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If the oulv information about the field f is a list of coordinates r,=(r,.0,.0,) with p=L......N (where NV is the umber of galaxies iu the latter example). the survey may be cousidered as a superposition of 3D Dirac deltas aud cach cocticient fri, can simply be estimated with a stun (2°22?) This formulation has been used for the analysis of shallow galaxy surveys such as the IRAS 1.213J survey Gkealaxics, 22?) and the 2\IRS survey (2\LASSRedshiftSurvey.~Lkgalaxies. ??).. | If the only information about the field $f$ is a list of coordinates $\textbf{r}_p = (r_p,\theta_p,\phi_p)$ with $p=1,\dots,N$ (where $N$ is the number of galaxies in the latter example), the survey may be considered as a superposition of 3D Dirac deltas and each coefficient $f_{\ell mn}$ can simply be estimated with a sum \citep{Heavens:1995,Fisher:1995,Erdogdu:2006dv,cmbbox} This formulation has been used for the analysis of shallow galaxy surveys such as the IRAS 1.2mJ survey \citep[$\sim6k$ galaxies,][]{IRAS,Fisher:1995,Heavens:1995}, and the 2MRS survey \citep[2MASS Redshift Survey, $\sim 45$k galaxies,][]{2MRS,Erdogdu:2006dv,Erdogdu:2005wi}. |
Since the time to calculate equation { is proportional to Niastfiax|d Equation { will become highly time-consumingwhen applied to larger surveys or when precise decomposition is required. (large Haws. 2010 (6,). | Since the time to calculate equation \ref{rawestimate} is proportional to $N n_{\rm max}(\ell_{\rm max}+1)^2/2$, Equation \ref{rawestimate} will become highly time-consumingwhen applied to larger surveys or when precise decomposition is required (large $n_{max}$ and $\ell_{max}$ ). |
Iu spherical coordinates. since 3D space eiu be viewed as an infinite series of closed shells Q(r). the spherical Fouricr-Bessel decomposition may also arise from repeated 2D spherical harmonic transforms to which spherical Bessel transforms are applied (?).. | In spherical coordinates, since 3D space can be viewed as an infinite series of closed shells $\Omega(r)$, the spherical Fourier-Bessel decomposition may also arise from repeated 2D spherical harmonic transforms to which spherical Bessel transforms are applied \citep{cmbbox}. |
Formally. the field fF even on each shell (7) is first expanded into spherical havuoics for which the inversion formula eives harmonics cocficients τω) depending ou the radius + It is then possible to perform: a spherical Besscl transforiu leading to the final Fouricr-Bessel coeffideuts ἔτι) This formulation/ hence extends the notion of 2D spherical harmonies to three-dimoeusional fields. | Formally, the field $f$ given on each shell $\Omega(r)$ is first expanded into spherical harmonics for which the inversion formula gives harmonics coefficients $f_{\ell m}(r)$ depending on the radius $r$ It is then possible to perform a spherical Bessel transform leading to the final Fourier-Bessel coefficients $f_{\ell m}(k)$ This formulation hence extends the notion of 2D spherical harmonics to three-dimensional fields. |
It is also possible to conceive the reverse approach. ie. fo perform the spherical Bessel trausform first and subsequeutlv expand the resulting cocfücieuts iuto spherical harmonics. | It is also possible to conceive the reverse approach, i.e. to perform the spherical Bessel transform first and subsequently expand the resulting coefficients into spherical harmonics. |
Formally. the (th order spherical Bessel traustorm of f (similar to its Hankel transform) is for which the inversion foruula elves The result is then expanded iuto spherical harmonics but with an uuusual formulation since fr(h.0.0) aud Vii(O00) (as well as the basis functions dike) and Vi, (0.0))have now the ( parameter iun colon: Again. using the inversion formula. we obtain the Fourier-Bessel coefficieuts Due to the closed domains of shells O(rj aud thus the relative iudependeuce of aneular aud radial dinieusious. the (equations 1. and 2)) the (denoted by SUB for spherical-Hurinonic-Dessel. equatious 8. to 11)) aud the(denoted bv SDII for Desscl-HÉHarionic.. equatious 12. to 15) methods are equivalent formulations of the spherical Fourier-Dessel decomposition of any threc-diuensioual feld νι δω]. | Formally, the $\ell$ -th order spherical Bessel transform of $f$ (similar to its Hankel transform) is for which the inversion formula gives The result is then expanded into spherical harmonics but with an unusual formulation since $f_\ell (k,\theta,\phi)$ and $Y_{\ell m}(\theta,\phi)$ (as well as the basis functions $j_\ell (kr)$ and $Y_{\ell m}(\theta,\phi)$ )have now the $\ell$ parameter in common: Again, using the inversion formula, we obtain the Fourier-Bessel coefficients Due to the closed domains of shells $\Omega(r)$ and thus the relative independence of angular and radial dimensions, the (equations \ref{thraw1} and \ref{thraw2}) ), the (denoted by SHB for , equations \ref{thdirect1} to \ref{thdirect4}) ) and the(denoted by SBH for , equations \ref{threverse1} to \ref{threverse4}) ) methods are equivalent formulations of the spherical Fourier-Bessel decomposition of any three-dimensional field $f(r,\theta,\phi)$ . |
This is stummarised in the following schematic description of cach method: | This is summarised in the following schematic description of each method: |
ININIIOI-21: Galactic nebulositv. with a sort of milky appearance. | KKH01-21: Galactic nebulosity, with a sort of milky appearance. |
WIL JO337+75: Spiralish galaxy. sliehtly huupy. | WHI J0337+75: Spiralish galaxy, slightly lumpy. |
UGCA 86: A nearby galaxy. (Ixarachentsevοἱal.2003). | UGCA 86: A nearby galaxy \citep{KSD03}. |
.. Not faint: included because of resolution on the survey plate. | Not faint; included because of resolution on the survey plate. |
Sienifieant areas of Πα emission. | Significant areas of $\alpha$ emission. |
WIL JOJ01--30: Galactic nebulositv: the figures in the table refer just to the brightest knot. | WHI J0401+80: Galactic nebulosity; the figures in the table refer just to the brightest knot. |
There is more spread throughout the whole frame. | There is more spread throughout the whole frame. |
ZOAG G135.05+16.10: Probably a heavily extincted. face-on spiral. | ZOAG G135.05+16.10: Probably a heavily extincted, face-on spiral. |
ZOAG G135.234-16.04: A lumpy bit of nebulositv? ( | ZOAG G135.23+16.04: A lumpy bit of nebulosity? ( |
Though it could possibly be a heavily extincted galaxy). | Though it could possibly be a heavily extincted galaxy). |
WII JOJ4314-44: Galactic nebulosity. | WHI J0431+44: Galactic nebulosity. |
WII 01-03: A Iumpy galaxy: it appeared in Whiting.Tan.&Irwin.(2002). | WHI B0441+02: A lumpy galaxy; it appeared in \citet{WHI02}. |
. ZOAG GLIG6T.44-04.35: Heavily extineted spiral. | ZOAG G167.44-04.85: Heavily extincted spiral. |
IXIXIIO1-29: Galactic nebulosity. involved wilh an apparent star cluster. | KKH01-29: Galactic nebulosity, involved with an apparent star cluster. |
The stars make it even more clifficult than normal to get a good surface brightness. | The stars make it even more difficult than normal to get a good surface brightness. |
WII J0512-00: Galactic nebulosity (superimposed on some distant. galaxies). | WHI J0512-00: Galactic nebulosity (superimposed on some distant galaxies). |
WII J05142-55: Nearly [ace-0n spiral. | WHI J0514+55: Nearly face-on spiral. |
WII J05154-56: Curve of galactic nebulositv. | WHI J0515+56: Curve of galactic nebulosity. |
WILL JO529+72: Of irregular shape. [rom the morphology. possibly an imegular galaxy or a Galactic rellection nebula. | WHI J0529+72: Of irregular shape, from the morphology possibly an irregular galaxy or a Galactic reflection nebula. |
The latter initially appeared more likely in light of the lack of H-alpha emission. | The latter initially appeared more likely in light of the lack of H-alpha emission. |
Huchtmeier&Skillman(1998) [find a redshift of 1089. so its probably ihe former. | \citet{HS98} find a redshift of 1089, so it's probably the former. |
WILL J06202-49: Wispy nebulosityv. against a background of distant galaxies. | WHI J0620+49: Wispy nebulosity, against a background of distant galaxies. |
The center is mottled. though. in a way thal suggests a dwarf galaxy. possibly nearing resolution. | The center is mottled, though, in a way that suggests a dwarf galaxy possibly nearing resolution. |
WIL J06232-09: Detected by IRAS. | WHI J0623+09: Detected by IRAS. |
Catalogued and analvzed by as a galaxy in the Zone of Avoidance. and listed as such in on-line databases. | Catalogued and analyzed by \citet{SSW96} as a galaxy in the Zone of Avoidance, and listed as such in on-line databases. |
However. it was not found in IHE by Pantojaetal.(1997): and was detected in a Cs line at 98 GIIz by Dronfnan.Nvman&May(1996). al a redshift of 35 km/s. Those data together with its morphology clearly identily it as Galactic nebulositv. apparently associated wilh a group of slars. | However, it was not found in HI by \citet{PAG97}; and was detected in a CS line at 98 GHz by \citet{BNM96} at a redshift of 35 km/s. Those data together with its morphology clearly identify it as Galactic nebulosity, apparently associated with a group of stars. |
IXIXIIO1-38: Probably a face-on barred spiral. | KKH01-38: Probably a face-on barred spiral. |
DIuish in color. with no apparent I-alpha | Bluish in color, with no apparent H-alpha |
dominating the spectrum at the softer energies (<2 keV) and the other one being more prominent above 2 keV. The properties of the soft component (<2 keV) could be reasonably well constrained i the case wwhere the absorption column density was relatively low C107? 2), whereas in the case of tthe detection of this component is less significant. | dominating the spectrum at the softer energies $\lesssim$ 2 keV) and the other one being more prominent above 2 keV. The properties of the soft component $\lesssim$ 2 keV) could be reasonably well constrained in the case where the absorption column density was relatively low $<$ $^{22}$ $^{-2}$ ), whereas in the case of the detection of this component is less significant. |
However. the similarity in the timing and spectral behavior observed in the quiescent state of the two sources argues in favor of adopting the same spectral model for both of them. | However, the similarity in the timing and spectral behavior observed in the quiescent state of the two sources argues in favor of adopting the same spectral model for both of them. |
We suggested that the model comprising a CUTOFFPL component at the higher energies plus a MKL component woulc provide a reasonable description of the data and a plausible physical explanation of the properties observed in the two sources (Sect. 5)). | We suggested that the model comprising a CUTOFFPL component at the higher energies plus a MKL component would provide a reasonable description of the data and a plausible physical explanation of the properties observed in the two sources (Sect. \ref{sec:discussion}) ). |
According to this interpretation. the MKL component would represent the contribution to the total X-ray emission of the shocks in the wind of the supergiant companion. | According to this interpretation, the MKL component would represent the contribution to the total X-ray emission of the shocks in the wind of the supergiant companion. |
The results of the fits with this model to the data of the three observations inferred a temperature of the MKL component and an emitting region comparable with the values found also in the case of the SFXT AXJJ1845.0-0433. | The results of the fits with this model to the data of the three observations inferred a temperature of the MKL component and an emitting region comparable with the values found also in the case of the SFXT J1845.0-0433. |
Similar soft spectral components have been detected in many other HMXBs and SGXBs. | Similar soft spectral components have been detected in many other HMXBs and SGXBs. |
In a few cases. the detection of a number of prominent emission lines in the high resolution X-ray spectra of these sources carried out with the gratings onboard aand the RGS onboard (seee.g.Watanabeetal.2006) have convincingly demonstrated that these components are produced by the stellar wind around the NS. and proved to be a powerful diagnostic to probe the structure and composition of the stellar wind in these systems. | In a few cases, the detection of a number of prominent emission lines in the high resolution X-ray spectra of these sources carried out with the gratings onboard and the RGS onboard \citep[see e.g.,][]{watanabe06} have convincingly demonstrated that these components are produced by the stellar wind around the NS, and proved to be a powerful diagnostic to probe the structure and composition of the stellar wind in these systems. |
The statistics of the present oobservations ts far too low to permit a similar in-depth studvá of the stellar wind in the case of παπά Furthermore. because of the relatively low luminosity and the high absorption that characterize the emission of these sources in quiescence. observations at higher spectroscopic resolution are probably too challenging for the present generation of X-ray satellites. and the improved sensitivity of the X-ray spectrometers planned for future X-ray missions (e.g. IXO) is probably required to firmly establish the presence of a soft spectral component in the quiescent emission of the SFXT sources and shed light on its nature. | The statistics of the present observations is far too low to permit a similar in-depth study of the stellar wind in the case of and Furthermore, because of the relatively low luminosity and the high absorption that characterize the emission of these sources in quiescence, observations at higher spectroscopic resolution are probably too challenging for the present generation of X-ray satellites, and the improved sensitivity of the X-ray spectrometers planned for future X-ray missions (e.g. IXO) is probably required to firmly establish the presence of a soft spectral component in the quiescent emission of the SFXT sources and shed light on its nature. |
If the harder X-ray emission (2-10 keV) detected from the oobservations of παπά wwas really produced by residual aceretion as we argued in the previous section. then the accretion process in these sources would take place over more than 4 orders of magnitude of X-rayluminosity?. | If the harder X-ray emission (2-10 keV) detected from the observations of and was really produced by residual accretion as we argued in the previous section, then the accretion process in these sources would take place over more than 4 orders of magnitude of X-ray. |
. This is similar to the results reported for the SEXT JJ17544-2619 (Rampyetal.2009) and. possibly. for the SFXT JJ1818.6-1703 (in.thelatterunclear.Bozzoetal..2009). | This is similar to the results reported for the SFXT J17544-2619 \citep{rampy09} and, possibly, for the SFXT J1818.6-1703 \citep[in the latter case the origin of the lowest quiescent emission
remains unclear,][]{bozzo09}. |
. In the case of JJ17544-2619. Rampyetal.(2009) ascribed the high dynamie range in the X-ray luminosity to the accretion of clumps from the wind of the supergiant star with a high density contrast with respect to the surrounding homogeneous wind. | In the case of J17544-2619, \citet{rampy09} ascribed the high dynamic range in the X-ray luminosity to the accretion of clumps from the wind of the supergiant star with a high density contrast with respect to the surrounding homogeneous wind. |
However. it was also suggested that a similar variability might result from the transition across different accretion regimes onto the NS (Bozzoetal..2008). | However, it was also suggested that a similar variability might result from the transition across different accretion regimes onto the NS \citep{bozzo08}. |
. We note that a similar scenario can be envisaged for interpreting the variations in the X-ray flux observed during the multiple small flares detected in the present observations. | We note that a similar scenario can be envisaged for interpreting the variations in the X-ray flux observed during the multiple small flares detected in the present observations. |
Even though they took place at a much lower luminosity level than the brightest outbursts (a factor of ~10°-10!). our analysis showed that all these events shared a number of similar timing and spectral properties. | Even though they took place at a much lower luminosity level than the brightest outbursts (a factor of $\sim$ $^3$ $^4$ ), our analysis showed that all these events shared a number of similar timing and spectral properties. |
In particular. the timescales on which the smaller flares develop ts comparable with the decay timescale of the source luminosity during the outbursts (see Sect. 1)). | In particular, the timescales on which the smaller flares develop is comparable with the decay timescale of the source luminosity during the outbursts (see Sect. \ref{sec:intro}) ), |
and the spectral photon indices and absorption column densities measured from the oobservations are also in. qualitative agreement with those reported previously when the sources were observed at a much higher X-ray luminosity level (see Sect. 1)). | and the spectral photon indices and absorption column densities measured from the observations are also in qualitative agreement with those reported previously when the sources were observed at a much higher X-ray luminosity level (see Sect. \ref{sec:intro}) ). |
It is thus most likely that the transitions between the lower quiescent states and the small flares detected by ftrom aand might have been triggered by the same mechanism that sometimes gives rise to the brightest outbursts .δ.. the aceretion of clumps from the stellar wind and/or the transition between different accretion regimes of the NS. see Sect. 1)). | It is thus most likely that the transitions between the lower quiescent states and the small flares detected by from and might have been triggered by the same mechanism that sometimes gives rise to the brightest outbursts (i.e., the accretion of clumps from the stellar wind and/or the transition between different accretion regimes of the NS, see Sect. \ref{sec:intro}) ). |
In contrast to the case for the SFXT JJ18483-0311. we did not detect any pulsation in the quiescent emission of either oor aand provided in Sects. | In contrast to the case for the SFXT J18483-0311, we did not detect any pulsation in the quiescent emission of either or and provided in Sects. |
4.1 and 4.2 the correspondg upper limits to the spin periods and pulsed fractions we were able to infer from the present data. | \ref{sec:xteresults} and \ref{sec:igrresults} the corresponding upper limits to the spin periods and pulsed fractions we were able to infer from the present data. |
Deep pointed observations of SFXTs in quiescence are still required in order to understand the origin of the peculiar X-ray variability of these sources and distinguish between different models proposed to interpret their behavior. | Deep pointed observations of SFXTs in quiescence are still required in order to understand the origin of the peculiar X-ray variability of these sources and distinguish between different models proposed to interpret their behavior. |
EB thanks N. Schartel and the sstaff for the rapid schedule of the oobservation of aafter the outburst occurred on 2009 March 10. and R. Farinelli for helpful discussions. | EB thanks N. Schartel and the staff for the rapid schedule of the observation of after the outburst occurred on 2009 March 10, and R. Farinelli for helpful discussions. |
We thank the anonymous referee for useful comments. | We thank the anonymous referee for useful comments. |
where wy)220.5. | where $x_{0}\approx0.5$. |
The inverse Compton spectrum above pla is similar to the svnchrotron one. which can be approximated by several power law segments. i.e. where we have neglected the logarithmic term for v>1ACnmt | The inverse Compton spectrum above $\nu_{a}^{\rm{IC}}$ is similar to the synchrotron one, which can be approximated by several power law segments, i.e. where we have neglected the logarithmic term for $\nu>\nu_{m}^{\rm{IC}}$. |
The SSC flux density begins to dominate over that of the svnchrotron radiation in the overall svnchrotron + SSC spectrum at the crossing point. which corresponds to zl. (Sari Esin 2001). | The SSC flux density begins to dominate over that of the synchrotron radiation in the overall synchrotron $+$ SSC spectrum at the crossing point, which corresponds to $\nu_{\times}^{\rm{IC}}$ (Sari Esin 2001). |
Using equation (38)) and the standard svnchrotvon spectrum. and assuming pK>vy,ve. we obtain the crossing point [requency for two cases. vf<pF and cyu.le where the coefficient is οι=J9r5(4225(1c—kUue(p-(p 1y*—. | Using equation \ref{eqn:fc:SSC-spectrum}) ) and the standard synchrotron spectrum, and assuming $\nu_{\times}^{\rm{IC}}>\nu_{m}>\nu_{c}$, we obtain the crossing point frequency for two cases, $\nu_{\times}^{\rm{IC}}<\nu_{c}^{\rm{IC}}$ and $\nu_{c}^{\rm{IC}}<\nu_{\times}^{\rm{IC}}<\nu_{m}^{\rm{IC}}$ , i.e. where the coefficient is $c_1=\displaystyle\frac{225(1-\epsilon)^2(p-1)^2}{49x_{0}^2(4-k-\epsilon)^2(p-2)^2}$. |
To derive the above equation we have used the relation Since L/(p—1) is always larger than 3/(2+3p). one can determine pl directlyby without judging whether pl<vl or not. | To derive the above equation we have used the relation Since $1/(p-1)$ is always larger than $3/(2+3p)$, one can determine $\nu_{\times}^{\rm{IC}}$ directlyby without judging whether $\nu_{\times}^{\rm{IC}}<\nu_{c}^{\rm{IC}}$ or not. |
We have calculated numerically the temporal evolution of vf and pl for typical physical parameters. c | We have calculated numerically the temporal evolution of $\nu_{\times,<}^{\rm{IC}}$ and $\nu_{\times,>}^{\rm{IC}}$ for typical physical parameters. , |
ase. (he expression lor vi is | the expression for $\nu_{\times,<}^{\rm{IC}}$ is |
1997: Pearce 2000: Dialek. Evrard Mohr 2001). | 1997; Pearce 2000; Bialek, Evrard Mohr 2001). |
Allen Fabian (1998) have shown that for hot (KZxz5 keV). relaxed: clusters Ly.X17 is recovered. once the elfeets of cool. central components are accounted for in the spectral rav analysis. suggesting (in agreement with the later mass-temperature results) that pre-heating may only significantly alfect the properties of cooler. less-Iuminous clusters. | Allen Fabian (1998) have shown that for hot $kT \approxgt 5$ keV), relaxed clusters $L_{\rm Bol} \approxpropto T^2$ is recovered once the effects of cool, central components are accounted for in the spectral X-ray analysis, suggesting (in agreement with the later mass-temperature results) that pre-heating may only significantly affect the properties of cooler, less-luminous clusters. |
A major goal of studies with the new generation of missions including the Chandra Observatory and XNMM-Newton. which permit the first direct spatiallv-resolved. X- spectroscopy of hot. distant clusters. is the verification and accurate calibration of the virial relations for galaxy clusters. | A major goal of studies with the new generation of X-ray missions including the Chandra Observatory and XMM-Newton, which permit the first direct spatially-resolved X-ray spectroscopy of hot, distant clusters, is the verification and accurate calibration of the virial relations for galaxy clusters. |
In particular. detailed stuclies of svstems for which precise mass measurements have been mace using other. independent methods: are required. | In particular, detailed studies of systems for which precise mass measurements have been made using other, independent methods are required. |
An carly attempt at combining X-ray and gravitational lensing data for clusters observed. with the ASCA satellite to. study the mass-temperature relation was presented. by Hjorth. Oukbir van Kampen (1998). | An early attempt at combining X-ray and gravitational lensing data for clusters observed with the ASCA satellite to study the mass-temperature relation was presented by Hjorth, Oukbir van Kampen (1998). |
In this letter we use new Chandra observations to determine the X-ray. virial relations for a sample of luminous. relatively relaxed clusters spanning the redshift range 0.1κz«0.45. for which lensing mass measurements are available and have been shown to be in &ood agreement with the Chandra results (Section 2: see Allen 1998. Bobhringer 1998 for earlier results). | In this letter we use new Chandra observations to determine the X-ray virial relations for a sample of luminous, relatively relaxed clusters spanning the redshift range $0.1 <z<0.45$, for which lensing mass measurements are available and have been shown to be in good agreement with the Chandra results (Section 2; see Allen 1998, Böhhringer 1998 for earlier results). |
We present gas mass-weighted temperatures. bolometric luminosities and total mass measurements within radii corresponding to a [fixed overdensity A=2500 at the redshifts of the clusters. and compare the observed. scaling relations between thesequantities with those predicted. by simulations. | We present gas mass-weighted temperatures, bolometric luminosities and total mass measurements within radii corresponding to a fixed overdensity $\Delta = 2500$ at the redshifts of the clusters, and compare the observed scaling relations between thesequantities with those predicted by simulations. |
Results are given for two cosmologies: SCDM with f=Hy/l00kms 1—(0.5. Q4,=1 and O4=0. and ACDAL with /—0.7. O4,=0.3 and Oy=0.7. | Results are given for two cosmologies: SCDM with $h = H_0/100$ $= 0.5$, $\Omega_{\rm m} = 1$ and $\Omega_\Lambda = 0$, and $\Lambda$ CDM with $h=0.7$, $\Omega_{\rm m} = 0.3$ and $\Omega_\Lambda = 0.7$. |
The Chandra observations were carried out using the back-illuminated 83 detector on the Advanced CCD Imaging Speetrometer (ACES) between 1999. August 30 and 2001 June 16. | The Chandra observations were carried out using the back-illuminated S3 detector on the Advanced CCD Imaging Spectrometer (ACIS) between 1999 August 30 and 2001 June 16. |
For our analysis we have used the the level-2 event lists provided by the standard Chandra pipeline processing. | For our analysis we have used the the level-2 event lists provided by the standard Chandra pipeline processing. |
These lists were cleaned. for periods of background. Daring using the CLXO. software package resulting in. the net exposure times summarized in Table 1.. | These lists were cleaned for periods of background flaring using the CIAO software package resulting in the net exposure times summarized in Table \ref{table:targets}. |
The Chandra. cata have been analysed using the methods described by Allen (2001b.c) ancl Sehimidt. Allen Fabian (2001). | The Chandra data have been analysed using the methods described by Allen (2001b,c) and Schmidt, Allen Fabian (2001). |
In brief. concentric annular spectra were extracted from the cleaned event Lists. centred on the peaks of the X-ray emission from the The spectra were analysed using NSPEC (version 11.0: Arnaud. 1996). the AIEINAL plasma emission code (Ixaastra Mewe 1993: incorporating the Fe-L caleulations of Liedhal. Osterheld Goldstein 1995). and the photoelectric absorption models of Balucinska-Church AleCammon (1992). | In brief, concentric annular spectra were extracted from the cleaned event lists, centred on the peaks of the X-ray emission from the The spectra were analysed using XSPEC (version 11.0: Arnaud 1996), the MEKAL plasma emission code (Kaastra Mewe 1993; incorporating the Fe-L calculations of Liedhal, Osterheld Goldstein 1995), and the photoelectric absorption models of Balucinska-Church McCammon (1992). |
Pwo separate models were applied to the data. the first of which was fitted to cach annular spectrum inciviclually in order to measure the projected temperature profiles. | Two separate models were applied to the data, the first of which was fitted to each annular spectrum individually in order to measure the projected temperature profiles. |
The second. model was applied to all annuli simultaneously. in order to determine the deprojected temperature profiles under the assumption of spherical symmetry. | The second model was applied to all annuli simultaneously, in order to determine the deprojected temperature profiles under the assumption of spherical symmetry. |
Only data in the 0.57.0 keV range were used. | Only data in the $0.5-7.0$ keV range were used. |
For the mass mocdelling. azimuthally-averaged surface brightness profiles were constructed. from background subtracted. Dat-Delded images with a 0.9840.984. arcsec? pixel scale (2«2 raw detector pixels). | For the mass modelling, azimuthally-averaged surface brightness profiles were constructed from background subtracted, flat-fielded images with a $0.984\times0.984$ $^2$ pixel scale $2\times2$ raw detector pixels). |
When combined with the cleprojectecl spectral temperature profiles. the surface brightness profiles can be used to determine the X-ray eas mass and total mass profiles in the clusters. | When combined with the deprojected spectral temperature profiles, the surface brightness profiles can be used to determine the X-ray gas mass and total mass profiles in the clusters. |
For this analvsis we have used an enhanced version of the image deprojection code described by White. Jones Forman with clistances calculated using the code of Ixavser. Ilelbig Schramm(1997). | For this analysis we have used an enhanced version of the image deprojection code described by White, Jones Forman with distances calculated using the code of Kayser, Helbig Schramm(1997). |
We have parameterized the mass profiles using a Navarro. Frenk White (1997: hereafter NEW) model with p(r)Ξpe(z)ocf(rfr)tllorny. where p(r) is the mass density. pols)=BH(2)(Sa is the eritical density for closure at. recshilt and ὃς=200e7In(1ο| ej]. | We have parameterized the mass profiles using a Navarro, Frenk White (1997; hereafter NFW) model with $\rho(r) = \rho_{\rm c}(z) \delta_{\rm c} / [({r/r_{\rm s}})
\left(1+{r/r_{\rm s}} \right)^2]$, where $\rho(r)$ is the mass density, $\rho_{\rm c}(z) = 3H(z)^2/ 8 \pi G$ is the critical density for closure at redshift $z$, and $\delta_{\rm c} = {200 c^3
/ 3 \left[ {{\rm ln}(1+c)-{c/(1+c)}}\right]}$ . |
The normalizations of the mass profiles may also be expressed in terms of an equivalent velocity dispersion. 7=v5ürsell(z) (with vr in units of Alpe). | The normalizations of the mass profiles may also be expressed in terms of an equivalent velocity dispersion, $\sigma = \sqrt{50} r_{\rm s} c H(z)$ (with $r_{\rm s}$ in units of Mpc). |
The best-fit NEW model parameter values ancl 68 per cent confidence limits are summarized in Table 2.. | The best-fit NFW model parameter values and 68 per cent confidence limits are summarized in Table \ref{table:nfw}. . |
In cterminine the results on the virial properties. we adopt A= 2500. since reso is wellematehecl to the outermost radii at which reliable temperature measurements can be made from the Chandra S3 cata. ( | In determining the results on the virial properties, we adopt $\Delta = 2500$ , since $r_{2500}$ is well-matched to the outermost radii at which reliable temperature measurements can be made from the Chandra S3 data. ( |
The rozoo | The $r_{2500}$ |
ol the 10.7 em flux. | of the 10.7 cm flux. |
It is clear that both GONG and MDI show very similar trends. which is encouraging since these are independent projects. | It is clear that both GONG and MDI show very similar trends, which is encouraging since these are independent projects. |
A straight line fit to all the points shows that the slope is 5.50 [rom zero. a result that is reasonably statistically significant. | A straight line fit to all the points shows that the slope is $\sigma$ from zero, a result that is reasonably statistically significant. |
We therefore. can conclude with a degree of confidence that the region of the Πο II ionization zone changes wilh change in solar activitv. | We therefore, can conclude with a degree of confidence that the region of the He II ionization zone changes with change in solar activity. |
In particular. the magnitude of the dip in Py. which is what causes the signal we are looking at. decreases with increasing activitv. | In particular, the magnitude of the dip in $\Gamma_1$, which is what causes the signal we are looking at, decreases with increasing activity. |
14 should be noted however. that there should be a correlated. noise component between the GONG and MIDI data sets since thev are observing the same object. hence the increase in the significance may be smaller than what we find on combining the results. | It should be noted however, that there should be a correlated noise component between the GONG and MDI data sets since they are observing the same object, hence the increase in the significance may be smaller than what we find on combining the results. |
A combination of all the results (GONG and MDI. low and intermeciale-cleeree set) gives a slope significant al (he 4.70 level. | A combination of all the results (GONG and MDI, low and intermediate-degree set) gives a slope significant at the $\sigma$ level. |
The easiest interpretation of a change in the magnitude of (he depression of P4 at the He II ionization zone is a change in the abundance of helium. | The easiest interpretation of a change in the magnitude of the depression of $\Gamma_1$ at the He II ionization zone is a change in the abundance of helium. |
However. (hat interpretation does nol apply in this case. since we are talking of evclical changes over very short. (me-scales. | However, that interpretation does not apply in this case, since we are talking of cyclical changes over very short time-scales. |
The only change in helium that is expected in (he CZ is à monotonic decrease due to the eravitational settling of helium that takes place over very long (ime-scales. | The only change in helium that is expected in the CZ is a monotonic decrease due to the gravitational settling of helium that takes place over very long time-scales. |
We need to look al (he (he equation of state (EOS) to understand the changes. | We need to look at the the equation of state (EOS) to understand the changes. |
The presence of magnetic fields can change the effective EOS because of contributions of the magnetic fields to energy and pressure. | The presence of magnetic fields can change the effective EOS because of contributions of the magnetic fields to energy and pressure. |
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