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arp 220 is the nearest ( @xmath3 77 mpc ) example of an ultraluminous infrared galaxy ( ulirg ) that supports star formation at extreme levels .
it contains two nuclei separated by 350 pc , both surrounded by massive discs of dense molecular gas ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
radio detections of supernovae at a rate of 13 yr@xmath4 @xcite confirm that huge populations of massive stars are present with an implied star formation rate ( sfr ) of @xmath5 yr@xmath4 .
although arp 220 could contain active galactic nuclei ( agns ) , particularly in the western nucleus , the observed supernova rates indicate that star formation provides a substantial fraction of the power radiated by the nuclei .
the nuclei of arp 220 provide access to the high - intensity mode of star formation in dense molecular media that appears to have been more common in young galaxies .
these types of environments are of special interest from a range of perspectives , including the information they can provide regarding the role of galactic winds , cosmic rays , and magnetic fields in feedback processes that influence galaxy evolution .
previous investigations show that arp 220 is likely to be a hadronic cosmic ray calorimeter where all of the power in cosmic rays is absorbed within the nuclear starburst zones ( e.g. , * ? ? ?
both nuclei also contain extremely intense far - infrared ( fir ) radiation fields ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , and the west nucleus is optically thick in the fir to wavelengths of @xmath6 @xcite .
the production of the observed radio synchrotron emission then requires magnetic fields of milligauss strength ( e.g. , * ? ? ?
* ; * ? ? ?
llllc physical parameters & east nucleus & west st & west cnd & references + distance & 77.0 mpc & 77.0 mpc & 77.0 mpc & + cmz radius & 70 pc & 90 pc & 30 pc & 1,2,3 + cmz disc scale height@xmath7 & 40 pc & 40 pc & 40 pc & 4 + molecular gas mass & @xmath8 @xmath9 & @xmath10 @xmath9 & @xmath8 @xmath9 & 2,5 + ionized gas mass@xmath11 & @xmath12 @xmath9 & @xmath13 @xmath9 & @xmath12 @xmath9 & + average ism density@xmath14 & @xmath157700 @xmath16 & @xmath153500 @xmath16 & @xmath1542 000 @xmath16 & + fir luminosity & @xmath17 @xmath18 & @xmath17 @xmath18 & @xmath19 @xmath18 & 2 + fir radiation field energy density@xmath20 & 40 000 ev @xmath16 & 27 000 ev @xmath16 & 440 000 ev @xmath16 & + dust temperature & 90 k & 50 k & 170 k & 2,6 + sn explosion rate ( @xmath21 ) & 0.7 yr@xmath4 & 0.7 yr@xmath4 & 1.3 yr@xmath4 & 7 + star formation rate ( sfr)@xmath20 & 65 @xmath9 yr@xmath4 & 65 @xmath9 yr@xmath4 & 120 @xmath9 yr@xmath4 & + sn explosion energy@xmath22 & 10@xmath23 erg & 10@xmath23 erg & 10@xmath23 erg & + sn energy in cosmic ray protons@xmath22 & 5 20% & 5 20% & 5 20% & + ratio of primary protons to electrons ( @xmath24/@xmath25 ) & 50 & 50 & 50 & + slope of primary cosmic ray source function & 2.1 2.3 & 2.1 2.3 & 2.1 2.3 & + + + + + + + + + + in this paper , we study cosmic ray interactions in the arp 220 starburst nuclear regions using an updated version of the @xcite models , hereafter yegz .
we develop a model with two spatial zones to accurately represent the inner and outer regions of the western nucleus as defined by its molecular gas properties @xcite .
we incorporate photopion energy losses and photon photon interactions to account for the extreme fir radiation field .
we calculate the hadronic calorimetry fraction for each nucleus for the best - fitting radio models , and we predict the total @xmath2-ray and neutrino fluxes . in section 2 , we review the physical parameters which we selected for the models .
section 3 details the basic assumptions of the models and our findings for the arp 220 starburst nuclei .
we present concluding remarks in section 4 .
due to its extreme properties , arp 220 has been extensively studied across the electromagnetic spectrum .
the nuclei of arp 220 are of particular interest as they contain more than half of the total bolometric infrared luminosity of the galaxy ( @xmath26 ; e.g. , * ? ? ?
* ; * ? ? ?
* and references therein ) .
as the nuclei are less than 100 pc in radius , the presence of an agn or a ` hot ' starburst is required to explain the extraordinarily large surface brightness in the western nucleus @xcite ; however , the existence of an agn has yet to be definitively established ( e.g. , * ? ? ?
further , the submillimetre observations suggest that whether or not agns are present , they are not the main heating source of the dust ( e.g. , * ? ? ?
estimates of the fir luminosities of the eastern and western nuclei range from @xmath27 to @xmath28 and from @xmath29 to @xmath30 , respectively ( e.g. , * ? ? ?
* ; * ? ? ?
the range on these luminosities is quite large due to uncertainty in the true sizes , inclinations , and opacities of the nuclei and their associated molecular disc . to keep our adopted fir luminosity in rough agreement with the observed supernova rate , we assume values of @xmath31 and @xmath32 for the eastern and western nuclei ( see table 1 ) . assuming similar ratios between the nuclei for the supernova rate and molecular gas content , we adopt values of @xmath33 yr@xmath4 , @xmath34 for the eastern nucleus and @xmath35 yr@xmath4 , @xmath36 for the western nucleus .
while our assumed molecular gas masses favour conservative estimates , other estimates of the gas content suggest the masses are as high as @xmath37 @xcite . [
cols="<,^,^,^,^,^,^,^ " , ] co observations of the western nucleus imply a temperature gradient increasing towards the centre and indicate significant differences in the physical conditions between the two nuclei @xcite . @xcite model the western nucleus as two distinct dust sources a cooler ( 50 k ) ring surrounding a hotter ( 170 k ) , dense dust core .
we use this two - zone model for the western nucleus and have adjusted our single - zone model to account for the differences in temperature and density between the two regions ( see section 3 ) .
for the eastern nucleus , we assume a single dust temperature of 90 k @xcite .
previously , we developed and tested a model for cosmic ray interactions in the central molecular zones ( cmzs ) of star - forming and starburst galaxies ( yegz ; * ? ? ?
* ; * ? ? ?
our single - zone model accounts for a variety of energy losses via interactions with the interstellar medium ( ism ) , magnetic fields , and radiation fields and for energy - independent advective escape via a galactic wind ( see fig .
1 ) . the resulting cosmic ray energy spectrum depends on both the total cosmic ray lifetime and a power - law injection spectrum which is directly proportional to the volume integrated supernova rate ( see yegz for further details ) .
accounting for the production of secondary cosmic rays , we use our calculations of the population of energetic particles to predict the radio , @xmath2-ray , and neutrino spectra . for the @xmath2-ray spectrum
, we include both leptonic ( bremsstrahlung , inverse compton ) and hadronic ( neutral pion decay ) emission mechanisms . for the radio spectrum
, we incorporate the effects of free free emission and absorption @xcite . as in our previous models , we assume that the ionized gas in the nuclei acts as a foreground screen that some fraction ( @xmath38 ) of the emitted synchrotron radiation passes through
. when the covering fraction is low ( @xmath39 ) , the radio spectrum flattens at low frequencies @xcite , and when the covering fraction is high ( @xmath40 ) , the radio spectrum turns down at low frequencies ( yegz ) .
as noted above , the western nucleus in arp 220 is best modelled with two separate regions : an inner circumnuclear disc ( cnd ) with a surrounding torus ( st ) .
we model the cosmic ray populations of the two regions independently , treating each region as a uniform slab .
however , the effects of absorption ( free free and @xmath2@xmath2 ) on the resulting radio and @xmath2-ray emission must be considered more carefully .
absorption occurs within each emission region , and in the case of the inner cnd , absorption also occurs as the emitted radiation moves through the external , st ( see the appendix for further details ) .
we perform @xmath41 tests following the approach described in yegz @xcite . comparing against radio observations for each nucleus , we vary magnetic field strength ( @xmath42 ) , wind speed ( @xmath43 ) , ionized gas density ( @xmath44 ) , and absorption fraction ( @xmath38 ) .
while magnetic field strength and wind speed both directly affect the total cosmic ray lifetimes , the ionized gas density and the absorption fraction only affect the emitted radio spectrum .
the free free emission and absorption coefficients are both directly proportional to the square of @xmath44 , and so , the frequency at which the radio spectrum flattens or turns down and the amount of free free emission at high frequencies both increase with @xmath44 .
observations in @xcite , @xcite and @xcite separate the integrated fluxes of the eastern and western nuclei from the total flux , allowing us to constrain parameters for each nucleus individually . as we do not have radio observations which are separable between the two regions of the western nucleus , we can not constrain the magnetic field strength in each region separately .
we therefore assume that the ratio between the magnetic field strength of the inner and outer regions of the western nucleus is equal to the square root of the ratio of the average gas densities , @xmath45 @xcite . thus , our magnetic field strength determination for the innermost western nucleus is an estimate that is guided by milky way observations .
when assuming the standard 10% cosmic ray acceleration efficiency , we find that agreement between the models and the observed radio data occurs only in a very narrow area of parameter space ( see fig .
the best - fitting models for the nuclei have magnetic field strengths limited to 1.0 mg for the western nucleus ( 3.5 mg in the cnd , estimated from scaling ) and 2.0 2.5 mg for the eastern nucleus ( see figs 2 and 3 and table 2 ) . as seen in fig .
3 , the total radio emission in both nuclei flattens at low frequencies , and in the eastern nucleus , the radio spectrum may be turning over completely .
this flattening of the radio spectra requires moderate to high absorption fractions of 50
100% in the eastern nucleus and low to moderate absorption fractions of 10 70% in the western nucleus .
in addition to moderate absorption fractions in each nucleus , we also find a high contribution from thermal emission to the total radio spectrum ( see fig .
3 ) , particularly in the western cnd where the majority of the radio emission is thermal above @xmath155 ghz . in part , this unusually high fraction of thermal emission is due to the inability of the model to effectively fit for free free absorption and free free emission simultaneously as seen in the eastern nucleus and in previous work ( see yegz ; * ? ? ?
the ability of the models to accurately fit the fraction of thermal emission is further strained by the complicated nature of the western nucleus and the lack of separable radio observations .
thus , in this particular case , the fractions of thermal emission in the best - fitting models have limited significance and do not necessarily contradict observations by @xcite which indicate more modest amounts of thermal emission ( @xmath46 ) .
the western nucleus arp 220 has a very complex structure which we greatly simplified .
while our two - zone density distribution reproduces the observed peak column density of @xmath47 @xmath48 , the mass is lower than that estimated by @xcite who derive the western nucleus gas mass from observations by the atacama large millimeter array ( alma ) of the submillimetre dust luminosity .
this yields a total gas mass of @xmath49 for the western nucleus or four times our adopted value .
we therefore explored the effect of increased gas mass on our model by tripling the mass in the western torus and ran a limited suite models with fixed parameters .
we set the spectral index to @xmath50 and ran @xmath41 tests over the entire range of magnetic field strengths and wind speeds but over a subset of the previously tested ionized gas densities and absorption fractions .
we ran tests on the western nucleus for acceleration efficiencies of 5% and 10% . in comparing the results of these models with a larger gas mass , we find that none of the tested models are within @xmath51 of the best - fitting model at the lower assumed gas mass .
further more , these results yield extremely short cosmic ray electron lifetimes such that the physical validity of the models are in question .
in addition to the higher gas mass estimates , @xcite also propose a geometry where the molecular gas in both nuclei is confined to a thin ( @xmath5210 pc ) disc .
the evolution of supernovae and cosmic ray interactions in this type of high molecular mass structure is beyond the scope of this study which is designed to estimate cosmic ray interaction rates in arp 220 but will need to be considered when arp 220 is detected in @xmath2-rays . in our earlier works
, we demonstrated that for a given ism , the yegz models are highly sensitive to the total flux of cosmic rays ( yegz ; * ? ? ? * ) .
this flux is primarily effected by the original energy input into cosmic rays and the advective time - scale , or escape fraction . the energy input into cosmic rays
is determined by the supernova rate and the assumed acceleration efficiency ( @xmath53 ) . within the uncertainty in the supernova rate , we vary acceleration efficiency from 5 to 20% . as shown above , for the standard 10% efficiency , the resulting best - fitting models are highly constrained in magnetic field strength , and we find that this is also true for an acceleration efficiency of 20% ( see table 2 ) .
however , for a lower acceleration efficiency of 5% , equivalent to a lower supernova rate , we find a much larger range of acceptable fits in the eastern nucleus with magnetic field strengths ranging from 4 to 7.5 mg and wind speeds spanning our entire tested range . as such , the best - fitting models for arp 220 are essentially independent of wind ( advection ) speed ( see figs 1 and 2 ) .
in contrast , a galactic wind was a vital component in modelling the cosmic ray populations of the starburst galaxies m82 and ngc 253 such that an extremely limited range of wind speeds resulted in fits within @xmath51 of the best - fitting models .
the wind speed determines the advective timescale for a galaxy and the fraction of cosmic rays which escape .
thus , wind speed is intrinsically tied to the proton calorimetry fraction for a galaxy which is closely related to the total radio and @xmath2-ray emission from a galaxy . other models for arp 220 have assumed fixed advection time - scales , thus ensuring proton calorimetry with the high gas densities in arp 220 @xcite . while our models agree with others in finding that the starburst regions of m82 and ngc 253 are only @xmath1540
60% proton calorimeters , we find that arp 220 s nuclei are 65 100% ( eastern ) and 90 100% ( western cnd ) proton calorimeters ( see fig .
lcccccc & supernova & average gas & cosmic ray & radiation field & magnetic field & magnetic field + & power & density & energy density & energy density & energy density & strength + & ( erg yr@xmath4 ) & ( @xmath16 ) & ( ev @xmath16 ) & ( ev @xmath16 ) & ( ev @xmath16 ) & ( @xmath54 g ) + milky way & @xmath55 & 1 & 1.4 & 0.3 & 0.9 & 6 + m82 & @xmath56 & 260 & 470 & 490 & 2200 & 300 + arp 220 east & @xmath57 & 7700 & 1100 & 40 000 & @xmath58 & 6500 + arp 220 west cnd & @xmath59 & 42 000 & 2500 & 440 000 & @xmath60 & 7000 + + despite the uncertainty in the calorimetry fraction and the total cosmic ray flux in the eastern nucleus , we can still use our best - fitting models to make a prediction on the emitted @xmath2-ray and neutrino fluxes from arp 220 . to calculate the possible @xmath2-ray flux ,
we apply the parameters of models within @xmath51 from our best - fitting radio model . combining each possible set of models from the eastern and western nucleus , we find that the resulting @xmath2-ray spectra peak around @xmath150.3 gev with a maximum flux of @xmath61 gev @xmath62 s@xmath4 ( see fig
. 4 ) . while this is roughly an order of magnitude lower than previous upper limits for arp 220 @xcite and _ fermi _ s differential sensitivity for four years of observations , it is only a factor of @xmath152 - 3 times smaller than the flux level of the recently detected ngc 2146 @xcite .
we also compared our @xmath2-ray flux with the differential sensitivity 50 h of observations with the future southern cta array ( see fig .
4 ) and find it to be only a factor of a few larger than our maximum flux .
arp 220 may still be detectable by _ fermi _ within the next several years and is a good target for cta , especially for energies near 1 tev .
in addition to making a prediction for the @xmath2-ray spectrum , we can use our same results from the radio emission to predict the neutrino flux from arp 220 .
proton interactions are responsible for the creation of secondary pions , both neutral and charged . while the neutral pions decay into @xmath2-rays , the charged pions decay into a neutrino and a muon which further decays into a secondary electron or positron and two more neutrinos .
the spectrum of the first neutrino from the decay of the charged pion is what we calculate here , as the calculation of the spectra of neutrinos produced during muon decay is more complex ( see * ? ? ?
the flux of our maximum model is roughly @xmath63 gev @xmath62 s@xmath4 at 0.1 pev , and at this energy , the range of possible models spans an order of magnitude in flux ( see fig .
current point source sensitivity limits for the northern sky for icecube are @xmath64 gev @xmath62 s@xmath4 , assuming a spectrum of @xmath65 @xcite .
thus , it seems unlikely that arp 220 will be detected as a point source during a similar time frame by icecube .
however , extreme ulirgs such as arp 220 should make a significant contribution to a diffuse neutrino background @xcite .
in addition to accounting for @xmath2-ray and neutrino emission in arp 220 , we have also take into account the effects of @xmath2@xmath2 absorption due to the intense radiation fields in the nuclei @xcite . at tev energies and above
, @xmath2-rays and infrared photons can interact to produce an electron / positron pair @xcite .
the resulting electrons will be of tev energies and most of their energy will be lost to emission of synchrotron x - rays @xcite .
beginning at @xmath152 5 tev , the opacity for @xmath2@xmath2 absorption in both nuclei is significantly greater than 1 .
this results in a steepening of the predicted @xmath2-ray spectrum at high energies ( see fig .
we find no such increase in slope in the neutrino flux as the steepening is an effect of interactions between the @xmath2-ray and the ambient radiation field and not the cosmic ray proton population .
therefore , in the case of arp 220 and other such ulirgs , the tev @xmath2-ray flux is an unreliable indicator of neutrino flux .
if the effects of spectral steepening by @xmath2@xmath2 absorption are accurate , then arp 220 is unlikely to be detected by cta or other ground based cherenkov telescopes above @xmath1510 tev .
in applying the yegz models to arp 220 , we find that the central starburst regions of arp 220 are moderate to complete cosmic ray proton calorimeters . as such ,
the leptonic cosmic ray population is dominated by secondary electrons and positrons .
the majority of these secondaries are produced at low energies ( e.g. , * ? ? ?
* ) and are likely a major contributor to heating of the ism via ionization @xcite .
based on our best - fitting models for the radio spectrum , we make predictions for both the @xmath2-ray and neutrino fluxes .
our maximum @xmath2-ray spectrum is a factor of a 2 5 lower than previous predictions by @xcite and less than a factor of 2 lower than those by @xcite . while the predicted @xmath2-ray flux will likely be detected by _
fermi _ in the future , under our model assumptions arp 220 is unlikely to be detected as a high energy neutrino point source with the current icecube observatory .
additionally , @xmath2@xmath2 absorption of the tev energy @xmath2-rays make the tev @xmath2-ray flux a poor indicator of the neutrino flux in ulirgs and other such systems with extremely intense infrared radiation fields .
in addition , we find that milligauss strength magnetic fields are still necessary to reproduce the observed radio fluxes from the starburst nuclei , even having assumed larger supernova rates than previous models by factors of 2 5 @xcite .
differences in assumed volume and radiation field energy density across the various models account for the similar best - fitting magnetic field strengths despite the range in assumed supernova rates .
the origins of milligauss strength magnetic fields in extreme starbursts and their impact on the evolution of these systems merit further examination . while the energy density in both magnetic and radiation fields is up from starbursts like m82 by two to three orders of magnitude
, the change in the ratio of their energy densities is up by less than an order of magnitude ( see table 3 ) .
conversely , we see a much larger change in the ratio of magnetic field energy density to cosmic ray energy density .
because the cosmic ray energy density depends on the particle energy loss rate , it does not increase at the same rate as the magnetic and radiation field energy densities ( yoast - hull , gallagher , zweibel , in preparation ) .
thus , the magnetic fields exceed energy equipartition with the cosmic rays by more than two orders of magnitude ( see table 3 ) .
this work was supported in part by nsf ast-0907837 , nsf phy-0821899 ( to the center for magnetic self - organization in laboratory and astrophysical plasmas ) , and nsf phy-0969061 ( to the icecube collaboration ) .
part of this research was carried out during jsg s appointment as a jubileumsprofessor at the chalmers university of technology .
we thank susanne aalto , kazushi sakamoto , dave sanders , nick scoville , and eskil varenius for conversations on arp 220 , justin vandenbroucke and reinhard schlickeiser for discussions regarding the modelling , and francis halzen for his help and support .
additionally , we thank the referee for their helpful comments .
our single - zone model uses a simple solution to the radiative transfer equation of ( e.g. , * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) @xmath66 where @xmath67 is the radiative flux prior to absorption , @xmath68 is the radiative flux after absorption , and @xmath69 is the optical depth for either free - free absorption or @xmath2-@xmath2 absorption .
this is still the solution for the eastern nucleus and the surrounding torus in the western nucleus . in the western cnd ,
we must account for a standard emission and absorption region with an additional , external absorbing region .
this observed flux is given by @xmath70 where @xmath67 is still the radiative flux prior to absorption , @xmath71 is the optical depth for @xmath2-@xmath2 or free - free absorption in the emission region , and @xmath72 is the optical depth in the external , absorbing region .
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we find that @xmath0 of cosmic rays are absorbed in these regions due to their huge molecular gas contents , and thus , the nuclei of arp 220 nearly complete proton calorimeters .
as the cosmic ray protons collide with the interstellar medium , they produce secondary electrons that are also contained within the system and radiate synchrotron emission . using results from @xmath1 tests between the model and the observed radio spectral energy distribution
, we predict the emergent @xmath2-ray and high - energy neutrino spectra and find the magnetic field to be at milligauss levels . because of the extremely intense far - infrared radiation fields , the @xmath2-ray spectrum steepens significantly at tev energies due to @xmath2@xmath2 absorption .
neutrinos cosmic rays galaxies : individual : arp 220 galaxies : starburst gamma rays : galaxies radio continuum : galaxies | introduction
arp 220 physical properties
models & results
discussion and conclusions
acknowledgements
two zone models | arxiv |
this study was supported by the danish national research foundation through the center for models of life and by physics of geological processes , a center of excellence at the university of oslo .
email data from the university of oslo were collected with the help and support of ingar vindenes and knut borge at the univ . of oslo center for information technology ( usit ) .
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lin , h. sundaram , k. s. candan , l. xie , and a. kelliher , in proceedings of the 4th international aaai conference on weblogs and social media , 34 ( 2010);http://konect.uni - koblenz.de / networks / munmun_twitterex_at h. chun _ et al . _ , imc08 , vouliagmeni , greece , october 2022 , 2008 . | in communication networks structure and dynamics are tightly coupled .
the structure controls the flow of information and is itself shaped by the dynamical process of information exchanged between nodes . in order to reconcile structure and dynamics , a generic model , based on the local interaction between nodes ,
is considered for the communication in large social networks . in agreement with data from a large human organization ,
we show that the flow is non - markovian and controlled by the temporal limitations of individuals .
we confirm the versatility of our model by predicting simultaneously the degree - dependent node activity , the balance between information input and output of nodes and the degree distribution . finally , we quantify the limitations to network analysis when it is based on data sampled over a finite period of time .
limitations on the processing capacities of nodes and links have a profound impact on the flow of information in online communication networks @xcite , the spreading of diseases in human encounter networks @xcite , and in social networks @xcite , where links between interacting individuals can be highly volatile @xcite .
it is often assumed that communication takes place in an unrestrained way on a set of established connections , thereby neglecting , that structure and dynamics are interdependent .
here we consider the evolution of a network where links form as a result of non - markovian interaction between nodes . in a time - limited environment , communication demands prioritization which is evident from the analysis of correspondence patterns @xcite .
hence , information flow on a network is a result of individuals choices which are influenced by the state of surrounding nodes . in natural @xcite and online @xcite social networks ,
the nodes activity is a non - trivial function of their degree .
the activity level can be quantified by the number of social relationships simultaneously maintained by an individual .
this number has been suggested to reflect basic cognitive capabilities of primates @xcite and humans @xcite .
here we model a network of individuals acting under time constraints and compare with a complete dataset of email communication in a large organization .
the model is discussed in the context of other communication networks .
we predict the information processing capacity of individuals as well as the structure of the network that they form .
we use representative communication data from a large social organization , the university of oslo .
the data comprise a complete time - ordered list of @xmath0 emails between 5600 employees , 30 000 students and approximately @xmath1 people outside the organization over a period of three months ( sep - nov 2010 ) . the email content was not recorded and identities of individuals were encrypted .
we limit the influence of unsolicited bulk emails by disregarding those simultaneously sent to more than five recipients
. however , the results are not sensitive to the filtering of bulk emails @xcite .
previous work on email data has considered static network structures @xcite .
_ results _ we show that the communication is non - markovian by comparing random and directed information flow : ( i ) random flow is given by random walks on the network .
the walker follows an empirical time - independent jump - probability @xmath2 from node @xmath3 to node @xmath4 .
the sum is taken over all nodes and @xmath5 is the number of emails sent from @xmath3 to @xmath4 during the timespan of the data .
( ii ) directed flow is given by the chronological email exchange .
starting from a random node @xmath3 , we wait for @xmath3 to send an email , say to @xmath4 .
we then jump to @xmath4 and wait for the next message @xmath4 sends either back to @xmath3 or to a new node @xmath6 . repeating this ,
we obtain a finite trajectory within the timespan of the data .
the number of unique nodes visited by the directed and random flow as function of the number of jumps are compared by averaging over trajectories originating from all nodes ( fig .
[ fig : markov ] ) . on average ,
directed flow visits relatively fewer nodes than random flow , indicating a significant correlation between sent and received messages . '' .
the solid line is a best fit by eq .
( [ eq : delta ] ) .
the dotted lines mark the peak and the dashed diagonal line shows @xmath7 .
inset : out - degree distribution for model and empirical data .
the dashed line denotes the scale - break @xmath8 .
mean degree is @xmath9 ( twitter data yields a mean degree of @xmath10 and a similar exponent for the degree distribution @xcite ) .
note the double - log scales.,title="fig : " ] + our model requires nodes to perform a trade - off between replying to others and initiating new conversations .
specifically , consider @xmath11 nodes , each initially connected to one other node .
the nodes have a limited capacity and can send a maximum of @xmath12 messages in a timestep @xmath13 day .
the dynamics follows from three possible actions for a node @xmath3 of out - degree @xmath14 : ( a ) @xmath3 processes received emails and if @xmath3 has sent less than @xmath12 messages , any received email is replied to with a probability proportional to the sender s degree .
emails not replied to within @xmath15 are subsequently deleted . in total , @xmath16 replies are sent by this action .
\(b ) if less than @xmath12 emails have been sent in ( a ) , the remaining capacity @xmath17 is available for sending messages , called @xmath18 , to previously established contacts .
the probability of sending a message to a contact is given by a constant @xmath19 .
hence , granted sufficient capacity on average @xmath20 messages are initiated by @xmath3 .
nodes with low @xmath14 will generally not reach their full capacity .
\(c ) nodes establish new contacts by sending requests with a probability @xmath21 .
the probability that a request is sent to a node @xmath4 is proportional to the degree of @xmath4 , @xmath22 .
a link is established between @xmath3 and @xmath4 , if @xmath4 in the next timestep according to ( a ) replies to @xmath3 . in reality , contacts might as well be established by face - to - face encounters , i.e. via channels not recorded explicitly in our data .
the total number of messages @xmath23 sent by a node in @xmath15 is the sum @xmath24 .
analogously , messages received by a node in the same timestep are termed @xmath25 .
nodes have an average lifetime @xmath26 and are therefore removed from the network with a probability @xmath27 . for every node removed ,
a new node with a single random connection to an existing node is introduced .
@xmath26 is estimated to be @xmath28 years from the known mean email user turnover time in the organization .
the parameters @xmath19 , @xmath21 and @xmath12 are determined below . according to ( c ) ,
a link is established between @xmath3 and @xmath4 if one of the nodes sends a message to the other and receives a reply .
the probability , @xmath29 , that a message is sent from @xmath3 to @xmath4 in @xmath15 is proportional to @xmath22 , @xmath30 where we in the approximation assume that @xmath31 . according to ( a ) , the mean number of requests that @xmath4 receives during a timestep is proportional to @xmath21 and @xmath22 .
the probability for @xmath4 to reply to a request from nodes of degree @xmath6 is proportional to @xmath32 , where @xmath33 is a constant and @xmath34 is the number of nodes with degree @xmath6 .
the number of replies written by @xmath4 is the product of eq .
( [ eq1 ] ) and the integral over nodes @xmath35 since nodes reply to requests and therefore establish new links with a probability proportional to the sender degree , @xmath36 , the mean degree @xmath37 of a node s contacts is @xmath38 , a number generally larger than the mean degree @xmath39 ( fig .
[ fig : rec_degree_vs_k ] ) . ) and weighted by the number of messages sent to recipients ( @xmath40 ) .
the horizontal line shows @xmath41 .
the curves marked by `` @xmath42 '' and `` @xmath43 '' are analogous to the unweighted case but for half , respectively , one quarter of the observational period .
dashed lines show projection of nodes with two values of @xmath6 for a varying observation window .
note the double - log scale . ]
consequently the average degree - increase of nodes of degree @xmath6 per timestep becomes @xmath44 .
the factor of 2 reflects the symmetry of sending and replying .
the rate of losing links is inversely proportional to @xmath26 , @xmath45 .
hence , the net degree - growth rate becomes @xmath46 , where @xmath47 .
as long as a node has sufficient capacity to reply to all requests its degree increases approximately exponentially , @xmath48 .
the degree distribution follows from the consideration that during @xmath15 , a fraction of nodes @xmath34 of degree @xmath6 changes their degree , @xmath49 $ ] , and a fraction @xmath50 is removed .
a continuum - limit approximation yields @xmath51-\frac{n(k)}{\tau}\;. \label{eq : time_evol}\ ] ] the steady - state solution has the form @xmath52 , where @xmath53 .
the constant @xmath54 is fixed by integrating eq .
( [ eq : time_evol ] ) over @xmath6 and by demanding that the total number of nodes @xmath55 be constant
. this yields @xmath56 .
the condition @xmath57 bounds the power - law exponent : @xmath58 .
the data yield @xmath59 ( fig .
[ fig : sent_vs_rec ] inset ) .
so far we have assumed that nodes have infinite capacity . as a node s degree increases , it receives more messages and this assumption becomes invalid .
consider the number of messages received by @xmath3 per timestep .
contact requests from other nodes amount to @xmath60 messages .
the senders of these messages are drawn from a distribution @xmath61 . the probability for @xmath3 to receive a message from its contacts is proportional to @xmath19 and @xmath14 , hence @xmath62 .
analogously , as defined in ( a ) , @xmath3 issues @xmath63 requests to recipients distributed according to @xmath64 ( where @xmath65 ) due to the weighting of probabilities by the recipient degree . in the same timestep
@xmath3 sends @xmath66 messages to its contacts .
finally we consider back - and - forth communication .
for every message sent by @xmath3 to @xmath4 , a response is returned with a probability @xmath67 ( eq . [ eq : prob_send_reply ] ) . in steady - state ,
the number of messages sent is identical for all timesteps and therefore @xmath3 receives @xmath68 replies to messages sent in the previous timestep .
@xmath16 is the number of messages @xmath3 sends in response to messages received from others which again is a sum over contributions from the actions ( a)-(c ) : @xmath69 the terms on the right are respectively , requests from any node in the network ( distributed as @xmath70 ) , messages from existing contacts ( distributed as @xmath71 ) , and back - and - forth messages ( distributed as @xmath72 ) .
each iteration of back - and - forth communication acts as a shift in the distribution of recipients relative to the distribution of senders @xmath73 .
the distribution @xmath72 accounts for all high - order shifts . to close the equations for @xmath74 and @xmath16
, we use that the reply probability for each iteration is reduced by a factor @xmath33 to approximate @xmath75 . inserting eq .
( [ eq : alpha2 ] ) , @xmath76 and @xmath77 in eq .
( [ eq5 ] ) yields @xmath78 where we introduce @xmath79 .
summing over @xmath80 , @xmath81 and @xmath82 we get @xmath83 here the first three terms ( referred to as @xmath84 ) are messages sent to recipients selected according to @xmath71 and with mean degree @xmath37 .
the other terms , @xmath85 , are messages to recipients distributed according to the higher order distribution @xmath86 which has a mean @xmath87 and contribute significantly only for large @xmath14 .
the mean of the weighted recipient degree ( weighted by number of messages received ) is @xmath88 , which departs from @xmath37 when @xmath85 becomes appreciable ( fig .
[ fig : rec_degree_vs_k ] ) . for low @xmath14 ( @xmath89 ) ,
the ratio of sent to received messages becomes @xmath90 .
conversely , @xmath91 when @xmath92 , hence an average node has a `` balanced '' email account .
when @xmath14 becomes larger than @xmath39 , @xmath3 will increasingly receive requests and responses to its messages ( fig .
[ fig : sent_vs_rec ] ) . the _ dunbar number _
@xmath93 is the degree where @xmath23 reaches the capacity limit ( @xmath94 ) and @xmath95 is maximal .
the scale break in the degree distribution ( @xmath8 ) , fig .
[ fig : sent_vs_rec ] ( inset ) , and @xmath96 , fig .
[ fig : dunbar ] , nearly coincide .
in fact @xmath97 is related to @xmath98 because nodes beyond @xmath98 have a reduced probability to form new links . to determine @xmath97 , consider the evolution of the nodes degree in the limit where all capacity is used for replying , hence @xmath99 . using that @xmath100 , we get @xmath101 which in turn yields @xmath102 . @xmath97 is found by solving this implicit equation .
@xmath98 then follows from eq .
( [ eq : delta ] ) .
, @xmath103 ( eq . [ eq : delta ] ) and @xmath95 is a superposition of a term @xmath104 due to the final quadratic term and a decaying term @xmath105 from the constant . at @xmath106 , nodes limited to @xmath12 messages per day , hence @xmath107 . ]
the parameters @xmath108 , @xmath109 and @xmath110 are determined by the data in fig .
[ fig : sent_vs_rec ] . from @xmath21 and @xmath111
we obtain @xmath112 .
larger @xmath12 increases the limit of @xmath23 .
@xmath21 is constrained by the offset at low @xmath113 and @xmath19 effects the skewness of the curve which follows from analysis of eqs .
( [ eq : alpha2 ] ) and ( [ eq : delta ] ) . fig .
[ fig : dunbar ] shows the model prediction of @xmath114 and the corresponding email data .
we complement our analysis with numerical computations . using a large number of nodes , @xmath115 , we iterate actions ( a)-(c ) until steady - state is reached . while the mean - field prediction ( figs .
[ fig : rec_degree_vs_k ] and [ fig : dunbar ] ) is close to the numerical solution , some differences exist , e.g. at small @xmath6 , @xmath116 is not a strict power - law in the numerical solution due to the discreteness of @xmath6 .
further , the simulation gives a smooth peak in @xmath95 ( fig .
[ fig : dunbar ] ) which is narrower than in the empirical data .
this is due to slight overestimation of the repeated back - and - forth communication between well - connected nodes ( @xmath117 ) relative to the data .
we have also simulated the information flow ( fig .
[ fig : markov ] ) and achieve similar results .
finally , the average local clustering coefficient of the empirical and simulated networks is relatively small , @xmath118 for both ( similar clustering coefficient @xmath119 @xcite and @xmath120 to @xmath121 have been reported for other communication networks @xcite ) .
we further checked the robustness of the model to variations @xcite .
_ discussion _ the data were recorded over three months and the communication network is therefore a finite - time projection of the real network .
the projection reduces the number of links .
more active links will more likely persist through the projection than less active links .
fig .
[ fig : rec_degree_vs_k ] shows the mean recipient degree @xmath122 as function of the sender degree @xmath14 for three observation time intervals .
consider again eq .
( [ eq : delta ] ) and remember that recipients of the @xmath84 ( @xmath85 ) messages are distributed as @xmath71 ( @xmath86 ) .
when observing only a single day , the probability for an out - link between @xmath3 to @xmath4 not to be active is @xmath123 . for @xmath124 days
we obtain @xmath125 . to produce the projected curves in fig .
[ fig : rec_degree_vs_k ] , @xmath126 is applied to both axes , @xmath6 and @xmath127 .
averaging w.r.t .
all recipients @xmath4 ( distributed as @xmath71 ) , the projected sender out - degree becomes @xmath128 .
similarly one can consider the projection of the mean recipient degree leading to a similar reduction in the degree for finite - time data .
for example , consider the data for the quarter period ( @xmath129 ) in fig .
[ fig : rec_degree_vs_k ] .
we have @xmath130 and therefore @xmath131 hence less than half the links persist . _ concluding remarks _ the finite capacity of agents in social networks induces an upper limit on the number of possible interactions @xcite .
we propose a comprehensive model that reconciles structure and dynamics of networks with finite capacity agents that dynamically form or lose links . in agreement with a complete set of email data and results from other social networks @xcite
, our model predicts a scale - free degree distribution up to a distinct scale - break induced by the capacity limit .
further , as agents gain importance in the network , the per - link - activity first increases with node - degree , peaks at intermediate degrees and declines at large degrees . the model and data therefore support the hypothesis of a general limit on the number ( 150 - 250 ) of active social relations that an individual can maintain @xcite and is in agreement with empirical observations on social networks @xcite . | acknowledgments | arxiv |
the author expresses special thanks to h. horiuchi and a. tohsaki for their valuable and helpful discussion and suggestions on the present work . the fruitful discussion with m. itoh
is highly appreciated .
thanks are also due to g. rpke , p. schuck , t. yamada , and b. zhou for their stimulating discussion .
this work was partially performed with the financial support by hpci strategic program of japanese mext , jsps kakenhi grant number 25400288 , and riken incentive research projects .
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120 * , 1139 ( 2008 ) . | the excited states in @xmath0 are investigated by using an extended version of the so - called tohsaki - horiuchi- schuck - rpke ( thsr ) wave function , where both the @xmath1 condensate and @xmath2 cluster asymptotic configurations are included .
a new method is also used to resolve spurious continuum coupling with physical states .
we focus on the structures of the `` hoyle band '' states , @xmath3 , and @xmath4 states , which are recently observed above the hoyle state and of the @xmath5 and @xmath6 states , which are also quite recently identified in experiment .
their resonance parameters and decay properties are reasonably reproduced .
all these states have gaslike configurations of the @xmath1 clusters with larger root mean square radii than that of the hoyle state .
the hoyle band is not simply considered to be the @xmath7 rotation as suggested by previous cluster model calculations , nor to be a rotation of a rigid - body triangle - shaped object composed of the @xmath1 particles .
this is mainly due to the specificity of the hoyle state , which has the @xmath1 condensate structure and gives a rise to the @xmath5 state with a prominent @xmath8 structure as a result of very strong monopole excitation from the hoyle state .
nuclear clustering is one of the fundamental degrees of freedom in nuclear excitation @xcite .
the hoyle state , the second @xmath9 state at @xmath10 mev in @xmath0 , as a typical example of the cluster states , has a long history ever since it was predicted by f. hoyle @xcite and subsequently observed by cook _
et al . _
@xcite as a key state in a synthesis of @xmath0 in stellar evolution .
the microscopic and semi - microscopic cluster models have clarified that the hoyle state has the structure of the @xmath11 particle loosely coupling in an @xmath12-wave with the @xmath13 core @xcite , not like a linear - chain structure of the @xmath1 particles proposed by morinaga in 1950 s @xcite . in the last decade , however , the aspect of the @xmath11 condensate , where the @xmath1 clusters occupy an identical @xmath12-orbit , has triggered a great interest , since the so - called tohsaki - horiuchi - schuck - rpke ( thsr ) wave function @xcite , which has the @xmath11 condensate character , was shown to be equivalent to the hoyle state wave function obtained by solving the equations of the full @xmath1 resonating group method ( rgm ) or generator coordinate method ( gcm ) @xcite .
on the other hand , the excited states of the hoyle state have been highlighted by recent great developments in experimental studies .
the second @xmath14 state @xmath15 , which had been predicted at a few mev above the hoyle state by the cluster model calculations , was recently confirmed by many experiments @xcite .
the gcm and rgm calculations propose that the @xmath3 state is built on the hoyle state as a rotational member with a @xmath8 configuration .
freer _
et al .
_ quite recently reported a new observation of the @xmath16 state at @xmath17 mev , which they consider to compose the `` hoyle band '' @xcite , together with the @xmath18 and @xmath3 states .
it is proposed that this band is formed by a rotation of a rigid @xmath1 cluster structure with an equilateral triangle shape based on @xmath19 symmetry @xcite , which is , however , not consistent with the picture of loosely coupled @xmath8 structure or the @xmath1 gaslike structure . besides the @xmath14 and @xmath16 states ,
a @xmath20 state at @xmath21 mev with a broad width , @xmath22 mev , has been known for a long time . however , quite recently itoh _ et al .
_ decomposed the broad @xmath20 state into the @xmath5 and @xmath6 states at @xmath23 mev and @xmath24 mev , with the widths of @xmath25 mev and @xmath26 mev , respectively @xcite .
this observation of the two @xmath20 states is consistent with theoretical prediction done by using the orthogonality condition model ( ocm ) combined with the complex scaling method ( csm ) and the analytical continuation of coupling constant ( accc ) method @xcite .
this was later on confirmed by another theoretical calculation using the ocm and csm with higher numerical accuracy @xcite .
on the other hand , in the antisymmetrized molecular dynamics ( amd ) @xcite , fermionic molecular dynamics ( fmd ) @xcite , and gcm calculations @xcite , the observed @xmath5 state seems to be missing . the @xmath5 state given by the amd and fmd , which may correspond to the observed @xmath6 state ,
is dominated by a linear - chain - like configuration of the @xmath1 clusters and is not inconsistent with the @xmath5 state obtained by the gcm calculation @xcite , or with the @xmath6 state in ref .
@xcite , where @xmath27_0 $ ] configuration is dominant .
it should also be mentioned that the authors in ref .
@xcite claimed that the @xmath5 state has an @xmath12-wave dominant structure with more dilute density than that of the hoyle state .
these are also consistent with the observed decay properties of the @xmath5 and @xmath6 states that the former only decays into the @xmath28_0 $ ] channel and the latter decays into the @xmath27_0 $ ] channel with a sizable partial @xmath11-decay width @xcite . in this letter , we investigate the structures of the positive parity excited states above the @xmath1 threshold by using an extended version of the thsr wave function @xcite so as to include @xmath29 asymptotic configurations with a treatment of resonances . in particular , we focus on the `` hoyle band '' ( the @xmath18 , @xmath3 , and @xmath4 states state at @xmath17 mev as the @xmath4 state , for simplicity , though it is located lower than the @xmath16 state at @xmath30 mev which forms the ground state rotational band . ] ) , and the @xmath5 and @xmath6 states , together with the corresponding experimental data , though we also obtained some other positive parity excited states . the extended version of the thsr wave function is written as follows : @xmath31 , \label{eq:1}\end{aligned}\ ] ] where the @xmath32 is an intrinsic wave function of the @xmath11 particle , where the @xmath33 configuration of the four nucleons is assumed with the size parameter @xmath34 , which is kept fixed at @xmath35 fm as almost the same value as at free space .
@xmath36 is the jacobi coordinates between the @xmath1 particles and @xmath37 , for @xmath38 .
this is a fully microscopic wave function and every nucleons are antisymmetrized by @xmath39 .
@xmath40 is a usual angular - momentum - projection operator .
this wave function is characterized by the parameters @xmath41 and @xmath42 , which correspond to the sizes of the @xmath43 core and the remaining @xmath11 particle center - of - mass ( c.o.m . )
motion , respectively .
in the subsequent calculations , the axial symmetric deformation is assumed , i.e. @xmath44 @xmath45 , for simplicity .
we should note that the case of @xmath46 results in the original thsr wave function , where the c.o.m .
motions of the @xmath1 particles relative to the total c.o.m . position are condensed into a lowest energy @xmath47 orbit , reflecting the bosonic feature @xcite .
thus this new thsr wave function is a natural extension of the original version , so that taking @xmath48 allows for the @xmath2 cluster structure , deviating from the identical @xmath1-cluster motion for @xmath46 .
this new wave function still gives gaslike cluster structure , as the original thsr wave function does , not being featured by the relative distance parameter between the @xmath43 and @xmath11 clusters .
it should also be mentioned that the thsr - type wave functions were recently shown to give the best description for various cluster states such as the @xmath49 inversion doublet band in @xmath50 @xcite , @xmath1- and @xmath51-linear - chain states @xcite , and @xmath52 cluster states in @xmath53 @xcite , etc . for the excited states above the @xmath1 threshold
, it is well known that the application of the bound state approximation gives accidental mixing between spurious continuum states and resonances . by using the fact that the root mean square ( r.m.s . ) radii of spurious continuum states
are calculated to be extremely large within the bound state approximation , we developed a new method to remove the spurious components @xcite .
first we diagonalize the operator of mean square radius as follows : @xmath54 where @xmath55 is the total c.o.m . position .
we then remove out of the present model space the eigenstates belonging to unphysically large eigenvalues . by taking the following bases , @xmath56 with @xmath57 satisfying @xmath58 , we diagonalize hamiltonian as follows : @xmath59 for hamiltonian
, we adopt volkov no . 2 force @xcite , with the strength parameters slightly modified @xcite , as effective nucleon - nucleon interaction . the cutoff radius is now taken to be @xmath60 fm . for diagonalizing the operator of r.m.s .
radius in eq .
( [ eq : cutoff1 ] ) , we adopt @xmath61 mesh points for the four - parameter set , @xmath62 , up to around @xmath63 fm .
since the present extended thsr wave function can include @xmath2 asymptotic form by taking the large values of the two width parameters @xmath41 and @xmath42 , the @xmath2 continuum components , as well as the @xmath1 continuum components , can be successfully removed by imposing the cut off for the mean square radius @xmath58 .
the more details will be shown in a forthcoming full paper .
although we could not obtain the excited states except for the @xmath18 and @xmath3 states by using the original thsr wave function @xcite , we can now obtain the other observed @xmath5 , @xmath6 , and @xmath4 states by using the present extended thsr wave function with a treatment of resonances . since all these states are resonance states above the @xmath1 threshold , we then calculate the partial widths of the @xmath11 particle decaying into @xmath64_j$ ] channel , which we simply denote as @xmath65_j$ ] , based on the @xmath66-matrix theory @xcite , where we use the following relations , @xmath67_j}= 2p_l(ka ) \gamma^2_{[i , l]_j } , \
\gamma^2_{[i , l]_j}=\frac{\hbar^2}{2\mu a } |a{\cal y}_{[i , l]_j}(a)|^2,\label{eq : width}\ ] ] where @xmath68 is the penetrability calculated from the coulomb wave functions , and @xmath69 , @xmath70 , and @xmath71 are the wave number of the relative motion , the channel radius , and the reduced mass , respectively .
@xmath72_j}(r)$ ] is the @xmath11 reduced width amplitude ( rwa ) defined below , @xmath73_j}(r)\hspace{-0.1cm}=\hspace{-0.1cm}\sqrt{\frac{12!}{4!8!}}\langle [ \phi_{i}(\be),y_{l}(\vc{\hat \xi}_2)]_{jm}\frac{\delta(\xi_2-r)}{\xi_2 ^ 2}\phi(\alpha ) | \psi^{(\lambda)}_{jm } \rangle , \label{eq : rwa}\ ] ] where @xmath74 is the eigenfunction in eq .
( [ eq : eigenwf ] ) , @xmath75 .
[ cols="^,^,^,^,^,^,^,^ " , ] we show in table [ tab:3 ] the calculated @xmath12-factors of the @xmath76 components , which can be defined below , @xmath77}(j)=\int dr \big ( r{\cal y}_{[i , l]_j}(r ) \big)^2 .
\label{eq : sfact}\ ] ] we can see that except for the @xmath6 state , all the states have the largest contribution from the @xmath78_j$ ] channel .
this supports the idea of @xmath2 rotation for the hoyle band , where the @xmath43 core is in the @xmath20 ground state . on the other hand ,
the hoyle state is considered to be the @xmath1 condensate state , where the @xmath1 clusters mutually move in an identical @xmath12-wave . since the ground state of @xmath43
is composed of weakly interacting @xmath79 clusters coupled loosely in a relative @xmath12-wave , it is natural that the hoyle state , with the @xmath11 condensate structure , also has a large overlap with the @xmath8 structure .
this is the same situation as for the @xmath51 condensate state in @xmath80 discussed in refs .
@xcite , which has a large overlap with the @xmath81 structure . however , the @xmath1 condensate structure in the hoyle state is not the same as the usual @xmath8 rotation , in which the remaining @xmath11 cluster orbits outside the @xmath43 core .
namely in the hoyle state , the remaining @xmath11 cluster also orbits inside the @xmath43 core and the independent @xmath1-cluster motion in an identical @xmath47-orbit is realized . as a result
, the hoyle state gains an extra binding , and hence its energy position is considered to be made lower than the @xmath82 line , as shown in fig .
[ fig:1 ] . the same effect is also argued in the study of the @xmath51 condensate and @xmath81 rotational band @xcite , where the @xmath51 condensate is mentioned as `` complete condensate '' and the @xmath81 state as `` local condensate '' . due to the existence of the `` complete condensate '' , a higher @xmath20 excited state , which has the prominent @xmath8 structure , with the remaining @xmath11 cluster orbiting outside the @xmath43 core , appears as a higher nodal state , that is the @xmath5 state .
in fact , we can see in table [ tab:2 ] that the @xmath5 state is strongly connected with the hoyle state by a monopole excitation .
the calculated strength @xmath83 @xmath84 is much larger than the other transitions , in spite of the fact that the @xmath85 strength between the hoyle and ground states @xmath86 @xmath84 is still strong enough as to be comparable to the single nucleon strength @xcite . _
j=[0,0]_0 $ ] channel , @xmath87_0}(r)$ ] in eq .
( [ eq : rwa ] ) , for the @xmath88 , @xmath18 , and @xmath5 states . ] in fig
. [ fig:2 ] , the rwas of @xmath89_0 $ ] channel for the @xmath18 and @xmath5 states are shown together with that for the ground state . while the rwa for the ground state has two nodes , that for the @xmath5 state has four nodes and for the hoyle state the nodal behaviour almost disappears and only a remnant of three nodes can be seen as an oscillatory behaviour .
since the outmost nodal position corresponds to a radius of repulsive core between the core @xmath43 and the @xmath11 cluster , due to the effect of the pauli principle , the disappearance of the nodes for the hoyle state indicates a dissolution of the @xmath43 core , and hence formation of the @xmath1 condensate . on the other hand ,
the @xmath5 state , which is excited from the hoyle state by the monopole transition , recovers the distinct nodal behaviour and , with one additional node , forms a higher nodal @xmath8 structure . in table
[ tab:3 ] , the @xmath6 state is shown to have the component of @xmath90_0 $ ] channel dominantly , which gives a rise to non - negligible partial decay width into this channel , consistently with the experimental information , as mentioned above .
we also mention that the @xmath3 state also includes non - negligible mixture from the @xmath91_2 $ ] channel and @xmath4 states from the @xmath90_4 $ ] channel and smaller amount from the @xmath92_4 $ ] channel .
these mixtures also deviate the @xmath3 and @xmath4 states from a pure @xmath29 rotational structure .
we will discuss this point of view in a forthcoming paper . in conclusion ,
the use of the extended thsr wave function allows us to obtain the wave functions of the hoyle band and @xmath5 and @xmath6 states , which are recently confirmed by experiments .
the calculated @xmath11-decay widths and the decay properties of these states are in good agreement with the experimental data .
all these states are shown to have large r.m.s .
radii and hence gaslike @xmath1-cluster structures .
the @xmath18 , @xmath3 , and @xmath4 states are not considered to form a simple @xmath8 rotational band , due to the specificity of the hoyle state with the @xmath1 condensate feature , which allows the @xmath5 state to have a prominent @xmath8 structure as a result of the strong monopole excitation . | acknowledgements | arxiv |
i zw 18 remains the most metal - poor blue compact dwarf ( bcd ) galaxy known since its discovery by sargent & searle ( 1970 ) .
later spectroscopic observations by searle & sargent ( 1972 ) , lequeux et al . (
1979 ) , french ( 1980 ) , kinman & davidson ( 1981 ) , pagel et al .
( 1992 ) , skillman & kennicutt ( 1993 ) , martin ( 1996 ) , izotov & thuan ( 1998 ) , vlchez & iglesias - pramo ( 1998 ) , izotov & thuan ( 1999 ) and izotov et al .
( 1999 ) have confirmed its low metallicity with an oxygen abundance of only @xmath5 1/50 the solar value .
zwicky ( 1966 ) described i zw 18 as a double system of compact galaxies , which are in fact two bright knots of star formation with an angular separation of 58 .
these two star - forming regions ( fig .
[ fig1 ] ) are referred to as the brighter northwest ( nw ) and fainter southeast ( se ) components .
later studies by davidson , kinman & friedman ( 1989 ) and dufour & hester ( 1990 ) have revealed a more complex optical morphology with additional diffuse features .
the most prominent diffuse feature , hereafter i zw 18c ( fig .
[ fig1 ] ) , is a blue irregular star - forming region @xmath5 22 northwest of the nw component .
dufour , esteban & castaeda ( 1996a ) , izotov & thuan ( 1998 ) and van zee et al .
( 1998 ) have shown i zw 18c to have a systemic radial velocity equal to that of the ionized gas in the nw and se components , thus establishing its physical association to i zw 18
. furthermore , van zee et al . (
1998 ) have shown that i zw 18c is embedded in a common h i envelope with the nw and se components .
many studies have been focused on the evolutionary state of i zw 18 .
searle & sargent ( 1972 ) and hunter & thronson ( 1995 ) have suggested that it may be a young galaxy , recently undergoing its first burst of star formation .
the latter authors concluded from _
hubble space telescope _
( _ hst _ ) images that the colors of the diffuse unresolved component surrounding the se and nw regions are consistent with a population of b and early a stars , i.e. with no evidence for older stars .
ongoing massive star formation in i zw 18 is implied by the discovery of a population of wolf - rayet stars in the nw component ( izotov et al .
1997a ; legrand et al . 1997 ) . from the analysis of color - magnitude diagram ( cmd ) based on _ hst _ images , dufour et al .
( 1996b ) concluded that star formation in i zw 18 began at least 30 50 myr ago and continuing to the present .
martin ( 1996 ) and dufour et al .
( 1996b ) have discussed the properties of expanding superbubbles of ionized gas driven by supernova explosions and have inferred dynamical ages of 15 27 myr and 13 15 myr respectively .
recently , aloisi , tosi & greggio ( 1999 ) have discussed the star formation history in i zw 18 using the same _ hst _ wfpc2 archival data ( i.e. those by hunter & thronson ( 1995 ) and dufour et al .
( 1996b ) ) .
they compared observed cmds and luminosity functions with synthetic ones and concluded that there were two episodes of star formation in i zw 18 , the first one occuring over the last 0.5 1 gyr , an age more than 10 times larger than that derived by dufour et al .
( 1996b ) , and the second one with more intense activity taking place between 15 and 20 myr ago .
no star formation has occurred within the last 15 myr .
stlin ( 2000 ) from _ hst _ nicmos @xmath10 and @xmath11 observations concluded that a 1 5 gyr old stellar population is present .
the component i zw 18c has not been studied in such detail mainly because of its faintness .
its flux - calibrated optical spectra ( izotov & thuan 1998 ; van zee et al .
1998 ) reveal a blue continuum with weak balmer absorption features and faint h@xmath0 and h@xmath1 in emission .
such spectral features suggest that the h ii region is probably ionized by a population of early b stars .
dufour et al .
( 1996b ) found in a @xmath12 vs. @xmath13 cmd analysis of i zw 18c a well - defined upper stellar main sequence indicating an age of the blue stars of @xmath5 40 myr .
however , numerous faint red stars were also present in the cmd , implying an age of 100 300 myr .
those authors concluded that i zw 18c consists of an older stellar population with an age of several hundred myr , but which has experienced recently a modest starburst in its southeastern half as evidenced by the presence of blue stars and h@xmath0 emission .
aloisi et al .
( 1999 ) estimated an age for i zw 18c not exceeding 0.2 gyr .
we use here high signal - to - noise 4.5 m mmt , 4 m kpno and keck ii spectroscopy to study the evolutionary status of i zw 18c .
we also discuss the nature of the extended emission in the outermost regions of i zw 18 .
our motivation for this study is the following .
until now , age estimates for the stellar populations in i zw 18c are based solely on _ hst _ cmds . while in principle cmd studies are powerful tools for studying stellar populations , they critically depend on the adopted distance to the galaxy and the interstellar extinction , which are a priori unknown .
we use here distance - independent techniques based on spectroscopic observations to derive the age of the stellar populations in i zw 18c . concerning the outermost regions of i zw 18 , while gaseous emission is an important contributor to the total light in the vicinity of the star - forming regions
, there was no evidence for extended stellar emission at distances as far as 20 from the h ii regions ( e.g. , dufour et al . 1996b ; izotov et al .
however , in some recent papers ( e.g. , legrand 2000 ; legrand et al .
2000 ; kunth & stlin 2000 ) such an old extended stellar population has been postulated .
we use deep mmt and keck ii spectroscopic observations to clarify the origin of the extended emission and estimate the contribution of the ionized gas to it .
the observations and data reduction are described in sect .
the properties of the stellar population in i zw 18c are discussed in sect .
3 . in sect .
4 we discuss the properties of the outlying regions of i zw 18 .
our results are summarized in sect .
the keck ii spectroscopic observations of i zw 18c were carried out on january 9 , 2000 with the low - resolution imaging spectrograph ( lris ) ( oke et al .
1995 ) , using the 300 groove mm@xmath14 grating which provides a dispersion 2.52 pixel@xmath14 and a spectral resolution of about 8 in first order .
the slit was 1@xmath15180 , centered on the brightest central region ( region c ) of i zw 18c and oriented with a position angle p.a .
= 80@xmath16 ( slit orientation `` 1 '' in fig . [ fig1 ] ) .
no binning along the spatial axis has been done , yielding a spatial sampling of 02 pixel@xmath14 .
the total exposure time was 60 min , broken into three 20 min exposures .
all exposures were taken at airmass of 1.42 .
the seeing was 09 .
mmt spectroscopic observations of i zw 18 and i zw 18c were carried out in the nights of 1997 april 29 and 30 .
a signal - to - noise ratio s / n @xmath9 50 was reached in the continuum of the bright nw region of i zw 18 .
observations were made in the blue channel of the mmt spectrograph using a highly optimized loral 3072 @xmath15 1024 ccd detector .
a 15 @xmath15 180 slit was used along with a 300 groove mm@xmath14 grating in first order and an l-38 second - order blocking filter .
this yields a spatial resolution along the slit of 03 pixel@xmath14 , a scale perpendicular to the slit of 1.9 pixel@xmath14 , a spectral range 3600 7500 , and a spectral resolution of @xmath5 7 ( fwhm ) . to improve the signal - to - noise ratio
, ccd rows were binned by a factor of 2 , yielding a final spatial sampling of 06 pixel@xmath14 .
the total exposure time was 180 minutes broken up in six subexposures , 30 minutes each .
all exposures were taken at airmasses @xmath6 1.1 1.2 .
the seeing during the observations was 07 fwhm .
the slit was oriented in a position angle p.a .
= 41@xmath16 to permit observations of the nw and se regions in i zw 18 and the eastern region ( region e ) in i zw 18c simultaneously ( slit orientation `` 2 '' in fig .
[ fig1 ] ) .
the kitt peak 4 m observations have been obtained on 18 march 1994 with the ritchey - chrtien rc2 spectrograph used in conjunction with the t2 kb 2048@xmath152048 ccd detector . we use a 2@xmath15300 slit with the kpc10a grating ( 316 lines mm@xmath14 ) in first order , with a gg 385 order separation filter .
this filter cuts off all second - order contamination for wavelengths blueward of 7400 which is the wavelength region of interest here .
the above instrumental set - up gave a spatial scale along the slit of 0.69 arcsec pixel@xmath14 , a scale perpendicular to the slit of 2.7 pixel@xmath14 , a spectral range of 35007500 and a spectral resolution of @xmath5 5 .
all exposures were taken at airmass of 1.1 .
the seeing was 15 .
the slit was oriented along the se
nw direction at a position angle of 41@xmath17 , the same as that during the mmt observations ( slit orientation `` 2 '' in fig . [ fig1 ] ) .
the total exposure time was 60 minutes and was broken up into 3 subexposures .
the spectrophotometric standard stars feige 34 and hz 44 were observed for flux calibration during each of three sets of the observations .
spectra of hg - ne - ar ( keck ii ) and he - ne - ar ( mmt and 4 m kpno ) comparison lamps were obtained before and after each observation to provide the wavelength calibration .
data reduction of spectral observations was carried out using the iraf software package .
this included bias subtraction , cosmic - ray removal and flat - field correction using exposures of a quartz incandescent lamp .
after wavelength calibration , night - sky background subtraction , and correcting for atmospheric extinction , each frame was calibrated to absolute fluxes .
one - dimensional spectra were extracted by summing , without weighting , different numbers of rows along the slit depending on the exact region of interest .
we have extracted spectra of two regions in i zw 18c ( fig .
[ fig1 ] ) : ( 1 ) the brightest region c ( keck ii observations ) and ( 2 ) the eastern region e ( all observations ) . the extracted spectra are shown in fig .
additionally , spectra of outlying regions of i zw 18 at different distances from it have been extracted .
one of the key problems discussed over the last three decades is the evolutionary status of i zw 18 : is this galaxy young or old ?
the evolutionary status of i zw 18c has not been discussed in comparable detail .
high signal - to - noise spectra of i zw 18c reveal blue continua and show only emission and absorption hydrogen balmer lines .
heavy element emission lines are not detected in the spectra , which precludes a metallicity determination of i zw 18c .
for the sake of simplicity we assume the heavy element mass fraction in i zw 18c to be @xmath18/50 , the same value as in i zw 18 .
however , the spectra obtained for i zw 18c allow to study stellar populations and constrain their age with various techniques .
a useful technique for determining the age of a galaxy is to fit its observed spectral energy distribution ( sed ) by theoretical seds calculated for various stellar population ages and star formation histories .
this method ( alone or in combination with photometric data ) has been applied to some extremely metal - deficient bcds with @xmath19 = ( 1/20 1/40)@xmath18 ( e.g. , sbs 0335052 ( izotov et al .
1997b ; papaderos et al .
1998 ) , sbs 1415 + 437 ( thuan et al .
1999a ) , tol 1214277 ( fricke et al .
it was shown that , after subtraction of ionized gas emission , the underlying stellar components of these galaxies are consistent with populations not older than a few hundred myr .
however , the spectral energy distribution fitting method is subject to uncertainties in the extinction , resulting in an age overestimate , if the adopted extinction is too low .
therefore , other methods are desirable to constrain the stellar population ages .
we discuss in this section two such methods , one relying on the balmer nebular emission line equivalent widths and the other on the balmer stellar absorption line equivalent widths . on the assumption of a dust - free ionization - bounded h ii region , the strongest hydrogen recombination emission lines h@xmath0 and h@xmath1
provide an estimate of the age of the young stellar population when o and early b stars are still present .
however , even if dust is present in h ii regions , the age estimate is quite robust .
this is because the ionizing flux from such a young stellar population and hence the equivalent widths of the balmer emission lines have a very strong temporal evolution .
therefore , the dating method based on the h@xmath0 and h@xmath1 emission lines is relatively insensitive to dust extinction .
the h@xmath0 and h@xmath1 emission lines are detected in i zw 18c in both regions c and e ( fig .
[ fig2 ] ) .
their fluxes and equivalent widths are listed in table [ tab1 ] . the exception is the keck ii spectrum of region e , where h@xmath1 emission was not detected .
this non - detection is probably due to the patchy distribution of the ionized gas . in fig .
[ fig3]a we compare the measured h@xmath0 and h@xmath1 emission line equivalent widths with those predicted for an instantaneous burst as a function of age .
the theoretical dependences have been kindly calculated for us by d. schaerer using the schaerer & vacca ( 1998 ) code with the @xmath19 = 0.0004 geneva evolutionary tracks from lejeune & schaerer ( 2001 ) .
they are shown by solid lines .
the age derived from different hydrogen nebular emission lines in the various spectra is in a narrow range around @xmath5 15 myr .
hence , the gas in i zw 18c is likely to be ionized by early b stars .
another method of stellar population age determination is based on the equivalent widths of absorption features .
this method probes larger ages as compared to the previous method because the most prominent hydrogen absorption lines are formed in the longer - lived a stars .
gonzalez delgado & leitherer ( 1999 ) and gonzalez delgado , leitherer & heckman ( 1999 ) have calculated a grid of synthetic profiles of stellar hydrogen balmer absorption lines for effective temperatures and gravities , characteristics of galaxies with active star formation .
they developed evolutionary stellar population synthesis models , synthesizing the profiles of the hydrogen balmer absorption lines from h@xmath1 to h13 for an instantaneous burst with an age ranging from @xmath20 to @xmath21 yr .
the calculations were made for a stellar initial mass function with salpeter slope and with mass cutoffs @xmath22 = 1 @xmath23 and @xmath24 = 80 @xmath23 .
the h@xmath2 , h@xmath3 and higher order hydrogen absorption lines due the underlying stellar populations are clearly seen in the keck ii spectra of i zw 18c ( fig .
[ fig2]a - [ fig2]b ) .
some hydrogen absorption lines are also seen in the 4 m kpno and mmt spectra ( fig .
[ fig2]c - [ fig2]d ) .
however , the signal - to - noise ratio is not high enough in the latter two spectra to measure equivalent widths .
although higher - order hydrogen balmer absorption lines are seen in the spectrum of i zw 18c , they are not suitable for age determination because of ( a ) the relatively low signal - to - noise ratio at short wavelengths and uncertainties in the placement of the continuum in the blue region caused by many blended absorption features , and ( b ) the weak dependence of their equivalent widths on age ( gonzalez delgado et al . 1999 ) . in table
[ tab2 ] we show the equivalent widths of the h@xmath3 and h@xmath2 absorption lines measured in the spectra of regions c and e in i zw 18c .
we need to correct the equivalent widths of the absorption lines for the contamination by nebular emission . for region
c we use the intensity of the h@xmath1 emission line to calculate the intensity of the h@xmath3 emission line adopting case b at the electron temperature of @xmath5 20000k ( e.g. , aller 1984 ) .
we do not use the h@xmath2 absorption line in the spectrum of this region because of the strong contamination by nebular emission .
the h@xmath1 emission line is not definitely detected in the keck ii spectrum of region e. for this region , the intensity of h@xmath0 emission line is used to correct equivalent widths of the absorption lines for the same effect .
the corrected equivalent widths of h@xmath3 and h@xmath2 absorption lines are shown in table [ tab2 ] . in fig .
[ fig3]b we show by solid lines the predicted behaviour of the equivalent widths of the h@xmath2 and h@xmath3 absorption lines with age for an instantaneous burst at a metallicity @xmath19 = 1/20 @xmath18 ( gonzalez delgado et al . 1999 ) .
the measured equivalent widths are shown for region c by filled circles and for region e by stars .
their values are consistent with an age of @xmath5 15 myr .
this age estimation is in excellent agreement with that obtained from the nebular emission line analysis implying that the light of i zw 18c is dominated by a young stellar population .
however , the age we derive here is significantly lower than the value of 40 myr derived by dufour et al .
( 1996b ) for the brightest stars from
_ hst _ cmds of i zw 18c .
although our two age estimates are consistent with each other , there are a number of uncertainties which may affect the result .
a major uncertainty is the unknown metallicity of the stars in i zw 18c .
we have assumed for simplicity the metallicity to be equal to that of the ionized gas in i zw 18 .
note , however , that a lower stellar metallicity would increase the age and vice versa . to estimate an age from the emission lines
we have assumed a dust - free ionization - bounded h ii region .
if the h ii region is density - bounded or dust is present , then some of the ionizing photons escape the h ii region or are absorbed .
equivalent widths of the balmer emission lines give in this case an upper limit to the age .
another source of uncertainty comes from the small number of massive stars in i zw 18c .
our age estimates are based on models where the stellar initial mass function is well - behaved and can be approximated by an analytical function .
however , small number statistics can introduce stochastic fluctuations at the high star mass end of the imf .
recently cervio , luridiana & castander ( 2000 ) have analyzed how such stochastic effects influence the observed parameters of young stellar clusters with solar metallicity such as the h@xmath1 equivalent width and the number of the ionizing photons .
the number of ionizing photons in i zw 18c derived from the h@xmath1 emission line ( table 1 ) is @xmath5 2 @xmath15 10@xmath25 s@xmath14 and 4 @xmath15 10@xmath25 s@xmath14 for the e and c regions respectively . with an equivalent width @xmath5 6 7 of the h@xmath1 emission line , this corresponds to the case of a 10@xmath26 @xmath27 stellar cluster ( cervio et al .
for such a cluster the age variations at a fixed h@xmath1 equivalent width can be as high as 15 percent at the 90% confidence level .
hence , the age of i zw 18c derived from the emission lines can lie in the range @xmath5 10 25 myr , with a central value of 15 myr .
similarly , age estimates based on absorption lines can also be slightly modified by stochastic effects .
however , calculations are not yet available in the literature .
gonzalez delgado et al . ( 1999 ) do not calculate the temporal evolution of the equivalent widths of the balmer absorption lines for the heavy element mass fraction @xmath19 = @xmath18/50 .
therefore , we use models with @xmath19 = @xmath18/20 .
extrapolation to the metallicity of i zw 18 would result in a @xmath6 1 decrease of the equivalent widths at a fixed age , or an age increase of up to 25 myr . finally , the age determination depends on the star formation history in the galaxy which we consider next .
our estimates for the stellar population age in i zw 18c in sect .
[ ageem ] and [ ageab ] are based on the assumption of an instantaneous burst of star formation .
now we discuss how that age changes if continuous star formation is considered .
we adopt a constant star formation rate in the interval between the initial time @xmath28 when star formation starts and the final time @xmath29 when it stops .
time is zero at the present epoch and increases into the past .
using model equivalent width of the emission and absorption lines and spectral energy distributions for instantaneous bursts ( schaerer , private communication ; lejeune & schaerer 2001 ; gonzalez delgado et al .
1999 ) , we calculate the temporal evolution of the hydrogen emission and absorption line equivalent widths for continuous star formation . the results of calculations are presented in figure [ fig3 ] . by dashed , dot - dashed and dotted lines are shown the temporal dependences of the equivalent widths of the h@xmath1 and h@xmath0 emission lines ( fig .
[ fig3]a ) , and of the h@xmath3 and h@xmath2 absorption lines ( fig .
[ fig3]b ) for continuous star formation starting at time @xmath28 , as defined by the abscissa value , and stopping at @xmath29 = 5 , 8 and 12.5 myr , respectively .
in other words , the equivalent widths of the above four lines in the spectrum of the stellar population formed between @xmath28 and @xmath29 have a value @xmath30 at time @xmath28 in fig .
[ fig3]a and [ fig3]b . at a fixed @xmath30 , the general trend seen from fig .
[ fig3 ] for continuous star formation is that the younger the youngest stars , the larger the time interval @xmath31 , and the older the oldest stars .
another feature is that , at a fixed age @xmath29 of the youngest stars , the age @xmath28 of the oldest stars derived from the observed emission line equivalent widths , differs from that derived from the observed absorption line equivalent widths . in particular , in the model where star formation stopped 5 myr ago ( dashed lines ) , the age of the oldest stars derived from hydrogen emission lines exceeds 100 myr , while the age of the oldest stars derived from hydrogen absorption lines is only @xmath5 50 myr .
this model seems to be excluded by consideration of the luminosity of the ionizing radiation .
the most massive stars in the stellar population with age 5 myr would have masses as high as 40 @xmath27 ( meynet et al .
the number of the ionizing photons produced by a single 40 @xmath27 star is equal to @xmath32(lyc ) @xmath33 1.5 @xmath15 10@xmath34 s@xmath14 ( vacca , garmany & shull 1996 ) , larger than that derived from the observed flux of the h@xmath1 emission line in i zw 18c ( table [ tab1 ] ) , assuming an ionization - bounded h ii region .
there can be an upward correction factor of @xmath6 2 due to the extinction , but the corrected @xmath32(lyc ) would still be below the value for a single 40 @xmath27 star .
these estimates can however be modified by massive star small number statistics caused by the stochastic nature of star formation . though smaller , the difference between the age of the oldest stars derived from the equivalent widths of emission lines ( 50 myr ) and that derived from the equivalent widths of absorption lines ( 40 myr ) , is present in the continuous star formation model with age of the youngest stars equal to 8 myr ( dot - dashed lines in fig .
[ fig3 ] ) .
however , this difference is small in the continuous star formation model which stopped 12.5 myr ago ( dotted lines ) . in this model
, the age of the oldest stars should be @xmath5 25 myr to consistently explain the observed hydrogen line equivalent widths . hence , similarly to the case of an instantaneous burst ,
we conclude that the observations of i zw 18c are best reproduced by a short star formation episode which occurred continuously between @xmath5 10 myr and @xmath5 25 myr ago .
uncertainties in the observations and models may extend this range to between @xmath5 10 myr and @xmath6 100 myr ago .
a useful constraint on the stellar population age can be obtained from the spectral energy distribution .
this method , as already noted , is subject to interstellar extinction .
however , when used in conjunction with the methods discussed in sect .
[ age ] it provides a powerful tool for studying stellar populations by allowing to derive simultaneously the age and the extinction of the same region . to fit the observed spectral energy distributions we use model seds calculated by d. schaerer using the schaerer & vacca ( 1998 ) code and the @xmath19 = 0.0004 geneva evolutionary tracks of lejeune & schaerer ( 2001 ) .
the contribution of the ionized gas was also included .
this contribution is small because the equivalent widths of hydrogen emission lines in i zw 18c are low . because the observed spectral energy distribution is extinction - dependent
, the extinction can be obtained for regions with known ages as derived from the equivalent widths of the hydrogen emission and absorption lines .
we consider the case of the 15 myr instantaneous burst stellar population discussed in sect .
first assume @xmath35(h@xmath1 ) = 0 , where @xmath35(h@xmath1 ) = @xmath36(@xmath37)/1.47 ( aller 1984 ) .
comparison of the observed keck ii spectra of regions c and e in i zw 18c with the theoretical seds ( bottom spectra in fig . [ fig4 ] ) shows that theoretical seds are bluer than the observed extinction - uncorrected spectra .
evidently , interstellar extinction is present in i zw 18c and it modifies the observed sed .
we derive @xmath35(h@xmath1 ) = 0.3 for region c and @xmath35(h@xmath1 ) = 0.1 for region e to achieve the best agreement between the extinction - corrected observed seds and the theoretical seds ( upper spectra in fig .
[ fig4 ] ) . for comparison
, we show in fig .
[ fig4]a by a dotted line the theoretical sed for a 40 myr stellar population which does not provide as good a fit .
a theoretical 15 myr stellar population sed also fits well the 4 m kpno and mmt spectra of region e extinction - corrected for @xmath35(h@xmath1 ) = 0.1 ( fig .
[ fig5 ] ) .
the theoretical 40 myr stellar population sed with @xmath35(h@xmath1 ) = 0.1 fits less well ( bottom solid line ) .
some support for a larger extinction in region c than in region e comes from the observed h@xmath0-to - h@xmath1 flux ratios ( table [ tab1 ] ) .
they are respectively equal to 4.5 and 3.8 , corresponding to @xmath35(h@xmath1 ) @xmath5 0.7 and 0.4 .
however , correction for underlying stellar absorption results in lower extinction coefficients .
we note that we have not corrected the observed seds for the effect of atmospheric refraction .
if such an effect were to be important , it can selectively decrease the blue light relatively to the red light , leading us to derive erroneously high extinction for regions c and e. indeed one may suspect that such an effect would be important for the keck ii spectrum which was obtained with a narrow slit of 1 at an airmass of 1.4 .
filippenko ( 1982 ) has shown that atmospheric dispersion can produce an offset as high as @xmath5 12 of the blue region near [ o ii ] @xmath383727 relative to the red region near h@xmath0 @xmath386563 at this airmass . however
, his calculations have been done for an altitude of 2 km , while the keck ii spectrum was obtained at an altitude about twice that .
furthermore , emission from the c component is extended and originates in a region significantly larger than the width of the slit , reducing the effect of the atmospheric dispersion .
perhaps the best argument for such an effect not to be important comes from the comparison of our different spectra of the same region .
although the keck , mmt and 4 m spectra of region e were obtained with different slit widths at different airmasses , they are all well fitted by the same 15 myr single stellar population model .
we have thus reached two important conclusions for i zw 18c : ( 1 ) the stellar population responsible for its observed sed is very young , with an age of @xmath5 15 myr and ( 2 ) the region is characterized by a varying interstellar extinction implying the presence of non - uniformly distributed absorbing material .
i zw 18c is not the only very metal - deficient object to have a clumpy dust distribution .
earlier similar conclusions have been reached for the metal - deficient bcd sbs 0335052 by izotov et al .
( 1997b , 1999 ) and thuan et al .
( 1997 , 1999b ) . while the brightest and youngest star - forming region in sbs 0335052 is relatively free of dust
, extinction is higher at the location of the fainter and older super star clusters .
clumpy regions with large extinctions are clearly seen in the _ hst _ @xmath39 image of sbs 0335052 ( thuan et al .
1997 ) .
we have arrived at the conclusion that the light from i zw 18c is dominated by stars @xmath5 15 myr old .
is this conclusion consistent with the photometric data ? in this section ,
we compare the predicted colors for the young stellar population with integrated broad - band colors of i zw 18c obtained from ground - based and _ hst _ photometric observations .
we also discuss the consistency between the properties of the stellar population in i zw 18c obtained from the spectroscopic data with those obtained from analysis of _ hst _ wfpc2 cmds . in table
[ tab3 ] we show the observed integrated @xmath13 magnitude and colors of i zw 18c .
the second column shows these quantities without correction for interstellar extinction .
because the spectroscopic data imply the presence of extinction in i zw 18c , we also show in the third column the colors corrected for interstellar extinction with @xmath35(h@xmath1 ) = 0.3 or @xmath4 = 0.65 mag .
these values are for the brightest region c. the extinction is lower in the fainter region e. the faint northwestern region of i zw 18c appears to be redder as compared to other regions ( dufour et al .
1996b ) , but the lack of spectroscopic data prevents us from determining the interstellar extinction in that region .
we assume that the extinction derived for region c to be representative for the whole galaxy .
we compare the observed integrated colors of i zw 18c with those predicted by instantaneous burst models for different ages .
the first set of models shown in table [ tab3 ] is the same as the one used for fitting the seds with an heavy element mass fraction @xmath19 = @xmath18/50 and based on geneva stellar evolutionary tracks .
another set of predicted colors based on the padua stellar evolution models has been calculated by tantalo et al .
( 1996 ) for a single stellar population and a heavy element mass fraction of @xmath18/50 .
comparison of the two sets of models shows that colors based on the padua stellar evolution models are systematically redder at a fixed age as compared to those based on the geneva ones .
consequently , the use of padua models results in younger ages as compared to geneva models . in the following
we compare the observed colors to the modeled ones based on geneva tracks .
it is seen from table [ tab3 ] that the colors uncorrected for extinction are well reproduced by the model with a 100 myr stellar population .
however , with this age the predicted equivalent widths of the hydrogen emission lines are too small ( @xmath30(h@xmath1 ) @xmath6 0.1 , @xmath30(h@xmath0 ) @xmath6 0.3 ) as compared to the observed ones ( fig . [ fig3]a ) . on the other hand , the predicted equivalent widths of the hydrogen absorption lines ( @xmath30(h@xmath3 ) @xmath9 10 , @xmath30(h@xmath2 ) @xmath9 8 )
are too high ( fig .
[ fig3]b ) .
again , to put observations into agreement with models , interstellar extinction has to be invoked .
indeed , all observed colors corrected for an extinction with @xmath35(h@xmath1 ) = 0.3 are in fair agreement with predicted ones for a 15 20 myr single stellar population .
our conclusions do not change significantly in the case of continuous star formation . in fig .
[ fig6]a [ fig6]c we show by solid lines the theoretical dependences on age of the ( @xmath40 ) , ( @xmath37 ) and ( @xmath39 ) colors in the case of constant continuous star formation , for different choices of @xmath28 and @xmath29 .
the observed colors uncorrected for extinction ( dashed lines ) can be fitted by models with star formation starting at @xmath28 = 100 300 myr . however , these models predict too low an equivalent width for the h@xmath0 emission line and too large an equivalent width for the h@xmath3 absorption line ( fig .
[ fig6]d [ fig6]e ) .
furthermore , models with star formation stopping at @xmath29 @xmath9 40 myr are excluded for the whole range of @xmath28 ( fig .
[ fig6]d [ fig6]e ) . to have the observed colors come into agreement with the observed equivalent widths of the balmer lines
, a non - negligible extinction must be assumed .
we show by dotted lines in fig .
[ fig6]a [ fig6]c the extinction - corrected colors with two values of the reddening , @xmath36(@xmath37 ) = 0.1 and 0.15 . in the latter case ,
the colors are explained by models with constant star formation starting at an age @xmath28 @xmath5 30 100 myr ( filled and open circles ) and stopping at an age @xmath29 = 8 12 myr .
observational uncertainties will only slightly increase this age range .
we conclude that our broad - band photometric data are consistent with a young stellar population and a non - negligible interstellar extinction in i zw 18c .
we emphasize that the ages derived above hold only for the brightest regions of the c component .
we can not exclude the possibility that the age of the stellar population in regions of i zw 18c , not covered by our spectroscopic observations , may be larger .
cmd analysis is a powerful tool for studying stellar populations . however , as already pointed out , this method is sensitive to the adopted extinction and distance .
while the extinction can be derived from spectroscopic observations , the determination of the distance is more uncertain . a distance of @xmath5 10 mpc to i
zw 18 has generally been adopted for analyzing the cmds ( hunter & thronson 1995 , dufour et al .
1996b and aloisi et al .
this assumes that the observed heliocentric radial velocity of the galaxy @xmath5 740 km s@xmath14 is a pure hubble flow velocity , and a hubble constant @xmath41 = 75 km s@xmath14 mpc@xmath14 .
adopting this distance would lead to a conflict with the well - observed ionization state of i zw 18c . at 10 mpc the brightest stars observed in i zw 18c would have absolute @xmath13 magnitudes fainter than 6 mag ( dufour et al .
1996b ; aloisi et al .
in that case , comparison with evolutionary tracks implies that the most massive stars in i zw 18c ( called the c component by dufour et al .
( 1996b ) ) would have masses less than 9 @xmath27 .
the age of the stellar population with such an upper mass limit is at least 40 myr ( e.g. , dufour et al .
1996b ) , larger than the one derived from the equivalent widths of hydrogen emission and absorption lines ( @xmath5 10 25 myr ) .
if the upper stellar mass limit of 9 @xmath27 derived by dufour et al .
( 1996b ) and aloisi et al .
( 1999 ) for i zw 18c is correct , then ionized gas should not be present in it because of the absence of early b stars . but
h@xmath0 and h@xmath1 are clearly observed . our derived age of @xmath5 15 myr implies that early b stars with masses as high as @xmath5 15 @xmath27 are present in i zw 18c , if an instantaneous burst of star formation is assumed . in the case of continuous star formation , the age of the youngest stars would be smaller and the upper mass limit larger to account for the presence of the ionized gas .
we argue therefore that the stellar absolute magnitudes derived by dufour et al .
( 1996b ) and aloisi et al .
( 1999 ) from their cmds are too faint because they are based on too small an adopted distance .
stlin ( 2000 ) assumed a distance of 12.6 mpc to analyze his _ hst _ nicmos cmd .
however , even this distance is not enough to explain the ionization state of i zw 18c .
an additional effect is due to extinction , with @xmath4 = 0.65 mag for the region c. correcting for extinction and increasing the distance by a factor of @xmath5 1.5 to @xmath5 15 mpc would make the most massive stars more luminous by a factor of @xmath5 4 and push the mass upper limit to @xmath5 15 @xmath27 .
a stellar population with such an upper mass limit would provide enough ionizing photons to account for the observed emission lines in i zw 18c .
furthermore , the age of the brightest stars in the cmds of i zw 18c would be @xmath5 15 myr , consistent with that derived from the hydrogen emission and absorption lines .
ground - based and _ hst _ h@xmath0 and broad - band imaging have revealed filamentary structure around i zw 18 , inside a 15 radius ( e.g. , hunter & thronson 1995 ; stlin , bergvall & rnnback 1996 ; dufour et al .
1996b ) . because of the presence of ionizing young stars , the light in these outlying regions is likely to be dominated by the emission of the ionized gas .
this conclusion is supported by spectroscopic observations .
dufour et al . ( 1996a ) and izotov & thuan ( 1998 ) have detected h@xmath0 emission at distances as large as 20 from i zw 18 in the nw direction of slit `` 2 '' ( fig .
[ fig1 ] ) .
izotov et al .
( 1999 ) have found that at a distance of @xmath5 5 to the northwest of the brightest nw region of i zw 18 , the equivalent width of the h@xmath1 emission line is @xmath5 300 . in this case
the contribution of the gaseous continuum near h@xmath1 is @xmath5 30 percent of the total continuum .
the contribution of the gaseous continuum near h@xmath0 is even larger , being @xmath5 50 percent of the total continuum .
therefore , when analyzing stellar populations with the use of photometric data , it is essential to correct broad - band colors for ionized gas emission . however , in some recent papers ( e.g. , kunth & stlin 2000 ) the extended emission around i zw 18 has been attributed to an old stellar population , while the contribution of the ionized gas is assumed to be not dominant . that this can not be true is seen in fig .
[ fig7 ] where we show a map of the h@xmath0 equivalent width distribution as obtained from _ hst _ narrow - band and broad - band images . while the h@xmath0 equivalent width is small in the direction of the stellar clusters , it exceeds 1000 in the outer regions .
in the following , we analyze mmt and keck ii spectroscopic observations of the outer regions around i zw 18 to clarify two issues : a ) how important is the contribution of the ionized gas in the outer regions of i zw 18 ? and b ) is stellar emission present at large distances ? in figure [ fig8 ]
we show the mmt spectrum of the region with a high h@xmath1 equivalent width . the spectrum is extracted within an aperture 15 @xmath15 3 ( slit `` 2 '' ) , centered at a distance 5 to the northwest from the nw component of i zw 18 .
it is characterized by strong emission lines .
we refer to this region as the `` h@xmath0 arc '' ( square box in fig .
[ fig1 ] ) .
a synthetic spectrum with a 2 myr stellar population combined with ionized gas emission fits best the observed sed of this region .
the observed and extinction - corrected emission - line intensities in the h@xmath0 arc together with their equivalent widths are listed in table [ tab4 ] .
the ionic and elemental abundances have been derived following izotov et al .
( 1994 , 1997c ) . the extinction coefficient @xmath35(h@xmath1 ) and the absorption equivalent width @xmath30(abs ) for the hydrogen lines
are obtained by an iterative procedure .
they are shown in table [ tab4 ] together with the observed flux @xmath42 of the h@xmath1 emission line . the electron temperature @xmath43(o iii )
was determined from the [ o iii ] @xmath384363 / ( @xmath384959 + @xmath385007 ) flux ratio and the electron number density @xmath44(s ii ) from the [ s ii ] @xmath386717/@xmath386731 flux ratio . the ionic and elemental abundances are shown in table [ tab5 ] together with ionization correction factors ( icfs ) .
they are in good agreement with the abundances derived by skillman & kennicutt ( 1993 ) , izotov & thuan ( 1998 ) , vlchez & iglesias - pramo ( 1998 ) and izotov et al .
( 1999 ) for the nw and se components of i zw 18 .
we have shown that the contribution of the ionized gas is large in the h@xmath0 arc , at @xmath5 5 from the nw component of i zw 18 .
a similar situation prevails at significantly larger distances , as evidenced by deep keck ii spectroscopic observations .
the slit during these observations ( slit `` 1 '' in fig .
[ fig1 ] ) crossed the outer regions of i zw 18 including the expanding supershell of ionized gas best seen in h@xmath0 images ( e.g. , hunter & thronson 1995 ; dufour et al .
the latter feature located at @xmath5 15 from i zw 18 is labeled in figure [ fig1 ] as `` loop '' .
we can thus study with deep spectroscopy the extended diffuse emission in i zw 18 all the way from i zw 18c to the bright star at the edge of fig .
[ fig1 ] and located @xmath5 40 from i zw 18 . in fig .
[ fig9]a [ fig9]b we show respectively the flux distributions along the slit of the continuum near h@xmath1 and of the line + continuum emission at the wavelength of the h@xmath1 emission line .
the origin is taken to be at region c. the locations of the bright star and loop are marked . in fig .
[ fig9]c we show the continuum - subtracted flux distribution of the h@xmath1 emission line .
the negative values of the flux in some regions of i zw 18c and around the star are probably caused by underlying stellar h@xmath1 absorption .
no appreciable continuum emission is seen in fig .
[ fig9]a between i zw 18c and the star .
however , nebular emission of h@xmath0 is present nearly everywhere between i zw 18c and the star ( fig .
[ fig9]c [ fig9]d ) , suggesting that the contribution to the total flux of the nebular emission from ionized gas is important in the outermost regions , as far as 30 from i zw 18 . in fig .
[ fig9]e we show the intensity distribution of the continuum at the wavelength of 4200 approximating the @xmath7 band .
we also plot by dotted lines the surface brightness levels in steps of 1 mag arcsec@xmath8 .
it is seen that the continuum surface brightness is fainter than 27 mag arcsec@xmath8 everywhere between i zw 18c and the star , including the loop region .
however , we can not exclude the presence of stellar emission at the level of 28 mag arcsec@xmath8 , as postulated by legrand ( 2000 ) .
the contamination by extended ionized gas emission makes the detection of such an extremely faint hypothetical stellar background problematic . in fig .
[ fig10 ] we show the distributions along slit `` 2 '' of : a ) the continuum intensity at 4200 , b ) the continuum - subtracted flux and c ) the equivalent width of the h@xmath0 emission line .
the distribution of h@xmath0 emission ( fig .
[ fig10]b ) around the main body is more extended as compared to the continuum ( fig .
[ fig10]a ) , the latter being confined in a region with radius less than 12 around the nw component .
the equivalent width of h@xmath0 is very high to the northwest of the nw component ( fig .
[ fig10]c ) and must be taken into account when photometric properties of the stellar population in i zw 18 are analyzed .
we also point out that the continuum distribution of region e in i zw 18c is narrower than that of the h@xmath0 emission line .
the maximum h@xmath0 equivalent width in region e ( fig .
[ fig10]c ) is offset to the northwest by @xmath5 2 relative to the continuum distribution ( fig .
[ fig10]a ) . in fig .
[ fig11 ] we show the spectrum of the loop . despite its faintness , several emission lines are seen . however , the sensitivity in the blue region was not sufficient to detect the [ o ii ] @xmath383727 emission line .
the continuum is very weak and can be significantly affected by uncertainties in the sky subtraction .
this makes the measurements of line equivalent widths difficult .
the fluxes and equivalent widths of the detected lines are given in table [ tab6 ] .
the flux errors include uncertainties in the placement of the continuum level and in the fitting of the lines by gaussian profiles .
however , these errors do not take into account the uncertainties in the sky subtraction which might be large . indeed , the loop flux in the continuum is only @xmath5 1% above the night sky flux , while that number is as high as 50% for the continuum flux in i zw 18c . even with these large uncertainties the emission line equivalent widths in the loop spectrum are very high . in particular , the equivalent width of the h@xmath1 emission line is 471 or about half of the value expected for pure gaseous emission at the electron temperature @xmath43 = 20000k .
hence , half of the flux in the continuum comes from the ionized gas , emphasizing again the importance of the correction of the spectral energy distribution and broad - band colors for gaseous emission .
this goes contrary to the assumption of kunth & stlin ( 2000 ) that the contribution of gaseous emission does not affect the colors of the outlying regions of i zw 18 .
if errors in the night sky subtraction are @xmath5 1% , then the equivalent width of the h@xmath1 emission line in the loop spectrum is in the range @xmath5 250 1000 . within the uncertainties ,
the emission of the loop is quite consistent with pure gaseous emission .
kunth & stlin ( 2000 ) have derived radial distributions of the surface brightness in the @xmath7 band and of the @xmath45 and @xmath46 colors of i zw 18 ( their fig .
they find that the colors rise continuously with increasing radius and reach @xmath45 = 0.6 mag and @xmath46 = 1.6 mag at a radius of 10 . assuming a purely stellar emission , they conclude that the observed colors can be reproduced by a single stellar population model with a metallicity of 1/50 @xmath18 and an age of log @xmath47 = 9.1 @xmath48 0.1 ( @xmath47 in yr ) , irrespective of the imf ( bruzual & charlot 2000 , unpublished ) .
however , this age estimate is rather uncertain and is dependent on the particular population synthesis model used .
for example , tantalo et al . ( 1996 ) using padua stellar evolutionary tracks give values @xmath45 = 0.8 mag and @xmath46 = 1.7 mag for a 1 gyr single stellar population with a metallicity of 1/50 @xmath18 .
the bluer colors derived by kunth & stlin ( 2000 ) would give an age of 100 300 myr according to tantalo et al .
( 1996) models . on the other hand , leitherer et al .
( 1999) models using the geneva stellar evolutionary tracks predict @xmath45 = 0.5 mag and @xmath46 = 1.1 mag for a 1 gyr single stellar population with a metallicity of 1/20 @xmath18 , bluer than those derived by kunth & stlin ( 2000 ) .
leitherer et al .
( 1999) models do not go beyond 1 gyr , but to reproduce the colors derived by kunth & stlin ( 2000 ) , the age of the stellar population in the outer regions of i zw 18 , if present , must be older than 1 gyr .
their models are calculated for a metallicity of 1/20 @xmath18 , but colors with a metallicity of 1/50 @xmath18 are expected to be bluer , further increasing the derived age .
these age estimates are very uncertain .
the models by leitherer et al .
( 1999 ) are less reliable for ages greater 100 myr because they do not include asymptotic giant branch ( agb ) star evolution .
tantalo et al .
( 1996 ) do include agb star evolution , but the little known mass loss processes in the agb phase introduce uncertainties in the predicted colors ( girardi & bertelli 1998 ) . the next source of uncertainties comes from the photometric observations themselves . beyond a radius of @xmath5 5 from i zw 18 , the @xmath45 color profile derived by kunth & stlin ( 2000 ) increases monotonously while the @xmath46 color profile shows discontinuous jumps .
these discontinuities are difficult to understand if the same stellar population is responsible for both colors .
kunth & stlin ( 2000 ) do not show the uncertainties of their photometry .
however , similar deep @xmath10-band photometry of another galaxy sbs 0335052 ( vanzi et al . 2000 ) with ukirt shows that at the @xmath10-band surface brightness of 24 25 mag arcsec@xmath8 the errors are already @xmath5 0.5 mag or more .
new recent @xmath7 and @xmath10 photometric observations of i zw 18 ( papaderos et al .
2001 ) do not confirm the large reddening of the @xmath46 color observed by kunth & stlin ( 2000 ) between radii 6 and 8 , nor the discontinuous jumps . to investigate whether the @xmath45 and @xmath46 colors of the extended emission can be explained by pure gaseous emission
, we calculate the spectral energy distribution of the ionized gas emission in the corresponding wavelength range . the contribution of the free - bound
, free - free and two - photon continuum emission is taken into account for the spectral range from 0 to 5 @xmath49 m ( aller 1984 ; ferland 1980 ) .
as for the electron temperature , we adopt the value of 19000k , which is the mean value between the electron temperatures in the nw and se components of i zw 18 .
emission lines are superposed on the gaseous continuum sed with intensities derived from the observed spectrum of the loop at the distance of @xmath5 15 from i zw 18 ( table [ tab6 ] ) , in the spectral range @xmath383700 7500 . outside this range
, the intensities of emission lines ( mainly hydrogen lines ) have been calculated from the extinction - corrected flux of h@xmath1 with reddening @xmath4 = 0.16 mag .
the reddening in the loop was calculated from the observed h@xmath0/h@xmath1 flux ratio ( table [ tab6 ] ) , assuming an electron temperature @xmath43 = 20000k .
we derive @xmath45 = 0.8 mag and @xmath46 = 0.9 mag .
if instead of the relative intensities of the emission lines observed in the loop , we use those seen in the nw or se regions of i zw 18 ( izotov et al . 1999 ) , we obtain slightly bluer colors , @xmath45 = 0.6 mag and @xmath46 = 0.7 mag .
the color difference is mainly due to a smaller contribution in the outer regions of some emission lines , e.g. [ ne iii ] @xmath383869 to the @xmath7 band . from this comparison
we conclude that colors become redder at larger distances , even in the case of pure gaseous emission .
while the @xmath45 color of gaseous emission is similar to the asymptotic value of @xmath5 0.7 mag derived by kunth & stlin ( 2000 ) at distances @xmath5 15 , the predicted @xmath46 color of gaseous emission is considerably bluer than the value they obtained .
however , that value is consistent with @xmath46 @xmath5 0.6 mag derived by papaderos et al .
( 2001 ) in the 6 9 arcsec radius range .
we note that the @xmath45 and @xmath46 colors are not ideal for constraining the existence of a possible extended low - surface - brightness 1 gyr underlying stellar population in i zw 18 , the latter being uncertain , and the former being very similar to the color of ionized gas .
the @xmath50 color is more useful because it can discriminate better between gaseous and a 1 gyr stellar population emission . indeed , adopting the relative line intensities in the h@xmath0 arc or in the loop , the @xmath50 colors of the ionized gas emission are 0.1 mag and + 0.1 mag respectively .
the expected @xmath50 color for a 1 gyr stellar population is much redder , @xmath5 + 1.2 mag ( tantalo et al . 1996 ) .
observations give @xmath50 @xmath5 0 at radii 8 10 arcsec ( papaderos et al .
2001 ) , strongly suggesting that the emission in the outer parts of i zw 18 is gaseous in origin .
we conclude that there is no convincing observational evidence for the presence of an extended underlying low - surface - brightness stellar component in i zw 18 .
its existence , as postulated by kunth & stlin ( 2000 ) , legrand ( 2000 ) and legrand et al .
( 2000 ) , is neither supported by spectroscopic nor photometric observations .
we use spectroscopic and photometric data to constrain the age of the stellar population in the c component of i zw 18 ( @xmath51 i zw 18c ) and to study the origin of the extended emission around i zw 18 .
we have arrived at the following main conclusions : \1 .
deep 4 m kpno , mmt and keck ii spectra of i zw 18c show h@xmath1 and h@xmath0 hydrogen lines in emission , and h@xmath3 and h@xmath2 hydrogen lines in absorption . using their equivalent widths we derive an age of the stellar population of @xmath5 15 myr if an instantaneous burst is assumed .
if star formation is continuous , then the equivalent widths are best reproduced by a short star formation episode continuously occurring between @xmath5 10 myr and @xmath5 25 myr ago .
uncertainties in the observations and models may extend this range to between @xmath5 10 myr and @xmath6 100 myr ago .
spectral energy distributions of the central ( c ) and eastern ( e ) regions of i zw 18c are used to derive extinction .
the equivalent widths of the hydrogen emission and absorption lines and the spectral energy distributions are modeled by a 15 myr single stellar population if the extinction coefficient @xmath35(h@xmath1 ) = 0.1 0.3 , corresponding to @xmath4 = 0.20 0.65 mag . \3 . with the usually assumed distance of @xmath5 10 mpc the stellar population age derived from _
color - magnitude diagrams is too large as compared to the young age derived from the spectroscopic data .
one possible source of the difference is interstellar extinction .
furthermore , to have agreement between the cmds and the ionization state of i zw 18c , the distance to the bcd should be increased to @xmath5 15 mpc .
concerning the extended emission around i zw 18 , keck ii spectra show h@xmath0 emission as far as 30 from the main body .
the equivalent widths of emission lines are particularly strong in the extended envelope ( @xmath30(h@xmath1 ) = 471 ) , implying a dominant contribution of the ionized gas emission in the outermost regions of i zw 18 . within the large uncertainties of the continuum level ,
the emission at @xmath5 15 from i zw 18 is consistent with pure ionized gas emission .
we do not find evidence for an old extended low - surface - brightness stellar component in the outlying regions of i zw 18 down to the surface brightness level @xmath7 @xmath5 27 mag arcsec@xmath8 , contrary to suggestions by kunth & stlin ( 2000 ) .
it will be very difficult to detect the extended stellar emission at @xmath7 @xmath5 28 mag arcsec@xmath8 postulated by legrand ( 2000 ) , because of the important ionized gas emission at large distances .
and n.g.g .
thank the universitts sternwarte of gttingen for warm hospitality .
we are grateful to d. schaerer for making available his stellar evolutionary synthesis models in electronic form and for valuable comments on the manuscript . y.i.i .
thanks the gttingen academy of sciences for a gauss professorship .
we acknowledge the financial support of the volkswagen foundation grant no .
i/72919 ( y.i.i . , n.g.g .
, p.p . and k.j.f .
) , of dfg grant 436 ukr 17/1/00 ( n.g.g . ) , deutsche agentur fr raumfahrtangelegenheiten ( dara ) gmbh grants 50 or 9407 6 and 50 or 9907 7 ( k.j.f . and p.p .
) , and of the national science foundation grants ast-9616863 ( t.x.t . and
y.i.i . ) and
ast-9803072 ( c.b.f . ) .
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the interplay between massive star formation , the ism and galaxy formation .
gif - sur - yvette : edition frontires , p. 605
lcrccrccrccr h@xmath1 & [email protected] & [email protected] & & & & & [email protected] & [email protected]&&[email protected] & [email protected] + h@xmath0&[email protected]&[email protected]&&[email protected]&[email protected] & & [email protected]&[email protected]&&[email protected]&[email protected] + lrrcrrrrcrrr @xmath13 & 19.20 & & & & & + @xmath40 & 0.50&0.72&&0.83&0.77&0.59&0.40&&0.81&0.75&0.37 + @xmath37 & 0.00&0.20&&0.17&0.15&0.08&0.01&&0.18&0.12 & 0.02 + @xmath39 & 0.10&0.17&&0.10&0.08 & 0.04 & 0.13&&0.16 & 0.08 & 0.21 + @xmath52 & & & & 0.15&0.10 & 0.25 & 0.47&&0.25 & 0.29 & 0.67 + lccr 3727 [ o ii ] & [email protected] & [email protected]&53 + 3868 [ ne iii ] & [email protected] & [email protected]&20 + 3889 he i + h8 & [email protected] & [email protected]&31 + 3968 [ ne iii ] + h7 & [email protected] & [email protected]&39 + 4101 h@xmath3 & [email protected] & [email protected]&49 + 4340 h@xmath2 & [email protected] & [email protected]&106 + 4363 [ o iii ] & [email protected] & [email protected]&13 + 4471 he i & [email protected] & [email protected]&9 + 4686 he ii & [email protected] & [email protected]&7 + 4861 h@xmath1 & [email protected] & [email protected]&292 + 4959 [ o iii ] & [email protected] & [email protected]&179 + 5007 [ o iii ] & [email protected] & [email protected]&551 + 5876 he i & [email protected] & [email protected]&40 + 6563 h@xmath0 & [email protected] & [email protected]&1683 + 6678 he i & [email protected] & [email protected]&21 + 6717 [ s ii ] & [email protected] & [email protected]&25 + 6731 [ s ii ] & [email protected] & [email protected]&15 + + @xmath35(h@xmath1 ) & + @xmath53(h@xmath1 ) & + @xmath30(abs ) & lc @xmath43(o iii)(k ) & 18200@xmath484000 + @xmath43(o ii)(k ) & 15100@xmath483200 + @xmath43(s iii)(k ) & 16800@xmath483400 + @xmath44(s ii)(@xmath54 ) & 10@xmath4810 + + o@xmath55/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o@xmath57/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o@xmath58/h@xmath55(@xmath1510@xmath56 ) & [email protected] + o / h(@xmath1510@xmath56 ) & [email protected] + 12
+ log(o / h ) & [email protected] + + ne@xmath57/h@xmath55(@xmath1510@xmath56 ) & [email protected] + icf(ne ) & 1.35 + log(ne / o ) & [email protected] lrcr 3889 h@xmath59+he i & [email protected]&0.183 & 54 + 4101 h@xmath3 & [email protected]&0.228 & 87 + 4340 h@xmath2 & [email protected]&0.392 & 125 + 4861 h@xmath1 & [email protected]&1.000 & 471 + 4959 [ o iii ] & [email protected]&0.148 & 69 + 5007 [ o iii ] & [email protected]&0.487 & 229 + 6563 h@xmath0 & [email protected]&2.902&1396 | long - slit keck ii , 4 m kitt peak , and 4.5 m mmt spectrophotometric data are used to investigate the stellar population and the evolutionary status of i zw 18c , the faint c component of the nearby blue compact dwarf galaxy i zw 18 .
hydrogen h@xmath0 and h@xmath1 emission lines are detected in the spectra of i zw 18c , implying that ionizing massive stars are present .
high signal - to - noise keck ii spectra of different regions in i zw 18c reveal h@xmath2 , h@xmath3 and higher order hydrogen lines in absorption .
several techniques are used to constrain the age of the stellar population in i zw 18c .
ages derived from two different methods , one based on the equivalent widths of the h@xmath0 , h@xmath1 emission lines and the other on h@xmath2 , h@xmath3 absorption lines are consistent with a 15 myr instantaneous burst model . we find that a small extinction in the range @xmath4 = 0.20 0.65 mag is needed to fit the observed spectral energy distribution of i zw 18c with that model . in the case of constant star formation , all observed properties are consistent with stars forming continuously between @xmath5 10 myr and @xmath6 100 myr ago .
we use all available observational constraints for i zw 18c , including those obtained from _ hubble space telescope _ color - magnitude diagrams , to argue that the distance to i zw 18 should be as high as @xmath5 15 mpc .
the deep spectra also reveal extended ionized gas emission around i zw 18 .
h@xmath0 emission is detected as far as 30 from it . to a @xmath7 surface brightness limit of @xmath5 27 mag arcsec@xmath8
we find no observational evidence for extended stellar emission in the outermost regions , at distances @xmath9 15 from i zw 18 . | introduction
observations and data reduction
the stellar population in i zw 18c
the extended emission in i zw 18
conclusions | arxiv |
several atoms play basic roles in modern physics and , in fact , very different roles .
a unit of time , the second , is defined via the hyperfine interval in the cesium atom , while the atomic mass unit and the avogadro number are defined via the mass of a carbon atom .
these two atoms are significant for our system of units , si . in addition , there are some favorite atomic systems where the basic laws of nature find their expression in the most transparent way . these simple atoms , most of which consist of two bound particles , have been crucial for our understanding of key moments of modern physics and their study is still of essential interest and importance .
the simplicity and harmony of the theory of bound systems have been tempting and challenging for a while .
johannes kepler believed the solar planetary system to be governed by a harmony of discrete numbers via geometry , trying with the so - called _ platonic _ or _
regular solids_. he failed to verify that kind of the harmony and discovered instead some regularities in the planetary orbital motion known now as kepler s laws .
his discovery became a milestone in the development of theory of gravitation and mechanics and eventually led to the establishment of new mechanics ( classical mechanics ) .
three centuries after the kepler s time , a planetary model was suggested for atoms .
meantime certain regularities in the spectrum of the hydrogen atom were discovered .
those regularities , like the kepler s laws , led once again to the establishment of new mechanics ( quantum mechanics ) and simultaneously realized the kepler s dream of the harmony of numbers governing the orbital motion .
by now we know that a quantity describing a classical object can be of an arbitrary value while in the quantum case only discrete values are possible for some quantities . and
this is how certain integer numbers enter the basic equations of modern physics .
working on a new mechanics , a new dynamics or a new model , one used to try first to apply it to some ` simple ' objects .
the simplest object is a free particle .
however , only a limited number of properties of a free particle can be derived _
ab initio _ and studied with a high accuracy .
study of simple atoms opens a broad field for possible effects and applications .
a two - body atomic system such as the hydrogen atom is a natural object to verify a new model or to test a new approach .
studies of the properties of the hydrogen atom have already served to establish the so - called ` old quantum mechanics ' ( the bohr theory ) , modern nonrelativistic quantum mechanics , relativistic quantum mechanics ( based on the dirac equation ) and quantum electrodynamics ( qed ) , which was the first successful quantum field theory .
perhaps , we should even say that qed is the only quantum field theory which is successful for a really broad range of problems from atomic spectra to scattering , from low energy , related to microwave radiation , to high energy phenomena with hard annihilation and bremsstrahlung , from nano- to giga- electronvolt . figure [ 00hydr ] shows several crucial contributions to hydrogen energy levels .
we note here that one of reasons for choosing a non - relativistic equation by schrdinger over a relativistic klein - gordon - fock equation was an incorrect description by the latter of the fine structure effects in the latter .
another remark on importance of the hydrogen atom for qed is that the anomalous magnetic moment of an electron was first discovered by rabi and his colleagues @xcite as an anomaly in the hyperfine structure of hydrogen .
immediately that was interpreted as a possible anomaly related to a free electron and only afterwards was that confirmed by a direct experiment .
a historic overview of the ` contribution ' of the hydrogen atom to modern physics can be found in @xcite .
levels are labelled by the values of the principal quantum number @xmath1 , orbital moment @xmath2 , electron angular momentum @xmath3 and atomic angular momentum @xmath4 , where @xmath5 is the nuclear spin .
the gross structure ( @xmath6 ) is well explained by the bohr theory ( so - called ` old quantum theory ' ) and schrdinger theory which also predicts the hyperfine structure ( @xmath7 ) . the fine structure ( @xmath8 )
is explained by the dirac theory while the lamb shift ( @xmath9 ) is a result of bound state qed effects .
meanwhile effects of quantum electrodynamics for free particles are responsible for the @xmath10 anomaly in the hyperfine structure .
[ 00hydr ] ] one can expect that the simplest atoms are the easiest for a theoretical prediction .
that is true only in part .
a simple atom usually possesses a simple spectrum which has relatively simple and transparent properties .
the atomic energy levels are often perturbed during measurements by various external factors such as a residual magnetic or electric field .
because of the simplicity of the spectra , there is a good chance to understand the influence of those external factors on simple atoms . as examples of the highest - precision experiments on simple atoms , we remind here that the hyperfine interval in the ground state of hydrogen was the most accurately measured physical quantity for a few decades and today the @xmath11 interval in the hydrogen atom is among the most precisely measured values . decade after decade , theorists and experimentalists investigated simple atoms . as a result
their theory is the most advanced atomic theory and it has to compete with very sophisticated experiments .
the theory of simple atoms goes now far beyond non - relativistic quantum mechanics with the reduced mass of an electron .
one has also to take into account relativistic effects , recoil effects , quantum electrodynamics , effects of the nuclear spin and nuclear structure . in an early time of modern physics a list of simple atoms consisted of hydrogen only and later
deuterium , neutral helium and helium ion were added .
now the list is much longer and quite impressive .
it also contains tritium , hydrogen - like ions of almost all elements ( and other few - electron ions ) .
artificial two - body atoms are the easiest to produce and the list also includes muonium and positronium , muonic atoms , pionic , kaonic and antiprotonic atoms , exotic bound systems of two unstable particles ( such as pionium and @xmath12-atoms ) and antihydrogen .
often accuracies of theory and experiment are not compatible .
however , there is a broad range of effects , for which theory and experiment approach the same high level of accuracy .
the study of such effects forms a field called _ precision tests of bound state qed _ , which is reviewed in part here .
a number of sources have contributed to uncertainty of such tests , and the current accuracy of qed calculations for free particles and two - body atoms is not a limiting factor for qed tests .
the accuracy of the tests is limited by one of the three other sources : * an experimental uncertainty ; * an inaccuracy of taking into account effects of the strong interactions ; * an uncertainty due to a determination of the fundamental constants .
the latter enters consideration because theory is not in a position to give itself any quantitative predictions .
it provides us with some expressions containing values of certain fundamental constants , such as the rydberg constant @xmath13 , the fine structure constant @xmath14 , the electron - to - proton mass ratio etc . to make a prediction
, one needs first to determine the fundamental constants by extracting their values from some other experiments .
thus , theory serves as a bridge between different experiments .
that makes the determination of fundamental physical constants to be another important part of precision physics of simple atoms .
the contemporary situation with the qed uncertainty being below the total uncertainty of any qed test is a result of significant theoretical progress for the two last decades .
twenty or thirty years ago the qed uncertainty was often the dominant one .
the _ precision tests of qed _ form a _
multidisciplinary _ field involving atomic , nuclear and particle physics , laser spectroscopy , frequency metrology , accelerator physics , advanced quantum mechanics , quantum field theory etc .
those tests are rather not to test qed itself , but to check the overall consistency of the results and methods from different fields and in part to search for possible new physics beyond the standard model . in doing more and more advanced qed calculations , we also need to verify our approaches to deal with infrared and ultraviolet divergences , renormalization and bound state problem for few - body atoms in the case of high order terms of perturbation theory .
as already mentioned , the simplest object to test any theory is a free particle .
a study with free leptons provides us with a possibility of testing the qed lagrangian .
the most accurate data for a free particle are related to the anomalous magnetic moments of an electron and a muon . in the former case the limitation of the accuracy of a comparison of theory versus
experiment is due to a determination of the fine structure constant @xmath14 while the latter qed test is limited by effects of strong interactions entering via hadronic intermediate states for the vacuum polarization effects and by the experimental accuracy .
the qed theory of the anomalous magnetic moment is quite advanced including many hundreds of multiloop diagrams ( up to the four - loop level for the electron and five - loop level for the muon ) . that differs from a so called
_ bound state qed _ , a quantum electrodynamics theory of simple atoms , which deals mainly with relatively few one - loop and two - loop diagrams , but those are not for free particles , but for the particles bound by the coulomb field .
three - loop contributions are rather below the uncertainty of most of the bound qed calculations and if necessary can be calculated neglecting binding effects .
these two qed theories , the free qed and the bound state qed , are very different in their approaches , problems and applications and it is worth to consider their tests separately .
the bound state problem makes all calculations more complicated . intending to perform a calculation within the free qed
, one can in principle find needed basic expressions in a number of textbooks . on the contrary ,
the bound state qed is not a well - established theory and there are no published common prescriptions for the relativistic quantum bound problem .
it involves different effective approaches to solve the two - body bound problem .
precision tests of the bound state qed offer a number of different options : * the approaches for the bound problem can be essentially checked with the low-@xmath15 two - body atomic systems like hydrogen and deuterium , neutral helium and helium ions , muonium , positronium , etc . at low value of the nuclear charge @xmath15 the binding energy is of order of @xmath16 and it is essentially smaller than the energy related to the rest mass @xmath17 of the orbiting particle .
that is a regime of a weak coupling in contrast to the high-@xmath15 physics .
the weak coupling allows efficient use of a perturbation expansion over the coulomb strength @xmath18 .
many contributions of higher - order in @xmath18 are calculated diagrammatically and thus are closely related to other qed diagrams ( e.g. , for scattering theory ) .
* studies of high-@xmath15 ions are related to a strong coupling regime , however , it is unlikely to provide us with more information on bound state qed because of substantial contributions due to the nuclear structure .
such an investigation is rather useful for testing different nuclear models .
however , in some particular cases , atomic systems with not too high @xmath15 can give some important information on higher order terms of the qed @xmath19 expansion .
importance of particular atomic systems also depends on the energy interval under study .
for instance , the hyperfine structure ( hfs ) interval depends more strongly on the nuclear - structure effects than the lamb shift , and the related calculations involve more details of the nuclear structure .
the accuracy of the nuclear - finite - size contribution is often claimed to be very high , however , their uncertainty is customarily not well verified .
it is often estimated from a variation of the result for an energy shift after application of certain models of the charge distribution while the charge radius is fixed .
however , to claim such a calculation as an _ ab initio _ result , one has to reconsider first the original scattering and spectroscopy data on the nuclear form factor and related nuclear theory and to check how strongly they constrain the shape of the distribution in general and its parameters and afterwards to question how much the nuclear - size correction varies within allowed shapes and parameters .
that may be done only on a nucleus - by - nucleus basis .
lack of such a consideration in study of high-@xmath15 study reduces importance of the bound - state qed calculations , which are in such a case one more confirmation of applicability of the phenomenological model of the charge distribution .
* studies of few - electron atoms involve electron - electron interactions . in the case of high @xmath15 ions ,
the electron - electron interaction is proportional to @xmath14 , while the interaction between an electron and the nucleus is proportional to @xmath18 .
if the ion is highly charged with only few electrons remaining , the electron - electron interaction can be treated as a perturbation . as a result , at high @xmath15
the electron must be treated relativistically , i.e. , no expansion over @xmath18 can be used , however , the treatment of the electron - electron interaction as a perturbation leads to the @xmath20 expansion .
in the case of light atoms , electrons are essentially nonrelativistic but the electron - electron interaction is compatible with the nucleus - electron interaction .
the few - electron atoms ( like , e.g. , the neutral helium atom or heavy lithium - like ions ) is the only case when the uncertainty of the qed calculations used to be a limiting factor for a comparison of theory versus experiment . * there are some other two - body atoms under investigation .
they contain a muon or a hadron as an orbiting particle .
the orbit in a muonic atom is much closer to the nucleus than in the case of a conventional atom with an orbiting electron , and the muonic energy levels are much more sensitive to the nuclear structure . in the case of a hadronic atom ,
the orbit lies even lower than in a muonic atom , while the interaction of the orbiting particle and the nucleus is dominated by the strong interaction , in contrast to the electromagnetic interaction in conventional ( i.e. , electronic ) and muonic atoms .
different exotic or muonic atoms offer a unique opportunity to study certain particle properties by spectroscopic means with high precision .
frequently high-@xmath15 spectroscopy is quoted as a qed test _ at a strong field_. however , that is not exactly the case .
a value of macroscopic meaning , such as the electric field strength * e * , has not much sense inside an atom .
other details are more important . for example , the strength of the field can be characterized by the average potential energy @xmath21 which increases with the nuclear charge @xmath15 and the mass of the orbiting particle @xmath22 and decreases with the principal quantum number @xmath1 .
the strongest field is related to high-@xmath15 atoms with an orbiting particle , heavier than electron , at the ground state .
muonic atoms have been studied for a while and with @xmath23 they offer a test at a field stronger than in electronic atoms and at shorter distances @xmath24 the distance @xmath25 ( or a related characteristic value of the momentum transfer @xmath26 ) is another important characteristic of the electric field inside an atom .
we also note that a characteristic value of the potential @xmath27 , the distance @xmath28 and the strength of the field @xmath29 strongly depends on a particular quantity under study .
what we present above in eqs .
( [ eaver ] ) and ( [ raver ] ) is related to the leading contributions to atomic energy levels .
higher order corrections , such as qed corrections , may involve various short - distance effects with characteristic distance @xmath30 and momentum transfer of about @xmath31 .
they correspond to higher field than long - distance effects .
the case of _ strong fields _ at short distances in which a characteristic momentum transfer is higher than @xmath32 leads to an enhancement of polarization effects of the electron - positron vacuum . that makes the physics of muonic atoms quite different from that of conventional atoms .
high-@xmath15 ions offer another opportunity a _ strong coupling _ regime with the binding energy comparable to @xmath17 which implies a completely relativistic consideration .
the strong - coupling regime is very different from perturbative weak - coupling one .
the calculations are more complicated
. they may be of interest for study of methods to be applied in the case of strong interactions .
some of the high-@xmath15 results are important also for high - accuracy low-@xmath15 physics .
however , one has to remember that the strong coupling is rather not a problem of qed , but a problem of its application . in this sense ,
the stronger the coupling , the less ` simple ' the system . to study pure qed problems in more detail
we should prefer a weak - coupling situation
. in muonic atoms the coupling constant and other parameters can take quite different values . while for @xmath33 the states of interest are @xmath34 , the principal quantum number @xmath1 for medium @xmath15 may be higher than 1 or 2 , which used to be common for the high-@xmath15 experiments with conventional atoms .
however , in both cases ( muonic / exotic atoms and high-@xmath15 ions ) , understanding the nuclear properties is needed for any theoretical predictions .
for instance , let us look at studies of the @xmath35 lamb shift in hydrogen - like uranium .
recently , the experimental result was improved @xcite .
the experimental uncertainty allows to check one - loop radiative corrections , but not two - loop effects calculated few years ago @xcite .
those two - loop corrections are an object of intensive theoretical study and are of great interest ( see sects . 5 , 7 and 13 ) .
the finite - nuclear - size uncertainty for the lamb shift in u@xmath36 is estimated at a level approximately tenfold below the experimental uncertainty .
however , the uncertainty of the result was obtained @xcite ( see also @xcite ) by comparison of two distributions of the nuclear charge , which were the homogenous spherical distribution and the fermi distribution .
the value of the mean square radius was fixed as @xmath37fm for both distributions . however , this result was obtained in ref .
@xcite from muonic uranium spectroscopy suggesting a modified fermi distribution , which is different from both distributions applied in @xcite .
it was stated @xcite that the uncertainty presented there was of pure statistical nature , while the model - dependence had not been studied and related systematic error was not included .
apparently , the characteristic atomic momentum in muonic uranium is much higher than in the conventional hydrogen - like uranium ion and muonic spectra are substantially more sensitive to the nuclear - shape effects than conventional ones . if one expects that a comparison of the homogenous spherical distribution and the fermi distribution leads to a plausible estimation of the finite - nuclear - size effects ( which should be verified ) , that should be applied first to muonic atoms to learn the systematic error of the mean square radius
then , with a value of the radius related to each distribution , one should calculate the energy levels .
that should substantially increase uncertainty .
this example shows that how fragile qed tests with high-@xmath15 ions can be and how much work should be additionally done to develop them .
a purpose of this paper is to give a brief review on _ precision _ physics of simple atoms related to the _ accurate _ tests of quantum electrodynamics for bound states and the determination of fundamental constants .
because of that , we focus our considerations mainly on light hydrogen - like atoms ( hydrogen , deuterium , helium ion , muonium , positronium ) and some medium @xmath15 ions , where the nuclear structure and hadronic effects are not too significant and allow a crucial test of advanced qed theory with high order contributions .
we distinguish here qed and bound state qed , which example is an application of qed to the simplest atoms .
studying less simple atoms we deal not with just bound state qed , but with its realization for not - too - simple atoms .
the additional problem may be related to strong field , strong coupling , crucial effects due to the nuclear structure , electron - electron interaction in few - electron atoms etc .
definitely , a number of investigations of less simple atoms are of interest . however , dealing with them is to go beyond the simplest bound state qed .
we note , that the light hydrogen - like atoms are the most attractive from a theoretical point of view .
they involve neither electron - electron interactions , nor strong - coupling effects or so .
we consider in the next section , what is the most favorite choice for experimental accuracy options in testing bound state qed for hydrogen - like atoms .
that is related to the light atoms .
they are also favorite in principle for theoretical accuracy , being the simplest atomic systems .
that does not mean that study of other atoms are out of interest .
first of all , what is important is not just an atomic system , but a certain transition there .
as one can see in this review , certain transitions , or combinations of certain quantities , related to different transitions , may offer various theoretical or experimental advantages . because of the simplicity of simple atoms and multidisciplinary nature of the _ precision tests of the bound state qed _ we have tried to review the results as simply and briefly as possible in order to make the paper readable by non - experts .
detailed reference information on crucial theoretical contributions is collected in tables . while considering qed , there is always the problem of selecting units . from a theoretical point of view
one should like to simplify equations by applying relativistic units in which @xmath38 , or using the atomic units .
meanwhile , from an experimental point of view the equations should be expressed in units convenient for measurements . in our paper
we choose a compromise .
all results are presented in the units of the si .
however , we present , when possible , most of the results using natural constants as a kind of units . for example , to define the so - called fermi energy which is a result of the non - relativistic interactions of the electron and nuclear magnetic moments , we write for hydrogen @xmath39and thus the proton and electron magnetic moments are explicitly expressed in units of the bohr magneton .
the other factors do not directly contain electron charge @xmath40 , but only the fine structure constant @xmath14 , which does not depend on a choice of ` macroscopic ' units ( in which it may be defined as @xmath41 , @xmath42 , @xmath43 depending on the definition of the unit for the electric charge ) .
still , here we make an exception for numerical values of atomic energy , which are always expressed in frequency units , i.e. , in terms of @xmath44 .
this is because it is widely preferred to write equations for energy , while the actually measured quantities are the transition frequencies , spectroscopic linewidths and decay rates .
the most frequently used notations are explained in appendix [ s : not ] .
more details on physics of hydrogen - like atoms can be found in : * various basic questions in books @xcite ; * an overall review with an extended comparison of theory and experiments related to sixties and early seventies in @xcite ; * minireviews on particular questions in @xcite ; * review on theory of light hydrogen - like atoms in @xcite ; * original results presented at _ hydrogen atom _ conferences and on international conferences of _ precision physics of simple atomic systems _ ( psas ) in @xcite .
the books @xcite published in series _ lecture notes in physics _ , volumes 570 and 627 , are also available on - line .
the recent psas conference on simple atoms took place in early august 2004 in brazil as a satellite meeting to the international conference on atomic physics ( icap ) .
the coming psas meeting is scheduled for june 2006 in venice .
a few problems related to our paper are not presented here in detail .
* we consider the fundamental constants only in connection to simple atoms and qed . more detail on fundamental constants can be found in @xcite .
* heavy few - electron ions , which are of a great interest for study of application of bound state qed to strong - coupling and few - electron systems , are reviewed in , e.g. , @xcite . * recent progress with exotic and muonic atoms
is presented in detail in @xcite .
study of such atoms are not of a big interest because of qed , on contrary , they deliver us a crucial information on other part of physics , namely , particle and nuclear physics
. most of this review was ready before new results on the fundamental constants @xcite became available and through out the paper we compare the qed - related results on the fundamental constants with a previous set of the recommended constants @xcite .
we also note that a substantial part of qed results under review appeared between the publications of two recommended sets ( their deadlines for collecting the input data were 1998 and 2002 ) and most of recent results have been accommodated in @xcite .
we start our review with an introductory discussion of spectra of simple atoms and related basic qed phenomena and next consider qed tests with hydrogen ( the lamb shift and hyperfine structure ) and other light atoms .
we discuss studies of pure leptonic atoms such as muonium and positronium .
in addition to spectra we consider the magnetic moments of bound particles in various two - body atomic systems .
the concluding sections of the paper are devoted to fundamental constants and problems of the bound state qed .
let us discuss the spectrum of simple two - body atoms in more detail .
the gross structure of atomic levels in a hydrogen - like atom comes from the schrdinger equation with the coulomb potential and the result is well known : @xmath45 where @xmath15 is the nuclear charge in units of the proton charge , @xmath46 is the reduced mass of the atomic orbiting particle ( mostly , an electron ) @xmath47 here , @xmath22 and @xmath48 are masses of the orbiting particle and the nucleus . *
relativistic corrections ( one can find them from the dirac equation ) ; * hyperfine structure ( due to the nuclear magnetic moment ) ; * recoil corrections ; * radiative ( qed ) corrections ; * nuclear - structure corrections .
a structure of levels with the same value of the principal quantum number @xmath1 is a kind of signature of any atomic system .
for most of the precision applications the substructure of interest is related to @xmath49 .
the corrections decrease with a value of the principal quantum number as @xmath50 or faster .
the only exception is the uehling correction for muonic and exotic atoms which scales as @xmath51 for medium and high @xmath15 .
.various contributions to the energy levels .
the results are in units of @xmath52 , where @xmath22 is the mass of the orbiting particle . here
: @xmath48 is the nuclear mass and @xmath53 is the proton mass which enters equations if one measure the nuclear magnetic moment in units of the nuclear magneton .
a contribution of the nuclear magnetic moment , i.e. , the hyperfine structure , appears if the nuclear spin is not zero .
@xmath54 stands for the nuclear ( charge ) radius.[tmainqed ] [ cols="<,^,^,^ " , ] a. gubmeridze , th .
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@xcite , p. 314 . | quantum electrodynamics is the first successful and still the most successful quantum field theory .
simple atoms , being essentially qed systems , allow highly accurate theoretical predictions . because of their simple spectra , such atoms have been also efficiently studied experimentally frequently offering the most precisely measured quantities .
our review is devoted to comparison of theory and experiment in the field of precision physics of light simple atoms .
in particular , we consider the lamb shift in the hydrogen atom , the hyperfine structure in hydrogen , deuterium , helium-3 ion , muonium and positronium , as well as a number of other transitions in positronium . additionally to a spectrum of unperturbed atoms , we consider annihilation decay of positronium and the @xmath0 factor of bound particles in various two - body atoms .
special attention is paid to the uncertainty of the qed calculations due to the uncalculated higher - order corrections and effects of the nuclear structure .
we also discuss applications of simple atoms to determination of several fundamental constants . simple atoms , precision measurements , bound states , quantum electrodynamics ( qed ) , hydrogen - like atoms , fundamental constants 12.20.fv , 12.20.ds , 31.30.jv , 06.02.jr , 31.30.gs , 36.10.dr , 13.40.em , 13.40.gp , 27.10.+h | introduction
spectrum of simple atoms and main qed phenomena | arxiv |
"as a consequence of the interaction of galactic cosmic ray particles with air nuclei in the upper a(...TRUNCATED) | " this paper presents first results of an ongoing study of a possible association between muon enhan(...TRUNCATED) | "introduction\nthe tupi muon telescope\nmethod of observation\ngle associated flare\nresults\nconclu(...TRUNCATED) | arxiv |
"for an integrable function @xmath0 on the complex unit circle @xmath1 the _ toeplitz matrix _ @xmat(...TRUNCATED) | " we consider the asymptotic behavior of the eigenvalues of toeplitz matrices with rational symbol a(...TRUNCATED) | "introduction\nstatement of results\nproof of theorem\nproofs of proposition\nexample\nacknowledgmen(...TRUNCATED) | arxiv |
"it has been known for many decades that an accelerated detector , moving in a quantum field prepare(...TRUNCATED) | " we propose an experiment in which the phonon excitation of ion(s ) in a trap , with a trap frequen(...TRUNCATED) | introduction
trapped ion model.
finite chirp, general expression
discussion and conclusion. | arxiv |
"measurement of the anomalous magnetic moment of the muon is very important because it can in princi(...TRUNCATED) | " a new high - precision experiment to measure the muon _ \n g_-2 factor is proposed . \n the develo(...TRUNCATED) | "introduction\n@xmath11-2 ring with a noncontinuous magnetic field and magnetic focusing\navoiding a(...TRUNCATED) | arxiv |
"when illuminated by the galactic interstellar radiation field , dust in the interstellar medium giv(...TRUNCATED) | " in 1992 the far - ultraviolet space telescope ( faust ) provided measurements of the ultraviolet ((...TRUNCATED) | "introduction\nthe data\nthe radiative transfer model\nderivation of scattering properties\ndiscussi(...TRUNCATED) | arxiv |
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