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The total length of the network at this scale is therefore given by where ΔΝ. is the total number of filament elements found and A is the step size taken by the filament finder. | The total length of the network at this scale is therefore given by where $N_e$ is the total number of filament elements found and $\Delta$ is the step size taken by the filament finder. |
Non-filamentary regions of space have already been excluded by the criteria in Equation 1, so an optimum set of parameters will maximize Ly while minimizing R. | Non-filamentary regions of space have already been excluded by the criteria in Equation \ref{eq:CellRemove}, so an optimum set of parameters will maximize $L_f$ while minimizing $R$. |
In the left panel of Fig. 4,, | In the left panel of Fig. \ref{fig:Crit1Tots}, |
we plot both the fraction of repeat detections (R/N., dashed lines) and the total length of the network (Ly, solid lines) as a function of C. | we plot both the fraction of repeat detections $R/N_e$, dashed lines) and the total length of the network $L_f$, solid lines) as a function of $C$. |
On all smoothing scales, the fraction of false positives rises steadily with increasing C, with no obvious breaks or minima. | On all smoothing scales, the fraction of false positives rises steadily with increasing $C$, with no obvious breaks or minima. |
The total length, however, tends to rise until it reaches a maximum, after which point it either flattens or falls slowly. | The total length, however, tends to rise until it reaches a maximum, after which point it either flattens or falls slowly. |
This suggests that, as long as the curvature criterion is above a critical value, the algorithm will trace out the entire filament network. | This suggests that, as long as the curvature criterion is above a critical value, the algorithm will trace out the entire filament network. |
Since the fraction of false positives rises with C, we will hereafter use a curvature criterion near this value; that is, C—50, 40, and 30? for |—15, 10, and 5Mpc,, respectively. | Since the fraction of false positives rises with $C$ , we will hereafter use a curvature criterion near this value; that is, $C=50$, $40$, and $30$ for $l=15$, $10$, and $5$, respectively. |
As each filament is found,we wish to remove from the grid as much of it as possible without preventing the detection of further real filaments. | As each filament is found,we wish to remove from the grid as much of it as possible without preventing the detection of further real filaments. |
Using the previously-determined critical values of C, we ran the filament-finder with a range of K and computed the total length of the filament network and the fraction of repeat detections as a function of K. | Using the previously-determined critical values of $C$, we ran the filament-finder with a range of $K$ and computed the total length of the filament network and the fraction of repeat detections as a function of $K$. |
The results are shown in the right panel of Fig. 4.. | The results are shown in the right panel of Fig. \ref{fig:Crit1Tots}. . |
All of the curves are monotonic, with repeat detections and the network length decreasing with increasing K. | All of the curves are monotonic, with repeat detections and the network length decreasing with increasing $K$. |
Hereafter, we will set K=1 because it yields R/N,€20 per cent. | Hereafter, we will set $K=1$ because it yields $R/N_e\lesssim 20$ per cent. |
Chemical inhomogeneities in light elements (C. N. O. Ε. Na. Al. Mg. and even S1) are intrinsic to globular clusters (GCs: see Grattonetal.2004. for an extensive review and references). | Chemical inhomogeneities in light elements (C, N, O, F, Na, Al, Mg, and even Si) are intrinsic to globular clusters (GCs; see \citealt{araa04} for an extensive review and references). |
In particular. the striking anticorrelation between Na and O abundances in GC red giant branch (RGB) stars discovered and studied by the Texas-Lick group (see Kraft190 for a review on those pioneering efforts) is the most notable signature observed with high resolution spectroscopy. | In particular, the striking anticorrelation between Na and O abundances in GC red giant branch (RGB) stars discovered and studied by the Texas-Lick group (see \citealt{kraft94} for a review on those pioneering efforts) is the most notable signature observed with high resolution spectroscopy. |
The pivotal discovery of the Na-O (and the Mg-Al) anticorrelation in unevolved cluster stars (Grattonetal..2001) led to the unambiguous conclusion that stars of different generations co-exist in the currently observed GCs. | The pivotal discovery of the Na-O (and the Mg-Al) anticorrelation in unevolved cluster stars \citep{gratton01} led to the unambiguous conclusion that stars of different generations co-exist in the currently observed GCs. |
The reason is that the high temperatures required for proton-capture reactions (Denisenkov&Denisenkova.1989:Langeretal..1993) to produce matter depleted in O. Mg and enriched in Na. AI are never reached in the interior of low-mass stars (temperature in excess of 25 or 70x10? K are required for the NeNa and MgAI cycles. respectively). | The reason is that the high temperatures required for proton-capture reactions \citep{denden89,langer93} to produce matter depleted in O, Mg and enriched in Na, Al are never reached in the interior of low-mass stars (temperature in excess of 25 or $\times10^6$ K are required for the NeNa and MgAl cycles, respectively). |
As a consequence. the observations by Gratton et al. | As a consequence, the observations by Gratton et al. |
(laterconfirmedbye.g..Cohenetal..2002;Car-rettaetal..2004:D'Orazi2010) require the existence of a previous stellar generation of more massive stars with higher internal temperatures sufficient to activate the necessary nucleosynthesis. providing the ejecta from which the second-generation stars formed. | \citep[later confirmed by
e.g.,][]{cohen02,carretta04,dorazi10} require the existence of a previous stellar generation of more massive stars with higher internal temperatures sufficient to activate the necessary nucleosynthesis, providing the ejecta from which the second-generation stars formed. |
Dilution processes with pristine gas left in the cluster may then reproduce the whole length of the Na-O anticorrelation (see Pranztos&Charbonnel20061: see however Gratton&Carretta 2010)). | Dilution processes with pristine gas left in the cluster may then reproduce the whole length of the Na-O anticorrelation (see \citealt{pc06}; see however \citealt{gra010}) ). |
The Na-O anticorrelation is so widespread among GCs (seee.g..Carrettaetal..2009a.b) that this feature is probably associated to the very same mechanism of cluster formation (Carretta.2006) and may be considered the main criterion to discriminate between GCs (Carrettaetal..2010) and other type of clusters. regardless of their old ages or even large mass (e.g.NGC6791.Bragagliaetal..2010b.inprep.). | The Na-O anticorrelation is so widespread among GCs \citep[see e.g.,][]{carretta09a,carretta09b}
that this feature is probably associated to the very same mechanism of cluster formation \citep{carretta06} and may be considered the main criterion to discriminate between GCs \citep{carretta10} and other type of clusters, regardless of their old ages or even large mass \citep[e.g. NGC~6791,][in prep.]{bragaglia6791}. |
. However. while the overall pattern of the chemical composition of first and second-generation GC. stars. is currently. well assessed. several issues are still left open. the principal being the nature of the actual polluters. either intermediate-mass Asymptotic Giant Branch stars (IM-AGBs.Venturaetal..2001) or fast-rotating massive stars (FRMSs.Decressinetal.. 2007). | However, while the overall pattern of the chemical composition of first and second-generation GC stars is currently well assessed, several issues are still left open, the principal being the nature of the actual polluters, either intermediate-mass Asymptotic Giant Branch stars \citep[IM-AGBs,][]{ventura01} or fast-rotating massive stars \citep[FRMSs,][]{decressin07}. |
. One of the main questions concerns the possible link between chemical signature and dynamical evolution of different stellar generations in GCs. | One of the main questions concerns the possible link between chemical signature and dynamical evolution of different stellar generations in GCs. |
This issue is puzzling and still poorly explored in a systematic way by theoretical nodels. | This issue is puzzling and still poorly explored in a systematic way by theoretical models. |
Models where a cooling-flow feeds gas enriched in IM-AGB ejecta to form second generation stars (D'Ercoleetal.. 2008).. intrinsically predict that this generation should be nore centrally concentrated. since the gas is re-collected at the cluster centre. | Models where a cooling-flow feeds gas enriched in IM-AGB ejecta to form second generation stars \citep{dercole08}, intrinsically predict that this generation should be more centrally concentrated, since the gas is re-collected at the cluster centre. |
On the other hand. also second-generation stars formed by matter polluted by FRMS are expected to be nore centrally concentrated. | On the other hand, also second-generation stars formed by matter polluted by FRMS are expected to be more centrally concentrated. |
They should have the same radial distribution of their progenitors. that are assumed to be born hear the cluster centre. being very massive objects. | They should have the same radial distribution of their progenitors, that are assumed to be born near the cluster centre, being very massive objects. |
Even if the second generation formed at the cluster centre. there is the action of the dynamical evolution over a Hubble time to be taken into account. | Even if the second generation formed at the cluster centre, there is the action of the dynamical evolution over a Hubble time to be taken into account. |
For instance. Decressinetal.(2008) compute that the second-generation stars are progressively spread out by dynamical encounters. | For instance, \cite{decressin08} compute that the second-generation stars are progressively spread out by dynamical encounters. |
As a result. the radial distributions of first and second-generation stars can no longer be distinguished from their dynamics alone at the present time. | As a result, the radial distributions of first and second-generation stars can no longer be distinguished from their dynamics alone at the present time. |
In this Note we combine information from our ongomg FLAMES survey of chemical abundances in giants in GCs (seeCarretta.2006) with newly derived photometry for the GC (Kravtsovetal..2009.2010) and we provide new insights on the radial distribution of first and second-generation stars in this GC. whose relaxation time at half-mass radius is about 1.6 Gyr Harris(1996). | In this Note we combine information from our ongoing FLAMES survey of chemical abundances in giants in GCs \cite[see][]{carretta06} with newly derived photometry for the GC \citep{kravtsov09,kravtsov10} and we provide new insights on the radial distribution of first and second-generation stars in this GC, whose relaxation time at half-mass radius is about 1.6 Gyr \cite{har96}. |
. In Carrettaetal.(2009a) we showed that it is possible to separate a stellar component of first-generation (or primordial. P) stars in all GCs observed. | In \cite{carretta09a} we showed that it is possible to separate a stellar component of first-generation (or primordial, P) stars in all GCs observed. |
The remaining second-generation stars can be further separated into intermediate (1) and extreme (E) components. according to the degree of O-depletion and Na-enhancement along the Na-O anticorrelation. | The remaining second-generation stars can be further separated into intermediate (I) and extreme (E) components, according to the degree of O-depletion and Na-enhancement along the Na-O anticorrelation. |
losses have already. strougly depleted the umber of highly relativistic electrous when inverse Compton scattering becomes important. | losses have already strongly depleted the number of highly relativistic electrons when inverse Compton scattering becomes important. |
The assumed shape of the density distribution of the ICM implies that central deusity. p,;. auc core radius. e,. are not independent parameters. | The assumed shape of the density distribution of the IGM implies that central density, $\rho
_o$, and core radius, $a_o$, are not independent parameters. |
The evolutionary tracks ouly depeud on a combination. pa). of the two. | The evolutionary tracks only depend on a combination, $\rho _o
a_o^{\beta}$, of the two. |
This means that the effects of a ‘denser cuviromment are indistinguishable from those of a cmore extended’ euvironiieut aud Figure 5. may also be interpreted in this wav. | This means that the effects of a `denser' environment are indistinguishable from those of a `more extended' environment and Figure \ref{fig:rhoredcom} may also be interpreted in this way. |
The model for the iutrinsic source evolution described iu the previous section depends on source parameters (jet power. aspect ratio of the cocoou. distribution of energies of he relativistic clectrous at injection. source age. redshift) aud parameters cescribing its cuvirouuent (external density parameter pa). power law expoucut 3). | The model for the intrinsic source evolution described in the previous section depends on source parameters (jet power, aspect ratio of the cocoon, distribution of energies of the relativistic electrons at injection, source age, redshift) and parameters describing its environment (external density parameter $\rho _o a_o^{\beta}$, power law exponent $\beta$ ). |
All of these wraleters are cither not directly observable or can ouly be imferred from observations at comparatively low redshift. | All of these parameters are either not directly observable or can only be inferred from observations at comparatively low redshift. |
The "birth function! of ERIT sources. ic. the comoving iuuber density of progenitors of radio galaxies starting to produce powerful jets as a "uction of redshift. is of course also unknown. | The `birth function' of FRII sources, i.e. the comoving number density of progenitors of radio galaxies starting to produce powerful jets as a function of redshift, is of course also unknown. |
In the following we will assume reasonable distribution functious of these source aud euviroument parameters which initially are asstuned to be indepeudent of cach other. | In the following we will assume reasonable distribution functions of these source and environment parameters which initially are assumed to be independent of each other. |
For the birth fiction we assume a power aw of the form (1|2)". | For the birth function we assume a power law of the form $(1+z)^n$. |
Fora eiven cosimology it is then possible with the help of the uodel for the intrinsic radio huninositv-linear size evolution described iu the previous section to calculate a contiuuous distribution function iu the P-D plane. | For a given cosmology it is then possible with the help of the model for the intrinsic radio luminosity-linear size evolution described in the previous section to calculate a continuous distribution function in the P-D plane. |
This is theu conipared with the observed. binned source distribution of the complete. flux Iuuited sample of Laing et al. ( | This is then compared with the observed, binned source distribution of the complete, flux limited sample of Laing et al. ( |
1983) using αν) Τοντ, | 1983) using a $\chi
^2$ -test. |
Using this technique we find that the steepenuing of the evolutionary tracks of FRI sources due to inverse Compton scattering of the CAIBR at large linear sizes alouc is not sufficicut to explain the observed decrease of the median linear size with redshift and/or radio huuinositv. | Using this technique we find that the steepening of the evolutionary tracks of FRII sources due to inverse Compton scattering of the CMBR at large linear sizes alone is not sufficient to explain the observed decrease of the median linear size with redshift and/or radio luminosity. |
More Iuuniuous sources at hieher redshift tend to host more powerful jets and this also implies higher hot spot advance speeds in these sources. | More luminous sources at higher redshift tend to host more powerful jets and this also implies higher hot spot advance speeds in these sources. |
The stecpening of their evolutionary tracks therefore occurs ouly at larger linear sizes: the opposite of what is observed. | The steepening of their evolutionary tracks therefore occurs only at larger linear sizes; the opposite of what is observed. |
Sources in denser enviromuents will not oulv be more Ipuuinous than those 1n more rarefied surroundings but their expiusiou speed will be lower as well. | Sources in denser environments will not only be more luminous than those in more rarefied surroundings but their expansion speed will be lower as well. |
The euviromucuts of sources at high redshift must have decoupled from the Ifubble flow earlier than those of low redshift objects; | The environments of sources at high redshift must have decoupled from the Hubble flow earlier than those of low redshift objects. |
This very simple picture implies that p,x(E|2)? which leads to a shortenine of the mean size of sources at high redshift. | This very simple picture implies that $\rho _o \propto
(1+z)^3$ which leads to a shortening of the mean size of sources at high redshift. |
However. we fiud that this effect is not strong enough. | However, we find that this effect is not strong enough. |
The single power law assumed for the birth fuuction predicts an monotonously Increasing nuniber of sources with increasing redshift. | The single power law assumed for the birth function predicts an monotonously increasing number of sources with increasing redshift. |
For low radio luninositics the Hux limit of the comparison sample iieaus that these sources will not be included in the sample. | For low radio luminosities the flux limit of the comparison sample means that these sources will not be included in the sample. |
For high radio liuinesities. however. we find that this birth fiction predicts nany more racio huuinous sources at high redshift than are observed. | For high radio luminosities, however, we find that this birth function predicts many more radio luminous sources at high redshift than are observed. |
This implies at cast a flattening. if not a turn over. of the radio hunuiuositv function at redshifts of around 2 which is also iudicated by observations (0.8. Dunlop Peacock 1990). | This implies at least a flattening, if not a turn over, of the radio luminosity function at redshifts of around 2 which is also indicated by observations (e.g. Dunlop Peacock 1990). |
We find some evideuce for a population of eiat radio galaxies distinct from tle main »pulatiou by either their exceptionally high age aud/or τον rarefied environnmieuts. | We find some evidence for a population of giant radio galaxies distinct from the main population by either their exceptionally high age and/or very rarefied environments. |
There are three. possibly four. sources m the sample of Laing et al. ( | There are three, possibly four, sources in the sample of Laing et al. ( |
1983) which have ear sizes close to or above 1.5 Mype aud radio luminosities below 107 W bt t which belong to this class. | 1983) which have linear sizes close to or above 1.5 Mpc and radio luminosities below $^{26}$ W $^{-1}$ $^{-1}$ which belong to this class. |
The probability for finding sources in this region of the P-D diaeran is extremely low in any of the models discussed here and their inclusion in he models by allowing for extremely high lite times (>10 years) leads to au excess | The probability for finding sources in this region of the P-D diagram is extremely low in any of the models discussed here and their inclusion in the models by allowing for extremely high life times $> 10^9$ years) leads to an excess |
For each bin. we calculate the structure functions [or all five bands of the quasars in that bin. | For each bin, we calculate the structure functions for all five bands of the quasars in that bin. |
ALL thirty structure functions (six bins times 5 bands) are shown in Figure &.. | All thirty structure functions (six bins times 5 bands) are shown in Figure \ref{Fig3.3}. |
Each structure function demonstrates the familiar relation between wavelength and variability: the & band in each bin shows the largest amplitude in its structure function.while the z-band. measurements show the least variability. | Each structure function demonstrates the familiar relation between wavelength and variability; the $u$ band in each bin shows the largest amplitude in its structure function,while the $z$ -band measurements show the least variability. |
The structure functions shown in Figure S. have only nine points in Ar. rather than the ten seen in Figure 6: the high-redshift nature of these quasars (which is necessary to observe C 1v)) results in the largest rest-frame time lag bin containing no observations. after one translates from the observed. [rame to the quasar’s rest. frame. | The structure functions shown in Figure \ref{Fig3.3} have only nine points in $\Delta{\tau}$, rather than the ten seen in Figure \ref{Fig3.1}; the high-redshift nature of these quasars (which is necessary to observe C ) results in the largest rest-frame time lag bin containing no observations, after one translates from the observed frame to the quasar's rest frame. |
One quickly. notices the large level of uncertainty in virtually all of these 30. structure functions in the filth bin in Ar. which is at approximately 00 days. | One quickly notices the large level of uncertainty in virtually all of these 30 structure functions in the fifth bin in $\Delta{\tau}$, which is at approximately 60 days. |
This is due to the lack of observations separated by 180. clays in the observed frame: this bin spans 150davs/(1|i22). where (22 is the mean redshift at whieh € is observable (ic. z2 2.5). | This is due to the lack of observations separated by 180 days in the observed frame; this bin spans $180\ \mathrm{days}/(1 + \langle z\rangle)$, where $\langle z\rangle$ is the mean redshift at which C is observable (i.e., $z \approx 2.5$ ). |
Additionallv. in certain time-lae bins. a reliable measurement of the variability cannot be mace. as the average uncertainty is greater than the average variability. | Additionally, in certain time-lag bins, a reliable measurement of the variability cannot be made, as the average uncertainty is greater than the average variability. |
Γης is seen most often in v- and z-band structure functions. as those bands have the lowest signal-to-noise [Lux determinations. | This is seen most often in $u$ - and $z$ -band structure functions, as those bands have the lowest signal-to-noise flux determinations. |
ὃν comparing the structure functions of quasars from adjacent bins in Figure 7.. we can isolate the dependences of variability upon luminosity ancl black hole mass. | By comparing the structure functions of quasars from adjacent bins in Figure \ref{Fig3.2}, we can isolate the dependences of variability upon luminosity and black hole mass. |
l'or example. the left-hancl panel of Figure 9 shows the g-band structure functions for the quasars from bins 1. 2 ancl 3. | For example, the left-hand panel of Figure \ref{Fig3.4} shows the $g$ -band structure functions for the quasars from bins 1, 2 and 3. |
Din 1 quasars are clearly more variable than those in bin 2. which are. in turn. more variable that those in bin 3. | Bin 1 quasars are clearly more variable than those in bin 2, which are, in turn, more variable that those in bin 3. |
Table 3 shows the results of the power-law fits to these structure 'unctions (às well as those representing the quasars in bins 4. 5 and 6). | Table 3 shows the results of the power-law fits to these structure functions (as well as those representing the quasars in bins 4, 5 and 6). |
Phe progression from high to low variability. as one ravels from bin 1 to bin 3. seen in Figure 7. is rellected in he values for V(Nr=100) for those bins. | The progression from high to low variability, as one travels from bin 1 to bin 3, seen in Figure \ref{Fig3.2} is reflected in the values for $V(\Delta{\tau}=100)$ for those bins. |
In the right-hand xuiel of Figure 7.. the same relation is observed for quasars at higher black hole mass. | In the right-hand panel of Figure \ref{Fig3.2}, the same relation is observed for quasars at higher black hole mass. |
Quasars in bin 4 are of lower uminositv than those in bin 5. and are also more variable. | Quasars in bin 4 are of lower luminosity than those in bin 5, and are also more variable. |
These results are not surprising. in that an anticorrclation between luminosity and variability has en. known for decades. | These results are not surprising, in that an anticorrelation between luminosity and variability has been known for decades. |
However. this shows. for the first ime. that this dependence exists independent of black hole mass. à property known to be correlated with Iuminositv. | However, this shows, for the first time, that this dependence exists independent of black hole mass, a property known to be correlated with luminosity. |
By comparing bins with quasars of similar Luminosity. rut clifferent black hole mass. one can isolate the dependence of variability on black hole mass. | By comparing bins with quasars of similar luminosity, but different black hole mass, one can isolate the dependence of variability on black hole mass. |
This is seen with bins 2 and 4. as they cover the same range in luminosity. but un 2 contains objects with Mgg«510Εν while bin 4 contains quasars with between 510M.«Mage10 NL... | This is seen with bins 2 and 4, as they cover the same range in luminosity, but bin 2 contains objects with $M_{BH} < 5 \times 10^{8} {\rm M}_{\sun}$, while bin 4 contains quasars with between $5 \times 10^{8} {\rm M}_{\sun} < M_{\rm BH} < 10^{9} {\rm M}_{\sun}$ . |
Ehe left-hand. panel of Figure 10. shows these two xus g-band structure functions. which indicate that. the objects in bin 4or those with the higher average black hole massesare more variable than those in bin 2. | The left-hand panel of Figure \ref{Fig3.5} shows these two bins' $g$ -band structure functions, which indicate that the objects in bin 4–or those with the higher average black hole masses–are more variable than those in bin 2. |
This is also reflected in their respective values of V(Nr=100) listed in ‘Table 3. | This is also reflected in their respective values of $V(\Delta{\tau}=100)$ listed in Table 3. |
This same trend. can be seen by comparing the three highest-luminosity bins: 3. 5 and 6. | This same trend can be seen by comparing the three highest-luminosity bins: 3, 5 and 6. |
In the right-hand. panel of Figure 10. and. Table 3. it can be seen that variability appears to increase with increasing black hole mass. | In the right-hand panel of Figure \ref{Fig3.5} and Table 3, it can be seen that variability appears to increase with increasing black hole mass. |
The increase is especially clear when one compares bin 3 with bin 6. the highest-black-hole-mass bin in our sample. | The increase is especially clear when one compares bin 3 with bin 6, the highest-black-hole-mass bin in our sample. |
ὃν isolating the cependence of variability. upon luminosity and. black hole mass. we are. in cllect. able to probe the dependence of variability upon the LEclclineton ratio. Liafleas | By isolating the dependence of variability upon luminosity and black hole mass, we are, in effect, able to probe the dependence of variability upon the Eddington ratio, $L_{bol}/L_{Edd}$. |
Vhe Eddington ratio of a quasar is a comparison of the actual bolometric luminosity. {ρω t0 the I5ddington luminosity. Lea. which is the maximum stable luminosity at which accretion can occur. | The Eddington ratio of a quasar is a comparison of the actual bolometric luminosity, $L_{bol}$, to the Eddington luminosity, $_{\rm Edd}$, which is the maximum stable luminosity at which accretion can occur. |
Llowever. as we are measuring the optical luminosity. we can recast this as: where s represents the fraction of the bolometric uminosityv emitted in the optical. | However, as we are measuring the optical luminosity, we can recast this as: where $\varepsilon$ represents the fraction of the bolometric luminosity emitted in the optical. |
This is likely to. be a function. of the bolometric Iuminosity: however. recent measurements for quasars with Lane2107L.; have shown his value to be approximately 0.1. (Mlopkins.Richarels.&Llernquist.2007:Richardsctal. 2006). | This is likely to be a function of the bolometric luminosity; however, recent measurements for quasars with $L_{bol} > 10^{10} L_{\sun}$ have shown this value to be approximately 0.1 \citep{hopkins07,richards06}. . |
Furthermore. since the Eclclington luminosity is directly. proportional to Xack hole mass (Rees1984).. we have that Lawflewcο | Furthermore, since the Eddington luminosity is directly proportional to black hole mass \citep{rees84}, we have that $L_{bol}/L_{Edd} \sim L_{opt}/M_{BH}$ . |
Characteristic Edclington ratios have been calculated or each bin and are provided in Table 5.. | Characteristic Eddington ratios have been calculated for each bin and are provided in Table \ref{bininfotab}. |
Phese values do not represent an average Liawfleas Lor the bin. but rather he Eddington ratio one obtains from the average values or AL\(145004) and Myg also given in Table 5.. | These values do not represent an average $L_{bol}/L_{Edd}$ for the bin, but rather the Eddington ratio one obtains from the average values for $\lambda L_{\lambda} (1450\AA)$ and $_{\rm BH}$ also given in Table \ref{bininfotab}. |
The black 1ole mass is converted to an Eddington luminosity through he familiar Lg;=1.3I107(AI/MYz erg +. | The black hole mass is converted to an Eddington luminosity through the familiar $L_{Edd} = 1.3 \times 10 ^{38} (M/{\rm M)_{\sun}}$ erg $^{-1}$. |
To get the Dolometric luminosity. we use the νι9AL\(5100.4) relation used in Waspietal.(2000) and Wollmeierctal.2106) and combine it with the a,=0.44 quasar spectral slope of VandenBerketal.(2001) to ect a new relation for he continuum near the € line: Liu.~5AL\(45024). | To get the Bolometric luminosity, we use the $L_{bol} \sim 9\times \lambda L_{\lambda} (5100\AA)$ relation used in \citet{kaspi00} and \citet{kollmeier06} and combine it with the $\alpha_{\nu} = 0.44$ quasar spectral slope of \citet{vandenberk01} to get a new relation for the continuum near the C line: $L_{bol} \sim 5\times\lambda L_{\lambda} (1450\AA)$. |
Five of the six bins have LogLg between 0.1 and 1. as did the vast majority of objects in Ixollmeierctal.(2006). | Five of the six bins have $L_{bol}/L_{Edd}$ between 0.1 and 1, as did the vast majority of objects in \citet{kollmeier06}. |
. Even Bin 3. with a value of νι greater than 1 is not unreasonable: a number of objects stuclied in Ixollmeieretal.(2006) were calculated to have stuper-Eddington luminosities. | Even Bin 3, with a value of $L_{bol}/L_{Edd}$ greater than 1 is not unreasonable; a number of objects studied in \citet{kollmeier06} were calculated to have super-Eddington luminosities. |
At any rate. the Edcington ratios calculated in ‘Table 5. should. primarily be used as a means for comparing the relative Eddington ratios of the quasars in cillerent bins. | At any rate, the Eddington ratios calculated in Table \ref{bininfotab} should primarily be used as a means for comparing the relative Eddington ratios of the quasars in different bins. |
ὃν combining the established. (ancl herein reproduced) inverse dependence of variability upon optical luminosity with the newly demonstrated correlation of variability with black hole mass. we find that variability appears to be inversely. related to the Edelington ratio. | By combining the established (and herein reproduced) inverse dependence of variability upon optical luminosity with the newly demonstrated correlation of variability with black hole mass, we find that variability appears to be inversely related to the Eddington ratio. |
Quasars with higher Eddington ratios are less variable than those with lower Edclington ratios. | Quasars with higher Eddington ratios are less variable than those with lower Eddington ratios. |
This suggests that the well-known anticorre[ation of variability with luminosity may in fact simply be a side ellect of a primary anticorrelation between variability and the Exldington ratio. | This suggests that the well-known anticorrelation of variability with luminosity may in fact simply be a side effect of a primary anticorrelation between variability and the Eddington ratio. |
In Figure 6.. lines of constant. Eddington ratio are simplv lines with intercept zero. | In Figure \ref{Fig3.1}, lines of constant Eddington ratio are simply lines with intercept zero. |
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