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is computationof the distribution P(D) of resonance widths. There is ampleamount ofworkon computing (D) in one-dimensional disordered | Upon elimination of the leads, one can reformulate the problem in terms of an effective non-hermitian hamiltonian, which depends only on the degrees of freedom of the disordered system. |
chains[0. 0. 0. 0.0].. Numerical resultspresented in someof these works indicate(hat P(DP)DP ina large rangeof values of E.where theexp | In this effective description, the outgoing-wave boundary condition in the original system is translated into a local non-hermitian, energy dependent boundary condition at the contact points (or more generally, contact regions) of the system and the leads. |
onent 5 isveryclos | This paper is organized as follows. |
eto 1. A moregeneral quantitythan P(D)is the! density of labeldor » oly Πιό,).(2) ptr.y)—[; δα’~~ Re€,,m | In Section 2 we discuss resonances in a generic quantum system coupled to the external world by a single one-dimensional lead ( a single channel lead). |
It the averaged inthe complex plane contains is widely believedthat [0. DOR0. | We derive a general expression for the DOR in terms of an appropriate diagonal matrix element of the resolvent of the original closed system. |
0].. This ? based information aboutthe Anderson | From this expression, we derive an integral representation for the averaged DOR of the disordered system. |
transition 0. expectationis avoid forumias.wedo notuse resonance widthD,in(2?) Minargmueutorder to | In Section 3 we specialize to the case of an open one dimensional disordered chain, derive the corresponding effective hamiltonian, and obtain its continuum limit. |
cluttering ofourT,,/2. the as anP7Ilearned ofthisHp.but aremimentratherabout the expected. scalingbehavior+ of. | The resulting continuum effective non-hermitian hamiltonian differs from the hermitian one of the closed system by a complex energy dependent boundary condition. |
the DOR- fromD.η Shapiro.. | The structure revealed in this way is quite generic, |
In Figure 14 we again plot e against Τομ, but this time adding three F stars for which mode identifications are ambiguous: Procyon, 449933 and 1181420. | In Figure \ref{fig14} we again plot $\epsilon$ against $T_\mathrm{eff}$, but this time adding three F stars for which mode identifications are ambiguous: Procyon, 49933 and 181420. |
For 449933, as previously mentioned, Scenario B is now considered to be correct, and indeed this is the identification that falls along the observed ε-Τεῃ trend. | For 49933, as previously mentioned, Scenario B is now considered to be correct, and indeed this is the identification that falls along the observed $\epsilon$ $T_\mathrm{eff}$ trend. |
Scenario 1 is the preferred identification in 1181420, and again this matches the trend. | Scenario 1 is the preferred identification in 181420, and again this matches the trend. |
Unfortunately the situation in Procyon is not completely clear. | Unfortunately the situation in Procyon is not completely clear. |
The trend with which we hope to identify the correct e is still loosely defined due to the scarcity of F stars for which we have measured ε unambiguously. | The trend with which we hope to identify the correct $\epsilon$ is still loosely defined due to the scarcity of F stars for which we have measured $\epsilon$ unambiguously. |
Scenario B appears to lie towards the top of any range we could expect for e for a star of its effective temperature. | Scenario B appears to lie towards the top of any range we could expect for $\epsilon$ for a star of its effective temperature. |
On the other hand, Scenario A appears to be at the minimum. | On the other hand, Scenario A appears to be at the minimum. |
Expecting a correction for near-surface effects, we are inclined to believe Scenario B is more likely, but we cannot rule out the alternative. | Expecting a correction for near-surface effects, we are inclined to believe Scenario B is more likely, but we cannot rule out the alternative. |
We anticipate that a measurement of e in many stars observed byKepler could help clearly define the observationalF ε-Τοῃ relation, clarifying the correct mode identification in Procyon. | We anticipate that a measurement of $\epsilon$ in many F stars observed by could help clearly define the observational $\epsilon$ $T_\mathrm{eff}$ relation, clarifying the correct mode identification in Procyon. |
2001) as well as for the SZ, produced. bx the hot X- οπλο eas. | 2004) as well as for the $_{th}$ produced by the hot X-ray emitting gas. |
According to these results. the spectral distortion of the CMD spectrin induced by a population of electron with momentum distribution f(y} cau be written as Ty is the CAIB temperature aud σ in terms of the pressure Z7. contributed by the specific electron | According to these results, the spectral distortion of the CMB spectrum induced by a population of electron with momentum distribution $f_e(p)$ can be written as where $T_0$ is the CMB temperature and in terms of the pressure $P_{e}$ contributed by the specific electron population. |
The spectral function gor). with ο τω. canbo popmtation.written as in terms of the photon redistribution function. (5) and of /j60)=2(kpοa?(e1) | The spectral function $\tilde{g}(x)$, with $x \equiv h \nu / k_{\rm B} T_0$ , can be written as in terms of the photon redistribution function $P(s)$ and of $i_0(x) = 2 (k_{\rm B}
T_0)^3 / (h c)^2 \cdot x^3/(e^x -1)$. |
Were 7= is the optical depth of the electrous with umber deusityns n. and £e;=c[Pdl=fyΡΟης is the average energy of the electronic plasina. | Here $\tau = \int d \ell n_e$ is the optical depth of the electrons with number density $n_e$ , and $ \langle \epsilon \rangle \equiv \frac{\sigma_{\rm T}}{\tau}\int P_{\rm e}
d\ell
= \int_0^\infty dp f_{\rm e}(p) \frac{1}{3} p v(p) m_{\rm e} c
$ is the average energy of the electronic plasma. |
The photon redistribution fiction Ps)fdpGPosp) with s=Inv py, in terms of the CAIB photon frequency increase factor m/v. de)ouds on the electron moment distribution f.(p). where the momentum p is normalized to mec. | The photon redistribution function $P(s)= \int dp f_{\rm e}(p) P_{\rm s}(s;p)$ with $s =
\ln(\nu'/\nu)$ , in terms of the CMB photon frequency increase factor $\nu' / \nu$, depends on the electron momentum distribution $f_{\rm e}(p)$, where the momentum $p$ is normalized to $m_e c$. |
The CAB temperature change produced by the SZE is finally eiven by "A | The CMB temperature change produced by the SZE is finally given by =. |
TnT TyThe specific SZpa; aud. SZ), effects for the various electronic components in the cluster are computed following the approach previously described. | The specific $_{DM}$ and $_{th}$ effects for the various electronic components in the cluster are computed following the approach previously described. |
The calculation of the secondary. clectron spectrum frou VA aunihilation iu ealaxv clusters has Όσοι already presented in de ailsby Colafrancesco Mole (2001) aud Colafrancesco ett al. ( | The calculation of the secondary electron spectrum from $\chi \chi$ annihilation in galaxy clusters has been already presented in details by Colafrancesco Mele (2001) and Colafrancesco et al. ( |
2006). aud here we will ouly recall the relevant steo necessary for the preseut We assu. for simplicity. a spherical DAL halo model for cach DAL clump of the cluster as indicated bv the lensing maps derived by Clowe et al. ( | 2006), and here we will only recall the relevant steps necessary for the present We assume, for simplicity, a spherical DM halo model for each DM clump of the cluster , as indicated by the lensing maps derived by Clowe et al. ( |
2006). with DAL density profile given bv gor)soe(bpou)?5ο ΝΕ oeSrry. | 2006), with DM density profile given by $ g(x) = x^{-\eta} (1+ x)^{\eta - \xi}$, with $x \equiv r/r_s$. |
Values 4=1 ancl &£=3 reproduce theNΝΑΤΟ. Frenk White (1997) density The neutralino umber density profiles n,(E.r)= of the two DM chuups havebeen calculated followiug the approach described in Colatrancesco et al. ( | Values $\eta = 1$ and $\xi = 3$ reproduce the Navarro, Frenk White (1997) density The neutralino number density profiles $n_{\chi}(E,r) = n_{\chi,0}(E) g(r)$ of the two DM clumps have been calculated following the approach described in Colafrancesco et al. ( |
2006) with à NFW DAL density profile aud the following structure parameters: Ay,=LOMAL. Ru,=197 Moc aud ον.=Rey"wea.fre5.66 (for the larger East DM chuup) AM,=6.25 Πρι=0781 Mpe aud Corp=7.46 (for the "bullet West DM The X annihilation rate in the DM clumps ds R=ny(rico. where iGUg is the yy anniiation Cross section averaged over a thermal velocity distribution at. freeze-out temperature. | 2006) with a NFW DM density profile and the following structure parameters: $M_{vir}=
10^{15} M_{\odot}$, $R_{vir} = 1.97$ Mpc and $c_{vir}= R_{vir}/r_s = 5.66$ (for the larger East DM clump); $M_{vir} = 6.25 \cdot 10^{13} M_{\odot}$, $R_{vir} = 0.784$ Mpc and $c_{vir}= 7.56$ (for the ”bullet” West DM The $\chi$ annihilation rate in the DM clumps is $ R = n_{\chi}(r) \langle \sigma v
\rangle_0 ~,$ where $\langle \sigma v \rangle_0$ is the $\chi \chi$ annihilation cross section averaged over a thermal velocity distribution at freeze-out temperature. |
The range of ueutralino lnasses and pair annihilation cross sections dji the inost eeneral supersvnuuetric DM setup is extremely wide (see discussion in Colafrancesco e al. | The range of neutralino masses and pair annihilation cross sections in the most general supersymmetric DM setup is extremely wide (see discussion in Colafrancesco et al. |
2006.lols 2007). | 2006, 2007). |
We consider here. specifically. the neuralino iioc worked out in Colatrancesco ot al. ( | We consider here, specifically, the neutralino models worked out in Colafrancesco et al. ( |
2006) with AL,=20.LO and SI GeV aud with their specific vales of (a0). The electron source unctions Q.GE.r)xUATEDr) for the specific neutralino model considered here have been derived in Colafraucesco et al. ( | 2006) with $M_{\chi}= 20, 40$ and $81$ GeV and with their specific values of $\langle
\sigma v \rangle_0$ The electron source functions $Q_{\rm e}(E,r) \propto \langle\sigma v\rangle_0
n^2_{\chi}(E,r)$ for the specific neutralino model considered here have been derived in Colafrancesco et al. ( |
2006) and the time evolution of the elecron spectrun is eiven bv the equatiou GE.r)biE) | = QUE.r).(5 where spatial diffusion can be safelv neglected iu cluster-size DM. chumps (Colafraucesco et al. | 2006) and the time evolution of the electron spectrum is given by the equation - (E,r) ] = (E,r), where spatial diffusion can be safely neglected in cluster-size DM clumps (Colafrancesco et al. |
2006). | 2006). |
The fiction eives the euergv loss per unit time at enecrev EL where ng, is the mean nuuber density of thermal clectrous iu cmὃν +=)ο and We~(25. cm0.0251. 09,cG3. Dou,cm151. all in units of Di10!GeVs1, | The function gives the energy loss per unit time at energy $E$ where $n_{th}$ is the mean number density of thermal electrons in $\rm{cm}^{-3}$, $\gamma \equiv E/m_e c^2$ and $b_{IC}^0
\simeq 0.25$, $b_{syn}^0 \simeq 0.0254$, $b_{Coul}^0 \simeq 6.13$, $b_{brem}^0 \simeq
1.51$, all in units of $10^{-16}\; \rm{GeV}\, \rm{s}^{-1}$. |
The equilibriun spectrum n.(L.46) obtained solving eq.(5)) allows to calculate the SZpa; effect. | The equilibrium spectrum $n_e(E,r)$ obtained solving \ref{eq.diffusion}) ) allows to calculate the $_{DM}$ effect. |
Εἶσ shows the CAMB temperature change. AT evaluated at the centers of the two DM clumps for differcu values of ALY. | \ref{fig.sz_dm} shows the CMB temperature change $\Delta T$ evaluated at the centers of the two DM clumps for different values of $M_{\chi}$. |
The SZpay signals are overcome by the SZ, signals a low Gv&200 GITZ) aud hieh ἐνZ230 GITz) frequencies. while they dominate in the frequency range where the zero of SZg, is found. Le. at zz223 GIIz for the Eas chunp and at Ézc219 CGIIz for the West chup. | The $_{DM}$ signals are overcome by the $_{th}$ signals at low $\nu \simlt 200$ GHz) and high $\nu \simgt 230$ GHz) frequencies, while they dominate in the frequency range where the zero of $_{th}$ is found, i.e. at $\nu \approx 223$ GHz for the East clump and at $\nu \approx 219$ GHz for the West clump. |
A such frequencies the SM temperature decrement takes values &21.1.10.63 a μῖν(East cmp) and values xNl.L2.Ol nd ο) for A,=s20.10.5]GeV. respectively. | At such frequencies the $_{DM}$ temperature decrement takes values $\approx -21.1, -10.6, -0.3$ $\mu$ K (East clump) and values $\approx -8.4, -4.2, -0.1$ $\mu$ K (West clump) for $M_{\chi} = 20, 40, 81$GeV, respectively. |
TheSZ, from the two ταν chuups of the clusteris also computed from the general approach delineated in eqs.(1-l). | The$_{th}$ from the two X-ray clumps of the clusteris also computed from the general approach delineated in eqs.(1-4). |
Following Markeviteh et al. rites( | Following Markevitch et al. ( |
2002. 2001) aud Tucker etal. ( | 2002, 2004) and Tucker etal. ( |
1998). we use the assuniptiou that the IC eas distribution cau be described by isothermal spheres fitted by a .+profile. | 1998), we use the simplifying assumption that the IC gas distribution can be described by isothermal spheres fitted by a $\beta$ -profile. |
For the East N-ray chunp we adopt AZ,=Ll keV (Alarkevitch et al. | For the East X-ray clump we adopt $kT_e = 14$ keV (Markevitch et al. |
as a single extended particle (Yangctal.901101. | as a single extended particle \citep{2010arXiv1011.0176Y}. |
This allows us to describe mereiug objects in the statistical mechanical theory. | This allows us to describe merging objects in the statistical mechanical theory. |
These properties mean that we can represent many clusters of galaxies απ eroups of imostly virialized subclusters. | These properties mean that we can represent many clusters of galaxies as groups of mostly virialized subclusters. |
The shapes of such clusters are thus given by the positions of their subclusters which reduces the many-body problem of describing all the salaxies iu a cluster to a few-body problem that describes how subchisters interact with cach other. | The shapes of such clusters are thus given by the positions of their subclusters which reduces the many-body problem of describing all the galaxies in a cluster to a few-body problem that describes how subclusters interact with each other. |
Based on the probability that a cell is not completely bound aud virialized as discussed in the previous section. we expect that most cells will usually contain fewer than about LO of these subchisters. depending on the value of 5b. | Based on the probability that a cell is not completely bound and virialized as discussed in the previous section, we expect that most cells will usually contain fewer than about 10 of these subclusters, depending on the value of $b$. |
Some cells. however. can be dominated by a single virialized cluster. | Some cells, however, can be dominated by a single virialized cluster. |
Subclusters can be described bw the positions of their individual galaxies with respect to the subclusters center-of-mass and the positions of these ceuters-of-mass. | Subclusters can be described by the positions of their individual galaxies with respect to the subcluster's center-of-mass and the positions of these centers-of-mass. |
To illustrate this. we use an example of a cluster composed of two subclusters. | To illustrate this, we use an example of a cluster composed of two subclusters. |
The total potential energy of such a cluster is given bv the stm of the internal potential energies of its subclusters and their mutual potential energies such that where the superscripts indicate particles frou different subclusters. | The total potential energy of such a cluster is given by the sum of the internal potential energies of its subclusters and their mutual potential energies such that where the superscripts indicate particles from different subclusters. |
The first two terms are the internal potential energies of the subchisters. which are inaccessible to the rest of the cuscmble if these subclusters are virialized. | The first two terms are the internal potential energies of the subclusters, which are inaccessible to the rest of the ensemble if these subclusters are virialized. |
The last teria represcuts the potential energw between the subchisters. | The last term represents the potential energy between the subclusters. |
We can write this term as where £C?D isH the separationB between the ceuters-of-.mass of cach subcluster. | We can write this term as where $r^{(1,2)}$ is the separation between the centers-of-mass of each subcluster. |
The detailed iuternal structure of the subclusters modifiestheir interaction potential. | The detailed internal structure of the subclusters modifiestheir interaction potential. |
This modification is where AL) and AZC5 ave the masses of. subelusters ] and 2 respectively | This modification is where $M^{(1)}$ and $M^{(2)}$ are the masses of subclusters 1 and 2 respectively. |
Then the potential betwoeeu subclusters is where jpP2?) deseribes the modification to the point-lass potential bv a pair of extended structures. | Then the potential between subclusters is where $\kappa(r^{(1,2)})$ describes the modification to the point-mass potential by a pair of extended structures. |
This modification term enters as a coefficient. το ο) (Almadetal.2002:Yang2011) aud if iip0) is close to unity its effect will be sniall. | This modification term enters as a coefficient to $\beta$ \citep{2002ApJ...571..576A,2010arXiv1011.0176Y} and if $\kappa(r^{(1,2)})$ is close to unity its effect will be small. |
We can write the positions of galaxies as the vector siu of the position of their subcluster's ceuter-ofanass and their position within the subcluster: where x!) is the eeuter-ofanass of subceluster 1 and xU is the position of particle / with respect to the center-ofmass of subcluster 1. | We can write the positions of galaxies as the vector sum of the position of their subcluster's center-of-mass and their position within the subcluster: where $\mathbf{x}^{(1)}$ is the center-of-mass of subcluster 1 and $\tilde{\mathbf{x}}_i^{(1)}$ is the position of particle $i$ with respect to the center-of-mass of subcluster 1. |
With this notation. the modification terii iu equation has the limits Iu the limit⋅⋅ where [x;-(2):«pO the modification term is 1. | With this notation, the modification term in equation has the limits In the limit where $|\tilde{\mathbf{x}}_i^{(1)} - \tilde{\mathbf{x}}_j^{(2)}| \ll r^{(1,2)}$ the modification term is 1. |
This meansxl that we can approximate subclusters that are widely separated as point masses when computing their iuteraction potential. | This means that we can approximate subclusters that are widely separated as point masses when computing their interaction potential. |
For subclusters that are touching.- &(p)Dm0,5 because |~(1)JXi~(2) is on average the radius of a subcluster. | For subclusters that are touching, $\kappa(r^{(1,2)}) \approx 0.5$ because $|\tilde{\mathbf{x}}_i^{(1)}-\tilde{\mathbf{x}}_j^{(2)}|$ is on average the radius of a subcluster. |
Subclusters that are closer to each other may have a sanaller value of &(:AL.2) j. but such pairs may be mereine. in which case we can treat such pairs as single subclusters with a different iuterual structure (Yangctal.2011). | Subclusters that are closer to each other may have a smaller value of $\kappa(r^{(1,2)})$ , but such pairs may be merging, in which case we can treat such pairs as single subclusters with a different internal structure \citep{2010arXiv1011.0176Y}. |
This means that the many-body problem of studving all the ealaxies in a cluster is reduced to the few-body problem of studyiug the positions and velocities of the subclusters iu the cluster. | This means that the many-body problem of studying all the galaxies in a cluster is reduced to the few-body problem of studying the positions and velocities of the subclusters in the cluster. |
This makes the problem considerably easier. since there are fewer “particles” to deal with. | This makes the problem considerably easier, since there are fewer “particles” to deal with. |
Because clusters with more than 10 particles are likely to be vintalized. we cousider the case where cells have less than 10 subclusters. | Because clusters with more than 10 particles are likely to be virialized, we consider the case where cells have less than 10 subclusters. |
These cells have a uon-neelieible probabilitv of having a positive specific heat. | These cells have a non-negligible probability of having a positive specific heat. |
Although subclusters may have different masses aud different iuternal structures. there are analyses that take iuto account these more general cases. | Although subclusters may have different masses and different internal structures, there are analyses that take into account these more general cases. |
The analysis for particles of different internal structure is described iu Yaneetal.(2011)... aud as suggested by equation enters as a coefficient to 2. | The analysis for particles of different internal structure is described in \citet{2010arXiv1011.0176Y}, and as suggested by equation enters as a coefficient to $\beta$ . |
The analysis for multiple masses is cousiderably more complicated (Alanaetal. 2006a).. so to simplify it we make the reasonable approximation that “particles” lave the same mass. | The analysis for multiple masses is considerably more complicated \citep{2006IJMPD..15.1267A}, so to simplify it we make the reasonable approximation that “particles” have the same mass. |
Iu Appendix A owe show that the masses of individual particles are typically within an order of maguitude of cach other. | In Appendix \ref{app-mmass} we show that the masses of individual particles are typically within an order of magnitude of each other. |
The detailed configuration of a cellis directly related to its energy. | The detailed configuration of a cell is directly related to its energy. |
We consider its potential aud kinetic energies separately, aud relate them to W and 2. | We consider its potential and kinetic energies separately, and relate them to $W_*$ and $T_*$ . |
We work with the instautancous values of the energies since these quantities are well-defined andcau bedeteriuined iu principle bv taking a snapshot of a cluster at a given | We work with the instantaneous values of the energies since these quantities are well-defined andcan bedetermined in principle by taking a snapshot of a cluster at a given |
No narrow spectral features are apparent. | No narrow spectral features are apparent. |
The absorption “feature” near 2 keV is likely to be an instrumental artifact(81). | The absorption “feature” near 2 keV is likely to be an instrumental artifact. |
. The 2-0 upper limit on the [lux of a persistent 0.2-keV FWIIM emission line in (he 57 keV range ⋈∙is 10photonsem>s.|. corresponding. to an equivalent. width. at 6.5. keVT of. less (han 150 eV; narrower lines with similar equivalent widths would have been readily apparent in the data. | The $\sigma$ upper limit on the flux of a persistent 0.2-keV FWHM emission line in the 5–7 keV range \\citealt{si00}) ) is $10^{-5}\mbox{ photons cm}^{-2}\mbox{ s}^{-1}$, corresponding to an equivalent width at 6.5 keV of less than 150 eV; narrower lines with similar equivalent widths would have been readily apparent in the data. |
We attempted to extract time-averaged spectral information from the cddata as well. | We attempted to extract time-averaged spectral information from the data as well. |
However. with the faintness of the source (which we estimate [rom our lits at σος for three active PCUS of the PPCA). the high background (100 !)) — some of which is likely due to unresolved sources near the Galactic plane ancl the non-imaging nature of the PCA. we have been unable so far to obtain meaningful results. | However, with the faintness of the source (which we estimate from our fits at 2 for three active PCUs of the PCA), the high background $\approx$ 100 ) – some of which is likely due to unresolved sources near the Galactic plane – and the non-imaging nature of the PCA, we have been unable so far to obtain meaningful results. |
To perform a phase-resolved spectral analvsis we divided the eevenis into six phase bins. according to the best fit period for each observation. | To perform a phase-resolved spectral analysis we divided the events into six phase bins, according to the best fit period for each observation. |
We constructed a spectrum for each spectral bin independently. | We constructed a spectrum for each spectral bin independently. |
With the reduced counts of the phase-binned speclra we were unable to discriminate between multicomponent spectral models and fit a PL only. fixing Ny, to the best-fit value for the PL-onlx fits to the phase-averaged. cata set (2.75x107?em7: see Table 1)). | With the reduced counts of the phase-binned spectra we were unable to discriminate between multicomponent spectral models and fit a PL only, fixing $N_{H}$ to the best-fit value for the PL-only fits to the phase-averaged data set $2.75\times 10^{22}\mbox{
cm}^{-2}$; see Table \ref{tab:spec}) ). |
Whether or not this model is accurate. the fits illustrate (he gross variations in spectral shape (hardness) with phase exhibited by (the source. | Whether or not this model is accurate, the fits illustrate the gross variations in spectral shape (hardness) with phase exhibited by the source. |
We see inFigue 4 (hat there are moderate variations across the phase. with the beginning of the evele harder than the end. and with an additional soltening at pulseanaxinunm. | We see in Figure \ref{fig:ps} that there are moderate variations across the phase, with the beginning of the cycle harder than the end, and with an additional softening at pulse-maximum. |
The shape remains similar over the two observations. | The shape remains similar over the two observations. |
As mentioned previously. we were not able to make a direct comparison with the sspectral results due to unresolved background emission in the cddata that corrupted the absolute flux levels. | As mentioned previously, we were not able to make a direct comparison with the spectral results due to unresolved background emission in the data that corrupted the absolute flux levels. |
ILowever. we were able to compare the fluxes for pulse ON-OFF. | However, we were able to compare the fluxes for pulse $-$ OFF. |
Specilicallv. we extracted spectral datasets for the 1/3 of the phase around the maxinum (ON) ancl the minimum (OFF) of the pulse for both aand {for the second epoch of (overlapping) observations. | Specifically, we extracted spectral datasets for the $1/3$ of the phase around the maximum (ON) and the minimum (OFF) of the pulse for both and for the second epoch of (overlapping) observations. |
Our goal was to use the (svo datasets in combination to make an independent test of the reasonabilitv of the PL+BB fits. | Our goal was to use the two datasets in combination to make an independent test of the reasonability of the PL+BB fits. |
This can be verified from the data in Table 1 rere we compare the activity indices. logHis aud logtfx/fy). in the chromospheric activity aud. rvevs. respectively. for the for rsars 1 conunon to these rvevs, | This can be verified from the data in Table \ref{einstein} where we compare the activity indices, $\log R'_{\rm HK}$ and $\log(f_{\rm X}/f_{\rm V})$, in the chromospheric activity and surveys, respectively, for the four stars in common to these surveys. |
The bulk of the active stars. according to the distributionDoa: fuuctiou:a4 \(logSP!fer:|. hasao dog4.PinsJorPOI~r1.50. which from the values im Tabe would correspond to logfx/fv)xm3.9 or lower. | The bulk of the active stars, according to the distribution function $\chi(\log R'_{\rm HK})$, has $\left\langle\log R'_{\rm HK}\right
\rangle\approx -4.50$, which from the values in Table \ref{einstein} would correspond to $\log(f_{\rm X}/f_{\rm V})\approx -3.9$ or lower. |
TMs. our Fig. | Thus, our Fig. |
does not ταle out the conclusions by Mox:i( et al. (19963). | \ref{m1active} does not rule out the conclusions by Morale et al. \cite{morale}) ). |
Note that our most active stars. that wou ive Ίοςἐνδν)τι2.8 if we extrapolate the relation for he stars from Table 1.. have daryzz(.07 in good agreeneat with Fiewure 3 bx Morale et al.. We can see that a ogfx/fy)=3.0. the CG aud Is cawarts still preseut siniar om, dudices. | Note that our most active stars, that would have $\log(f_{\rm X}/f_{\rm V})\approx
-2.8$ if we extrapolate the relation for the stars from Table \ref{einstein}, have $\delta m_1\approx 0.07$ in good agreement with Figure 3 by Morale et al.. We can see that at $\log(f_{\rm X}/f_{\rm V})\approx -3.0$, the G and K dwarfs still present similar $\delta m_1$ indices. |
Frou the considerations above. we can couchide that. oulv for the most active dwar. the cooler stars will preseut larger A compared to he €i dwarfs. | From the considerations above, we can conclude that, only for the most active dwarfs, the cooler stars will present larger $\Delta$ compared to the G dwarfs. |
From the fuuctio1 \(logA). these τον active stars comprise arouud ofthe active stars wie are clealing with (that is. O.05eNor stars). so that their influence on the metallicity distrition will be negligible. aud our hypothesis for equal c and A is fairly reasonable. | From the function $\chi(\log R'_{\rm HK})$, these very active stars comprise around of the active stars we are dealing with (that is, $0.05cN_{\rm tot}$ stars), so that their influence on the metallicity distribution will be negligible, and our hypothesis for equal $c$ and $\bar\Delta$ is fairly reasonable. |
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