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For emissivities dc7, additional power appears at high frequencies as ϱ rises, with the particular values of the slopes and cuttoffs scaling with the cooling mechanism (Fig.3)). | For emissivities $\dot{q}\propto T^{\beta}$, additional power appears at high frequencies as $\beta$ rises, with the particular values of the slopes and cuttoffs scaling with the cooling mechanism \ref{fig:pds}) ). |
This result thus provides in principle a discriminant for GRB central engines powered by neutrino-cooled accretion flows, and illustrates how any local mechanism can be used in the same way to test its viability. | This result thus provides in principle a discriminant for GRB central engines powered by neutrino-cooled accretion flows, and illustrates how any local mechanism can be used in the same way to test its viability. |
The reason for the difference when modifying the cooling mechanism is fundamentally related to the local energy balance. | The reason for the difference when modifying the cooling mechanism is fundamentally related to the local energy balance. |
For dος7? (model T9)and a given internal energy supply, e, the local cooling time ἔσοοι=e/q is shorter under a temperature perturbation AT than when dxT? (model T6). | For $\dot{q}\propto
T^{9}$ (model $T9)$and a given internal energy supply, $e$, the local cooling time $t_{\rm cool} = e/\dot{q}$ is shorter under a temperature perturbation $\Delta T$ than when $\dot{q}\propto T^{6}$ (model $T6$ ). |
The power is thus higher at such frequencies, leading to the observed displacement in Fig. 3.. | The power is thus higher at such frequencies, leading to the observed displacement in Fig. \ref{fig:pds}. |
The argument holds when comparing model T6 with the test run at 6=1, and gives a way to characterize the accretion flow and discriminate between competing mechanisms, if the variations in the neutrino luminosity are directly reflected in the accretion power output which drives a relativistic flow. | The argument holds when comparing model $T6$ with the test run at $\beta=1$, and gives a way to characterize the accretion flow and discriminate between competing mechanisms, if the variations in the neutrino luminosity are directly reflected in the accretion power output which drives a relativistic flow. |
Two possible scenarios in which the difference between the cooling regimes studied here may be of relevance are worthy of note. | Two possible scenarios in which the difference between the cooling regimes studied here may be of relevance are worthy of note. |
First, the mass of the black hole introduces a scaling into the problem when combined with neutrino cooling. | First, the mass of the black hole introduces a scaling into the problem when combined with neutrino cooling. |
If Mgg is too large, the density and temperature in the accretion flow can be too low for neutrino cooling to operate. | If $M_{\rm BH}$ is too large, the density and temperature in the accretion flow can be too low for neutrino cooling to operate. |
As the BH mass is reduced, pair annihilation, with dος7? first becomes effective as an energy sink, followed by pair captures, with dοςT9. | As the BH mass is reduced, pair annihilation, with $\dot{q}\propto T^{9}$ first becomes effective as an energy sink, followed by pair captures, with $\dot{q}\propto
T^{6}$. |
A signature of the BH mass is thus in principle available in the variability of the flow. | A signature of the BH mass is thus in principle available in the variability of the flow. |
The second case is related to the time evolution of the flow, assuming a certain amount of mass Mais is initially available for accretion, and no further feeding of the central engine takes place. | The second case is related to the time evolution of the flow, assuming a certain amount of mass $M_{\rm
disk}$ is initially available for accretion, and no further feeding of the central engine takes place. |
Global disk simulations show that the density and temperature drop as the disk drains into the BH on the viscous time scale. | Global disk simulations show that the density and temperature drop as the disk drains into the BH on the viscous time scale. |
Given enough time, the whole disk will lie on the branch cooled by pair annihilation where o particles have formed, below log[p(g cm?)|~6.5 and log[T(?K)] ~10 for the adopted black hole mass (see also Figure 2 in Lee,Ramirez-Ruiz&Lopez-C | Given enough time, the whole disk will lie on the branch cooled by pair annihilation where $\alpha$ particles have formed, below $\log[\rho(\mbox{g~cm$ $})]\simeq 6.5$ and $\log[T(\mbox{$ $K})]\simeq 10$ for the adopted black hole mass (see also Figure 2 in \citet{lrrlc09}) ). |
amara 'Thus the variability may initially behave as in (2009))).case T'6, and end as in case T9 (the luminosity will have decreased substantially by then, along with the accretion rate). | Thus the variability may initially behave as in case $T6$, and end as in case $T9$ (the luminosity will have decreased substantially by then, along with the accretion rate). |
'The associated GRB, if one occurs, need not necessarily be powered by neutrinos in order for these effects to be apparent. | The associated GRB, if one occurs, need not necessarily be powered by neutrinos in order for these effects to be apparent. |
The fact that neutrinos are responsible for the cooling process allowing accretion makes them relevant in this context. | The fact that neutrinos are responsible for the cooling process allowing accretion makes them relevant in this context. |
A number of limitations apply to this study, and can be matters for further study. | A number of limitations apply to this study, and can be matters for further study. |
First, the scaling we have used to infer the variability at different radii is strictly valid if the flow is adiabatic. | First, the scaling we have used to infer the variability at different radii is strictly valid if the flow is adiabatic. |
Explicit cooling thus violates this assumption. | Explicit cooling thus violates this assumption. |
However, given the choice of ¢ and ἔσοοι the associated time scales are such that t¢oo1>> tayn, making this a reasonable approximation. | However, given the choice of $\zeta$ and $t_{\rm cool}$ the associated time scales are such that $t_{\rm cool} \gg t_{\rm dyn}$ , making this a reasonable approximation. |
Second, we have used the neutrino luminosity as a proxy for the manifestation of central engine activity, which could have a neutrino | Second, we have used the neutrino luminosity as a proxy for the manifestation of central engine activity, which could have a neutrino |
a Evicani. also known as Achernar (LID 10144). is one of the brightest stars in the Southern hemisphere. | $\alpha$ Eridani, also known as Achernar (HD 10144), is one of the brightest stars in the Southern hemisphere. |
With an apparent magnitude equal to 0.46. it is the brightest and one of the nearest Be stars to Earth (?).. | With an apparent magnitude equal to 0.46, it is the brightest and one of the nearest Be stars to Earth \citep{2007NewAR..51..706K}. |
3e stars are non-supergiant B-twpe stars that show. or have shown at one time or another. emission in the Balmer line series. | Be stars are non-supergiant B-type stars that show, or have shown at one time or another, emission in the Balmer line series. |
The first Be star was reported in 1866 by. Padre Angelo Secchi. where Balmer lines were observed in emission rather than in absorption (?).. | The first Be star was reported in 1866 by Padre Angelo Secchi, where Balmer lines were observed in emission rather than in absorption \citep{2003PASP..115.1153P}. |
Lor Be stars. the rotational velocity is 70-804. of the critical limit (2).. | For Be stars, the rotational velocity is $\%$ of the critical limit \citep{2003PASP..115.1153P}. |
Ehe rapid rotation causes two ellects on the structure of the star: rotational flattening and equatorial cdarkening (?).. | The rapid rotation causes two effects on the structure of the star: rotational flattening and equatorial darkening \citep{2007NewAR..51..706K}. |
2ο stars have pulsation modes that are typical of > Cophei and/or SPD stus. with frequencies. roughly between 0.4 d (eveles per dav) and 4 1 (7?).. | Be stars have pulsation modes that are typical of $\beta$ Cephei and/or SPB stars, with frequencies roughly between 0.4 $^{-1}$ (cycles per day) and 4 $^{-1}$ \citep{2008CoAst.157...70G}. |
X more complete review of Be stars may be found in ?.. | A more complete review of Be stars may be found in \citet{2003PASP..115.1153P}. |
In this paper we present an analvsis of the tempora variation of the two main oscillation frequencies. detectec in Xchernar. | In this paper we present an analysis of the temporal variation of the two main oscillation frequencies detected in Achernar. |
A description of the SALEL instrument. used to collect the data is presented in Section 2. | A description of the SMEI instrument used to collect the data is presented in Section 2. |
Xn overview of the data analysis procedure is given in Section 3. | An overview of the data analysis procedure is given in Section 3. |
The results of the amplitude. frequency and. phase analysis are presentec in Section 4 and possible theories for the nature of the uncovered variation in oscillation amplitude are cliscusse in Section 5. | The results of the amplitude, frequency and phase analysis are presented in Section 4 and possible theories for the nature of the uncovered variation in oscillation amplitude are discussed in Section 5. |
Finallv. concluding remarks are in Section 6. | Finally, concluding remarks are in Section 6. |
Launched. on 2003 January 6. the Solar Mass. Ejection Imager (SMEL on board the Coriolis satellite was designed primarily to detect ancl forecast. Coronal Mass. Ejections (CMES) from the Sun moving towards the Earth. | Launched on 2003 January 6, the Solar Mass Ejection Imager (SMEI) on board the Coriolis satellite was designed primarily to detect and forecast Coronal Mass Ejections (CMEs) from the Sun moving towards the Earth. |
However. as a result. of the satellite being. outside the Earth's atmosphere anc having a wide angle of view it has been able to obtain photometric lighteurves for most of the bright stars in the skv. | However, as a result of the satellite being outside the Earth's atmosphere and having a wide angle of view it has been able to obtain photometric lightcurves for most of the bright stars in the sky. |
These data have been used. to study the oscillations of a number of stars. for example: Arcturus (?).. Shedir (2).. Polaris (?).. 7 Vrsae Minoris (7).. 4. Doradus (?).. 3 ορ stars (Stevens et al. | These data have been used to study the oscillations of a number of stars, for example: Arcturus \citep{2007MNRAS.382L..48T}, Shedir \citep{2009arXiv0905.4223G}, Polaris \citep{2008MNRAS.388.1239S}, $\beta$ Ursae Minoris \citep{2008A&A...483L..43T}, $\gamma$ Doradus \citep{2008A&A...492..167T}, $\beta$ Cephei stars (Stevens et al. |
2010. in. prep.) | 2010, in prep.) |
and Cepheid variables (?).. | and Cepheid variables \citep{2010vsgh.conf..207B}. |
SMEL consists of three cameras each with a field of view of 60 37. which are sensitive over the optical wavebane. | SMEI consists of three cameras each with a field of view of $^{\circ}$ $\times$ $^{\circ}$, which are sensitive over the optical waveband. |
Phe optical svstem is ΠΠσος, so the pass band is determined. by the spectral response of the CCD. | The optical system is unfiltered, so the pass band is determined by the spectral response of the CCD. |
The quantum. cllicieney of the CCD is 45% at TOOnm. falling to LO% at roughly 460nm and 990nm. | The quantum efficiency of the CCD is $\%$ at 700nm, falling to $\%$ at roughly 460nm and 990nm. |
The cameras are mounted such that they sean most of the sky every 101 minutes. therefore the notional Nvquist. frequcney for the data is 7.086 I. | The cameras are mounted such that they scan most of the sky every 101 minutes, therefore the notional Nyquist frequency for the data is 7.086 $^{-1}$. |
Photometric results from Camera L ancl Camera 2 are used in the analysis of Achernar. | Photometric results from Camera 1 and Camera 2 are used in the analysis of Achernar. |
Camera 53 is ina higher temperature environment than the other two cameras and as a result the photometric data is highly degracecd. | Camera 3 is in a higher temperature environment than the other two cameras and as a result the photometric data is highly degraded. |
The photometric timeseries for Achernar djs. shown in Figure 1.. | The photometric timeseries for Achernar is shown in Figure \ref{whole_timeseries}. |
Note that the pronounced. u-shapes in the lighteurve are due to elfects from the SALEL instrumentation. | Note that the pronounced u-shapes in the lightcurve are due to effects from the SMEI instrumentation. |
Tidal dissipation iu the approximately svuchronously rotating mnuer plauct decreases its orbital enerev without sienificautly chaneine its orbital angular ΤΠ ΠΕ, | Tidal dissipation in the approximately synchronously rotating inner planet decreases its orbital energy without significantly changing its orbital angular momentum. |
follows from equations aud that this causes a, to vary as dluodfc203r. | It follows from equations and that this causes $a_1$ to vary as $d\ln a_1/dt \approx - 2 e_1^2/\tau$. |
Since secular mteractions transfer augular momentum but uot enerev between the planets” orbits. e» remains constant. | Since secular interactions transfer angular momentum but not energy between the planets' orbits, $a_2$ remains constant. |
Conservation of total angular momentum leads to. where J;—Vl>ο» ΗΗΗΠ. | Conservation of total angular momentum leads to, where $J_i = \sqrt{1-e_i^2}
m_i n_i a_i^2$ . |
Apse aliguincut. imyplics. €4=flay.e). from which we deduce that where f,,=OluΕπ aud f.,=Olnf/O lies. | Apse alignment implies $e_1 = f(a_1, e_2)$, from which we deduce that where $f_{a_1} \equiv \partial\ln f/\partial\ln a_1$ and $f_{e_2}
\equiv \partial\ln f/\partial \ln e_2$ . |
Auunenealh. fi,m8 aud fr,m Lb. so currently ey aud e» are damping on time-scales of 8.67 and. £07. respectively: ος damips | Numerically, $f_{a_1}
\approx 8$ and $f_{e_2} \approx 1$ , so currently $e_1$ and $e_2$ are damping on time-scales of $8.6\tau$ and $40\tau$, respectively; $e_1$ damps faster than $e_2$ because the inner planet's semi-major axis is shrinking. |
faster than e» | In \ref{sec:nonlinear} we assumed that tidal dissipation had delivered the planets' orbits into a state of periapse alignment. |
because the immer | Here we justify this assumption by studying the secular dynamics and tidal evolution under the somewhat inaccurate approximation of small orbital eccentricities. |
planct’s | Our starting point is the expansion of the secular disturbing function to second order in $e_1$ and $e_2$. |
seniüanajor axi | Substituting the expanded disturbing function into Lagrange's equationsfor the variations of the $e$ 's and the $\pomega$ 's, and adding a term to represent tidal damping of $e_1$, yields the following set of linear equations (MD2000) for the complex variable $I_i \equiv e_i \exp(i\pomega_i)$. |
s i | The coefficients $A_{ij}$ read: with $\alpha = a_1/a_2$, $B_1 = b_{3/2}^{(1)}$, and $B_2 = b_{3/2}^{(2)}$; $b_s^{(j)}$ is the usual Laplace coefficient. |
null | The symbols $\Delta$ and $\tau$ retain their definitions from \ref{sec:nonlinear}. . |
x sliailkine. | In the absence of tidal damping $\tau\to \infty$ ), the general solution of equation can be expressed as a super-position of two eigenvectors ${\cal I}_*$ with where We distinguish the two eigensolutions by $*=p$ and $*=m$ according to the plus or minus sign that appears before the square root in the expression for $g_*$. |
Tu 52 | Since $(e_{1p}/e_{2p})\, (e_{1m}/e_{2m}) =
A_{12}/A_{21} \approx 1.0$, one solution satisfies $e_{1} < e_{2} $ and the other $e_{1} > e_{2}$. |
we asstuned that tid | If we take $\Delta = 5\times
10^{-11}\s^{-1}$, the $*=m$ solution reproduces the observed eccentricity ratio $e_{1m}/e_{2m} = 0.19$ and apse The other solution has $e_{1p}/e_{2p} = 5.26$ and anti-aligned apses. |
al dissipation had de | If both eigenvectors have non-zero lengths, the orbital eccentricities oscillate out of phase on time-scales of thousands of years $\sim 1/g_*$ ) as depicted in Mayor \cite{mayor83443}) ). |
null | When $\tau$ is finite, $a_1$ and therefore $A_{ij}$ vary with time. |
livered | Since $\tau \gg 1/g_*$, the solution for the eigenvectors given by equation remains valid. |
the planet | However, the eigenvector components, $e_{i*}$, decay slowly with time as where $\gamma_{i*} \equiv - d\ln e_{i*}/dt$. |
s’ orbits iuto a sta | In thelimit of small eccentricity, the expressions for the nonlinear damping rates given by equation reduce to those in equation provided $f_{e_2}=1$ as is appropriate for the linear problem. |
te o | Each of the four eigenvector components has its individual damping rate. |
f poriapse | Both components of the $p$ eigenvector decay more rapidly than those of the $m$ eigenvector. |
aliguiucut. | If the current state consists mostly the $m$ eigenvector with a small admixture of the $p$ eigenvector, both components in the latter would decay on time-scale $\approx \tau$, while the dominant components $e_{1m}$ and $e_{2m}$ would decay much more slowly, as $\gamma^{-1}_{1m}\approx 7\tau$ and $\gamma^{-1}_{2m}\approx 32\tau$. |
Tere we justi | It is notable that the asymmetry in damping rates between the aligned and the anti-aligned solutions only applies if the inner planet has an orbital period $\leq 5$ days. |
f | For longer periods, the additional precession rates (eqs. \ref{eq:pretide}] ] - \ref{eq:preGR}] ]) |
null | are unimportant and $(e_{1m}/e_{2m}) \sim (e_{1p}/e_{2p}) \sim 1$. |
y this assum | Both the survival of HD $83443b$ 's orbital eccentricity and the locking of its apse to that of HD $83443c$ are straightforward consequences of secular and tidal dynamics. |
ption b | The ratio $e_1/e_2$ provides compelling evidence for the tidal and rotational distortion of HD $83443b$. |
y stu | It constrains unknown system parameters by relating $C\equiv (k_{2}/k_{2J})(R_1/R_J)^5$ to $\sin i$ . |
c | In particular, we find $C > 0.9$. |
vin | Apse alignment indicates that a minimum amount of tidal dissipation has occurred over the life-time of the system, taken to be$N$ Gyrs. |
g the sec | By requiring that thisisat least 3 times longer than $\tau$ , we deduce that $Q < 5 \times 10^5\, N C \sin i$ .It may also be possible toinfer a lower limit for $Q$ by integrating the orbits backward in time. |
u | At the present time $n_1/n_2 = 9.99 \pm 0.06$ , suggestive of a $10:1$ mean motion resonance. |
la | This is probably just a coincidence. |
r 4o4 " 4 N | Moreover, no physically important effects would be associated withthis resonance should it exist. |
inodel in NSPEC (Tiarchuk.199L).. | model in XSPEC \cite{tita94apj}. |
This model is andiprovement over the [oruer model as the tjeory is extended to iucluce relativistics efTects. and O WOFLς for both he optically thin aud thick Cases (Titachuk.LOOL).. | This model is an improvement over the former model as the theory is extended to include relativistics effects, and to work for both the optically thin and thick cases \cite{tita94apj}. |
The [ree parameters of the 1jodel are: the teuyeratur'e ol the seed. pliotous |J). assunec to follow a Wieu law. the optic: depth of the scatte‘iL[n]0 region (7). the electron leimperalttre (KT«)). :d a parameter deining t eeometry (a disk or a sphee). | The free parameters of the model are: the temperature of the seed photons ), assumed to follow a Wien law, the optical depth of the scattering region $\tau$ ), the electron temperature ), and a parameter defining the geometry (a disk or a sphere). |
In the following. we cousider the spherical case. | In the following, we consider the spherical case. |
Obviously in t abseuce of siguilicant cutoffs in the specrun. (d electron temperature canuot be easily constraiued N the ASCA data. | Obviously in the absence of significant cutoffs in the spectrum, the electron temperature cannot be easily constrained by the ASCA data. |
Tierefore. to begin with. we lave selTu. 7. to their bes fit values isted in Cuainazzi et al. (1008): | Therefore, to begin with, we have set, $\tau$, to their best fit values listed in Guainazzi et al. \cite*{tz2:guainazzi98aa}; |
ie. 1. | i.e. 1. |
keV. 3.. | keV, 3., |
ali 30 keV respectively (see Table 1). | and 30 keV respectively (see Table 1). |
We thus it he normalization of the LOCel. and the soft excess modeled either by a BB or a DBB (a it is not possible when such a coiipolent is not added). | We thus fit the normalization of the model, and the soft excess modeled either by a BB or a DBB (a fit is not possible when such a component is not added). |
The results of the fit are listed in Table 2, | The results of the fit are listed in Table 2. |
Τιe lowest. reducec V> value is. obtained" when the soft component is fit by a BB. | The lowest reduced $\chi^2$ value is obtained when the soft component is fit by a BB. |
A comparison with the values listed in Table 1 indicates hat our bes ib parameters a'e consistent with SAN for the ‘ompll++BB model. but cillers iu the luner disk emperature [or the Compl++DBB model. | A comparison with the values listed in Table 1 indicates that our best fit parameters are consistent with SAX for the +BB model, but differs in the inner disk temperature for the +DBB model. |
Asstunine a disk geometry insead of a spherical one leads to the same conclusio1 | Assuming a disk geometry instead of a spherical one leads to the same conclusion. |
As a further tes of the validity of the latter 1iodel. we have le and 7 as [ree parauete ‘sof tje fit in the Compl++BB model. | As a further test of the validity of the latter model, we have let and $\tau$ as free parameters of the fit in the +BB model. |
The paraneters so recovered =0.940.03»107?7e F=X2329Ulqur kTyy=L.1Dlo» keV while the paunetlers of the BB componerC remained unchanged. | The parameters so recovered $0.94\pm0.03 \times
10^{22}$, $\tau=3.2^{+0.4}_{-0.7}$, $1.1^{+0.1}_{-0.2}$ keV while the parameters of the BB component remained unchanged. |
We have shown above that the ASCA spectra alone could be fit by tie SAN and RNTE mode. | We have shown above that the ASCA spectra alone could be fit by the SAX and RXTE model. |
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