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We can also calculate the run of the optical depthMEN which. shows that 7 decreases as sC/7.34) | We can also calculate the run of the optical depth, which shows that $\tau$ decreases as $r^{-3/2}$. |
Thisqu. result conibined with equation (58)) also demonstrates that as r Micreases the relative contribution of the iuternal dissipation to the midplane teniperature rapidly goes down- sinceJuvenit dois proportional⋅⋅⋅. to -02τοςXO0?OQ", | This result combined with equation \ref{eq:T_m}) ) also demonstrates that as $r$ increases the relative contribution of the internal dissipation to the midplane temperature rapidly goes down sinceit is proportional to $\tau\Omega^2\propto\Omega^3$. |
Optically thin ivracliated disk can cmeree as an extension of an optically thin selt-Iuninous disk. which happens when Z eiven by equation (18)) becomes equal to Zy. or at∙ | Optically thin irradiated disk can emerge as an extension of an optically thin self-luminous disk, which happens when $T$ given by equation \ref{eq:T_thin}) ) becomes equal to $T_0$, or at. |
Such an optically thin transition is possible when au Inequality opposite to (55)) is satisfied. | Such an optically thin transition is possible when an inequality opposite to \ref{eq:tran_thick}) ) is satisfied. |
An irradiated optically thin disk can also appear as a continuation of the ivradiated optically thick disk considered in §3.2.. | An irradiated optically thin disk can also appear as a continuation of the irradiated optically thick disk considered in \ref{subsect:thin}. |
According to equation (72)) this 7=1 transition occus at Optically thin disk is roughly isothermal vertically and its thermal balance requiresfracMO?ro. which is different from equation (58)) by a factor rE in the second term on the right-hand side accounting for the inetiicicney of radiative cooling aud absorption iu the optically thin disk. | According to equation \ref{eq:opt}) ) this $\tau=1$ transition occurs at Optically thin disk is roughly isothermal vertically and its thermal balance requires, which is different from equation \ref{eq:T_m}) ) by a factor $\tau^{-1}$ in the second term on the right-hand side accounting for the inefficiency of radiative cooling and absorption in the optically thin disk. |
There are two obvious regimes to consider. | There are two obvious regimes to consider. |
First. whenMfr. radiationdoes uot affect disk propertics and we eo back to the case studied in §3.2.. | First, when, irradiationdoes not affect disk properties and we go back to the case studied in \ref{subsect:thin}. |
Second. when the condition opposite to (78)) is satisfied irradiation sets the disk temperature. | Second, when the condition opposite to \ref{eq:case4}) ) is satisfied irradiation sets the disk temperature. |
Iu this case all results (except for the equation [10 different from |58]]) obtained in the optically thick iradiated case equations (68))-(72)) relmain valid since in deriving them we did not use any assuniptious about the value of 7. | In this case all results (except for the equation \ref{eq:th_bal}] ] different from \ref{eq:T_m}] ]) obtained in the optically thick irradiated case – equations \ref{eq:surf}) \ref{eq:opt}) ) – remain valid since in deriving them we did not use any assumptions about the value of $\tau$. |
The most important result resardiug externally radiated constant Af disks is that they can remain eravitoturbuleut independent of their optical depth aud that their effective viscosity 06;; remadus constant. | The most important result regarding externally irradiated constant $\dot M$ disks is that they can remain gravitoturbulent independent of their optical depth and that their effective viscosity $\alpha_{GI}$ remains constant. |
If the backgrouud viscosity does not dominate angular mionmentun transport at least in some sclf-IJuninous parts of the eravitoturbuleut disk it will uot dominate the transport in the iradiated part either since 665 18 constant there. | If the background viscosity does not dominate angular momentum transport at least in some self-luminous parts of the gravitoturbulent disk it will not dominate the transport in the irradiated part either since $\alpha_{GI}$ is constant there. |
Thus. torque needed for trausportiug mass through the disk must be due to the CT. | Thus, torque needed for transporting mass through the disk must be due to the GI. |
When fj, becomes comparable to Q+ disk can no longer sustain the eravitoturbulence and las to fragment iuto bound objects (Camunic 2001). | When $t_{cool}$ becomes comparable to $\Omega^{-1}$ disk can no longer sustain the gravitoturbulence and has to fragment into bound objects (Gammie 2001). |
As mentionedbefore fragmentation condition Of,,,;&1 can be recast in terms of the ec; threshold according to equation (21)). | As mentionedbefore fragmentation condition $\Omega t_{cool}\lesssim 1$ can be recast in terms of the $\alpha_{GI}$ threshold according to equation \ref{eq:frag}) ). |
This formmlation now allows us to directly apply our results for oc; derived iu previous sections. | This formulation now allows us to directly apply our results for $\alpha_{GI}$ derived in previous sections. |
In particular. scltbluniuous optically thück eravitoturbuleut disk starts fragmoentiug at which follows from demanding oc; given by equation (11)) to be larger than x. the critical value of à needed or fragiuentation. | In particular, self-luminous optically thick gravitoturbulent disk starts fragmenting at , which follows from demanding $\alpha_{GI}$ given by equation \ref{eq:alpha_thick}) ) to be larger than $\chi$ – the critical value of $\alpha$ needed for fragmentation. |
Radius rye at which fragmentation first ους is given bv rpg—rpH20/75 and is rather insensitive to either (j| or i. so that fragmentatiou always occur not too far from ry. | Radius $r_{frag}$ at which fragmentation first occurs is given by $r_{frag}=r_f \dot m^{-4(2-\beta)/27}$ and is rather insensitive to either $\beta$ or $\dot m$, so that fragmentation always occur not too far from $r_f$. |
It is rather interestius hat for 9= 2. corresponding to the low temperature dust opacity the location of fragmentation edee in the optically thick limit is completely independent of m: ragnentation occurs exactlyat Q=O,. | It is rather interesting that for $\beta=2$ , corresponding to the low temperature dust opacity the location of fragmentation edge in the optically thick limit is completely independent of $\dot m$: fragmentation occurs exactlyat $\Omega=\Omega_f$. |
This fact has con. first noticed by Matzuer Levin (2005). | This fact has been first noticed by Matzner Levin (2005). |
Clearly. constant AY self-Iunünous gravitoturbuleut disk caunot e fed by a source located outside | Clearly, constant $\dot M$ self-luminous gravitoturbulent disk cannot be fed by a source located outside $r_{frag}$. |
According to equation (11)) efrag Γεν.corresponds to the optically thick part of the disk only if a21. | According to equation \ref{eq:opt_tran}) ) $\omega_{frag}$ corresponds to the optically thick part of the disk only if $\dot m >1$. |
Thus. whenever niz1 the exavitoturbulent disk stays optically thick all the wav to the fragmentation edge located inside of ry. | Thus, whenever $\dot m>1$ the gravitoturbulent disk stays optically thick all the way to the fragmentation edge located inside of $r_f$. |
Then external disk feeding must necessarily occu interior to ry. | Then external disk feeding must necessarily occur interior to $r_f$ . |
Iu the optically thin case one finds frou equation (51)) that | In the optically thin case one finds from equation \ref{eq:alpha_thin}) ) that . |
Fragiieutation boundary lies in theoptically thin part of the disk onlyif <1. in which case it is located outside of the fiducial radius ry. | Fragmentation boundary lies in theoptically thin part of the disk onlyif $\dot m<1$ , in which case it is located outside of the fiducial radius $r_f$ . |
Fragmentation radius Fieay=ag(2|61/9 is a rather sensitive function of | Fragmentation radius $r_{frag}=r_f \dot m^{-(2\beta+6)/9}$ is a rather sensitive function of |
prescription does uot allow the precession frequecy to depend on other quantities. | prescription does not allow the precession frequecy to depend on other quantities. |
Ilowever. appropriate matching can be carried out for a particular case mucder consideration. | However, appropriate matching can be carried out for a particular case under consideration. |
The equations of motion are derived frou: dE;dt=nlOX;(nyt|D)óg. dhj;fdt=OIl/ON—Oll'füzΤομ. dA;dt=n;|n;(OE;OII'füh;. didt=OLfüh;. with I//—Ifl»|IT;,;. These can be obtained from Tamiltows equations (eg. | The equations of motion are derived from: ${dE_i}/{dt} =-n_i\partial H' / \partial \lambda_i -(n_1 T +D)\delta_{i1},
$ ${dh_i}/{dt} =-{\partial H' / \partial \lambda_i} -{\partial H' / \partial \varpi_i} -T\delta_{i1},$ $ {d\lambda_i / dt} = n_i +n_i {\partial H' / \partial E_i} + {\partial H' / \partial h_i}, $ ${d\varpi_i / dt} ={\partial H' / \partial h_i}, $ with $H' = H_{12} + H_{int}.$ These can be obtained from Hamilton's equations (eg. |
e Drowwer Clemence 1961) to which we have added. for the outer planet 5. au additional external torque LT with an associated orbital enerev loss rate ny together with additional orbital enerev dissipation rate D. The torque and dissipatiou rate could be produced by tidal interaction with the disc leading to inward migration aud orbital circularization. | Brouwer Clemence 1961) to which we have added, for the outer planet $m_1,$ an additional external torque $- T$ with an associated orbital energy loss rate $n_1 T$ together with additional orbital energy dissipation rate $D.$ The torque and dissipation rate could be produced by tidal interaction with the disc leading to inward migration and orbital circularization. |
We thus obtain to lowest order in the planetary eccenutriciics and perturbing masses. | We thus obtain to lowest order in the planetary eccentricities and perturbing masses. |
— 2n no. a — 5ZH| Hr° dH° When uo migration or circularization occurs (T—D= 0) equilibriu solutions may exist such that c and o are either zero or π. Each of 94.02.64.609 are then constant. | = 2n_1- n_2 - n_1 = 2n_1- n_2 + n_2 When no migration or circularization occurs $T=D=0$ ) equilibrium solutions may exist such that $\psi $ and $\phi$ are either zero or $\pi.$ Each of $n_1,n_2,e_1,e_2$ are then constant. |
A relation between the ecceutricities then follows fron (8)) aud (9)) in the form This condition matches the precession rates of the orbits of the two plaucts. | A relation between the eccentricities then follows from \ref{last2}) ) and \ref{last}) ) in the form This condition matches the precession rates of the orbits of the two planets. |
Also 254Do. Noting that the ecceutricities are positive. when they are of very simall magnitude. the precessional terms become uceligible aud there is a solution with ο=Q.0=7 ort =OQ.0=7. In either case we have μμ. Mae he... For larecr ecceutricities the precessioual terius way become important in (10)) aud then solutious with o=Qo—0. may occur. | Also $2n_1=n_2.$ Noting that the eccentricities are positive, when they are of very small magnitude, the precessional terms become negligible and there is a solution with $\psi=0, \phi =\pi$ or $\psi=0, \phi =\pi.$ In either case we have m_2 a_1 e_2 n_1 B = m_1 a_2 e_1n_2 C For larger eccentricities the precessional terms may become important in \ref{eccp}) ) and then solutions with $\psi=0, \phi =0,$ may occur. |
Then (10)) gives Hye =Cpe. Mi, For stable solutions. when perturbed. the augles maynudergo librations about them equilibrium points (ce. | Then \ref{eccp}) ) gives m_2 a_1 e_2 n_1 B + m_1 a_2 e_1n_2 C= e_1 e_2 a_1 M_* ( For stable solutions, when perturbed, the angles mayundergo librations about their equilibrium points (eg. |
Sinclair 1975). | Sinclair 1975). |
There are two frequencies of oscillation μη.ο. beige eiven for anv c.o im the Lit of small ecceutricities by vi=(monBy?/(M.e4). aud v3=GnynadaP,flayAL.es)?. We look for a solutions with nueration which are close to stable solutious of this type. | There are two frequencies of oscillation $\nu_1,\nu_2$ being given for any $\psi, \phi $ in the limit of small eccentricities by $\nu_1^2 = (m_2 n_1 B)^2/(M_*e_1)^2,$ and $\nu_2^2 =(m_1n_2a_2C)^2/(a_1M_* e_2)^2.$ We look for a solutions with migration which are close to stable solutions of this type. |
We look for solutions of (1—-9)) correspondius to the situation where the two planets muegrate inwards locked in resonance with yf. maintained nearly equal to 1/2 while the eccentricities reluain nearly constant. | We look for solutions of \ref{first}- \ref{last}) ) corresponding to the situation where the two planets migrate inwards locked in resonance with $n_1/n_2$ maintained nearly equal to $1/2$ while the eccentricities remain nearly constant. |
The teudeucv for the resonant coupling to excite the ecceutricifies is counterbalanced bw circularization through the action⋅ of. D—(GMinyet)ftnt.)1» which defines the circularization time for ej. Suuibulv (1)) defines an dward uueration tiuescale f,,;,,=GALanfaq). We beein by supposing that the angle ce executes a libration about zero such that the mean rate of chanec of c» is zero. | The tendency for the resonant coupling to excite the eccentricities is counterbalanced by circularization through the action of $D \equiv (GM_* m_1 e_1^2) /(a_1 t_c)$ which defines the circularization time for $e_1.$ Similarly \ref{first}) ) defines an inward migration timescale $t_{mig}= GM_* m_1/(3 T a_1n_1).$ We begin by supposing that the angle $\psi$ executes a libration about zero such that the mean rate of change of $e_2$ is zero. |
Similarly the mean rates of change of ni aud 2» mduced by ce are zero. | Similarly the mean rates of change of $n_1$ and $n_2$ induced by $\psi$ are zero. |
Such a libration is seen in «αμαος, | Such a libration is seen in simulations. |
We also suppose the angle o either librates or circulates but in such a wav that the correspoudinely iucuced iiean rates of change are not zero. | We also suppose the angle $\phi$ either librates or circulates but in such a way that the correspondingly induced mean rates of change are not zero. |
The simplest exanuple is when the angle executes a very πα or even zero amplitude libration about a value slelthy offset from zero or & (es. | The simplest example is when the angle executes a very small or even zero amplitude libration about a value slightly offset from zero or $\pi$ (eg. |
Lin Papaloizou 1979). | Lin Papaloizou 1979). |
We suppose the circulation/libration periods to be short compared to the timescale of migration so that averaging is possible. | We suppose the circulation/libration periods to be short compared to the timescale of migration so that averaging is possible. |
We denote the average of ieysing by à. Then averaging (1--9)) eives Unt T|D A | We denote the average of $n_1 e_1\sin\varphi $ by $\delta.$ Then averaging \ref{first}- \ref{last}) ) gives (n_1 T ) |
We denote the average of ieysing by à. Then averaging (1--9)) eives Unt T|D AD | We denote the average of $n_1 e_1\sin\varphi $ by $\delta.$ Then averaging \ref{first}- \ref{last}) ) gives (n_1 T ) |
Three new high-: CO systems have been reported receuth. | Three new $z$ CO systems have been reported recently. |
All three detections were made with interferometers (which are more reliable than single dishes for measurement of weak. broac lines). but. all. as of this writing. have vet to be confirmed using another iustrtient or iu another CO transition. | All three detections were made with interferometers (which are more reliable than single dishes for measurement of weak, broad lines), but all, as of this writing, have yet to be confirmed using another instrument or in another CO transition. |
The first is the 2=2.39 radio galaxy S3WO02. with a dtection of CO(32) using he OVRO interferometer reported by Scoville e al (1997). | The first is the $z=2.39$ radio galaxy 53W002, with a detection of CO(3–2) using the OVRO interferometer reported by Scoville et al (1997). |
This object is 1iterestiug because it appears to lie witlin a cluster of roughly 20 Ίσα ciuission liue objects. the most distant such cluster known (Pascarclle et al 1996). | This object is interesting because it appears to lie within a cluster of roughly 20 $\alpha$ emission line objects, the most distant such cluster known (Pascarelle et al 1996). |
Previously. a possibe detection of CO(L0) had been reported frou 53002 by Yamada e al (1995). | Previously, a possible detection of CO(1–0) had been reported from 53W002 by Yamada et al (1995). |
The Scoville et al result is particularly intriguing because of :v claiuxd extension of the CO source by about 3” or 15 kpc. with a veocity eradieut along the major axis. | The Scoville et al result is particularly intriguing because of a claimed extension of the CO source by about $3''$ or 15 kpc, with a velocity gradient along the major axis. |
Such a aree. luminous CO source. if real. would be unprecedented. all previously known ultraluminous CO sources have been confined to the inner kiloparsec or so of the host eaaxv uucleus. | Such a large, luminous CO source, if real, would be unprecedented – all previously known ultraluminous CO sources have been confined to the inner kiloparsec or so of the host galaxy nucleus. |
Iu coutrast o the three confirued CO sources discussed above. 53N002 does not show anv evidence for eraviational lensing effects. nor has it been detected as a dust source in the far-IR or subi coitin. | In contrast to the three confirmed CO sources discussed above, 53W002 does not show any evidence for gravitational lensing effects, nor has it been detected as a dust source in the far-IR or submm continuum. |
The secoud new source is BRIL3350115. a Quasar a DoLU detected in COS1) bv Cuilloteau et al (1997) using the PdDI. | The second new source is BRI1335–0415, a quasar at $z=4.41$ detected in CO(5–4) by Guilloteau et al (1997) using the PdBI. |
This object was previously deteced as a dust source in the continua at 1 nuu: it shows uopriori ονdeuce for lensing. | This object was previously detected as a dust source in the continuum at 1 mm; it shows no evidence for lensing. |
Tle authors state hat. given the uncertainties in the CO-to-M(ID) COMVCLSIO1i factor. the molecular nass iu this system coukl be as hieh as lott AL:. | The authors state that, given the uncertainties in the $_2$ ) conversion factor, the molecular mass in this system could be as high as $10^{11}$ $_{\sun}$. |
This can be compared to the case ¢Xf the Cloverleaf where because of lensing. aud i1e modelling which sueeostsoo a low CO-to-M(LD) COMVCLSIOi factor. the estimated. IT» mass may be as low as afew 10 AL. (Barvainis et al 1997). | This can be compared to the case of the Cloverleaf where because of lensing, and line modelling which suggests a low $_2$ ) conversion factor, the estimated $_2$ mass may be as low as a few $\times 10^9$ $_{\sun}$ (Barvainis et al 1997). |
A more likev value for DRII3350415 is MCIIo) ~ a cw ον1jlo. using the conversion factor appropriate for ultraluni10us IR ealaxies (Solomon ct al 1997) a1 asstning no lensing boost. | A more likely value for BRI1335–0415 is $_2$ ) $\sim$ a few $\times 10^{10}$, using the conversion factor appropriate for ultraluminous IR galaxies (Solomon et al 1997) and assuming no lensing boost. |
The hird new detection of CO at high redshift is frox ALG ΘΕΑ0531. a lensed quasar at z=2.61. | The third new detection of CO at high redshift is from MG 0414+0534, a lensed quasar at $z= 2.64$. |
Iu the centimeter radio coutiuuua i has four conr)011C1ls. With a παπα separation of 2” (Πατ et al 1992). | In the centimeter radio continuum it has four components, with a maximum separation of $2''$ (Hewitt et al 1992). |
We detect¢à the CO(32) ine at 95 Czwith Heh SNR using the PdBI (Barvainis et al 1995). | We detected the CO(3–2) line at 95 GHzwith high SNR using the PdBI (Barvainis et al 1998). |
The line is quite broad. at Διην—hs0 kins ο παcho is about as broad as tιο very broadest lines seen in low redshift IB-hunünous eaaxies. | The line is quite broad, at $\Delta v_{\rm FWHM}
= 580$ km $^{-1}$, which is about as broad as the very broadest lines seen in low redshift IR-luminous galaxies. |
Although the svuthesized beam size of Du27 was comparable to the separation between the A { Al|A2) and D radio iages. if proved possible to produce iudividual CO spectra for these components by fitting in the UV plane. | Although the synthesized beam size of $2''$ was comparable to the separation between the A (= A1+A2) and B radio images, it proved possible to produce individual CO spectra for these components by fitting in the UV plane. |
These spectra. plus hei suu. are shown iu Figure 1. | These spectra, plus their sum, are shown in Figure 1. |
When higher SNR aud higjer resolution Observations become available. 1 mmy )e possible to play the same trick with AICGOLIL051 as Ikucib et al (1997) did with the Cloverleat: deterwining the verv stuallscale kinematic structure of the CO source by deprojection. using a reasonabv oncctrate model for the lensing potential (such as that of Faco. Lelixr. Shapiro 1991). | When higher SNR and higher resolution observations become available, it may be possible to play the same trick with MG0414+0534 as Kneib et al (1997) did with the Cloverleaf: determining the very small-scale kinematic structure of the CO source by deprojection, using a reasonably accurate model for the lensing potential (such as that of Falco, Lehárr, Shapiro 1997). |
Finally. I report here a non-detection (see also Barvaius et al 1998). | Finally, I report here a non-detection (see also Barvainis et al 1998). |
The radio qiict object PC. 1631]706 is an optically huuinous quasar at 2=1.3:?3. with detectious in all four IRAS wavebancds. | The radio quiet object PG 1634+706 is an optically luminous quasar at $z=1.33$, with detections in all four IRAS wavebands. |
It is the most distaut unleused IRAS source. aud amoung the most hninous. | It is the most distant unlensed IRAS source, and among the most luminous. |
Its spectral energy distribution in the TR /subnuuis very similar to those of the Cloverleaf and F102111£721. νο PG 1631]706 seeined like a very eood candidate for CO detections. | Its spectral energy distribution in the IR/submm is very similar to those of the Cloverleaf and F10214+4724, so PG 1634+706 seemed like a very good candidate for CO detections. |
We used the adjusted systemic redshift estimate of Tytler Fan (1992). >= 1.337. for the CO(21) | We used the adjusted systemic redshift estimate of Tytler Fan (1992), $z=1.337$ , for the CO(2–1) |
with the photometric minor axis of (he galaxy. attributed at least partly. to incomplete cluster survevs in these regions. | with the photometric minor axis of the galaxy, attributed at least partly to incomplete cluster surveys in these regions. |
These objects shoukl. therelore. not be dropped [rom the GC catalog without further spectroscopic analvsis. | These objects should, therefore, not be dropped from the GC catalog without further spectroscopic analysis. |
We determine the rotation amplitude and axis of the GCS of NGC 5128 [rom (seeCotéetal.2001:Richtler2004:Woodley2006). | We determine the rotation amplitude and axis of the GCS of NGC 5128 from \citep[see][]{cote01,richtler04,w06}. |
. In Equation 1.. e, is the observed radial velocity of the GCs in the system. c4, is the galaxys svstenic velocity. His the projected radial distance of each GC from the center of the svstem assuming a distance of 3.9 AIpe to NGC 5128. and O is the projected azimuthal angle of the GC measured in degrees east of north. | In Equation \ref{eqn:kin}, $v_r$ is the observed radial velocity of the GCs in the system, $v_{sys}$ is the galaxy's systemic velocity, $R$ is the projected radial distance of each GC from the center of the system assuming a distance of 3.9 Mpc to NGC 5128, and $\Theta$ is the projected azimuthal angle of the GC measured in degrees east of north. |
The systemic velocity of NGC 5128 is held constant at 0.=541 kin ? (thietal.1995) [or all kinematic calculations. | The systemic velocity of NGC 5128 is held constant at $v_{sys}=541$ km $^{-1}$ \citep{hui95}
for all kinematic calculations. |
The rotation axis of the GCs. O,. and the product OR. the rotation amplitude of the GC's in the svstem. are (he values obtained from (he numerical solution. | The rotation axis of the GCs, $\Theta_o$, and the product $\Omega R$, the rotation amplitude of the GCs in the system, are the values obtained from the numerical solution. |
We use a Marquarcdt-Levenbere non-linear fitting routine 1992). | We use a Marquardt-Levenberg non-linear fitting routine \citep{press92}. |
Equ. | Eqn. |
l1 assumes spherical svimmetry. | \ref{eqn:kin} assumes spherical symmetry. |
While (his may. be a decent. assumption for the inner 12 kpe region (ithasalowellipticityof~0.2:Pene.Ford&Freeman2004b).. true elliplicilies for (he outerregions of the svstem are not well known because of the sample bias (see Fig. 3)). | While this may be a decent assumption for the inner 12 kpc region \citep[it has a low ellipticity of
$\sim0.2$;][]{peng04}, true ellipticities for the outerregions of the system are not well known because of the sample bias (see Fig. \ref{fig:gc_thetar}) ). |
Future studies to remove these biases are vital to obtaining a sound kinematic solution for the entire svstem. | Future studies to remove these biases are vital to obtaining a sound kinematic solution for the entire system. |
Equ. | Eqn. |
1 also assumes (hat is only a function of the projected racius and that the rotation axis lies in (hie plane of the sky. | \ref{eqn:kin} also assumes that $\Omega$ is only a function of the projected radius and that the rotation axis lies in the plane of the sky. |
It is not entirely clear how (hese assumptions. discussed thoroughly in Cotéοἱal.(2001).. apply to the GC and PN svstenis ol NGC: 5128. | It is not entirely clear how these assumptions, discussed thoroughly in \cite{cote01}, apply to the GC and PN systems of NGC 5128. |
The O we solve for is. therefore. only a lower limit to the true © if the true rotation axis is not in the plane of the sky. | The $\Omega$ we solve for is, therefore, only a lower limit to the true $\Omega$ if the true rotation axis is not in the plane of the sky. |
The projected velocity dispersion is also calculated [rom the normal condition. where Vis the number of clusters in the sample. vy, is the GC's radial velocity. 1. aud a, is (he projected velocity dispersion. | The projected velocity dispersion is also calculated from the normal condition, where $N$ is the number of clusters in the sample, $v_{f_i}$ is the GC's radial velocity , and $\sigma_v$ is the projected velocity dispersion. |
the derived. properties from the different observations could be compared ancl contrasted. | the derived properties from the different observations could be compared and contrasted. |
Furthermore. (he analvsis presented here will be applicable to future observations of shocks. | Furthermore, the analysis presented here will be applicable to future observations of shocks. |
nights we obtained diatawwasrethl(A) 4500A,, which with the 600B cross-disperser corresponds to a wavelength resolution of ~6.5A.. | nights we obtained data was $\sim$ at $\sim$ 4500, which with the 600B cross-disperser corresponds to a wavelength resolution of $\sim$. |
On the first night (when both chips were working) we covered the 3500-4950 ((chip 2) and 5100-6600 ((chip 1) ranges, and opted for the bluer 3700-5200 rregime after chip 2 failed. | On the first night (when both chips were working) we covered the 3500-4950 (chip 2) and 5100-6600 (chip 1) ranges, and opted for the bluer 3700-5200 regime after chip 2 failed. |
That way, we included the Balmer lines from Ha upwards on night 1, and from Hf upwards on nights 3 and 4 for our target star. | That way, we included the Balmer lines from $\alpha$ upwards on night 1, and from $\beta$ upwards on nights 3 and 4 for our target star. |
We also monitored a F7V comparison star, HD 196286, picked purely for its location ffrom the target) and brightness (V~ 10.1, from Simbad). | We also monitored a F7V comparison star, HD 196286, picked purely for its location from the target) and brightness $V\sim$ 10.1, from Simbad). |
The spectral range covered for the comparison star is shifted ~400 bbluewards with respect to the target star due to its different location on the chip. | The spectral range covered for the comparison star is shifted $\sim$ 400 bluewards with respect to the target star due to its different location on the chip. |
We set the exposure time for each pair of spectra to 25 s, yielding a total integration time of 1025 s (for 41 measurements) on the CCD before each 36 s readout. | We set the exposure time for each pair of spectra to 25 s, yielding a total integration time of 1025 s (for 41 measurements) on the CCD before each 36 s readout. |
This means that only of the time-series is taken up by dead time in between exposures! | This means that only of the time-series is taken up by dead time in between exposures! |
Given that the HIT-MS mode is not supported by the FORS pipeline, we wrote our own IRAF script to batch-process the data. | Given that the HIT-MS mode is not supported by the FORS pipeline, we wrote our own IRAF script to batch-process the data. |
This script essentially treats the 82 (41 each for the target and comparison star) spectra on each image as echelle orders, and writes the reduction products out as individual FITS files with a time-stamp corresponding to the middle of the exposure calculated for each pair of spectra. | This script essentially treats the 82 (41 each for the target and comparison star) spectra on each image as echelle orders, and writes the reduction products out as individual FITS files with a time-stamp corresponding to the middle of the exposure calculated for each pair of spectra. |
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