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With an initial ΤοZ 2400, the blast wave decelerates at <0.3 s (eq. Pll) | With an initial $\G_0 \gtrsim
2400$ , the blast wave decelerates at $\lesssim 0.3$ s (eq. \ref{dec_time}] ]) |
for the parameters used here: Esse 2, τι £z3cm *. | for the parameters used here: $E_{55}\approx
2$ , $n\approx 3$ $^{-3}$ . |
errors) as £ ll km J| in reflex velocity magnitude and one or (wo degrees in direction: and 3-5 kms + !1 in Unbble tensor components. | errors) as $\pm$ 11 km $^{-1}$ in reflex velocity magnitude and one or two degrees in direction; and 3-5 km $^{-1}$ $^{-1}$ in Hubble tensor components. |
We have al hand three dillerent wavs of estimating the uncertaitv in our numbers: the variation in parameters between calculations: (he formal standard error: ancl (hie error tensor. | We have at hand three different ways of estimating the uncertainty in our numbers: the variation in parameters between calculations; the formal standard error; and the error tensor. |
Unfortunately. they do not agree. | Unfortunately, they do not agree. |
Considering (he solar reflex velocity (magnitude and direction) aud out-ol-plane Hubble lensor eigenvectors and eigenvalues. variations between the ealeulations are much larger Chan the standard error allows. | Considering the solar reflex velocity (magnitude and direction) and out-of-plane Hubble tensor eigenvectors and eigenvalues, variations between the calculations are much larger than the standard error allows. |
They are. however. consistent with estimates based on the error tensor. | They are, however, consistent with estimates based on the error tensor. |
The stancarel error assumes a rather strict Gaussian distribution of deviations [rou the model. and that the sample at hand is an unbiased description of it. | The standard error assumes a rather strict Gaussian distribution of deviations from the model, and that the sample at hand is an unbiased description of it. |
The error tensor also assumes a Gaussian distribution. but is less strict. in Chat it takes into account. just how the variance changes with this particular data set when changing a parameter. | The error tensor also assumes a Gaussian distribution, but is less strict, in that it takes into account just how the variance changes with this particular data set when changing a parameter. |
We must conclude. then. that the data at hand do not satislv the necessary conditions for the standard error: an unbiased sample of an underlving model. with a strictly Gaussian distribution of errors around it. | We must conclude, then, that the data at hand do not satisfy the necessary conditions for the standard error: an unbiased sample of an underlying model, with a strictly Gaussian distribution of errors around it. |
Exactly how the samples fail is not clear [rom calculations to this point. but two general areas are identifiable. | Exactly how the samples fail is not clear from calculations to this point, but two general areas are identifiable. |
First. (he samples could be chosen with some bias which prevents a good representation ol the kinematies of the Local Volume. | First, the samples could be chosen with some bias which prevents a good representation of the kinematics of the Local Volume. |
A good fit to the data may then translate to a poor fit to the real Kinematics. | A good fit to the data may then translate to a poor fit to the real kinematics. |
This is indicated bv (he fact that some parameters show a very strong dependence on the particular data set used. ancl that in places where (he data sets overlap (such as in the direction of the 82-84 kms ! ! Hubble tensor component) the results are more similar (han the error tensor indicates they should be. | This is indicated by the fact that some parameters show a very strong dependence on the particular data set used, and that in places where the data sets overlap (such as in the direction of the 82-84 km $^{-1}$ $^{-1}$ Hubble tensor component) the results are more similar than the error tensor indicates they should be. |
second. (he assumed kinematic models could be inappropriate. in that (here are systematic motions not considered. | Second, the assumed kinematic models could be inappropriate, in that there are systematic motions not considered. |
This would make the peculiar velocity distribution about the models non-Gaussian. | This would make the peculiar velocity distribution about the models non-Gaussian. |
Deviations which are svstematic in space will be investigated in (he next section. followed by other possibilities. | Deviations which are systematic in space will be investigated in the next section, followed by other possibilities. |
In Figures D. through 12. are plotted the deviations of each galaxy from (he various solutions (that is. how much its radial velocity differs [rom what the model woulcl predict) against the various spatial coordinates. | In Figures \ref{fig:99X} through \ref{fig:35W} are plotted the deviations of each galaxy from the various solutions (that is, how much its radial velocity differs from what the model would predict) against the various spatial coordinates. |
The first figures use Supergalactic NX. Y and Z: the | The first figures use Supergalactic X, Y and Z; the |
the two dust filaments in this field (0~—24.57) and are absent [rom the southern filement (0~—24. 17). even though the latter contains a somewhat higher concentration of T Tari stars. | the two dust filaments in this field $\delta\sim-24.5^\circ$ ) and are absent from the southern filement $\delta\sim-24.7^\circ$ ), even though the latter contains a somewhat higher concentration of T Tauri stars. |
This contrasts with the findings of Luhman(2006).. whose study of the Taurus star-forming region indicated no significant dillerence between the distributions of stars ancl brown clwarls. | This contrasts with the findings of \citet{luh06}, whose study of the Taurus star-forming region indicated no significant difference between the distributions of stars and brown dwarfs. |
The situation is. however. somewhat reminiscent of spatial segregation effects in Taurus found by Guieuetal.(2007).. whereby brown dwarls with disks ave prelerentially located in one particular filament: no such segregation was evident for the T Tauri stars. | The situation is, however, somewhat reminiscent of spatial segregation effects in Taurus found by \citet{guieu07}, whereby brown dwarfs with disks are preferentially located in one particular filament; no such segregation was evident for the T Tauri stars. |
since (he presence of a disk suggests voull. the spatial segregation might be interpreted in terms of age dillerences between different aggregates of objects in the region. | Since the presence of a disk suggests youth, the spatial segregation might be interpreted in terms of age differences between different aggregates of objects in the region. |
Similarly. in the case of p Oph. the relatively compact aggregate of low-mass candidates in the northern filament in Figure 5 may have resulted [rom a star formation event more recent (han for some or all of the T Tauri stus. | Similarly, in the case of $\rho$ Oph, the relatively compact aggregate of low-mass candidates in the northern filament in Figure \ref{fig5}
may have resulted from a star formation event more recent than for some or all of the T Tauri stars. |
Therefore. the spatial compactness of the low-mass ageregale does not necessarily argue against the ejection moclel, | Therefore, the spatial compactness of the low-mass aggregate does not necessarily argue against the ejection model. |
AlvesdeOliveiraetal.(2010) have recently compared our photometry with their own data. obtained in 2006. for the subset of 7 sources observed spectroscopically by (2010).. xl found cliserepancies in 3 cases. | \citet{alv10} have recently compared our photometry with their own data, obtained in 2006, for the subset of 7 sources observed spectroscopically by \cite{mar10}, and found discrepancies in 3 cases. |
In. particular. thev. found 444450 to be 1.42 magnitudes fainter than our estimate of A=17.71. | In particular, they found 4450 to be 1.42 magnitudes fainter than our estimate of $K_s=17.71$. |
After hither examination of our images. we find the source to be slightly extended. with FWIIM ~2"—3". | After further examination of our images, we find the source to be slightly extended, with FWHM $\sim2''-3''$. |
Its estimated fIux will therefore depend (to some extent on the aperture or beanmsize used. | Its estimated flux will therefore depend to some extent on the aperture or beamsize used. |
We have estimated its Αν band aperture magnitude from the 2\IASS Deep Field data using apertures of various radii. making appropriate corrections lor (rumeation of the point-source response. and obtain A,=18.3423:0.15 in a 1.5" aperture. | We have estimated its $K_s$ -band aperture magnitude from the 2MASS Deep Field data using apertures of various radii, making appropriate corrections for truncation of the point-source response, and obtain $K_s = 18.34\pm0.15$ in a $1.5''$ aperture. |
We have compared this result with that obtained [rom the A-band "peak-up image during (he spectroscopic observations of (2010).. which vield A=18.5720.15 for a 1.5" aperture: the image was unfortunately loo noisy for larger apertures. | We have compared this result with that obtained from the $K$ -band “peak-up" image during the spectroscopic observations of \citet{mar10}, which yield $K=18.57\pm0.15$ for a $1.5''$ aperture; the image was unfortunately too noisy for larger apertures. |
The (wo magnitudes are nevertheless consistent within the error bars. and therefore suggest the absence of anv significant source variabilitv over the | The two magnitudes are nevertheless consistent within the error bars, and therefore suggest the absence of any significant source variability over the |
for the best-fitting σι by fitting the smeared PSF to the binned radial profile of ffrom the H2 observation using a least-squares method. | for the best-fitting $\sigma_+$ by fitting the smeared PSF to the binned radial profile of from the H2 observation using a least-squares method. |
The best description. shown in Fig. 1. | The best description, shown in Fig. \ref{psf}, |
is achieved ford,~1.5 aresec. | is achieved for $\sigma_+\approx 1.8$ arcsec. |
This value is well within the range determined from numerous other analyses of point source observations. | This value is well within the range determined from numerous other analyses of point source observations. |
Given the mentioned uncertainties in the determination of the correct HRI PSF for an individual observation. we conclude that the X-ray emission of iis point like to the limit of the HRI resolution. | Given the mentioned uncertainties in the determination of the correct HRI PSF for an individual observation, we conclude that the X-ray emission of is point like to the limit of the HRI resolution. |
Source counts were extracted from a circular region with radius 1.5 aremin around the centroid position of the X-ray source. | Source counts were extracted from a circular region with radius 1.5 arcmin around the centroid position of the X-ray source. |
The background was determined from a source free annulus with inner radius 3 and outer radius 5 aremin. | The background was determined from a source free annulus with inner radius 3 and outer radius 5 arcmin. |
Since wwas observed on-axis. no vignetting correction was applied to the data. | Since was observed on-axis, no vignetting correction was applied to the data. |
Finally. the observations were split into individual observation intervals with durations between 1000 to 3000 s. The intervals were chosen by hand to ensure sufficient photon statistics in each data point and to match as closely as possible the distribution of the observation intervals in "real time". | Finally, the observations were split into individual observation intervals with durations between 1000 to 3000 s. The intervals were chosen by hand to ensure sufficient photon statistics in each data point and to match as closely as possible the distribution of the observation intervals in “real time”. |
The resulting light curves are shown in Fig. 2.. | The resulting light curves are shown in Fig. \ref{light}. |
The average count rate in both. HI and H2. ts z0.3 ets/s. Assuming a simple power law spectrum with a photon index of [=2.8 (see Sect. | The average count rate in both, H1 and H2, is $\approx 0.3$ cts/s. Assuming a simple power law spectrum with a photon index of $\Gamma = 2.8$ (see Sect. |
3.3.1) and Galactic absorption =(1.240.2)«1079 7: Murphy et al. 1996)). | 3.3.1) and Galactic absorption $= (1.1\pm 0.2)\times 10^{20}$ $^{-2}$; Murphy et al. \cite{murphy}) ), |
this count rate yields an unabsorbed 0.1-2.4 keV flux of 5.2«10.1?i. | this count rate yields an unabsorbed 0.1-2.4 keV flux of $8.2\times 10^{-12}$. |
The corresponding rest frame 0.1-2.4 keV luminosity is LG«I0 ergs. !. | The corresponding rest frame 0.1-2.4 keV luminosity is $4.6\times 10^{44}$ erg $^{-1}$. |
cclearly displays rapid and large amplitude variability in both HRI observations. | clearly displays rapid and large amplitude variability in both HRI observations. |
At the beginning of H2 we obviously observed the decline of a larger outburst and the count rate decreases by almost a factor of two within one day. | At the beginning of H2 we obviously observed the decline of a larger outburst and the count rate decreases by almost a factor of two within one day. |
The fastest variability seen is a decline by within about 3 hours. which is truly remarkable given the high X-ray luminosity of3349+2438. | The fastest variability seen is a decline by within about 3 hours, which is truly remarkable given the high X-ray luminosity of. |
. Following Lawrence Papadakis (1993)). who find a correlation between luminosity and doubling time scale from EXOSAT variability power spectrum analysis. a variability time scale of the order of 10° to 10° sec would have been expected for13349+2438. | Following Lawrence Papadakis \cite{lawrence}) ), who find a correlation between luminosity and doubling time scale from EXOSAT variability power spectrum analysis, a variability time scale of the order of $^{\rm 6}$ to $^{\rm 7}$ sec would have been expected for. |
. For illustrative purposes we show in Fig. | For illustrative purposes we show in Fig. |
3. the historical X-ray light curve for uncluding all available data from the ROSAT All-Sky Survey to the latest HRI observations. | \ref{historic} the historical X-ray light curve for including all available data from the ROSAT All-Sky Survey to the latest HRI observations. |
All measurements have been converted to PSPC count rates assuming a simple power law spectrum (I— 2.8) and Galactic absorption. | All measurements have been converted to PSPC count rates assuming a simple power law spectrum $\Gamma = 2.8$ ) and Galactic absorption. |
X-ray variability was detected in all previous observations of1334942438. | X-ray variability was detected in all previous observations of. |
.. BFP96 note a factor of z| change between the two PSPC observations separated by about one year (PI and P2). | BFP96 note a factor of $\approx 4$ change between the two PSPC observations separated by about one year (P1 and P2). |
During P2 in December 1992. oobviously happened to be in a historical high state. | During P2 in December 1992, obviously happened to be in a historical high state. |
Unfortunately. P2 only lasted z1570 s and no variability was detected within this time interval (BFP96). | Unfortunately, P2 only lasted $\approx
1570$ s and no variability was detected within this time interval (BFP96). |
In the ASCA observation a increase in flux within about 5 hours was | In the ASCA observation a increase in flux within about 5 hours was |
lt is four vears now since the kilohertz quasi-periodic oscillations (kllz QPOs) were discovered in the persistent Hux of Scorpius X-1. (vanderΊανctal.199Ga) ancl 1172834 (Strohmaver.Zhang&Swank1996). | It is four years now since the kilohertz quasi-periodic oscillations (kHz QPOs) were discovered in the persistent flux of Scorpius X-1 \cite{vdk_iauc} and 1728–34 \cite{stroh_iauc}. |
. In the meantime. similar kIlz QPOs have been seen in some 20 other low-mass X-rav binaries (LAINBs: see van der lis 2000 for a review). | In the meantime, similar kHz QPOs have been seen in some 20 other low-mass X-ray binaries (LMXBs; see van der Klis 2000 for a review). |
These QPOs often appear in pairs. with frequencies £j and vo (vo> £0) between 400 and ~1300 Iz. which in à given source can shift by a few hundred Lz. ipparently as a function of mass accretion ra | These QPOs often appear in pairs, with frequencies $\nu_{1}$ and $\nu_{2}$ $\nu_{2} >
\nu_{1}$ ) between $\sim 400$ and $\sim 1300$ Hz, which in a given source can shift by a few hundred Hz, apparently as a function of mass accretion rate. |
Most of the models proposed so far assume that one of 1 kHz QPOs reflects the Ixeplerian orbital motion at some preferred radius in the accretion dise (e.g... Miller. Lamb Psaltis 1998: Stella Vietri 1999: Osherovich Titarchuk 1999). but there are other explanations as well (Ixlein et al. | Most of the models proposed so far assume that one of the kHz QPOs reflects the Keplerian orbital motion at some preferred radius in the accretion disc (e.g., Miller, Lamb Psaltis 1998; Stella Vietri 1999; Osherovich Titarchuk 1999), but there are other explanations as well (Klein et al. |
19962.b: Jernigan. Wlein Arons 2000). | 1996a,b; Jernigan, Klein Arons 2000). |
In. recent discussions (Lamb Miller 1999: Stella 1999: Psaltis 1999) it was emphasized that. empirical discrimination. between two of the leading classes of models is possible in. principle by studying the harmonic and sideband structure of the kllz QPOs. | In recent discussions (Lamb Miller 1999; Stella 1999; Psaltis 1999) it was emphasized that empirical discrimination between two of the leading classes of models is possible in principle by studying the harmonic and sideband structure of the kHz QPOs. |
Lore we concentrate only on these two mocel classes In the "sonic-point model (SPM: Miller et al. | Here we concentrate only on these two model classes In the `sonic-point' model (SPM; Miller et al. |
1998). the QPO at ve (the upper QPO) is produced at the radius where the racial How. velocity in the disc turns from subsonic to supersonic (the sonic radius). and the ΟΡΟ at vj (the lower QPO) originates by a beat between. the upper QPO and the spin frequency of the neutron star. | 1998), the QPO at $\nu_{2}$ (the upper QPO) is produced at the radius where the radial flow velocity in the disc turns from subsonic to supersonic (the sonic radius), and the QPO at $\nu_{1}$ (the lower QPO) originates by a beat between the upper QPO and the spin frequency of the neutron star. |
In the "relativistic-precession” model (IPM: Stella Vietri 1999) the QPO at vis also assumed. to be Ixeplerian. but the ΟΡΟ at £j is produced by the apsidal precession of a slightly non-circular inner aceretion disc. | In the `relativistic-precession' model (RPM; Stella Vietri 1999) the QPO at $\nu_{2}$ is also assumed to be Keplerian, but the QPO at $\nu_{1}$ is produced by the apsidal precession of a slightly non-circular inner accretion disc. |
Phe OPO frequencies in the RPAL are calculated. for test. particles in. purely geoclesic relativistic motion. Lc.. neglecting the hvdrodynamical ancl radiative ellects of the accretion Low. | The QPO frequencies in the RPM are calculated for test particles in purely geodesic relativistic motion, i.e., neglecting the hydrodynamical and radiative effects of the accretion flow. |
However. Psaltis Norman (2000) have recently proposed a dynamical model in which the QPOs are produced by oscillations in the accretion clisk. | However, Psaltis Norman \shortcite{psaltis_norman}
have recently proposed a dynamical model in which the QPOs are produced by oscillations in the accretion disk. |
1n this mocel which we will call ‘transition-raclius’ moce (TRAD. there is a transition racüus in the accretion disc tha acts às à band-pass filter with resonances near the orbita ancl periastron-precession frequencies. | In this model, which we will call `transition-radius' model (TRM), there is a transition radius in the accretion disc that acts as a band-pass filter with resonances near the orbital and periastron-precession frequencies. |
Besides the main peaks at οι and £5. the SPM and the TRAL predict other (weaker) harmonics ancl sidebancds of these QPOs. at specific frequencies. | Besides the main peaks at $\nu_{1}$ and $\nu_{2}$, the SPM and the TRM predict other (weaker) harmonics and sidebands of these QPOs, at specific frequencies. |
For instance. the SPA predicts a relatively strong harmonic of the lower QPO a Py, (see Table 3 of Miller et al. | For instance, the SPM predicts a relatively strong harmonic of the lower QPO at $2 \nu_1$ (see Table 3 of Miller et al. |
1998 for a list of other sidchancl peaks predicted by the SPAL). whereas the TRÀ predicts a sideband at 2v.νι (c. | 1998 for a list of other sideband peaks predicted by the SPM), whereas the TRM predicts a sideband at $2 \nu_{2} - \nu_{1}$ (cf. |
cq. | eq. [ |
29] in Psaltis Norman 2000). | 29] in Psaltis Norman 2000). |
In principle. the detection of a QPO at 21 and a non-detection ofa QPO at 212νι would tend to rule out the TIUM. whereas the detection of a QPO at ονι and a non-detection of à QDPO at 2/9, would tend to rule out the SPAL (Aliller 2000).. | In principle, the detection of a QPO at $2 \nu_1$ and a non-detection of a QPO at $2 \nu_{2} - \nu_{1}$ would tend to rule out the TRM, whereas the detection of a QPO at $2
\nu_{2} - \nu_{1}$ and a non-detection of a QPO at $2 \nu_1$ would tend to rule out the SPM \cite{miller_bologna}. . |
im danI] = —20 — DIJO nun where (he upper sign is for the liquid side and the lower sign is lor the gas side (1,= —iy). | = )^2 ] = -2 - 1) f_2(t) (1 _l)^2 where the upper sign is for the liquid side and the lower sign is for the gas side $\eta_g = - \eta_l$ ). |
This indicates that {ο must of course be negative for (<Q. | This indicates that $f_2$ must of course be negative for $t < 0$. |
The formula [ον the heat capacity along the coexistence curve is ec m dg Uu pata- i, ”..". | The formula for the heat capacity along the coexistence curve is c_V = - _0(t) + + ^2 + ]. |
The singular part comes Irom the terms which are zero and second order in j. | The singular part comes from the terms which are zero and second order in $\eta$ . |
This leads to ο—€(—/)" where Τιο —(2—a)(1—a)q ο I) ((tb (4 | This leads to $c_V \rightarrow c_- (-t)^{-\alpha}$ where T_c c_- = )a_- + - 1) ( b_-. |
9) Accorcling5 to Ref. | According to Ref. |
(23) the thermal compressibility hyp1 diverges5 as af when /—0 and as &(—/) when /—0. witha /e&5 a universal ralio. | \cite{Guida} the thermal compressibility $\kappa_T$ diverges as $\kappa_+ t^{-\gamma}$ when $t \rightarrow 0^+$ and as $\kappa_- (-t)^{-\gamma}$ when $t \rightarrow 0^-$, with $\kappa_+/\kappa_- \approx 5$ a universal ratio. |
Also. the heat capacity al 1—0 diverges as c/" when /—0. and as c(—/1)" when /—0. wilh efe20.5 another universal ratio. | Also, the heat capacity at $\eta \rightarrow 0$ diverges as $c_+ t^{-\alpha}$ when $t \rightarrow 0^+$ and as $c_- (-t)^{-\alpha}$ when $t \rightarrow 0^-$, with $c_+/c_- \approx 0.5$ another universal ratio. |
The former leads to the constraint hb = while the latter leads lo D.=a mu= « | (run. b(51 | The former leads to the constraint b_+ = while the latter leads to 2 a_+ = a_- + ( b_-. |
) If we are not too [ar from (he critical point wecan use the following parameterizalions. | If we are not too far from the critical point wecan use the following parameterizations. |
We can take f, to be a constant. | We can take $f_{\sigma}$ to be a constant. |
The function bu)(93) | The function f_2(t) =. |
where /o(/) is a smooth [uncetion which vanishes at /=0 as a power bigger (han 5. | where $\bar{f}_2(t)$ is a smooth function which vanishes at $t=0$ as a power bigger than $\gamma$. |
The Function. /y(/) is the chemical potential along (he critical curve. whichmay be parameterized like (his. | The function $f_1(t)$ is the chemical potential along the critical curve, whichmay be parameterized like this. |
Assume a quaclratic relationship between T. and µ,. | Assume a quadratic relationship between T and $\mu_x$ . ) |
null | ^2 + )^2 = 1 |
in refvelos.. | in \\ref{velos}. |
For LE8. Olszewski et ((1991)) give a velocity of 2484S5kkmss7!. in excellent agreement with our value. | For LE8, Olszewski et \cite{OSSH91}) ) give a velocity of $\pm$ $^{-1}$, in excellent agreement with our value. |
The mean velocity of all investigated AGB stars in NGC 1846 is 248x5.5 ss... which is in good agreement with the published values for the cluster. | The mean velocity of all investigated AGB stars in NGC 1846 is $\pm$ $^{-1}$, which is in good agreement with the published values for the cluster. |
The only star with a larger deviation from the mean velocity is LET7. | The only star with a larger deviation from the mean velocity is LE17. |
The C/O ratio and the C/C ratio were determined for all six O-rich stars in our sample. | The C/O ratio and the $^{12}$ $^{13}$ C ratio were determined for all six O-rich stars in our sample. |
The C/O ratios between 0.2 and 0.65 were found with an uncertainty between +0.05 and +£0.1. and 'C/UC isotopic ratios between 12 and 60 were found. | The C/O ratios between 0.2 and 0.65 were found with an uncertainty between $\pm$ 0.05 and $\pm$ 0.1, and $^{12}$ $^{13}$ C isotopic ratios between 12 and 60 were found. |
Temperatures of the best fit model spectra agree closely with the Tay values derived from near infrared photometry (Lebzelter Wood 2007)). | Temperatures of the best fit model spectra agree closely with the $T_{\rm eff}$ values derived from near infrared photometry (Lebzelter Wood \cite{LW07}) ). |
The result for each star is listed in Table 2. with the basic parameters of the stellar models used. | The result for each star is listed in Table \ref{t:coetc} with the basic parameters of the stellar models used. |
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