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When increasing the base density. the deusity drop with height decreases and the volume between the stellar surface aud the Alfvénn surface is filled with more mass and as a result. more torque is being applied ou the rotating star. thus increasing the aneulay momoeutun loss. | When increasing the base density, the density drop with height decreases and the volume between the stellar surface and the Alfvénn surface is filled with more mass and as a result, more torque is being applied on the rotating star, thus increasing the angular momentum loss. |
For a given distribution of pa. the angular moment loss rate is directly proportional O,. soit is not surprising that J varies with the rotation rate. | For a given distribution of $\rho u$, the angular momentum loss rate is directly proportional $\Omega_\star$, so it is not surprising that $\dot{J}$ varies with the rotation rate. |
We fine the mass loss rate also depends on the stellar rotation rate. | We find the mass loss rate also depends on the stellar rotation rate. |
This effect is not apparent if the rotation rate is nof very high. | This effect is not apparent if the rotation rate is not very high. |
However. the case of the extremely short rotation period of 0.54 has a significant effect. | However, the case of the extremely short rotation period of $0.5\;d$ has a significant effect. |
The reason for this behavior cau be found iu the azinuthal taneling of the coronal field. | The reason for this behavior can be found in the azimuthal tangling of the coronal field. |
When the rotation is slow. the elobal topology of the coronal feld is radial (as seen in Fieure 9)). | When the rotation is slow, the global topology of the coronal field is radial (as seen in Figure \ref{fig:f4}) ). |
Iu this case. the coronal deusitv profile essentially drops like +7. | In this case, the coronal density profile essentially drops like $r^{-2}$. |
When strong rotation is present. the azimuthal component of both the magnetic field and the flow become nuportaut aud the radial component of the velocity is reduced. | When strong rotation is present, the azimuthal component of both the magnetic field and the flow become important and the radial component of the velocity is reduced. |
The increase iu density in the slow wind case is ereater than the decrease in speed. aud as a result. the total value of pe increases. | The increase in density in the slow wind case is greater than the decrease in speed, and as a result, the total value of $\rho u$ increases. |
Another contributor to the augulu momentum loss increase with rotation is the shape of the Alfvéóun surface for cach case. | Another contributor to the angular momentum loss increase with rotation is the shape of the Alfvénn surface for each case. |
Figures 10-12 show the shape of the Alfvénn surface for cach test case with different rotation period. | Figures \ref{fig:f5}- \ref{fig:f7} show the shape of the Alfvénn surface for each test case with different rotation period. |
When the rotational period is small. the shape is modified. to account for the azimuthal componcut. aud the surface is enlarged. | When the rotational period is small, the shape is modified to account for the azimuthal component, and the surface is enlarged. |
Iu the case of fast rotation. the shape of the Alfvénn surface is modified and sigus of the azimuthal conrponeut of the coronal field cam be seen. | In the case of fast rotation, the shape of the Alfvénn surface is modified and signs of the azimuthal component of the coronal field can be seen. |
It also seenis like the size of the Alfvénn surface with faster rotation is sliehtlv bigger. | It also seems like the size of the Alfvénn surface with faster rotation is slightly bigger. |
ao The simulations presented | The simulations presented |
Solar flares produce TNR and continu -rav wission. ecnerally attributed to thermal or non-jiecrinal bremsstrabhme cmission, | Solar flares produce HXR and continuum $\gamma$ -ray emission, generally attributed to thermal or non-thermal bremsstrahlung emission. |
As such. TIXR wission provides kev diagnostics of plasma heating. ectron acceleration. and electron transport. | As such, HXR emission provides key diagnostics of plasma heating, electron acceleration, and electron transport. |
luteuse Tiromosphieric TNR thick target enission is produced at 16 foot poiuts of coronal magnetic loops. | Intense chromospheric HXR thick target emission is produced at the foot points of coronal magnetic loops. |
The relatively fain coronal TINR aud continu οταν onudssion is ΟΠΟΥ more difficult to observe iu the prescuce of iteuse foot point enussiou eiven the limited dvuauiic range of N-rav Huaging iustrunients, | The relatively faint coronal HXR and continuum $\gamma$ -ray emission is generally more difficult to observe in the presence of intense foot point emission given the limited dynamic range of X-ray imaging instruments. |
Iu most CHSCR. tverefore. coronal ΠΝΗ or οταν ciission ds observed i1 flares that occur iu active regions behind the solar linx the intense foot poiut emission is occulted. thereby Y'evealiug the relatively faint coronal emission. | In most cases, therefore, coronal HXR or $\gamma$ -ray emission is observed in flares that occur in active regions behind the solar limb; the intense foot point emission is occulted, thereby revealing the relatively faint coronal emission. |
While reports of coronal INR cinission data back ‘0 the carly LsO70s imaging observations. first by audthen by— thehuager?).. have led to renewed interest "miu scopiccoronal ΗΝ sources. | While reports of coronal HXR emission data back to the early 1970s imaging observations, first by and then by the, have led to renewed interest in coronal HXR sources. |
As discussed in the review by?.. coronal IINR sources 'veal a diverse phenomenology. | As discussed in the review by, coronal HXR sources reveal a diverse phenomenology. |
These include sources rat precede the impulsive phase as well as a variety of corona IINR sources that may occur dunug the upulsive phase: "over-the-loop-top sources(?).. double sources(2). :xd coronal thick target sources(2). | These include sources that precede the impulsive phase as well as a variety of coronal HXR sources that may occur during the impulsive phase: “over-the-loop-top" sources, double sources, and coronal thick target sources. |
. During tιο late phase of flares c"superhot thermal sources7).. eradual sources that display a "soft-hard-liurder PApectral evolution(??).. aud. nouthermal sources that isplav lard. ου οταν cluission may occur. | During the late phase of flares “superhot" thermal sources, gradual sources that display a “soft-hard-harder" spectral evolution, and non-thermal sources that display hard, continuum $\gamma$ -ray emission may occur. |
Coronal INR sources have been observed over a rauge of heights in the corona aud cau be associated with stationary or moving sources, | Coronal HXR sources have been observed over a range of heights in the corona and can be associated with stationary or moving sources. |
We refer the reader to?.. and refereuces therein. for a more detailed discussion of he types of coronal INR sources. their properties. aud he circumstances under which they occu. | We refer the reader to, and references therein, for a more detailed discussion of the types of coronal HXR sources, their properties, and the circumstances under which they occur. |
We focus here on non-thermal coronal INR anc continui 5-ray sources that occur during the impulsive yhase of flares. | We focus here on non-thermal coronal HXR and continuum $\gamma$ -ray sources that occur during the impulsive phase of flares. |
These have beeu interpreted iu terms of liu-target. nou-thermal bremestrahline. | These have been interpreted in terms of thin-target, non-thermal bremsstrahlung. |
This may well ο correct but iu the case of certain coronal INR. continuuni y-ray. sources the parameters required can vc extreme. | This may well be correct but in the case of certain coronal HXR, or continuum $\gamma$ -ray, sources the parameters required can be extreme. |
For example. three powerful N-class flares - those on 2003 Oct 28. 2005 Jan 20. aud 2005 Sep 7 - were aeconipaniedby continuun 5-ray euidsson >200 sev(C. These flares. observedby RIIESSI. were not occultedby the Bib. and both foot poiut and coronal enissions were observed. | For example, three powerful X-class flares - those on 2003 Oct 28, 2005 Jan 20, and 2005 Sep 7 - were accompanied by continuum $\gamma$ -ray emission $>200$ keV. These flares, observed by RHESSI, were not occulted by the limb and both foot point and coronal emissions were observed. |
Iu the case of 2005 Jan 20 foot oot enuission doniünated during times uecar the 5-raxv uaxiuun but the coronal source became increasingly xoniüneut duriug the decay of the οταν cussion. | In the case of 2005 Jan 20 foot point emission dominated during times near the $\gamma$ -ray maximum but the coronal source became increasingly prominent during the decay of the $\gamma$ -ray emission. |
The »ower-Inmw iudex o of the photon spectrum of the coronal source from 200-800 keV was siguificautlv harder 1.5) than that of the foot points (az 2.9). | The power-law index $\alpha$ of the photon spectrum of the coronal source from 200-800 keV was significantly harder $\alpha\approx 1.5$ ) than that of the foot points $\alpha\approx 2.9$ ). |
The other wo flares displaved simular propertics. | The other two flares displayed similar properties. |
Tuterpreting the cnussion in terms of non-thermal. thin-target. electrou- brenisstralluung ciission inplies the spectral iudices | Interpreting the emission in terms of non-thermal, thin-target, electron-ion bremsstrahlung emission implies the spectral indices |
too long. | too long. |
“Phe ejected mass would clissipate before a visible PN could be observed. | The ejected mass would dissipate before a visible PN could be observed. |
This is known as a "Iazv planetary nebula (Renzini1979). | This is known as a “lazy” planetary nebula \citep{ren79}. |
. The masses of the central stars of the globular cluster PN are larger than the typical measured. masses of the voungest white chvarls (WDs) at the top of the cooling sequence. | The masses of the central stars of the globular cluster PN are larger than the typical measured masses of the youngest white dwarfs (WDs) at the top of the cooling sequence. |
The cooling sequences of white dwarfs in globular clusters have been determined. by a number of authors (ltenzinictal.1996:Cool.Piotto&IxingRicherοἱCalamidactal. 2008). | The cooling sequences of white dwarfs in globular clusters have been determined by a number of authors \citep{renwd,cool,richwd,zocwd,han02,han04,han07,calwd}. |
. Phe find that the majority of main sequence stars evolve into whitecdwarls of mass M;. | They find that the majority of main sequence stars evolve into whitedwarfs of mass $\,{\rm M}_{\sun}$. |
and the average of a voung GC white dwarf is probably between 0.50 and NL;. | and the average of a young GC white dwarf is probably between 0.50 and $\,{\rm M}_{\sun}$. |
All of the GC PNe have central star (CSPN) masses above this average. | All of the GC PNe have central star (CSPN) masses above this average. |
Alvesetal.(2000) and Bianchietal.(2001). find that the mass of the central star o£ Ps Lis 0.58-0.60 AL.. (Jac | \citet{al} and \citet{bitwo} find that the mass of the central star of Ps 1 is 0.58-0.60 $_{\sun}$. |
obyetal.1997). determined the CSPN of Jaku 1 and Jabu 2 have masses of 0.55 | \citep{jac}
determined the CSPN of JaFu 1 and JaFu 2 have masses of 0.55 $_{\sun}$. |
The estimated mass of €JJ€-1 is around 0.56 NL; (Penaetal. 1992). | The estimated mass of GJJC-1 is around 0.56 $_{\sun}$ \citep{pen92}. |
. Both the CSPN masses higher than the white dwarls and the unusual abundances seem to require unusual stars as progenitors. | Both the CSPN masses higher than the white dwarfs and the unusual abundances seem to require unusual stars as progenitors. |
GC's contain a variety of unusual star tvpes in addition to the standard tvpes of stars which should. be considered as potential progenitors of GC Όλα | GCs contain a variety of unusual star types in addition to the standard types of stars which should be considered as potential progenitors of GC PNe. |
GCs contain blue stragelers which are thought to be the mergers of two main sequence stars and hence act [ike main sequence stars with masses higher then the turn-olf mass. | GCs contain blue stragglers which are thought to be the mergers of two main sequence stars and hence act like main sequence stars with masses higher then the turn-off mass. |
Bianchietal.(2001) ancl Alvesctal.(2000). suggested. the projenitor of Ps Lis a blue strageler because of its high. core mass and evidence of at least one third. dredge up CEDU) event as evidenced by the very high. C/O ratio in the nebula. | \citet{bitwo} and \citet{al}
suggested the projenitor of Ps 1 is a blue straggler because of its high core mass and evidence of at least one third dredge up (TDU) event as evidenced by the very high C/O ratio in the nebula. |
Theoretical models of thermally pulsing-AGB (LP-ACB) stars sugeest a core mass of 0.58AL. is required to get a TDU event. | Theoretical models of thermally pulsing-AGB (TP-AGB) stars suggest a core mass of $\sim0.58\,{\rm M}_{\sun}$ is required to get a TDU event. |
Another tvpe of star found. in globular clusters are second generation stars which incorporate material from the more massive stars of the first generation (primordial component or P). | Another type of star found in globular clusters are second generation stars which incorporate material from the more massive stars of the first generation (primordial component or P). |
The first generation stars have abundances which rellect. the abundances of the interstellar medium material from which they formed and hence have a normal amount of helium (Y= 0.25). | The first generation stars have abundances which reflect the abundances of the interstellar medium material from which they formed and hence have a normal amount of helium $Y\approx0.25$ ). |
The second generation often incorporates material with a higher fraction of helium than normal (Y.= 0.30). | The second generation often incorporates material with a higher fraction of helium than normal $Y\approx0.30$ ). |
Some clusters show evidence for additional populations with even higher Y values (Ίο, C'aloi&D'Antona (2007))). | Some clusters show evidence for additional populations with even higher $Y$ values (E.g. \citet{cal07}) ). |
These multiple populations show up observationally in a number of wavs. | These multiple populations show up observationally in a number of ways. |
H shows up as an Na- anticorrelation in both red giants and in main sequence stars. | It shows up as an Na-O anticorrelation in both red giants and in main sequence stars. |
This was [first observed. by Grattonetal.(2001) when they noted the Na-O anticorrelation shows up in main-sequence stars in addition to red giants in several clusters. | This was first observed by \citet{grat} when they noted the Na-O anticorrelation shows up in main-sequence stars in addition to red giants in several clusters. |
Some clusters have distinct. multiple main sequences (e.g. wen (Bedinetal.2004).. NGC 2808 (Plottoetal.2007).. ete.) | Some clusters have distinct multiple main sequences (e.g. $\omega$ \citep{bed04}, NGC 2808 \citep{pio07}, etc.) |
which can be fit by having a second main sequence with a higher Y. | which can be fit by having a second main sequence with a higher $Y$. |
Phere are GC with multiple subgiant branches (c.g. M22 (Milonectal.20102.b)... NGC 104. etc). | There are GC with multiple subgiant branches (e.g. M22 \citep{mil10a,mil10b}, NGC 104, etc.). |
See Diotto(2009). for a review of the evidence. | See \citet{sgrev09} for a review of the evidence. |
These second. generation. stars. form a substantial portion of all stars in GC (up to 60-70 percent of the total number of stars in a GC (Carrettaetal. 2009))) and should have a substantial impact on what is observed. during the AGB and PN phases of evolution. | These second generation stars form a substantial portion of all stars in GC (up to 60-70 percent of the total number of stars in a GC \citep{caret}) ) and should have a substantial impact on what is observed during the AGB and PN phases of evolution. |
In this paper I1 look at the expectations of the PN phase from all the generations of GC stars. | In this paper I look at the expectations of the PN phase from all the generations of GC stars. |
E: also model the expected. PN. phase of the blue strageler stars. | I also model the expected PN phase of the blue straggler stars. |
In section 2 I describe the TP-ACGD model used. | In section 2 I describe the TP-AGB model used. |
La section 3 Lb describe the results of these models. | In section 3 I describe the results of these models. |
In section 4 LE discuss some implications. | In section 4 I discuss some implications. |
In section 5 | summarize the results. | In section 5 I summarize the results. |
Alost of the relevant details of this model are explained in Suellοἱal.(1997)... Buell(1997)... and Gavilan.Buell.&Alolla(2005). | Most of the relevant details of this model are explained in \citet{bu97}, \citet{bue97}, and \citet{gbm}. |
. In this section L concentrate on mocifieations of this model. | In this section I concentrate on modifications of this model. |
Of. particular importance is the mass-Ioss on the ROB and. E-AGB which can be significant input. especially at the low ZAAIS masses of globular cluster stars. | Of particular importance is the mass-loss on the RGB and E-AGB which can be significant input, especially at the low ZAMS masses of globular cluster stars. |
Particular attention is paid to the cllect of enhanced helium abundances on mass-loss during these stages. | Particular attention is paid to the effect of enhanced helium abundances on mass-loss during these stages. |
The mass-loss in these stages is found. by integrating the mass-loss rate formula over the Padua tracks. | The mass-loss in these stages is found by integrating the mass-loss rate formula over the Padua tracks. |
There is an important point to note. in the pre-EP-ACGD mass-loss model the stellar evolution and the mass-loss are not coupled and hence the equations for mass-loss derived: below should: be used with care. | There is an important point to note, in the pre-TP-AGB mass-loss model the stellar evolution and the mass-loss are not coupled and hence the equations for mass-loss derived below should be used with care. |
The mass-loss shift is probably a small effect as explained below but the reader should be aware of it. | The mass-loss shift is probably a small effect as explained below but the reader should be aware of it. |
‘The mass-loss which occurs on the red giant branch (ROB) is very important for low mass stars found in elobular clusters. in some extreme cases jb maw sometimes prevent the star [rom even reaching the PP-AGB. | The mass-loss which occurs on the red giant branch (RGB) is very important for low mass stars found in globular clusters, in some extreme cases it may sometimes prevent the star from even reaching the TP-AGB. |
Most of the ρολ nmiass-loss occurs during the RD. | Most of the pre-TP-AGB mass-loss occurs during the RGB. |
The standard method to determine the amount of mass-loss is to use Reimers’ Law (Reimers1975). given hy where L. lt and Al ave the stellar luminosity. raclius and miss. respectively in solar units. | The standard method to determine the amount of mass-loss is to use Reimers' Law \citep{rei} given by where L, R and M are the stellar luminosity, radius and mass, respectively in solar units. |
However. to calculate the pre-PP-AGB mass-loss the modified version of the Reimers formula of SchróderandCuntz2005. eiven by: where Zig is the effective stellar temperature. g, is the surface eravity of the star in ces units. | However, to calculate the pre-TP-AGB mass-loss the modified version of the Reimers formula of \citealt{sch}
given by: where $T_{\rm eff}$ is the effective stellar temperature, $g_{\star}$ is the surface gravity of the star in cgs units. |
Values of 27400cnis.? for gs and 8.0.10.H for yp were adopted (Value recommended by SehróderandCuntz 2005)). | Values of ${\rm
cms^{-2}}$ for $g_{\sun}$ and $8.0\times10^{-14}$ for $\eta$ were adopted (Value recommended by \citealt{sch}) ). |
This new niass-Ioss rule appears to give better results for horizonatal branch masses then the Reimer’s rate (Schróderand 2005). | This new mass-loss rule appears to give better results for horizonatal branch masses then the Reimer's rate \citep{sch}. |
. This mass-loss law is applied to the variable Y stellar evolution tracks [rom the Padova stellar evolutionary library (http://pleadi.pd.astro.it) described in detail in Bertellietal.(2008) and Bertellietal. (2009). | This mass-loss law is applied to the variable $Y$ stellar evolution tracks from the Padova stellar evolutionary library (http://pleadi.pd.astro.it) described in detail in \citet{berta} and \citet{bertb}. . |
. To determine the red giant mass-loss the mass-loss rate was integrated from the beginning of the red. giant branch. (encoded. in the Padova files asbrebs) up to the tip of the red giant branch (encoded as (reb) using the trapezoidal rule. | To determine the red giant mass-loss the mass-loss rate was integrated from the beginning of the red giant branch (encoded in the Padova files asbrgbs) up to the tip of the red giant branch (encoded as trgb) using the trapezoidal rule. |
Phe amount of mass-loss between time steps in the models is given by: | The amount of mass-loss between time steps in the models is given by: |
phase curves can in principle be explained by the light time effect caused by an invisible companion orbiting the star in a nearly circular orbit. | phase curves can in principle be explained by the light time effect caused by an invisible companion orbiting the star in a nearly circular orbit. |
However. this orbital motion would be revealed by considerable radial velocity variations ARV. | However, this orbital motion would be revealed by considerable radial velocity variations $\Delta\mathit{RV}$. |
In the case of tthe observed change in the period AP of 3.8 s corresponds to a variation ARV=cAP/P25 kmss7!. while the observed change in pperiod of AP=20 s predicts a ARV variation of even 45 kmss7!! | In the case of the observed change in the period $\Delta P$ of 3.8 s corresponds to a variation $\Delta \mathit{RV}=c \Delta P/P= 25$ $^{-1}$, while the observed change in period of $\Delta P\simeq 20$ s predicts a $\Delta\mathit{RV}$ variation of even 45 $^{-1}$! |
However no such long-term RV variations synchronised with period changes have been found (Pyperetal.1997;Mikuláseketal. 2008). | However, no such long-term $RV$ variations synchronised with period changes have been found \citep{pyper97,mik901}. |
. Additionally. the light time effect cannot explain the complex medium-term period. variations observed in ((see Sec. 4.1.. refFigdetail)). | Additionally, the light time effect cannot explain the complex medium-term period variations observed in (see Sec. \ref{cuvir}, \\ref{Figdetail}) ). |
We conclude that the observed period variations are caused by uneven rotation of surface layers of the stars (Stepien1998;Mikuláseketal.2008. | We conclude that the observed period variations are caused by uneven rotation of surface layers of the stars \citep{step,mik901,unstead}. |
2011).. However. the physical explanation of the observed rotation period variations is not straightforward. | However, the physical explanation of the observed rotation period variations is not straightforward. |
Rotational braking by angular momentum (AM) loss via a magnetised stellar wind (Mikuláseketal.2008) cannot be the sole cause of the period variations because we detected intervals when the rotational period decreases. | Rotational braking by angular momentum (AM) loss via a magnetised stellar wind \citep{mik901} cannot be the sole cause of the period variations because we detected intervals when the rotational period decreases. |
Consequently. the AM loss may affect only the outer stellar envelope that is hardened by the global magnetic field. leaving the faster rotating core unaffected. | Consequently, the AM loss may affect only the outer stellar envelope that is hardened by the global magnetic field, leaving the faster rotating core unaffected. |
Intervals of rotational deceleration may then alternate with intervals of angular momentum exchange between the slowly rotating envelope and the faster rotating core. | Intervals of rotational deceleration may then alternate with intervals of angular momentum exchange between the slowly rotating envelope and the faster rotating core. |
The latter manifests itself by the rotational acceleration of the outer envelope. | The latter manifests itself by the rotational acceleration of the outer envelope. |
However. this explanation is theoretically challenging. | However, this explanation is theoretically challenging. |
Given the intermediate age ofVir.. the detected rotation variations are likely not connected with the transient behaviour that is connected with settling into the MS stage. | Given the intermediate age of, the detected rotation variations are likely not connected with the transient behaviour that is connected with settling into the MS stage. |
Therefore. we can expect a stable inner configuration in both stars. | Therefore, we can expect a stable inner configuration in both stars. |
As follows from Flowers&Ruderman(1977) and Braithwaite&Norelund(2006).. the magnetic fields confined to the outer stellar layers only are unstable. | As follows from \citet{flora} and \citet{brano}, the magnetic fields confined to the outer stellar layers only are unstable. |
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