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Tιο made of the N-rav spectrum for cach burst. therefore. depends om onlv three piranueters: the color ten]eraure of the blackbody. το. the normalization of the blackbody spectrum. ο, and he equivalent hydrogen colum1 densitv. AY of the 1iterstellar extinction. | The model of the X-ray spectrum for each burst, therefore, depends on only three parameters: the color temperature of the blackbody, $T_{\rm c}$, the normalization of the blackbody spectrum, $A$, and the equivalent hydrogen column density, $N_{\rm H}$ of the interstellar extinction. |
Allowing the hydrogcu coluun deusity ΑΙ to be a free xuanmeter iu the fittime procedure leads to correlaed uncertainties between the aukmut of 1iterstellar extiuction and the temperature of the blackbody. | Allowing the hydrogen column density $N_{\rm H}$ to be a free parameter in the fitting procedure leads to correlated uncertainties between the amount of interstellar extinction and the temperature of the blackbody. |
Moreover. for every 107 2 overestiniio1 Cuuderestimatio1) of the colin deusitv. tie luferred flux of he tjcrnial eniissioiLis systematically larger (smaler) by ~ () ere P78 ο (Galoway et 220082). | Moreover, for every $^{22}$ $^{-2}$ overestimation (underestimation) of the column density, the inferred flux of the thermal emission is systematically larger (smaller) by $\approx$ $^{-9}$ erg $^{-2}$ $^{-1}$ (Galloway et 2008a). |
Reducing t1o uncertainties relaed to the interstellar extinctio1 for each Inrstor 19. herefore. verv nuportaut 1 controlling svstcmatic effects. | Reducing the uncertainties related to the interstellar extinction for each burster is, therefore, very important in controlling systematic effects. |
A recent analvsis of high resolution erating spcνα from a miuber of N-rav binaries (Miller. Cacsett. Reis 209) showed that the tudividua plotoclectzje absortion. edges cyoserved in tl1ο N-vay spectra do nuo slow sienific:uat varatious with source luminosity or spectral state. | A recent analysis of high resolution grating spectra from a number of X-ray binaries (Miller, Cackett, Reis 2009) showed that the individual photoelectric absorption edges observed in the X-ray spectra do not show significant variations with source luminosity or spectral state. |
This resit stronely πιeeests that the neutral livdrogeu colim density is dominated by absorption in the iuOCLstellay medium aud does 1Xt change on short tinescaes. | This result strongly suggests that the neutral hydrogen column density is dominated by absorption in the interstellar medium and does not change on short timescales. |
Iu order to reduce this svstematic uncertainty aud giveu the fact that there is no evideice of variable5 neutral asorption for czcli system. we fixed the lycheeech column density for cach source to a constant value tha we obtained iu one of the ollowing ways. | In order to reduce this systematic uncertainty and given the fact that there is no evidence of variable neutral absorption for each system, we fixed the hydrogen column density for each source to a constant value that we obtained in one of the following ways. |
(7) For a uuuber of sources. t1e equivalent hydrogen colum density was 1ferred indepeudently usiug ieli-resolution spectrographs. | For a number of sources, the equivalent hydrogen column density was inferred independently using high-resolution spectrographs. |
Di cases wlwre only au optical extinction or reddening ueasurement exists; we uscd he relation given by CivY Ozzel (2009) ο convert it to the equivalent bydrogen coltuu density. | In cases where only an optical extinction or reddening measurement exists, we used the relation given by Güvver Özzel (2009) to convert it to the equivalent hydrogen column density. |
(7) Finally. for hnree sources (LU 129. KS 260. SAN 2900) there are no indepeudeut hvdrogeu colum density ucasureimoeuts published iu the literature. | Finally, for three sources (4U $-$ 429, KS $-$ 260, SAX $-$ 2900) there are no independent hydrogen column density measurements published in the literature. |
Iu hese cases. we fit the X-ray spectra obtaimecL during all the N-rayv bursts 6ot cach sowree allowing the Ny value to vary in a wide range. | In these cases, we fit the X-ray spectra obtained during all the X-ray bursts of each source allowing the $_{\rm H}$ value to vary in a wide range. |
We then found tle resiΠιο mean value of Nyy and used this as a constant iu Qur secoud set of fits to all the N-ray bursts. | We then found the resulting mean value of $_{\rm H}$ and used this as a constant in our second set of fits to all the X-ray bursts. |
In Table . we show the adopted values or the hydrogen cohnun deidtv together with the neasunreiient uncertaimties. the method with which the values were estimated. and t10 appropriate references. | In Table \ref{sourcestable}, we show the adopted values for the hydrogen column density together with the measurement uncertainties, the method with which the values were estimated, and the appropriate references. |
Futur' observations of these sources with N-rav erating spectrometers 6miboard Chaudra iux| AL can help decise the uncertainty ariung frou the lack of knowledge of the xoperties of the interstella rauatter towards these soirees, | Future observations of these sources with X-ray grating spectrometers onboard Chandra and $-$ can help decrease the uncertainty arising from the lack of knowledge of the properties of the interstellar matter towards these sources. |
Our eval in this aricle is o study potential svstcmatic uncertainties m the meastroiuent of the apparent area of cach neutron star duriug the cooling tails of N-rav πανί», | Our goal in this article is to study potential systematic uncertainties in the measurement of the apparent area of each neutron star during the cooling tails of X-ray bursts. |
Hereafter. we adopt the following working definition of he cooling tail. | Hereafter, we adopt the following working definition of the cooling tail. |
It is the time interval during which he interred flux is lower than the peak flux of the burst or the ouchdown flux for pliotos]xierie radius expansion biists. | It is the time interval during which the inferred flux is lower than the peak flux of the burst or the touchdown flux for photospheric radius expansion bursts. |
For the purpose of this definifion. we use as a touchdown xut the first moment at which the blackbody temperature reaches its highest value aud the iuferred appareit radius is lowest. | For the purpose of this definition, we use as a touchdown point the first moment at which the blackbody temperature reaches its highest value and the inferred apparent radius is lowest. |
In order to coitrol the countrate statistics. we also set a lower limit ou the thermal flux during eac1 cooling ail of 510 Pere tem ? (oy «10 10 for the exceptionally faint sources LU 37 and [U 101). | In order to control the countrate statistics, we also set a lower limit on the thermal flux during each cooling tail of $\times$ $^{-9}$ erg $^{-1}$ $^{-2}$ (or $\times$ $^{-10}$ for the exceptionally faint sources 4U $-$ 37 and 4U $-$ 401). |
Iu 83 and lL. we discuss in detail our approach of quautifving the svstematic uncertainties in the 1icasureimoents of he apparent radi duriis the cooling tails of thermonuclear bursts. using the sources KS 260. IU 1728 51. and IU 536 as case studies. | In 3 and 4, we discuss in detail our approach of quantifying the systematic uncertainties in the measurements of the apparent radii during the cooling tails of thermonuclear bursts, using the sources KS $-$ 260, 4U $-$ 34, and 4U $-$ 536 as case studies. |
For another analysis of the cooling tails of X-ray bursts from IU. 53Y που Zhaug. Mendez. Altanirano (2011). | For another analysis of the cooling tails of X-ray bursts from 4U $-$ 536, see Zhang, Mendez, Altamirano (2011). |
In 5. we repeat this procedure svstematically for seven additional sources youn Table 1.. | In 5, we repeat this procedure systematically for seven additional sources from Table \ref{sourcestable}. |
Finally. t1¢ cooling tails of the bursts observed frou £U 10 aud Aql X-1 show imegular yoliavior. and we report our analysis of thei in the Appendix. | Finally, the cooling tails of the bursts observed from 4U $-$ 40 and Aql X-1 show irregular behavior, and we report our analysis of them in the Appendix. |
Qur first working hypothesis is that the spectra of neutron stars diving the cooling tails of thermonuclear bursts cau be modeled by blackbody functious iu the observed energy ranec. | Our first working hypothesis is that the spectra of neutron stars during the cooling tails of thermonuclear bursts can be modeled by blackbody functions in the observed energy range. |
The fact that ταν spectra can be described well with blackbody fuuctious has been established since the first time resolved N-rav. spectral studies of thermonuclear bursts (sec. ce... Swank et 11977: Lewin. van Paradijs. Tiuuu 1993: Galloway ct 22008a and references therein). | The fact that X-ray spectra can be described well with blackbody functions has been established since the first time resolved X-ray spectral studies of thermonuclear bursts (see, e.g., Swank et 1977; Lewin, van Paradijs, Taam 1993; Galloway et 2008a and references therein). |
originally located at an outer edge of disk, extending to 10 AU. | originally located at an outer edge of disk, extending to 10 AU. |
Figure 7 shows the terrestrial planetary accretion becomes more difficult when the mutual inclination of the gas-giants increases. | Figure \ref{fig7} shows the terrestrial planetary accretion becomes more difficult when the mutual inclination of the gas-giants increases. |
One may note that in the final evolution there exist several survivals in the outer region of the disk up to 7 AU. | One may note that in the final evolution there exist several survivals in the outer region of the disk up to 7 AU. |
In Figure 8,, the runs of 0.1 ~ 10 AU configuration would relatively have less and smaller final planets than those in the 0.3 ~ 10 AU cases. | In Figure \ref{fig8}, the runs of 0.1 $\sim$ 10 AU configuration would relatively have less and smaller final planets than those in the 0.3 $\sim$ 10 AU cases. |
In addition, in the cases of initial distribution up to 10 AU, there also exist several survivals in the outer realm of the planetary disk at 7 AU. | In addition, in the cases of initial distribution up to 10 AU, there also exist several survivals in the outer realm of the planetary disk at 7 AU. |
Some survivals experienced several accretion, at a limited rate, in comparison with those in the inner disk; others, like Mar-sized protoplanets, emerged around the region of 7.5 ~ 11 AU in the end of simulations. | Some survivals experienced several accretion, at a limited rate, in comparison with those in the inner disk; others, like Mar-sized protoplanets, emerged around the region of 7.5 $\sim$ 11 AU in the end of simulations. |
Compared with our solar system, these formed bodies may be in a rescaled position of Uranus, Neptune and the Kuiper belt objects, implying that Neptunian-mass planet may be formed if more bodies were initial placed there and abundant mass were considered over much longer timescale evolution. | Compared with our solar system, these formed bodies may be in a rescaled position of Uranus, Neptune and the Kuiper belt objects, implying that Neptunian-mass planet may be formed if more bodies were initial placed there and abundant mass were considered over much longer timescale evolution. |
As mentioned, the OGLE-2006-BLG-109L was the first double planet system, discovered by gravitational microlensing method (Gaudietal.2008). | As mentioned, the OGLE-2006-BLG-109L was the first double planet system, discovered by gravitational microlensing method \citep{gaudi08}. |
. Two companions occupy the masses of ~0.71 and ~0.27M;, respectively, and each orbits its central star at about 2.3 and 4.6 AU. | Two companions occupy the masses of $\sim0.71$ and $\sim0.27 M_J$, respectively, and each orbits its central star at about 2.3 and 4.6 AU. |
The mass of the star is ~0.5 Mo, and this system is resemblance to a rescaled solar system, for the mass ratio and the separation between them is comparable to those of Jupiter and Saturn. | The mass of the star is $\sim$ 0.5 $M_{\odot}$, and this system is resemblance to a rescaled solar system, for the mass ratio and the separation between them is comparable to those of Jupiter and Saturn. |
The habitable zone around a star is defined as the region where a terrestrial planet with N2-CO2-H2O atmosphere could maintain liquid oceans on its surface (Kastingetal. 1993).. | The habitable zone around a star is defined as the region where a terrestrial planet with $N_{2}$ $CO_{2}$ $H_{2}O$ atmosphere could maintain liquid oceans on its surface \citep{kasting93}. . |
As for this system, the HZ is about 0.25—0.36 AU from the star (Migaszewskietal. 2009). | As for this system, the HZ is about $\sim$ 0.36 AU from the star \citep{miga09}. |
. Since the OGLE-2006-BLG-109L planetary system bears very close similarity to solar system, one may wonder to know whether it is possible to harbor other planets, especially for potential Earth-like planets in the HZ. | Since the OGLE-2006-BLG-109L planetary system bears very close similarity to solar system, one may wonder to know whether it is possible to harbor other planets, especially for potential Earth-like planets in the HZ. |
Recently, Malhotra&Minton(2008) analyzed secular dynamics for OGLE-2006-BLG-109L and presented astable | Recently, \citet{malhotra08} analyzed secular dynamics for OGLE-2006-BLG-109L and presented astable |
Only the relative amplitude of the two terms is important: Apowerlaw@30GHz/Adustpeak=867+400. | Only the relative amplitude of the two terms is important: $A_{power law 30\,GHz}/A_{dust peak} = 867 \pm 400$. |
We also include uncertainty in the laboratory measurement of each experiment’s bandpass. | We also include uncertainty in the laboratory measurement of each experiment's bandpass. |
The mean value and uncertainty in Ἡ, is estimated using 100,000 realizations of the above parameters, and found to be 1.008+0.021 (See in Table | The mean value and uncertainty in R is estimated using 100,000 realizations of the above parameters, and found to be $1.008 \pm 0.021$ (See in Table \ref{tab:calerr}. .) |
Given that our integration radius is larger then RCW38’s size, the flux contribution of diffuse emission near RCWS38 A6..)can be significant. | Given that our integration radius is larger then RCW38's size, the flux contribution of diffuse emission near RCW38 can be significant. |
The spectrum of this extended structure may be different from that of RCW38, in which case the calibration ratio would depend on the integration radius. | The spectrum of this extended structure may be different from that of RCW38, in which case the calibration ratio would depend on the integration radius. |
We estimate this uncertainty from the observed variability of the calibration ratio with integration distance. | We estimate this uncertainty from the observed variability of the calibration ratio with integration distance. |
The calibration value from the real map is normalized by the spectral correction for RCW38 and the signal-only transfer functions estimated for each experiment. | The calibration value from the real map is normalized by the spectral correction for RCW38 and the signal-only transfer functions estimated for each experiment. |
The result of this analysis is that the temperature scale for ACBAR’s CMB fields in 2002 should be multiplied by 1.128+0.066 relative to the planet-based calibration given in ?.. | The result of this analysis is that the temperature scale for ACBAR's CMB fields in 2002 should be multiplied by $1.128 \pm 0.066$ relative to the planet-based calibration given in \citet{runyan03a}. |
Table A6 tabulates the contributing factors and error budget. | Table \ref{tab:calerr} tabulates the contributing factors and error budget. |
We now proceed to propagate this RCW38-based calibration to the CMB2 observation done in 2001. | We now proceed to propagate this RCW38-based calibration to the CMB2 observation done in 2001. |
We carry the 2002 RCWS3S calibration into 2001 by comparing the 2001 observations of the CMB2 field to the overlapping 2002 CMB4 field. | We carry the 2002 RCW38 calibration into 2001 by comparing the 2001 observations of the CMB2 field to the overlapping 2002 CMB4 field. |
The fields are reduced to the overlapping region and a power spectrum is calculated for each field. | The fields are reduced to the overlapping region and a power spectrum is calculated for each field. |
The bands are widened (A4~ 200) to avoid large noise correlations between the band-powers. | The bands are widened $\Delta \ell \sim 200$ ) to avoid large noise correlations between the band-powers. |
Care was taken to insure the filtering of the two maps only occurs in the overlapped region. | Care was taken to insure the filtering of the two maps only occurs in the overlapped region. |
However, differences in scan patterns and array configurations between the two seasons cause differences in filtering. | However, differences in scan patterns and array configurations between the two seasons cause differences in filtering. |
We assume that the band-powers from field o and field 6 are (f=a, B). | We assume that the band-powers from field $\alpha$ and field $\beta$ are ${\bf q}_f$ $f=\alpha\,,\beta$ ). |
If the relative calibration factor between fields a and 6 is η, we can find the value 7 that maximizes the likelihoodq; function: The quantity is the variance of (quy;—48.1) for band i, which can be found by Monte Carlo analysis. | If the relative calibration factor between fields $\alpha$ and $\beta$ is $\eta$, we can find the value $\eta_0$ that maximizes the likelihood function: The quantity $\sigma_{D,i}^2$ is the variance of $(q_{\alpha,i}-q_{\beta,i})$ for band $i$, which can be found by Monte Carlo analysis. |
A separate Monte Carlo simulationση, was carried out to confirm that 5o is an unbiased estimator, and to calculate its uncertainty. | A separate Monte Carlo simulation was carried out to confirm that $\eta_0$ is an unbiased estimator, and to calculate its uncertainty. |
We find the calibration factor to be CMB2/CMB4=1.238+0.067 (Πο, in units of temperature). | We find the calibration factor to be $= 1.238 \pm 0.067$ $\sqrt{\eta_0}$, in units of temperature). |
Approximating the uncertainties as Gaussian, it implies the CMB2 temperature scale should be multiplied by 0.911+0.074 relative to the scale used for the analysis of K04. | Approximating the uncertainties as Gaussian, it implies the CMB2 temperature scale should be multiplied by $0.911 \pm 0.074$ relative to the scale used for the analysis of K04. |
»f introduced. at. the outset to cover places where it is anticipated that extra resolution will be required. (e.g. Burkert Bodenheimer 1993): or they can be introduced and removed during the course of the simulation. whenever and wherever they are required. as in Acaptive Mesh tefinement (AMIR: Bereer Colella 1989. Truclove ct al. | be introduced at the outset to cover places where it is anticipated that extra resolution will be required (e.g. Burkert Bodenheimer 1993); or they can be introduced and removed during the course of the simulation, whenever and wherever they are required, as in Adaptive Mesh Refinement (AMR; Berger Colella 1989, Truelove et al. |
997. 1998). | 1997, 1998). |
With AMI. the resolution can. in principle. be increased indefinitely in regions where this is necessary. | With AMR, the resolution can, in principle, be increased indefinitely in regions where this is necessary. |
Io standard SPLL the linear resolution is (inp) where m ids the mass of an individual particle and p is the local density. | In standard SPH, the linear resolution is $\sim
(m/\rho)^{1/3}$, where $m$ is the mass of an individual particle and $\rho$ is the local density. |
In. this paper we describe a rather straightforward algorithm whereby the resolution of an SPILL code can also. in principle. be increased. indefinitely. by splitting particles. | In this paper we describe a rather straightforward algorithm whereby the resolution of an SPH code can also, in principle, be increased indefinitely, by splitting particles. |
We test the method on the standard Boss DBodenheimer (1979) test. using the Jeans condition to trigeer splitting. | We test the method on the standard Boss Bodenheimer (1979) test, using the Jeans condition to trigger splitting. |
Particle splitting and merging were used bv Meglicki. Wickramasinghe Bicknell (1993) in their simulations of accretion clises. and particle merging had been used. earlier by Monaghan Varnas (LOSS) in simulations of interstellar cloud. complexes. | Particle splitting and merging were used by Meglicki, Wickramasinghe Bicknell (1993) in their simulations of accretion discs, and particle merging had been used earlier by Monaghan Varnas (1988) in simulations of interstellar cloud complexes. |
However. in these earlier implementations splitting was used to maintain resolution in ow-density regions. and merging was used to avoid following aree numbers of particles in high-density regions (essentially he motivation for sink particles: Date. Bonnell Price. 1995). | However, in these earlier implementations splitting was used to maintain resolution in low-density regions, and merging was used to avoid following large numbers of particles in high-density regions (essentially the motivation for sink particles; Bate, Bonnell Price, 1995). |
Therefore. the philosophy was very dillerent. fron hat adopted here. where high resolution is advocated in ueh-censity regions to avoid violating the Jeans condition (i.e. to ensure resolution of the local Jeans mass). | Therefore, the philosophy was very different from that adopted here, where high resolution is advocated in high-density regions to avoid violating the Jeans condition (i.e. to ensure resolution of the local Jeans mass). |
Moreover. no Losts were reported in these earlier papers. | Moreover, no tests were reported in these earlier papers. |
Section 2 gives a brief review of standard SPILL. | Section 2 gives a brief review of standard SPH. |
Section 3 sketches the implementation of Particle Splitting. at the microscopic level. and Section. 4 describes how the best parameters for Particle Splitting are determined. | Section 3 sketches the implementation of Particle Splitting, at the microscopic level, and Section 4 describes how the best parameters for Particle Splitting are determined. |
Sections 5 and 6 deseribe two algorithms for Particle Splitting at the macroscopic level. | Sections 5 and 6 describe two algorithms for Particle Splitting at the macroscopic level. |
In. Nested Splitting (Section 5) all the particles in α- prescribed: sub-clomain are split. from some predetermined. time /split onwards. so that in cllect a high-resolution simulation is performed. inside the sub-domain using initial anc boundary. conditions supplied. by the coarser simulation in the overall computationa lomain. | In Nested Splitting (Section 5) all the particles in a prescribed sub-domain are split from some predetermined time $t\spl$ onwards, so that in effect a high-resolution simulation is performed inside the sub-domain using initial and boundary conditions supplied by the coarser simulation in the overall computational domain. |
In On-The-FIv Splitting (Section 6). particles are split only when this is necessary according to some local criterion such as the Jeans Condition. | In On-The-Fly Splitting (Section 6), particles are split only when this is necessary according to some local criterion such as the Jeans Condition. |
On-VPhe-Fly Splitting is the more computationally ellieient procedure. requiring no manual intervention ancl no prior knowledge of where high resolution will be required: it. corresponds closely to ΑΔΗ. | On-The-Fly Splitting is the more computationally efficient procedure, requiring no manual intervention and no prior knowledge of where high resolution will be required; it corresponds closely to AMR. |
Section 7 introduces the Jeans Condition. and Section & demonstrates how effectively ancl economically SPILL performs the standard Boss Bodenheimer (1979). rotating collapse test. when Particle Splitting is included. | Section 7 introduces the Jeans Condition, and Section 8 demonstrates how effectively and economically SPH performs the standard Boss Bodenheimer (1979) rotating collapse test when Particle Splitting is included. |
Section 9 discusses the results and compares them with those obtained using AAR. | Section 9 discusses the results and compares them with those obtained using AMR. |
Our conclusions are summarized in Section 10. | Our conclusions are summarized in Section 10. |
In SPIEL the evolution of the gas is predicted. by following the motions of an ensemble of ια] particles which act as discrete. sampling points (e.g. Monaghan. 1992). | In SPH the evolution of the gas is predicted by following the motions of an ensemble of $i\ttl$ particles which act as discrete sampling points (e.g. Monaghan 1992). |
Lach particle has associated with it a mass m;. smoothing length h;. position r;. velocity v;. ancl values for any other local intensive thermodynamic variables required to describe the state of the gas. for exaniple density. ρε. pressure £7. specific internal encrey η. magnetic field D;. etc. | Each particle has associated with it a mass $m_i$, smoothing length $h_i$, position ${\bf r}_i$, velocity ${\bf v}_i$, and values for any other local intensive thermodynamic variables required to describe the state of the gas, for example density $\rho_i$, pressure $P_i$ , specific internal energy $u_i$, magnetic field ${\bf B}_i$, etc. |
Ato an arbitrary position r. the value of any physical variable ct can be obtained by interpolating over the values of ch associated with nearby particles. using a smoothing kernel Wo. viz. | At an arbitrary position ${\bf r}$ , the value of any physical variable $A$ can be obtained by interpolating over the values of $A$ associated with nearby particles, using a smoothing kernel $W$, viz. |
The kernel is normalized. and has compact support (specifically VW(s2)=0). so that the sum is over a finite number Aq, of neighbouring particles. | The kernel is normalized, and has compact support (specifically $W(s>2) = 0$ ), so that the sum is over a finite number ${\cal N}\nei$ of neighbouring particles. |
The gradient of “A can then be evaluated [rom where M(3)=dMdas. | The gradient of $A$ can then be evaluated from where $W'(s) \equiv dW/ds$. |
The motions of the particles are driven by interparticle forces representing the local pressure gradient. viscosity. eravitv anc magnetic field. | The motions of the particles are driven by interparticle forces representing the local pressure gradient, viscosity, gravity and magnetic field. |
Lnterparticle [οσο are formulated. svnimetrically so that lincar anc angular momentum are conserved. | Interparticle forces are formulated symmetrically so that linear and angular momentum are conserved. |
In our basic implementation we use the MA. kernel introduced. by Monaghan Lattanzio (1985). | In our basic implementation we use the M4 kernel introduced by Monaghan Lattanzio (1985). |
Phe particles all have the same mass. | The particles all have the same mass. |
We omit magnetic fields. | We omit magnetic fields. |
The smoothing length Ph; is varied. so that the kernel contains JAoj,50(35) particles. | The smoothing length $h_i$ is varied so that the kernel contains ${\cal N}\nei \sim 50 \; (\pm 5)$ particles. |
This ensures that the spatial resolution becomes finer with increasing density. | This ensures that the spatial resolution becomes finer with increasing density. |
The density at the position of particle 7 is given hy where The sum in Eqn. (4)) | The density at the position of particle $i$ is given by where The sum in Eqn. \ref{SPHDENSITY}) ) |
is over the ~Nus, neighbours within 2h;;. and it includes particle # itself. | is over the $\sim {\cal N}\nei$ neighbours within $2 \bar{h}_{ij}$, and it includes particle $i$ itself. |
In many applications. the pressure at particle ὁ is given by a barotropic equation of state £7=P(p;). | In many applications, the pressure at particle $i$ is given by a barotropic equation of state $P_i = P(\rho_i)$. |
In this case there is no need to solve an energy equation. | In this case there is no need to solve an energy equation. |
The gravitational acceleration.of aparticle can be obtained by a direct sum of the contributions from all the other particles. | The gravitational accelerationof aparticle can be obtained by a direct sum of the contributions from all the other particles, |
We chose to investigate the 6.6 hour orbital period CV. SS Cyveni. | We chose to investigate the 6.6 hour orbital period CV, SS Cygni. |
Since the secondary star in SS Cve is an early Ix star (Harrisonetal.(2000) ancl references therein). the mere detection of M-tvpe TiO features would be a positive indication of spot activity on ils surface. | Since the secondary star in SS Cyg is an early K star \citet{h.etal} and references therein), the mere detection of M-type TiO features would be a positive indication of spot activity on its surface. |
Four SS (νο spectra totaling 24 mins exposure (ime were taken within 0.3 in phase of superior conjunction of the white dwarf (i.e. when (he secondary star is al ils closest to the Earth). using the Intermediate Dispersion Spectrograph on the 2.5m Isaac Newton Telescope. | Four SS Cyg spectra totaling 24 mins exposure time were taken within 0.3 in phase of superior conjunction of the white dwarf (i.e. when the secondary star is at its closest to the Earth), using the Intermediate Dispersion Spectrograph on the 2.5m Isaac Newton Telescope. |
With the 1.6 aresec slit used. the spectrograph vielded a resolution of2. | With the 1.6 arcsec slit used, the spectrograph yielded a resolution of. |
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