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In the simplest case, we measure the delay difference between the antennas on the reference baseline (i.e., baseline A3—A4 in Figure [) as a function of time. | In the simplest case, we measure the delay difference between the antennas on the reference baseline (i.e., baseline $A_3-A_4$ in Figure \ref{C-PACS}) ) as a function of time. |
Assuming a non-dispersive atmosphere, the delay on the science baseline (i.e., baseline A,—Ag) is corrected by applying the delay difference on the reference baseline to the visibility measurement. | Assuming a non-dispersive atmosphere, the delay on the science baseline (i.e., baseline $A_1-A_2$ ) is corrected by applying the delay difference on the reference baseline to the visibility measurement. |
The reference and science delays are not identical since the two baselines are not exactly co-located. | The reference and science delays are not identical since the two baselines are not exactly co-located. |
The relevant distances that determine the efficacy of the delay corrections are not the baseline lengths at the ground, but the distances between the radio beams as they traverse the turbulent layer (i.e. A;—A5 and A5—Αι in Figure [1]. | The relevant distances that determine the efficacy of the delay corrections are not the baseline lengths at the ground, but the distances between the radio beams as they traverse the turbulent layer (i.e. $A_1' - A_3'$ and $A_2' - A_4'$ in Figure \ref{C-PACS}) ). |
'The beam separation at the turbulent layer depends upon the relative positions of the target and reference source in the sky, the height of the turbulent layer, and the configuration of the antennas on the ground. | The beam separation at the turbulent layer depends upon the relative positions of the target and reference source in the sky, the height of the turbulent layer, and the configuration of the antennas on the ground. |
The upper limit to the beam separation is given by where a is the angular separation between the science target and the atmospheric calibrator, h the height of the turbulence, and 6 is the source elevation. | The upper limit to the beam separation is given by where $\alpha$ is the angular separation between the science target and the atmospheric calibrator, $h$ the height of the turbulence, and $e$ is the source elevation. |
Assuming that the turbulent layer is at a height of 1 km (continuous line) or at 2 km (dashed line), Figure shows the trajectories of the 3.5-m beam locations relative to the 6-m and 10-m beams for an 8 h observation of PP 13S* centered on transit with 3C111 as the atmospheric calibrator. | Assuming that the turbulent layer is at a height of 1 km (continuous line) or at 2 km (dashed line), Figure \ref{fig:ant} shows the trajectories of the 3.5-m beam locations relative to the 6-m and 10-m beams for an 8 h observation of PP 13S* centered on transit with 3C111 as the atmospheric calibrator. |
As shown in this figure, the choice of 3C111 as the atmospheric calibrator is particularly fortuitous for PP 19593 since the science and reference beams for most antennas nearly cross within the turbulent zone. | As shown in this figure, the choice of 3C111 as the atmospheric calibrator is particularly fortuitous for PP 13S* since the science and reference beams for most antennas nearly cross within the turbulent zone. |
Using phase closure (Jennison]{1958),, the difference between the actual delay for the target beams ΑΙ(Ar)—A5 and the atmospheric reference beams A5—Αι is given by Ar=(A,—A5)7(ASA)vr(A15)—7(45 A1). | Using phase closure \citep{Jennison58}, the difference $\Delta\tau$ ) between the actual delay for the target beams $A_1'-A_2'$ and the atmospheric reference beams $A_3'-A_4'$ is given by $\Delta\tau = \tau(A_1'-A_2') - \tau(A_3'-A_4') =
\tau(A_1'-A_3') - \tau(A_2'-A_4')$ . |
In a favorable configuration, the beam separations A,— and — will be much less than either the targetA$ beam AjseparationAj, Ai—A or the reference beam separation A;—Aj. | In a favorable configuration, the beam separations $A_1'-A_3'$ and $A_2'-A_4'$ will be much less than either the target beam separation $A_1'-A_2'$ or the reference beam separation $A_3'-A_4'$. |
The RMS of the corrected visibilities will be 2 worse than for an array with beam separation Aj— which implies that C-PACS will have a performance equivalentA5, to an array that has baseline lengths larger than the Aj—A5 beam separation, depending on the structure function exponent. | The RMS of the corrected visibilities will be $\sqrt{2}$ worse than for an array with beam separation $A_1'-A_3'$, which implies that C-PACS will have a performance equivalent to an array that has baseline lengths larger than the $A_1'-A_3'$ beam separation, depending on the structure function exponent. |
The complete analysis contains additional correlation terms and added uncertainties caused by the finite signal-to-noise for the atmospheric calibrator observations. | The complete analysis contains additional correlation terms and added uncertainties caused by the finite signal-to-noise for the atmospheric calibrator observations. |
The ability of C-PACS to correct the atmospheric delays is limited by the delays from the short beam spacings A!— and A,—Aj (see Figure [I]), instrumental phase A$drifts on the reference array, and the radiometer noise. | The ability of C-PACS to correct the atmospheric delays is limited by the delays from the short beam spacings $A_1'-A_3'$ and $A_2'-A_4'$ (see Figure \ref{C-PACS}) ), instrumental phase drifts on the reference array, and the radiometer noise. |
The delay errors caused by differences in the beam spacings between the science and reference arrays are given statistically by the structure function R(|A;—Aj|). | The delay errors caused by differences in the beam spacings between the science and reference arrays are given statistically by the structure function $R(|A_i'-A_j'|)$. |
The instrumental errors can be removed by removing a box-car average over the length of the observation, and will contribute negligible delay errors as long as the timescale for the instrumental drifts are large compared to the box-car width. | The instrumental errors can be removed by removing a box-car average over the length of the observation, and will contribute negligible delay errors as long as the timescale for the instrumental drifts are large compared to the box-car width. |
The delay errors due to radiometric phase noise depend on the strength of the source being observed, the receiver properties, and the atmospheric characteristics etal.|001). | The delay errors due to radiometric phase noise depend on the strength of the source being observed, the receiver properties, and the atmospheric characteristics \citep{TMS2001}. |
. 'The uncertainty in the measured(Thompson phase from radiometer noise is given by where kg is the Boltzmann’s constant, Tyys is the system temperature, ng is the correlator quantization efficiency, Aeg is the effective collecting area of the antennas, S is the flux density of the atmospheric calibrator, B is the bandwidth of the observations, and t is the integration time. | The uncertainty in the measured phase from radiometer noise is given by where $k_{\mathrm{B}}$ is the Boltzmann's constant, $T_\mathrm{sys}$ is the system temperature, $\eta_Q$ is the correlator quantization efficiency, $A_\mathrm{eff}$ is the effective collecting area of the antennas, $S$ is the flux density of the atmospheric calibrator, $B$ is the bandwidth of the observations, and $t$ is the integration time. |
The measurement uncertainty in the delay is then A@/(27v). | The measurement uncertainty in the delay is then $\Delta \phi / (2\pi\nu)$. |
The net variance in the delay after applying C-PACS is given by The structure function is generally described by(3) a power law with exponents varying from 5/3 to 2/3 depending upon the spacing and thickness of the turbulent layer. | The net variance in the delay after applying C-PACS is given by The structure function is generally described by a power law with exponents varying from 5/3 to 2/3 depending upon the spacing and thickness of the turbulent layer. |
The scaling coefficient of the power law also varies depending upon the weather conditions. | The scaling coefficient of the power law also varies depending upon the weather conditions. |
In order for C-PACS to improve the image quality, the target and reference beams must be close at the turbulent layer such that R(|A—+R(|ASA4|)«X. | In order for C-PACS to improve the image quality, the target and reference beams must be close at the turbulent layer such that $R(|A_1'-A_3'|) + R(|A_2'-A_4'|) \ll \lambda^2$. |
This requires angular separationsΑς) ffor the A and B configuration C-PACS pairings and typical winter weather conditions at the CARMA site. | This requires angular separations for the A and B configuration C-PACS pairings and typical winter weather conditions at the CARMA site. |
A future publication will use actual measurements to quantify how the quality of the C-PACS correction varies with angular separation between the science target and the atmospheric calibrator (Zaudereral.,, in preparation). | A future publication will use actual measurements to quantify how the quality of the C-PACS correction varies with angular separation between the science target and the atmospheric calibrator (Zauderer, in preparation). |
The radiometer noise should also contribute much less than a wavelength of delay error for the C-PACS corrections to be successful. | The radiometer noise should also contribute much less than a wavelength of delay error for the C-PACS corrections to be successful. |
For the characteristics of the 3.5-m telescopes and the 1-cm receivers, the radiometer delay error is given by Thus, 1.3-mm observations with integration times of t=4 s enough to measure and correct most of the atmosphere(short fluctuations) require a reference source brighter than S~1 Jy in the 1 cm band. | For the characteristics of the 3.5-m telescopes and the 1-cm receivers, the radiometer delay error is given by Thus, 1.3-mm observations with integration times of $t=4$ s (short enough to measure and correct most of the atmosphere fluctuations) require a reference source brighter than $S\sim1$ Jy in the 1 cm band. |
When several atmospheric calibrators are available, the optimum choice between calibrator separation and brightness can be found by minimizing Equation B] for the expected weather conditions. | When several atmospheric calibrators are available, the optimum choice between calibrator separation and brightness can be found by minimizing Equation \ref{eq:dtau} for the expected weather conditions. |
We combined the SZA 30 GHz calibrator list, the GBT catalog at 1.4 and 5.0 GHz and the WMAP point source catalog (Condon(Wright&Yin||2001),,etal.||2009] to estimate the density of potential C-PACS calibrators. | We combined the SZA 30 GHz calibrator list, the GBT catalog at 1.4 and 5.0 GHz \citep{Condon2001}, and the WMAP point source catalog \citep{Wright2009} to estimate the density of potential C-PACS calibrators. |
For each source in the GBT catalog, we extrapolated the flux density from 5.0 GHz to 30 GHz by measuring the spectral index a between 1.4 and 5.0 GHz v*). | For each source in the GBT catalog, we extrapolated the flux density from 5.0 GHz to 30 GHz by measuring the spectral index $\alpha$ between 1.4 and 5.0 GHz $S_{\nu} \propto
\nu^{\alpha}$ ). |
We find that 50% of the sky is within 5? of a | We find that $50 \%$ of the sky is within $5\degr$ of a |
transformed into other MIID waves depending on complex configurations of magnetic fields (Tarbell.Rvutova.&Covington1999:Sakaietal.2000). | transformed into other MHD waves depending on complex configurations of magnetic fields \citep{trc99,sky00}. |
. We would like to remark that other mechanisms also predict production of longitudinal waves al the coronal height. alühough we have taken up the chromospheric reconnection model as a (vpical process in (his paper. | We would like to remark that other mechanisms also predict production of longitudinal waves at the coronal height, although we have taken up the chromospheric reconnection model as a typical process in this paper. |
Generation of spicules in an open magnetic flux tube have been investigated by various authors (Hollweg.Jackson.&Galloway1982:Hollweg1992:Ixudoh&Shibata 1999). | Generation of spicules in an open magnetic flux tube have been investigated by various authors \citep{hjg82,hol92,ks99}. |
. They found that Alfvénn waves excited by random motionsad the photosphere (Ulrich1996) effectively transport their energy to the corona. | They found that Alfvénn waves excited by random motionsat the photosphere \citep{ulr96} effectively transport their energy to the corona. |
The nonlinear effect of torsional Alfvénn waves produces longitudinal waves along the vertical flux (tube in the corona. | The nonlinear effect of torsional Alfvénn waves produces longitudinal waves along the vertical flux tube in the corona. |
A sizable fraction of the initial energy of the transverse waves at the photosphere is converted to energv of the longitudinal waves al the coronal height. and these waves could become acoustic waves propagating upwardly, | A sizable fraction of the initial energy of the transverse waves at the photosphere is converted to energy of the longitudinal waves at the coronal height, and these waves could become acoustic waves propagating upwardly. |
They could heat the surrounding plasma in the verv same wav as those triggered by (lie reconnection events above. | They could heat the surrounding plasma in the very same way as those triggered by the reconnection events above. |
Thus. acoustic waves are expected to be universally generated by various mechanisms in the corona far above the photosphere. and therefore. it is quite worth studying their role in coronal heating. | Thus, acoustic waves are expected to be universally generated by various mechanisms in the corona far above the photosphere, and therefore, it is quite worth studying their role in coronal heating. |
In (his section. we describe our method of treating the outward propagation of acoustic waves, alter thev have been excited in the corona. | In this section, we describe our method of treating the outward propagation of acoustic waves, after they have been excited in the corona. |
We first. estimate a distance acoustic waves (ravel before lormine shocks on a plane-parallel geometry. | We first estimate a distance acoustic waves travel before forming shocks on a plane-parallel geometry. |
Then. we derive an equation dealing with variation of N-wave amplitude on spherical geometry. | Then, we derive an equation dealing with variation of N-wave amplitude on spherical geometry. |
For simplicity’s sake. in the following discussions. we neglect the effects of magnetic field on the propagation οἱ the waves. | For simplicity's sake, in the following discussions, we neglect the effects of magnetic field on the propagation of the waves. |
This simplification is valid when the circumstantial magnetic configuration is perpendicular above (he flux tube generating (he waves. | This simplification is valid when the circumstantial magnetic configuration is perpendicular above the flux tube generating the waves. |
Anv acoustic wave having a finite amplituce inevitably changes its shape. makes the wave front steepen and eventually forms (he shock front (e.g. Landau Lifshitz 1959). | Any acoustic wave having a finite amplitude inevitably changes its shape, makes the wave front steepen and eventually forms the shock front (e.g. Landau Lifshitz 1959). |
The distance the acoustic waves travel belore forming the shocks can be estimated on a given wave length. A. and initial amplitude. dey. | The distance the acoustic waves travel before forming the shocks can be estimated on a given wave length, $\lambda$, and initial amplitude, $\delta v_0$. |
Consider (he acoustic waves propagating in the upward direction. z. in isothermal atmosphere with density structure of p=poexpt—z/11,). | Consider the acoustic waves propagating in the upward direction, $z$, in isothermal atmosphere with density structure of $\rho = \rho_0 \exp(-z/H_{\rho})$ . |
Assuming waves having initially sinusoidal velocity profiles.ὃς=ovosin(2* Z/A). the wave | Assuming waves having initially sinusoidal velocity profiles,$\delta v=\delta v_0 \sin(2\pi Z/\lambda)$ , the wave |
(2001) suggest the activation of slow circulation mechanisms below the convective border of the envelope for of the observed stars. in order to explain the low observed ο ος values. | (2001) suggest the activation of slow circulation mechanisms below the convective border of the envelope for of the observed stars, in order to explain the low observed $^{12}$ $^{13}$ C values. |
In conclusion. it appears that at least a fraction of PN progenitors do have reduced CO 1Ο) ratios compared to standard model expectations. and that some extra-mixing should be required. | In conclusion, it appears that at least a fraction of PN progenitors do have reduced $^{12}$ $^{13}$ C ratios compared to standard model expectations, and that some extra-mixing should be required. |
In Fig. | In Fig. |
2 we show the ο ο ratio vs. progenitor mass in the total ejecta of LIAIS compared: to observations in Όλο (Palla et al. | 2 we show the $^{12}$ $^{13}$ C ratio vs. progenitor mass in the total ejecta of LIMS compared to observations in PNe (Palla et al. |
2000). | 2000). |
Stellar models are either from van den Hoek CGroenewegen (1997). or Ventura et al. ( | Stellar models are either from van den Hoek Groenewegen (1997) or Ventura et al. ( |
2002)penct). and have been computed. for different) values of the initial stellar metallicity. | 2002), and have been computed for different values of the initial stellar metallicity. |
Ventura et al. ( | Ventura et al. ( |
2002) predict sistematically lower ratios than van den Lloek CGroenewegen (1097) in the mass range 66 AJ... | 2002) predict sistematically lower ratios than van den Hoek Groenewegen (1997) in the mass range 6 $M_\odot$. |
However. their computations do not extend to the 33 M. range. where model predictions could be usefully compared to observations. | However, their computations do not extend to the 3 $M_\odot$ range, where model predictions could be usefully compared to observations. |
Ες is the reason why we do not adopt the carbon vields of Ventura et al. ( | This is the reason why we do not adopt the carbon yields of Ventura et al. ( |
2002). | 2002). |
Notice that in this figure we are comparing theoreticalfofad vields with 12C/0€ patios observed. in. PNe. | Notice that in this figure we are comparing theoretical yields with $^{12}$ $^{13}$ C ratios observed in PNe. |
I would be better to compare the observations with the vieldsAGB. | It would be better to compare the observations with the yields. |
We choose the total vields rather than those at the end of the AGB phase in order to compare the results from van den Lock Crocneweeen (1997) to those [ron Ventura et al. ( | We choose the total yields rather than those at the end of the AGB phase in order to compare the results from van den Hoek Groenewegen (1997) to those from Ventura et al. ( |
2002). | 2002). |
In. fact. these latter authors do not eive the vields at the tip of the AGB. but only the total ones. | In fact, these latter authors do not give the yields at the tip of the AGB, but only the total ones. |
Theoretical “C/C ratios at the tip of he AGB [rom van den Llock Croenewegen (1997) are in general higher than those in the total ejecta. | Theoretical $^{12}$ $^{13}$ C ratios at the tip of the AGB from van den Hoek Groenewegen (1997) are in general higher than those in the total ejecta. |
It is immediately seen that the adopted. vields overestimate the OC ο ratio in the ejecta of low-mass stars. | It is immediately seen that the adopted yields overestimate the $^{12}$ $^{13}$ C ratio in the ejecta of low-mass stars. |
However. as demonstrated by Palla et al. ( | However, as demonstrated by Palla et al. ( |
2000). as far as CEC is concerned. the fraction of low-mass stars experiencing deep mixing ¢oes not allect significantly the overall results of the chemical. evolution models. duc to the fact that the Galactic evolution of £C and 7€ is mainly governed by stars in which this process is not expected. to occur. | 2000), as far as $^{12}$ $^{13}$ C is concerned, the fraction of low-mass stars experiencing deep mixing does not affect significantly the overall results of the chemical evolution models, due to the fact that the Galactic evolution of $^{12}$ C and $^{13}$ C is mainly governed by stars in which this process is not expected to occur. |
Therefore. the main conclusions reached in the present. paper should not be aveely allected by the adoption of stellar. vields taking exra-müxing into account. | Therefore, the main conclusions reached in the present paper should not be largely affected by the adoption of stellar yields taking extra-mixing into account. |
On the other hand. the discrepancy ανασα model xedietions and. observations can be solved if the adopted stellar vields overestimate the effect. of IDD: running a model which adopts AGB vields computed with less LBB (tables from 22 to 31 of van den Lloek Crocsneweeen 1997) cads toa PCC ratio which decreases in he last 12 Gyr (Fie. | On the other hand, the discrepancy between model predictions and observations can be solved if the adopted stellar yields overestimate the effect of HBB: running a model which adopts AGB yields computed with less HBB (tables from 22 to 31 of van den Hoek Groenewegen 1997) leads to a $^{12}$ $^{13}$ C ratio which decreases in the last 12 Gyr (Fig. |
3). in agreement with observations. | 3), in agreement with observations. |
11owever. in this case the predicted solar abundance of IC is lower (PC. = 148 « 7). | However, in this case the predicted solar abundance of $^{13}$ C is lower $^{13}$ $_\odot$ = 1.48 $\times$ $^{-5}$ ). |
Notice also that the level of LLDD required by his mocel isobservations in stars. | Notice also that the level of HBB required by this model is in stars. |
issues described in BOT. including: resolution (Pickettetal.2003:Bolev&Durisen2003).. radiative (ransport algorithms (Boleyetal.2007).. irradiation (Caietal.2008).. and viscosity (Pickett&Durisen2007). | issues described in B07, including: resolution \citep{pickett03, bd08}, radiative transport algorithms \citep{boley07}, , irradiation \citep{cai08}, , and viscosity \citep{pickett07}. |
. Iu this paper. we locus on differences in the radiative schemes by using initial conditions and input physics Chat are as similar as possible to those described in DOT and by keeping as much of the numerical treatment as close to the techniques ancl conditions regularly used by Boss. | In this paper, we focus on differences in the radiative schemes by using initial conditions and input physics that are as similar as possible to those described in B07 and by keeping as much of the numerical treatment as close to the techniques and conditions regularly used by Boss. |
By emploving a different ancl. we think. better treatment of (he radiation physics. we find. as we have before. that realistic cooling is not nearly strong enough to initiate clump formation. | By employing a different and, we think, better treatment of the radiation physics, we find, as we have before, that realistic cooling is not nearly strong enough to initiate clump formation. |
This paper is organized as lollows. | This paper is organized as follows. |
In Section 2. we describe our numerical methodology and initial axisvimnetric equilibrium state. | In Section 2, we describe our numerical methodology and initial axisymmetric equilibrium state. |
We describe the results for a simulation lasting five outer rotation periods in Section 3. | We describe the results for a simulation lasting five outer rotation periods in Section 3. |
In Section 4. we compare our results with those in DOT and trv to isolate the causes for our differences. | In Section 4, we compare our results with those in B07 and try to isolate the causes for our differences. |
Section 5 is a brief summary. | Section 5 is a brief summary. |
We conducted our three-dimensional disk simulation using the CIIYMERA (Computational IEYdrodyvnamies with MultiplE Radiation Algorithms) code. developed αἱ Indiana University (e.g..Dolevetal.2007). | We conducted our three-dimensional disk simulation using the CHYMERA (Computational HYdrodynamics with MultiplE Radiation Algorithms) code, developed at Indiana University \citep[e.g.,][]{boley07}. |
. CIIYMERA uses an Eulerian. scheme that is fully second-order in space and time to solve the equations of hvdrodynanmies and Poisson's equation on a evlindrical grid. | CHYMERA uses an Eulerian scheme that is fully second-order in space and time to solve the equations of hydrodynamics and Poisson's equation on a cylindrical grid. |
The erid has a resolution of (256.512.64) in eviindrical coordinates (r. ©. 2). | The grid has a resolution of (256,512,64) in cylindrical coordinates $r$, $\phi$, $z$ ). |
The disk is initially located between radial cells 40 and 202. corresponding (to 4 and 20 AU. but is [ree to expand both racially and vertically. | The disk is initially located between radial cells 40 and 202, corresponding to 4 and 20 AU, but is free to expand both radially and vertically. |
Ivclrodvnamic boundary conditions are outflow along the top. sides ancl inner hole of the erid: material from the disk (hat moves inside the inner radius al radial cell number13is | Hydrodynamic boundary conditions are outflow along the top, sides and inner hole of the grid; material from the disk that moves inside the inner radius at radial cell number13is |
are adopted in our model of the dACN distribution. | are adopted in our model of the dAGN distribution. |
Under the assumption that significant nuclear activities can be triggered only in gasrich progenitor galaxies with central MDBIIs when their companion galaxies are sufficiently close to them. the dACN frequeuey aud their separation distribution are obtained in Section ??.. | Under the assumption that significant nuclear activities can be triggered only in gas-rich progenitor galaxies with central MBHs when their companion galaxies are sufficiently close to them, the dAGN frequency and their separation distribution are obtained in Section \ref{sec:results}. |
By comparison with observations. we show that the observational frequency of dACGNs is. cousistent with current observational constraints ou the merecr rates of ealaxies aud the scenario that major mergers of galaxies lead to significant uuclear activities. | By comparison with observations, we show that the observational frequency of dAGNs is consistent with current observational constraints on the merger rates of galaxies and the scenario that major mergers of galaxies lead to significant nuclear activities. |
Iu Section ??.. we also preseut our model predietious for the dACN distribution at hieher redshifts. | In Section \ref{sec:results}, we also present our model predictions for the dAGN distribution at higher redshifts. |
We further the uncertainties in our estimates of the dAGNdiscuss frequency due to different processes. which are uot iucluded iu our model. such as iiergers of progenitor galaxies whose uuclei have already been activated m previous niergers and also tidally-induced uuclear activities in two galaxies which are still far away from each other. | We further discuss the uncertainties in our estimates of the dAGN frequency due to different processes, which are not included in our model, such as mergers of progenitor galaxies whose nuclei have already been activated in previous mergers and also tidally-induced nuclear activities in two galaxies which are still far away from each other. |
Conclusions are eiven in Section ?7.. | Conclusions are given in Section \ref{sec:conclusions}. |
Iu this section. we introduce a phenomenological model to estimate the frequency of dACNs. by taking into account the factors mentioned in Section ??.. | In this section, we introduce a phenomenological model to estimate the frequency of dAGNs, by taking into account the factors mentioned in Section \ref{sec:intro}. |
The crucial elements of this model are described as follows. | The crucial elements of this model are described as follows. |
For galaxies with stellar mass zcAL. we describe their merecr rate af a eiven time bv the πανα: fraction of those galaxies for which a galaxy merger completed per uuit time. aud denote the merger rate by where f.=[|[dz ds the cosinie time at redshift :. as LOPLreseuts the mass ratio of two mereine galaxies. Deosο1)stellarM2) is the comoving nuniboer density of ealaxies with mass >AL. at redshift τν and (doyft.jdt. eives the comoving πο. deusitv of those galaxies that are products of mergers of two progenitor galaxies with mass ratio &c and with thei mergers being completed over a cosuuic time from f. tot.|dfi. | For galaxies with stellar mass $\geq M_*$, we describe their merger rate at a given time by the number fraction of those galaxies for which a galaxy merger completed per unit time, and denote the merger rate by where $t_{z}= \int^{\infty}_{z} |\frac{dt}{dz'}| dz'$ is the cosmic time at redshift $z$, $x$ $\le 1$ ) represents the mass ratio of two merging galaxies, $n\tot(\geq M_*,z)$ is the comoving number density of galaxies with stellar mass $\geq M_*$ at redshift $z$, and $(dn\mrg/dt_z)dt_z$ gives the comoving number density of those galaxies that are products of mergers of two progenitor galaxies with mass ratio $\ge x$ and with their mergers being completed over a cosmic time from $t_z$ to $t_z+dt_z$. |
Iu the past several vears. tremendous efforts have becoe made in estimating the mereer rate of galaxies cithe through close pairs of galaxies or throug[um morphological disturbances of galaxies fouud iu various deep surveys 20093. | In the past several years, tremendous efforts have been made in estimating the merger rate of galaxies either through close pairs of galaxies or through morphological disturbances of galaxies found in various deep surveys . |
. The dependence of the mereer rate R(2M...2) onu mass ratio ο can be absorbed im a function fle). which describes the fraction of mergers with mass ratio larger than c. ic. Moat) =fé Ms. | The dependence of the merger rate ${\cal R}(\geq M_*,x,z)$ on mass ratio $x$ can be absorbed in a function $f(x)$, which describes the fraction of mergers with mass ratio larger than $x$, i.e., M_*,x,z) =f(x) M_*,z). |
For major muerecrs (ustially defined by ia=>1/3). observational estimates can be M fitted by το shuuple formula. RlHnMSAMaajdolp", where huajor70.2|(ALEFICyr WaALhuajior71.65O15loe(AL"m and Afy=2«101"2010).. | For major mergers (usually defined by $x\geq 1/3$ ), observational estimates can be roughly fitted by the simple formula, $
R(\geq M_*,z)=A(M_*)_{\rm major}(1+z)^{\beta(M_*)_{\rm major}},
$ where $
A(M_*)_{\rm major} \approx 0.2[1+(M_*/M_0)^{0.5}]\Gyr^{-1},
$ $
\beta(M_*)_{\rm major} \approx1.65-0.15\log(M_*/M_0),
$ and $M_0=2\times 10^{10}\msun$. |
The normalization of the merecr rate ACAL) is uncertain by a factor of about 2 due to systematic errors and the unucertaüutv in the evolution slope (ALJ is A~OLS0.20. | The normalization of the merger rate $A(M_*)$ is uncertain by a factor of about $2$ due to systematic errors and the uncertainty in the evolution slope $\beta(M_*)$ is $\Delta\beta \sim
0.15-0.20$. |
The dependence of the merecr rates on dass ratio .0 can be approximated by fle)xePPLve)2010).. | The dependence of the merger rates on mass ratio $x$ can be approximated by $f(x)\propto x^{-0.3}(1-x)$. |
The stellar mass function of galaxies involved iu equation (1)). has been estimated over a large redslüft range from various galaxy redshift survevs. | The stellar mass function of galaxies, involved in equation \ref{eq:mrg}) ), has been estimated over a large redshift range from various galaxy redshift surveys. |
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