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For he purposes of this study we believe the small difference cau be ueglected. as we are atempting to illusrate a general principle rather than derive an exact match to the stellar profiles recorded ou the Diiberg plates. | For the purposes of this study we believe the small difference can be neglected, as we are attempting to illustrate a general principle rather than derive an exact match to the stellar profiles recorded on the Bamberg plates. |
We ecnerated a seres of svuthetic stellar radialflux profiles. based ou our fit | We generated a series of synthetic stellar radial–flux profiles, based on our fit |
In this section. we will first characterize the value funcGon V. | In this section, we will first characterize the value function $V$. |
Based on the properties of Vo we can estimate |V.—V| to prove1.. and |—V to prove3.. respectively. | Based on the properties of $V$, we can estimate $|V - V^\beta|$ to prove, and $|V - \bar V^\beta|$ to prove, respectively. |
We have seen (hat the option price V. of is one of the solutions of DS(Q.f). | We have seen that the option price $V$ of is one of the solutions of $BS(Q,f)$. |
To proceed. we need identify which solution corresponds to the option price V. among many. | To proceed, we need identify which solution corresponds to the option price $V$ among many. |
enables us (o establishthe connection between parabolic partial differential equation dus9(Q.f) and probability representation (2.3).. | This enables us to establishthe connection between parabolic partial differential equation $BS(Q,f)$ and probability representation . |
(For simplicity. we consider here only pure neutron matter.) | (For simplicity, we consider here only pure neutron matter.) |
At first. the ligure can appear surprising. since Luger skin does not necessarily imply Larger radius. | At first, the figure can appear surprising, since larger skin does not necessarily imply larger radius. |
On the other haud. one must keep in mind (hat: 1) The radius depends mostly on the pressure ab (he higher densities. whereas the skin probes normal or subnuclear densities: 2) As we argued in the previous paragraph lor the case of a nucleus. these models dilfer from one another in more than just the slope of the symmetry energy. | On the other hand, one must keep in mind that: 1) The radius depends mostly on the pressure at the higher densities, whereas the skin probes normal or subnuclear densities; 2) As we argued in the previous paragraph for the case of a nucleus, these models differ from one another in more than just the slope of the symmetry energy. |
Table I can give additional insight. | Table I can give additional insight. |
Even though UIX predicts a larger skin (han DDIIF. its predicted neutron star radius is 11.8 km as opposed to 12.4 km [rom DBIIF. | Even though UIX predicts a larger skin than DBHF, its predicted neutron star radius is 11.8 km as opposed to 12.4 km from DBHF. |
Notice. however. the much larger central energy density in the UIX case. | Notice, however, the much larger central energy density in the UIX case. |
In other words. UIX. produces a more compact 1.4M. neutron star. | In other words, UIX produces a more compact $_{\odot}$ neutron star. |
BOB and V18 are quite similar to DDIIF in both radius and central energy densitv. | BOB and V18 are quite similar to DBHF in both radius and central energy density. |
On the other haud. in a case like N93. where. as seen [rom Fig. | On the other hand, in a case like N93, where, as seen from Fig. |
4. high pressure is sustained pretty much ab all densities. the 14M. star is larger and more diffuse. | 4, high pressure is sustained pretty much at all densities, the $_{\odot}$ star is larger and more diffuse. |
In summary. for each moclel the available mass will distribute itself differently. depending on both the svimetric and the asvuumetric part of the Eosdensity. The different values of the predicted maximum masses. apparent [rom Fie. | In summary, for each model the available mass will distribute itself differently, depending on both the symmetric and the asymmetric part of the EoS. The different values of the predicted maximum masses, apparent from Fig. |
9. reflect differences among the | 9, reflect differences among the |
model for the Aquarius stream under the assumption of a single satellite dissolving in the Galaxy’s potential. | model for the Aquarius stream under the assumption of a single satellite dissolving in the Galaxy's potential. |
The model predicts that the stream is spread in XYZ away from the sun, with distance dmode)=3.2+0.8kpc in the direction 30°«|75°,—70°b<—50°. | The model predicts that the stream is spread in $XYZ$ away from the sun, with distance $d_\mathrm{model}=3.2\pm0.8\,\kpc$ in the direction $30^\circ<l<75^\circ, -70^\circ<b<-50^\circ$. |
The distance range derived above therefore probably reflects more on the distance errors than the real distribution for the stream. | The distance range derived above therefore probably reflects more on the distance errors than the real distribution for the stream. |
We assume that the Aquarius stream is a single, distinct object. | We assume that the Aquarius stream is a single, distinct object. |
The isochrone from Figure 4 has a J-band turn-off of M;=3.5. | The isochrone from Figure \ref{f4} has a $I$ -band turn-off of $M_I=3.5$. |
Hence, for the distance moduli above we could expect turn-off stars in the range J=11.8 - 18.5. | Hence, for the distance moduli above we could expect turn-off stars in the range $I=11.8$ - $18.5$. |
The lower magnitude falls within the RAVE magnitude limits (9<I 13). | The lower magnitude falls within the RAVE magnitude limits $9<I<13$ ). |
However, RAVE’s unbiased selection criteria mean that the thin disk dominates dwarf/turn-off stars, even at these higher magnitudes. | However, RAVE's unbiased selection criteria mean that the thin disk dominates dwarf/turn-off stars, even at these higher magnitudes. |
Our sample of halo dwarfs is therefore too small to detect the turn-off, and we only see giant stars in our Aquarius stream sample from RAVE. | Our sample of halo dwarfs is therefore too small to detect the turn-off, and we only see giant stars in our Aquarius stream sample from RAVE. |
The coherence of the group selection is shown by the reduced proper motion diagram (RPMD), which plots the reduced proper motion (RPM) against color. | The coherence of the group selection is shown by the reduced proper motion diagram (RPMD), which plots the reduced proper motion (RPM) against color. |
Described in detail in Seabrokeetal. the RPMD essentially creates a HR diagram from (2008),,the proper motions, where the absolute magnitude is smeared by the variation in the tangential speed of the stars. | Described in detail in \citet{Seabroke2008}, the RPMD essentially creates a HR diagram from the proper motions, where the absolute magnitude is smeared by the variation in the tangential speed of the stars. |
Halo stars have a large dispersion in tangental velocity and so this smearing is large. | Halo stars have a large dispersion in tangental velocity and so this smearing is large. |
In contrast, for a small, nearby section of a stream the transverse velocity spread is small and we effectively recover magnitudes for the stars. | In contrast, for a small, nearby section of a stream the transverse velocity spread is small and we effectively recover magnitudes for the stars. |
The RPM is given by where J and M, are the apparent and absolute magnitudes respectively, µ is the proper motion in arcsec yr-l, vp is the tangential velocity in kms!. | The RPM is given by where $J$ and $M_J$ are the apparent and absolute magnitudes respectively, $\mu$ is the proper motion in arcsec $\mathrm{yr}^{-1}$ , $v_T$ is the tangential velocity in $\kms$. |
Here we have again used 2MASS colors. | Here we have again used 2MASS colors. |
Thus, from the observables J and µ we can establish something about the more fundamental parameters M; and vp without requiring a either distance or a radial velocity. | Thus, from the observables $J$ and $\mu$ we can establish something about the more fundamental parameters $M_J$ and $v_\mathrm{T}$ without requiring a either distance or a radial velocity. |
Figure 5 gives the RPMD for the stars in our magnitude and latitude selected sample with the Aquarius stream candidates over-plotted, where for the latterthe more accurate PPMXL proper motions were used. | Figure \ref{f5} gives the RPMD for the stars in our magnitude and latitude selected sample with the Aquarius stream candidates over-plotted, where for the latterthe more accurate PPMXL proper motions were used. |
Note that | Note that |
(S. Mandel?)) and likely for the O6 [anf runaway star. Cep (2). | (S. ) and likely for the O6 I(n)f runaway star $\lambda$ Cep \citep{Gva11}. |
Although these are much hotter stars with faster stellar winds. the physical mechanism responsible for producing a curved tail is the same. | Although these are much hotter stars with faster stellar winds, the physical mechanism responsible for producing a curved tail is the same. |
In light of our simulations. such a feature implies either a very small space motion for the star relative to ambient ISM (unlikely given the strength of the bow shock emission) or that the bow shock is young. | In light of our simulations, such a feature implies either a very small space motion for the star relative to ambient ISM (unlikely given the strength of the bow shock emission) or that the bow shock is young. |
Thus. modelling the shape of the bow shock (head and tail. if present) is a promising avenue for constraining the age and the evolutionary stage of these systems. | Thus, modelling the shape of the bow shock (head and tail, if present) is a promising avenue for constraining the age and the evolutionary stage of these systems. |
Estimates for Betelgeuse's age are highly uncertain. ranging from 8-13 million years depending on the stellar evolution model and distance assumed (?).. | Estimates for Betelgeuse's age are highly uncertain, ranging from 8-13 million years depending on the stellar evolution model and distance assumed \citep{Harp08}. |
The core helium-burnitffe phase lasts about of the total lifetime. thus Ty.=0.8—I.2 Myr. | The core helium-burning phase lasts about of the total lifetime, thus $\tau_{\rm He} = 0.8 -1.3$ Myr. |
The star is expected to start and spend a significant fraction of this phase as an RSG. | The star is expected to start and spend a significant fraction of this phase as an RSG. |
According to current stellar models. the star may or may not have undergone a so-called blue loop in the HR diagram. spending a fraction of core helium burning as a blue supergiant. before returning to the RSG stage at core helium exhaustion (??).. | According to current stellar models, the star may or may not have undergone a so-called blue loop in the HR diagram, spending a fraction of core helium burning as a blue supergiant, before returning to the RSG stage at core helium exhaustion \citep{Mey00,Heg00}. |
A young age of Betelgeuse’s bow shock might imply that the star has entered the RSG stage only recently. | A young age of Betelgeuse's bow shock might imply that the star has entered the RSG stage only recently. |
It may be interesting to relate this idea with the recent finding that the diameter of Betelgeuse at jm has systematically decreased by about over the past 15 years (?).. | It may be interesting to relate this idea with the recent finding that the diameter of Betelgeuse at $\mu$ m has systematically decreased by about over the past 15 years \citep{Tow09}. |
While ? and ? debate whether this implies a real radius change. or rather a change in the density of certain layers in the envelope of the star. it is remarkable that the time scale of these changes is of the order of 100 years. | While \cite{Rav10} and \cite{Ohn11} debate whether this implies a real radius change, or rather a change in the density of certain layers in the envelope of the star, it is remarkable that the time scale of these changes is of the order of 100 years. |
This might imply that the envelope of Betelgeuse is not in thermal equilibrium. as might be the case when a star is entering the RSG stage. | This might imply that the envelope of Betelgeuse is not in thermal equilibrium, as might be the case when a star is entering the RSG stage. |
On the other hand. the radius of a star entering the RSG phase for the first time after core hydrogen exhaustion only increases. with time scales for the radius change (R/R) down to about 0000 years. | On the other hand, the radius of a star entering the RSG phase for the first time after core hydrogen exhaustion only increases, with time scales for the radius change $R/\dot R$ ) down to about 000 years. |
This 1s different. however. when the star enters the RSG stage after core helium exhaustion. because the igniting helium shell leads to an intermittent. episode of shrinkage. | This is different, however, when the star enters the RSG stage after core helium exhaustion, because the igniting helium shell leads to an intermittent episode of shrinkage. |
Since the star might have lost a substantial fraction of its hydrogen-rich envelope by that time. its thermal time scale might also have decreased. | Since the star might have lost a substantial fraction of its hydrogen-rich envelope by that time, its thermal time scale might also have decreased. |
Thus. while more observational and theoretical efforts are required to give this speculation more substance. a consistent picture might involve Betelgeuse recently having finished core helium burning and returning to the RSG stage from a previous blue supergiant excursion. | Thus, while more observational and theoretical efforts are required to give this speculation more substance, a consistent picture might involve Betelgeuse recently having finished core helium burning and returning to the RSG stage from a previous blue supergiant excursion. |
If Betelgeuse has only recently entered the RSG stage. it may not have had enough time to travel beyond its main sequence or blue supergiant wind bubble. | If Betelgeuse has only recently entered the RSG stage, it may not have had enough time to travel beyond its main sequence or blue supergiant wind bubble. |
Assuming a wind mass-loss rate of 107M. yr!. and a wind velocity of -[O kkmss7!. the stand-off distance for Betelgeuse’s main sequence or blue supergiant bow shock shell was around 1 pe. | Assuming a wind mass-loss rate of $10^{-7}\,\msun\,$ $^{-1}$, and a wind velocity of $\sim$$10^3$ $^{-1}$, the stand-off distance for Betelgeuse's main sequence or blue supergiant bow shock shell was around 1 pc. |
A RSG phase of a ~fewx 100000 years would bring the star close to the edge of such a bubble. | A RSG phase of a $\sim$ $\times$ 000 years would bring the star close to the edge of such a bubble. |
This raises the possibility that the mysterious “bar? ahead of the bow shock could be a remnant of this earlier phase of evolution. | This raises the possibility that the mysterious `bar' ahead of the bow shock could be a remnant of this earlier phase of evolution. |
A theoretical investigation of such a scenario is currently underway. | A theoretical investigation of such a scenario is currently underway. |
We presented the first 3D models of the interaction of Betelgeuse’s RSG wind with the ISM. | We presented the first 3D models of the interaction of Betelgeuse's RSG wind with the ISM. |
We took dust. atomic-. molecular-. and metal-line cooling into account. | We took dust, atomic-, molecular-, and metal-line cooling into account. |
The models cover a range of plausible ISM densities of 0.3 - 1.9 em? and stellar velocities of 28 - 73 kmss!. | The models cover a range of plausible ISM densities of 0.3 - 1.9 $^{-3}$ and stellar velocities of 28 - 73 $^{-1}$. |
We showed that the flow dynamies and morphology of the bow shocks in the models differed due to the growth of Rayleigh-Taylor or Kelvin-Helmholtz instabilities. | We showed that the flow dynamics and morphology of the bow shocks in the models differed due to the growth of Rayleigh-Taylor or Kelvin-Helmholtz instabilities. |
The former dominate the slow models. resulting in a clumpy substructure. whereas the latter are characteristic of the fast model and produce a more layered substructure. | The former dominate the slow models, resulting in a clumpy substructure, whereas the latter are characteristic of the fast model and produce a more layered substructure. |
In the fast model. gas is shocked to high temperatures. | In the fast model, gas is shocked to high temperatures. |
If gas as hot as this were to be detected at short wavelengths (e.g. UV). this would exclude the slow models as an explanation for Betelgeuse’s bow shock. | If gas as hot as this were to be detected at short wavelengths (e.g. UV), this would exclude the slow models as an explanation for Betelgeuse's bow shock. |
High spatial and spectral resolution observations (e.g.Herschel. and ALMA) particularly of the more dominant cooling/emitting species. e.g. rotational lines of H»O or CO. could also be used to further constrain the physical characteristics of the system. | High spatial and spectral resolution observations (e.g., and ) particularly of the more dominant cooling/emitting species, e.g. rotational lines of $_2$ O or CO, could also be used to further constrain the physical characteristics of the system. |
In addition. better determination of the molecular to atomic hydrogen fraction in the RSG wind. the abundances of various species. and temperature of the ISM would also reduce the number of free parameters in the model. | In addition, better determination of the molecular to atomic hydrogen fraction in the RSG wind, the abundances of various species, and temperature of the ISM would also reduce the number of free parameters in the model. |
The large fluxes in the infrared compared to the theoretical limit for the bolometric luminosity suggest that the stellar flux and/or flux from hotter gas in the bow shock is reprocessed by the dust and reemitted in the far-infrared. | The large fluxes in the infrared compared to the theoretical limit for the bolometric luminosity suggest that the stellar flux and/or flux from hotter gas in the bow shock is reprocessed by the dust and reemitted in the far-infrared. |
We showed that. if the bow shock shell mass is low. as is suggested by theAKARI fluxes. then Betelgeuse's bow shock is young. | We showed that, if the bow shock shell mass is low, as is suggested by the fluxes, then Betelgeuse's bow shock is young. |
The smoothness and circular nature of the bow shock would be consistent with this conclusion. | The smoothness and circular nature of the bow shock would be consistent with this conclusion. |
Furthermore. if the bow shock has not yet reached a steady state. we are less able to constrain the physical parameters of the system. e.g. the ram pressure of the ISM. | Furthermore, if the bow shock has not yet reached a steady state, we are less able to constrain the physical parameters of the system, e.g. the ram pressure of the ISM. |
This also raises the intriguing possibility that Betelgeuse has only recently become an RSG and that the mysterious “bar” ahead of the bow shock is a remnant of a wind shell created during an earlier phase of evolution. | This also raises the intriguing possibility that Betelgeuse has only recently become an RSG and that the mysterious `bar' ahead of the bow shock is a remnant of a wind shell created during an earlier phase of evolution. |
Ziypy defined as in Paper 1. col 5. 6. aud 7 give the mean nmaguitudes of the pairs. sinelets. and triplets. and col 58 gives the triplet age. | $_{(\rm r-b)}$, defined as in Paper I, col 5, 6, and 7 give the mean magnitudes of the pairs, singlets, and triplets, and col 8 gives the triplet age. |
The ages of triplets C aud D have con. calculated here frou their ejection angeles using the values derived for the ages of triplets A and D iu ΜΕ a rotation axis tip angele of 5; — 18 (where 5 is the rotation axis tip angle ucasured four the plane of the slaw towards the observer at the top in Fie. | The ages of triplets C and D have been calculated here from their ejection angles using the values derived for the ages of triplets A and B in paper I, assuming a rotation axis tip angle of $\gamma$ = $^{\arcdeg}$ (where $\gamma$ is the rotation axis tip angle measured from the plane of the sky towards the observer at the top in Fig. |
1). | 1). |
Tt has also been assumed. as in paper L that he direction of ejection remains fixed relative to he Sevtert galaxy frame. and that the change iu ejection angle is due to rotation of the ceutral object. | It has also been assumed, as in paper I, that the direction of ejection remains fixed relative to the Seyfert galaxy frame, and that the change in ejection angle is due to rotation of the central object. |
This is asstuned to be justified bv the act that the resulting rotation period calculated or the central object is in good agreciment with he rotation period determined from the results of Alloinctal.(2001). | This is assumed to be justified by the fact that the resulting rotation period calculated for the central object is in good agreement with the rotation period determined from the results of \citet{all01}. |
. Iu calculating the mean redshift only paired sources are used since the suelet redshüfts have not beeu measured for all triplets. | In calculating the mean redshift only paired sources are used since the singlet redshifts have not been measured for all triplets. |
Although the mean pair redshitts (2,444) likely coutain some Doppler compoucuts these were previously found to be small. | Although the mean pair redshifts $_{\rm mean}$ ) likely contain some Doppler components these were previously found to be small. |
Tn Πο, | In fig. |
2 the maeniudes of the two sources making up cach pair rave been plotted versus their respective measured redshifts. | 2 the magnitudes of the two sources making up each pair have been plotted versus their respective measured redshifts. |
The umubers iclentify sources as liste in Durbidee(1999).. and in paper L The heavy solid liue shows how Zien varies with the mean maeuitude of the paired sources. | The numbers identify sources as listed in \citet{bur99}, and in paper I. The heavy solid line shows how $_{\rm mean}$ varies with the mean magnitude of the paired sources. |
There are two lüues5 to note iu this plot. | There are two things to note in this plot. |
The dashed lines. which connect cach pair. all pass through the point near Ziau = 0.70. mag = 18.5. | The dashed lines, which connect each pair, all pass through the point near $_{\rm mean}$ = 0.70, mag = 18.5. |
The mean values vary systematically from A to D. with the magnitudes becoming brighter as tle Zucan redshifts decrease. | The mean values vary systematically from A to D, with the magnitudes becoming brighter as the $_{\rm mean}$ redshifts decrease. |
The brightening varies slowly at first. but increases With age (1.6. as Ziucan decreases). | The brightening varies slowly at first, but increases with age (i.e. as $_{\rm mean}$ decreases). |
Iu Fig. | In Fig. |
2 the mean triplet magnitudes aud nean ya redshifts (cols 7 and 3 respectively. in Table 1) are plotted versus age (col 8). | 3 the mean triplet magnitudes and mean pair redshifts (cols 7 and 3 respectively, in Table 1) are plotted versus age (col 8). |
From Fig. | From Fig. |
3(a). he triplets appear to be born with mean apparent nagnitudes near 19.5. | 3(a), the triplets appear to be born with mean apparent magnitudes near 19.5. |
Their brightening eraduallyIncreases with time. | Their brightening graduallyincreases with time. |
Iu Fig. | In Fig. |
3(b). zuian decreases roni L2 af age 24105 vrs to less than 0.5 at age [«109 vrs. | 3(b), $_{\rm mean}$ decreases from 1.2 at age $\times10^{6}$ yrs to less than 0.5 at age $4\times10^{6}$ yrs. |
If the change is linear (solid ine). the objects are born with intrinsic redshift conrponents near Zyean = 2. and this componct will have disappeared after ~ος109 vrs. | If the change is linear (solid line), the objects are born with intrinsic redshift components near $_{\rm mean}$ = 2, and this component will have disappeared after $\sim6\times10^{6}$ yrs. |
If the objects are born with zia, = 1.25 (see paper I. the change in µια nma progress slowly at first. proceed. through a more rapid stage [Age = (25).10° vis]. and then slow down as shown x the dashed curve. | If the objects are born with $_{\rm mean}$ = 1.25 (see paper I), the change in $_{\rm mean}$ may progress slowly at first, proceed through a more rapid stage [Age = $(2-5)\times10^{6}$ yrs], and then slow down as shown by the dashed curve. |
Since the pair-uglet position angles (col 2 of Table 1) rotate through approximately 120° from riplet A to D. both the magnitude aud redshift relations in Fie. | Since the pair-singlet position angles (col 2 of Table 1) rotate through approximately $\arcdeg$ from triplet A to D, both the magnitude and redshift relations in Fig. |
3 approximate cosine curves. | 3 approximate cosine curves. |
They welt both then be explained by changes iu the ejection angle as the galaxy rotates. | They might both then be explained by changes in the ejection angle as the galaxy rotates. |
Iowever. since cach singlet and pair is assumed to have con. ejected iu opposite directions. im a rotation uodel. magnitude changes with ejectiou angle iu he singlets would be 1505 out of phase with anv simular magnitude changes in the pairs. | However, since each singlet and pair is assumed to have been ejected in opposite directions, in a rotation model, magnitude changes with ejection angle in the singlets would be $\arcdeg$ out of phase with any similar magnitude changes in the pairs. |
In Fie. | In Fig. |
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