoionization modelling3- abundance determination with empirical methods

4 - the temperature fluctuation problem5 - abundances from recombination lines

physical processes in ionized nebulaeIonization

Ionic fractions

Recombination

Heating

Electron temperature

Cooling

• Photoionization• Collisions• Charge exchange

• Radiative recombination• Dielectronic recombination• Charge exchange

• Photoionization• Collisional ionization

• Free-free radiation• Free-bound radiation• Bound-bound radiation

What drives the electron temperature

The energy gain of the electrons are:

G = i, jni j i

j

where the i j are due to photoionization and collisional ionization

The energy losses of the electrons are:

L = i, jni j i

j

where the i j are due to recombination

and collisional excitation followed by photon emission

The net energy gain is:dE / dt = G - L

If thermal equilibrium is achieved, the temperature is determined by: G = L

Why is the gas temperature roughly uniform in photoionized nebulae

Energy gained by photoionization of H at a distance r from the source

G = n(H°) 4J/(h) a (h-13.6eV) derg cm-3 s-1]

Ionization equilibrium equation of H at distance r =f RS

n(H°) 4J/(h)a d= n(H+) ne B(H)

Substituting: G = n(H+) ne B(H) < E >

with < E > = 4J/(h) a (h-h°) d4J/(h)a d≈ A T*

Energy gains due to photoionization of H are • independent of distance to the star• proportional to the star temperature

comments on energy losses• The most important cooling process is collisionally excited line radiation

• For a given ion in a two-level approximation, the cooling rate is given byLcoll = n2 A21 h21 erg cm-3 s-1]

where n2 results from the equilibrium equation of levels 1 and 2 : n1 ne q12 = n2 (A21 + ne q21)

• In the limit of small ne one has Lcoll = n1 ne q12 h21 = (n1 /nH) nHne 8.629 10-6 (1,2) /1 Te

-0.5 exp (-E12/kTe) h21

For « normal abundances » the most important cooling ion is O++

(H and He, the most abundant elements, have too high excitation potentials to be significantly excited at “normal temperatures”)

Cooling by collisional line excitation is important in the case of• abundant ions • lines corresponding to large • levels that can be easily attained at the temperature of the medium

Other comments on the gas temperatureDependence of Te with distance to the ionizing source• no dependence to first order

Spatial variations of Te • are mostly determined by • the mean energy of the absorbed photons• the populations of the main cooling ions

• are generally small • except at high metallicities • in the O++ zone cooling is very efficient through emission of

[OIII]52, 88 lines which have very low excitation potentials• while in the O+ zone the cooling efficiency is smaller (O+ has

no low-lying levels )

General dependence of Te with the defining properties of the nebulae• for a given T*, Te is lower for higher metallicity• for a given metallicity, Te is lower for lower T*• for given T* , ionization state and metallicity Te is higher if n > ncrit

The Te structure in ionized nebulaePhotoionization models from Stasinska 1978

effect of metallicity---- Z subsolar__ Z over solar

Strong Te gradients are present at high metallicities due to important O++ cooling

comments on line intensities• Most nebular lines are optically thin

exception: resonance lines H Ly, CIV 1550A, Si IV 1391A, NV 1240A

• Optically thin lines are powerful tools for abundance determinations

• Ratios of collisionally excited IR lines are almost independent of Te

• Recombination line intensities are inversely proportional to Te

• Ratios of recombination lines are almost independent of Te

• The ratios of a collisionally excited optical line and a recombination line strongly depend on Te

The various methods for abundance determinations

• Photoionization model fitting

• Empirical methods• Individual (or Te-based) methods• Statistical (or strong line) methods

the process of model-fitting

Model Must

Match Observations

abundance derivation by model fitting1) Define the input parameters• The characteristics of the ionizing radiation field• The density distribution of the nebular gas• The chemical composition of the nebular gas • The distance

2) Use a photoionization code (e.g. CLOUDY) that solves• The system of ionization equations for each species• The energy balance equation• The transfer of the ionizing radiation

3) Compare model output with observations (corrected for extinction)• The total observed H flux• The SH distribution (and the angular size of the H zone)• The visual magnitude of the ionizing source• The line intensities

4) Go back to 1 and iterate until observations are reproduced

for a good quality model fitting1)use as many observational constraints as possible • not only line intensity ratios

2) keep in mind that some constraints are very important• eg: HeII/H, [OIII]4363/5007

3) some constraints are not independant1) eg if [OIII]5007/H is fitted => [OIII]4959/H should be fitted as well

because [OIII]5007/4959 is fixed by atomic physics2) if it is not, this may indicate an observational problem, eg that the

strong [OIII]5007 line is saturated3) so [OIII]4959/H should not be used as a test of the model but of

the observations

4) chose a good estimator for your goodness of fit• eg avoid using a 2 = (Imod i-Iobs i ) / i ]2 minimization criterion• better use ∀i, [(Imod i-Iobs i ) / i ] 2<1

possible conclusions from model fitting1) If all the observations are well fitted within the error bars • this may imply that the model abundances are the true

abundances• (within error bars not easy to determine)

• If the constraints are not sufficient,• the model abundances may be very different from the true

ones

2) If some observations cannot be fitted1) either the observations are not as good as thought2) or the model does not represent the object well

i.e. some assumptions are incorrect, eg• the nebular geometry (eg not spherical)• the assumed stellar radiation field• some important process is missing (eg heating by an

additional mechanism)=> the chemical composition is not known to the desired accuracy

an example of photoionization modelling with insufficient constraints

obs Ratag B1 B2 B3

T* 37500K 39000K 37000K 39000Kr* (cm) 5.00+10 5.75+10 4.90+10Rin 0.062 pc 0.050 pc 0.065 pcRout 0.10 pc 0.087 pc 0.081 pc 0.085 pcF(Hbeta) 3.9-12 3.92-12 3.90-12 3.88-12ne (cm-3) 2050 1800 1700 1800He 0.117 0.117 0.180 0.100C 1.50-3 1.00-3 1.20-3N 4.80-4 2.50-4 5.50-4 6.00-4O 2.20-4 2.40-4 1.00-3 1.20-3Ne 5.00-5 2.00-4 2.40-4S 2.30-5 3.00-6 6.00-6 7.00-6 [OII] 3727 0.596 a) 0.587 0.613 0.604[NeIII] 3869 0.014 0.0084 0.0096 [OIII] 4363 <0.0013 0.0006 0.0002 0.0001 HeII 4686 0.0004 0.0003 0.0003 HI 4861 1.00 1.00 1.00 1.00 [OIII] 5007 0.283 0.304 0.281 0.275 [NI] 5200 0.0149 0.0043 0.0093 0.0087 [NII] 5755 0.0071: 0.0151 0.0069 0.0060 HeI 5876 0.128 0.126 0.124 0.128 [OI] 6300 0.0054 0.0106 0.0116 [NII] 6584 2.85 2.79 2.87 2.81 [SII] 6717 0.0565 0.053 0.0602 0.0558 [SII] 6731 0.084 0.077 0.0868 0.0826 [OII] 7325 0.0091: 0.0126 0.0070 0.0063 T(NII) 6734 5422 5426 T(OIII) 7319 5876 5394

Is the PN M 2-5 O-poor or O-rich ?• Ratag 1992 claimed it to be O-poor• Stasinska, Malkov, Golovatyj 1995 found that both O-poor (B1) and O-rich (B2 abd B3) models can fit all the available data• factor 5 uncertainty in O/H

an example of photoionization modelling without a satisfactory solution

Detailed modelling of the giant HII region NGC 588 in M33Jamet et al 2005

many observational constraints• narrow band imaging• long slit optical spectra• infrared data from ISO• full characterization of the ionizing stars

failure of the models• the observed [OIII]4363/5007 and [OIII]88m/5007 cannot be

explained at the same time• this implies an uncertainty of at least a factor 2 in the oxygen

abundance

Empirical abundance determinationsIndividual (or Te-based) methods

1) Te and ne are obtained from plasma diagnostics

QuickTime™ et un décompresseurGIF sont requis pour visualiser

cette image.

Plasma diagnostic diagram for NGC 7027Energy level diagrams

Empirical abundance determinationsIndividual (or Te-based) methods

2) Ionic abundance ratios are determined from line intensity ratios eg: O++/H+ = ([OIII]5007/H) / ([OI II]5007(Te)/H(Te))

3) Elemental abundance ratios are obtained

• either by adding all the observed ions eg: O/H = O+/H+ + O++/H+ + O+++/H+ + … • or by using ionization correction factors (icfs)

a note on ionization correction factorsIonization correction factors based on ionization potentials

• a first approximation promoted by M. and S. Peimbert• but risky: eg (O+++ + ..)/O ≠ He++/He

• there is nothing which empedes O++ ions to be present in the He++ zone

Ionization correction factors based on model grids may be risky too

• observations often pertain only to a small fraction of the object• there is no robust formula to correct for He°

Cases when no icf is needed

• when all the expected ionization stages are observed however in this case beware of errors in determining ionic abundances

• from different spectral ranges • from lines extremely sensitive to Te (UV lines or transauroral lines)

a rough evaluation of Te-based methods

• the methods are easy to implement

• they depend on a very limited amount of assumptions • error bars are relatively easy to estimate

• the abundances of the most important elements are expected to be correct (within error bars)

• they are very close to abundances obtained from successfull tailored photoionization modelling

abundances from optical lines

1.many telescopes, large collectors2.all lines available in same spectrum3.spectra affected by extinction 4.lines optically thin5.intensities depend on Te, emitted in regions of Te>4000K6.recombination affects forbidden lines at low Te

7.no icf needed for O in HII regions8.icfs needed for N, Ne, S, Ar9.main available diagnostic lines[NII] 6584[OII] 3727, [OIII] 5007[NeIII 3869], [NeV]3426 [SII] 6720, [SIII] 9532,[Ar III] 7751, [Ar IV]4740, [ArV]6435Te:[ OIII] 4363/5007, [NII]5755/6884ne: [SII]6731/6717, [ArIV]4740/4711

abundances from FIR lines

1.few telescopes, small collectors2.beamsize and calibration problems3.FIR lines probe obscured regions4.lines may be optically thick 5.intensities of CE lines independent of Te

6.

7.icf needed for O8.no icf for N, small for Ne, S, Ar9.main available lines[NII] 121.7, [NIII]57.3m,[OIII] 51.8mu, [OIV] 25mu, [NeII] 12.8, [NeIII] 15.5, [NeV] 14.3[SIII] 18.6, [SIV ]10.5[ArII] 6.9, [ArIII] 9.0, [ArV], [ArVI]ne:[OIII]52/88,[SIII]18/33,[NeIII]15/36, [ArIII] 9/21,[NeV]14/24

a case of failure of Te-based abundancesmetal rich giant HII regions (Stasinska 2005)

• with VLTs [OIII]4363/5007, [NII]5755/6584, [SIII]6312/9532 become measurable even at high metallicity (eg Bresolin et al 2005)

the problem• at Z > Z strong Te gradients are predicted• Te sensitive ratios strongly overestimate Te in the emitting zones

• depending on what line is measured• and what relation is adopted between T(O+) and T(O++)

O/H is strongly biased !

a further problem at high metallicity

contamination of collisionally excited lines by recombination

• at low temperatures, collisionally excited lines such as [OII]7330 or [NII]5755 may be dominated by recombination

• this effect, very strong in the case of [OII]7330 is wrongly corrected in the literature

Empirical abundance determinationsstatistical (or strong line) methods

In many cases, the weak [OIII] 4363 or [NII]5755 lines are not available because

• the temperature is too low • the spectra are of low signal-to-noise • the data consist of narrow band images in the strongest lines only

It is possible, under certain conditions, to estimate the metallicity (i.e. O/H) using only “strong lines”

Strong line methods • are statistical • have to be calibrated

Best known strong line methods: the ones based on oxygen lines

• Pagel et al 1979 used ([OII]+[OIII])/Has an indicator of O/H • this method, la, has been calibrated many

times• Mc Gaugh 1994 refined the method to account for the ionization parameter U• Pilyugin (2000, 2001 ..., 2005) proposed the most sophisticated approach

Rationale of Mc Gaugh’s method

there are 4 independent strong line ratios• HH, [OII]/H, [OIII]/H, [NII] /H

there are 5 parameters determining them• C(H , <T*>, U, O/H, N/O

underlying hypothesis of the method• <T*> is related to O/H

• (this is expected statistically for giant HII regions)

the procedure• both O/H and U are derived simultaneously from• ([OII]+[OIII])/H, and [OIII]/[OII]

a problem• ([OII]+[OIII])/Hvs. O/H is double valued

a way out• [NII]/[OII] indicates whether O/H is high or low (“astrophysical” argument)

McGaugh diagrams for the O23+ method

versus /

N/ersu/

what lies behind the behaviour of [OIII]5007/H vs O/H

• Intensity ratio: [OIII]5007/H = A n(O++) / n(H+) Te0.5 exp (-28800/Te)

• Thermal balance equation: n(H+) ne T* ≈ B ni j ne Te

-0.5 exp (- Eexc/Te)

• For 12 + log O/H < 8.2 • cooling is due to H Ly , •Te is independent of O/H• [OIII]5007/H ≈ C T* O/H

• For 12 + log O/H > 9 • cooling is essentially due to [OIII]52,88•[OIII]5007/H ≈ C T* f(Te) •where f(Te) = Te exp (- 28800/Te)

•which decreases with increasing O/H

An evaluation of strong line methodsPerez-Montero & Diaz 2005

• uses a data base of 367 objects with measured Te

• including some giant HII regions in the inner parts of galaxies (expected to be metal rich)• but ignores the strong bias due to low Te evidenced by Stasinska 05

the strong line method recalibratedPilyugin Thuan 2005

upper branch calibration(ie high O/H)

lower branch calibration (ie low O/H)

• uses a data base of over 700 objects with measured Te• including some giant HII regions in the inner parts of galaxies (expected to be metal rich)• uses only Te-derived abundances• but ignores the strong bias due to low Te evidenced by Stasinska 05

=> the last word on abundances from strong line methods is not said

other problems and uncertainties in abundance determinations

atomic datastellar fluxesaperture correction and complex nebular geometryreddening correctiontemperature and density inhomogeneitiesthe mystery of optical recombination linesthe role dust grains

Aperture correctionWhen the studied objects are more extended than the observing beam• aperture correction is needed if the observing beams are not the same

for all wavelenghts. For example• combining optical and UV spectra from IUE• combining FIR measurements with optical measurements• combining various line intensities from ISO

Aperture correction can be made• using line ratios that have a known intrinsic value

(e.g. HeII 1640 / He II 4686)• using ratios of beam solid angles

Such procedures bear uncertainties• they do not take into account the ionization stratification of the

nebulae

relevance of elemental abundances in case of depletion

Mg, Si, Fe, Ni, Ca• these elements can be almost entirely in the form of grains• their abundances in the gas phase cannot be easily used as indicators

• of chemical evolution of galaxies • or nuclear processes in PN progenitors

C• can be heavily depleted by carbon-based grains (graphite, PAHs...)

O• can be slightly depleted (20% for Orion, estimated from depletion pattern of

metals, Esteban et al 1998)

He, Ne, Ar• rare gases, do not combine into grains

Correction for dust extinctionThe method:

• The “logarithmic extinction at H”, C, is derived from the observed H/H ratio by comparing it to the theoretical one for case B recombination (at the Te corresponding to the nebula) using an extinction law f()

C = [ log (FH/ FH)B - log (FH / FH)obs ] / (f- f)

• Emission line ratios are then dereddened using the formulalog (F1 / F2)corrB = log (F1 / F2)obs + C (f1- f2)

Problems:• The “extinction law” is not universal• The intrinsic H/H ratio may be different from the theoretical case B(collisional excitation, case C)• If some dust is mixed with the ionized gas and strongly contributes to

the extinction, no “extinction law” applies

the “extinction law” is not universal• the canonical extinction law corresponds to RV=AV / E(B-V) =3.2

•in Orion RV = 5.5

• towards the Galactic bulge, RV ~ 2.5 (Stasinska et al 94)

Histogram of RV for 95 galactic O stars Patriarchi et al 2001

• larger values of RV are found for lines of sight crossing molecular clouds where dust grains are expected to be larger

Extinction laws corresponding to various RV=AV / E(B-V).1989

checks on the reddening correctionBefore dereddening check that conditions for case B are likely satisfied• If not, consider building a photoionization model, and redden the

resulting the emission lines to fit the observed Balmer decrement.

If case B is relevant for the object under study• check that, after reddening correction , H / H is close to the case B

recombination value• If not, [OIII]4363/5007 is likely to be in error by the same amount as

H/H differs from the case B value

• If many Balmer lines are measured with good accuracy• Rather than using an “extinction law”, fit the observed spectrum to

the theoretical Balmer decrement for reddening. • This method is valid (in first approximation) also if dust is mixed

with the ionized gas

Temperature fluctuations

€

T0(Ni )=TeNiNedV∫NiNedV∫ t2 (Ni) =

(T −T0(N 2 NNdV

T02 NNdV

Do temperature fluctuations exist?• see reviews by Peimbert 1995, 2001, Mathis 1997, Stasinska 1998, Esteban 1998, 2001• Numerous studies point towards t2 ~ 0.04• But little direct evidence is seen

Were postulated by Peimbert (1967) to explain discrepancies betwen Te from various diagnostics

Peimbert’s formalism

example of indirect evidence for t2 ≠ 0

In planetary nebulae Te from Balmer discontinuity is smaller than T[OIII] 4363/5007 (Liu & Danziger 1993)

t2 ~ 0.04 is a representative value

If Te fluctuations exist they affect abundance determinations

e.g. abundance derived in M8 (Peimbert et al 1993) using Taylor series expansion of the line emissivites for various values of t2

t2=0 t2=0.02 t2=0.04 t2=0.06He 11.02 11.01 11.00 10.99C 8.21 8.31 8.48 8.77N 7.57 7.66 7.77 7.88O 8.50 8.60 8.71 8.84

Abundances derived from optical forbidden lines with respect to H are underestimated when ignoring t2

Abundances derived from recombination or FIR lines are not affected

Abundance ratios like N/O or C/O are less affected

visualizing the Peimbert formalism on a two-zone toy model

€

T0 =T1+fT21+f

€

t2 =(T1−T0)2 +f(T2 −T0)2

(1+f )To2

f = N2n2V2 / N1n1V1

V1

n1 N1 T1

V2

n2 N2 T2

in this case the values of T0 and t2 are simply :

variations of T1 and T2 with f for t2 = 0.04 and fixed T0

• f >> 1 corresponds to a photoionized nebula with small shock-heated regions of very high T1

• f << 1 corresponds to a nebula with high metallicity clumps of somewhat lower T2

volumeelectron densityionic densitytemperature

Is the Peimbert formalism adequate ?plots of log O++ as a function of f for two values of T0

• left: T0 = 8000K • right: T0 = 15000K

O++ is derived from [OIII]5007 using Te measured by [OIII]4363/5007

The amount by which O++ 5007 is underestimated depends on T0 and f

Even in a simple two-zone model, the situation requires• 3 parameters to be described (e.g. T0, t2 and f ) • not 2 (T0 and t2)

Visualisation of energy requirements

The simplest example:

for t2 = 0.04 and T0 = 10000K, f =1 implies T1 = 12000K and T2 = 8000K

-: log of heating rate in arbitrary units

-: log of cooling rate in the O+ zone

By shifting the heating curve up and down one understands how Te varies with energy input

t2 = 0.04 requires log = 0.3, ie a factor 2 difference in heating rates between regions 1 and 2 !

What fluctuates?

•Te ? • Natural gradients in photoionized nebulae are small

• except at high metallicities • (Stasinska 1980, Garnett 1992, Kingdon & Ferland 1995, Perez 1997)

• Ne ?

• In high density clumps collisional dexcitation increases Te with respect to the ambient medium Kholtygin 1998, Mathis et al. 1998

• (this is not sufficient to explain t2 ~ 0.04)

• densities above 105cm-3 boost [OIII] 4363/5007 (Viegas & Clegg 1994)• but there is no evidence of such high densities in the O++ zones

€

T0(Ni )=TeNiNedV∫NiNedV∫ t2 (Ni) =

(T −T0(N 2 NNdV

T02 NNdV

What fluctuates? • Ni ?

• Te is lower in C-rich zones (Torres-Peimbert et al 1990)

• The O++ discrepancy between collisional and optical lines requires the existence of O-rich zones (Stasinska 1998, Liu et al 2000, 2001, Péquignot 2001)

•Te ? • Natural gradients in photoionized nebulae are small

• except at high metallicities • (Stasinska 1980, Garnett 1992, Kingdon & Ferland 1995, Perez 1997)

• Ne ?

• In high density clumps collisional dexcitation increases Te with respect to the ambient medium Kholtygin 1998, Mathis et al. 1998

• (this is not sufficient to explain t2 ~ 0.04)

• densities above 105cm-3 boost [OIII] 4363/5007 (Viegas & Clegg 1994)• but there is no evidence of such high densities in the O++ zones

Is photoionization the only heating source in photoionized nebulae?

• In a number of nebulae, classical photoionization models produce T[OIII] lower than observed

• Giant HII regions: Campbell 1990, Garcia-Vargas et al 1997, Stasinska & Schaerer 1999, Luridiana et al 1999, Luridiana & Peimbert 2001

• PNe: Peña et al 1998

• Additional energy sources have been proposed:• Shocks (Peimbert et al 1991)

• Conduction fronts (Maciejewski et al 1996)

the ORL /CEL discrepancyExpected properties of optical recombination lines (ORLs) • their emissivity is roughly proportional to Te

-1

• they should give correct abundances with respect to H • even in presence of temperature fluctuations

ORL abundances versus CEL (collisionally excited lines) abundances • ORL abundances larger than CEL abundances by important factors• Wyse 1947, Peimbert et al 1993, Liu et al 1995 (O) , Kaler 1986 (C)• Esteban et al 1998, Liu et al 2000, 2001 (C,N,O)

C++ 1909 / O++ 5007 versusC++ 4267 /O++ 5007 in planetary nebulaeCompilation Rola & Stasinska 1994

ORL versus CEL abundancesionic abundances in the planetary nebula NGC 6153 Liu et al 2000

invoked causes of ORL-CEL discrepancyFaintness of the ORLs• biased measurements • flux calibration is difficult over a

large dynamical range • they may suffer from blends

Heavy element recombination coefficients are not reliable

Temperature fluctuations

Density condensations

• no (from high S/N spectroscopy)• no: [OIII]4931/[ [OIII]4959 agrees with

theory: 4 10-4 Mathis& Liu 1999• no (from echelle spectra )

• have been recomputed with the R-matrix method Storey 1994

• ORL abundances from numerous transitions are in agreement

• t2 explaining ORL-CEL discrepancy >> t2 explaining Te[OIII] -Te(BJ)

• IR-CEL abundances are consistent with optical -CEL abundances Liu 2000,2001

• no (high order Balmer lines) Liu 2000...

invoked causes of ORL-CEL discrepancyFaintness of the ORLs• biased measurements • flux calibration is difficult over a

large dynamical range • they may suffer from blends

Heavy element recombination coefficients are not reliable

Temperature fluctuations

Density condensations

• no (from high S/N spectroscopy)• no: [OIII]4931/[ [OIII]4959 agrees with

theory: 4 10-4 Mathis& Liu 1999• no (from echelle spectra )

• have been recomputed with the R-matrix method Storey 1994

• ORL abundances from numerous transitions are in agreement

• t2 explaining ORL-CEL discrepancy >> t2 explaining Te[OIII] -Te(BJ)

• IR-CEL abundances are consistent with optical -CEL abundances Liu 2000,2001

• no (high order Balmer lines) Liu 2000...

Recombination coefficients computed so far do not include dielectronic recombination for n > 10 which is likely to be efficient at Te > 2 104K

Chemical inhomogeneities: they require super metal rich inclusions with solar C/N/O/Ne Liu 2000, Tsamis 2003

Temperature of the ORL zones Tsamis et al 2004, Liu et al 2004

• In 4 PNe, it has been possible to derive Te from ratios of OII recombination lines that have a slightly different sensitivity to Te

• The zones emitting the OII recombination lines are found to be ultracold

• This goes against the dust+rec coeff explanation, at least for these objects• The ORL zones • must havedifferent metallicity (very high) compared to the rest of the nebula

a dual abundance photoionization model for 30 Dor

Tsamis & Péquignot 2005

The t2 problem and the ORL/CEL discrepancy

are still a subject of debate

quiz 1

quiz 1

Is HeII/H fitted?

quiz 2

quiz 2

Quiz 3

1rst lines: observations

2nd lines: models

Quiz 3

1rst lines: observations

2nd lines: models