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In particular. they tabulate median values of log Big. which we use in the followinganalyses. | In particular, they tabulate median values of $\log$ $^{\rm \prime}_{\rm HK}$ which we use in the followinganalyses. |
We cross-referenced their data with the ValentiandFischer data ancl exclucled stars with Fr« 5500 Ix. and | We cross-referenced their data with the \citet{vf05} data and excluded stars with $_{\rm eff} <$ 5500 K and |
where Ar is a distance about Plauck length scale. aud will be called as spread range of the horizon hereafter. | where $\Delta r$ is a distance about Planck length scale, and will be called as spread range of the horizon hereafter. |
With these cousicderations. the original classical horizon became to the so called which has a spread range aud its space-time is non-conmmautative in that ranec. | With these considerations, the original classical horizon became to the so called which has a spread range and its space-time is non-commutative in that range. |
Nou-comuttative relation eq.(12)) iius (ΔΑνοop. | Non-commutative relation \ref{trnr}) ) means $(\Delta t)(\Delta
r) |_{r\sim r_H}\sim l_p^2$. |
This space-time uucertaimty relation can be re-explained by other view. | This space-time uncertainty relation can be re-explained by other view. |
In eq.(37)). QLE of the black hole is the function of the radial coordinate 7 aud the parameters of the black hole my. | In \ref{39}) ), QLE of the black hole is the function of the radial coordinate $r$ and the parameters of the black hole $r_H$. |
Now. we consider a quanti fluctuation of the QLE where Ar and Ary represent the fluctuations of the radial and the ones of parameters respectively. | Now, we consider a quantum fluctuation of the QLE where $\Delta r$ and $\Delta r_H$ represent the fluctuations of the radial and the ones of parameters respectively. |
Near the horizon. i.e. r=lie«(eg|6). we lave where D(r)2Εμ) has been used. | Near the horizon, i.e., $r= \lim_{\epsilon\rightarrow 0} (r_H+\epsilon)$, we have where $D(r)=f(r)(r-r_H)$ has been used. |
If requiring the fluctuation of the QLE to befinite. then we lave By using the Heiseuberg uncertainty relation (ΔΑErg)~1. theu we get again. | If requiring the fluctuation of the QLE to befinite, then we have By using the Heisenberg uncertainty relation $(\Delta t)(\Delta
E_{QLE})\sim 1$, then we get $(\Delta t)(\Delta r)|_{r\sim
r_H}\sim G=l_p^2$ again. |
Iu this section. we will construct a model based on the to discuss the thermodynamic of the Schwarzschild black hole. | In this section, we will construct a model based on the to discuss the thermodynamic of the Schwarzschild black hole. |
The Sclyvarzsclild metric is given by the Eq.(11)}. | The Schwarzschild metric is given by the \ref{Schwarzchild}) ). |
Treating the black hole as quantum state. the cuerev of quanti states is the QLE at the horizon. and then we have the space-time nou-comuuutative 12)). | Treating the black hole as quantum state, the energy of quantum states is the QLE at the horizon, and then we have the space-time non-commutative \ref{trnr}) ). |
Now. we construct a model with nonconuuutative o-ficld in the range of tryArg|A} aud with conmuutative o-ficlds in the ranges of {Q.ryAb and try|ALL}. ( | Now, we construct a model with noncommutative $\phi$ -field in the range of $\{r_H -\Delta, r_H + \Delta\}$ and with commutative $\phi$ -fields in the ranges of $\{0, r_H -\Delta \}$ and $\{r_H +\Delta,L\}$. ( |
for simplicity we use A to denote Ar hereafter). | for simplicity we use $\Delta$ to denote $\Delta r$ hereafter). |
As the cliscussion in the Introduction. that A should be au trinsic quantity of the model. which characterizes the boundary between the noucommutative ranee and the conmuuutative space-time ranges. aud it could be determined by the dynamics of the model. | As the discussion in the Introduction, that $\Delta$ should be an intrinsic quantity of the model, which characterizes the boundary between the noncommutative space-time range and the commutative space-time ranges, and it could be determined by the dynamics of the model. |
We rewrite eq.(12)) as follows where 2 is antisvinmetrical with 591— 1. | We rewrite \ref{trnr}) ) as follows where $\varepsilon^{ij}$ is antisymmetrical with $\varepsilon^{01}=1$ . |
The star product of two function f(r} and g(r) is given by the Moval formula (which has been studied iu tikzinarkinainBodyCitationStart3317|12|)): | The star product of two function $f(x)$ and $g(x)$ is given by the Moyal formula (which has been studied in ): |
Results are presented for a at. low-density cosmology (Q)=0.35.040.65.0.71.06 0.9). using 160 cach of gas and. dark matter particles within a box of comoving ength. 100. Mpe. | Results are presented for a flat, low-density cosmology $\Omega_{0}=0.35, \Omega_{\Lambda} = 0.65, h=0.71, \sigma_8=0.9$ ), using $160^3$ each of gas and dark matter particles within a box of comoving length, $100h^{-1}$ Mpc. |
We consider 2 models: the model. includes radiative cooling of the gas. anc thePreheating model for which we additionally preheat the gas w raising its specific thermal energy by 1.5 keV per particle ab 2=d. | We consider 2 models: the model includes radiative cooling of the gas, and the model for which we additionally preheat the gas by raising its specific thermal energy by 1.5 keV per particle at $z=4$. |
In both models. we adopted a time-dependent elobal metallicitv. Z=O30l)Z.. where fy~13€ vr is he current age of the universe. | In both models, we adopted a time-dependent global metallicity, $Z=0.3(t/t_0)Z_{\odot}$, where $t_0 \sim 13$ Gyr is the current age of the universe. |
As discussed. in. M2002. roth moclels reproduce the X-ray cluster scaling relations. although the model contains a significantly higher cooled. fraction than the model (15 per cent. as opposed to 70.5 per cent). | As discussed in M2002, both models reproduce the X-ray cluster scaling relations, although the model contains a significantly higher cooled fraction than the model (15 per cent, as opposed to $\sim 0.5$ per cent). |
In the former model. cooling is limited by numerical resolution. | In the former model, cooling is limited by numerical resolution. |
The cooled gas fraction is larger than that determined by Balogh et al. ( | The cooled gas fraction is larger than that determined by Balogh et al. ( |
2001). using results from the 2ALASS and 2dE galaxy surveys (Cole ct al. | 2001), using results from the 2MASS and 2dF galaxy surveys (Cole et al. |
2001). but in close agreement with the observed. value fron the SDSS (Blanton et al. | 2001), but in close agreement with the observed value from the SDSS (Blanton et al. |
2001). | 2001). |
Simulatecl cluster catalogues were produced: using the same method as described in M2002. | Simulated cluster catalogues were produced using the same method as described in M2002. |
In summary. clusters were identified as clumps of particles within spheres of average overdensities compared. to the comoving critical density. centred on the position of the densest dark matter particle. | In summary, clusters were identified as clumps of particles within spheres of average overdensities compared to the comoving critical density, centred on the position of the densest dark matter particle. |
For this paper. we adopt an overdensity of 1000: the associated radius. Z/000 is approximately half the size of the virial radius. and is comparable to the extent currently probed byChandra. | For this paper, we adopt an overdensity of 1000; the associated radius, $R_{1000}$ is approximately half the size of the virial radius, and is comparable to the extent currently probed by. |
Maps of emission-weiehted temperature were made for each of the 9 most massive clusters (Adjouo=1.+.107. 1M.) in the catalogue. and for the corresponding clusters in the catalogue. | Maps of emission-weighted temperature were made for each of the 9 most massive clusters $M_{1000} = 1-4 \times 10^{14} h^{-1}
\mathrm{M_{\odot}}$ ) in the catalogue, and for the corresponding clusters in the catalogue. |
Each map was produced bv first locating all hot. (2° 101) gas particles within a cube of length. 272,000 along cach side. centred. on the cluster of interest. | Each map was produced by first locating all hot $T>10^5$ K) gas particles within a cube of length, $2R_{1000}$ along each side, centred on the cluster of interest. |
“Phe desired quantity was then smoothee onto a 3D array using the SPII interpolation method (c.g. Monaghan 1992) where The sums ; extend over all particles and. & over all voxels. | The desired quantity was then smoothed onto a 3D array using the SPH interpolation method (e.g. Monaghan 1992) where The sums $i$ extend over all particles and $k$ over all voxels. |
cl; ds the value of the quantity for. particle 7 at position r;. r; is the centroid: position of voxel j. Ari orf. 7 isa weight factor. VWGNA) is the same SPILL smoothing kernel used by (Thomas Couchman 1992) and ; is the smoothing length of particle ; (M0 as Aro» 2h). | $A_i$ is the value of the quantity for particle $i$ at position ${\bf r}_i$, ${\bf r}_j$ is the centroid position of voxel $j$, $\Delta r_{ij} = | {\bf r}_j - {\bf r}_i |$ , $w_i$ is a weight factor, $W(\Delta r, h)$ is the same SPH smoothing kernel used by (Thomas Couchman 1992) and $h_i$ is the smoothing length of particle $i$ $W \rightarrow 0$ as $\Delta r \rightarrow 2h$ ). |
Equation 2. normalizes the contribution of each particle to be zt; when summed. over all voxels. | Equation \ref{eqn:wnorm} normalizes the contribution of each particle to be $A_i$ when summed over all voxels. |
For all mass-weightecd quantities. w;,=am; and for emission-weighted quantities. πρ=m;n;NZ). for particle ¢ with mass mz. density n;. temperature 7; ancl X-ray cmiissivity. =neACL.Z): the cooling function. (7. is the same function adopted for our simulations. using tablesZ) published in Sutherland Doptia (1993). | For all mass-weighted quantities, $w_i = m_i$ and for emission-weighted quantities, $w_i = m_i n_i \Lambda(T_i,Z)$, for particle $i$ with mass $m_i$ , density $n_i$, temperature $T_i$ and X-ray emissivity, $\epsilon_i = n_i^2 \Lambda(T_i,Z)$; the cooling function, $\Lambda(T,Z)$ is the same function adopted for our simulations, using tables published in Sutherland Doptia (1993). |
The smoothed clistribution was then projected onto a G4.64 grid. with the width of cach pixel being approximately 7 to 10 5.tkpe. smaller than the gravitational softening length used in the simulations (25 h!kpe). | The smoothed distribution was then projected onto a $64 \times 64$ grid, with the width of each pixel being approximately 7 to 10 $h^{-1}\mathrm{kpc}$, smaller than the gravitational softening length used in the simulations (25 $h^{-1}\mathrm{kpc}$ ). |
lies. | Figs. |
1 2. show the emission-weighted temperature maps (with superimposed X-ray surface brightness contours and projected. velocity vectors) for thePrefeating and rans. respectively, | \ref{fig:phmap} \ref{fig:radmap} show the emission-weighted temperature maps (with superimposed X-ray surface brightness contours and projected velocity vectors) for the and runs respectively. |
Surface. brightness: contours are normalized to the maximum value. with cach contour representing variation bv an order of magnitude. | Surface brightness contours are normalized to the maximum value, with each contour representing variation by an order of magnitude. |
Velocities (in km/s) are mass-weighted and the length of cach velocity vector represents the tangential speed. (i.e. across the map rane) of the gas at that point. | Velocities (in km/s) are mass-weighted and the length of each velocity vector represents the tangential speed (i.e. across the map plane) of the gas at that point. |
Positive temperature [uctuations are evident of up o 3 times the average. associated with compression. of he eas. | Positive temperature fluctuations are evident of up to 3 times the average, associated with compression of the gas. |
There also exists bright clumps of cool gas. with emperatures less than half the mean value. | There also exists bright clumps of cool gas, with temperatures less than half the mean value. |
Phese features are far more abundant in the maps than in hePreheating maps because the energy. injection in thePrehealing run. was large. enough to erase a significant amount of substructure. | These features are far more abundant in the maps than in the maps because the energy injection in the run was large enough to erase a significant amount of substructure. |
Among the 9 clusters shown in Fig. L.. | Among the 9 clusters shown in Fig. \ref{fig:phmap}, |
several show clear bimoclal structure (clusters 1. 2 and S). | several show clear bimodal structure (clusters 1, 2 and 8). |
This is most clear for the 27 ane S maps. in each of which 2 cold elumps appear to be moving away from one another. | This is most clear for the $^{\rm nd}$ and $^{\rm th}$ maps, in each of which 2 cold clumps appear to be moving away from one another. |
In the maps (Fig. 2)) | In the maps (Fig. \ref{fig:radmap}) ) |
the bimodal structure of map 1 is not so clear. | the bimodal structure of map 1 is not so clear. |
The ΚΙ eluster shows evidence of a sharp boundary at the leacing edge of the cold clump in the upper right quadrant: when the maps for this cluster are re-centred to show the cold clump in the lower right of the map. a similar sharp edge can be seen. | The $^{\rm th}$ cluster shows evidence of a sharp boundary at the leading edge of the cold clump in the upper right quadrant; when the maps for this cluster are re-centred to show the cold clump in the lower right of the map, a similar sharp edge can be seen. |
These features are indicative of observed cold fronts and this is investigated in the next section. | These features are indicative of observed cold fronts and this is investigated in the next section. |
The velocity vectors superimposed on the cluster maps show clear evidence for rotation in some cases (For example maps 3. 5 and 9). | The velocity vectors superimposed on the cluster maps show clear evidence for rotation in some cases (for example maps 3, 5 and 9). |
Ritehie Thomas (2001) found. similar rotation of the velocity field in. their simulated: off-centre merger of two equal mass clusters. | Ritchie Thomas (2001) found similar rotation of the velocity field in their simulated off-centre merger of two equal mass clusters. |
Except for map 9. | Except for map 9. |
where the surface brightness contours are essentially round. the contours are gencrally elliptical. again as found for merging clusters. | where the surface brightness contours are essentially round, the contours are generally elliptical, again as found for merging clusters. |
The spiral patterns seen in some of the temperature maps (e.g. 3 5) may be due to heating by shocks produced during the merger as was seen in the simulations of Hitchie Thomas during the interaction of cluster cores. | The spiral patterns seen in some of the temperature maps (e.g. 3 5) may be due to heating by shocks produced during the merger as was seen in the simulations of Ritchie Thomas during the interaction of cluster cores. |
With its rouncer surface brightness contours. cluster 9 may be ina later. more relaxed state. | With its rounder surface brightness contours, cluster 9 may be in a later, more relaxed state. |
The centres of all. clusters show a decline. in temperature. | The centres of all clusters show a decline in temperature. |
This is in contrast to the simulated temperature maps of Loken et al. ( | This is in contrast to the simulated temperature maps of Loken et al. ( |
2002). produced. using an Adaptive Alesh-Refinement code. where the racial temperature profiles continue to rise at small ας. | 2002), produced using an Adaptive Mesh-Refinement code, where the radial temperature profiles continue to rise at small radii. |
The Santa Barbara cluster comparison project. (Frenk ct al.. | The Santa Barbara cluster comparison project (Frenk et al., |
1999) found the general result that for non-raciative gas. SPIEL codes produce a central Hat or slightly declining temperature profile. while all of the grid codes produce a temperature profile which continues to rise to the resolution limit. | 1999) found the general result that for non-radiative gas, SPH codes produce a central flat or slightly declining temperature profile, while all of the grid codes produce a temperature profile which continues to rise to the resolution limit. |
This conllict between results using the dillerent codes still needs to be resolved. | This conflict between results using the different codes still needs to be resolved. |
However. Fabian (2002) noted that all of the cores mapped. so far by Chandra: with radiative cooling times of a few Civr showsignificant central temperature drops. | However, Fabian (2002) noted that all of the cores mapped so far by Chandra with radiative cooling times of a few Gyr showsignificant central temperature drops. |
This temperature drop of afactor of about 3 or more in the central region of the clusters was | This temperature drop of afactor of about 3 or more in the central region of the clusters was |
secondly. we examined the perturbation amplitudes of the wave. | Secondly, we examined the perturbation amplitudes of the wave. |
In both methods we focused on a ssector on the solar surface. where all four waves were distinctly observable FFig. | In both methods we focused on a sector on the solar surface, where all four waves were distinctly observable Fig. |
1 panel 1: vellow curves). | \ref{fig1} panel 7; yellow curves). |
In the first method the wave fronts were tracked manually. and their center was obtained from a circular fit to the earliest wave Ironts carried out in spherical coordinates 2006). | In the first method the wave fronts were tracked manually, and their center was obtained from a circular fit to the earliest wave fronts carried out in spherical coordinates \citep[see][]{veronig06}. |
. The mean distauce of each wave front [rom the thus determined. center is caleulated along the solar surface. | The mean distance of each wave front from the thus determined center is calculated along the solar surface. |
In the second method we subdivided the solar surface into spherical segments of equal width concentric around the wave center obtained by method one. | In the second method we subdivided the solar surface into spherical segments of equal width concentric around the wave center obtained by method one. |
Plotting the average intensities versus mean distances of all segments gives one intensity profile per RR image (c£.Muhretal.2010). | Plotting the average intensities versus mean distances of all segments gives one intensity profile per RR image \citep[cf.][]{muhr10}. |
. In these perturbation profiles. shown in Fig. | In these perturbation profiles, shown in Fig. |
4 (Lop panels). the wave [ront presents itself as clistinet bump above the backeround intensity. | \ref{fig4} (top panels), the wave front presents itself as distinct bump above the background intensity. |
In each case. the perturbation amplitude reaches its maximum 5 minutes alter the onset ol the wave (see Fig. 4)). | In each case, the perturbation amplitude reaches its maximum $\sim$ 5 minutes after the onset of the wave (see Fig. \ref{fig4}) ). |
As example. the evolution of the perturbation amplitude of wave 4is plotted in Fig. | As example, the evolution of the perturbation amplitude of wave 4 is plotted in Fig. |
3. (bottom panel). | \ref{fig3} (bottom panel). |
From the wave perturbation profiles we extracted the foremost position of the wave front. defined as the point at which the gaussian fit to the profile [alls below the value of f/f=1.02 (blue dotted line in top panel of Fie. 4)). | From the wave perturbation profiles we extracted the foremost position of the wave front, defined as the point at which the gaussian fit to the profile falls below the value of $I/I{_0} = 1.02$ (blue dotted line in top panel of Fig. \ref{fig4}) ). |
We note that the visually tracked cdistauces match well the position of (he wave front obtained [rom the perturbation profiles (Fig. 4)). | We note that the visually tracked distances match well the position of the wave front obtained from the perturbation profiles (Fig. \ref{fig4}) ). |
In Fig. | In Fig. |
3 (top panel) we show the kinematics for wave 4 derived with both methods together with error bars. | \ref{fig3} (top panel) we show the kinematics for wave 4 derived with both methods together with error bars. |
Linear as well as «quadratic least squares fits were applied to the time-cistance data. both vielding similar velocities of ~340kams.![. | Linear as well as quadratic least squares fits were applied to the time-distance data, both yielding similar velocities of $\sim340\,\mathrm{km\,s}^{-1}$. |
The parabolic fit suggests a small deceleration of —10nmis7. | The parabolic fit suggests a small deceleration of $-10\,\mathrm{m\,s}^{-2}$. |
In order to distinguish whether the linear or parabolic fit better represents the data. we derived the confidence interval for the linear fit. | In order to distinguish whether the linear or parabolic fit better represents the data, we derived the confidence interval for the linear fit. |
The quadratic fit Bes within the error bars and the confidence interval. thus it is reasonable to represent the waves kinematics by (he linear fit with a constant velocity of 337£31kms1 over (he full propagation clistauce up to 800Man. | The quadratic fit lies within the error bars and the confidence interval, thus it is reasonable to represent the wave's kinematics by the linear fit with a constant velocity of $337 \pm 31\,\mathrm{km\,s}^{-1}$ over the full propagation distance up to $800\,\mathrm{Mm}$. |
We found similar results for the other three wave events under study. which is illustrated in Fig. | We found similar results for the other three wave events under study, which is illustrated in Fig. |
4 (bottom panels). | \ref{fig4} (bottom panels). |
The velocities ος of the four homologous waves are: 257422kins (wave 1). 219EI8kms.| (wave 2). 2494I8kis.l| (wave 3) and 33731kms.| (wave 4). | The velocities $v_{\mathrm{c}}$ of the four homologous waves are: $257\pm22\,\mathrm{km\,s}^{-1}$ (wave 1), $219\pm18\,\mathrm{km\,s}^{-1}$ (wave 2), $249\pm18\,\mathrm{km\,s}^{-1}$ (wave 3) and $337\pm31\,\mathrm{km\,s}^{-1}$ (wave 4). |
All peak perturbation profiles together with a gaussian fit are shown in Fig. | All peak perturbation profiles together with a gaussian fit are shown in Fig. |
4. (top panel). | \ref{fig4} (top panel). |
The perturbation profiles of all [our waves are steepening and show an increase in amplitude in (he early phase of their evolution until the peak perturbation amplitude lay is reached. | The perturbation profiles of all four waves are steepening and show an increase in amplitude in the early phase of their evolution until the peak perturbation amplitude $A_{\mathrm{max}}$ is reached. |
The values «μας of all four waves are: 1.15 (wave D). 1.1 (wave 2). 1.14 (wave 3) ancl 1.24 (wave 4). | The values $A_{\mathrm{max}}$ of all four waves are: $1.15$ (wave 1), $1.1$ (wave 2), $1.14$ (wave 3) and $1.24$ (wave 4). |
The bright fronts of coronal EUV waves are in general caused by a local temperature and | The bright fronts of coronal EUV waves are in general caused by a local temperature and |
ny. | mJy. |
More observatious between 10.6 GIIz aud 115 Giz would be needed to predict this contribution more accurately. | More observations between 10.6 GHz and 113 GHz would be needed to predict this contribution more accurately. |
Note that these values refer to the integrated flux density of the radio source. | Note that these values refer to the integrated flux density of the radio source. |
However. Figure 2 shows that the radio source is resolved iuto three componenuts. which appear to have roughly simular flux deusities at 8.1 CGIIz. | However, Figure \ref{radioKoverlay} shows that the radio source is resolved into three components, which appear to have roughly similar flux densities at 8.4 GHz. |
Unfortunately. we do not have spectral iudex information of the individual components. but im most radio galaxies. the outer lobes have steeper spectra than the core2000).. although sources with relatively steep spectiui cores have also been seen at hieh redshift1997). | Unfortunately, we do not have spectral index information of the individual components, but in most radio galaxies, the outer lobes have steeper spectra than the core, although sources with relatively steep spectrum cores have also been seen at high redshift. |
. The orientation of the 112.7 GIIz PdBI svuthesized beam is roughly perpendicular to the radio axis. which means we should slightly resolve the svuchrotron component at 112.7 CGIIz. | The orientation of the 112.7 GHz PdBI synthesized beam is roughly perpendicular to the radio axis, which means we should slightly resolve the synchrotron component at 112.7 GHz. |
Depending on the spectral iudex of the northeru hotspot. this further reduces the svuchrotrou contribution at the position of the ACN to ὃς0.5 ids. | Depending on the spectral index of the northern hotspot, this further reduces the synchrotron contribution at the position of the AGN to $\simlt 0.5$ mJy. |
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