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This is due to the fact that the overall impact. of (rapid) rotation on the neutron star structure is smaller for more centrally condensed models resulting from "softer" ος (Friedmanetal.1984).. | This is due to the fact that the overall impact of (rapid) rotation on the neutron star structure is smaller for more centrally condensed models resulting from “softer” EOSs \citep{1984Natur.312..255F}. |
Therefore. for such models (here is smaller deviation from properties and structure of static configurations. | Therefore, for such models there is smaller deviation from properties and structure of static configurations. |
In addition. al even higher densities matter is expected to undergo a transition to quark-gluon plasma (Weber1999:Baldoetal.2000).. which favors a [ast cooling through enhanced nucleonic direct Urea and quark direct Urea processes (seee.g..Pageetal.2006).. | In addition, at even higher densities matter is expected to undergo a transition to quark-gluon plasma \citep{Weber:1999a,2000PhRvC..61e5801B}, which favors a fast cooling through enhanced nucleonic direct Urca and quark direct Urca processes \citep[see e.g.,][]{Page:2005fq}. |
We have studied properties of (rapidly) rotating neutron stars emploving four nucleonic EOSs. | We have studied properties of (rapidly) rotating neutron stars employing four nucleonic EOSs. |
Rapid rotation affects the neutron star structure significantly. | Rapid rotation affects the neutron star structure significantly. |
It increases the maximum possible mass up to ~1776 and increases/decreases the equatorial/polar radius by several kilometers. | It increases the maximum possible mass up to $\sim 17\%$ and increases/decreases the equatorial/polar radius by several kilometers. |
Our findings. (hrough the application of EOSs with constrained svinmnetry energy by recent nuclear terrestrial laboratory data. allowed us to constrain the mass of the neutron star in NTE J1739-235 to be between 1.7 and 9.111... | Our findings, through the application of EOSs with constrained symmetry energy by recent nuclear terrestrial laboratory data, allowed us to constrain the mass of the neutron star in XTE J1739-285 to be between 1.7 and $2.1M_{\sun}$. |
Additionally. rotation reduces central clensityv and proton fraction in the neutron star core. anc depending on the exact stellar mass ancl rotational frequency. could effectively close the fast cooling channel in millisecond pulsars. | Additionally, rotation reduces central density and proton fraction in the neutron star core, and depending on the exact stellar mass and rotational frequency could effectively close the fast cooling channel in millisecond pulsars. |
This circumstance may have important consequences for both the interpretation of cooling data and the thermal evolution modeling. | This circumstance may have important consequences for both the interpretation of cooling data and the thermal evolution modeling. |
WWe would like to thank Nikolaos Stergioulas for making the RNS code available. | We would like to thank Nikolaos Stergioulas for making the RNS code available. |
We also (hank Wei-Zhou Jiang for helpful discussions. | We also thank Wei-Zhou Jiang for helpful discussions. |
This work was supported by the National science Foundation under Grant No. | This work was supported by the National Science Foundation under Grant No. |
PILY0652548 and the Research Corporation under Άννα No. | PHY0652548 and the Research Corporation under Award No. |
7123. | 7123. |
We can perhaps begin to understand why by considering the power spectra shown in figure 3. | We can perhaps begin to understand why by considering the power spectra shown in figure 3. |
Here is the energy. contained in the /th spherical harmonic mode. either poloidal or toroidal. and as in (24) the integration over i includes the energy in the exterior vacuum fickle. ( | Here is the energy contained in the $l$ -th spherical harmonic mode, either poloidal or toroidal, and as in (24) the integration over $r$ includes the energy in the exterior vacuum field. ( |
Amel of course the cross-ternis {B,,-Di,dV vanish by the orthogonality of the spherical harmonics. | And of course the cross-terms $\int{\bf B}_{l_1}\cdot
{\bf B}_{l_2}\,dV$ vanish by the orthogonality of the spherical harmonics. |
The total enerey is thus indeed just 1e sum Of these individual £7.) | The total energy is thus indeed just the sum of these individual $E_l$ .) |
‘Turning to the poloidal spectra first. we note that the Rey=200 curve follows an / sealing over the cntire range of /. whereas the lower Be curves start out much the same. but then crop olf somewhat more rapidly. exactly as one might expect. | Turning to the poloidal spectra first, we note that the $R_B=200$ curve follows an $l^{-5}$ scaling over the entire range of $l$, whereas the lower $R_B$ curves start out much the same, but then drop off somewhat more rapidly, exactly as one might expect. |
We note though that there is no sign of a definite dissipation scale. either at the /=O(Ry) appropriate to an /> spectrum. or the --OCR") appropriate to an /" spectrum. | We note though that there is no sign of a definite dissipation scale, either at the $l=O(R_B)$ appropriate to an $l^{-2}$ spectrum, or the $l=O(R_B^{2/5})$ appropriate to an $l^{-5}$ spectrum. |
As discussed. in section this suggests that the coupling is not. purely local in wavenumber space. | As discussed in section 2.2, this suggests that the coupling is not purely local in wavenumber space. |
We note also that this particular exponent ο is rather cillerent from. the 2 predicted by Goldreich. Reisenceecr (1992). but of course one should hardly expect the (wo to be the same. given that their result applies to 3D turbulence. whereas our calculations here are 2D laminar. | We note also that this particular exponent $-5$ is rather different from the $-2$ predicted by Goldreich Reisenegger (1992), but of course one should hardly expect the two to be the same, given that their result applies to 3D turbulence, whereas our calculations here are 2D laminar. |
‘Turning to the toroidal spectra next. for small / they too are of the form //. but now the exponent is around 3.5 rather than 5 or 2. | Turning to the toroidal spectra next, for small $l$ they too are of the form $l^p$, but now the exponent is around $-3.5$ rather than $-5$ or $-2$. |
The entire curves also shift upward slightly with increasing Ae. and show no sign of saturating for sulliciently large. values. | The entire curves also shift upward slightly with increasing $R_B$ , and show no sign of saturating for sufficiently large values. |
Probably more worrisome though is the behaviour for large /.. where the Re=200 curve actually rises ever so slightly. between |—60 and 100. | Probably more worrisome though is the behaviour for large $l$, where the $R_B=200$ curve actually rises ever so slightly between $l=60$ and 100. |
LLowever. runs done at truncations varving between SO and 120 all showed this same minimum at /=60. suggesting that it is real. anc not some numerical artifact. | However, runs done at truncations varying between 80 and 120 all showed this same minimum at $l=60$, suggesting that it is real, and not some numerical artifact. |
Furthermore. the 2g=100 and 50 curves also show slight rises but still fall againthereafter. so perhaps the Ay=200 curve would too. if only we could. include enough. moces. | Furthermore, the $R_B=100$ and 50 curves also show slight rises but still fall againthereafter, so perhaps the $R_B=200$ curve would too, if only we could include enough modes. |
]t is nevertheless not. quite clear what to make of this Ry=200 curve. and whether it really is fully resolved at the truncations we can alford. | It is nevertheless not quite clear what to make of this $R_B=200$ curve, and whether it really is fully resolved at the truncations we can afford. |
Based on these spectra though. we can certainly unclerstanc why attempting to increase /?g further still was not successful. | Based on these spectra though, we can certainly understand why attempting to increase $R_B$ further still was not successful. |
sriclly returning also to the results of Shalybkov Urpin. we have already noted that. they too obtained helicoidal oscillations much like those in. figure 2. | Briefly returning also to the results of Shalybkov Urpin, we have already noted that they too obtained helicoidal oscillations much like those in figure 2. |
Unfortunately. they did not plot power spectra at all. but simply stated that only the /<5 modes “give an appreciable contribution.” without further comment on what that means quantitatively, | Unfortunately, they did not plot power spectra at all, but simply stated that only the $l\le5$ modes “give an appreciable contribution,” without further comment on what that means quantitatively. |
With 40 [atitudinal finite cillerence points they were actually resolving considerably more than just the {<5 modes though — although of course considerably. less than the LOO modes we have resolved here. | With 40 latitudinal finite difference points they were actually resolving considerably more than just the $l\le5$ modes though – although of course considerably less than the 100 modes we have resolved here. |
I is nevertheless surprising that they obtained. such good. results with such a low resolution. ( | It is nevertheless surprising that they obtained such good results with such a low resolution. ( |
In contrast. the fact that they worked in a full sphere rather than a thin shell makes very. little difference: we also did a few runs with r;/r;=0.5 and 0.25. and obtained spectra quite similar to those in figure 3.) | In contrast, the fact that they worked in a full sphere rather than a thin shell makes very little difference; we also did a few runs with $r_i/r_o=0.5$ and 0.25, and obtained spectra quite similar to those in figure 3.) |
Finally. we would liketo know what the solutions actually look like. ancl in particular see whether we can identify the features corresponding to these ever. Latter spectra. | Finally, we would liketo know what the solutions actually look like, and in particular see whether we can identify the features corresponding to these ever flatter spectra. |
Figure 4 shows the field for Re=200 and / between 4 and 05. that is. covering the last two of these helicoida oscillations in figure 2 (and also precisely the time over which 10 spectra in figure 5 were averaged). | Figure 4 shows the field for $R_B=200$ and $t$ between 0.4 and 0.5, that is, covering the last two of these helicoidal oscillations in figure 2 (and also precisely the time over which the spectra in figure 3 were averaged). |
We see that these oscillations involve reversals in the sign of D. originating a )0 equator and. propagating to the poles. | We see that these oscillations involve reversals in the sign of $B$, originating at the equator and propagating to the poles. |
What we do no see. however. are any. small scale features corresponding to us part of the spectrum between 60 and. 100. | What we do not see, however, are any small scale features corresponding to this part of the spectrum between 60 and 100. |
In retrospec js is probably not surprising though. since this plateau is after all 6 orders of magnitude down from the large scale eatures. and therefore shouldn't be expected to be visible on a simple contour plot such as this. | In retrospect this is probably not surprising though, since this plateau is after all 6 orders of magnitude down from the large scale features, and therefore shouldn't be expected to be visible on a simple contour plot such as this. |
In some of our solutions xlow though we will see small scale features as well. at which point we will better understand why they break down or sulliciently large Lp. | In some of our solutions below though we will see small scale features as well, at which point we will better understand why they break down for sufficiently large $R_B$. |
The maximum toroidal field in figure 4 is only 0.1. and even in the earlier stages of evolution it never exceeds 0.25. | The maximum toroidal field in figure 4 is only 0.1, and even in the earlier stages of evolution it never exceeds 0.25. |
[t is therefore probably. not surprising that 65 never becomes comparable with 5,. since according to (12) the toroidal field is a crucial ingredient in inducing higher harmonics in the poloidal field. | It is therefore probably not surprising that $b_3$ never becomes comparable with $b_1$, since according to (12) the toroidal field is a crucial ingredient in inducing higher harmonics in the poloidal field. |
Lf sve did have a Iarger toroidal field though. it seems likely we would.also obtain a larger by. perhaps even comparable with οι. | If we did have a larger toroidal field though, it seems likely we wouldalso obtain a larger $b_3$ , perhaps even comparable with $b_1$ . |
To test this hypothesis. we add | To test this hypothesis, we add |
scattering, and to reproduce images and polarization maps at various viewing angles. | scattering, and to reproduce images and polarization maps at various viewing angles. |
The benchmark case consists. once again. of a central star surrounded by a disk. | The benchmark case consists once again of a central star surrounded by a disk. |
The central star has a radius of 2RRo and a temperature of KK, with a blackbody spectrum. | The central star has a radius of $_\odot$ and a temperature of K, with a blackbody spectrum. |
The disk extends from 0.1 to AAU (with cylindrical edges), and the density is given by Equation (9)), with ro=100 AAU, ho=10 AAU, a=2.625, and B=1.125. | The disk extends from 0.1 to AU (with cylindrical edges), and the density is given by Equation \ref{eq:disk}) ), with $r_0=100$ AU, $h_0=10$ AU, $\alpha=2.625$, and $\beta=1.125$. |
The disk is made up of dust grains with a | The disk is made up of dust grains with a |
different from the aligned m=1 case. | different from the aligned $m=1$ case. |
The reason is simply wavenumber parameter £ will also \ m=0, =0.45, n-1 in equation (71)) LIalter | The reason is simply that the additional radial wavenumber parameter $\xi$ will also alter values of coefficients in equation \ref{spiral}) ). |
ervalues : of H again use asymptotic expression (65)) (3 to estimate A and then userecursion relation. (64)) ' . . . analvtical expressionB for Ay (8) _E EE| namely'hto fragmentationthan-- 0.54.[= | In this case, we again use asymptotic expression \ref{asymN}) ) to estimate ${\cal N}_4(\xi)$ and then use recursion relation \ref{recurN}) ) to derive an approximate analytical expression for ${\cal N}_1(\xi)$ , namelywith relative error less than $0.5\%$. |
With 4, = with£7 |relative |s—25TerrorB,=4:37 ibLA1 according to definitions (69)). we immecdiately obtain which increases monotonically with increasing € [orfixed 3 values and attains its nininium | With ${\cal A}_1
=\xi^2+5/4+2\beta$ and ${\cal B}_1=4\beta^2-1$ according to definitions \ref{ABCHspiral}) ), we immediately obtain which increases monotonically with increasing $\xi$ forfixed $\beta$ values and attains its minimum |
Α natural assumption for describing multi-planet svstems is (hat the n-planet distribution [unction isseparable. (hat is. This assumption can only be approximately validlor example. it is inconsistent with (he observational finding Chat planets tend to be concentrated near nmtual orbital resonances. and with the theoretical finding that planets separated bv less than a few Lill radii are unstable. | A natural assumption for describing multi-planet systems is that the $n$ -planet distribution function is, that is, This assumption can only be approximately valid—for example, it is inconsistent with the observational finding that planets tend to be concentrated near mutual orbital resonances, and with the theoretical finding that planets separated by less than a few Hill radii are unstable. |
Nevertheless. we argue that the separability assumption is sufficiently. accurate {ο provide a powerful tool for analvzing the statistics of multi-planet svstems. | Nevertheless, we argue that the separability assumption is sufficiently accurate to provide a powerful tool for analyzing the statistics of multi-planet systems. |
We describe the evidence on its validitv in re[sectvalid.. | We describe the evidence on its validity in \\ref{sec:valid}. |
Let Q-*(w) be the probability that a planet with properties w is detected in the survey labeled by A if its host star is on the target list for this survey and the orientation of the observer is correct (we assume (hat whether or not a planet can be detected is independent of the presence or absence of other planets in the same svstem. which is a reasonable first approximation). | Let $\Theta^{A}(\mathbf{w})$ be the probability that a planet with properties $\mathbf{w}$ is detected in the survey labeled by A if its host star is on the target list for this survey and the orientation of the observer is correct (we assume that whether or not a planet can be detected is independent of the presence or absence of other planets in the same system, which is a reasonable first approximation). |
Thus the function Ον) describes the survey selection effects for A. but not the geometric selection effects. | Thus the function $\Theta^{A}(\mathbf{w})$ describes the survey selection effects for A, but not the geometric selection effects. |
The probability (hat a planet is detected. ignoring geometric selection effects. is then If the survey target list contains V;' stars with m planets. then using the separability assumption (4)) the expected number of svstems in which / planets will be detected is where the survev selection matrix S is a (A+1)x(IN4+ matrix whose entries are given by the binomial distribution. | The probability that a planet is detected, ignoring geometric selection effects, is then If the survey target list contains $N^{A}_m$ stars with $m$ planets, then using the separability assumption \ref{eq:sep}) ) the expected number of systems in which $k$ planets will be detected is where the survey selection matrix $\mathbf{S}$ is a $(K+1)\times(K+1)$ matrix whose entries are given by the binomial distribution, |
AMevlan (1994) for the whole globular cluster svstem. provides a good fit to the number density profile of the Old Lalo cluster svstem as well. | Meylan (1994) for the whole globular cluster system, provides a good fit to the number density profile of the Old Halo cluster system as well. |
Ixeeping all three parameters free. we obtain a steeper slope (>c 4.5) coupled. with a larger core. | Keeping all three parameters free, we obtain a steeper slope $-\gamma \simeq -4.5$ ) coupled with a larger core. |
The core reflects the Πατομής of the spatial distribution at small ealactocentric distances. presumably owing to the ereater cllicicney of disruptive processes. | The core reflects the flattening of the spatial distribution at small galactocentric distances, presumably owing to the greater efficiency of disruptive processes. |
lenoring this core region and focusing on the Old Lalo clusters located at galactocentric distances z 3kkpe. that is. where memory of the initial conditions has perhaps been better preserved. we find that both the mass and the number density. profiles of the Old. Halo are well-approximated: by pure power-laws wit1 slope zz3.5 (see Table 2)). | Ignoring this core region and focusing on the Old Halo clusters located at galactocentric distances $\gtrsim
3$ kpc, that is, where memory of the initial conditions has perhaps been better preserved, we find that both the mass and the number density profiles of the Old Halo are well-approximated by pure power-laws with slope $\simeq -3.5$ (see Table \ref{tab:fit_pure_pl}) ). |
The steepness of the Old Lalo spatial distribution is thus similar to that of the whole halo (Zinn 1985). | The steepness of the Old Halo spatial distribution is thus similar to that of the whole halo (Zinn 1985). |
While the mass and number density oofiles show very similar steepness at distances larger than 38kkpe. their overall shapes are also very similar. | While the mass and number density profiles show very similar steepness at distances larger than kpc, their overall shapes are also very similar. |
Fitting he Old Halo mass density profile with the same functions as used. for the number density. profile (Le. equation 1. and the (5. 2.) couples listed. in Table 1)) provides equally good. values of the incomplete gamma function (see the last column of Table 1)). | Fitting the Old Halo mass density profile with the same functions as used for the number density profile (i.e., equation \ref{eq:log_pl_core} and the $\gamma$ , $D_c$ ) couples listed in Table \ref{tab:fit_pl_core}) ) provides equally good values of the incomplete gamma function (see the last column of Table \ref{tab:fit_pl_core}) ). |
Therefore. the number and the mass density. profiles of the Old Lalo are incdistinguishable through the whole extent of the halo. | Therefore, the number and the mass density profiles of the Old Halo are indistinguishable through the whole extent of the halo. |
‘The previous section shows that theobserved mass density and number density. profiles of the Old Lalo are identical in shape. | The previous section shows that the mass density and number density profiles of the Old Halo are identical in shape. |
1 we that the elobular cluster ormation mechanism produced the same mass range an he same mass spectrum for the clusters. irrespective of heir galactocentric distance. then thetial mass anc number density. profiles were identical in shape as well. | If we that the globular cluster formation mechanism produced the same mass range and the same mass spectrum for the clusters, irrespective of their galactocentric distance, then the mass and number density profiles were identical in shape as well. |
Lf he mass density profile has been preserved. (ancl we show in this section that it is actually the case). all together. hese facts imply that the number density profile itself das remained fairly unaltered during evolution in the tida ielel of the Milky Way. | If the mass density profile has been preserved (and we show in this section that it is actually the case), all together, these facts imply that the number density profile itself has remained fairly unaltered during evolution in the tidal field of the Milky Way. |
In what follows. we evolve various outative elobular cluster systems. considering clilferen combinations of initial mass spectra ancl initial spatia distributions. | In what follows, we evolve various putative globular cluster systems, considering different combinations of initial mass spectra and initial spatial distributions. |
We then investigate in which case(s) has he number density profile been reasonably preserved. | We then investigate in which case(s) has the number density profile been reasonably preserved. |
We also compare in a least-squares sense the evolved. spatia distributions to the observed ones that we have derived in Section 2. | We also compare in a least-squares sense the evolved spatial distributions to the observed ones that we have derived in Section 2. |
To evolve the radial mass and numboer density. profiles ofa cluster system from the time of its formation up to an age of GCOwvr. we adopt the analytic formula of Vesperini llegeie (1997) which supplies at any time/ the mass m oL a star cluster with initial mass m; which is moving along a circular orbit. perpenclicular to the galactic disc at a galactocentric distance D. | To evolve the radial mass and number density profiles of a cluster system from the time of its formation up to an age of Gyr, we adopt the analytic formula of Vesperini Heggie (1997) which supplies at any time$t$ the mass $m$ of a star cluster with initial mass $m_i$ which is moving along a circular orbit perpendicular to the galactic disc at a galactocentric distance $D$. |
“Phe assumption of circular orbits is clearly a simplifving one since it implies that the time variations of the tidal field for clusters on elliptical orbits are not allowed for in our calculations. | The assumption of circular orbits is clearly a simplifying one since it implies that the time variations of the tidal field for clusters on elliptical orbits are not allowed for in our calculations. |
Yet. the svstem of relevance here is the Old Halo. | Yet, the system of relevance here is the Old Halo. |
This shows less extreme kinematics than the Younger Halo group of clusters. making this assumption less critical than if we have dealt with the whole halo svstem. | This shows less extreme kinematics than the Younger Halo group of clusters, making this assumption less critical than if we have dealt with the whole halo system. |
As for the inlluence of the cluster orbit inclination with respect to the Galactic disc. Murali Weinberg (1997). found hat. although [ow-inclination halo clusters evolve more rapidly than hieh-inclination ones. the dillerences are. not extreme. | As for the influence of the cluster orbit inclination with respect to the Galactic disc, Murali Weinberg (1997) found that, although low-inclination halo clusters evolve more rapidly than high-inclination ones, the differences are not extreme. |
Furthermore. our sample excluding disce clusters. the assumption of high inclination orbits is a reasonable one. | Furthermore, our sample excluding disc clusters, the assumption of high inclination orbits is a reasonable one. |
“Phe simulations of Vesperini llegeie (1997) were designed in the frame of a host galaxy modelled as a simple isothermal sphere with a constant circular velocity. | The simulations of Vesperini Heggie (1997) were designed in the frame of a host galaxy modelled as a simple isothermal sphere with a constant circular velocity. |
This actually constitutes a reasonable assumption for the Old. Halo system. whose racial extent is kkpc. that is. where the mass profile of the Milky Wav (Le. the total Galactic mass enclosed. within a radius D) grows linearly. with the galactocentric distance (Llarris POOL). | This actually constitutes a reasonable assumption for the Old Halo system whose radial extent is kpc, that is, where the mass profile of the Milky Way (i.e, the total Galactic mass enclosed within a radius $D$ ) grows linearly with the galactocentric distance (Harris 2001). |
We consider the ellect of non-circular orbits in more detail in Section 4. below. | We consider the effect of non-circular orbits in more detail in Section 4, below. |
The relations describing the temporal evolution of the mass of a elobular cluster have been obtained. by fitting the results of a laree set of N-bocly simulations in which Vesperini Llegeic (1997) take into account the effects of stellar evolution as well as two-bocly relaxation. which leads to evaporation through the cluster tical boundary. | The relations describing the temporal evolution of the mass of a globular cluster have been obtained by fitting the results of a large set of N-body simulations in which Vesperini Heggie (1997) take into account the effects of stellar evolution as well as two-body relaxation, which leads to evaporation through the cluster tidal boundary. |
Disc shocking can also be included. (see below). | Disc shocking can also be included (see below). |
In. order to take into account dynamical friction. globular clusters whose time-scale of orbital decay (sec. e.g. Anunev ‘Tremaine 1987) is smaller than / are removed Crom the cluster svstem at that time (see Vesperini: 1905. his Section 2. for Curther details). | In order to take into account dynamical friction, globular clusters whose time-scale of orbital decay (see, e.g., Binney Tremaine 1987) is smaller than $t$ are removed from the cluster system at that time (see Vesperini 1998, his Section 2, for further details). |
Le is important to note a specific assumption underlying the validity of this analysis. | It is important to note a specific assumption underlying the validity of this analysis. |
The large-scale Galactic gravitational potential is assumed constant. that. is. this mocel considers the evolution of a elobular cluster svstem only after it has been assembled in a time-independent Galaxy. | The large-scale Galactic gravitational potential is assumed constant, that is, this model considers the evolution of a globular cluster system only after it has been assembled in a time-independent Galaxy. |
That is the physical basis for a restriction to the svstem of Old Lalo elobular clusters. | That is the physical basis for a restriction to the system of Old Halo globular clusters. |
The temporal evolution of the mass of a cluster orbiting at constant galactocentric distance D is found to follow: imus£m; is the fraction of cluster mass lost due to stellar evolution (18 per cent in this particular moclel). | The temporal evolution of the mass of a cluster orbiting at constant galactocentric distance $D$ is found to follow: $\Delta m_{st,ev}/m_i$ is the fraction of cluster mass lost due to stellar evolution (18 per cent in this particular model). |
The time fis expressed in units of MMwyr ancl £54. à quantity proportional to the initial relaxation time. is definedas: | The time $t$ is expressed in units of Myr and $F_{cw}$ , a quantity proportional to the initial relaxation time, is definedas: |
of 0.25 dex) and -3xlog ms+2 (step of | dex) at a microturbulent velocity. € = 5 km s7!. | of 0.25 dex), and $-3\le$ $\le+$ 2 (step of 1 dex) at a microturbulent velocity, $\xi$ = 5 km $^{-1}$. |
For each model atmosphere with mes-]. we calculated a second model with (marked by © in Table 2)) following the prediction of the “flash mixing" scenario. adopting mass fractions of and for carbon and nitrogen. respectively (Lanz et al. 2004)). | For each model atmosphere with $\ge-1$, we calculated a second model with (marked by $^{\rm C}$ in Table \ref{Tab:Par_Rich}) ) following the prediction of the “flash mixing” scenario, adopting mass fractions of and for carbon and nitrogen, respectively (Lanz et al. \cite{labr04}) ). |
We adopted scaled-solar abundanees at co CCen's dominant metallicity ([Fe/H| = —].5). | We adopted scaled-solar abundances at $\omega$ Cen's dominant metallicity ([Fe/H] = $-$ 1.5). |
This abundance ratio by numbers was kept the same for all models. including helium-rich models. which implies that the heavy element mass fraction differs for models with different helium (and C. N) content. | This abundance ratio by numbers was kept the same for all models, including helium-rich models, which implies that the heavy element mass fraction differs for models with different helium (and C, N) content. |
We emphasize. however. that the abundance of tron-peak elements in. EHB stellar photospheres is unknown and probably affected by diffusion processes. | We emphasize, however, that the abundance of iron-peak elements in EHB stellar photospheres is unknown and probably affected by diffusion processes. |
Furthermore. the low abundance of heavy elements limits the effect of metal line blanketing on the atmospheric structure and the predicted emergent spectrum. | Furthermore, the low abundance of heavy elements limits the effect of metal line blanketing on the atmospheric structure and the predicted emergent spectrum. |
Therefore. the resulting uncertainty in our analysis caused by assuming the same [Fe/H] value remains small. | Therefore, the resulting uncertainty in our analysis caused by assuming the same [Fe/H] value remains small. |
Once the atmospheric structure of each model atmosphere converged. we calculated detailed emergent spectra in the 13800-4600 rrange with the spectrum synthesis code. SYNSPEC. using the NLTE populations calculated by TLUSTY. | Once the atmospheric structure of each model atmosphere converged, we calculated detailed emergent spectra in the $\lambda\lambda$ range with the spectrum synthesis code, SYNSPEC, using the NLTE populations calculated by TLUSTY. |
For the helium-poor stars above KK. we also used the TLUSTY models. | For the helium-poor stars above K, we also used the TLUSTY models. |
For the cooler stars. we used models (Moehler et al. 2000)). | For the cooler stars, we used models (Moehler et al. \cite{mosw00}) ). |
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