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We thank the referee for valuable comments that helped us to improve the paper. | We thank the referee for valuable comments that helped us to improve the paper. |
S.W., C.L., R.L., and H.W. were supported by NSF grants AGS 08-19662, AGS 08-49453, and AGS 09-36665, and NASA grants NNX 08AQ90G, NNX 08AJ23G, and NNX 11AC05G. N.D. was supported by NASA grant NNX 08AQ32G. | S.W., C.L., R.L., and H.W. were supported by NSF grants AGS 08-19662, AGS 08-49453, and AGS 09-36665, and NASA grants NNX 08AQ90G, NNX 08AJ23G, and NNX 11AC05G. N.D. was supported by NASA grant NNX 08AQ32G. |
It will be instructive to look briefly at Wari Dark Matter (WDMD). both to see that some variants of CDAD have less success than others iu fitting cosmological observations. and also because there is renewed interest in WDAL | It will be instructive to look briefly at Warm Dark Matter (WDM), both to see that some variants of CDM have less success than others in fitting cosmological observations, and also because there is renewed interest in WDM. |
Although CIIDM and WDALI ave similar in the seuse that doth are intermediate models between CDAL aud TIDAL CITDM aud WDAL are quite different in their implications. | Although CHDM and WDM are similar in the sense that both are intermediate models between CDM and HDM, CHDM and WDM are quite different in their implications. |
The problems with a pure Hot Dark Matter (IIDMD) adiabatic cosmology are well known: frec-streaming of the hot dark matter completely estrovs scale fluctuations. so that the first structures that can form are on the mass scale of clusters or superclusters. and galaxies nuust formi by fragmentation of these burger structures: but observations show that galaxies are much older than superchisters. which have low overdensity aud are still forming. | The problems with a pure Hot Dark Matter (HDM) adiabatic cosmology are well known: free-streaming of the hot dark matter completely destroys small-scale fluctuations, so that the first structures that can form are on the mass scale of clusters or superclusters, and galaxies must form by fragmentation of these larger structures; but observations show that galaxies are much older than superclusters, which have low overdensity and are still forming. |
Moreover. with the CODE upper limit to the normalization of IDF. hardly αν structure at all will form even by the preseut epoch. | Moreover, with the COBE upper limit to the normalization of HDF, hardly any structure at all will form even by the present epoch. |
WODAL is a simple modification of DAL obtained by changing the assumed average tuber deusitv » of the particles. | WDM is a simple modification of HDM, obtained by changing the assumed average number density $n$ of the particles. |
In the usual TIDAL. the dark matter particles are ucutrinos. cach species of which has 5,=108 7". with a corresponding mass of ny—QO,pg/n,0,9257 cV. with lL. | In the usual HDM, the dark matter particles are neutrinos, each species of which has $n_\nu=108$ $^{-3}$, with a corresponding mass of $m_0 =
\Omega_\nu \rho_0/n_\nu = \Omega_\nu 92 h^2$ eV, with $\Omega_\nu =
1- \Omega_b \approx 1$ . |
In WDAML there is a new parameter. m/m. the ratio of the mass of the warn particl to the above neutrino mass: correspoudinely. the ή, of the wari particles is reduced bv the inverse of this factor. so that thei otal contribution to the cosmological density is unchanged. | In WDM, there is a new parameter, $m/m_0$, the ratio of the mass of the warm particle to the above neutrino mass; correspondingly, the number density of the warm particles is reduced by the inverse of this factor, so that their total contribution to the cosmological density is unchanged. |
Pagels and I [55] xoposed perhaps the first WDAL particle candidate. a light exaxitiuo. which was ie lightest supersvuuuetric particle (LSP) in the earliest version of Πο phenomenoloey Gvhich was subsequeutly larecly abaudoned m favor of a hidden-sector gsvpeorsvnuuuetry breaking scheme. but which is now being reconsidered: 0). | Pagels and I \cite{PP82} proposed perhaps the first WDM particle candidate, a light gravitino, which was the lightest supersymmetric particle (LSP) in the earliest version of supersymmetric phenomenology (which was subsequently largely abandoned in favor of a hidden-sector sypersymmetry breaking scheme, but which is now being reconsidered: cf. \cite{Dine96}) ). |
Olive Turner [60] proposed left-handed ucutrinos as ΤΟΝΙ | Olive Turner \cite{OT82} proposed left-handed neutrinos as WDM. |
, In tli cases. these particles iuteract much more weakly than neutrinos. decouple earlier from the hot big bane. aud thus have diluted uuuber density compared o neutrinos since they do not share iu the entropy rekased by the subsequent aunihilation of species such as quarks. | In both cases, these particles interact much more weakly than neutrinos, decouple earlier from the hot big bang, and thus have diluted number density compared to neutrinos since they do not share in the entropy released by the subsequent annihilation of species such as quarks. |
This is analogous to (but more extreme han) the neutrinos themselves. which have lower uuuber deusitv today than photous because the neutrinos decouple before | annihilation (aud also vecause they are fermiouxs). | This is analogous to (but more extreme than) the neutrinos themselves, which have lower number density today than photons because the neutrinos decouple before $^+$ $^-$ annihilation (and also because they are fermions). |
Iu order to investigate the cosmnological duplications of any dark matter candidate, it is necessary to work out the gravitational clustering of these warticles. first in linear theory. and then after the amplitude of the fluctuations grows iuto the nonlinear regime. | In order to investigate the cosmological implications of any dark matter candidate, it is necessary to work out the gravitational clustering of these particles, first in linear theory, and then after the amplitude of the fluctuations grows into the nonlinear regime. |
Colombi. Dodelson. Widrow [61] recently did this for WDML aud Figure lin their paper compares the square of the lear vauster functions for WDM aud CIIDM. | Colombi, Dodelson, Widrow \cite{WDM} recently did this for WDM, and Figure 1 in their paper compares the square of the linear transfer functions for WDM and CHDM. |
One often can study huge scale structure just on the basis of such linear calculations. without the need to do computationally expensive sinulatious of he nou-lnear gravitational clusterimg. | One often can study large scale structure just on the basis of such linear calculations, without the need to do computationally expensive simulations of the non-linear gravitational clustering. |
Such studies have shown that matchinghe observed cluster aud ealaxy correlations on scales of about 20-30 72.1 Mpe | Such studies have shown that matchingthe observed cluster and galaxy correlations on scales of about 20-30 $h^{-1}$ Mpc |
system ds close to zero velocity in the source frame while the blue svsteià originates from immaterial which has an outflow velocity of ~L900 aus + (towards us with respect to the quasar). | system is close to zero velocity in the source frame while the blue system originates from material which has an outflow velocity of $\sim 1900$ km $^{-1}$ (towards us with respect to the quasar). |
Several factors the measurement of the absorption lines: iiodest. ο...S/N. thelimited resolution of FOS and uncertainties affecting profile of the eissiou lines. | Several factors hamper the measurement of the absorption lines: modest S/N, limited resolution of FOS and uncertainties affecting the profile of the emission lines. |
The absorption lues were measured assunius a plausible reconstruction of the top of the ομοι Ines. | The absorption lines were measured assuming a plausible reconstruction of the top of the emission lines. |
Still it is not possible to obtain sufficieutlv accurate Lya absorption profile. aid NW aud CTV doublet ratios to ascertain whether these lines are optically thin or thick and to estimate the covering factor (some of the galactic lines do not reach zero either). | Still it is not possible to obtain sufficiently accurate $\alpha$ absorption profile, and NV and CIV doublet ratios to ascertain whether these lines are optically thin or thick and to estimate the covering factor (some of the galactic lines do not reach zero either). |
Hieher spectral resolution is needed to elucidate these important points. | Higher spectral resolution is needed to elucidate these important points. |
With this caveat. the measures are eiven in Table | | With this caveat, the measures are given in Table 4. |
Particularly interesting is theabsorption line at 1282.5 (CEWIIM of and EW of z 0.7 Aj. which could be 1175 in the 2 = 0.091 system. | Particularly interesting is the absorption line at 1282.5 (FWHM of and EW of $\approx$ 0.7 ), which could be $^*$ 1175 in the $z$ = 0.091 system. |
The agreement in redshift is good. | The agreement in redshift is good. |
This line is preseut in the IUE aud | This line is present in the IUE and |
clouds where they formed. while as they become older. they progressively. eiierge from them. | clouds where they formed, while as they become older, they progressively emerge from them. |
In this picture. the spectra of SSPs of differcut ages are supposed to be dust-reddened by different amounts: dust is assunued to he distributed so to simulate a uuiforiui laver in frout of the stars. and the Calactic extinction curve (Cardellictal..1989) is adopted. | In this picture, the spectra of SSPs of different ages are supposed to be dust-reddened by different amounts; dust is assumed to be distributed so to simulate a uniform layer in front of the stars, and the Galactic extinction curve \citep{cardelli89}
is adopted. |
Building a selt£-cousisteut. clemical model. that svouk ake iuto account changes in the metal coutent of a ealaxy and its cliemical evolution as a fiction of mass and star formation history. was far bevoud the scope of lis work. | Building a self-consistent chemical model, that would take into account changes in the metal content of a galaxy and its chemical evolution as a function of mass and star formation history, was far beyond the scope of this work. |
Tlis is why we adopted a homogencous value or the metallicitv. for our theoretical spectra. aud lef it to the model free to choose between. three differeu sets of metallicity. namely Z=0.05. Z=0.02 aud Z=0.0 (super-solar. solar and sub-solar. respectively). | This is why we adopted a homogeneous value for the metallicity for our theoretical spectra, and left it to the model free to choose between three different sets of metallicity, namely Z=0.05, Z=0.02 and Z=0.004 (super-solar, solar and sub-solar, respectively). |
Fitting au observed spectrum with a single value of the metallicity ix equivalent to assuming that this value belongs to the stellar population that is dominating its light. | Fitting an observed spectrum with a single value of the metallicity is equivalent to assuming that this value belongs to the stellar population that is dominating its light. |
However. as described in F07. acceptable fits are obtained for most of he spectra adopting different metallicities. which means hat this kiud of analysis is often not able to provide a unique value for the metallicity. | However, as described in F07, acceptable fits are obtained for most of the spectra adopting different metallicities, which means that this kind of analysis is often not able to provide a unique value for the metallicity. |
It is clear lat. assuninueg a unique value for the SSP'« netallicity when reproducing an observed spectruni is a sinplifiug assumption since. iu practice. the stellar o»pulatious of a galaxy span a rauge iu metallicity values. | It is clear that, assuming a unique value for the SSP's metallicity when reproducing an observed spectrum is a simplifying assumption since, in practice, the stellar populations of a galaxy span a range in metallicity values. |
One could hence question the reliability of the nass aud of the SEIT determination done Ww usns one single metallicity value. | One could hence question the reliability of the mass and of the SFH determination done by using one single metallicity value. |
To etter undoerstaiu this possible das due to the mix of uctallicities that is expected in galaxies. we repeated the check already performed in FOF: we built template svuthetic spectra with 26 differen SFUs as in FU. but with values ofthe metallicity varving as a function of stellar age. to roughly simulate a chenucal evolution. and we analyzed them by means of our spectroplotometric fitting code. | To better understand this possible bias due to the mix of metallicities that is expected in galaxies, we repeated the check already performed in F07: we built template synthetic spectra with 26 different SFHs as in F07, but with values of the metallicity varying as a function of stellar age, to roughly simulate a chemical evolution, and we analyzed them by means of our spectrophotometric fitting code. |
The results clearly show that the wav we deal with the metallicity docs not introduce any bias in the recovered total stellar mass or SFU. | The results clearly show that the way we deal with the metallicity does not introduce any bias in the recovered total stellar mass or SFH. |
All of the stellar population properties that are derived are strictly related to the theoretical mocels that we use in our fitting aleorithlun. | All of the stellar population properties that are derived are strictly related to the theoretical models that we use in our fitting algorithm. |
It is hence of fouudamental Huportance to eive all the details of the plivsies aud of the paraueters that were used to build the. | It is hence of foundamental importance to give all the details of the physics and of the parameters that were used to build them. |
First of all. WE make use of he Padova evolutionary tracks (Bertellietal...1991) and use a standard Salpeter(1955). initial mass function (IATF). with masses in the range 0.15-120 AL... | First of all, WE make use of the Padova evolutionary tracks \citep{bertelli94} and use a standard \cite{salpeter55} initial mass function (IMF), with masses in the range 0.15-120 $_\odot$. |
Our optical spectra were obtained using two different sets of observed stellar atinosplieres: for ages vouuger than LO? vears we used Jacobyetal.(198L).. while for older SSPs we used spectra from the MILES library (Sauchez-Blazquez.2001:Sánchez-Dlázquezetal.Ww106) and both sets were degraded in spectral resolution. oeoi order to match that of our observed spectra (namely 3. 6 and 9 oof FWIAL see Sect.?? for details). | Our optical spectra were obtained using two different sets of observed stellar atmospheres: for ages younger than $10^9$ years we used \cite{jacoby84}, while for older SSPs we used spectra from the MILES library \citep{Psanchez04,Psanchez06} and both sets were degraded in spectral resolution, in order to match that of our observed spectra (namely 3, 6 and 9 of FWHM, see \ref{sec:data} for details). |
Using the theoretical libraries of I&urucz. the SSP spectra were extended to the ultra-violet aud iutrared. widening. iu this wav. the waveleneth range down to 90 am up to ~10? ((note that in hese intervals the spectral resolution is much lower. being ~20Α.. but im auv case outside the range of interest or the spectra used for our analysis). | Using the theoretical libraries of Kurucz, the SSP spectra were extended to the ultra-violet and infrared, widening, in this way, the wavelength range down to 90 and up to $\sim 10^9$ (note that in these intervals the spectral resolution is much lower, being $\sim 20$, but in any case outside the range of interest for the spectra used for our analysis). |
Cas enission. whose effect is visible through emission lines, was also computed and included in the theoretical spectra by ineans of the plhotoiouisation code (Ferlaud.1996). | Gas emission, whose effect is visible through emission lines, was also computed –and included in the theoretical spectra– by means of the photoionisation code \citep{ferland96}. |
.. The optical spectra of SSPs Vounecr than ~ὃς10° display. in this way. both permitted forbidden lines (typically. hydrogen. [On].Ομ. and ΓΗ). | The optical spectra of SSPs younger than $\sim 2\times 10^7$ display, in this way, both permitted and forbidden lines (typically, hydrogen, , and ). |
This nebular component was computed asstunine (seeOsterbrock.-150).. an electron teixperatine of 104 IS. aud an electron density of 100 cix7. | This nebular component was computed assuming \citep[see][]{osterbrock89}, an electron temperature of $10^4$ K, and an electron density of $100$ $^{-3}$. |
The radius of the ioniziug star cluster was asstuned to be 15 pe. and its mass 105 M. | The radius of the ionizing star cluster was assumed to be 15 pc, and its mass $10^4$ $_\odot$. |
Finally. cuuission from the circmusteclar euvelopes of ACB stars was computed aud added as described iu Bressan(1995). | Finally, emission from the circumstellar envelopes of AGB stars was computed and added as described in \cite{bressan98}. |
. The initial set of SSPs was composed of 105 theoretical spectra referring to stella ages ranging from LO? to 20«10? years. for each one of the three aforeaneutioned values of the metallicity. | The initial set of SSPs was composed of 108 theoretical spectra referring to stellar ages ranging from $10^5$ to $20\times 10^9$ years, for each one of the three afore-mentioned values of the metallicity. |
Determining the age of stellar »opulatious from an iuteerated optical spectrum with such a ligh temporal resolution is well bevoud the capabilities of auy spectral analysis. | Determining the age of stellar populations from an integrated optical spectrum with such a high temporal resolution is well beyond the capabilities of any spectral analysis. |
Weuce. as a first step. we reduced he stellar age resolution by binning the spectra. | Hence, as a first step, we reduced the stellar age resolution by binning the spectra. |
This was done by taking into account both the characteristics of the evolutionary plases of stars. aud the trends im spectral catures as a function of the SSP age (see both sectiou 2.1.1 and Fie. | This was done by taking into account both the characteristics of the evolutionary phases of stars, and the trends in spectral features as a function of the SSP age (see both section 2.1.1 and Fig. |
| in ΕΟΤ). | 1 in F07). |
After combining the spectra at lis first stage. we euded up with 13 stellar age bius. | After combining the spectra at this first stage, we ended up with 13 stellar age bins. |
As we describe in FO7. this set of theoretica Spectra originally included also a SSP whose age. namely ~17.5 Cyr. is older than the universe age. | As we describe in F07, this set of theoretical spectra originally included also a SSP whose age, namely $\sim 17.5$ Gyr, is older than the universe age. |
The use of his SSP was nierelv statistical: since the ouly appreciable difference between the three oldest SSPs of our set is. actually. the niass-to-hunuinositv ratio. using such an ok SSP would prevent our random search of the best fit model to be systematically biased towards the vounges of the old SSPs. | The use of this SSP was merely statistical: since the only appreciable difference between the three oldest SSPs of our set is, actually, the mass-to-luminosity ratio, using such an old SSP would prevent our random search of the best fit model to be systematically biased towards the youngest of the old SSPs. |
Nevertheless. the adoption of such an approach can lead some models to be dominated by lis verv old stellar population vielding. in this way. mass values that are too high. due to the higher mass-to-lieht ratio. | Nevertheless, the adoption of such an approach can lead some models to be dominated by this very old stellar population yielding, in this way, mass values that are too high, due to the higher mass-to-light ratio. |
To overcome this issue we decide to avoid the use of the oldest stellar populations. linütiug ourselves tostellar populations whose ages are consistent with that of the universe. | To overcome this issue we decide to avoid the use of the oldest stellar populations, limiting ourselves tostellar populations whose ages are consistent with that of the universe. |
We will hence refer. from now on. to these 12 SSP«. | We will hence refer, from now on, to these 12 SSPs. |
the reference lor the oxvgen abundances. | the reference for the oxygen abundances. |
If we examine the clusters with nominal ages corresponding to the progenitors of the three PN Types. namely. in the age intervals of (<1. 1<t<5. and (25 Gyr. we can compare directly the PN results with those age-appropriate clusters whose oxvgen abundances are available. | If we examine the clusters with nominal ages corresponding to the progenitors of the three PN Types, namely, in the age intervals of $\leq$ 1, $<$ $<$ 5, and $\geq$ 5 Gyr, we can compare directly the PN results with those age-appropriate clusters whose oxygen abundances are available. |
We have caleulated the absolute oxvgen abundances for the clusters by using the same solar oxvgen ratio used in the original papers. | We have calculated the absolute oxygen abundances for the clusters by using the same solar oxygen ratio used in the original papers. |
In cases where we could not find the solar oxvgen ratio used by the authors. we do not use the datum in the gradient determination. given that adopted values can differ bv large amounts. | In cases where we could not find the solar oxygen ratio used by the authors, we do not use the datum in the gradient determination, given that adopted values can differ by large amounts. |
Oxveen abundances are available [or a few very voung clusters (<1 Gyr) within a very limited. galactocentric distance range. (hus an estimate of the gradient. would not be meaningful (allhough the abundance distribution would be compatible. within the uncertainties. with that of Tvpe I PNe). | Oxygen abundances are available for a few very young clusters $\leq$ 1 Gyr) within a very limited galactocentric distance range, thus an estimate of the gradient would not be meaningful (although the abundance distribution would be compatible, within the uncertainties, with that of Type I PNe). |
A better comparison sample is available for the intermediate age clusters. corresponding approximatelv to (the ages of Type II PNe progenitors. where (he data span a range of ealactocentrie distances. ancl a gradient estimate is sensible. | A better comparison sample is available for the intermediate age clusters, corresponding approximately to the ages of Type II PNe progenitors, where the data span a range of galactocentric distances, and a gradient estimate is sensible. |
We find that the oxveen eradient in open clusters of ages between |. and 5 Gyr is Mog(O/ID) /.NAB;—-0.023 dex +. | We find that the oxygen gradient in open clusters of ages between 1 and 5 Gyr is $\Delta$ $\Delta$ $_{\rm G}$ =-0.028 dex $^{-1}$. |
As discussed further in the next section. NGC 6253 and NGC 6791 can be suspected of having an origin in (he inner disk. therefore not being truly representative ol the metallicity at the radius where Chev are observed today. | As discussed further in the next section, NGC 6253 and NGC 6791 can be suspected of having an origin in the inner disk, therefore not being truly representative of the metallicity at the radius where they are observed today. |
Only NGC 6253 [alls in the age interval considered here. | Only NGC 6253 falls in the age interval considered here. |
Lit is removed from the sample. (the gradient decreases to -0.022 dex 4. | If it is removed from the sample, the gradient decreases to -0.022 dex $^{-1}$. |
The two estimates of metallicity gradient for open clusters of ages corresponding to Type IL PN progenitors well encompass those derived directly from these PNe. | The two estimates of metallicity gradient for open clusters of ages corresponding to Type II PN progenitors well encompass those derived directly from these PNe. |
As shown in Figure 8. oxveen gradients from open clusters and PNe of intermediate ages are fully consistent. with one another. | As shown in Figure 8, oxygen gradients from open clusters and PNe of intermediate ages are fully consistent with one another. |
Finally. for clusters older (han 5 Gyr we have only three possible data points and | Finally, for clusters older than 5 Gyr we have only three possible data points and |
redshift 2. the maximum value of A. | redshift $z$, the maximum value of $K$. |
Similar to stars. non- disces have a maximum mass Al/p4 which increases with central redshift. | Similar to stars, non-rotating discs have a maximum mass $M/\rho\subscr{d}$ which increases with central redshift. |
In Fie. | In Fig. |
3 the total gravitational mass is plotted versus central redshift for the two dillerent equations of state. | 3 the total gravitational mass is plotted versus central redshift for the two different equations of state. |
The higher total masses are reached in the case of the simple equation (20)). dashed line. while the polvtropic relation (23)) has a maximum at a finite redshift οτε3.25. | The higher total masses are reached in the case of the simple equation \ref{eos0}) ), dashed line, while the polytropic relation \ref{eos1}) ) has a maximum at a finite redshift $z \approx 3.25$. |
The stronger relativistic cases have been caleulatecl here using a higher grid resolution of 256. 256. | The stronger relativistic cases have been calculated here using a higher grid resolution of $256 \times 256$ . |
Llowever. the last part of the curves for z/(1|]2)20.9 may nevertheless still be unreliable. | However, the last part of the curves for $z/(1+z) \ga 0.9$ may nevertheless still be unreliable. |
Phe problem arises from the strong central concentration of the surface density (Fig. | The problem arises from the strong central concentration of the surface density (Fig. |
4). | 4). |
Due to this problem AZ reaches its maximum for the first equation of state (20)) most likely at infinite central redshift. | Due to this problem $M$ reaches its maximum for the first equation of state \ref{eos0}) ) most likely at infinite central redshift. |
At larger and larger redshifts. the density increasingly peakes at the origin. (refer to the model with z= 17.0). | At larger and larger redshifts, the density increasingly peakes at the origin (refer to the model with $z=17.0$ ). |
The sequence terminates eventually at a black hole configuration. with all the mass located at the origin. | The sequence terminates eventually at a black hole configuration, with all the mass located at the origin. |
The strong increase of internal pressure with z leads also to a central drop in the metric potential D whieh is plotted. in Fig. | The strong increase of internal pressure with $z$ leads also to a central drop in the metric potential $B$ which is plotted in Fig. |
5. | 5. |
Por the pressure-free dust. disc. Bo=l everywhere: however. in the case of non-zero pressure. equation (28)) ereates a deviation from this simple relation. | For the pressure-free dust disc, $B=1$ everywhere; however, in the case of non-zero pressure, equation \ref{bjmp}) ) creates a deviation from this simple relation. |
With increasing z. the metric potential D decreases. (in all space) and its central value approaches zero in the extreme relativistic limit (2= 1). | With increasing $z$ , the metric potential $B$ decreases (in all space) and its central value approaches zero in the extreme relativistic limit $z=1$ ). |
The variation of the polvtropic constant AN with redshift is displaved in Fig. | The variation of the polytropic constant $K$ with redshift is displayed in Fig. |
6. | 6. |
With increasing redshift. A drops from the Newtonian limit and increases again for stronger relativistic disces. | With increasing redshift, $K$ drops from the Newtonian limit and increases again for stronger relativistic discs. |
Adding rotation (£2 0) will lower the value of A. In the following diagrams ἐν will be normalized. to this maximum value uas | Adding rotation $\Omega>0$ ) will lower the value of $K$ In the following diagrams $K$ will be normalized to this maximum value $K\subscr{max}$. |
Rotating clises were studied for the isentropic second equation of state only (23)). and the primary results are displaved in Fig. | Rotating discs were studied for the isentropic second equation of state only \ref{eos1}) ), and the primary results are displayed in Fig. |
7 in the A9 diagram. | 7 in the $K-\Omega$ diagram. |
The curves are labeled: with the corresponding central redshifts of the sequences. | The curves are labeled with the corresponding central redshifts of the sequences. |
Weaker relativistic disces (solid line. 2=0.025) follow the Newtonian curve (thin dotted Line. eq. 24)) | Weaker relativistic discs (solid line, $z=0.025$ ) follow the Newtonian curve (thin dotted line, eq. \ref{newtok}) ) |
closely up to a finite ο«O, and then biftweate continuously into a ring-like structure at οος20.84. | closely up to a finite $\Omega < \Omega\subscr{c}$ and then bifurcate continuously into a ring-like structure at $\Omega/\Omega_c \approx 0.84$. |
This continuous transition [rom disc to ring occurs only lor z&0.01. | This continuous transition from disc to ring occurs only for $z \la 0.01$. |
The details of the bifurcation process will be explained. below. | The details of the bifurcation process will be explained below. |
Lor intermediate central recshifts 0.01<z0.22 (dashed-dotted. line). the dises and rings coexist. with no apparent connecting branch between them. | For intermediate central redshifts $0.01 < z < 0.22$ (dashed-dotted line), the discs and rings coexist with no apparent connecting branch between them. |
Stronger relativistic. zc 0.22. (dashed. Lines) disces terminate in a mass shed limit. where at the outer edge of the disc (p= pa) gravity is balanced exactly by centrifugal forces. | Stronger relativistic, $z \ga 0.22$ , (dashed lines) discs terminate in a mass shed limit, where at the outer edge of the disc $\rho =\rho\subscr{d}$ ) gravity is balanced exactly by centrifugal forces. |
From the hydrostatic equation we obtain the criterion for reaching the mass-shed limit The end points of the dashed curves and the thick otted line mark the loci where the edge of the disc reaches is limit. | From the hydrostatic equation we obtain the criterion for reaching the mass-shed limit The end points of the dashed curves and the thick dotted line mark the loci where the edge of the disc reaches this limit. |
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