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Ro Neri and M. Crewing are gratefully acknowledged for their support of this program. and D. Downes for useful discussions. | R. Neri and M. Grewing are gratefully acknowledged for their support of this program, and D. Downes for useful discussions. |
We also thank R. Lucas for help with the cata reduction ancl the referee. AS. Evans. for comments which improved the contents of this paper. | We also thank R. Lucas for help with the data reduction and the referee, A.S. Evans, for comments which improved the contents of this paper. |
The LUAM. Plateau de Bure stall is kindly acknowledged for its cllicient assistance. | The IRAM Plateau de Bure staff is kindly acknowledged for its efficient assistance. |
HAM. is supported. by INSU/CNIUS (France). MPG (Germany). and IGN (Spain). | IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). |
The JCMT is operated by JAC. Hilo. on behalf of the parent organisations of the Particle Physics anc Astronomy Research Council in the Ulx. the National Research Council in Canada. and the Scientific Research Organisation of the Netherlands. | The JCMT is operated by JAC, Hilo, on behalf of the parent organisations of the Particle Physics and Astronomy Research Council in the UK, the National Research Council in Canada and the Scientific Research Organisation of the Netherlands. |
The National Radio Astronomy Observatory (NIUXO) is a facility of the National Science Foundation. operated under cooperative agreement by Associated Universities. Ine. SCD and SAIC acknowledge a partial support from the Bressler Foundation. | The National Radio Astronomy Observatory (NRAO) is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. SGD and SMC acknowledge a partial support from the Bressler Foundation. |
the ceutral stars in tle svstenis. based ou the carlyAepler data. | the central stars in the systems, based on the early data. |
We have analyzed data fronNepler for three. plauct-hosting stars usine a upeliue. developed. for fast aud robust analysis of a|Kepler pauodoe data (Cliistcnsen-Dalsgamrdetal.2008:Ihiber 2009). | We have analyzed data from for three planet-hosting stars using a pipeline developed for fast and robust analysis of all p-mode data \citep{CDetal2008, Huber2009}. |
.. Each time series contaius 6332 daa points. | Each time series contains 63324 data points. |
SC data characteristics and minor post-pipeline1 processing are discussed in Cullilandetal.(2010c). | SC data characteristics and minor post-pipeline processing are discussed in \citet{Gillil2010c}. |
. Tn addition a limib-darkened transit light curve model fit has been removed and σ clipping applied to remove outlvine data poiuts from each of the time series. | In addition a limb-darkened transit light curve model fit has been removed and $\sigma$ clipping applied to remove outlying data points from each of the time series. |
The frequency. analysis coutaius four main steps: For observations with lowv sigual-to-noise ratio it may not be possible to ideutiv the individual frequencies. | The frequency analysis contains four main steps: For observations with low signal-to-noise ratio it may not be possible to identify the individual frequencies. |
Notice that the spectrum we obtain lor STFA is different. from the power law result of Jones(1994). | Notice that the spectrum we obtain for STFA is different from the power law result of \citet{jones1994}. |
. This is a matter of regime: we discussed in the text the acceleration of non-relativistic particles in a region of non-relativistic turbulence. here we show that when the particles are relativistic. a power law spectrum emerges. | This is a matter of regime; we discussed in the text the acceleration of non-relativistic particles in a region of non-relativistic turbulence, here we show that when the particles are relativistic, a power law spectrum emerges. |
Recall that (5)) for fully relativistie electrons in a region of non-relativistic turbulence is given by (G)) where £ is the (total energy. kinetic plus rest. of the electron belore reflection. | Recall that \ref{Epm}) ) for fully relativistic electrons in a region of non-relativistic turbulence is given by \ref{Estep}) ) where $E$ is the total energy, kinetic plus rest, of the electron before reflection. |
Notice that if we the low velocity limit. e«&c where E=mc. the expression reduces to (5)). | Notice that if we the low velocity limit, $v \ll c$ where $E = mc^2$, the expression reduces to \ref{Epm}) ). |
The relative velocity between the compression ancl electron lor head-on and catch-up tvpe interactions are still given by (8)) so the steady acceleration rate is given bv Alternatively. we can find the mean acceleration per rellection by multiplving equation dS bv RO! In the highly relativistie limit. / is just the kinetic energy. and we recover the familiar result (Jones1994) that dE/dlxE. | The relative velocity between the compression and electron for head-on and catch-up type interactions are still given by \ref{erates}) ) so the steady acceleration rate is given by Alternatively, we can find the mean acceleration per reflection by multiplying equation \ref{rateHR} by $R^{-1}$ In the highly relativistic limit, $E$ is just the kinetic energy, and we recover the familiar result \citep{jones1994}
that $dE/dt \propto E$. |
This proportionality is expected to produce a power law. | This proportionality is expected to produce a power law. |
We derive the power law spectrum lor STFA of highly relativistic electrons by following the approach of Bell(1978) and assume (hat δεις is a constant in flare plasmas. independent of electron energv. | We derive the power law spectrum for STFA of highly relativistic electrons by following the approach of \citet{Bell} and assume that $p_{esc}$ is a constant in flare plasmas, independent of electron energy. |
We start bv integrating d£/dl to obtain E() The probability of an electron remaining in (he acceleration region for at least / rellections is given by | We start by integrating $dE/dl$ to obtain $E(l)$ The probability of an electron remaining in the acceleration region for at least $l$ reflections is given by |
to the gravitational energy of the initial density. (1)): The change of the magnetic energy. 011,, depends on the shape of the boundary. ο aud on the deformation fiekl tuside the magnetized regiou. | to the gravitational energy of the initial density \ref{dens}) ): The change of the magnetic energy $\delta W_m$ depends on the shape of the boundary $\psi$ and on the deformation field inside the magnetized region. |
But since the frozen-in deformations conserve the magnetic helicity. a lower bound ou the change of the magnetic energy can be obtained by imininiziug the magnetic euerey lor a eiven boundary (1)) aid. helicity. | But since the frozen-in deformations conserve the magnetic helicity, a lower bound on the change of the magnetic energy can be obtained by minimizing the magnetic energy for a given boundary \ref{boundary}) ) and helicity. |
The boundary coucditiou for this minimization is still the vanishing of the normal component of the maguetic Held. because this property is preserved by the frozen-in deformation of the Geld. | The boundary condition for this minimization is still the vanishing of the normal component of the magnetic field, because this property is preserved by the frozen-in deformation of the field. |
Thus we have where Hj, is the magnetic energy of the initial field (2)). and «wy[e] is the dimensionless change of the minimal magnetic energy [or a given helicity due to the variation of the boundary ο. | Thus we have where $W_m$ is the magnetic energy of the initial field \ref{mfield}) ), and $w_m[\psi ]$ is the dimensionless change of the minimal magnetic energy for a given helicity due to the variation of the boundary $\psi$ . |
The dimensionless chaugeoO of the oOgravitational energyo προ)4 is positive. and as we show in Appendix L up to the second order iu i. Here aud below C' denotes positive dimieusiouless coustauts. | The dimensionless change of the gravitational energy $w_g[\psi]$ is positive, and as we show in Appendix I, up to the second order in $\psi$, Here and below $C$ denotes positive dimensionless constants. |
Bt is assumed that the deformation v does not contain a dipole mode (trauslation). for this mode clearly does not contribute to the iuagnetic energy. aud will be stabilized by the non-zero outer deusity. p». | It is assumed that the deformation $\psi$ does not contain a dipole mode (translation), for this mode clearly does not contribute to the magnetic energy, and will be stabilized by the non-zero outer density $\rho _2$. |
The dimensionless change of the magnetic energy wy[v] can be negative. but as we show in Appendix IL. also up to the second order in e. The total enerevOe of the minimunrenergyOe configurationOm is sinaller than the initial energv.Oe It follows that the ninunum-euergy coufiguratiou is close to the spliere: As shown in Appendix IL the minimum-enereyOe coufigurationOm is axisviunetric. | The dimensionless change of the magnetic energy $w_m[\psi ]$ can be negative, but as we show in Appendix II, also up to the second order in $\psi$, The total energy of the minimum-energy configuration is smaller than the initial energy, It follows that the minimum-energy configuration is close to the sphere: As shown in Appendix II, the minimum-energy configuration is axisymmetric. |
The magnetice field oL thetinimum-enerey coufiguration is close to tlie initial field (2)). | The magnetic field of theminimum-energy configuration is close to the initial field \ref{mfield}) ). |
et al. | et al. |
2006: Ixilic et al. | 2006; Kilic et al. |
2007a). | 2007a). |
ILowever. only one of them has been searched for companions and found to be in a binary system (Ixilie et al. | However, only one of them has been searched for companions and found to be in a binary system (Kilic et al. |
2007b). | 2007b). |
A svstematic search for companions io WDs with masses 0.2—0.4 iis required {ο evaluate the binary [raction al different mass ranges. | A systematic search for companions to WDs with masses $-$ 0.4 is required to evaluate the binary fraction at different mass ranges. |
A prediction of this scenario is (hat we should see very few truly metal rich red clump eianls. | A prediction of this scenario is that we should see very few truly metal rich red clump giants. |
This effect is seen in the red giant branch huminositv. [uncetion of NGC 6791: the stars are (hinning out of the red giant branch ancl some stars never reach (he tip of the eiant branch (ναναί οἱ al. | This effect is seen in the red giant branch luminosity function of NGC 6791; the stars are thinning out of the red giant branch and some stars never reach the tip of the giant branch (Kalirai et al. |
2007a). | 2007a). |
A similar effect is also seen in the local population of ejants. | A similar effect is also seen in the local population of giants. |
Figure 9 in Luck Ileiter (2007) compares the metallicity histograms for clwarls and giants within 15 pe of the Sun. | Figure 9 in Luck Heiter (2007) compares the metallicity histograms for dwarfs and giants within 15 pc of the Sun. |
It is clear from their figure Chat even though the dwarl population has a high metallicity tail extending up to |Fe/1I]o--0.6 (for both of their dwarl samples including and excluding the planet-hosting stars: top tree panels). the giants show a significant drop in numbers after [Fe/II] = 40.2 and no giants with |Fe/Il] > --0.45 are observed in the field population. | It is clear from their figure that even though the dwarf population has a high metallicity tail extending up to $\sim$ +0.6 (for both of their dwarf samples including and excluding the planet-hosting stars; top three panels), the giants show a significant drop in numbers after [Fe/H] = +0.2 and no giants with [Fe/H] $>$ +0.45 are observed in the field population. |
Using Ilipparcos parallaxes and metallicities for 284 nearby red giant stars. Udalski (2000) showed that there is a very well defined red clump for lower metallicity elants. ie. Ῥοί κ —0.05. but above that the red clump becomes really broad. (see their Figure 2). | Using Hipparcos parallaxes and metallicities for 284 nearby red giant stars, Udalski (2000) showed that there is a very well defined red clump for lower metallicity giants, i.e. [Fe/H] $< -$ 0.05, but above that the red clump becomes really broad (see their Figure 2). |
The higher metallicity bin is poorly defined due to the small nunber of metal-rich red chup stars in (he solar neighborhood. | The higher metallicity bin is poorly defined due to the small number of metal-rich red clump stars in the solar neighborhood. |
Depending on the metallicity of the progenitors. (hie mass loss process is likely (o create red giants wilh a range of envelope masses which would create a broad distribution of red. clump stars. | Depending on the metallicity of the progenitors, the mass loss process is likely to create red giants with a range of envelope masses which would create a broad distribution of red clump stars. |
These observations also suggest that the mass loss can (hin out the envelopes of red giants and that most super-solar metallicity stars will nol reach the tip of the red giant branch. | These observations also suggest that the mass loss can thin out the envelopes of red giants and that most super-solar metallicity stars will not reach the tip of the red giant branch. |
In our scenario. the mass loss rate in cool giants increases dramatically with increased metallicitv. | In our scenario, the mass loss rate in cool giants increases dramatically with increased metallicity. |
This could impact the winds Irom more massive stars as well. which in turn could allect the Type Ia SNe production rate. | This could impact the winds from more massive stars as well, which in turn could affect the Type Ia SNe production rate. |
There are (wo main formation scenarios for Tvpe Ia SNe: accretion Irom a non-degenerate companion star onto a WD close to the Chnandrasekhar mass limit (e.g. Whelan Iben 1973) and mergers of two WDs (e.g. Webbink 1934). | There are two main formation scenarios for Type Ia SNe; accretion from a non-degenerate companion star onto a WD close to the Chandrasekhar mass limit (e.g. Whelan Iben 1973) and mergers of two WDs (e.g. Webbink 1984). |
In the former case. a finely tuned mass accretion rate is required to avoid either nova explosions | In the former case, a finely tuned mass accretion rate is required to avoid either nova explosions |
with current observations (Brownetal.2007). | with current observations \citep{Brown07}. |
. We ouly take those IIVSs in the surveved area with the racial distance from 25kpe to 130kpe aud radial velocities Cgc275kans as the simulated sample. which are then used to compare with the observations. | We only take those HVSs in the surveyed area with the radial distance from $25\kpc$ to $130\kpc$ and radial velocities $v_{\rm
rf}\geq275 \kms$ as the simulated sample, which are then used to compare with the observations. |
The effect of lianited lifetime of the ejected IIVSs is also considered in our calculations. | The effect of limited lifetime of the ejected HVSs is also considered in our calculations. |
The sclection effects are similarly considered for those models in the BBIT mechanisa iu Section 77. below. | The selection effects are similarly considered for those models in the BBH mechanism in Section \ref{sec:BBH} below. |
We note here that the OCDE is uot affected much but the eCDF max be siguificautlv affected by the selection effects. | We note here that the $\Theta$ CDF is not affected much but the $v$ CDF may be significantly affected by the selection effects. |
Figure 11 shows the OCDEs obtained from the “RW- model | Figure \ref{fig:f11} shows the $\Theta$ CDFs obtained from the ``RW-1'' model. |
As seen from this figure. the observational OCDE for both IIVS populations cau be well reproduced if the IIVSs were originated from two disk-like stellar structures with orieutations the same as the two best-fit planes and thickness of ~12° aud 13°. respectively (also see the top paucl of Figure 121). | As seen from this figure, the observational $\Theta$ CDF for both HVS populations can be well reproduced if the HVSs were originated from two disk-like stellar structures with orientations the same as the two best-fit planes and thickness of $\sim12\arcdeg$ and $13\arcdeg$, respectively (also see the top panel of Figure \ref{fig:f12}) ). |
Were. the thickuess of a stellar disk is defined by the standard deviation of the inchnation angele of stars in the disk from the disk ceutral plauc. | Here, the thickness of a stellar disk is defined by the standard deviation of the inclination angle of stars in the disk from the disk central plane. |
For these two populations of the detected IIVSs. our νο tests Bud the likeliboods of 0.896. aud 0.999 that the observational OCDE are drawn from the sale distribution as that obtained from the uuucrical simulations for the "BW-17 model. respectively. | For these two populations of the detected HVSs, our K-S tests find the likelihoods of 0.896 and 0.999 that the observational $\Theta$ CDF are drawn from the same distribution as that obtained from the numerical simulations for the “RW-1” model, respectively. |
All the other models (0. "LP-l'. ον πο "CD-2'. and "RW-27) cau reproduce the observational OCDEs for both IIVS populatious by choosing suitable thickuess (typically in the range of 77. 137) for the two disks. | All the other models (i.e., “LP-1”, “LP-2”, “UB-1”, “UB-2”, and “RW-2”) can reproduce the observational $\Theta$ CDFs for both HVS populations by choosing suitable thickness (typically in the range of $7\arcdeg$$-$$13\arcdeg$ ) for the two disks. |
The resulted OCDFs are mainly determuned by the thickness of the disks where the IIVS progenitors are originated. and also weakly depend on the mechanism adopted that leads to the injection of stellar binarics iuto the unediate vicinity of the MDIT. | The resulted $\Theta$ CDFs are mainly determined by the thickness of the disks where the HVS progenitors are originated, and also weakly depend on the mechanism adopted that leads to the injection of stellar binaries into the immediate vicinity of the MBH. |
Alternatively. αποιο that the stellar binaries. Lo. the ITVS progenitors. were isotropically distributed but with other initial settings the same as those in the above models. the ejected IWSs should also be isotropically distributed in the survey area. | Alternatively, assuming that the stellar binaries, i.e., the HVS progenitors, were isotropically distributed but with other initial settings the same as those in the above models, the ejected HVSs should also be isotropically distributed in the survey area. |
We do simular αποΊσα simulations aud obtain the OCDEs for those isotropically distributed IIVSs. relative to the two best-fit thin disk planes of the detected ITVSs. as shown by the dashed curves in Figure 11.. | We do similar numerical simulations and obtain the $\Theta$ CDFs for those isotropically distributed HVSs, relative to the two best-fit thin disk planes of the detected HVSs, as shown by the dashed curves in Figure \ref{fig:f11}. |
Hore. a simulated ITVS is assigned to oue of the two populations if it is closer to the best- plane of that population than to the other plane. | Here, a simulated HVS is assigned to one of the two populations if it is closer to the best-fit plane of that population than to the other plane. |
asina beta parameter. aud b) the increase of the proton-(ectron lass ratio n» n, makes computations longer. ut results are similar. | plasma beta parameter, and b) the increase of the proton-electron mass ratio $_p$ $_e$ makes computations longer, but results are similar. |
We compared the preseut simulation also with that iu ie nuinerical model which size was two times smaller (L,νLy, = GOOA « 2000A) and iu which ouly 5 jasmeoids were initiated (contrary to 10 plaswaoids iu ie present simulation). | We compared the present simulation also with that in the numerical model which size was two times smaller $L_x \times L_y$ = $\Delta$ $\times$ $\Delta$ ) and in which only 5 plasmoids were initiated (contrary to 10 plasmoids in the present simulation). |
In this case the final mean enerev of accelerated electrous was 5.3 tunes greater iui the initial one. compare with that of 10.7 times eoroni the iuitial temperature LO MIN to final teniperature WF MIS) iu the present case. | In this case the final mean energy of accelerated electrons was 5.3 times greater than the initial one, compare with that of 10.7 times (from the initial temperature 10 MK to final temperature 107 MK) in the present case. |
Namely. cach coalescence orocess ducreases the energv of accelerated clectrous. icrefore the nunmiber of successive coalescence processes is essential for their final enerex. | Namely, each coalescence process increases the energy of accelerated electrons, therefore the number of successive coalescence processes is essential for their final energy. |
For calculations of the hard X-ray spectra. preseuted iu Fig. | For calculations of the hard X-ray spectra, presented in Fig. |
bbb we used two methods. | \ref{figure5}b b we used two methods. |
The obtained results are simular. | The obtained results are similar. |
Sinall differences are due to differences in these methods and deviations of the computed distribution functions from the thermal onc. | Small differences are due to differences in these methods and deviations of the computed distribution functions from the thermal one. |
The plasiuoids in 2-D are in reality 3-D maguetic ropes. | The plasmoids in 2-D are in reality 3-D magnetic ropes. |
While in 2-D the trapping of energetic electrous is a natural consequence of a close magnetic feld structure of the plaxuoid. im 3-D. this structure is only seniclosed. | While in 2-D the trapping of energetic electrons is a natural consequence of a close magnetic field structure of the plasmoid, in 3-D, this structure is only semi-closed. |
Ilowever. we consider the inergng processes in the turbulent reconnection outflow therefore the magnetic trapping of electrous. similar to that proposed by Jaknuiecetal.(1998). is highly. probable. | However, we consider the merging processes in the turbulent reconnection outflow therefore the magnetic trapping of electrons, similar to that proposed by \citet{Jakimiec1998} is highly probable. |
Moreover. the coalescence fragmentation process. which generates the reverse clectric currents (which iu 3-D has to be closed in finite volume) will contribute to a full trapping of electrons. | Moreover, the coalescence fragmentation process, which generates the reverse electric currents (which in 3-D has to be closed in finite volume) will contribute to a full trapping of electrons. |
Tn agreement with the conclusious by IExruckerotal.(2010).. in the model the acceleration region is very close to the hard N-rav source. | In agreement with the conclusions by \citet{Kruckeretal2010}, in the model the acceleration region is very close to the hard X-ray source. |
It euables to re-accelerate energetic electrons. which loss their energy due to collisions. | It enables to re-accelerate energetic electrons, which loss their energy due to collisions. |
Acceleration regions are among interacting plasmoids and also between the plasimolds aud the arcade of flaring loops. | Acceleration regions are among interacting plasmoids and also between the plasmoids and the arcade of flaring loops. |
This model can explain uot only the ivove-the-loop-top hard X-ray sources. but also the loop-top sources because the arcade of loops is. ia principle. the ‘plasimoicd’ fixed in its half height at the plotosphere. | This model can explain not only the above-the-loop-top hard X-ray sources, but also the loop-top sources because the arcade of loops is, in principle, the 'plasmoid' fixed in its half height at the photosphere. |
Considering all aspects of the fragmentation process (power-law spatial scales of plasumioids. effective acceleration of olectrous. trapping of clectrous— in plasinoids. location in the reconnection plasina outflow) we think that this process can explain a generation of the narrowband dim-spikes. | Considering all aspects of the fragmentation process (power-law spatial scales of plasmoids, effective acceleration of electrons, trapping of electrons in plasmoids, location in the reconnection plasma outflow) we think that this process can explain a generation of the narrowband dm-spikes. |
We supported this idea by the radio spectrum observed duiug the 28 March 2001 showing Όρος drifting towards the uurowband diu-spikes. | We supported this idea by the radio spectrum observed during the 28 March 2001 showing DPSs drifting towards the narrowband dm-spikes. |
Furthermore. it is known that more than TUO % of all groups. of diispikes are observed during the COES-+visine-flare phases (Jizéiekaetal.2001). | Furthermore, it is known that more than 70 $\%$ of all groups of dm-spikes are observed during the GOES-rising-flare phases \citep{Jirickaeta2001}. |
. Although these arguments support the preseuted idea. further analysis of the narrowband cdiu-plkes aud their modelling is necessary. | Although these arguments support the presented idea, further analysis of the narrowband dm-spikes and their modelling is necessary. |
lu this first study; we cousidered only the neutral current shect. ie. 1.—0. | In this first study, we considered only the neutral current sheet, i.e. $B_z=0$. |
For more realistic description. we plan to extend our study also to cases with non-zero euidiue maenetic field. | For more realistic description, we plan to extend our study also to cases with non-zero guiding magnetic field. |
The presented model is a natural exteusion of our previous models explaining the plasimoicd formation. its ejection. aud corresponding DPS. | The presented model is a natural extension of our previous models explaining the plasmoid formation, its ejection, and corresponding DPS. |
The question arises why the above-the-loop-top hard X-rav sources are | The question arises why the above-the-loop-top hard X-ray sources are |
aud ors. with the 3«lirection taken to be orthogoual to the computational plane. | and $x_2$, with the 3-direction taken to be orthogonal to the computational plane. |
The mesh whose eric) lines mark zoue edges is labeled by coordinates ola; aud αρ while the mesh whose lines intersect at zone centers is labeled by coordinates lb; and 20. as shown in figure 2.. | The mesh whose grid lines mark zone edges is labeled by coordinates $x1a_i$ and $x2a_j$, while the mesh whose lines intersect at zone centers is labeled by coordinates $x1b_i$ and $x2b_j$ , as shown in figure \ref{fig:mesh}. |
On this mesh. the radiation variables are discretized according to The components of the Eddiugton tensor f are centered identically to the components of P. | On this mesh, the radiation variables are discretized according to The components of the Eddington tensor ${\mathsf f}$ are centered identically to the components of ${\mathsf P}$. |
The opacity. absorption ancl scattering coefficients. and Planck function are all zone-centered. | The opacity, absorption and scattering coefficients, and Planck function are all zone-centered. |
The radiation diffusion coefficient D is computed on zone 1-Laces (D1; j) aud 2-[aces (D2;;) along with the componeuts of the flux. | The radiation diffusion coefficient $D$ is computed on zone 1-faces $D1_{i,j}$ ) and 2-faces $D2_{i,j}$ ) along with the components of the flux. |
Under the FLD approximation. the [lus divergence te‘wu in the radiation energy equation (20)) is a function of the eracieut iu radiation enerey deusity. | Under the FLD approximation, the flux divergence term in the radiation energy equation \ref{eqn:erdivflux}) ) is a function of the gradient in radiation energy density. |
When implicitly dilfereuced. it therefore depeuds on the time-advauced values of E in adjacent eric zones. aud iuay be updated by a matrix inversion. | When implicitly differenced, it therefore depends on the time-advanced values of $E$ in adjacent grid zones, and may be updated by a matrix inversion. |
To simplify the form of the matrix. we operaor-split this from the remainiug source terms. auc integrate with D given by equation (11)). | To simplify the form of the matrix, we operator-split this from the remaining source terms, and integrate with $D$ given by equation \ref{eqn:dc}) ). |
The solution metloc is described here for a uniforimly-spaced square Cartesian grid. | The solution method is described here for a uniformly-spaced square Cartesian grid. |
Extension to the curvilinear coorditates used in ZEUS-2D is straightforward. | Extension to the curvilinear coordinates used in ZEUS-2D is straightforward. |
The implicit difference equations to be solved are of the fogae where Av is the grid spacing. and the spatial iudices span rauges 7=1....N aud j—1...M. | The implicit difference equations to be solved are of the form where $\Delta x$ is the grid spacing, and the spatial indices span ranges $i=1\ldots N$ and $j=1\ldots M$. |
Racliation energy deusities are to be determined at timeste putl. | Radiation energy densities are to be determined at timestep $n+1$. |
The difference equatious are kept linear by usiug diffusion coelficieuts from step η. | The difference equations are kept linear by using diffusion coefficients from step $n$. |
This is a good approximation when [V.EAXE) Changes little each timestep. but may be less appropriate when the radiation field variesrapidly with | This is a good approximation when $|\nabla E|/(\chi E)$ changes little each timestep, but may be less appropriate when the radiation field variesrapidly with |
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