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YT would like to thank RolfPeter I&uditzki aud Boh McLaren for their wari hospitality. | YT would like to thank Rolf-Peter Kudritzki and Bob McLaren for their warm hospitality. |
YS thauks the Japan Society for Promotion of Scicuce (ISPS) Research Fcolowships for Young Scicutist. | YS thanks the Japan Society for Promotion of Science (JSPS) Research Fellowships for Young Scientist. |
This work was supported in part by the Ministry of Education. Scicuce. Sports aud Culture in Japan uuderCrvant Nos. | This work was supported in part by the Ministry of Education, Science, Sports and Culture in Japan underGrant Nos. |
0705501. 10011052. and LO30L013. | 07055044, 10044052, and 10304013. |
velocity 3/Le5,,008). where the angle between the pulsar velocity and the shock normal ijj 20.tan1(4R/d()/RI. | velocity $3/4 v_{\rm psr} {\rm cos} \eta$, where the angle between the pulsar velocity and the shock normal is $\eta = \theta - {\rm tan}^{-1}[(dR/d\theta)/R]$. |
Only later does the post-shock flow converge to the tangent to the contact discontinuity. | Only later does the post-shock flow converge to the tangent to the contact discontinuity. |
This effect may be seen in the simulations of Cormeron&Kaper(1998). | This effect may be seen in the simulations of \citet{ck98}. |
. Thus immediately behind the ISM shock one expects a line-of-sight velocity Since cosy) is always positive. this velocity always has the sign of |cos. | Thus immediately behind the ISM shock one expects a line-of-sight velocity Since ${\rm cos} \eta$ is always positive, this velocity always has the sign of $-{\rm cos} i$. |
An initial question is whether the shock created by a equatorial wind can reproduce the flattened shape of this pulsar bow shock. | An initial question is whether the shock created by an equatorial wind can reproduce the flattened shape of this pulsar bow shock. |
One way to parametrize this shape ts the ratio of the perpendicular half angular size Ó,. measurec through the pulsar. to the angle from the pulsar to the projected limb of the wind shock in the forward directior 0j. | One way to parametrize this shape is the ratio of the perpendicular half angular size $\theta_\perp$, measured through the pulsar, to the angle from the pulsar to the projected limb of the wind shock in the forward direction $\theta_{||}$. |
Note that i4 osine. | Note that $\theta_{||} \ne \theta_0 {\rm sin} i$ . |
This ratio is very large for PSR 2054s bow shock. z3.7+0.3. | This ratio is very large for PSR $-$ 2054's bow shock, $\approx 3.7
\pm 0.3$. |
We have computed this ratio for à number of axisymmetric. aligned wind bow shock models. | We have computed this ratio for a number of axisymmetric, aligned wind bow shock models. |
Figure 6 shows that the projected shock limb has a ratio nearly independent of / for an isotropic (¢2= 0) wind. but that the ratio can reach the observed value for à αι (c=— 3/2) wind and a pulsar motion nearly in the plane of the sky. | Figure \ref{Shockshape} shows that the projected shock limb has a ratio nearly independent of $i$ for an isotropic $c_2=0$ ) wind, but that the ratio can reach the observed value for a ${\rm sin}^2\theta$ $c_2=-3/2$ ) wind and a pulsar motion nearly in the plane of the sky. |
Considering winds that are even more equatorially concentrated or including the flaring effect of the finite pressure in a thick bow shock allows somewhat smaller i to be accommodated. | Considering winds that are even more equatorially concentrated or including the flaring effect of the finite pressure in a thick bow shock allows somewhat smaller $i$ to be accommodated. |
Figure 7. shows the projected shape of an equatorial relativistic wind emanating from the pulsar position (circle) for an inclination /=80°. | Figure \ref{Mod_im} shows the projected shape of an equatorial relativistic wind emanating from the pulsar position (circle) for an inclination $i = 80^\circ$. |
For comparison. the line shows the limb of the isotropic wind bow shock: the standoff distance for the equatorial wind Is appreciably reduced. | For comparison, the line shows the limb of the isotropic wind bow shock; the standoff distance for the equatorial wind is appreciably reduced. |
Note that. since formallyg pyXπα00. the bow shock approaches the star in. the forward (7=0) direction. | Note that, since formally $p_w \propto {\rm sin}^2\theta$, the bow shock approaches the star in the forward $\theta=0$ ) direction. |
Since the pulsar wind likely has a jet component. a physical bow shock would be smooth at the apex. | Since the pulsar wind likely has a jet component, a physical bow shock would be smooth at the apex. |
Similar matching to the Ha limb was used by Gaensleretal.(2002) to argue that the wind of the millisecond pulsar PSR 3358 is anisotropic. | Similar matching to the $\alpha$ limb was used by \citet{get02} to argue that the wind of the millisecond pulsar PSR $-$ 3358 is anisotropic. |
For PSR 2054 we have additionally been able to connect the shock shape with anisotropy detected in the X-ray synchrotron nebula (32.2). | For PSR $-$ 2054 we have additionally been able to connect the shock shape with anisotropy detected in the X-ray synchrotron nebula 2.2). |
We next check if our model can explain the features of Figure 5.. | We next check if our model can explain the features of Figure \ref{SlitSpec}. |
Most remarkable is the dominance of blue-shifted emission at and in front of the pulsar. | Most remarkable is the dominance of blue-shifted emission at and in front of the pulsar. |
For the PA=50” slit. the bright knot at the apex is ate, =3lto Llkims | For the $50^\circ$ slit, the bright knot at the apex is at $v_r = -34$ to $-44{\rm \, km\, s}^{-1}$. |
‘The cross-axis slithas Ho offsetto e,=29to [Ims|. with no equivalent red-shifted component. | The cross-axis slit has $\alpha$ offset to $v_r = -29$ to $-44{\rm \, km\,s}^{-1}$, with no equivalent red-shifted component. |
This is significantly offset from the systemic (ISM) velocity. | This is significantly offset from the systemic (ISM) velocity. |
The lack of any red-shifted emission is at first surprising. given that mass flows around the bow shock. | The lack of any red-shifted emission is at first surprising, given that mass flows around the bow shock. |
However. if the pulsar is moving out of the plane of the sky (/«907). we see that the prompt post-shock broad-line emission (from neutral atom charge exchange) will show negative radial velocities from both sides of the nebula. | However, if the pulsar is moving out of the plane of the sky $i < 90^\circ$ ), we see that the prompt post-shock broad-line emission (from neutral atom charge exchange) will show negative radial velocities from both sides of the nebula. |
This ts illustrated 1n the top two panels of Figure 8.. which trace the velocity structure of the prompt emission. | This is illustrated in the top two panels of Figure \ref{Mod_spec}, which trace the velocity structure of the prompt emission. |
Here we plot a model for /=70" to better separate the velocity components. | Here we plot a model for $i=70^\circ$ to better separate the velocity components. |
The /=75NO" models (for a 3D space velocity c,2:150+50]ans ‘) that match the shock shape also provide the best match to the velocity shifts in the slit spectra. | The $i=75-80^\circ$ models (for a 3D space velocity $v_{\rm psr} \approx 150\pm 50{\rm \, km\,s}^{-1}$ ) that match the shock shape also provide the best match to the velocity shifts in the slit spectra. |
The immediate post-shock layers can dominate if the neutrals penetrate only partly into the shocked ISM. before suffering nearly complete ionization. | The immediate post-shock layers can dominate if the neutrals penetrate only partly into the shocked ISM, before suffering nearly complete ionization. |
This is equivalent to Case C of Bueciantini&Bandiera(2001). | This is equivalent to Case C of \citet{bb01}. |
. In fact PSR 2054 has a relatively large spindown luminosity among pulsars showing ISM Ha shocks (although below that of PSR BO740-28). so if the density and ionization fraction of the upstream medium are high. tonization may indeed be strong in the shocked ISM. | In fact PSR $-$ 2054 has a relatively large spindown luminosity among pulsars showing ISM $\alpha$ shocks (although below that of PSR B0740-28), so if the density and ionization fraction of the upstream medium are high, ionization may indeed be strong in the shocked ISM. |
Thus. ahead of the pulsar where the shock is nearly normal. neutral H may only exist in the immediate post-shock layer and blue shifted emission would dominate the observed spectrum. | Thus, ahead of the pulsar where the shock is nearly normal, neutral H may only exist in the immediate post-shock layer and blue shifted emission would dominate the observed spectrum. |
The pulsar space velocity c, is modest and as one moves behind the pulsar position. the shock obliquity jj increases sharply and the post-shock heating drops. | The pulsar space velocity $v_{\rm psr}$ is modest and as one moves behind the pulsar position, the shock obliquity $\eta$ increases sharply and the post-shock heating drops. |
As ionization drops. neutral H may reach the bulk flow along the contact discontinuity (Bucciantini Bandiera 2001 Case B). | As ionization drops, neutral H may reach the bulk flow along the contact discontinuity (Bucciantini Bandiera 2001 Case B). |
Charge exchange will continue and thus we expect components with negative radial velocity (near side) and positive radial velocity (far side). | Charge exchange will continue and thus we expect components with negative radial velocity (near side) and positive radial velocity (far side). |
Indeed in Figure 5.. across the body of the shell. the PA=50° spectrum shows two components with velocity extrema 125launsο and |25kims+ from the front and back sides of the flow. | Indeed in Figure \ref{SlitSpec}, across the body of the shell, the $50^\circ$ spectrum shows two components with velocity extrema $-125{\rm \, km\,s}^{-1}$ and $+25{\rm \, km\,s}^{-1}$ from the front and back sides of the flow. |
Figure 8 shows the computed radial velocity for this “mixed component’. | Figure \ref{Mod_spec} shows the computed radial velocity for this `mixed component'. |
The large positive and negative velocities expected at the pulsar position are not seen in the PA=1107 spectrum or in the PA=50° spectrum in front of the pulsar. suggesting that near the apex neutral H atoms do not penetrate to this ‘mixed’ tangential portion of the flow. | The large positive and negative velocities expected at the pulsar position are not seen in the $140^\circ$ spectrum or in the $50^\circ$ spectrum in front of the pulsar, suggesting that near the apex neutral H atoms do not penetrate to this `mixed' tangential portion of the flow. |
The lower right panel shows a heuristic merged model. where the ‘mixed’ component grows as @ increases downstream. | The lower right panel shows a heuristic merged model, where the `mixed' component grows as $\theta$ increases downstream. |
This smoothed model bears reasonable similarity to. the observed 2D spectrum. although it fails to reproduce the ‘closed off shape alongthe symmetry axis and the red- "mixed emission from the body of the nebula is less prominent in the data. | This smoothed model bears reasonable similarity to the observed 2D spectrum, although it fails to reproduce the `closed off' shape alongthe symmetry axis and the red-shifted `mixed' emission from the body of the nebula is less prominent in the data. |
Clearly additional effects are needed to reproduce the full velocity structure of this bow shock and additional imaging and spectroscopy will be needed to fully | Clearly additional effects are needed to reproduce the full velocity structure of this bow shock and additional imaging and spectroscopy will be needed to fully |
These equations of motion form asvstem of linear. nonhomosgenous differential equations. whose solution is a linear combination of a homogeneous and a particular solution. | These equations of motion form a system of linear, nonhomogenous differential equations, whose solution is a linear combination of a homogeneous and a particular solution. |
The homogeneous solution can be found by inspection. giving: where ¢ and es are constant coellicients. | The homogeneous solution can be found by inspection, giving: where $c_1$ and $c_2$ are constant coefficients. |
We use the method of variation of parameters io fined the particular. solution. | We use the method of variation of parameters to find the particular solution. |
ÀAecordinglv. we replace the constants e; and e» in the homogeneous solution with Dunctions lV) and D(/). to seek the particular solution of the form substituting this into the equations of motion we now have: therefore Equatious (17)) aud (18)) do not have a simple closed-form solution. but their solution can be expressed in terms of Fresnel integrals (Zwillinger1996).. | Accordingly, we replace the constants $c_1$ and $c_2$ in the homogeneous solution with functions $A(t)$ and $B(t)$, to seek the particular solution of the form Substituting this into the equations of motion we now have: therefore Equations \ref{e:dotA}) ) and \ref{e:dotB}) ) do not have a simple closed-form solution, but their solution can be expressed in terms of Fresnel integrals \citep{Zwillinger:1996CRC}. |
The Fresnel integrals are defined as follows: and have the following properties: | The Fresnel integrals are defined as follows: and have the following properties: |
of orbits/projection/age can certainly reproduce the properties of VirogHI21, and a galaxy flying away at ~1000 km s! during ~1 Gyr can be at a projected distance of 1 Mpc today, possibly even at the center of the Virgo Cluster. | of orbits/projection/age can certainly reproduce the properties of VirogHI21, and a galaxy flying away at $\sim 1000$ km $^{-1}$ during $\sim 1$ Gyr can be at a projected distance of 1 Mpc today, possibly even at the center of the Virgo Cluster. |
The number of massive interloper candidates is then large, making hard to identify the real culprit. | The number of massive interloper candidates is then large, making hard to identify the real culprit. |
This is anyway not required for our demonstration that VirgoHI21 can be a tidal debris, since we have shown that possible interlopers do exist. | This is anyway not required for our demonstration that VirgoHI21 can be a tidal debris, since we have shown that possible interlopers do exist. |
The presence of a strong velocity gradient in a tidal tail, if not due to streaming motions, may actually pinpoint the presence of a gravitationally bound object that need not be a pre-existing dark-matter dominated galaxy. | The presence of a strong velocity gradient in a tidal tail, if not due to streaming motions, may actually pinpoint the presence of a gravitationally bound object that need not be a pre-existing dark-matter dominated galaxy. |
Massive substructures in tidal tails often become kinematically decoupled, self-gravitating and form new stars, becoming rotating Tidal Dwarf Galaxies. | Massive substructures in tidal tails often become kinematically decoupled, self-gravitating and form new stars, becoming rotating Tidal Dwarf Galaxies. |
This is the case for VCC 2062, a TDG candidate in Virgo (Ducetal. 2007b). | This is the case for VCC 2062, a TDG candidate in Virgo \citep{Duc07b}. |
. VirgoHI21 appears as a gas condensation within a tidal tail; it is currently not a TDG since it is starless, but could be its gaseous progenitor. | VirgoHI21 appears as a gas condensation within a tidal tail; it is currently not a TDG since it is starless, but could be its gaseous progenitor. |
Whether stars will be formed in this structure later is questionable. | Whether stars will be formed in this structure later is questionable. |
If the surface column density has remained unusually very low during the first several hundreds of Myr after the formation of VirgoHI21, it is unlikely that star-formation is ignited later on. | If the surface column density has remained unusually very low during the first several hundreds of Myr after the formation of VirgoHI21, it is unlikely that star-formation is ignited later on. |
On the other hand, the dynamical collapse time of the cloud, 1/.,/Gp (Elmegreen2002),, is as large as 400 — 500 Myr for an initially resting system and an average volume mass density in our modeled cloud of p~107° Mo pc-? (this is also about the density of the real VirgoHI21 cloud assuming a vertical scale-height of ~ 300 | On the other hand, the dynamical collapse time of the cloud, $1/\sqrt{G \rho}$ \citep{elmegreen02}, is as large as 400 – 500 Myr for an initially resting system and an average volume mass density in our modeled cloud of $\rho \sim 10^{-3}$ $_{\sun}$ $^{-3}$ (this is also about the density of the real VirgoHI21 cloud assuming a vertical scale-height of $\sim$ 300 pc). |
Comparing this time scale to the age of the cloud, pc).about 500 Myr (it appears in the model at t=200—300 Myr), one may conclude that VirgoHI21 would barely have had the time to collapse and form stars even in the most favorable conditions and incidentally that the absence of stars today is not dependent on an arbitrary choice of the threshold. | Comparing this time scale to the age of the cloud, about 500 Myr (it appears in the model at $t=200-300$ Myr), one may conclude that VirgoHI21 would barely have had the time to collapse and form stars even in the most favorable conditions and incidentally that the absence of stars today is not dependent on an arbitrary choice of the threshold. |
In other words, the system could still be contracting under the effect of its internal gravity today, and begin to form stars later-onif its density comes to exceed the star formation threshold. | In other words, the system could still be contracting under the effect of its internal gravity today, and begin to form stars later-on its density comes to exceed the star formation threshold. |
However, the main argument against VirgoHI21 beingyet a TDG fully responsible for the observed velocity gradient is the large dynamical mass inferred from the rotation curve: in galaxies made out of collisional debris, the dynamical mass should be of the same order as the luminous one, even if the presence of dark baryons may cause some differences between them (Bournaudetal.2007). | However, the main argument against VirgoHI21 being a TDG fully responsible for the observed velocity gradient is the large dynamical mass inferred from the rotation curve: in galaxies made out of collisional debris, the dynamical mass should be of the same order as the luminous one, even if the presence of dark baryons may cause some differences between them \citep{B07}. |
. In the case of VirgoHI21, the dynamical mass inferred from the velocity curve is more than a factor of 3000 greater than the luminous one, i.e. that of the HI component. | In the case of VirgoHI21, the dynamical mass inferred from the velocity curve is more than a factor of 3000 greater than the luminous one, i.e. that of the HI component. |
Clearly, streaming motions provide a much more reasonable explanation for the kinematical feature observed near VirgoHI21, if indeed this object is of tidal origin. | Clearly, streaming motions provide a much more reasonable explanation for the kinematical feature observed near VirgoHI21, if indeed this object is of tidal origin. |
In the group environment, Bekkietal. proposed a scenario in which the group tidal(2005b) field is able to strip gas from Hl-rich galaxies, explaining the presence of isolated intergalactic HI clouds. | In the group environment, \cite{bekki05b} proposed a scenario in which the group tidal field is able to strip gas from HI-rich galaxies, explaining the presence of isolated intergalactic HI clouds. |
Following this idea, B05 presented a model in which the combined action of galaxy-galaxy interactions and the cluster tidal field produce debris with properties similar to Dark Galaxies. | Following this idea, B05 presented a model in which the combined action of galaxy-galaxy interactions and the cluster tidal field produce debris with properties similar to Dark Galaxies. |
Haynesetal.(2007) even suggested that VirgoHI21 and the whole HI structure would result from just the long-term harassment by the large-scale cluster potential. | \citet{haynes07} even suggested that VirgoHI21 and the whole HI structure would result from just the long-term harassment by the large-scale cluster potential. |
However, the tidal field exerted by a structure of mass M and typical scale R scales as M/I?. | However, the tidal field exerted by a structure of mass $M$ and typical scale $R$ scales as $M/R^3$. |
The tidal field of the Virgo Cluster (10? Mo, 1 Mpc) at the present distance of NGC 4254 is then more than ten times smaller than that of the interloper galaxy in our interaction model (2x1013 Mo, 50 kpc). | The tidal field of the Virgo Cluster $10^{15}$ $_\sun$, 1 Mpc) at the present distance of NGC 4254 is then more than ten times smaller than that of the interloper galaxy in our interaction model $2\times 10^{12}$ $_\sun$, 50 kpc). |
It is just unlikely that the cluster field can develop a tail as long as the galaxy interaction can do. | It is just unlikely that the cluster field can develop a tail as long as the galaxy interaction can do. |
The harassment process has a longer timescale than the galaxy pair interaction, but over long timescales the orientation changes, which hardly accounts for the single, thin and long tail around NGC 4254. | The harassment process has a longer timescale than the galaxy pair interaction, but over long timescales the orientation changes, which hardly accounts for the single, thin and long tail around NGC 4254. |
This structure is more typical of a short and violent interaction like a close galaxy encounter than a weaker and longer process like the harassment by the global cluster field. | This structure is more typical of a short and violent interaction like a close galaxy encounter than a weaker and longer process like the harassment by the global cluster field. |
Ram pressure exerted by the cluster hot gas may also expulse gas from the outer regions of spiral disks and create isolated HI clouds without any optical counterpart. | Ram pressure exerted by the cluster hot gas may also expulse gas from the outer regions of spiral disks and create isolated HI clouds without any optical counterpart. |
Yet, this scenario suffers fundamental concerns, also pointed out by M07, in particular: | Yet, this scenario suffers fundamental concerns, also pointed out by M07, in particular: |
the stress prescriptions which sield large surface deusity (a=0.0L. mean aud beta disks) show very simular behavior. | the stress prescriptions which yield large surface density $\alpha=0.01$, mean and beta disks) show very similar behavior. |
Onlv the alpha disk with a=0.1 gives sienificautly higher temperatures at high huuinosities. | Only the alpha disk with $\alpha=0.1$ gives significantly higher temperatures at high luminosities. |
However. at low luminosities. all the stress prescriptions give a fanlv good T! relation. showing clearly that the result of a substantial iucrease in color temperature at low luuinositics is indeed an artifact. and isnot representative of a eas pressure dominated beta diskimocdol). | However, at low luminosities, all the stress prescriptions give a fairly good $T^4$ relation, showing clearly that the result of a substantial increase in color temperature at low luminosities is indeed an artifact, and is representative of a gas pressure dominated beta disk. |
. The huninosity iu these plots is derivedasini the £L.TP plots of GDOL and DCTOT by correcting the disk flux and temperature for inclination angle aud relativistic effects. | The luminosity in these plots is derivedas in the $L-T$ plots of GD04 and DGK07 by correcting the disk flux and temperature for inclination angle and relativistic effects. |
The general relativistic corrections are taken from for a Sclavarzscluld black hole as this is closer to an a.=0.5 disk than the alternative tabulation for e.=0.998. | The general relativistic corrections are taken from for a Schwarzschild black hole as this is closer to an $a_*=0.5$ disk than the alternative tabulation for $a_*=0.998$. |
Tlowever. these have very little effect at 607. as this is where the Doppler blueshift and eravitational redshifts approximately cancel. thus the flux to huninosity conversion is very close to a simple disk area correction of L=2xD?F/cosiInD°F tor cos?=0.5. and the temperature correction is negligible. | However, these have very little effect at $60^\circ$, as this is where the Doppler blueshift and gravitational redshifts approximately cancel, thus the flux to luminosity conversion is very close to a simple disk area correction of $L=2\pi D^2 F/\cos i=4\pi
D^2 F$ for $\cos i =0.5$, and the temperature correction is negligible. |
However. the recovered flix is less than the input value (e.c. for the highest huninosity poiuts made frou 0.2. while the deuse disks give 0.26) as Hanh darkening is present in the radiation transfer. | However, the recovered flux is less than the input value (e.g. for the highest luminosity points made from $\log l=-0.2$ , while the dense disks give $-0.26$ ) as limb darkening is present in the radiation transfer. |
The solid lines show the predicted £ Z rolation for a constant color temperature of 1.6. 1.8. 2.0 and 2.2 again as iu GDOL but scaling the disk area to that expected from ana.=0.5 black hole. | The solid lines show the predicted $L-T$ relation for a constant color temperature of 1.6, 1.8, 2.0 and 2.2 again as in GD04, but scaling the disk area to that expected from an $a_*=0.5$ black hole. |
Clearly the CCD data are very close to a coustant value of foolL.7. while the proportional counter bandpass gives a small change in foo, from 1.8-2.0 for the dense disks. aud 1.8-2.2 for the ipha disk with a=0.1. | Clearly the CCD data are very close to a constant value of $f_{\rm col}=1.7$, while the proportional counter bandpass gives a small change in $f_{\rm col}$ from 1.8-2.0 for the dense disks, and 1.8-2.2 for the alpha disk with $\alpha=0.1$. |
The top paucls of Fig. | The top panels of Fig. |
2. show the effective optical depth of the disk (πμ=VEULa| Tan) as a function of radius for log?=1 aud —0.2 respectively. | \ref{f:l-1} show the effective optical depth of the disk $\tau_{\rm eff} \equiv \sqrt{ \tau_{\rm abs}(\tau_{\rm
es}+\tau_{\rm abs})}$ ) as a function of radius for $\log l=-1$ and $-0.2$ respectively. |
For colparison we also plot the surface density (iuiddle) and effective temperature (bottoni) for the same models. | For comparison we also plot the surface density (middle) and effective temperature (bottom) for the same models. |
For alpha disks the minima in Τομ closely corresponds to the miuinuunn iu X. | For alpha disks the minimum in $\tau_{\rm eff}$ closely corresponds to the minimum in $\Sigma$. |
Furthermore. these nuüninia are close to the radius of maxiumi Toy. so the majority of the flux will be produced in the regions of the disk which have the lowest opacity and hence the highest color temperature. | Furthermore, these minima are close to the radius of maximum $T_{\rm eff}$, so the majority of the flux will be produced in the regions of the disk which have the lowest opacity and hence the highest color temperature. |
Comparison of the top and micelle xuels show that τομ Is verv similar iu shape to X. | Comparison of the top and middle panels show that $\tau_{\rm eff}$ is very similar in shape to $\Sigma$. |
For a eiven alpha. the beta disk always vields a larger X (aud. herefore. a larger r4) than the alpha disk. with the nean disk midway betweenthem. | For a given alpha, the beta disk always yields a larger $\Sigma$ (and, therefore, a larger $\tau_{\rm eff}$ ) than the alpha disk, with the mean disk midway betweenthem. |
Due to the differeut scalines with J (see Eqs. | Due to the different scalings with $l$ (see Eqs. |
Lo and 53). the models become nore diserepaut as Iuninosityv increases. | \ref{eq:siggas} and \ref{eq:sigrad}) ), the models become more discrepant as luminosity increases. |
The alpha disk with a=0.1 (red. solid curve in Fig. 2)) | The alpha disk with $\alpha=0.1$ (red, solid curve in Fig. \ref{f:l-1}) ) |
always has the owest effective optical depth. aud so gives the largest color-teiiperature correction. | always has the lowest effective optical depth, and so gives the largest color-temperature correction. |
However. the effect of this is small at low ldhunünosities (Fig. 1)) | However, the effect of this is small at low luminosities (Fig. \ref{f:lt}) ) |
as there is still substantial absorption opacity (Fie. | as there is still substantial absorption opacity (Fig. |
2aa) aud the heating mainly occurs at hieh effective optical depth. | \ref{f:l-1}a a) and the heating mainly occurs at high effective optical depth. |
Thus. the photosphere acts simply as an atinosphiere. | Thus, the photosphere acts simply as an atmosphere. |
By contrast. at the highest Iuuinosities. the alpha disk with a=0.1 approaches rag1. | By contrast, at the highest luminosities, the alpha disk with $\alpha=0.1$ approaches $\tau_{\rm eff}=1$. |
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