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We have jxovided and discussed. two methods of GD measurement and calibration: 1). GD measurement wine white light combs (WLCs) generated by the interferometer iu a DEDI Doppler instrument: 2). GD calibration using au RV reference star (RS). | We have provided and discussed two methods of GD measurement and calibration: 1), GD measurement using white light combs (WLCs) generated by the interferometer in a DFDI Doppler instrument; 2), GD calibration using an RV reference star (RS). |
Table 5. summarizes the main results and he comparison between these two methods. | Table \ref{tab:Cal_Comp} summarizes the main results and the comparison between these two methods. |
The accuracy of GD meastuement is sufficient [or current RV precision achieved with instruments usiug the DEDI method (???).. | The accuracy of GD measurement is sufficient for current RV precision achieved with instruments using the DFDI method \citep{Fleming2010, Lee2011, Muirhead2011}. |
However. higher ueasuremenut and calibration precision is required iu the wear future as higher RV precision is achieved by DEDI instruments in search for exoplanets. | However, higher measurement and calibration precision is required in the near future as higher RV precision is achieved by DFDI instruments in search for exoplanets. |
RS aud WLC methods cau serve as complementary methods of CD measurement aud calibration for DEDI instruments. | RS and WLC methods can serve as complementary methods of GD measurement and calibration for DFDI instruments. |
The GD ineasurement using WLCs created by the interferometer provides a direct. way of calibrating the PV scale. | The GD measurement using WLCs created by the interferometer provides a direct way of calibrating the PV scale. |
In the region where combs are visible. effective S/N is relatively low (~15) because of low comb visibility ). | In the region where combs are visible, effective S/N is relatively low $\sim$ 15) because of low comb visibility ). |
In addition. GD cannot be measured in the region where combs are uot visible. which limits the application of this method. | In addition, GD cannot be measured in the region where combs are not visible, which limits the application of this method. |
We are able to measure GD iu a region that accounts for hall of the spectrum coverage. | We are able to measure GD in a region that accounts for half of the spectrum coverage. |
Extrapolation bevoucl the measurement range may result iu large uucertainties. | Extrapolation beyond the measurement range may result in large uncertainties. |
The major issue lacing the method is that the data reductiou pipeline may have introduced uuknown errors while correcting optical distortions such as spectrum curvature and slant. | The major issue facing the method is that the data reduction pipeline may have introduced unknown errors while correcting optical distortions such as spectrum curvature and slant. |
Gravitational leuses provide excellent tools for te study ol cosmology. | Gravitational lenses provide excellent tools for the study of cosmology. |
Iu particular. the uethod developed byBefsd:il(1961) can be used to dete‘mine the Hubble Coistant at ¢osmological distauces. | In particular, the method developed by\citet{refsdal} can be used to determine the Hubble Constant at cosmological distances. |
This method reqlives a leus systeu lor whic both lens and souree redshifts have been ueasured. for which a well-coustrained moclel of the οravitational potentia of tle lensiug 1Lass distribution has been determined. and for which the tije. delays between the leused images lave )een measured. | This method requires a lens system for which both lens and source redshifts have been measured, for which a well-constrained model of the gravitational potential of the lensing mass distribution has been determined, and for which the time delays between the lensed images have been measured. |
To date. measwements of time delays ave beeu Pee‘ted it the literature for 11 eus systems: 09574-5261. (Ix.uncdiéetal.1995.1997c).. PC 11154-0:0 (Sejechteretal. 1997).. JVAS BO0218--357T -etal.1999:Cohen20)0).. PIxX5 1830-211 (Loveleal. 2001).. CLASS Bl16084-656 neelal.19t)9).hereafterPayerI).. CLASS BI60( ciaelal.20xBurudet2000 e. JV.AS B11224231 Narasitula 2001).. nHE 1101-1305 {6AeioWisotzki.&Wiunbsgaiss2(002.see.Refsclal.&—2() s HE 2119-2715 (Buridetal.20tPa). RX J0911.12-05251. 2002).. allid SBS 153+53) n(Burudetal.2002b). | To date, measurements of time delays have been reported in the literature for 11 lens systems: 0957+561 \citep{tk0957_1,tk0957_2}, PG 1115+080 \citep{pls1115,barkana1115}, JVAS B0218+357 \citep{adb0218,cohen0218}, PKS 1830-211 \citep{lovell1830,tw1830}, CLASS B1608+656 \citep[][hereafter Paper I]{paper1}, CLASS B1600+434 \citep{lvek1600,burud1600}, , JVAS B1422+231 \citep{1422delay}, HE 1104-1805 \citep[see, however, Pelt, Refsdal, \& Stabell
2002]{gilmerino1104}, HE 2149-2745 \citep{burud2149}, RX J0911.4+0551 \citep{hjorth0911}, and SBS 1520+530 \citep{burud1520}. |
. Of these leus systeuis. CLASS B1608--656 is the on Oour-iÀnage svsten which all three iudependeut time delays have been unambiguously lueastir | Of these lens systems, CLASS B1608+656 is the only four-image system for which all three independent time delays have been unambiguously measured. |
Tle CLASS B1608+656 leus system consists of the core of a racio-loud »oststarbu‘st galaxy al a recdshilt of z=1.391 (Fassuachtetal.1996) being lensed by a z=0.630) pair of gal:inies ).. | The CLASS B1608+656 lens system consists of the core of a radio-loud poststarburst galaxy at a redshift of $z = 1.394$ \citep{zs1608} being lensed by a $z = 0.630$ pair of galaxies \citep{sm1608}. |
adio maps of the system show four images of the backeroun| source aTauged ina ivpica leus geometry (Figure 1)). | Radio maps of the system show four images of the background source arranged in a typical lens geometry (Figure \ref{fig_1608map}) ). |
Models of the leus system preclict that.if [the backeround source is variable. image B should show the variatiou first. Followed yy componens A. C. ala D in turn (Myersetal.1995:ουράς&Fassuacht1999.herealterPayer ΤΠ). | Models of the lens system predict that, if the background source is variable, image B should show the variation first, followed by components A, C, and D in turn \citep[hereafter
Paper~II]{sm1608,paper2}. . |
.. The une delays determined inPaper I were based ou radio-waveleugth light curves obtaine with the Very Large Arav (VLA!)) | The time delays determined inPaper I were based on radio-wavelength light curves obtained with the Very Large Array ) |
rellilies)). | ). |
More formally: one distributes small disks of area a=x7? randomly on a large surface 4=wR? RSr. with overlap allowed. | More formally: one distributes small disks of area $a=\pi r^2$ randomly on a large surface $A=\pi R^2$, $R\gg r$, with overlap allowed. |
With an increasing number of disks. clusters begin to form. | With an increasing number of disks, clusters begin to form. |
If the large surface were water and the small disks floating water lilies: how many lilies are needed for a cluster to connect the opposite sides. so Chat an ant could walk across the pond without getting its [eet wet? | If the large surface were water and the small disks floating water lilies: how many lilies are needed for a cluster to connect the opposite sides, so that an ant could walk across the pond without getting its feet wet? |
Given ;N disks. the disk density is n— N/A, | Given $N$ disks, the disk density is $n=N/A$ . |
Clearly. (he average cluster ο) size will increase with ». | Clearly, the average cluster $S(n)$ size will increase with $n$. |
The striking feature is that it does so ina verv sudden wav (see relcluster)): as 7 approaches some “critical value" n. 5(n) suddenly becomes large enough to span (he pond. | The striking feature is that it does so ina very sudden way (see \\ref{cluster}) ); as $n$ approaches some “critical value” $n_c$, $S(n)$ suddenly becomes large enough to span the pond. |
In fact. in the limit Nox and ο—oe al constant n. both S(n) and d5(n)/dn diverge for n—n: we have percolation as a geometric form of critical behaviour. | In fact, in the limit $N \to \infty$ and $A \to \infty$ at constant $n$, both $S(n)$ and $dS(n)/dn$ diverge for $n \to n_c$: we have percolation as a geometric form of critical behaviour. |
The critical density for the onset of percolation has been determined (numerically) for a variety. of different svstems. | The critical density for the onset of percolation has been determined (numerically) for a variety of different systems. |
In (wo dimensions. disks percolate αἱ i.c1.13/(ar7). Le. when we have a little more (han one disk per unit area. | In two dimensions, disks percolate at $n_c\simeq 1.13/(\pi r^2)$, i.e., when we have a little more than one disk per unit area. |
Decause of overlap. at (his point only of space is covered by disks. remain empty. | Because of overlap, at this point only of space is covered by disks, remain empty. |
Nevertheless. when our ant can walk across. a ship can no longer cross the pond. and vice versa. | Nevertheless, when our ant can walk across, a ship can no longer cross the pond, and vice versa. |
This is a special feature ol (wo dimensions (ihe "fence effect). and no longer holds for d>2. | This is a special feature of two dimensions (the “fence effect”), and no longer holds for $d>2$. |
In three dimensions. (he corresponding problem is one of overlapping spheres in a large volume. | In three dimensions, the corresponding problem is one of overlapping spheres in a large volume. |
Here the critical density for the percolating spheres becomes n,70.34/[((4x/3)77]. wilh r denoting the radius of the little spheres now taking the place of the small disks we had in two dimensions. | Here the critical density for the percolating spheres becomes $n_c \simeq 0.34/[(4\pi/3)r^3]$, with $r$ denoting the radius of the little spheres now taking the place of the small disks we had in two dimensions. |
At the critical point in three dimensions. however.only of space is covered by overlapping spheres. while remains empty. and here both spheres | At the critical point in three dimensions, however,only of space is covered by overlapping spheres, while remains empty, and here both spheres |
peaks at orbital phase approximately 0.4 and approximately 0.7 are due to gas stream impacts close to the central object. | peaks at orbital phase approximately $0.4$ and approximately $0.7$ are due to gas stream impacts close to the central object. |
The peaks and dips repeat on the svnodic period. as they did for X-1. ancl again most of the variability in dissipation is eenerated by the stream impact moving close to the compact object. | The peaks and dips repeat on the synodic period, as they did for X-1, and again most of the variability in dissipation is generated by the stream impact moving close to the compact object. |
The overall disc tilt out of the orbital plane was found to be 11.37 with a maximum warp tilt angle of 18.6" near the edge of the disc. | The overall disc tilt out of the orbital plane was found to be $\degrees{11.3}$ with a maximum warp tilt angle of $\degrees{18.6}$ near the edge of the disc. |
The tilt of the disc will lead to the radiation source being hidden to a high inclination observer as the (πο dise moves between the observer ancl the radiation source. | The tilt of the disc will lead to the radiation source being hidden to a high inclination observer as the tilted disc moves between the observer and the radiation source. |
This will occur as the warped disc precesses. as it did in theXX- simulation (see Figure 7)). | This will occur as the warped disc precesses, as it did in theX-1 simulation (see Figure \ref{HerX1:Figure:herx1_on_off}) ). |
The radiation source may als» be obscured by the material in the warp close to the cenre of the clise. | The radiation source may also be obscured by the material in the warp close to the centre of the disc. |
ὃν tuning he central luminosity we have been able to reproduce the «observed. long period. for 4433. | By tuning the central luminosity we have been able to reproduce the observed long period for 433. |
‘The tilt angle of the disc at the edge is very similar to the half- (20°) for he collimated) precessing jets. | The tilt angle of the disc at the edge is very similar to the half-angle $\degrees{20}$ ) for the collimated precessing jets. |
In the centre of the disc. whic his presumably where the jets are launched. the disc tilt angle was. however. about 12°. | In the centre of the disc, which is presumably where the jets are launched, the disc tilt angle was, however, about $\degrees{12}$. |
Lt is intriguing to wonder wheher this discrepanev might. be diminished with the svsteni parameters recently published by Blundell. (2008)... | It is intriguing to wonder whether this discrepancy might be diminished with the system parameters recently published by \cite{Blundell:2008}. . |
We note also. however. | We note also, however, |
as found by NT2007. | as found by NT2007. |
This form of presentation. namely the fraction of active objects with respect to the “local” F(Mpyjj). is particularly convenient because it can be directly used. as we do in Sect. | This form of presentation, namely the fraction of active objects with respect to the “local” $F(M_{BH})$, is particularly convenient because it can be directly used, as we do in Sect. |
6. without resorting to assumptions on the quantity g. Eq. (6)). | 6, without resorting to assumptions on the quantity $q$ , Eq. \ref{fraction}) ). |
It is also interesting to check whether these distributions can be represented by the function (1)). | It is also interesting to check whether these distributions can be represented by the function \ref{lorentz}) ). |
Since the curves in H2004. Fig. | Since the curves in H2004, Fig. |
3. are devoid of error bars. we could only adjust. as best as we could. the integral of this function to them. | 3, are devoid of error bars, we could only adjust, as best as we could, the integral of this function to them. |
The outcomes are in general quite satisfactory. and one example is shown in Fig. | The outcomes are in general quite satisfactory, and one example is shown in Fig. |
2. (where the curve from H2004 is represented by a selection of points). | \ref{Heck8lor} (where the curve from H2004 is represented by a selection of points). |
Typically the high Jt tails stay somewhat above the H2004 curves. | Typically the high $\lambda$ tails stay somewhat above the H2004 curves. |
One might require that the excess in the Lorentzian tail does to some extent take care of the lack of the brightest objects in the original sample. but we need not insist on this issue. because the difference is of little consequence for our final results. | One might require that the excess in the Lorentzian tail does to some extent take care of the lack of the brightest objects in the original sample, but we need not insist on this issue, because the difference is of little consequence for our final results. |
The parameters extracted from H2004 that we use in Sect. | The parameters extracted from H2004 that we use in Sect. |
6 are summarised in. Table [.. | 6 are summarised in Table \ref{tabHeckman}. |
The Z4,4,0 in the second column (with the corresponding vty in the third) represents a completeness limit valid for all masses between 107 and 10? M... | The $L_{bol,0}$ in the second column (with the corresponding $\lambda_0$ in the third) represents a completeness limit valid for all masses between $^7$ and $^9$ $M_{\odot}$. |
The fourth and fifth columns contain. repectively. ,b,,.. the maximum value that can be read in H2004. Fig. | The fourth and fifth columns contain, repectively, $\lambda_{max}$ , the maximum value that can be read in H2004, Fig. |
3. and qj. the fraction of active SMBH above ty from the same figure. | 3, and $q_H$, the fraction of active SMBH above $\lambda_0$ from the same figure. |
From the sixth column onward the quantities given. were obtained using the Lorentzian profile. as explained above. | From the sixth column onward the quantities given were obtained using the Lorentzian profile, as explained above. |
The quantity γω IS very low and badly determined. but the quantity that really matters is ww: the fraction gy; in the last column differs only marginally from qy;. | The quantity $\lambda_{peak}$ is very low and badly determined, but the quantity that really matters is $w$; the fraction $q_{Hl}$ in the last column differs only marginally from $q_H$. |
It should be noted that the fraction of active SMBH in H2004 ts given with respect to the local F(Mpjj). such as the one given by Marconi et al. ( | It should be noted that the fraction of active SMBH in H2004 is given with respect to the local $F(M_{BH})$, such as the one given by Marconi et al. ( |
2004) and described in the next section. | 2004) and described in the next section. |
The number density of SMBH as a function of mass. F(Mg). has been estimated in the local universe by several authors (see Marconi et al. | The number density of SMBH as a function of mass, $F(M_{BH})$, has been estimated in the local universe by several authors (see Marconi et al. |
2004. Shankar et al. | 2004, Shankar et al. |
2004 and references therein). | 2004 and references therein). |
The one adopted here is from Marconi et al. ( | The one adopted here is from Marconi et al. ( |
2004) and is shown in Fig. 3.. | 2004) and is shown in Fig. \ref{BHMF_Marconi}, |
where the MF which includes all morphological types (their Fig. | where the MF which includes all morphological types (their Fig. |
2b) 1s given along with the MF in late. F;. and early. Fe. type galaxies separately (A. Marconi. priv. | 2b) is given along with the MF in late, $F_L$, and early, $F_E$, type galaxies separately (A. Marconi, priv. |
comm.). | comm.). |
This F, is preferred to the one estimated by Shankar et al. ( | This $F_L$ is preferred to the one estimated by Shankar et al. ( |
2004) for the attention paid to the change in the bulge to the total luminosity ratio along the Hubble sequence of late type galaxies. | 2004) for the attention paid to the change in the bulge to the total luminosity ratio along the Hubble sequence of late type galaxies. |
The reason we keep the late separate from the early type galaxies will be explained in Sect. | The reason we keep the late separate from the early type galaxies will be explained in Sect. |
7. | 7. |
For our purpose. what does matter is the shape of the MF. | For our purpose, what does matter is the shape of the MF. |
Insofar as the growth of the SMBH ts a consequence of accretion and is therefore accompanied by the AGN type activity. it is most likely that the shape of the MF for the “active” SMBH changes with redshift and is different from that of the whole population as observed in the local universe. | Insofar as the growth of the SMBH is a consequence of accretion and is therefore accompanied by the AGN type activity, it is most likely that the shape of the MF for the “active” SMBH changes with redshift and is different from that of the whole population as observed in the local universe. |
In principle. the redshift dependence of the "active" SMBH mass function. F°CW,;;. 5). could be inferred from the AGN luminosity function as a function of z. provided that a simple relationship holds between Mj; and the luminosity (for instance. ο = constant). | In principle, the redshift dependence of the “active” SMBH mass function, $F^*(M_{BH}, z)$ , could be inferred from the AGN luminosity function as a function of $z$, provided that a simple relationship holds between $M_{BH}$ and the luminosity (for instance, $\lambda$ = constant). |
In practice. as discussed in Marconi et al. ( | In practice, as discussed in Marconi et al. ( |
2004) and as has become particularly evident after NT2007. the conversion is neither straightforward nor univocal. | 2004) and as has become particularly evident after NT2007, the conversion is neither straightforward nor univocal. |
Hence F'(Mggj) can only be reliably estimated through a "direct" evaluation of Mp; 1n a properly selected sample of AGN. with special care devoted to correcting for incompleteness selection effects. | Hence $F^{*}(M_{BH})$ can only be reliably estimated through a “direct” evaluation of $M_{BH}$ in a properly selected sample of AGN, with special care devoted to correcting for incompleteness selection effects. |
On the basis of SDSS samples. H2004 (z<0.1) for narrow emission line AGN and Greene Ho (2007. =<0.3. with masses derived as in NT2007) for broad emission line AGN. both find that in the local universe F(Mp) is steeper. above 10’ M. than F(OMjjj): a difference attributed to the downsizing In Sects. | On the basis of SDSS samples, H2004 $z\leq 0.1$ ) for narrow emission line AGN and Greene Ho (2007, $z\leq 0.3$, with masses derived as in NT2007) for broad emission line AGN, both find that in the local universe $F^{*}(M_{BH})$ is steeper, above $^7$ $M_{\odot}$, than $F(M_{BH})$: a difference attributed to the so-called downsizing In Sects. |
5 and 6 we. in a first instance. adopt the "shape" of the local Εμ): however. the impact on the results of other options for the downsizing 1s also discussed. | 5 and 6 we, in a first instance, adopt the “shape” of the local $F(M_{BH})$; however, the impact on the results of other options for the downsizing is also discussed. |
The basic feature of the model in Papl is the gravitational force exerted by the BH (and by the associated stellar bulge) on a rotationally supported dise made of molecular clouds. | The basic feature of the model in PapI is the gravitational force exerted by the $BH$ (and by the associated stellar bulge) on a rotationally supported disc made of molecular clouds. |
This force shapes the gas distribution profile. as a function of the radial distance R from the BH. in such a way as to determine an anticorrelation between Mp), and the "covering factor" determined by the disc. | This force shapes the gas distribution profile, as a function of the radial distance $R$ from the $BH$, in such a way as to determine an anticorrelation between $M_{BH}$ and the “covering factor” determined by the disc. |
Quantitatively this factor was calculated in Papl assuming that the distance out to which the discextends is at least equal to 2R;,j; (R5,;; 15 the distance from the BHat which the molecular gas distribution profile has a point of inflection. see Eq. ( | Quantitatively this factor was calculated in PapI assuming that the distance out to which the discextends is at least equal to $R_{infl}$ $R_{infl}$ is the distance from the BHat which the molecular gas distribution profile has a point of inflection, see Eq. ( |
9) inPapl). that ts. about 25 pc | 9) inPapI), that is, about 25 pc |
'The transition takes place at 10(1 z)ós S. | The transition takes place at (1+z) _A^8 . |
Note that the cooling Lorentz factor for ions 5;4=(A3/Z4)(mp/me)? can be larger than the saturation Lorentz factor (eq. i. [15]]). | Note that the cooling Lorentz factor for ions $\g^\p_{c,A} =
(A^3/Z^4)(m_p/m_e)^3 \g^\p_{c,e}$ can be larger than the saturation Lorentz factor (eq. \ref{A_sat_Lorentz}] ]). |
To calculate synchrotron spectra at different epoch and light curves at different frequencies arising from a forward shock, it is sufficient to calculate different spectral break frequencies and flux normalization along with their time evolution (e.g.Sarietal.1998;Chevalier 2002). | To calculate synchrotron spectra at different epoch and light curves at different frequencies arising from a forward shock, it is sufficient to calculate different spectral break frequencies and flux normalization along with their time evolution \citep[e.g.][]{spn98,cl00,pk00,gs02}. |
. The characteristic synchrotron frequencies for the electrons with the minimum, saturation and cooling Lorentz factors respectively are given by 7.T(1-- nae, GeV, 180(1 9/56δις) GeV, 0.5(112e(Eggn?)-2t;ev. | The characteristic synchrotron frequencies for the electrons with the minimum, saturation and cooling Lorentz factors respectively are given by 7.7 _e^2 _B , 180 , 0.5. |
A transition from the fast-cooling (voe« Vm,e) to (vc,«> Vm,e) takes place at1019 Bü.j2nEgs In both the fast- and slow- cooling cases the maximum e-synchrotron flux is given by (e.g.Sarietal.1998) 52!& BEssJy. Here dos/(10?5 is the luminosity distance. | A transition from the fast-cooling $\nu_{c,e} < \nu_{m,e}$ ) to slow-cooling $\nu_{c,e} > \nu_{m,e}$ ) takes place at _e)^2 In both the fast- and slow- cooling cases the maximum $e$ -synchrotron flux is given by \citep[e.g.][]{spn98}
52 _e _B. Here $d_{28}/(10^{28}~{\rm cm})$ is the luminosity distance. |
Note that the synchrotron cm)self-absorption frequency is in the radio band (e.g.Panaitescu&Kumar2000) and we ignore that while modeling optical to > GeV data. | Note that the synchrotron self-absorption frequency is in the radio band \citep[e.g.][]{pk00} and we ignore that while modeling optical to $>$ GeV data. |
The synchrotron break frequencies for the ions of minimum and cooling Lorentz factors can be expressed as scaling relations to the corresponding break frequencies for electrons as = AU ,= (49/2?(mpm. )*v,. and for the ions of saturation Lorentz factor (eq. [15]]) | The synchrotron break frequencies for the ions of minimum and cooling Lorentz factors can be expressed as scaling relations to the corresponding break frequencies for electrons as = _e)^2 (m_e/m_p)^3 = (A^6/Z^7) (m_p/m_e)^5 and for the ions of saturation Lorentz factor (eq. \ref{A_sat_Lorentz}] ]) |
as TeV; = (A/Z) A)Vsat,e? Note that the ion-synchrotron spectrum is always in the slow-cooling regime (νοA> Vm,a) as opposed to the e-synchrotron spectrum which can be in the fast-cooling regime early and changes to the slow-cooling regime later. | as ; = (A/Z)^2 _A); Note that the ion-synchrotron spectrum is always in the slow-cooling regime $\nu_{c,A} > \nu_{m,A}$ ) as opposed to the $e$ -synchrotron spectrum which can be in the fast-cooling regime early and changes to the slow-cooling regime later. |
'The maximum ion synchrotron flux is∣∁⊵↕⊓↗⇇↕↕↕≣∑⇘⋛≣↕∶⋣≽⊼⋯ for kyZ1 and kp>2. | The maximum ion synchrotron flux is, for $k_1 \ne 1$ and $k_2>2$. |
Figure 1. shows light curves at different energies from the combined leptonic-hadronicmodel of a decelerating adiabatic blast wave in constant density medium. | Figure \ref{fig:synchrotron} shows light curves at different energies from the combined leptonic-hadronicmodel of a decelerating adiabatic blast wave in constant density medium. |
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