Measurements of 4He in Metal-Poor Extragalactic HII Regions: the Primordial Helium Abundance and the Delta Y / Delta O Ratio
aa r X i v : . [ a s t r o - ph . C O ] D ec Light elements in the UniverseProceedings IAU Symposium No. 268, 2010C. Charbonnel, M. Tosi, F. Primas, & C. Chiappini, eds. c (cid:13) Measurements of He in Metal-PoorExtragalactic H ii Regions: the PrimordialHelium Abundance and the ∆ Y / ∆ O Ratio
M. Peimbert , A. Peimbert , L. Carigi , and V. Luridiana Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, Apdo. postal 70-264,M´exico D.F. 04510, Mexicoemail: [email protected] Instituto de Astrof´ısica de Canarias, c/ V´ıa L´actea s/n, 38205 La Laguna, Spain
Abstract.
We present a review on the determination of the primordial helium abundance Y p , based onthe study of hydrogen and helium recombination lines in extragalactic H ii regions. We alsodiscuss the observational determinations of the increase of helium to the increase of oxygenby mass ∆ Y / ∆ O , and compare them with predictions based on models of galactic chemicalevolution. Keywords.
ISM: abundances, H ii regions; galaxies: abundances; galaxies: evolution; galaxies:irregular; Galaxy: disk; early universe
1. Overview
The determinations of the helium abundance by mass, Y , from metal-poor extragalac-tic H ii regions provide the best method to obtain the primordial helium abundance.During their evolution galaxies produce a certain amount of helium and oxygen per unitmass that we will call ∆ Y and ∆ O . The galaxies less affected by chemical evolution arethose that present a large fraction of their baryonic mass in gaseous form and a smallfraction of their baryonic mass in stellar form. These galaxies when experiencing burstsof star formation present bright metal poor H ii regions that have been used to determinetheir chemical composition.From a set of Y and O values and assuming a linear relationship it is possible to obtain Y p and ∆ Y / ∆ O from the following equation: Y p = Y − O ∆ Y ∆ O . (1.1)The determinations of Y p and ∆ Y / ∆ O are important for at least the following reasons:(a) Y p is one of the pillars of Big Bang cosmology and an accurate determination of Y p permits to test the Standard Big Bang Nucleosynthesis (SBBN), (b) the modelsof stellar evolution require an accurate initial Y value; this is given by Y p plus theadditional ∆ Y produced by galactic chemical evolution, which can be estimated basedon the observationally determined ∆ Y / ∆ O ratio, (c) the combination of Y p and ∆ Y / ∆ O is needed to test models of galactic chemical evolution.Recent reviews on primordial nucleosynthesis have been presented by Steigman (2007),Olive (2008), Weinberg (2008), and Pagel (2009). A review on the primordial heliumabundance has been presented by Peimbert (2008) and a historical note on the primordialhelium abundance has been presented by Peimbert & Torres-Peimbert (1999).1 Peimbert et al.
2. Recent Y p determinations The best Y p determinations are those by Izotov, Thuan, & Stasi´nska (2007) thatamounts to 0 . ± . . ± . Y p are very dif-ferent and it is not easy to make a detailed comparison of all the steps carried out byeach of them. We consider that the error presented by Izotov et al. is a lower limit to thetotal error because it does not include estimates of some systematic errors. The differencebetween the central Y p values derived by both groups is mainly due to the treatment ofthe temperature structure of the H ii regions. Izotov et al. adopt temperature variationsthat are smaller than those derived by Peimbert et al. To be more specific we can define the temperature structure of the H ii regions bymeans of an average temperature, T , and a mean square temperature fluctuation, t (Peimbert 1967). The value of t derived by Izotov et al. (2007) for their sample is about0.01; while Peimbert et al. (2007a) obtain a t of about 0.026. From the observationsadopted by Peimbert et al. and assuming t = 0.000 we obtain Y p = 0 . ± . t = 0 .
01 we obtain Y p = 0 . t derived by Izotov et al. (2007) is due to the parameter spaceused in their Monte Carlo computation where they permitted T (He i ) to vary from 0.95to 1.0 times the T (4363/5007) value, which yields a t of about 0.01. By allowing their T (He ii ) to vary from 0.80 to 1.0 times the T (4363/5007) value their t result wouldhave become higher. This can be seen from their Table 5 where 71 of the 93 spectracorresponded to the lowest T (He i ) allowed by the permitted parameter space of theMonte Carlo computation. A higher t value for the Izotov et al. (2007) sample producesa lower Y p , reducing the difference with the Peimbert et al. (2007a) Y p value.Two other systematic problems with the Izotov et al. (2007) determination related withthe temperature structure are: (a) that to compute the N (O ++ ) abundance they adoptedthe T (4363/5007) value which is equivalent to adopt t = 0 .
00, and (b) to compute theonce ionized oxygen and nitrogen abundances, N (O + ) and N (N + ), they adopted the T (4363/5007) value, but according to photoionization models and observations of O-poor objects, the temperature in the O + regions, T (O + ), is considerably smaller thanin the O ++ regions, typically by about 2000 K for objects with T (O ++ ) = 16000 K,and reaching 4000 K for the metal poorest H ii regions ( e. g. Peimbert, Peimbert, &Luridiana 2002; Stasi´nska 1990).2.1.
Recombination coefficients of the helium I lines
There are two recent line emissivity estimates to derive the He abundance from recom-bination lines: one due to Benjamin, Skillman, & Smits (1999, 2002), and another byBauman et al. (2005) and Porter et al. (2007, 2009). The difference in the Y values de-rived from both sets of data amount to about 0.0040, the emissivities by the first groupyielding values smaller than those of the second group. According to Porter, Ferland, &MacAdam (2007), the error introduced in their emissivities by interpolating in temper-ature the equations provided by them is smaller than 0.03%, which translates into anerror in Y p considerably smaller than 0.0001. Moreover according to Porter et al. (2009)the expected error in their line emissivities amounts to about 0.0010 in the Y derivedvalues. In this review we are adopting the Bauman et al. and Porter et al. emissivities inthe presented Y p values. 2.2. Beyond case B
There are at least four processes that modify the level populations of H and He atomsrelative to case B and consequently the Y p determination: the optical depth of the He i He in Extragalactic H ii Regions Table 1.
COSMOLOGICAL PREDICTIONS BASED ON SBBN AND OBSERVATIONSFOR τ n = 885.7 ± Y p D p η Ω b h Y p . ± . a .
93 + 2 . − . b . ± . b . ± . b D p . ± . b . ± . a . ± . b . ± . b W MAP . ± . b . ± . b . ± . b . ± . aa Observed value. b Predicted value. References: τ n Arzumanov et al. (2000); Y p Peimbert et al. (2007a); D p O’Meara etal. (2006);
W MAP
Dunkley et al. (2009).
Table 2.
COSMOLOGICAL PREDICTIONS BASED ON SBBN AND OBSERVATIONSFOR τ n = 878.5 ± Y p D p η Ω b h Y p . ± . a .
22 + 1 . − . b . ± . b . ± . b D p . ± . b . ± . a . ± . b . ± . b W MAP . ± . b . ± . b . ± . b . ± . aa Observed value. b Predicted value. References: same as in Table 1 with the exception of τ n that comes from Serebrov et al. (2005, 2008). lines, the collisional excitation from the He 2 S metastable level, the collisional excita-tions from the H ground level, and the fluorescent excitation of the H i and the He i lines, (case D of Luridiana et al. 2009). The last two processes are the least studied ofthe four.Luridiana et al. (2009) have introduced case D into the study of gaseous nebulae. CaseD increases slightly the emissivities of the H i and He i lines affecting the accuracy ofthe Y p determination. There are no published estimates of the importance of this effectbut it depends on the spectra of the ionizing stars, particularly in the region of the H i and He i lines, the spatial distribution of the ionizing stars, the gaseous electron densitydistribution, the fraction of ionizing photons that escape the nebula, the radial velocity ofthe gas relative to that of the stars, the nebular turbulence, and the region of the nebulaobserved. Therefore it requires tailor-made models for each observed object. Peimbert et al. (2007a) presented a list of thirteen sources of error in the Y p determination, eachof them with systematic and statistical components but dominated by one or the other.The systematic error produced by not having considered Case D should be added to thatlist.
3. Comparison of the directly determined Y p with the Y p valuescomputed under the assumption of the SBBN and theobservations of D p and WMAP To compare the Y p value with the primordial deuterium abundance D p (usually ex-pressed as 10 ( D/H ) p ) and with the WMAP results, we will use the framework of theSBBN. The ratio of baryons to photons multiplied by 10 , η , is given by (Steigman2006, 2007): η = (273 . ± . b h , (3.1)where Ω b is the baryon closure parameter, and h is the Hubble parameter. In the range4 < η < . < Y p < . Y p is related to η by (Steigman Peimbert et al. Y p = 0 . ± . . η − . (3.2)In the same η range, the primordial deuterium abundance is given by (Steigman2006, 2007): 10 ( D/H ) p = D p = 46 . ± . η ) − . . (3.3)From the Y p value by Peimbert et al. (2007a), the D p value by O’Meara et al. (2006),the Ω b h value by Dunkley et al. (2009), and the previous equations we have producedTable 1. From this table, it follows that within the errors Y p , D p , and the WMAPobservations are in very good agreement with the predicted SBBN values.Equations (2), (3), and (4) were derived under the assumption of a neutron lifetime, τ n , of 885 . ± . et al. et al. (2005, 2008)yielded a τ n of 878 . ± . ± . Y p value of 0.2470for the WMAP Ω b h determined value (Steigman 2007, Mathews et al. et al. obtain for τ n = 881 . ± . et al. (2000) and Serebrov et al. (2005, 2008), a Y p value 0.0009 smaller than for τ n =885 . ± . σ difference between both τ n determinations probably indicates that at least oneof them includes systematic errors that have not yet been sorted out.From the Y p by Peimbert et al. (2007a), the D p by O’Meara et al. (2006), and SBBN itis found that for τ n = 881 . ± . N eff , is equalto 3 . ± .
23. Based on the production of the Z particle by electron-positron collisionsin the laboratory and taking into account the partial heating of neutrinos produced byelectron-positron annihilations during SBBN Mangano et al. (2002) find that N eff =3 .
04. The N eff value derived from Y p , D p , and SBBN is in excellent agreement with thevalue derived by Mangano et al. (2002).The Y p derived from WMAP is based on the very strong assumption of SBBN. It isalso possible to derive Y p from the study of the microwave radiation without assumingSBBN: from the cosmic microwave background radiation ( i.e. WMAP + ACBAR +CBI + BOOMERANG), Ichikawa, Sekiguchi and Takahashi (2008a,b) obtain that Y p < .
44, when they also include the information obtained from BAO + SN + HST (baryonacoustic oscillations in the distribution of galaxies, the distance measurements from typeIa supernovae, and the HST value for H ) the constrain improves to Y p = 0 . +0 . − . .Including the expected data from the Planck satellite they predict a reduction on the Y p error of about a factor of four to seven, an error still about four to six times higherthan the one estimated from the best Y p determinations based on metal poor H ii regionobservations.
4. The ∆ Y / ∆ O ratio To determine the Y p value from a set of metal poor H ii regions it is necessary toestimate the fraction of helium, present in the interstellar medium of the galaxy whereeach H ii region is located, produced by galactic chemical evolution. From observations ofmetal poor extragalactic H ii regions it has been found that the Y versus O observationscan be fitted with a straight line given by ∆ Y / ∆ O and equation (1) has been used oftento derive Y p . A straight line is predicted by chemical evolution models of metal poorgalaxies with the same initial mass function, a given set of stellar yields, and differentstar formation rates. To obtain different ∆ Y / ∆ O from the models it is necessary tochange the initial mass function (for example the maximum mass allowed or the slope athigh masses), or the adopted yields. He in Extragalactic H ii Regions Y p , ∆ Y , and ∆ O are affected by different amounts in the presence oftemperature variations, while Y p diminishes by 0.0046 due to temperature variations inthe sample of Peimbert et al. (2007a), ∆ Y is slightly affected and ∆ O is strongly affectedby temperature variations ( e. g. Carigi & Peimbert 2008, Table 3). In addition a fractionof O is embedded in dust grains, fraction that needs to be estimated to compare withmodels of galactic chemical evolution.The importance of ∆ Y / ∆ O is two fold: it permits us to obtain a more accurate Y p value, and permits us to test for the presence of large temperature variations in gaseousnebulae when comparing nebular values with stellar ones.4.1. The gaseous O/H determination
There are two methods to derive the gaseous O/H ratio, from the I (4363) /I (5007) ratiotogether with the I (3727) /I (H β ) and the I (5007) /I (H β ) line ratios, the so called T (4363)method, and from the intensity ratio of O ii recombination lines to H i recombinationlines that has been called the O ii RL method by Peimbert et al. (2007b). The O ii RL method usually provides higher O/H ratios by 0.15 to about 0.3 dex, this difference isdue to temperature variations inside the observed volume and is smaller for metal poorH ii regions and higher for metal rich H ii regions. The T (4363) method in the presenceof temperature variations produces a systematic effect that lowers the O/H abundancesrelative to the real ones. On the other hand the O ii RL method is independent of thetemperature structure. 4.2. The total O/H determination
Another factor that has to be taken into account to obtain the total O/H ratio is thefraction of O atoms trapped in dust grains. Esteban et al. (1998), based on the depletionof Fe, Mg and Si in the Orion nebula, estimated that the fraction of oxygen atoms trappedin dust grains amounts to 0.08 dex. Mesa-Delgado et al. (2009) have estimated that thefraction of O atoms trapped in dust grains in the Orion nebula amounts to 0.12 ± ii regions in the gaseous phase, the depletions derived for the differentobjects define a trend of increasing depletion at higher metallicities; while the GalacticH ii regions show less than than 5% of their Fe atoms in the gas phase, the extragalacticones (LMC 30 Doradus, SMC N88A, and SBS 0335-052) have somewhat lower depletions.Izotov et al. (2006) find a slight increase of Ne/O with increasing metallicity, which theyinterpret as due to a moderate depletion of O onto grains in the most metal-rich galaxies,they conclude that this O/Ne depletion corresponds to ∼
20% of oxygen locked in thedust grains in the highest-metallicity H ii regions of their sample, while no significantdepletion would be present in the H ii regions with lower metallicity. Peimbert & Peimbert(2010) based on the Fe/O abundances of Galactic and extragalactic H ii regions estimatethat for objects in the 8 . <
12 + log O/H < . . ± .
03 dex, for objects in the 7 . <
12 + log O/H < . . ± .
03 dex, and for objects in the 7 . <
12 + log O/H < . . ± .
03 dex. Peimbert et al.
Extrapolation of the Y determinations to the value of Y p , or the O ( ∆ Y / ∆ O )correction From chemical evolution models of different galaxies it is found that ∆
Y / ∆ O dependson the initial mass function (IMF), the star formation rate, the age, and the O value ofthe galaxy in question. Peimbert et al. (2007b) have found that ∆ Y / ∆ O is well fittedby a constant value for objects with the same IMF, the same age, and an O abundancesmaller than 4 × − . This result is consistent with the custom of using a constant valuefor ∆ Y / ∆ O to fit observational data.To obtain an accurate Y p value, a reliable determination of ∆ Y / ∆ O for O-poor ob-jects is needed. The ∆ Y / ∆ O value derived by Peimbert, Peimbert, & Ruiz (2000) fromobservational results and models of chemical evolution of galaxies amounts to 3 . ± . . ± .
85 from observationsof 30 Dor and NGC 346, and by Izotov & Thuan (2004) who, from the observations of82 H ii regions, find ∆ Y / ∆ O = 4 . ± .
7. Peimbert et al. (2007a) have recomputed thisvalue by taking into account two systematic effects not considered by Izotov & Thuan:the fraction of oxygen trapped in dust grains, and the increase in the O abundances dueto the presence of temperature variations. From these considerations they obtained forthe Izotov & Thuan sample that ∆
Y / ∆ O = 3 . ± . et al. (2007b) from chemical evolution models with dif-ferent histories of galactic inflows and outflows for objects with O < × − find that2 . < ∆ Y / ∆ O < .
0. From the theoretical and observational results Peimbert et al.(2007a) adopted a value of ∆
Y / ∆ O = 3 . ± .
7, that they used with the Y and O determinations from each object to obtain the Y p value.4.4. Comparison of the oxygen abundances of the ISM of the solar vicinity with those ofthe Sun and F and G stars of the solar vicinity
In addition to the evidence presented in section 2 in favor of large t values, and con-sequently in favor of the O ii RL method, there is another independent test that can beused to discriminate between the T (4363) method and the O ii RL method that consistsin the comparison of stellar and H ii region abundances of the solar vicinity.Esteban et al. (2005) determined that the gaseous O/H value derived from H ii regionsof the solar vicinity amounts to 12 + log (O/H) = 8.69, and including the fraction of Oatoms tied up in dust grains it is obtained that 12 + log (O/H) = 8.81 ± .
04 for theO/H value of the ISM of the solar vicinity. Alternatively from the protosolar value byAsplund et al. (2009), that amounts to 12 + log(O/H) = 8.71, and taking into account theincrease of the O/H ratio due to galactic chemical evolution since the Sun was formed,that according to the chemical evolution model of the Galaxy by Carigi et al. (2005)amounts to 0.13 dex, we obtain an O/H value of 8.84 ± .
04 dex, in excellent agreementwith the value based on the O ii RL method. In this comparison we are assuming thatthe solar abundances are representative of the abundances of the solar vicinity ISM whenthe Sun was formed.There are two other estimates of the O/H value in the ISM that can be made fromobservations of F and G stars of the solar vicinity. According to Allende-Prieto et al. (2004) the Sun appears deficient by roughly 0.1 dex in O, Si, Ca, Sc, Ti, Y, Ce, Nd, andEu, compared with its immediate neighbors with similar iron abundances; by assumingthat the O abundances of the solar immediate neighbors are more representative of thelocal ISM than the solar one, and by adding this 0.1 dex difference to the protosolar solarvalue by Asplund et al. (2009) we obtain that the present value of the ISM has to behigher than 12 + log O/H = 8.81. A similar result is obtained from the data by Bensby& Feltzing (2005) who obtain for the six most O-rich thin-disk F and G dwarfs of the He in Extragalactic H ii Regions ii RL method.4.5. Comparison of ∆ Y / ∆ Z values of the Galactic H ii region M17 with K dwarfs ofthe solar vicinity The best Galactic H ii region to determine the He/H ratio is M17 because it containsa very small fraction of neutral helium and the error introduced by correcting for itspresence is very small. Carigi & Peimbert (2008) obtained from observations for M17(and adopting a Y p value) a value of ∆ Y / ∆ Z = 1 . ± .
41 for t = 0 . ± . Y and Z are the helium and heavy elements by unit mass; by correcting this valueconsidering that the fraction of O trapped in dust amounts to 0.12 dex instead of 0.08dex (Mesa-Delgado et al. Y / ∆ Z = 1 . ± .
37. This ∆
Y / ∆ Z value isin agreement with two independent ∆ Y / ∆ Z determinations derived from K dwarf starsof the solar vicinity that amount to 2 . ± . et al. . ± . et al. Y / ∆ Z = 3 . ± .
68 derived from collisionally excitedlines of M17 under the assumption of t = 0 .
00 by Carigi and Peimbert (2008) (valuecorrected for a fraction of O trapped by dust grains of 0.12 dex) is not in agreement withthe values derived from K dwarf stars of the solar neighborhood.4.6.
Comparison of Y p and ∆ Y / ∆ O with models of chemical evolution for the disk ofthe Galaxy To compare Galactic chemical evolution models of Y and O with observations we need touse the best determinations available of the abundances of these elements. We consider Figure 1.
In the two panels the origin corresponds to Y p = 0.2477, the triangles represent the Y and O values derived by Peimbert et al. (2007a) for five metal poor extragalactic H ii regions,the open and filled circles represent the M17 values for t = 0 .
00, and t = 0 .
036 respectivelyfrom Carigi & Peimbert (2008), the large open circle with a dot in the center corresponds to thepresolar values by Asplund et al. (2009). The left panel presents two chemical evolution modelsby Carigi & Peimbert (2010), the solid line is a model for the Galactic disk with a time spanfrom the formation of the Galaxy to the formation of the Sun at a galactocentric distance of 8kpc, while the dashed line corresponds to a model for the Galactic disk with a time span fromthe formation of the Galaxy to the present at a galactocentric distance of 6.75 kpc, the distanceof M17 to the galactic center; the stellar yields for the two panels are different, see subsection4.6 for further details.
Peimbert et al. that the two most accurate Galactic Y and O determinations are the presolar values(Asplund et al. ii region values (Carigi & Peimbert 2008).Carigi et al. (2005) presented chemical evolution models for the disk of the Galaxy thatfit the slope and the absolute value of the O/H gradient, these models are also successfulin reproducing the C/O gradient derived from H ii regions and the C/O versus O/Hevolution history of the solar vicinity obtained from stellar observations. In Figure 1we present chemical evolution models by Carigi & Peimbert (2010) for the Y and O abundances of the Galactic disk for two sets of stellar yields where they have added thepresolar values by Asplund et al. (2009) for comparison. These models are based on thesame assumptions than those adopted by Carigi and Peimbert (2008). The only differencebetween the models in the left panel and those in the right one, is that for massive starswith Z > .
004 the ones in the left use the yields by Maeder (1992) with high mass loss,while the ones on the right use the yields by Hirschi, Meynet, & Maeder (2005) withlow mass loss. There are at least three conclusions that we can extract from the figure:(a) the fit for the presolar values is very good for the two sets of yields, (b) the fit forM17 for t = 0 . ± .
013 is good, but it can be improved by assuming a set of yieldsintermediate between the two sets used; a similar result in favor of an intermediate setof yields was obtained by Cescutti et al. (2009) based on C/O observations for stars inthe bulge of the Galaxy, (c) the models do not fit M17 for t = 0 .
00, this result is inagreement with those by Esteban et al. (2005, 2009) and Peimbert et al. (2007a) whofind large t values for Galactic and extragalactic H ii regions. Moreover the O and Y abundances for the presolar material and for M17 are derived from independent methods,and they are fitted by the same chemical evolution model. This fit provides us with aconsistency check on the gaseous nebulae helium and oxygen abundance determinationsbased on large t values.
5. Conclusions
During the last 50 years the determination of Y p has been very important for the studyof cosmology, stellar evolution, and the chemical evolution of galaxies. To determine Y p itis necessary to determine accurate atomic parameters and the physical conditions insideionized gaseous nebulae.During the last five decades the accuracy of the Y p determination has increased consid-erably, and during the last two decades the differences among the best Y p determinationshave been due to systematic effects ( e. g. Olive & Skillman 2004, Peimbert 2008). Thesesystematic effects have been gradually understood, particularly during the last few years.The best Y p determination available, that by Peimbert et al. (2007a), is in agreementwith the D p determination and with the WMAP observations under the assumption ofSBBN. The errors in the Y p determination are still large and there is room for non-standard physics.Similarly to improve the accuracy of the Y p value derived under the assumption ofSBBN and the Ω b h derived from the background radiation, a new determination ofthe neutron lifetime is needed to sort out the difference between the τ n obtained byArzumanov et al. (2000) and the τ n obtained by Serebrov et al. (2005, 2008).To improve the accuracy of the Y p determination based on metal poor H ii regions, thefollowing steps should be taken in the near future: (a) to obtain new observations of highspectral resolution of metal poor H ii regions, those with 0 . < Z < .
001 to reducethe effect of the collisional excitation of the Balmer lines, which for the present day Y p determinations is one of the two main sources of error; (b) to determine the temperatureof the H ii regions based on a large number of He i lines observed with high accuracy, He in Extragalactic H ii Regions T (4363/5007) temperatures that weigh preferentially theregions of higher temperature than the average one, effect that artificially increases the Y p determinations; (c) additional efforts should be made to understand the mechanismsthat produce temperature variations in giant H ii regions and once they are understoodthey should be incorporated into photoionization models; (d) the He i recombinationcoefficients should be computed again with an accuracy higher than that of the last twodeterminations; (e) the computation of tailor-made photoionization models for extra-galactic H ii regions including the effect of case D on the intensity of the H i and He i lines should be carried out.The observed ∆ Y / ∆ O ratios provide us with strong constrains for models of galacticchemical evolution. The gaseous O abundances have to be corrected by the fraction ofoxygen embedded in dust grains and by the effect of temperature variations, these twocorrections together amount to about 0.2 to 0.4 dex. Chemical evolution models for theGalactic disk are able to reproduce the observed Y and O presolar values and the Y and O values derived for M17 based on H, He and O recombination lines, but not the M17 Y and O values derived from T (4363/5007) and O collisionally excited lines under theassumption of t = 0 .
00. This result provides a consistency check in favor of the presenceof large temperature variations in H ii regions and in favor of the Y p determinationsbased on the T (He i ) and t values derived from He i recombination lines. Acknowledgements
We wish to thank Gary Ferland, Evan Skillman and Gary Steigman for several fruitfuldiscussions. We are grateful to the SOC for an outstanding meeting and to the LOC fortheir warm hospitality. We would also like to acknowledge partial support received fromCONACyT grant 46904.
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