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Equating Eq. (20)) | Equating Eq. \ref{prima}) ) |
to Eq. (21)). | to Eq. \ref{primac}) ), |
indeed one finds that £4,«RF. so that a value of7 1n the lower end of the range of values expectec for short GRBs would require a higher value of Fj,.y to ensure that /,,;z2.5 scooling).. These estimates. however. are the most robust that can be derived. from the publicly available data. | indeed one finds that $t_{cool}\propto n F_{1 \rm keV}^{2/3}$, so that a value of $n$ in the lower end of the range of values expected for short GRBs would require a higher value of $F_{1 \rm keV}$ to ensure that $t_{cool}\gtrsim 2.5$ s. These estimates, however, are the most robust that can be derived from the publicly available data. |
They are also sufficient to show that a solution does indeed exist for a reasonable set of parameters. which ts the aim of this work. | They are also sufficient to show that a solution does indeed exist for a reasonable set of parameters, which is the aim of this work. |
We emphasize that scenarios 2. 3. and 4. which are related to the emission from a lately emitted shell (2) or from the ES deceleration phase (3 and 4). all offer a natural explanation of the observed temporal delay between the high energy tail and the main burst. | We emphasize that scenarios 2, 3, and 4, which are related to the emission from a lately emitted shell (2) or from the ES deceleration phase (3 and 4), all offer a natural explanation of the observed temporal delay between the high energy tail and the main burst. |
Moreover. scenario 2 (emission from a lately emitted shell) may be consistent with the steeply declining emission from an extended X-ray tail that has been observed in association with some short GRBs before 100 s after the trigger time | Moreover, scenario 2 (emission from a lately emitted shell) may be consistent with the steeply declining emission from an extended X-ray tail that has been observed in association with some short GRBs before 100 s after the trigger time |
value. | value. |
It does not change the predicted differential luminosity distribution. | It does not change the predicted differential luminosity distribution. |
The critical difference. however. between the soft X-ray cooling-flow problem and the classic cooling-flow problem is that the latter requires a clear explanation for why X-ray cooling does not appear to be carried to completion. | The critical difference, however, between the soft X-ray cooling-flow problem and the classic cooling-flow problem is that the latter requires a clear explanation for why X-ray cooling does not appear to be carried to completion. |
It is difficult to find a relevant heating. mixing. or cooling time scale that would be comparable to the X-ray cooling time. | It is difficult to find a relevant heating, mixing, or cooling time scale that would be comparable to the X-ray cooling time. |
If the time scale for a given process is too short it will overwhelm the cooling-flow (n2 x). and if the time-scale is too long. it is dynamically unimportant (à= 0). | If the time scale for a given process is too short it will overwhelm the cooling-flow $\alpha=\infty$ ), and if the time-scale is too long, it is dynamically unimportant $\alpha=0$ ). |
Below. we discuss the proposed physical processes and whether they can account. both energetically and dynamically. for the missing soft X-ray luminosity. | Below, we discuss the proposed physical processes and whether they can account, both energetically and dynamically, for the missing soft X-ray luminosity. |
There are three requirements for additional heating mechanisms to be compatible with the observations. | There are three requirements for additional heating mechanisms to be compatible with the observations. |
First. the total time averaged heating power. <P>. has to roughly cancel the radiative losses so that the expression. is near unity. | First, the total time averaged heating power, $<$ $>$, has to roughly cancel the radiative losses so that the expression, is near unity. |
The denominator in the expression varies among the clusters in our sample by four orders of magnitude. so the proposed heating process must operate on a variety of scales. | The denominator in the expression varies among the clusters in our sample by four orders of magnitude, so the proposed heating process must operate on a variety of scales. |
Second. the heating has to be distributed spatially throughout the cooling-flow volume to cancel cooling everywhere. | Second, the heating has to be distributed spatially throughout the cooling-flow volume to cancel cooling everywhere. |
Third. the process has to be self-regulating. so that the time scale for heating remains comparable to the cooling time for all clusters. | Third, the process has to be self-regulating, so that the time scale for heating remains comparable to the cooling time for all clusters. |
Time-dependent AGN outflow heating models have been considered by à variety of authors (e.g. Rosner&Tucker 1989.. Tabor&Binney1993.. Churazoval. 2001.. Brüggen&Kaiser 2001.. Quilis 2001.. Davidetal. 2001.. Nulsen 2002)). | Time-dependent AGN outflow heating models have been considered by a variety of authors (e.g. \citealt{rosner}, , \citealt{tabor}, \citealt{churazov}, \citealt{brueggen}, \citealt{quilis}, \citealt{david}, \citealt{nulsen3}) ). |
Buoyant bubbles carrying relativistic plasma appear to be à common phenomena in clusters with central AGNs. | Buoyant bubbles carrying relativistic plasma appear to be a common phenomena in clusters with central AGNs. |
The thermal energy seems to be enough to heat cooling-flows through cosmic ray interactions and mechanical heating. but it is unclear whether this energy gets properly channeled into the cooling volume (e.g. Loewenstein.Zweibel.Begelman 1991.. Fabianetal. 2001)). | The thermal energy seems to be enough to heat cooling-flows through cosmic ray interactions and mechanical heating, but it is unclear whether this energy gets properly channeled into the cooling volume (e.g. \citealt{loewenstein}, \citealt{fabian3}) ). |
Note that it is essential that the heat be distributed evenly throughout the region which is thermally unstable. | Note that it is essential that the heat be distributed evenly throughout the region which is thermally unstable. |
In addition these models must be made self-regulating to counteract cooling at a rate proportional to the mass deposition rate. and with periods of heating roughly as long as periods of cooling. | In addition these models must be made self-regulating to counteract cooling at a rate proportional to the mass deposition rate, and with periods of heating roughly as long as periods of cooling. |
Clearly. it requires a significant degree of fine-tuning. | Clearly, it requires a significant degree of fine-tuning. |
There is considerable thermal energy in the outer regions of clusters that can destroy any existing cooling-flow through electron thermal conduction (e.g. Tucker&Rosner 1983.. Stewartetal. 1984.. Bertschinger&Meiksin 1986)). | There is considerable thermal energy in the outer regions of clusters that can destroy any existing cooling-flow through electron thermal conduction (e.g. \citealt{tucker}, \citealt{stewart}, \citealt{bertschinger}) ). |
The size of cooling-flows are only a few electron mean free paths in the absence of magnetic fields. | The size of cooling-flows are only a few electron mean free paths in the absence of magnetic fields. |
The critical question is to what level is conduction suppressed by tangled magnetic fields. an issue which continues to be debated theoretically (Chandran&Cowley1998.. Narayan&Medvedev 2001)). | The critical question is to what level is conduction suppressed by tangled magnetic fields, an issue which continues to be debated theoretically \citealt{chandran}, \citealt{narayan}) ). |
Observationally. conduction is suppressed by factors near 100 in identified cold fronts (Ettorietal.2002.. Markeviteh 2000.. Vikhlinin.Markevitch&Murray 2001)). Voigtetal.(2002).. Zakamska&N | Observationally, conduction is suppressed by factors near 100 in identified cold fronts \citealt{ettori}, \citealt{markevitch}, \citealt{vikhlinin}) ). \cite{voigt}, \cite{zakamska}, |
arayan (2001).. Fabian.Voigt.&Morris(2002) have demonstrated that the heat flow from the outer regions of clusters with a small suppression (> 107!) in the Spitzer conductivity would appear to cancel radiative losses 1n many clusters. | \cite{fabian5} have demonstrated that the heat flow from the outer regions of clusters with a small suppression $>$ $10^{-1}$ ) in the Spitzer conductivity would appear to cancel radiative losses in many clusters. |
The spatial distribution of the heating and overall energetic requirements appear to be satisfied by conduction models. but there is no explanation for why the cluster would cool to their current temperature distribution. | The spatial distribution of the heating and overall energetic requirements appear to be satisfied by conduction models, but there is no explanation for why the cluster would cool to their current temperature distribution. |
Since conduction suppresses temperature gradients by definition. this mechanism alone does not solve the dynamical problem presented here. | Since conduction suppresses temperature gradients by definition, this mechanism alone does not solve the dynamical problem presented here. |
Markevitch(2001) hàs show- that clusters previously thought to be fully relaxed. exhibi=a temperature fronts consistent with plasma exhibiting large bulk motions im the gravitational potential. | \cite{markevitch2} has shown that clusters previously thought to be fully relaxed exhibit temperature fronts consistent with plasma exhibiting large bulk motions in the gravitational potential. |
This provides another source of energy that has not been dissipated and therefore leads to an increase of the cooling time above previous estimates (Gomezetal.2002)). so that the time scales for radiative cooling and dynamical relaxation may be similar. | This provides another source of energy that has not been dissipated and therefore leads to an increase of the cooling time above previous estimates \citealt{gomez}) ), so that the time scales for radiative cooling and dynamical relaxation may be similar. |
There is no reason for this process to be self-regulating. however. and we would expect a much larger difference in the temperature distributions. depending on each cluster’s merger history. | There is no reason for this process to be self-regulating, however, and we would expect a much larger difference in the temperature distributions, depending on each cluster's merger history. |
This explanation also requires a conspiracy of factors to both cancel radiative cooling in global energetics. and ensure that the mergers occur with a frequency that allows some cooling. | This explanation also requires a conspiracy of factors to both cancel radiative cooling in global energetics, and ensure that the mergers occur with a frequency that allows some cooling. |
Future numerical simulations may test this further. and presumably observations of cooling-flows at a different epoch would not show the same effects. | Future numerical simulations may test this further, and presumably observations of cooling-flows at a different epoch would not show the same effects. |
The radiative isobaric cooling-flow model assumes that all of the thermal energy Is released in X-rays at high temperatures. | The radiative isobaric cooling-flow model assumes that all of the thermal energy is released in X-rays at high temperatures. |
There may. however. be additional contributions from other cooling processes. | There may, however, be additional contributions from other cooling processes. |
There are three main requirements. for additional cooling channels to explain. the observations. which are similar to. but slightly different from the heating requirements. | There are three main requirements for additional cooling channels to explain the observations, which are similar to, but slightly different from the heating requirements. |
The first ts that the total power in the coolant be comparable to the missing soft X-ray luminosity. | The first is that the total power in the coolant be comparable to the missing soft X-ray luminosity. |
Any coolant with power. Pooofun. reduces the total X-ray emission by a factor. The second requirement is that the ratio a needs to have a temperature dependence consistent with Equation (3) or that the cooling channel is self-regulating in the same sense as the discussed heating models. | Any coolant with power, $P_{coolant}$, reduces the total X-ray emission by a factor, The second requirement is that the ratio $\frac{P_{coolant}}{L_{x}}$ needs to have a temperature dependence consistent with Equation (3) or that the cooling channel is self-regulating in the same sense as the discussed heating models. |
The third requirement is that the energy should be released with a similar spatial distribution to the lowest temperature X-rays. | The third requirement is that the energy should be released with a similar spatial distribution to the lowest temperature X-rays. |
Begelman&Fabian(1990) and Fabianetal.(2001) discussed the possibility that hot electrons are cooled conductively by interfaces with cold clouds. | \cite{begelman} and \cite{fabian3} discussed the possibility that hot electrons are cooled conductively by interfaces with cold clouds. |
This leads to emission in the UV where the cooling function ts the highest. and is consistent energetically with the large observed Ha luminosities (Heckmanetal.1989.. Crawfordetal. 1999)) of 1077 to 1077 ergs/s in cooling-flows. | This leads to emission in the UV where the cooling function is the highest, and is consistent energetically with the large observed $\alpha$ luminosities \citealt{heckman}, \citealt{crawford}) ) of $10^{42}$ to $10^{44}$ ergs/s in cooling-flows. |
However. by itself. this model does not explain the observed temperature distribution in the soft X-ray band. since high temperature electrons are cooled by this process as well. | However, by itself, this model does not explain the observed temperature distribution in the soft X-ray band, since high temperature electrons are cooled by this process as well. |
In addition. in such a picture highly charged ions should also impact the cloud interfaces. resulting in charge exchange. which produces copious soft X- line emission. | In addition, in such a picture highly charged ions should also impact the cloud interfaces, resulting in charge exchange, which produces copious soft X-ray line emission. |
The spatial distribution of Ha is remarkably similar to the coolest X-rays (Ettorietal. 2002)) and the total luminosity ismarginally sufficient to account for the missing | The spatial distribution of $\alpha$ is remarkably similar to the coolest X-rays \citealt{ettori}) ) and the total luminosity ismarginally sufficient to account for the missing |
lack of detectable thermal emission from the disk. the best-fit power-law index (C=1.7) is consistent with recent Chandra and XMM-Newton results and similar to the low hard state observed in galactic BHBs (MeClintock Remillard 2004) The power-law spectrum of the ULX in NGC 3379 suggests that the emission is dominated by Compton up-scattering of soft disk photons in a optically thin corona. | lack of detectable thermal emission from the disk, the best-fit power-law index $\Gamma=1.7$ ) is consistent with recent Chandra and XMM-Newton results and similar to the low hard state observed in galactic BHBs (McClintock Remillard 2004) The power-law spectrum of the ULX in NGC 3379 suggests that the emission is dominated by Compton up-scattering of soft disk photons in a optically thin corona. |
The variability in the light curve could arise from a partial eclipse or absorption by cooler material in the outer parts of the accretion disk. | The variability in the light curve could arise from a partial eclipse or absorption by cooler material in the outer parts of the accretion disk. |
The slow rise and fall times m the light curve (see Fig. | The slow rise and fall times in the light curve (see Fig. |
8). along with the lack of any variation in hardness ratio. suggest that the intensity variation 1s due to a partial eclipse of the extended coronal emission with a period of 8-10 hours. | 8), along with the lack of any variation in hardness ratio, suggest that the intensity variation is due to a partial eclipse of the extended coronal emission with a period of 8-10 hours. |
While variability on the scales of months to years is well known for ULXs. there are only a few published cases of periodic. variability on the time scale of hours. all of which are in late-type galaxies (Sugiho et al. | While variability on the scales of months to years is well known for ULXs, there are only a few published cases of periodic variability on the time scale of hours, all of which are in late-type galaxies (Sugiho et al. |
2001. Cireinus; Bauer 2001: MS] Liu et al. | 2001, Circinus; Bauer 2001; M51 Liu et al. |
2002: NGC 628: Liu et al. | 2002; NGC 628; Liu et al. |
2005). | 2005). |
The ULX in NGC 3379 is the only known ULX in an early-type galaxy with possible periodic behavior in its light curve on the time scale of hours. | The ULX in NGC 3379 is the only known ULX in an early-type galaxy with possible periodic behavior in its light curve on the time scale of hours. |
Orbital periods of 8-10 hr are very common for LMXBs (Verbunt 1993). | Orbital periods of 8-10 hr are very common for LMXBs (Verbunt 1993). |
Assuming that the secondary star is filling its Roche lobe and transferring mass to the primary. we can estimate the mass of the secondary. | Assuming that the secondary star is filling its Roche lobe and transferring mass to the primary, we can estimate the mass of the secondary. |
Using the expression for the Roche lobe radius from Paezynski (1967) and Kepler's law gives P=89(R»/R.(M./M») hr (Verbunt 1993). | Using the expression for the Roche lobe radius from Paczynski (1967) and Kepler's law gives $P=8.9(R_2/R_{\odot})(\Mo / M_2)$ hr (Verbunt 1993). |
For main sequence stars (Ro/R.)=(ΜΜ.). indicating that if the period is 8-10 hr. then the secondary star is approximately a solar mass. | For main sequence stars $(R_2/R_{\odot}) = (M_2/ \Mo)$, indicating that if the period is 8-10 hr, then the secondary star is approximately a solar mass. |
The mass-radius relation for a He core burning star ora white dwarf predicts a much larger mass for the secondary which is unlikely in an early-type galaxy. | The mass-radius relation for a He core burning star or a white dwarf predicts a much larger mass for the secondary which is unlikely in an early-type galaxy. |
The Chandra observation of the intermediate luminosity elliptical galaxy NGC 3379 shows that only a small fraction of the gas shed by evolving stars still resides within the hot ISM. | The Chandra observation of the intermediate luminosity elliptical galaxy NGC 3379 shows that only a small fraction of the gas shed by evolving stars still resides within the hot ISM. |
A wavelet detection algorithm resolves of the emission within the central 5 kpe into point sources. | A wavelet detection algorithm resolves of the emission within the central 5 kpc into point sources. |
The luminosity function of the point sources detected at greater than 46 significance is consistent with that found for other ellipticals observed by Chandra (Kim Fabbiano 2004). | The luminosity function of the point sources detected at greater than $4 \sigma$ significance is consistent with that found for other ellipticals observed by Chandra (Kim Fabbiano 2004). |
Unlike other ellipticals observed by Chandra. only of the point sources are associated with globular clusters. which is comparable to the fraction of LMXBs in our galaxy. | Unlike other ellipticals observed by Chandra, only of the point sources are associated with globular clusters, which is comparable to the fraction of LMXBs in our galaxy. |
The low specific frequency of globular clusters and the low fraction of X-ray point sources associated with globulars clusters in NGC3379 is actually more similar to Chandra observations of SO galaxies rather than ellipticals (e.g.. Blanton. Sarazin Irwin 2001). | The low specific frequency of globular clusters and the low fraction of X-ray point sources associated with globulars clusters in NGC3379 is actually more similar to Chandra observations of S0 galaxies rather than ellipticals (e.g., Blanton, Sarazin Irwin 2001). |
Spectral analysis of the unresolved emission within. the central 15” (770 pe) indicates that of the emission probably arises from point sources with fluxes below the detection limit in the Chandra observation. | Spectral analysis of the unresolved emission within the central $15^{\prime\prime}$ (770 pc) indicates that of the emission probably arises from point sources with fluxes below the detection limit in the Chandra observation. |
If the luminosity function of the detected point sources is valid at lower luminosities. then the diffuse power-law emission can be accounted for by approximately 40 sources with luminosities below 3.0«1077 ergs s7!. | If the luminosity function of the detected point sources is valid at lower luminosities, then the diffuse power-law emission can be accounted for by approximately 40 sources with luminosities below $3.0 \times 10^{37}$ ergs $^{-1}$. |
The remaining of the unresolved emission from the central 770 pe is well described by thermal emission with kT=0.6 keV. Assuming a uniform eas density in this region gives a gas mass of 5«10°M.. which can be supplied by stellar mass loss in 107 years. | The remaining of the unresolved emission from the central 770 pc is well described by thermal emission with $kT=0.6$ keV. Assuming a uniform gas density in this region gives a gas mass of $5 \times 10^5 \Mo$, which can be supplied by stellar mass loss in $10^7$ years. |
Such a small amount of gas indicates that the stellar mass loss ts either being expelled from the central regions in a wind or ts being accreted by the central black hole. | Such a small amount of gas indicates that the stellar mass loss is either being expelled from the central regions in a wind or is being accreted by the central black hole. |
The X-ray luminosity of the central AGN is 8«10° eres s7!. | The X-ray luminosity of the central AGN is $8 \times 10^{38}$ ergs $^{-1}$. |
If gas is being acereted by the central black hole at either the Bondi accretion rate or mass cooling rate. then the radiative efficiency of the black hole must be 107°. as in the ADAP or RIAF models models (Narayan Yi 1994 and Yuan Narayan 1995). | If gas is being accreted by the central black hole at either the Bondi accretion rate or mass cooling rate, then the radiative efficiency of the black hole must be $~\sim 10^{-6}$, as in the ADAF or RIAF models models (Narayan Yi 1994 and Yuan Narayan 1995). |
If the gas is flowing out of the system in a wind. the energy outflow rate would be 5<10°? eres s7!. which is 4 orders of magnitude greater than the 1.4 GHz radio power of the AGN. | If the gas is flowing out of the system in a wind, the energy outflow rate would be $5 \times 10^{39}$ ergs $^{-1}$, which is 4 orders of magnitude greater than the 1.4 GHz radio power of the AGN. |
Such a large ratio between AGN mechanical and radio power is commonly found among cluster cooling flows with X-ray cavities and shocks (Birzan et al. | Such a large ratio between AGN mechanical and radio power is commonly found among cluster cooling flows with X-ray cavities and shocks (Birzan et al. |
2004: Nulsen et al. | 2004; Nulsen et al. |
200δα: Nulsen et al. | 2005a; Nulsen et al. |
2005b: MeNamara et al. | 2005b; McNamara et al. |
2005). | 2005). |
a" The most luminous source in NGC3379 ts located 360 pc from the central AGN with a peak luminosity of 3.5\IO? eres s7!. corresponding to the Eddington luminosity of a 30M. object. | The most luminous source in NGC3379 is located 360 pc from the central AGN with a peak luminosity of $3.5 \times 10^{39}$ ergs $^{-1}$, corresponding to the Eddington luminosity of a $30 \Mo$ object. |
The spectrum of this source ts well fitted with an absorbed power-law model with P=1.7. which is similar to other ULXs observed by Chandra and XMM-Newton and the low-hard state of galactic black hole binaries. | The spectrum of this source is well fitted with an absorbed power-law model with $\Gamma=1.7$, which is similar to other ULXs observed by Chandra and XMM-Newton and the low-hard state of galactic black hole binaries. |
Examining the archival ROSAT HRI observation of NGC 3379 shows that the ULX was at a comparable luminosity 5 years prior to the Chandra observation. | Examining the archival ROSAT HRI observation of NGC 3379 shows that the ULX was at a comparable luminosity 5 years prior to the Chandra observation. |
The long term stability of the ULX may pose a problem for the micro-quasar interpretation of ULXs in early-type galaxies. | The long term stability of the ULX may pose a problem for the micro-quasar interpretation of ULXs in early-type galaxies. |
The light curve of the ULX in NGC 3379 varies smoothly by a factor of two during the Chandra observation. | The light curve of the ULX in NGC 3379 varies smoothly by a factor of two during the Chandra observation. |
The slow rise and fall times in the light curve and the consistency of the power-law spectrum during the observation all suggest that the ULX is undergoing a partial eclipse of the extended corona surrounding an accretion disk with a period of 8-10 hr. | The slow rise and fall times in the light curve and the consistency of the power-law spectrum during the observation all suggest that the ULX is undergoing a partial eclipse of the extended corona surrounding an accretion disk with a period of 8-10 hr. |
Assuming the secondary ts a main sequence star filling its Roche lobe gives a mass for the secondary of approximately IM.. | Assuming the secondary is a main sequence star filling its Roche lobe gives a mass for the secondary of approximately $1 \Mo$. |
Variability has been observed in other ULXs. but the ULX in NGC 3379 is the only ULX in an elliptical galaxy with possible periodic behavior. | Variability has been observed in other ULXs, but the ULX in NGC 3379 is the only ULX in an elliptical galaxy with possible periodic behavior. |
Due to the high surface brightness density of sources in the central 1 kpe of NGC 3379. only a long Chandra observation can determine if the lightcurve of the ULX ts truly periodic. | Due to the high surface brightness density of sources in the central 1 kpc of NGC 3379, only a long Chandra observation can determine if the lightcurve of the ULX is truly periodic. |
In simulatioi projects on galaxies 1t is customary to run a relatively large number of simulations to inter-compare and understand the effect of various. parameters. | In simulation projects on galaxies it is customary to run a relatively large number of simulations to inter-compare and understand the effect of various parameters. |
Thus. creating the initial conditions can be a considerable part of the work and it makes sense to streamline it. | Thus, creating the initial conditions can be a considerable part of the work and it makes sense to streamline it. |
In particular. the amount of CPU involved depends on the number of iteration steps made. | In particular, the amount of CPU involved depends on the number of iteration steps made. |
This number should be sufficiently large. so that the iteration procedure can converge. but not excessively large. so as not to needlessly waste time. | This number should be sufficiently large, so that the iteration procedure can converge, but not excessively large, so as not to needlessly waste time. |
It is thus necessary to be able to assess whether the iteration has converged or not. | It is thus necessary to be able to assess whether the iteration has converged or not. |
The most straightforward way is of course to plot the evolution in time of various radial profiles (such as the density. the mean velocities and dispersions ete) and check by eye whether the variation between the two last iteration times Is sufficiently small. | The most straightforward way is of course to plot the evolution in time of various radial profiles (such as the density, the mean velocities and dispersions etc) and check by eye whether the variation between the two last iteration times is sufficiently small. |
This. however. can be very tedious. particularly if it is carried out a number of times for each initial conditions. | This, however, can be very tedious, particularly if it is carried out a number of times for each initial conditions. |
It is thus useful to prepare tools that can give information on whether a rough convergence has been achieved. before starting the visual examination. | It is thus useful to prepare tools that can give information on whether a rough convergence has been achieved, before starting the visual examination. |
In this appendix we will describe how this can be carried out in. practice. | In this appendix we will describe how this can be carried out in practice. |
We aim to compare the system in the beginning and in the end of a short-term evolution during a single iterative step. | We aim to compare the system in the beginning and in the end of a short-term evolution during a single iterative step. |
So we need tools to compare two N-body models (in this context. a gaseous disk consisting of SPH particles can also be considered as an N-body model). | So we need tools to compare two $N$ -body models (in this context, a gaseous disk consisting of SPH particles can also be considered as an $N$ -body model). |
We note that the following algorithm ts fairly similar to a test of the statistical hypothesis that two N-body systems are just two random realizations of the same distribution function (hereafter DF). | We note that the following algorithm is fairly similar to a test of the statistical hypothesis that two $N$ -body systems are just two random realizations of the same distribution function (hereafter DF). |
Our algorithm is based on comparisons of profiles of different quantities. | Our algorithm is based on comparisons of profiles of different quantities. |
We wish to compare profiles of some quantity Q along some axis A for both systems. | We wish to compare profiles of some quantity $Q$ along some axis $A$ for both systems. |
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