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The blazar PINS 014 is classified as à PSRQ. | The blazar PKS $-$ 014 is classified as a FSRQ. |
H has been observed. in optical bands for four decades. | It has been observed in optical bands for four decades. |
Several papers have reported multiple. optically active ancl bright phases of the source ancl perhaps regular major Πανάς eveles (e.g. Villata ct 11997: Webb et 11998: Raiteri et 11998a and references therein). | Several papers have reported multiple optically active and bright phases of the source and perhaps regular major flaring cycles (e.g., Villata et 1997; Webb et 1998; Raiteri et 1998a and references therein). |
phases An increase of — 3 magnitudes curing the active of this blazar was reported by Webb ct "a curing observations that stretched from December 1969 otto January 1986. | An increase of $\sim$ $-$ 3 magnitudes during the active phases of this blazar was reported by Webb et (1998) during observations that stretched from December 1969 to January 1986. |
Variations of 2.8 mag with a time scale ~22 vears have been reported (Clements et 11995). | Variations of 2.8 mag with a time scale of $\sim$ 22 years have been reported (Clements et 1995). |
Pwo mocles of variability at radio wavelengths superimposed on a 13 months nearly regular evele were suggested through 5 vears of radio monitoring in the Hamburg Quasar Monitoring Program (Britzen et 22000). | Two modes of variability at radio wavelengths superimposed on a $\sim$ 13 months nearly regular cycle were suggested through 5 years of radio monitoring in the Hamburg Quasar Monitoring Program (Britzen et 2000). |
During our observations PIS 014. exhibited significant flux and. colour variations 33). | During our observations PKS $-$ 014 exhibited significant flux and colour variations 3). |
Raiteri et (01905) reported that the historical peak in the light curve of this source reached Ro = 14.15 mae after which it decaved to 16.9 mag. | Raiteri et (1998) reported that the historical peak in the light curve of this source reached R = 14.15 mag after which it decayed to 16.9 mag. |
The light curve of the source during our observing run varies from 16.9 to 16.3 mag in Ro with an average of —16.6 valuemag. which is ~2.5 mag fainter than brightest reported for this source. | The light curve of the source during our observing run varies from 16.9 to 16.3 mag in R with an average of $\sim$ 16.6 mag, which is $\sim$ 2.5 mag fainter than brightest reported value for this source. |
Since our values are close to the faintest reported for this source. we conclude that PIAS 014 was probably in a low state in the period we observed it. | Since our values are close to the faintest reported for this source, we conclude that PKS $-$ 014 was probably in a low state in the period we observed it. |
This blazar is classified as a BL Lac object. | This blazar is classified as a BL Lac object. |
]t is important to note that this source has been classified as an IBL by Giomnmi et ((1999) since the frequency of the first SED peak varies between and LON Lz. and thus does not fall into the wavebands specified hy the usual clefinitions of LBLs and HBLs. | It is important to note that this source has been classified as an IBL by Giommi et (1999) since the frequency of the first SED peak varies between $^{14}$ and $^{15}$ Hz, and thus does not fall into the wavebands specified by the usual definitions of LBLs and HBLs. |
More recentIy. however. deppola ct ((2006) studied the SIZD distribution of a arge sample of BL Lac objects and categorized 16] 714 as a LBL: we adopt that classification in this paper. | More recently, however, Nieppola et (2006) studied the SED distribution of a large sample of BL Lac objects and categorized $+$ 714 as a LBL; we adopt that classification in this paper. |
The optical continuum of the source is so featureless that it was hard Oo estimate its recshift but there is a very recent claim of >=0.31-60.08 by Nilsson et ((2008). | The optical continuum of the source is so featureless that it was hard to estimate its redshift but there is a very recent claim of $z = 0.31\pm0.08$ by Nilsson et (2008). |
This source has been extensively studied at all observable wavelengths from racio ο 5-ravs on diverse time scales(6.&.. Wagner et 11990 Heidt Wagner 1996: Ciommi et 11999: Villata ct 22000: Itaiteri et 22003: Montagni et 22006: Foschini et 22006: Ostorero et 22006: Gupta et 22008a. 200sc and references therein). | This source has been extensively studied at all observable wavelengths from radio to $\gamma$ -rays on diverse time scales (e.g., Wagner et 1990; Heidt Wagner 1996; Giommi et 1999; Villata et 2000; Raiteri et 2003; Montagni et 2006; Foschini et 2006; Ostorero et 2006; Gupta et 2008a, 2008c and references therein). |
bandaThis source is one of the brightest BL Lacs in optical and has an LDV duty evele of nearly 1. | This source is one of the brightest BL Lacs in optical bands and has an IDV duty cycle of nearly 1. |
Unsurprisingly. 85 161 714 has been the subject of several optical noneeins campaigns on LDV timescales (c.g.. Wagner ct 1199€ Bar et 11999: Montagni et 22006: Gupta et 22008c. and. references therein). | Unsurprisingly, S5 $+$ 714 has been the subject of several optical monitoring campaigns on IDV timescales (e.g., Wagner et 1996; Sagar et 1999; Montagni et 2006; Gupta et 2008c, and references therein). |
This source has shown live major optical outbursts (Gupta et αν.SNe) separated by vears. | This source has shown five major optical outbursts (Gupta et 2008c) separated by $\sim$ $\pm$ 0.3 years. |
High optical of ~ 20% — 29'4 .have also been rved in the source (CL"kalo et 119€n Fan et 11997). | High optical polarizations of $\sim$ $\%$ – $\%$ have also been observed in the source (Takalo et 1994; Fan et 1997). |
veRecently. Gupta. Srivastava Wiita (2009) analysed the excellent intracay optical LCs of the source obtained by Montagni et ((2006) ancl reported σου evidence for nearly periodic oscillations ranging between 25 and τὸ minutes on severa different nights. | Recently, Gupta, Srivastava Wiita (2009) analysed the excellent intraday optical LCs of the source obtained by Montagni et (2006) and reported good evidence for nearly periodic oscillations ranging between 25 and 73 minutes on several different nights. |
Our LCs of $5 714. showed. very significan variations of ~2 mag in each of the observed bands: however. no significant colour variation is observed in the source except for B. L 44) | Our LCs of S5 $+$ 714 showed very significant variations of $\sim$ 2 mag in each of the observed bands; however, no significant colour variation is observed in the source except for $-$ I 4). |
AX [arge[arg increase⋅ in the brightness of the source occurred between the two carly observations anc the seven later ones: over 230 days the source brightenec by 2.2 magnitudes (15.4 to 13.2) in lt. becoming only ~0.65 magnitude fainter than the brightest M (1 = 12.55 mag) reported for the source (Ciupta et "n22008c). | A large increase in the brightness of the source occurred between the two early observations and the seven later ones: over 230 days the source brightened by 2.2 magnitudes (15.4 to 13.2) in R, becoming only $\sim$ 0.65 magnitude fainter than the brightest magnitude (R = 12.55 mag) reported for the source (Gupta et 2008c). |
Assuming that the source continued its brightening a linear trend. it should. have reached. R=12.55 mae in September 2009. | Assuming that the source continued its brightening following a linear trend, it should have reached $=$ 12.55 mag in September 2009. |
The calculated: time difference between he last known outburst in January 2007 and this possible outburst is ~2.7 vears. | The calculated time difference between the last known outburst in January 2007 and this possible outburst is $\sim$ 2.7 years. |
This temporal gap is consistent with the optical outburst time 3.00.3. vears in the long erm optical period that was earlier reported for this source (Gupta ct 22008e). | This temporal gap is consistent with the optical outburst time $\pm$ 0.3 years in the long term optical period that was earlier reported for this source (Gupta et 2008c). |
Pherefore we organized. a radio-optical observing campaign for five davs in December 2000 o ascertain the source behaviour and state: the analysis of his cata is underway. | Therefore we organized a radio-optical observing campaign for five days in December 2009 to ascertain the source behaviour and state; the analysis of this data is underway. |
Vhis blazar has been classified. as a BL Lac object (Carswell ct 11974) | This blazar has been classified as a BL Lac object (Carswell et 1974). |
There have been several papers concerning its redshift determination (c.g. Carswell et 11974: Palomo Ulrich 2000. and references herein) with the most recent result. of z=0.424 for PAS | 718 found using a HIST snapshot image (Sbarufatti et 22005). | There have been several papers concerning its redshift determination (e.g., Carswell et 1974; Falomo Ulrich 2000, and references therein) with the most recent result of $z = 0.424$ for PKS $+$ 718 found using a HST snapshot image (Sbarufatti et 2005). |
Since it was optically identified. this source has oen extensively observed: across the whole LEAL spectrum (Vorassranta οἱ 22004: Fan Lin 2000: Gu et 22006: Gupta et 22008c: Ciprini et 22007 and references. therein.). | Since it was optically identified, this source has been extensively observed across the whole EM spectrum (Terässranta et 2004; Fan Lin 2000; Gu et 2006; Gupta et 2008c; Ciprini et 2007 and references therein.). |
Radio [frequency observations show slow variability with some outbursts (Vorassranta οἱ 11992. 2004). but hardly any correlation jw been [found between the pio iud optical [lares (6fenments et periodicity11995: Lanski ct "002: Ciaramella e 22004). | Radio frequency observations show slow variability with some outbursts (Terässranta et 1992, 2004), but hardly any correlation has been found between the radio and optical flares (Clements et 1995; Hanski et 2002; Ciaramella et 2004). |
A of ~14 vears been suggested to »* present in the source using a century lone optical ligh curve (Fanet 11997). | A periodicity of $\sim$ 14 years has been suggested to be present in the source using a century long optical light curve (Fan et 1997). |
ally Optical variability on LDV and S'TV imescales has nat been observed for | 178 (Xie e 11992: Massaro et 11995: Fan et un91: Zhang et 22004: Ciprini et 22007: Gupta et 22008c). | Optical variability on IDV and STV timescales has naturally been observed for $+$ 178 (Xie et 1992; Massaro et 1995; Fan et 1997; Zhang et 2004; Ciprini et 2007; Gupta et 2008c). |
Signilican ractional polarizations ( 14 to 30% ) have been observec roth at optical and LIV bands (Mead et 11990 Takalo e 11991. 1992b: Valtaoja ct 1991a. 1993: 'l'ommasi οἱ 22001). | Significant fractional polarizations $\sim$ $\%$ to $\%$ ) have been observed both at optical and IR bands (Mead et 1990; Takalo et 1991, 1992b; Valtaoja et 1991a, 1993; Tommasi et 2001). |
We found strong [ux variations in all the well observec passbancs for PINS | 178 (Fig.55ji) however. as there are only two data points in the 3 band LO. we cannot discuss the nature of its variation. | We found strong flux variations in all the well observed passbands for PKS $+$ 178 5); however, as there are only two data points in the B band LC, we cannot discuss the nature of its variation. |
Except for LI. no significan colour variation was seen in the source. | Except for $-$ I, no significant colour variation was seen in the source. |
The average It44 of PINS | 178 during our observing run was 15.80 which is L8 mag fainter than the brightest (1t — 14.0 mag) anc ].2 mag brighter than the faintest magnitude (Rave ~ 17.0) reported. in the source (Ciprini et 22007). | The average $_{mag}$ of PKS $+$ 178 during our observing run was 15.80 which is $\sim$ 1.8 mag fainter than the brightest (R $\sim$ 14.0 mag) and $\sim$ 1.2 mag brighter than the faintest magnitude $_{mag}$ $\sim$ 17.0) reported in the source (Ciprini et 2007). |
Thus we have probably observed the source in either a pre- or post-outburst state as long as there have been no long-term changes in the underlving light-curve. | Thus we have probably observed the source in either a pre- or post-outburst state as long as there have been no long-term changes in the underlying light-curve. |
This BL Lac object is one of the most extensively observed. for variability: it is also among the very [ew | This BL Lac object is one of the most extensively observed for variability; it is also among the very few |
and other parameters. | and other parameters. |
Dillerent approaches lead. to small systematic differences in the inferred Cepheid parameters. first ofall. in the radii. | Different approaches lead to small systematic differences in the inferred Cepheid parameters, first of all, in the radii. |
Based on ecometrical considerations. ltastorguev(2010). derived phase-dependent PEs as simple three-parametric analytic expressions depending on the pulsation velocity. limb darkening coefficient. and. spectral line broadening. adjusted to COILAVIZL radial velocities of Copheids. | Based on geometrical considerations, \citet{Ras10}
derived phase-dependent PFs as simple three-parametric analytic expressions depending on the pulsation velocity, limb darkening coefficient, and spectral line broadening, adjusted to CORAVEL radial velocities of Cepheids. |
We suspect that the period. dependence rellects mainly the dependence of the PE. on limb darkening. | We suspect that the period dependence reflects mainly the dependence of the PF on limb darkening. |
To compare our results with other calculations. we finalA adopted a moderate: dependence. of PE on. the. period advocated by Nardettoetal.(2007):: though we repeated all caleulations with other variants of PE dependence on the period ancl pulsation phase. to assure the stability of the calculated colour excess. | To compare our results with other calculations, we finally adopted a moderate dependence of PF on the period advocated by \citet{Nar07}: though we repeated all calculations with other variants of PF dependence on the period and pulsation phase to assure the stability of the calculated colour excess. |
To test the new method. we used the maximum likelihood echnique to solve Eq. ( | To test the new method, we used the maximum likelihood technique to solve Eq. ( |
7) for the V-band light curve and D.V colour curve for several classical Cepheics residing in voung na clusters: SZ Tau. (NGC. 1647). CE Cas (NGC 77! Ser (IC 4725). DL Cas (NGC 129). GY See (anonymous Obnian (Forbes 1982))) as well as for approximately 30 field Cepheids from our sample. | 7) for the $V$ -band light curve and $B-V$ colour curve for several classical Cepheids residing in young open clusters: SZ Tau (NGC 1647), CF Cas (NGC 7790), U Sgr (IC 4725), DL Cas (NGC 129), GY Sge (anonymous OB-association \citep{Forb82}) ) as well as for approximately 30 field Cepheids from our sample. |
We founc wo ος) calibrations those of Flower(1996)— anc Desselletal.(1998) combined with the DC'(V) calibration asa facion of normal colour (D.Vo proposed by. (1996 to ve the best fit to the observed Y -band lieh curve via Iq. A | We found two $log (T_{eff})$ calibrations – those of \citet{F96} and \citet{BCP98} – combined with the $BC(V)$ calibration as a function of normal colour $(B-V)_0$ proposed by \citet{F96} – to yield the best fit to the observed $V$ -band light curve via Eq. ( |
op | 7). |
tedweak sensitivity of caleulated reddening. Ep v.tothe (7). PE value (constant or period/phasedependent) and to the derived <<£2 value can be explainec bv very strong dependence of the light curve’s amplitude on the effective temperature. ~LOου). and. as a consequence. on the dereddened. colour. | A weak sensitivity of calculated reddening, $E_{B-V}$, to the adopted PF value (constant or period/phase--dependent) and to the derived $<R>$ value can be explained by very strong dependence of the light curve's amplitude on the effective temperature, $\sim 10
\times log(T)$, and, as a consequence, on the dereddened colour. |
"Though the internal errors of the reddening Jg.«e seen to be very small. the values determined. using the two best calibrations. Flower(1996). and. Bessellctal. (1998).. may differ by as much as 0.080.050. due to the systematic shift. between these two calibrations (Fig. | Though the internal errors of the reddening $E_{B-V}$ seem to be very small, the values determined using the two best calibrations, \citet{F96} and \citet{BCP98}, , may differ by as much as $0.03 -
0.05^m$, due to the systematic shift between these two calibrations (Fig. |
2 e). | 2 e). |
Table summarizes the inferred parameters for the cluster Cepheids studied. | Table \ref{tab1} summarizes the inferred parameters for the cluster Cepheids studied. |
Fie. | Fig. |
1 shows the observed and smoothed data and the final fit to the V-band light curve for U Ser Cepheid. | 1 shows the observed and smoothed data and the final fit to the $V$ -band light curve for U Sgr Cepheid. |
Our reddenings seem to agree well with the corresponding WEBDA values. particularly if we remember that the errors of the adopted cluster reddening estimates are as high as +0.05'". | Our reddenings seem to agree well with the corresponding WEBDA values, particularly if we remember that the errors of the adopted cluster reddening estimates are as high as $\pm 0.05^m$. |
Our next step will be to make use of the calibrations of Topp and DC' as a function of red. ancl infrared. colours (VR.V[VN) and to compare derived reddening ratios with the conventional extinction laws. | Our next step will be to make use of the calibrations of $T_{eff}$ and $BC$ as a function of red and infrared colours $(V-R, ~V-I, ~V-K)$ and to compare derived reddening ratios with the conventional extinction laws. |
Note that the inferred racdus ane luminosity of SZ Tau are too large for its short period: this Cepheid probably pulsates in the or even in: the ud2' Overtone. as nw be indirectly evidenced by its low colour amplitude (about 0.157). | Note that the inferred radius and luminosity of SZ Tau are too large for its short period; this Cepheid probably pulsates in the $1^{st}$ or even in the $2^{nd}$ overtone, as may be indirectly evidenced by its low colour amplitude (about $0.15^m$ ). |
Fig. | Fig. |
2 shows the observed. data. the fit to the Y -band light curve. and the inferredcalibration £=10.οσα"rr)| vs (B. Vo) calculated for PR Aql Cephe (as adrbι mer expansion in the normal colour). | 2 shows the observed data, the fit to the $V$ -band light curve, and the inferred calibration $F = 10\times log (T_{eff}) +
BC(V)$ vs $(B-V)_0$ ) calculated for TT Aql Cepheid (as a $5^{th}$ -order expansion in the normal colour). |
The inferred calibration is very close to that of Flower(1996). | The inferred calibration is very close to that of \citet{F96}. |
. We used a Per as the "standard? star. with 275zz(6240+20)Wy. Fefll]z0.28£0.06 (Leeetal.2006). (2V)zQ4" and Apvc0.00" (AWWEBDA. for a Per cluster). | We used $\alpha$ Per as the “standard” star, with $T^{st} \approx
(6240\pm20)~K$, $[Fe/H]\approx -0.28\pm0.06$ \citep{Lee06}, $(B-V)^{st}\approx 0.48^m$ and $E_{B-V}\approx 0.09^m$ (WEBDA, for $\alpha$ Per cluster). |
To take into account the elect of metallicity on the zero-point FUCLIGGY νο estimated the gradient d(C44)Fell][0.24 from the calibrations by Alonsoetal.(1999)Bonifacio(2009). | To take into account the effect of metallicity on the zero-point $F(CI_0)^{st}$, we estimated the gradient $dF(CI_0)^{st} / d[Fe/H]\approx +0.24$ from the calibrations by \citet{AAMR99, SF00, GHB09}. |
. For TP ql. £gy8(0.65+0.03)'". | For TT Aql, $E_{B-V}\approx (0.65\pm0.03)^m$. |
In some cases (with large amplitude of color variation) the “free” calibration (Iq. | In some cases (with large amplitude of color variation) the “free” calibration (Eq. |
10) can markedly improve the moce fit to the observed light curve of the Cepheicl variable. | 10) can markedly improve the model fit to the observed light curve of the Cepheid variable. |
Eig. | Fig. |
2 [ shows the example of calibrations of the £ functions derived from nine Cepheids with dillerent. metallicity anc surface gravity values. | 2 f shows the example of calibrations of the $F$ functions derived from nine Cepheids with different metallicity and surface gravity values. |
Temperature difference at. 66005100ÁN dis amounted {ο 54. | Temperature difference at $T_{eff} \sim 6600 - 5100~K$ is amounted to $3 - 5\%$. |
When applied to an extensive sample of Cepheic variables with homogeneous photometric data and cdetaile radial velocity curves. the new method is expected to give a completely independent scale of reddenings. a new Period - Colour - Luminosity relation. and a new distance scale for the Alilky-Way Cepheids. | When applied to an extensive sample of Cepheid variables with homogeneous photometric data and detailed radial velocity curves, the new method is expected to give a completely independent scale of reddenings, a new Period - Colour - Luminosity relation, and a new distance scale for the Milky-Way Cepheids. |
We erateful to ALY. Zabolotskikh for her assistance in data preparation and to L.N. Berdnikov. Yu. | We grateful to M.V. Zabolotskikh for her assistance in data preparation and to L.N. Berdnikov, Yu. |
N. Eremov. ALIS. Sachkov. Woe. Panchuk and A.B. Fokin for comments ancl helpful discussions. | N. Efremov, M.E. Sachkov, V.E. Panchuk and A.B. Fokin for comments and helpful discussions. |
This research. has mace use of the WEBDA database operated at the Institute for Astronomyof the University of Vienna. | This research has made use of the WEBDA database operated at the Institute for Astronomyof the University of Vienna. |
Our work is supported by the Russian Foundation for Basie Research (projects nos. 07- | Our work is supported by the Russian Foundation for Basic Research (projects nos. 08-02-00738-a, |
02-00380-a. and06-02-16077-a). | 07-02-00380-a, and 06-02-16077-a). |
Lastly. we Gun to the hardest question to answer al this stage: how much eas there should be as a [function of radius. here parameterized as r(r). | Lastly, we turn to the hardest question to answer at this stage: how much gas there should be as a function of radius, here parameterized as $\tau(r)$. |
Particularly in the inner region. ib would take very little gas to create a large optical depth: even integrated out to rír,=100. a disk with constant 7=LOO would require only 10.1/2... | Particularly in the inner region, it would take very little gas to create a large optical depth: even integrated out to $r/r_g=100$, a disk with constant $\tau = 100$ would require only $10^{-4}M_7^2 M_{\odot}$. |
Even though there is no reason to think the disk is anywhere near a conventional state of inflow ecquilibrium. one could use the optical depth of such a disk as a standard of comparison. | Even though there is no reason to think the disk is anywhere near a conventional state of inflow equilibrium, one could use the optical depth of such a disk as a standard of comparison. |
The thermocwuamics of equilibrium disks creates a characteristic scale For the surface density: (he maximum at which thermal equilibrium can be achieved. | The thermodynamics of equilibrium disks creates a characteristic scale for the surface density: the maximum at which thermal equilibrium can be achieved. |
One of the predictions of the 7? model is that in a steady-state disk in whieh the vertically-integrated r-ó stress is a limes (he verlically-inleeratecd (total. pressure. the accretion rate al any. particular. radius increases as the surface density increases. but only up to a point. | One of the predictions of the \cite{SS73} model is that in a steady-state disk in which the vertically-integrated $r$ $\phi$ stress is $\alpha$ times the vertically-integrated total pressure, the accretion rate at any particular radius increases as the surface density increases, but only up to a point. |
Larger surlace density (aid accretion rate) lead {ο a larger ratio of radiation to gas pressure. | Larger surface density (and accretion rate) lead to a larger ratio of radiation to gas pressure. |
If radiation pressure exceeds gas pressure. increasing accretion rate can only be accommodated by adecreasing surface density. | If radiation pressure exceeds gas pressure, increasing accretion rate can only be accommodated by a surface density. |
In other words. there is a possible surface density. | In other words, there is a possible surface density. |
Although recent work on explicit simulation of disk thermocdsynanmies under the influence of MIID turbulence driven by the magneto-rotational instability has shown that this phenomenological moclel’s prediction about the thermal stabilitw of disks is wrong (?).. thev also show that disk properties averaged over times long compared to a thermal time match (hose predicted by the a model (2): when radiation pressure dominates. the surface density and accretion rate are inversely related. | Although recent work on explicit simulation of disk thermodynamics under the influence of MHD turbulence driven by the magneto-rotational instability has shown that this phenomenological model's prediction about the thermal stability of disks is wrong \citep{Hirose09a}, they also show that vertically-integrated disk properties averaged over times long compared to a thermal time match those predicted by the $\alpha$ model \citep{Hirose09b}: when radiation pressure dominates, the surface density and accretion rate are inversely related. |
The Thomson optical depth corresponding to this maximum surface density is where we have scaled the stress/pressure ratio to 0.1. | The Thomson optical depth corresponding to this maximum surface density is where we have scaled the stress/pressure ratio to 0.1. |
Close to the black hole. it occurs al a comparatively low accretion rate in Eddington units: such a state might be consistent with a eas supply rate al large radius capable of feeding an AGN (1.e.. me 0.1). but reduced two orders of magnitude by the effects of binary torques and the inability of internal stresses in (he disk (o drive its inner edge inward as [ast as gravitational wave emission compresses (he black hole binary. | Close to the black hole, it occurs at a comparatively low accretion rate in Eddington units: Such a state might be consistent with a gas supply rate at large radius capable of feeding an AGN (i.e., $\dot m \sim 0.1$ ), but reduced two orders of magnitude by the effects of binary torques and the inability of internal stresses in the disk to drive its inner edge inward as fast as gravitational wave emission compresses the black hole binary. |
It is sienilicant in this respect (hal even such a strong suppression of accretion still vields an inner disk optical depth that is quite large. | It is significant in this respect that even such a strong suppression of accretion still yields an inner disk optical depth that is quite large. |
A smaller accretion rate would produce a smaller optical depth. but only ox5m. when gas pressure dominates and the disk remains radiative. | A smaller accretion rate would produce a smaller optical depth, but only $\propto \dot m^{3/5}$, when gas pressure dominates and the disk remains radiative. |
short duration events pose strong constraints on their contribution to the halo mass budget (Alcocketal. 1996;; Alcocketal.1998;; Renaultetal. 1998)). | short duration events pose strong constraints on their contribution to the halo mass budget \cite{A96}; ; \cite{A98}; \cite{R98}) ). |
Following these works, we have chosen two values for µι given by 0.001 and 0.01 respectively. | Following these works, we have chosen two values for $\mu_l$ given by 0.001 and 0.01 respectively. |
In all our analysis we are assuming that the MF is the same in the mass range (ju,Wu) Mo, i.e. that the slope o does not change in this range, which seems quite reasonable as a first approximation. | In all our analysis we are assuming that the MF is the same in the mass range $(\mu_l, \mu_u) \ M_{\odot}$ , i.e. that the slope $\alpha$ does not change in this range, which seems quite reasonable as a first approximation. |
To integrate Eqs.(18)) and (21)) we need the detection efficiency of the MACHO collaboration for their monitoring campaign towards LMC since in our analysis we will use their results of the first 5.7 years of observations. | To integrate \ref{eq: iev}) ) and \ref{eq: itau}) ) we need the detection efficiency of the MACHO collaboration for their monitoring campaign towards LMC since in our analysis we will use their results of the first 5.7 years of observations. |
This function has been carefully evaluated by the MACHO group itself (Alcocketal. 2000b)), but they give no analytical formula for it. | This function has been carefully evaluated by the MACHO group itself \cite{epsMACHO}) ), but they give no analytical formula for it. |
That is why we have built up an approximated expression of £(fg) interpolating the data taken from Fig. | That is why we have built up an approximated expression of $\varepsilon(t_E)$ interpolating the data taken from Fig. |
5 of Alcock (20002), obtaining being log the decimal logarithm; the first expression holds for 2d€tg300 and the second for 300d<tgX900 d. Eq.(24)) | 5 of Alcock (2000a), obtaining being $\log$ the decimal logarithm; the first expression holds for $2 \ d \le t_E \le 300 \ d$ and the second for $300 \ d < t_E \le 900 \ d$ . \ref{eq: epst}) ) |
differs from the measured e(tg) less than in the range examined, the error being larger for events lasting more than 900 days. | differs from the measured $\varepsilon(t_E)$ less than in the range examined, the error being larger for events lasting more than 900 days. |
This is not a problem since the thirteen observed events which we use in our analysis last approximately from 34 to 102 days, so we are confident that no serious systematic error is induced by our approximation for &(tg). | This is not a problem since the thirteen observed events which we use in our analysis last approximately from 34 to 102 days, so we are confident that no serious systematic error is induced by our approximation for $\varepsilon(t_E)$. |
For the same reason also the discontinuity in tg=300 dd has no effect on our analysis. | For the same reason also the discontinuity in $t_E = 300$ d has no effect on our analysis. |
Now we have all we need to estimate the functions defined in Eqs.(18)) and (21)). | Now we have all we need to estimate the functions defined in \ref{eq: iev}) ) and \ref{eq: itau}) ). |
Without entering in details, the numerical integration and the following interpolation of the results have shown us that it is possible to write where the values of thecoefficients (dev,bev,Cev,dev) and (a,,0;,c;,d,) depend on the model considered. | Without entering in details, the numerical integration and the following interpolation of the results have shown us that it is possible to write where the values of thecoefficients $(a_{ev}, b_{ev}, c_{ev}, d_{ev})$ and $(a_{\tau}, b_{\tau}, c_{\tau}, d_{\tau})$ depend on the model considered. |
We report them in Tables 2and 3. | We report them in Tables 2and 3. |
These approximations | These approximations |
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