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They varied bv «230€ for mdices rangiue youn 1 to 5 and o /p ratios frou 0.1 to 0.5. | They varied by $<$ for indices ranging from 4 to 5 and $\alpha$ /p ratios from 0.1 to 0.5. |
We sunnuuize the results of these fits for the flare spectra acctuntlated at the 5 hehoceutric angles iu Table 2 for accelerated power-law particle spectra vine an index of —L2 aud à 'p ratio of ~0.3 (typical of flare observations). | We summarize the results of these fits for the flare spectra accumulated at the 5 heliocentric angles in Table \ref{tbl-1} for accelerated power-law particle spectra having an index of $\sim$ 4.2 and $\alpha$ /p ratio of $\sim$ 0.3 (typical of flare observations). |
Results from the Ctaussiana fits shown in Figure 11.. where the line width aud enerev were free to vary. are listed for comparison. | Results from the Gaussian fits shown in Figure \ref{fig3}, where the line width and energy were free to vary, are listed for comparison. |
A downward bea of accelerated particles is ruled out bv the fits. | A downward beam of accelerated particles is ruled out by the fits. |
Both isotropic and broad fau beam distributions also ail to provide acceptable fits to the data. | Both isotropic and broad fan beam distributions also fail to provide acceptable fits to the data. |
Iu contrast. a predominately dowisvard distribution of isotropic interacting particles provides accepable fts. comparable to that obtained by the Carssian fits. | In contrast, a predominately downward distribution of isotropic interacting particles provides acceptable fits, comparable to that obtained by the Gaussian fits. |
There is a small iuprovenient in some of the fits when an upward isotropic distribution with ~20% of the amplitude of the downward isotrovic distribution is added (see probabilities listed in the parentheses) iu column 6. | There is a small improvement in some of the fits when an upward isotropic distribution with $\sim$ of the amplitude of the downward isotropic distribution is added (see probabilities listed in the parentheses) in column 6. |
The augular distributions that we have used iu the fits above are approxinatious to distributions tha are derived under more realistic conditions. | The angular distributions that we have used in the fits above are approximations to distributions that are derived under more realistic conditions. |
Ta.Biuuaty.&Lingentelter(1989) and Murplis.etal.(1990). calculated the angular distributions of oeiteracting particles after transport iu maeuetic loops where the feld is constant in the corona and couverges frou, the chromosphere to the photosphere. | \citet{hua89} and \citet{murphy90} calculated the angular distributions of interacting particles after transport in magnetic loops where the field is constant in the corona and converges from the chromosphere to the photosphere. |
This calculation takes iuto account both magnetic mirroring and pitch anele scattering. | This calculation takes into account both magnetic mirroring and pitch angle scattering. |
Ta.Ramaty.&Liuseufelter(1089) «define a parameter A as the scattering mean free path clivicles by the half leneth of he coronal segment of the uaenetic loop. | \citet{hua89} define a parameter $\lambda$ as the scattering mean free path divided by the half length of the coronal segment of the magnetic loop. |
They show that pitch angle scattcring saturates for A = 25: particles are not scattered for A» oc. | They show that pitch angle scattering saturates for $\lambda$ = 25; particles are not scattered for $\lambda \rightarrow \infty$ . |
In the left panel of Figure 15 we plot the angular distributions of selected interacting particle distributions. | In the left panel of Figure \ref{fig10} we plot the angular distributions of selected interacting particle distributions. |
They are plotted μυ... cos 0, where 0 is the angele relative tothe solu radius. | They are plotted versus cos $\theta$ , where $\theta$ is the angle relative tothe solar radius. |
We see that the dowisvard isotropic distribution is a step function iu this representation aud the fan beam (siu"0) shows a sinoothlv falling svuuuetric distribution about the parallel to the solar surface. | We see that the downward isotropic distribution is a step function in this representation and the fan beam $^{6} \theta$) shows a smoothly falling symmetric distribution about the parallel to the solar surface. |
The weal pitch- scattering distribution. A = 300. beeius to | The weak pitch-angle scattering distribution, $\lambda$ = 300, begins to |
classification. we first removed the emission lines from the spectra by interpolating to the adjacent ‘continuum’. | classification, we first removed the emission lines from the spectra by interpolating to the adjacent `continuum'. |
We then cross-correlated the resulting. spectrum. (using the IRAP task) with a sample of digital. spectra of AL chwarls from the Cliese catalog (Llenry. Kirkpatrick. Simons. 1994). | We then cross-correlated the resulting spectrum (using the IRAF task) with a sample of digital spectra of M dwarfs from the Gliese catalog (Henry, Kirkpatrick, Simons, 1994). |
By far the best correlation. and a very close match was with the M3.0V star Gliese 251 (cross-correlation peak height 0.99). | By far the best correlation, and a very close match was with the M3.0V star Gliese 251 (cross-correlation peak height 0.99). |
Lor an M3.0V. star. we find Αι=11.7x0.6 from equation 1 of Henry ct al. ( | For an M3.0V star, we find $M_V = 11.7\pm 0.6$ from equation 1 of Henry et al. ( |
1994: note that this error is the empirical rms value. more representative of the actual error than lo). | 1994; note that this error is the empirical rms value, more representative of the actual error than $1\sigma$ ). |
Bessell (1991) tabulates V.R=1.1 and D211.55 for an M3.0V. star. | Bessell (1991) tabulates $V-R=1.1$ and $B-V=1.55$ for an M3.0V star. |
From this we derive Alp=10.6 for comparison to the USNO red. magnitude. to obtain a distance of 52 pc. | From this we derive $M_R= 10.6$ for comparison to the USNO red magnitude, to obtain a distance of 52 pc. |
The PSPC light curve (Figure 4)) shows prominent X-rav [lare activity on 19 June 1990. | The PSPC light curve (Figure \ref{pspc_flare}) ) shows prominent X-ray flare activity on 19 June 1990. |
Using the parameters or the best fit spectral model (section 4.3). the peak Dare uminositv is Lx 107 ere toa 24 fold. increase over the mean quiescent luminosity Lx —4.6.-1077 erg st (Section 4.2). | Using the parameters for the best fit spectral model (section 4.3), the peak flare luminosity is $_{X}$ $\times10^{30}$ erg $^{-1}$ , a 24 fold increase over the mean quiescent luminosity $_{X}$ $\times10^{28}$ erg $^{-1}$ (Section 4.2). |
The outburst can be characterized by an olding rise time 7,2:2.2 hours and decay time r,/277 hours. | The outburst can be characterized by an e-folding rise time $\tau$$_{r}$$\simeq$ 2.2 hours and decay time $\tau$$_{d}$$\simeq$ 7 hours. |
The total rise ancl decav. time for the [lare is z2M 5:426 rours andl 2M q4,uuu,2230 hours as measured from the quiescent o peak count rate. | The total rise and decay time for the flare is $\Delta$ $_{rise}$$\simeq$ 6 hours and $\Delta$$t_{decay}$$\simeq$ 30 hours as measured from the quiescent to peak count rate. |
The tail end of the Hare significantly departs [rom the exponential decay of the [are (Figure. 4)). | The tail end of the flare significantly departs from the exponential decay of the flare (Figure \ref{pspc_flare}) ). |
‘This Hare is evidently a long duration event. | This flare is evidently a long duration event. |
Most long decav [lares on eMe stars have 7,41 hour (Pallavicini et al.. | Most long decay flares on dMe stars have $\tau$$_{d}\sim1$ hour (Pallavicini et al., |
1990). | 1990). |
Recently. long duration Hares have been detected on EV Lac (74,210.5 hours: Schmitt. 1904). and AD Leo (r,—2.2 hours Favata. Alicela Reale. 2000). | Recently, long duration flares have been detected on EV Lac $\tau$$_{d}$ =10.5 hours; Schmitt 1994) and AD Leo $\tau$$_{d}$ =2.2 hours; Favata, Micela Reale, 2000). |
Continual icing of the Uarine region during the decay has been ooposed to explain such long decaving events. | Continual heating of the flaring region during the decay has been proposed to explain such long decaying events. |
Due to he long cleeay time. the total energy. Eyor=s.5 get ore (Table 3) released is similar to the [lare seen on EV Lac (γω 107? erg: Schmitt 1994) and. large as compared o other dMoe flare stars (23We - 1oft: pPallavicini et al. | Due to the long decay time, the total energy $_{TOT}$ $\times10^{34}$ erg (Table 3) released is similar to the flare seen on EV Lac $_{TOT}$ $\times10^{33}$ erg; Schmitt 1994) and large as compared to other dMe flare stars $\approx3\times10^{30}$ - $1\times10^{34}$; Pallavicini et al., |
1990). | 1990). |
These long decay [lares on clAle stars are. iowever. 2 - 3 orders of magnitude less energetic than elant X-ray. [lares on RS CVn stars such as Algol (7,,28.4 jours: Eyor=7. 1075 erg: Ottmann Schmitt. 1996). and CE Tucanae (7,222 hours: Eyor=l.4. 1077 ere: Ixtuster Schmitt. 1996). | These long decay flares on dMe stars are, however, 2 - 3 orders of magnitude less energetic than giant X-ray flares on RS CVn stars such as Algol $\tau$$_{d}$ =8.4 hours; $_{TOT}$ $\times10^{36}$ erg; Ottmann Schmitt, 1996), and CF Tucanae $\tau$$_{d}$ =22 hours; $_{TOT}$ $\times10^{37}$ erg; $\rm\ddot{u}$ rster Schmitt, 1996). |
On 04 July. 1990 at 21:11:35. UT. à flare with a peak uminositv. about the same magnitude as the PSPC outburst (Figure. 5)). was detectedwith the IRI. | On 04 July 1990 at 21:11:35 UT, a flare with a peak luminosity, about the same magnitude as the PSPC outburst (Figure \ref{hri1_flare}) ), was detectedwith the HRI. |
Since the HEU has extremely limited. spectral resolution. we input the mocel it to the PSPC [lare and the HIR count rate into the IAE/PROS task to convert counts to luminosity. | Since the HRI has extremely limited spectral resolution, we input the model fit to the PSPC flare and the HRI count rate into the IRAF/PROS task to convert counts to luminosity. |
ωνPhe peak flare luminosity.+ was Lxy-2.9- 10°20 erg 1 a actor of 54 larger than the mean quiescent luminosity Έντο 1077D erg . Loo | The peak flare luminosity was $_{X}$ $\times10^{30}$ erg $^{-1}$ a factor of 54 larger than the mean quiescent luminosity $_{X}$ $\times10^{28}$ erg $^{-1}$. |
pThe outburst can be characterized. bv an e-folcling rise time 7,215 minutes and decay time Tecel.2 hours. | The outburst can be characterized by an e-folding rise time $\tau$$_{r}$$\simeq$ 15 minutes and decay time $\tau$$_{d}$$\simeq$ 1.2 hours. |
The total rise and decay time for the [are is 2M 4, 040 minutes and 2M quuuuuc4.6 hours. | The total rise and decay time for the flare is $\Delta$ $_{rise}$$\simeq$ 40 minutes and $\Delta$$t_{decay}$$\simeq$ 4.6 hours. |
The total energy released during the Hare is E-ror=1.6. 1075 erg Clable 3). | The total energy released during the flare is $_{TOT}$ $\times10^{34}$ erg (Table 3). |
We also discovered the tail end. of an additional [lare observed. on 03 June 1996 with a two-fold. rise above the background. level. | We also discovered the tail end of an additional flare observed on 03 June 1996 with a two-fold rise above the background level. |
Due to our incomplete light. curve. we can onlv providean approximation to the decay time ( ~9 hours). | Due to our incomplete light curve, we can only providean approximation to the decay time ( $\sim$ 9 hours). |
The X-ray characteristics of these Dares are listed in ‘Table 3.. | The X-ray characteristics of these flares are listed in Table \ref{flare_tab}. . |
The quiescent” X-ray. emission. outside the large [lares is clearly itself variable. | The “quiescent" X-ray emission outside the large flares is clearly itself variable. |
Fie. | Fig. |
4 shows a factor of 3 change in | 4 shows a factor of 3 change in |
Figure 10. shows the line broadening for the stars [rom Table 5 (left). and those stars plus the results from C2003 (right). | Figure \ref{fig10} shows the line broadening for the stars from Table 5 (left), and those stars plus the results from C2003 (right). |
The 45 stars lor which we have obtained new results behave (he same as the 91 stars studied by C2003. | The 45 stars for which we have obtained new results behave the same as the 91 stars studied by C2003. |
Specifically, stars with My<—2 show svstematically higher levels of line broadening than the fainter stars. | Specifically, stars with $M_{\rm V} < -2$ show systematically higher levels of line broadening than the fainter stars. |
As noted earlier. C2003 speculated that while some of the cases might be due to tidal locking in a binarysvstem”.. some of the apparently elevated line broadening might be due to increased rotation due to the absorption of a giant. planet that had. an orbital separation of about one AU. | As noted earlier, C2003 speculated that while some of the cases might be due to tidal locking in a binary, some of the apparently elevated line broadening might be due to increased rotation due to the absorption of a giant planet that had an orbital separation of about one AU. |
This idea was also consistent with the generally high levels of line broadening seen in the RIID stars. which are. of course. the direct. descendents of some of the RGB Up stars. | This idea was also consistent with the generally high levels of line broadening seen in the RHB stars, which are, of course, the direct descendents of some of the RGB tip stars. |
The significant line broadening of many of the RIID stars in our program 1s also apparent in Figure 10.. | The significant line broadening of many of the RHB stars in our program is also apparent in Figure \ref{fig10}. |
Line broadening mav also be due (to macroturbulence. | Line broadening may also be due to macroturbulence. |
C2003 dismissed this as a cause because (he levels of macroturbulence necessary (o provide a total line broadening; of 9 to 10 aad higher seemed improbable. | C2003 dismissed this as a cause because the levels of macroturbulence necessary to provide a total line broadening of 9 to 10 and higher seemed improbable. |
Grav (1982) and Gray Pallavicini (1989) reported macroturbulent velocities for IX. giants (Iuminositv class HD). and all values were smaller than 6.5. | Gray (1982) and Gray Pallavicini (1989) reported macroturbulent velocities for K giants (luminosity class III), and all values were smaller than 6.5. |
!.. Even considering luminosity classes H-ILE and IH. Grav Toner (1986) found macroturbulent velocities to be smaller than 8ο | Even considering luminosity classes II-III and II, Gray Toner (1986) found macroturbulent velocities to be smaller than 8. |
, However. these results were derived [rom studies of metal-rich disk stars. | However, these results were derived from studies of metal-rich disk stars. |
An obvious question is whether lower-mass. older. more metal-poor halo giants have elevated levels of macroturbulence compared to disk giants. | An obvious question is whether lower-mass, older, more metal-poor halo giants have elevated levels of macroturbulence compared to disk giants. |
Carnev οἱ ((2007) do not. in fact. find significant differences in Qyp between metal-rich disk and halo red giants. | Carney et (2007) do not, in fact, find significant differences in $\zeta_{\rm RT}$ between metal-rich disk and metal-poor halo red giants. |
What about velocity jitter? | What about velocity jitter? |
Let us consider only the RGB stars and exclude the known binary svstems. where tical interactions may have contributed to the rotation and line broadening. | Let us consider only the RGB stars and exclude the known binary systems, where tidal interactions may have contributed to the rotation and line broadening. |
We have noted (hat velocity jitter beeins to appear at only (he highest Iuminosities. Af<—ld. and especially lor Ady<—2.0. | We have noted that velocity jitter begins to appear at only the highest luminosities, $M_{\rm V} \leq\ -1.4$, and especially for $M_{\rm V} \leq -2.0$. |
These are also the Iuminosity levels for which line broadening measures are highest. | These are also the luminosity levels for which line broadening measures are highest. |
In Figure 11. we replace luminosity with log P(\7) and compare directly with Έτ. | In Figure \ref{fig11} we replace luminosity with log $\chi^{2}$ ) and compare directly with $V_{\rm broad}$. |
Open red. circles are stus wilh —2.0<yx—1.5. | Open red circles are stars with $-2.0 < M_{\rm V} \leq\ -1.5$. |
Filled red circles are stars with A/yx —-2.0. As the Figure shows. some of the huninous RGB stars have normal P(\7) and modest but typical line broadening. | Filled red circles are stars with $M_{\rm V} \leq\ -2.0$ As the Figure shows, some of the luminous RGB stars have normal $\chi^{2}$ ) and modest but typical line broadening. |
However. a significant nunber of the most luminous stars have very low values of P(\7). and thev also tend to have much hieher than average values of Ἑτυμα. | However, a significant number of the most luminous stars have very low values of $\chi^{2}$ ), and they also tend to have much higher than average values of $V_{\rm broad}$. |
In other words. it appears that there is a strong correlation between velocity jitter and line broadening. | In other words, it appears that there is a strong correlation between velocity jitter and line broadening. |
This is not a matter of "velocity | This is not a matter of “velocity |
due to ? aud to 2.. were originally designed to treat strong shock waves in the non-isotropic mecitun. | due to \cite{Komp} and to \cite{LaumbachProbstein}, were originally designed to treat strong shock waves in the non-isotropic medium. |
These two complimentary methods have been exteusively applied in astropliysics to treat supernova explosions (?) aud noin-isotropic winds (e.g...2).. | These two complimentary methods have been extensively applied in astrophysics to treat supernova explosions \citep{Bisnovatyi-KoganSilich95} and non-isotropic winds \citep[\eg][]{Icke}. |
In the Ixouipaneets approximation the internal pressure of the gas is assumed to be coustaut. | In the Kompaneets approximation the internal pressure of the gas is assumed to be constant. |
Then the BRaukin-Hugonio couditions determine the normal velocity of the shock in the external inhomogeneous inecium. | Then the Rankin-Hugonio conditions determine the normal velocity of the shock in the external inhomogeneous medium. |
A moclilication ol the Ixompaneets approximation - a thin or suowplow shell approximation - has also been used exteusively (e.g...2??).. | A modification of the Kompaneets approximation - a thin or snowplow shell approximation - has also been used extensively \cite[\eg][]{1978ApJ...221...41W,1988ApJ...324..776M,1989Ap&SS.154..229B}. |
In a complimentary Latumbach-Probstein approach (?) the streamlines ol the shocked material are assumed to be racial. thus neglecting the late‘al pressure forces. | In a complimentary Laumbach-Probstein approach \citep {LaumbachProbstein} the streamlines of the shocked material are assumed to be radial, thus neglecting the lateral pressure forces. |
The relativistic generalization of the Ixoimpaneets aud the Laumbach-Probstein methods have been discussed by ?.. | The relativistic generalization of the Kompaneets and the Laumbach-Probstein methods have been discussed by \cite{1979ApJ...233..831S}. |
Relativistic dynamics provide extra support for the tlin shell inethod. since in the relativistic blast waves the shocked material is concentrated in even uarower region. R/T>τὸ than in the nou-relativistic Seclov solution. | Relativistic dynamics provide extra support for the thin shell method, since in the relativistic blast waves the shocked material is concentrated in even narrower region $R/\Gamma^2$ than in the non-relativistic Sedov solution. |
Ln addition. the limited causal coirection (over the auele 1/1) provides a justification for the Laumbach-Probsteiu method on the angle scale comparable to L/L. | In addition, the limited causal connection (over the angle $\sim 1/\Gamma$ ) provides a justification for the Laumbach-Probstein method on the angle scale comparable to $1/\Gamma$. |
As has been pointed out by ?.. the two methods - Ixompaneets adà Laumbach-Probstein - become very similar in the relativistic regime. | As has been pointed out by \cite{1979ApJ...233..831S}, the two methods - Kompaneets and Laumbach-Probstein - become very similar in the relativistic regime. |
This is due to the fact the in a relativistic wave. the typical augle that a shock wave makes with the direction of the velocity is of the order à~yojp?1/17. | This is due to the fact the in a relativistic quasi-spherical wave, the typical angle that a shock wave makes with the direction of the velocity is of the order $ \alpha \sim 1/\Gamma^2$. |
Thus the post shock pressure along the shock differstq ouly by one part iu⋅ I7.EY so that both approxiuiations of coustaut post-slock pressure aud racial post-shoek motion become ecuivalent. | Thus the post shock pressure along the shock differs only by one part in $\Gamma^2$, so that both approximations of constant post-shock pressure and radial post-shock motion become equivalent. |
Iu some sense the propagation of strongly relativistic uou-spherical shocks becomes trivial: relativistic kinematic effect freeze out. the lateral dyuaimics of the flow so that different parts of the flow behave virtually indepeudenutly. | In some sense the propagation of strongly relativistic non-spherical shocks becomes trivial: relativistic kinematic effect freeze out the lateral dynamics of the flow so that different parts of the flow behave virtually independently. |
Inu this section we re-clerive the relativistic lvompaueets equation ?? allowing for the arbitrary velocity of the shock aud arbitrary (angle-dependent) luminosity and/or external density. | In this section we re-derive the relativistic Kompaneets equation \cite{Komp,1979ApJ...233..831S} allowing for the arbitrary velocity of the shock and arbitrary (angle-dependent) luminosity and/or external density. |
Consider ashock propagating with a three-velocity V. at an angle a to its normal. | Consider a shock propagating with a three-velocity $V$ at an angle $\alpha$ to its normal. |
There are three generic rest frames in the problem: laboratory [fraiue A. a frame where the shock is nor.ual to the flow /v4 aud a shock rest frame Ax. A [rame A4 is related to the lab frame A by a Lorentz |00st along y axis with a Lorentz factor Py=1/v1-—V2sin?a. | There are three generic rest frames in the problem: laboratory frame $K$ , a frame where the shock is normal to the flow $K_1$ and a shock rest frame $K_0$ A frame $K_1$ is related to the lab frame $K$ by a Lorentz boost along $y$ axis with a Lorentz factor $\Gamma_\parallel =
1/\sqrt{1- V^2 \sin^2 \alpha}$. |
In A4 the velocity of the shock is Vi=PyWVcosa (along the . ↽∣↝≺⇂⊔⋅≺↲∢∙ | In $K_1$ the velocity of the shock is $V_1 = \Gamma_\parallel V \cos \alpha $ (along the $x$ direction). |
⊔∩∐⋅⊺↥⋯⊳∖⋅↕↕∶↥↙∕∕⊔≓⇖↓−⋝∶↕−∢∙∩⊳∖−∩↲↱⊳∖∐−∩∩∐≺↵⊳∖∐∩∢∙↕⊆∣⋈↵∢∙∩⋯≺↵⊳∖∐∩∐−↓⋅≺↵⋜⋯∖⇁↥⊳∖⊔∢∙∖∖↽∐≺↵∐32yfeero>κο9D / ⋅⋅⋅ ∣⊤↙∕∕∣⊇⇥↘∿↕↙∕∣∣ | Thus, $\Gamma_1^2 = 1/(1-V_1^2) = \Gamma^2 \cos^2 \alpha + \sin ^2 \alpha$ (the shock becomes non-relativistic when $\pi/2- \alpha \sim 1/\Gamma$ ). |
⊔⋅↕∐↕∐≺↵∐⋅⋜⋃∐↩∫↘⋟∣∣⇖⊽↓⋜↕∐≼⊔↴↓⋜⋃⋅↩↕∐≺↵∖⇁≺↵∩∢∙∐⊽∖⊽⋜↕∐≺⇂↕∐≺↵∟∩↥⋅≺↵∐↕∠↥∎⋜↕∢∙↕∩↥⋅∩↥∎↕∐≺↵⋃∐⊳∖∐∩∢∙↥⊆≺↵≺⇂ medium. | In the frame $K_0$ $V_1$ and $\Gamma_1$ are the velocity and the Lorentz factor of the unshocked medium. |
In the lab frame the vr component of the shocked velocity V?E—ιαC+V(Vsina) eenerally has a completed form. but simple relatious cau be obtained iu the stronglyrelativistic limit (see below).We introduce uext ai acceleration parameter A. (Ixompaneets 1960. Icke LOSS) as a Lorentz [actor of the shock in the ἐν frame | In the lab frame the $x$ component of the shocked velocity $V_x ' = V_1' /( \Gamma_\parallel (1+ V_1' V \sin \alpha)$ generally has a completed form, but simple relations can be obtained in the stronglyrelativistic limit (see below).We introduce next an acceleration parameter $K$ (Kompaneets 1960, Icke 1988) as a Lorentz factor of the shock in the $K_1$ frame |
The: wind. isη ejected. at speed -Vay. so las mean enerevy por unitη mass of] Ey.=21-)ay We- | The wind is ejected at speed $V_{\rm SW}$, so has mean energy per unit mass of $E_{\rm av} = \frac{1}{2}V_{\rm SW}^2$. |
asst that to escape the halo itη needs energy per unit. lass Agwwt)Vig. | We assume that to escape the halo it needs energy per unit mass $\lambda_{\rm SW}V_{\rm
disk}^2$. |
Assuuiug. a thermal distribution. ofB euergiesI iu. the wind. (Le.. f(E)xοκE/ E). the fraction of the mass flowing out of the disk which also escapes from the halo is given by where c=EfEy aud sw=Aw-)VafLav. | Assuming a thermal distribution of energies in the wind (i.e. $f(E) \propto \exp(-E/E_{\rm av})$ ), the fraction of the mass flowing out of the disk which also escapes from the halo is given by where $x = E/E_{\rm av}$ and $x_{\rm SW} = \lambda_{\rm SW} V_{\rm
disk}^2/E_{\rm av}$. |
For the material. which. escapes the halo. the mean energy excess over that necded to eject it from the halo is | For the material which escapes the halo, the mean energy excess over that needed to eject it from the halo is |
two IMEs. | two IMFs. |
The metal vields are taken from vaudeuIock&Croenewegen(1997) for stellar masses < 8 AL. aud from Woosley&Weaver(1995). for the &LO AD. mass range. | The metal yields are taken from \citet{vangroen} for stellar masses $<$ 8 $_\odot$ and from \citet{ww95}
for the 8–40 $_\odot$ mass range. |
Note that for Z,=0.001 massive stars. we use the Z=0.17... case frou Woosley&Weaver(1995). | Note that for $Z_\star=0.001$ massive stars, we use the $Z=0.1 Z_\odot$ case from \citet{ww95}. |
.. Since there are no cletailed vields to date for Z,=0 IMSs. we use those for Z,=0.001 IMSs. which are not substantially different (A. Chicfi 2002. private communication). | Since there are no detailed yields to date for $Z_\star=0$ IMSs, we use those for $Z_\star=0.001$ IMSs, which are not substantially different (A. Chieffi 2002, private communication). |
For Z,=0 VMSs. ouly the stars of mass 110.260 AL.. avoid complete collapse into a black hole (Ileger&Woosley2002) and contribute to the uucleosvuthetic output. | For $Z_\star=0$ VMSs, only the stars of mass 140–260 $M_\odot$ avoid complete collapse into a black hole \citep{heger02} and contribute to the nucleosynthetic output. |
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