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At later times. the desity is low enough that the Ilemholtz equation of state no longer applies. | At later times, the density is low enough that the Helmholtz equation of state no longer applies. |
The code was coufigured to use pressure. density. ‘Tle. and 1O as its refinement variables. | The code was configured to use pressure, density, $^4$ He, and $^{16}$ O as its refinement variables. |
An eror estimate for a block was computed using the second derivative of the chosen refinement variables. | An error estimate for a block was computed using the second derivative of the chosen refinement variables. |
If the estimated normalized error 11 one or more of these variables was greater thu a given value. regions were refined until a normalized error is reached that is less than the acceptable value or the maxinuun level of refucment is reached. | If the estimated normalized error in one or more of these variables was greater than a given value, regions were refined until a normalized error is reached that is less than the acceptable value or the maximum level of refinement is reached. |
Reeious of the simulations with steep eracients im one or more of these variables were likely to be refined. | Regions of the simulations with steep gradients in one or more of these variables were likely to be refined. |
If the normalized error estimate was below a certain value. Ίνοι, the absolute value of the second derivative of one or more variables was πα, then the region was "de-crefiued. | If the normalized error estimate was below a certain value, i.e., the absolute value of the second derivative of one or more variables was small, then the region was “de-refined”. |
The FLASH22.5 release was customized to include a nodule that inserts a roughly circular zero-gracieut iuner voundary around the origin. | The 2.5 release was customized to include a module that inserts a roughly circular zero-gradient inner boundary around the origin. |
This preveuted iufalliug uatter near the origin in the simulations from backing up aud affecting the outer regions. | This prevented infalling matter near the origin in the simulations from backing up and affecting the outer regions. |
Matter was allowed o fall though the zero-eradicut. quasi-circular boundary at the center of the model aud accunulate on the ont amass at the origi. | Matter was allowed to fall though the zero-gradient, quasi-circular boundary at the center of the model and accumulate on the point mass at the origin. |
If the radius for the iuner )oundary passed through one of the iuucer zones. the zones interior to that zone were set to be duplicates of he zone on the bouudary. | If the radius for the inner boundary passed through one of the inner zones, the zones interior to that zone were set to be duplicates of the zone on the boundary. |
Wlile the Cartesian nature of the axisviunuetrie coordinate svstem meant that the inner boundary was only as close to circular as oue can reproduce with square components. it did not iutroduce a sienificant amount of error iuto the caleulation. | While the Cartesian nature of the axisymmetric coordinate system meant that the inner boundary was only as close to circular as one can reproduce with square components, it did not introduce a significant amount of error into the calculation. |
The inner boundaries were chosen to be within the sonic radius. censure that simall numerical errors at the interior boundary dil not accumulate aud affect the flow of fluid upstream from the boundary. | The inner boundaries were chosen to be within the sonic radius, ensuring that small numerical errors at the interior boundary did not accumulate and affect the flow of fluid upstream from the boundary. |
The sonic point moved outward. not iuvard. as the simulation time progressed. so the inner boundary was always within the sonic radius. | The sonic point moved outward, not inward, as the simulation time progressed, so the inner boundary was always within the sonic radius. |
The falline temperature caused the souud speed to decline with time. while the velocity near the inner boundary increased with tine. | The falling temperature caused the sound speed to decline with time, while the velocity near the inner boundary increased with time. |
The eravitational potential resulting from this poiut mass was updated and added to the gravitational forces computed by themultipole solver at cach time step. | The gravitational potential resulting from this point mass was updated and added to the gravitational forces computed by themultipole solver at each time step. |
A module was also included that locally deposited enerey frou radioactive decay of “°Ni to °CFe. | A module was also included that locally deposited energy from radioactive decay of $^{56}$ Ni to $^{56}$ Fe. |
The amount of cacrey The decay rate of Ni. Ass is 1.315&10 ον5 and the amount of energy released per era of decaying tis q(09Ni). for which we took the value 2.96«1010 erg Ἐν | The amount of energy The decay rate of , $\lambda_{\Ni}$, is $1.315\times10^{-6}$ $^{-1}$, and the amount of energy released per gram of decaying is $q(\Ni)$, for which we took the value $2.96 \times 10^{16}$ erg $^{-1}$. |
Xeox is the fraction of in the block. | $X_{\Ni}$ is the fraction of in the block. |
The iuount of aat a given time cold be found asa function of the amouut of initial by so that theenerey deposition rate frou aas a function of tine was eiven by We assumed a decay rate for ος, Asics. of LOL& and an energv per erai) ofdecaviug "9Co.. q(9 Co). for which we took the value of 6.1«1019 ore | The amount of at a given time could be found asa function of the amount of initial by so that theenergy deposition rate from as a function of time was given by We assumed a decay rate for , $\lambda_{\Co}$ , of $1.042
\times10^{-7}$ $^{-1}$ , and an energy per gram ofdecaying , $q(\Co)$ , for which we took the value of $6.4 \times 10^{16}$ erg $^{-1}$ . |
Jupiter using several different hydrogen EOSs that span (he range of data obtained. [rom LLNL laser (Collinsetal.1998) and Sandia Z Uxunedsonetal.2004) data. | Jupiter using several different hydrogen EOSs that span the range of data obtained from LLNL laser \citep{Collins98} and Sandia Z \citep{Knudson04} data. |
These different EOSs predict temperatures than can differ bv as much as at 1 Mbar. | These different EOSs predict temperatures than can differ by as much as at 1 Mbar. |
They find that Jupiter models cool to the planets known luminosity in ~3 (o0 5.5 Gyr using these various EOSs. | They find that Jupiter models cool to the planet's known luminosity in $\sim$ 3 to 5.5 Gyr using these various EOSs. |
This 2.5 Gvr uncertainty is rather significant. | This 2.5 Gyr uncertainty is rather significant. |
The atmospheres of Jupiter and Saturn ave both depleted in helium relative to protosolar composition (Atrevaetal.2003). | The atmospheres of Jupiter and Saturn are both depleted in helium relative to protosolar composition \citep{Atreya03}. |
. This observation. together with theoretical work indicating (hat helium has a limited solubility in metallic hydrogen at planetary interior temperatures ol ~10! K (Stevenson1975;IIubbard&Dewitt1985;Pfaffenzellerοἱal.1995)... indicates helium is phase separating [rom hwdrogen aud being lost to deeper lavers in each planet. | This observation, together with theoretical work indicating that helium has a limited solubility in metallic hydrogen at planetary interior temperatures of $\sim$ $^4$ K \citep{Stevenson75,HDW,Pfaff}, indicates helium is phase separating from hydrogen and being lost to deeper layers in each planet. |
The evolution of Saturn. aud perhaps Jupiter. must be able to accommodate the substantial additional energv source due to dillerentiation within the planet. | The evolution of Saturn, and perhaps Jupiter, must be able to accommodate the substantial additional energy source due to differentiation within the planet. |
This "helium rain.” if present. has been shown to be the dominant energy source lor several-Gyr-old giant planets (Stevenson&SalpeterLOTT:FortueyIIubbard2003.2004). | This “helium rain,” if present, has been shown to be the dominant energy source for several-Gyr-old giant planets \citep{SS77b, FH03, FH04}. |
. In order to understad to what degree helium phase separation has progressed in Jupiter and Saturn. and how far down into the planet the helium has rained to. we mast understand (he deep interior temperature of these planets. | In order to understand to what degree helium phase separation has progressed in Jupiter and Saturn, and how far down into the planet the helium has rained to, we must understand the deep interior temperature of these planets. |
To date. temperature measurements have been published by Holmesetal.(1995). and Collinsetal.(2001). | To date, temperature measurements have been published by \citet{Holmes95} and \citet{Collins01}. |
. These experiments were performed using gas gun and laser apparatuses. respectively. | These experiments were performed using gas gun and laser apparatuses, respectively. |
Both found temperatures generally lower than most calculated hydrogen EOSs. which if indeed correct. would lead to shorter cooling timescales [for giant planets. | Both found temperatures generally lower than most calculated hydrogen EOSs, which if indeed correct, would lead to shorter cooling timescales for giant planets. |
This faster cooling would) more easily accommodate the additional energy source due to helium rain. | This faster cooling would more easily accommodate the additional energy source due to helium rain. |
Additional data. especially at the high pressures ancl “cool” temperatures of planetary interest. (off of the single-shock Hugoniot) would be of great interest. | Additional data, especially at the high pressures and “cool” temperatures of planetary interest (off of the single-shock Hugoniot) would be of great interest. |
If we are to understand eiant planets as a class of astronomical objects. we must understand how similar other giant planets are to Jupiter and Saturn. | If we are to understand giant planets as a class of astronomical objects, we must understand how similar other giant planets are to Jupiter and Saturn. |
The mass-radius relation of exoplanets allows us. in principle. to. understand if these planets have heavy element enrichments (hat are similar to. Jupiter and Saturn. | The mass-radius relation of exoplanets allows us, in principle, to understand if these planets have heavy element enrichments that are similar to Jupiter and Saturn. |
shows the mass and radius of Jupiter. Saturn. and the 10 known transiting hot Jupiters. | shows the mass and radius of Jupiter, Saturn, and the 10 known transiting hot Jupiters. |
It is interesting to note while Jupiter and Saturn differ in mass by a factor of 3.3. their radii only differ by1894. | It is interesting to note while Jupiter and Saturn differ in mass by a factor of 3.3, their radii only differ by. |
.. llowever. while the hot Jupiters differ in mass bv a similar factor (of 4) they differ in radius by a factor of 2. | However, while the hot Jupiters differ in mass by a similar factor (of 4) they differ in radius by a factor of 2. |
This large spread is presumably due to lareedifference in the interior heavy element abundanunces of these planets (Fortneyοἱal.2006:Guillot2005:et 2006).. | This large spread is presumably due to largedifference in the interior heavy element abundances of these planets \citep{Fortney06, Guillot05,Guillot06}. . |
N-body | $N$ |
distance). | distance). |
Although the errors are very larec. these slopes are the same within their errors. | Although the errors are very large, these slopes are the same within their errors. |
Therefore. the burst rate may be a unique function of the persistent flux. | Therefore, the burst rate may be a unique function of the persistent flux. |
Another nuportaut burst paramcter is the c-folding decay time. | Another important burst parameter is the e-folding decay time. |
As discussed in 11. this diagnoses the composition of the burst fuel. | As discussed in 1, this diagnoses the composition of the burst fuel. |
To derive the decay time for each burst we generated liehteurves with a 1 s time resolution. | To derive the decay time for each burst we generated lightcurves with a 1 s time resolution. |
A runing average of 5 s was used to determine the moment of the peak flux. | A running average of 5 s was used to determine the moment of the peak flux. |
The persistent cuiission level aud the decay time are then simmiltancously fitted with a constant and exponential. respectively. | The persistent emission level and the decay time are then simultaneously fitted with a constant and exponential, respectively. |
We took the biu in which the peak flux was reached as the first data point. | We took the bin in which the peak flux was reached as the first data point. |
Iu refdecay owe show the decay times as a function of persistent cunission for the nine sources. | In \\ref{decay} we show the decay times as a function of persistent emission for the nine sources. |
We here discuss 0. 260 and 21 la nire detail aud compare them with the other sources. | We here discuss $-$ 0, $-$ 260 and $-$ 24 in more detail and compare them with the other sources. |
O only shows bursts with decay times shorter than z10 seconds at all ux levels. | $-$ 0 only shows bursts with decay times shorter than $\simeq$ 10 seconds at all flux levels. |
The same applies to 536. 33] 1 and 30. | The same applies to $-$ 536, $+$ 1 and $-$ 30. |
21 shows a large range of decay times at all porsisteut. fiux | $-$ 24 shows a large range of decay times at all persistent flux |
and Na-rich. but O-poor (yields from a combination between pristine gas and massive stars before they explode). with a small enrichment in Fe (contribution from a small fraction of FG SN II material). | and Na-rich, but O-poor (yields from a combination between pristine gas and massive stars before they explode), with a small enrichment in Fe (contribution from a small fraction of FG SN II material). |
TG stars. on the other hand. are expected to be He-. Na-. and Fe-rich. but O-poor. | TG stars, on the other hand, are expected to be He-, Na-, and Fe-rich, but O-poor. |
Even though this model predicts separate distributions in the O-Na plane for the FG. SG and TG stars — a gap which becomes "bridged" when the 4G stars are formed (see Fig. 5.. | Even though this model predicts separate distributions in the O-Na plane for the FG, SG and TG stars – a gap which becomes “bridged” when the 4G stars are formed (see Fig. \ref{FIGMassiveGC}, |
panel h) — one should bear in mind that this is only a schematic representation. and the real chemical evolution can be more complex. especially due to the fact that 4G stars are formed with material processed by previous generations plus a fraction of pristine gas (f the latter has not been completely expelled). | panel h) – one should bear in mind that this is only a schematic representation, and the real chemical evolution can be more complex, especially due to the fact that 4G stars are formed with material processed by previous generations plus a fraction of pristine gas (if the latter has not been completely expelled). |
In fact. as is observed in the O-Na anticorrelation of c) Cen(??).. there is an overlap between metal-poor and metal-intermediate branches in the O-Na plane. | In fact, as is observed in the O-Na anticorrelation of $\omega$ Cen, there is an overlap between metal-poor and metal-intermediate branches in the O-Na plane. |
However. metal-rich stars ({Fe/H]> —1.3) are all Na-rich. which can also be associated to some degree of He enrichment. | However, metal-rich stars ${\rm [Fe/H]}>-1.3$ ) are all Na-rich, which can also be associated to some degree of He enrichment. |
In this section. we have presented a scenario for the formation of multiple populations in GCs which makes several non-standard assumptions. | In this section, we have presented a scenario for the formation of multiple populations in GCs which makes several non-standard assumptions. |
The purpose of this subsection is to explain why these assumptions were made. | The purpose of this subsection is to explain why these assumptions were made. |
First. we incorporate into the cluster's evolving chemistry the ejecta of massive stars. | First, we incorporate into the cluster's evolving chemistry the ejecta of massive stars. |
Most frequently. however. the contribution of massive stars 1s ignored. due to the fact that their winds are very fast2). | Most frequently, however, the contribution of massive stars is ignored, due to the fact that their winds are very fast. |
. We decided to take these winds into account for four main reasons. namely: It should also be emphasized that stars with extreme He abundance may naturally have extreme abundances of other elements. such as high Na and low O. Therefore. stars at the high-Na. low-O end of the O-Na anticorrelation (panel h in Fig. 5.. | We decided to take these winds into account for four main reasons, namely: It should also be emphasized that stars with extreme He abundance may naturally have extreme abundances of other elements, such as high Na and low O. Therefore, stars at the high-Na, low-O end of the O-Na anticorrelation (panel h in Fig. \ref{FIGMassiveGC}, |
and panels g in Figs. | and panels g in Figs. |
3 and 1)) are those which are more naturally expected to have high He abundances. | \ref{FIGFairlyMassiveGC} and \ref{FIGNonMassiveGC}) ) are those which are more naturally expected to have high He abundances. |
This is in the same sense as recently observed by?. | This is in the same sense as recently observed by. |
. This would also be qualitatively consistent with the observed correlation between the presence and extent of these abundace anomalies and position along the horizontal branch222). | This would also be qualitatively consistent with the observed correlation between the presence and extent of these abundance anomalies and position along the horizontal branch. |
. However. to more properly test this scenario. models of Nassive stars with mass loss for low metallicities and with time-dependent | However, to more properly test this scenario, models of massive stars with mass loss for low metallicities and with time-dependent |
turbulence was later refined by Scalo&/Puuiplirev(1982).. which they called ‘turbulent vinalization'. | turbulence was later refined by \citet{ScaloPumphrey1982}, which they called `turbulent virialization'. |
Fleck(1983). suggested that the injection of turbulence bv eravitational coutraction 1s iniportant inthe interstellar uediuu. aud Beeehuan&Shlosan(2009)/ conclude hat augularnonmieutum transport duriug the turbuleu collapse ofa gaseous xvsteni lay suppress fragmentation. | \citet{Fleck1983} suggested that the injection of turbulence by gravitational contraction is important inthe interstellar medium, and \citet{BegelmanShlosman2009} conclude that angular-momentum transport during the turbulent collapse of a gaseous system may suppress fragmentation. |
Tere we show with numerical simulations that urbulence and immaenetic field erowth are inclece cficicutly driven by the eravitational cucrey releascc during the collapse of a dense eas cloud. | Here we show with numerical simulations that turbulence and magnetic field growth are indeed efficiently driven by the gravitational energy released during the collapse of a dense gas cloud. |
The conuection jetween eravity-driven turbulence and magnetic fik aüuplification has also been suggested recentlyl iu a mode w Schleicheretal.(2010). ancl ΕΕ 1 in Suretal.(2010). | The connection between gravity-driven turbulence and magnetic field amplification has also been suggested recently in a model by \citet{SchleicherEtAl2010} and confirmed numerically in \citet{SurEtAl2010}. |
. In this process. poteutial cucrey is converted into turbulent motions. which iu turni amplify he magnetic euergv via the turbulent dvnamo. | In this process, potential energy is converted into turbulent motions, which in turn amplify the magnetic energy via the turbulent dynamo. |
Thus. he driving of turbulence auc magnetic field growth by eravitational iufall nav be the consequence of a selfreenlating instabilitv. | Thus, the driving of turbulence and magnetic field growth by gravitational infall may be the consequence of a self-regulating instability. |
At the bottom of this cascade. close to the sonic scale (Federrathetal.2010b).. eas is expected to become subsonie as a cousequeuce of a steep rise in the temperature when the gas becomes optically thick. | At the bottom of this cascade, close to the sonic scale \citep{FederrathDuvalKlessenSchmidtMacLow2010}, gas is expected to become subsonic as a consequence of a steep rise in the temperature when the gas becomes optically thick. |
During this process. compressible iiodes will be converted iuto solenoidal turbulent motions. uutil a natural euergv ratio of πω9c2/3 Is reached. (Ehnuceereen&Scalo2001:Federvrathetal.2008b). | During this process, compressible modes will be converted into solenoidal turbulent motions, until a natural energy ratio of $\solratio\approx2/3$ is reached, \citep{ElmegreenScalo2004,FederrathKlessenSchmidt2008}. |
. Iu this paper. we first discuss the plivsies of maeuetic field amplification by eravityv-driven turbulence. aud subsequently derive a new resolution criterion required to resolve these processes. | In this paper, we first discuss the physics of magnetic field amplification by gravity-driven turbulence, and subsequently derive a new resolution criterion required to resolve these processes. |
We aim to address the following key questions: Ou what scales does the maguetic field erow cling the collapse of a deuse. maguctized gas cloud? | We aim to address the following key questions: On what scales does the magnetic field grow during the collapse of a dense, magnetized gas cloud? |
What is the effective kinetic energy imjection scale of eravitv-divenu turbulence? | What is the effective kinetic energy injection scale of gravity-driven turbulence? |
Tow are the compressible motions in a contracting svsteni couverted into turbulent randoni notions. and what is the asviuptotic fraction of solenoidal motions generated during the coutraction? | How are the compressible motions in a contracting system converted into turbulent random motions, and what is the asymptotic fraction of solenoidal motions generated during the contraction? |
After presenting our methods in section 2.. we address these physical questions in section 3. with the use of maenetolivdrodvuamical (MIID) «αμαος, | After presenting our methods in section \ref{sec:methods}, we address these physical questions in section \ref{sec:gravitydriventurb} with the use of magnetohydrodynamical (MHD) simulations. |
We find that the magnetic field is most οποιοτν amplified ou the samallest resolvable scales in the simulations. aud erows exponentiallv fast due to the small-scale dynamo. | We find that the magnetic field is most efficiently amplified on the smallest resolvable scales in the simulations, and grows exponentially fast due to the small-scale dynamo. |
The effective energy injection scale of eravity-criven turbulence is close to the local Jeans scale during the contraction. | The effective energy injection scale of gravity-driven turbulence is close to the local Jeans scale during the contraction. |
Finally. we show in section L2. that about 2/3 of the total kinetic cucrev released durus the collapse is converted iuto solenoidal. turbulent motions. cficiently driving magnetic field amplification. | Finally, we show in section \ref{sec:rot_ratio} that about 2/3 of the total kinetic energy released during the collapse is converted into solenoidal, turbulent motions, efficiently driving magnetic field amplification. |
Iu the secoud part of the paper (section. 19). we discuss the nuuercal resolution requirenieuts for modeling turbulent. selferavitatius svstenis and for mimi, dynamo action to set iu. | In the second part of the paper (section \ref{sec:newjeansresol}) ), we discuss the numerical resolution requirements for modeling turbulent, self-gravitating systems and for minimum dynamo action to set in. |
To study the resolution dependence of our results. we use a sequence of sinuulatious in which we resolve the Jeans leugth with S. 16. 32. GI. and 128 exid cells (sec.Suretal.2010.pa-per D. | To study the resolution dependence of our results, we use a sequence of simulations in which we resolve the Jeans length with 8, 16, 32, 64, and 128 grid cells \citep[see,][paper I]{SurEtAl2010}. |
. With a Fourier analysis of the maguetic energy. we confirm our earlier findings in paper L showing that 30 exid cells per Jeaus leneth is the threshold resolution for minimi dvnamo amplification of the maguetic field to occur. | With a Fourier analysis of the magnetic energy, we confirm our earlier findings in paper I, showing that 30 grid cells per Jeans length is the threshold resolution for minimum dynamo amplification of the magnetic field to occur. |
In addition. we show that a Jeans resolution of about 30 exid cells is required to obtain couvereccl values of the turbulent cnerey. m particular of the solenoidal (rotational) component. which drives dyna auplification. | In addition, we show that a Jeans resolution of about 30 grid cells is required to obtain converged values of the turbulent energy, in particular of the solenoidal (rotational) component, which drives dynamo amplification. |
Iu contrast. we find that the turbulent energy. density. be. the turbulent pressure on the Jeans scale is underestimated. if the Jeaus leugth is resolved with 16 exid cells or less during the collapse. | In contrast, we find that the turbulent energy density, i.e., the turbulent pressure on the Jeans scale is underestimated, if the Jeans length is resolved with 16 grid cells or less during the collapse. |
Apart from a few exceptions (e.g.Abeletal.90051. the Jeans leneth is resolved with less than 16 exid cells in almost all munerical simulations today. | Apart from a few exceptions \citep[e.g.,][]{AbelBryanNorman2002}, the Jeans length is resolved with less than 16 grid cells in almost all numerical simulations today. |
We speculate that this is because in the study by Trucloveetal.(L997).. it was found that oulv four eril cells per Jeaus leneth are enough to avoid artificial fragmentation. | We speculate that this is because in the study by \citet{TrueloveEtAl1997}, it was found that only four grid cells per Jeans length are enough to avoid artificial fragmentation. |
Au equivalent resolution criterion for the Jeans mass in smoothed particle livdrodvuaimics (SPIT) simulations was fouud bv Bate&Burkert(1997). | An equivalent resolution criterion for the Jeans mass in smoothed particle hydrodynamics (SPH) simulations was found by \citet{BateBurkert1997}. |
. Also. the computational expenses inerease strongly. if one ains to resolve the Jeans length with more than a few cells (or few particles in SPU). | Also, the computational expenses increase strongly, if one aims to resolve the Jeans length with more than a few cells (or few particles in SPH). |
Thus. most existing bydrodvuamical aud MITD simulations of sclberavitating media have typically usec about ten grid cells per Jeans leugth or less. | Thus, most existing hydrodynamical and MHD simulations of self-gravitating media have typically used about ten grid cells per Jeans length or less. |
Some modification of this criterion was recently also sueecstec bv Cunvrvszezaketal.(2010) to better resolve selferavitating disks. | Some modification of this criterion was recently also suggested by \citet{GawryszczakEtAl2010} to better resolve self-gravitating disks. |
ere we find that the turbulence and maguetic field structure are under-aesolved. if the Jeans length is resolved with 16 cells or less. | Here we find that the turbulence and magnetic field structure are under-resolved, if the Jeans length is resolved with 16 cells or less. |
Moreover. turbulent dynamo amplification of the magnetic field is completely absent in this case. | Moreover, turbulent dynamo amplification of the magnetic field is completely absent in this case. |
To avoid this problem. we sugeest a new resolution criterion for simulations of selt-eravitatiug eascous inedia: iu order to resolve turbulence on the Jeans scale aud to account for the turbulent pressure ou that scale. as well as to capture mimi dvuauo action in MIID simulations. 30 exrid cells per Jeaus leugth are required. which is discussed in detail iu section L.. | To avoid this problem, we suggest a new resolution criterion for simulations of self-gravitating gaseous media: in order to resolve turbulence on the Jeans scale and to account for the turbulent pressure on that scale, as well as to capture minimum dynamo action in MHD simulations, 30 grid cells per Jeans length are required, which is discussed in detail in section \ref{sec:newjeansresol}. |
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