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The plots have been generated by randomly sampling (in time) the accreting transition phase of each disc model several times, and should therefore provide a reasonable estimate of both the general form of the correlation plus the associated scatter.
The plots have been generated by randomly sampling (in time) the accreting transition phase of each disc model several times, and should therefore provide a reasonable estimate of both the general form of the correlation plus the associated scatter.
Clearly, since discs with higher X-ray luminosities open gaps earlier and at higher accretion rates a strong positive correlation between
Clearly, since discs with higher X-ray luminosities open gaps earlier and at higher accretion rates a strong positive correlation between
population. with the Se being widely dominant in the 100 bband.
population, with the Sc being widely dominant in the 100 band.
Despite their high luminosities. we predict that starburst galaxies do not provide a significant contribution to the IR emission. due to their low expected number.
Despite their high luminosities, we predict that starburst galaxies do not provide a significant contribution to the IR emission, due to their low expected number.
We obtain a contribution of 15.4 Jy at 60 aand 12.1 Jy at 100 corresponding respectively to about and of total predicted signal.
We obtain a contribution of 15.4 Jy at 60 and 12.1 Jy at 100 corresponding respectively to about and of total predicted signal.
Only in our extreme scenario. their contribution becomes non-negligible in the 60 bband. reaching the of the total predicted flux.
Only in our extreme scenario their contribution becomes non-negligible in the 60 band, reaching the of the total predicted flux.
This low contribution agrees with the low rate of starburst galaxies as found in the field by LeFloc'hetal.(2005). at the redshift range of the SDSS-maxBCG catalogue (1.6. 0.]<zΒ« 0.3).
This low contribution agrees with the low rate of starburst galaxies as found in the field by \citet{lefloch05} at the redshift range of the SDSS-maxBCG catalogue (i.e. $0.1<z<0.3$ ).
The total fluxes associated to the galactic emission predicted by our reference model are 684.5|546.5.895.8] Jy at 60 and 1904.8|1475.7.2521.9] Jy at 100 ((the bracketed interval indicate the values derived from our conservative and extreme models. see Sect. 3.4)).
The total fluxes associated to the galactic emission predicted by our reference model are $684.5 \ [546.5,895.8]$ Jy at 60 and $1904.8 \ [1475.7,2521.9]$ Jy at 100 (the bracketed interval indicate the values derived from our conservative and extreme models, see Sect. \ref{ssec:unc}) ).
It appears that the reconstructed IR emission due to the galactic dust emission of the cluster members can explain the entire signal measured by Giardetal.(2008)... with an indication that our reference model overestimates the total flux. particularly. at 100um.
It appears that the reconstructed IR emission due to the galactic dust emission of the cluster members can explain the entire signal measured by \cite{giard08}, with an indication that our reference model overestimates the total flux, particularly at 100.
. Indeed. these authors obtained 570.1+36.1 Jy and 1359.9+249.1 Jy at 60 aand 100ΓΌum.. respectively.
Indeed, these authors obtained $570.1\pm36.1$ Jy and $1359.9\pm249.1$ Jy at 60 and 100, respectively.
We will propose an explanation of this discrepancy in the next sections.
We will propose an explanation of this discrepancy in the next sections.
However. when considering our conservative scenario. the predicted emission is in good agreement with the total measured signal in. both bands.
However, when considering our conservative scenario, the predicted emission is in good agreement with the total measured signal in both bands.
Given these results. modulo the uncertainties. of our modelisation. we obtain that the IR emission of the galaxy members is consistent with the total observed emission of our clusters sample. leaving little space to the possible presence of other components like intracluster dust.
Given these results, modulo the uncertainties of our modelisation, we obtain that the IR emission of the galaxy members is consistent with the total observed emission of our clusters sample, leaving little space to the possible presence of other components like intracluster dust.
inviscid.
inviscid.
In this work we examine low Mach. number Lows which. taken in concert with the isothermal assumption. means that features in the Low such as shocks are unlikely to create regions of high ionisation.
In this work we examine low Mach number flows which, taken in concert with the isothermal assumption, means that features in the flow such as shocks are unlikely to create regions of high ionisation.
The HYDILA code solves the following equations for a system of NV Ξ Ξ±Β».
The HYDRA code solves the following equations for a system of $N$ fluids.
Phe simulations described in this paper consist of three Duids. indexed. by 7=0 for the neutral Iluid and ;=1 and ;=2 for the electron. anc ion fluids respectively.
The simulations described in this paper consist of three fluids, indexed by $i=0$ for the neutral fluid and $i=1$ and $i=2$ for the electron and ion fluids respectively.
The equations to be solved are where p;. ΞΏ. B. and J are the mass. densities. velocities. magnetic field ancl current. density. respectively.
The equations to be solved are where $\rho_i$, $\mathbf{q}_i$, $\mathbf{B}$, and $\mathbf{J}$ are the mass densities, velocities, magnetic field and current density, respectively.
e denotes the sound speed. anc a; and. Ajo) are the charge-to-mass ratios and the collision coellicients. between the charged species and the neutral Duid. respectively.
$a$ denotes the sound speed, and $\alpha_i$ and $K_{i0}$ are the charge-to-mass ratios and the collision coefficients between the charged species and the neutral fluid, respectively.
These equations lead to an expression for the electric ficld in the frame of the Εις. E/. given by the generalised Ohm's Lawwhere the components of the field are given by using the definitions ay=foB. ay=fuB. ay=f/,B. where foVrofD. fuΓΗ and fyβ€”
These equations lead to an expression for the electric field in the frame of the fluid, $\mathbf{E}^\prime$, given by the generalised Ohm's Lawwhere the components of the field are given by using the definitions $\mathbf{a}_{\rm O} \equiv f_{\rm O}\mathbf{B}$, $\mathbf{a}_{\rm H} \equiv f_{\rm H}\mathbf{B}$, $\mathbf{a}_{\rm A} \equiv f_{\rm A}\mathbf{B}$, where $f_{\rm O} \equiv \sqrt{r_{\rm O}}/B$, $f_{\rm H} \equiv r_{\rm H}/B$ and $f_{\rm A} \equiv \sqrt{r_{\rm A}}/B$.
The pesistivities given here are the Ohmic. Hall ancl VrxX/D.ambipolar resistivities. respectively. and are defined by where the conductivities are given hy where the Hall parameter 3; for a charged. species is eiven by To solve these equations numerically we use three diferent operators: These operators are applied. using Strangoperator splitting in order to maintain the second order accuracy of the overall scheme.
The resistivities given here are the Ohmic, Hall and ambipolar resistivities, respectively, and are defined by where the conductivities are given by where the Hall parameter $\beta_i$ for a charged species is given by To solve these equations numerically we use three different operators: These operators are applied using Strangoperator splitting in order to maintain the second order accuracy of the overall scheme.
We refer the reader to O'SullivanDownes(2006.2007) for Γ  more detailed description.
We refer the reader to \cite{osd06, osd07} for a more detailed description.
The simulations are carried out on a DD slab grid in the Àày-plane.
The simulations are carried out on a D slab grid in the $xy$ -plane.
The grid consists of 6400.Β«2001 cells. in the .r. jy. and 2 directions respectively.
The grid consists of $6400 \times 200 \times 1$ cells, in the $x$, $y$, and $z$ directions respectively.
This resolution was chosen on the basis that it reproduces the initial linear. growth. of the ideal MED system in Ixeppensetal.(1999).
This resolution was chosen on the basis that it reproduces the initial linear growth of the ideal MHD system in \cite{keppens99}.
.. Resolution stucies were performed to confirm the resolution as being appropriate (see and 4.3.1)).
Resolution studies were performed to confirm the resolution as being appropriate (see \\ref{sec:ambi-res-study} and \ref{sec:hall-res-study}) ).
The initial set-up used was that of two plasmas Ilowing anti-parallel side-by-side on a grid of size ΞΌΞΉ0.32Β£] and y=0.1).
The initial set-up used was that of two plasmas flowing anti-parallel side-by-side on a grid of size $x = [0, 32L]$ and $y = [0, L]$.
The plasma velocities are given by. |i and m in the g-direction. with a tangential shear laver of width 2Β« at the interface at c=16L.
The plasma velocities are given by $+ \frac{V_0}{2}$ and $-\frac{V_0}{2}$ in the $y$ -direction, with a tangential shear layer of width $2a$ at the interface at $x = 16L$.
This velocity profile is described by The width of the shear layer is chosen to be.=0.05. or approximately 20 erid zones.
This velocity profile is described by The width of the shear layer is chosen to be $\frac{a}{L} = 0.05$, or approximately 20 grid zones.
Ehe magnetic field is initially set to be uniform and aligned with the plasma flow.
The magnetic field is initially set to be uniform and aligned with the plasma flow.
The initial background for all three IEuids in the svsteni is now an exact equilibrium.
The initial background for all three fluids in the system is now an exact equilibrium.
Phe initial neutral velocity field. Vo is then augmented with a perturbation given by where 8l; is set to 10LY.
The initial neutral velocity field, $V_0$ is then augmented with a perturbation given by where $\delta V_0$ is set to $10^{-4}\,V_0$.
The wavelength of the perturbation is set equal to the characteristic length scale. A=5= L. so that a single wavelength fits exactly into the computational domain.
The wavelength of the perturbation is set equal to the characteristic length scale, $\lambda = \frac{2 \pi}{k_y} = L$ , so that a single wavelength fits exactly into the computational domain.
This maximises the possibility of resolving structures that are small relative to. the initial perturbecl wavelength. (Frankctal. 1996)...
This maximises the possibility of resolving structures that are small relative to the initial perturbed wavelength, \citep{frank96}. .
β€œLhe
The
was the first millisceone pulsar (AISP1)) discovered in the ongoing Arecibo L-band Feed Array (ALFA) pulsar survey 7..
was the first millisecond pulsar ) discovered in the ongoing Arecibo L-band Feed Array (ALFA) pulsar survey \cite{cfl+06}.
In the discovery. paper ὃν, we presented the results of shase-coherent radio timing of this pulsar carried out with he Green Bank and Arecibo radio telescopes.
In the discovery paper \cite{crl+08}, we presented the results of phase-coherent radio timing of this pulsar carried out with the Green Bank and Arecibo radio telescopes.
These quickly revealed that the pulsar was in a 05-day orbit around. a 1 solar mass CM.) companion.
These quickly revealed that the pulsar was in a 95-day orbit around a 1 solar mass $M_{\odot}$ ) companion.
This object is remarkable for ing the first (and thus Far. the only) disk. MSP known o have an eccentric (e= 0.44) orbit.
This object is remarkable for being the first (and thus far, the only) disk MSP known to have an eccentric $e = 0.44$ ) orbit.
In globular. clusters (GCs) there are several binary AISPs with eccentric orbits: rut those are thought to be caused by perturbations of the unarv svstemis by occasional close interactions with other stars.
In globular clusters (GCs) there are several binary MSPs with eccentric orbits; but those are thought to be caused by perturbations of the binary systems by occasional close interactions with other stars.
Coincident with the pulsar position derived. from the iming. a star was found. whose near-infrared) magnitudes were consistent with a 14. main-sequence star at the distance and reddening estimated for PSR. 1903]0327.
Coincident with the pulsar position derived from the timing, a star was found whose near-infrared magnitudes were consistent with a $1\,M_{\odot}$ main-sequence star at the distance and reddening estimated for PSR J1903+0327.
1 was not known then whether this was just an unlikely. (~ 2.6%)) chance alignment or whether the star is genuinely associated withis and if so whether i is the binary companion responsible for the 95-day orbi of the pulsar.
It was not known then whether this was just an unlikely $\sim$ ) chance alignment or whether the star is genuinely associated with, and if so whether it is the binary companion responsible for the 95-day orbit of the pulsar.
Such a finding would be surprising. as the conventional understanding of AISP evolution posits tha such a neutron star (NS) is spun up to high spin frequencies bv aceretion of matter and angular momentum from a Companion star while the companion passes through a gian phase ?:: this circularises the system. and a recycled. MSP is left orbiting a low-mass white dwarl (the remnant core of the donor) in a low-ceecentricity orbit (ΞΏ<10.7: Phinney 992)).
Such a finding would be surprising, as the conventional understanding of MSP evolution posits that such a neutron star (NS) is spun up to high spin frequencies by accretion of matter and angular momentum from a companion star while the companion passes through a giant phase \cite{bv91}; this circularises the system, and a recycled MSP is left orbiting a low-mass white dwarf (the remnant core of the donor) in a low-eccentricity orbit $e < 10^{-3}$; Phinney \nocite{phi92}) ).
Until the discovery of all known AISPs in the Galactic disk hac such low-eccentricitvy orbits.
Until the discovery of all known MSPs in the Galactic disk had such low-eccentricity orbits.
Lor Leviews. see Phinney Wulkarni (1904)... Stairs. (2004)... Tauris van cen Lleuvel (2006).
For reviews, see Phinney Kulkarni \nocite{pk94}, Stairs \nocite{sta04}, Tauris van den Heuvel \nocite{th06}.
. For these reasons. Champion ct al.
For these reasons, Champion et al.
proposed hat βŠ³βŠ”β†“β‹œβ†§βˆ™βˆ–β‡βˆ£β‹ˆβ‹…β†“β‰»β‹œβŠ”β‹…β†₯βˆͺβ‡‚β‹…β‹œβ†§β‡‚β†“β‹…βŠ²β†“β†“β‰»β†“βˆ’βŠΎβŠ³βˆ–βˆ™βˆ–β‡βŠ³βˆ–βˆ©βŠΎβŠ”β†“βˆ–βˆ–βŠ½β‡‚β†₯βˆ’β‹…β†“β‹…β‹–β‹…β†₯⇂β†₯βˆ’β‹…β‰€β‹—β‹…βˆβˆ’βˆ β‡‚β‹œβ†§βˆ™βˆ–β‡β‹–β‹Ÿβ†“β‹…βˆ£β‹‘β†•β†Ώβˆͺβ‡‚βˆŽ he pulsar is caused by a massive unseen WD and the third member is the star detected in the near-infrared.
\nocite{crl+08} proposed that may be part of a triple system where the 95-day orbit of the pulsar is caused by a massive unseen WD and the third member is the star detected in the near-infrared.
The latter is in Γ  long-period orbit and. drives the eccentricity of the inner pair through the Ixozai mechanism ?..
The latter is in a long-period orbit and drives the eccentricity of the inner pair through the Kozai mechanism \cite{koz62}.
An alternative possibility. also discussed in ? is that the companion to .in the 95-dav orbit is the star detected in the near-infrared. but that this eccentric. unusual svstem originated in an exchange interaction in a dense stellar environment. like a elobular cluster.
An alternative possibility, also discussed in \cite{crl+08} is that the companion to in the 95-day orbit is the star detected in the near-infrared, but that this eccentric, unusual system originated in an exchange interaction in a dense stellar environment, like a globular cluster.
In this paper. we present new optical measuremoents and further radio timing of obtained with the aim of testing these scenarios.
In this paper, we present new optical measurements and further radio timing of obtained with the aim of testing these scenarios.
The plan for the rest of this paper is as follows.
The plan for the rest of this paper is as follows.
The optical and racio observations are described 2..
The optical and radio observations are described \ref{sec:observations}.
Phe immediate results from. these observations are described in 3..
The immediate results from these observations are described in \ref{sec:results}.
In +4 we discuss the implications of these results regarding the formation and evolution of this svstem.
In \ref{sec:discussion} we discuss the implications of these results regarding the formation and evolution of this system.
In 5 we discuss how this svstenm might have formed.
In \ref{sec:formation} we discuss how this system might have formed.
We sumnmarise our main conclusions in ETT.
We summarise our main conclusions in \ref{sec:conclusions}.
Long-slit spectroscopy of the suspected. counterpart to was obtained. with FORS2 7.. the low dispersion spectrograph of ESO's Very. Large Telescope.
Long-slit spectroscopy of the suspected counterpart to was obtained with FORS2 \cite{aff+98}, the low dispersion spectrograph of ESO's Very Large Telescope.
Four spectra were obtained on 2008 June 21. three on 2008 August 23 and one a day Later. on August 24.
Four spectra were obtained on 2008 June 21, three on 2008 August 23 and one a day later, on August 24.
All spectra had exposure times of mminutes. and used a 17 slit combined. with the 10254 holographic grism. providing wavelength coverage over ttoAA.
All spectra had exposure times of minutes, and used a $1\asec$ slit combined with the 1028Z holographic grism, providing wavelength coverage over to.
.. The detectors were read out with 2.2 binning. vielding a resolution ofAA.. sampled at 0.86 ppix+.
The detectors were read out with $2\times2$ binning, yielding a resolution of, sampled at $0.86$ $^{-1}$.
The slit was placed such that both the pulsar companion and a bright nearby star were centred on the slit.
The slit was placed such that both the pulsar companion and a bright nearby star were centred on the slit.
The observations were taken during clear and photometric nights. with the seeing between 0748 and 0772.
The observations were taken during clear and photometric nights, with the seeing between $0\farcs48$ and $0\farcs72$.
Phe spectral observations were corrected. for bias and. Uat-Llicldecl using lamp fats.
The spectral observations were corrected for bias and flat-fielded using lamp flats.
Spectral extraction is complicated by the bright star. henceforth star A. located 273 from the pulsar counterpart (sce Lig. 2.1).
Spectral extraction is complicated by the bright star, henceforth star A, located $2\farcs3$ from the pulsar counterpart (see Fig. \ref{fig:finder}) ).
The star is brighter by about manag in the {ρα and as a result. about of the detected counts at the spatial position of the pulsar counterpart are
The star is brighter by about mag in the $I$ -band and as a result about of the detected counts at the spatial position of the pulsar counterpart are
and the mass fraction i4.=0.3.
and the mass fraction $m_1 = 0.3$.
For the smaller source radii (p,=0.05 and p,= 0.1. the source transits the caustic and moves completely inside. so that separated fold-caustic passages produce distinct characteristic peaks that can be described by a generic profile function (e.g.?)..
For the smaller source radii $\rho_\star = 0.05$ and $\rho_\star = 0.1$ ), the source transits the caustic and moves completely inside, so that separated fold-caustic passages produce distinct characteristic peaks that can be described by a generic profile function \citep*[e.g.][]{Do:fold}.
In these cases. he insertion of a seed image mapping to a local minimum of he distance of the source centre from the caustic was essential.
In these cases, the insertion of a seed image mapping to a local minimum of the distance of the source centre from the caustic was essential.
In contrast. the source never moves completely inside the caustic for p,=0.2 or f=0.5.
In contrast, the source never moves completely inside the caustic for $\rho_\star = 0.2$ or $\rho_\star = 0.5$.
For p,=0.2. there are epochs for which the source crosses the same fold line (as for the smaller racii) or different adjoining fold lines. with or without the cusp in between.
For $\rho_\star = 0.2$, there are epochs for which the source crosses the same fold line (as for the smaller radii) or different adjoining fold lines, with or without the cusp in between.
Moreover. or p,=0.5 there are epochs for which a various number of up to hree cusps are enclosed by the source.
Moreover, for $\rho_\star = 0.5$ there are epochs for which a various number of up to three cusps are enclosed by the source.
For p,=05. a light curve for a brightness profile corresponding to maximal limb darkening. is shown along with that for a uniformly bright source.
For $\rho_\star = 0.5$, a light curve for a brightness profile corresponding to maximal limb darkening, is shown along with that for a uniformly bright source.
With a smaller fraction of the total brightness in the outer parts for the imb-darkened source. the source magnification shows a smaller rise as the source enters or exits the caustic but is larger in between.
With a smaller fraction of the total brightness in the outer parts for the limb-darkened source, the source magnification shows a smaller rise as the source enters or exits the caustic but is larger in between.
The application of Green's theorem to replace the integration over the image area by an integration along its boundary is a very efficient approach if the images are moderately distorted. so that a large area for the given boundary length is enclosed (a circle is optimal).
The application of Green's theorem to replace the integration over the image area by an integration along its boundary is a very efficient approach if the images are moderately distorted, so that a large area for the given boundary length is enclosed (a circle is optimal).
There is signiticantly less gain. however. for extremely strong distortions leading to the enclosed area resembling a line.
There is significantly less gain, however, for extremely strong distortions leading to the enclosed area resembling a line.
This case is indeed approached if a source star gets very closely aligned with a lens star that is only associated with much less massive companions such as orbiting planets.
This case is indeed approached if a source star gets very closely aligned with a lens star that is only associated with much less massive companions such as orbiting planets.
Resulting in large oak magnifications. such configurations. are of some specific interest due to their planet-detection potential (22)..
Resulting in large peak magnifications, such configurations are of some specific interest due to their planet-detection potential \citep{GS:HME,Ratt:high}.
For modelling he event with the largest peak magnification recorded so far. OGLE 2004-BLG-343 with 244~3000. 2. have derived a more efficient variant of the ray-shooting technique.
For modelling the event with the largest peak magnification recorded so far, OGLE 2004-BLG-343 with $A_0 \sim 3000$, \citet{Dong:ray} have derived a more efficient variant of the ray-shooting technique.
In fact. the current version of he adaptive contouring algorithm is significantly slowed down for very small impact angles between source and planet-surrounded ens star. but. for uniformly bright sources. the computation of a single magnification with a relative uncertainty of 5107 can still be carriedout in ~200 ms on a 600 MFlops machine for magnifications in the range ef~ 100β€”1000. while a result is
In fact, the current version of the adaptive contouring algorithm is significantly slowed down for very small impact angles between source and planet-surrounded lens star, but, for uniformly bright sources, the computation of a single magnification with a relative uncertainty of $5\times 10^{-4}$ can still be carriedout in $\sim 200$ ms on a 600 MFlops machine for magnifications in the range $A \sim 100$ $1000$ , while a result is
in a given virtual detector is given by summing over all those trajectories whose velocity vector lies within the acceptance cone of that detector.
in a given virtual detector is given by summing over all those trajectories whose velocity vector lies within the acceptance cone of that detector.
Formally, ,t) ,t) ,PP(Q,,c,,t) = ,t) ,t) and the integrations over x and p reduce in the PIC case to summations over all trajectories that illuminate the specified detector.
Formally, ,t) ,t) ,t) = ,t) ,t) and the integrations over $\bm{x}$ and $p$ reduce in the PIC case to summations over all trajectories that illuminate the specified detector.
For the high-energy emission of particles accelerated at a relativistic shock front, the restriction imposed by this procedure is not important, because the anisotropy of the particle distribution is expected to be on a scale larger than the beaming angle.
For the high-energy emission of particles accelerated at a relativistic shock front, the restriction imposed by this procedure is not important, because the anisotropy of the particle distribution is expected to be on a scale larger than the beaming angle.
Thus, the angular dependence of the emitted radiation found by Sironi&Spitkovsky and Frederiksenetal. should just reflect the(2009b) angular dependence of the(2010) distribution function at the relevant particle energy, and would be preserved in this approach.
Thus, the angular dependence of the emitted radiation found by \citet{sironispitkovsky09b} and \citet{frederiksenetal10} should just reflect the angular dependence of the distribution function at the relevant particle energy, and would be preserved in this approach.
As pointed out by Hededal(2005) the computation of synthetic spectra from trajectories taken from PIC simulations inevitably involves interpolation.
As pointed out by \citet{hededalphd} the computation of synthetic spectra from trajectories taken from PIC simulations inevitably involves interpolation.
Specifically, the algorithm presented in (18))-(26)) transforms the integration variable from time to phase.
Specifically, the algorithm presented in \ref{ptotal}) \ref{p2}) ) transforms the integration variable from time to phase.
In order to split the contributions to the integral into an alternating series (59)), the discrete trajectory must be interpolated.
In order to split the contributions to the integral into an alternating series \ref{instpower2}) ), the discrete trajectory must be interpolated.
Interpolation is not a sensitive procedure provided many points are contained within a photon formation time, a constraint that will be made more precise below.
Interpolation is not a sensitive procedure provided many points are contained within a photon formation time, a constraint that will be made more precise below.
An accurate evaluation of the instantaneous power at any time step can, for example, be obtained simply by linearly interpolating the functions g, y, 6, g, 6B and 6A, which are known at all neighboring grid points.
An accurate evaluation of the instantaneous power at any time step can, for example, be obtained simply by linearly interpolating the functions $g$, $\gamma$, $\beta$, $\dot{g}$, $\delta\bm{\beta}$ and $\delta\Delta$, which are known at all neighboring grid points.
When the photon formation length drops to only a few time steps, this procedure fails.
When the photon formation length drops to only a few time steps, this procedure fails.
However, the validity of the PIC simulation requires that the electromagnetic fields vary slowly between time steps, which is precisely the condition for applicability of the generalized synchrotron formula (56)).
However, the validity of the PIC simulation requires that the electromagnetic fields vary slowly between time steps, which is precisely the condition for applicability of the generalized synchrotron formula \ref{instsyncheq}) ).
Therefore, in a valid simulation, the instantaneous power can safely be evaluated using this method, if the formation time is not long compared to the time step.
Therefore, in a valid simulation, the instantaneous power can safely be evaluated using this method, if the formation time is not long compared to the time step.
It follows that, for a given frequency, the method of evaluating the instantaneous power at each of the discrete set of particle positions a(t,), depends on the value of the photon formation time at that point.
It follows that, for a given frequency, the method of evaluating the instantaneous power at each of the discrete set of particle positions $\bm{x}(t_n)$, depends on the value of the photon formation time at that point.
At high frequencies, the formation time is short, and can be much shorter than the typical time-step used in PIC simulations, which is a fraction of a plasma cycle.
At high frequencies, the formation time is short, and can be much shorter than the typical time-step used in PIC simulations, which is a fraction of a plasma cycle.
It is straightforward to find for each time-step (labeled by n) the values JA of the deviation of the displacement at the neighboring points n+ 1.
It is straightforward to find for each time-step (labeled by $n$ ) the values $\delta\Delta^\pm_n$ of the deviation of the displacement at the neighboring points $n\pm1$ .
For a given frequency, the photon formation lengths in the forward and backward directions follow.
For a given frequency, the photon formation lengths in the forward and backward directions follow.
Alternatively, two critical frequencies wzTL can be found such that atthese frequencies the neighboring points lie precisely one
Alternatively, two critical frequencies $\omega^\pm_n$ can be found such that atthese frequencies the neighboring points lie precisely one
Under the influence of gravity. disk galaxies are expected o assemble in an β€œinside-out” fashion: stars form first from uigh-density gas in the central region of the galaxy where the yotential is deepest. and subsequently at increasing galacto- radii (e.g.Larson1976).
Under the influence of gravity, disk galaxies are expected to assemble in an β€œinside-out” fashion: stars form first from high-density gas in the central region of the galaxy where the potential is deepest, and subsequently at increasing galacto-centric radii \citep[\eg][]{Larson76}.
. An immediate consequence of his formation scenario is that stars born at the same time and in the same region of a galaxy should have similar chemical compositions.
An immediate consequence of this formation scenario is that stars born at the same time and in the same region of a galaxy should have similar chemical compositions.
However. observations in our Galaxy suggest that hese initial conditions are not maintained.
However, observations in our Galaxy suggest that these initial conditions are not maintained.
Wielenetal.(1996) argued that the Sun was substantially more metal rich than nearby solar age stars and the local interstellar medium (ISM).
\citet{Wielen96} argued that the Sun was substantially more metal rich than nearby solar age stars and the local interstellar medium (ISM).