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Alternatively. given the present values of r aud 0. one may solve for the ratio Ep. or equivalently Lf where vis the ambient number density. aud compare this with the value estimatedusing the X-ray spectral modeling aud Tea surface brightness techuiqucs as described above. | Alternatively, given the present values of $r$ and $u$, one may solve for the ratio $E/\rho$, or equivalently $E/n$ where $n$ is the ambient number density, and compare this with the value estimatedusing the X-ray spectral modeling and $\alpha$ surface brightness techniques as described above. |
sshows the simplest structure of the three reninaunts presented here. | shows the simplest structure of the three remnants presented here. |
Its well defined sshell smrounds a smoothelliptical distribution of soft N-rav. emission with radii of 1:14«0.757 or 19.1.«11.25 pe. | Its well defined shell surrounds a smoothelliptical distribution of soft X-ray emission with radii of $1.3'\times0.75'$ or $19.4\times11.25$ pc. |
The shell has typical values for oof ~ 0.6. as opposed to πιαπα values ~0.2 elsewhere. | The shell has typical values for of $\sim0.6$ , as opposed to maximum values $\sim0.2$ elsewhere. |
The bright eenissiou to the cast of | The bright emission to the east of |
1992))). | ). |
Llere. we want briellv to analyze the biases and the errors due to the use of an “incorrect” parametric mocel to fit the observable data. | Here, we want briefly to analyze the biases and the errors due to the use of an “incorrect” parametric model to fit the observable data. |
A wav to face this important question consists in to build simulated. svstems for a lens model and then to fit the image positions and time delays eencrated by it. using an other functional form. studving the change in the lens parameters ancl above all in . | A way to face this important question consists in to build simulated systems for a lens model and then to fit the image positions and time delays generated by it, using an other functional form, studying the change in the lens parameters and above all in $h$. |
We can furnish a qualitative estimate of he effect. of he model depen:ence on Z£o. | We can furnish a qualitative estimate of the effect of the model dependence on $H_0$. |
At the same time the procedure allows to quantify this "systeniatic errors. | At the same time the procedure allows to quantify this “systematic” errors. |
)v means of a simulated system built using t1e LEubble model. we fit to the image positions and the time cdelavs so obtained the and2. | By means of a simulated system built using the Hubble model, we fit to the image positions and the time delays so obtained the and. |
The analysis of the most significant. parameters shows interesing trends which are partially already well Known. | The analysis of the most significant parameters shows interesting trends which are partially already well known. |
In particular. in our simulated system f=0.7. and the fitting of the other models gives us the means values =0.35 for the anc h0.46 for the3. respectively with a percentage change of 50% and 344. | In particular, in our simulated system $h=0.7$, and the fitting of the other models gives us the means values $h=0.35$ for the and $h=0.46$ for the, respectively with a percentage change of $50\%$ and $34\%$. |
This percentage obviously. changes if we simulate other lens svstenms. | This percentage obviously changes if we simulate other lens systems. |
This shows that if the "correct? model for a lens is one with constant mass-to-light ratio and it tries to shape it with a separable model we obtain a lower estimate of 5 than that obtained with the first one: this veriies some previous results in Ditcrature (see (Ixochanek202) for an analysis mace on Leal systems). | This shows that if the “correct” model for a lens is one with constant mass-to-light ratio and it tries to shape it with a separable model we obtain a lower estimate of $h$ than that obtained with the first one; this verifies some previous results in literature (see \cite{Koch02} for an analysis made on real systems). |
Similar trends are obtained if we create a simuaded system. using a de Vatσυ(ος model. | Similar trends are obtained if we create a simulated system using a de Vaucouleurs model. |
Leis also possib eto analyze he uncertainies introduced. by the lack of the internal ellipticitv of t1e lens galaxy. | It is also possible to analyze the uncertainties introduced by the lack of the internal ellipticity of the lens galaxy. |
IHE we try to fit wih the the observades generated with the2 we obtain a ower mean / rut i£ we consider the errors this «stinate is in agreement wih the simulated: value. instead t10 estimated value for à raises. | If we try to fit with the the observables generated with the we obtain a lower mean $h$ but if we consider the errors this estimate is in agreement with the simulated value, instead the estimated value for $\alpha$ raises. |
ὃν means of simulated systems we can also obtain statistical correlations anions paranietCrs in order to | By means of simulated systems we can also obtain statistical correlations among parameters in order to |
General Relativity predicts redshift of photons due to a gravitational field. | ] General Relativity predicts redshift of photons due to a gravitational field. |
When a photon with wavelength Ais cluitted iu a gravitational potential &. it will lose energv when it climbs up in the eravitational field aud will consequently be redshitted. | When a photon with wavelength $\lambda$ is emitted in a gravitational potential $\Phi$, it will lose energy when it climbs up in the gravitational field and will consequently be redshifted. |
The redshift observed at infinity is given in the weak field linüt by: where AA. Ad are respectively the difference in wavelength. and difference im potential between where the photon is cluitted and where it is observed. | The redshift observed at infinity is given in the weak field limit by: where $\Delta \lambda$, $\Delta \Phi$ are respectively the difference in wavelength, and difference in potential between where the photon is emitted and where it is observed. |
If we consider galaxies as sources of the photons. the eravitational redshift effect is so tiny that we take it for eranted that a measurement of the total galaxy redshift can be asstuned to be the stun of Thibble expansion and peculiar velocities. | If we consider galaxies as sources of the photons, the gravitational redshift effect is so tiny that we take it for granted that a measurement of the total galaxy redshift can be assumed to be the sum of Hubble expansion and peculiar velocities. |
In this paper we examine whether this is always the case. aud in particular whether galaxies im galaxy clusters could have measurable values of τρ. | In this paper we examine whether this is always the case, and in particular whether galaxies in galaxy clusters could have measurable values of $z_{g}$. |
Since the eravitational potential depends ou the mass distribution around galaxies. the eravitational redshift. if observable. should be most evident iu dense environments. | Since the gravitational potential depends on the mass distribution around galaxies, the gravitational redshift, if observable, should be most evident in dense environments. |
Iu an early study by Nottale (1976). the redshitt difference between pairs of clusters was compared to the richness difference, | In an early study by Nottale (1976), the redshift difference between pairs of clusters was compared to the richness difference. |
À supposed strong effect was found. with the pair ου. of higher richness having a svstcmatically lnieh redshift. | A supposed strong effect was found, with the pair member of higher richness having a systematically high redshift. |
ILlowever. when Rood Struble (1982) rexanuned this with a larger sample. their result showed no such correlation. | However, when Rood Struble (1982) rexamined this with a larger sample, their result showed no such correlation. |
Nottale (19900). discussed. that. the effect should be looked for in galaxies at the centers of ealaxv clusters. by comparing them redshifts with those of galaxies at the cluster edges. | Nottale (1990) discussed that the effect should be looked for in galaxies at the centers of galaxy clusters, by comparing their redshifts with those of galaxies at the cluster edges. |
Stiavelli Setti (1993) carried out a related test in iudividual elliptical ealaxies. finding at 99.9% confidence that elliptical galaxy cores are redshifted with respect to the galaxw outer regions. explaining this as a result of eravitational redshift. | Stiavelli Setti (1993) carried out a related test in individual elliptical galaxies, finding at $99.9\%$ confidence that elliptical galaxy cores are redshifted with respect to the galaxy outer regions, explaining this as a result of gravitational redshift. |
The study of gravitational redshifts in galaxy clusters was taken further by Cappi (1995). who modelled clusters using differcut density profiles iucludiug a de Vaucouleurs law. | The study of gravitational redshifts in galaxy clusters was taken further by Cappi (1995), who modelled clusters using different density profiles including a de Vaucouleurs law. |
It was predicted that the eravitational redshift is non-ueelieible in very rich clusters. | It was predicted that the gravitational redshift is non-negligible in very rich clusters. |
For example. the centers of clusters of masses 10195.EXD. should be vedshifted by as nich as 300nis with respect to infinity. | For example, the centers of clusters of masses $10^{16} \msun$ should be redshifted by as much as $300 \kms$ with respect to infinity. |
Broadlurst Scannapieco (2000) modelled the effect using a Navarro Freuk White (1997) (hereafter NEW) density profile. and sugeested that the gravitational redshift of ictal lunes iu the cluster eas could eventually be used to nap out the potential directly. | Broadhurst Scannapieco (2000) modelled the effect using a Navarro Frenk White (1997) (hereafter NFW) density profile, and suggested that the gravitational redshift of metal lines in the cluster gas could eventually be used to map out the potential directly. |
As the gravitational redshitt is sensitive to the distribution of mass in the mnerost regions of clusters. it could be used as à probe to coustrain the amount of dark matter there. | As the gravitational redshift is sensitive to the distribution of mass in the innermost regions of clusters, it could be used as a probe to constrain the amount of dark matter there. |
Ciavitational redshifts would provide complinentary information to eravitational lensing (e... Sand 2003). as uulike leusiug they do uot depend on the mass deusity projected along the line of sight. | Gravitational redshifts would provide complimentary information to gravitational lensing (e.g., Sand 2003), as unlike lensing they do not depend on the mass density projected along the line of sight. |
Tere we use an N-body saulation mace publically available by the Virgo Consortimu (Freak et al. | Here we use an $N$ -body simulation made publically available by the Virgo Consortium (Frenk et al. |
2000) to estimate the magnitude of the effect of gravitational redshifts on ealaxy clusters in a ACDAL universe. | 2000) to estimate the magnitude of the effect of gravitational redshifts on galaxy clusters in a $\rm{\Lambda CDM}$ universe. |
We examine possible observational strategies and determine if ealaxv eravitational redshifts could be detected with a reasonable uunuber of clusters. | We examine possible observational strategies and determine if galaxy gravitational redshifts could be detected with a reasonable number of clusters. |
By using tle umucrical simulation. we will be able to study the effect of substructure in the density aud the potentially complex velocity field of realistic clusters. | By using the numerical simulation, we will be able to study the effect of substructure in the density and the potentially complex velocity field of realistic clusters. |
We will see if Nottale’s suggestion of measuring the difference iu eravitational redshift between the ceutral galaxy aud galaxies at the οσο of the cluster is realizable iu practice. | We will see if Nottale's suggestion of measuring the difference in gravitational redshift between the central galaxy and galaxies at the edge of the cluster is realizable in practice. |
The poteutial wells should be deeper for the most massive clusters. which mieaus Ilurger eravitational redshitts. | The potential wells should be deeper for the most massive clusters, which means larger gravitational redshifts. |
Towever. massive clusters are rare. | However, massive clusters are rare, |
REFERENCES Riess. A.aL.. 2001. ApJ.. 560. 49-71. “ | REFERENCES Riess, A. 2001, 560, 49-71. “ |
The Farthest Known Supernova: Support for an Accelerating Universe and a Glimpse of the Epoch of Deceleration™” quote [rom abstract: "It is inconsistent with erev dust or simple luminosity - Regos. E.. Tout. C.. Wickramasinghe. D.. Hurley. J. Pols. O. 2001. astro-ph/0112355. "Could Edge-Lit Type Ia Supernovae be Standard Candles" quote from abstract: "we find à svstematie shift in Chis relation that would make distant SNe Ia fainter than those nearby" Perlmutter. S. 11999.ÀpJ.. 517. 565-586. | The Farthest Known Supernova: Support for an Accelerating Universe and a Glimpse of the Epoch of Deceleration” quote from abstract: “It is inconsistent with grey dust or simple luminosity ' Regos, E., Tout, C., Wickramasinghe, D., Hurley, J. Pols, O. 2001, astro-ph/0112355, “Could Edge-Lit Type Ia Supernovae be Standard Candles” quote from abstract: “we find a systematic shift in this relation that would make distant SNe Ia fainter than those nearby” Perlmutter, S. 1999, 517, 565-586. |
deusities exist iu the outer regions of these disks εrere ds stil sjenificaut amounts of gas cing transported ot js region (see Figure 5)). | densities exist in the outer regions of these disks there is still significant amounts of gas being transported to this region (see Figure \ref{fig:mass_transfer}) ). |
ενis that solids :we carried along with the eas as] is transported outward. there is sufficient hass present for saollie formation. | Assuming that solids are carried along with the gas as it is transported outward, there is sufficient mass present for satellite formation. |
Caven a solar abuudaice of solids. wὉ calculated that there is rearly wice as mich nass in solids transported outwar¢ over a 10" vr ine period than is needed o fonu Callisto. | Given a solar abundance of solids, we calculated that there is nearly twice as much mass in solids transported outward over a $10^5$ yr time period than is needed to form Callisto. |
There are reasons το belicve hat the solids-to-gas mass ratio would be higher han soar and therefore we take lis value as an tunerCRuate. | There are reasons to believe that the solids-to-gas mass ratio would be higher than solar and therefore we take this value as an underestimate. |
We plan to investigate the actual clistyinion and transport of solids in the near "ture wath a more comprehensive model that is currently in developiieut. | We plan to investigate the actual distribution and transport of solids in the near future with a more comprehensive model that is currently in development. |
Another aspect of our simulations that may Lave COlsCQ(quenceos oli satellite formation is the density cuhancement seen i our sinulations at ~25F» the location of peak mass infall. | Another aspect of our simulations that may have consequences on satellite formation is the density enhancement seen in our simulations at $\sim 25\ r_{\rm p}$, the location of peak mass infall. |
A simular density cauhaucement was secu in ?.. but was nof preseut in more recent simulations wuc1 include radiative trauster (7?).. | A similar density enhancement was seen in \citet{machida08}, but was not present in more recent simulations which include radiative transfer \citep{ayliffe09}. |
Tt is unclear at this time if these density euhlhaucenienuts woukl be present in a more realistic model in whi11011 the viscosity was determined locally, | It is unclear at this time if these density enhancements would be present in a more realistic model in which the viscosity was determined locally. |
This is an interesting question aud one which we plan to vesieate in the near future. | This is an interesting question and one which we plan to investigate in the near future. |
If it is real. fLOSC chhawcelments would have a siguificaut impact on satelite formation. | If it is real, these enhancements would have a significant impact on satellite formation. |
Density eulianceiments suc las these are accolupanied by pressure maxima. | Density enhancements such as these are accompanied by pressure maxima. |
It has been shown that migrating solids cau be rapped Histch pressure niaxinia and rapidly erow iuto satelitesimials (7). | It has been shown that migrating solids can be trapped in such pressure maxima and rapidly grow into satellitesimals \citep{kretke09}. |
Iu an effort to test what effect the ocation at which iufalliug material intersects ιο disk las ou steady-state disk morphology. we have yerformecd oιο test sinuulatiou ia which 1e peak of the infalliue material occured at 35+, rather han at 25+» | In an effort to test what effect the location at which infalling material intersects the disk has on steady-state disk morphology, we have performed one test simulation in which the peak of the infalling material occurred at $35\ r_{\rm p}$ rather than at $25\ r_{\rm p}$. |
In the test simulation. the ocation of the disk outer edee was shifted farther out aud the otal disk iuass ducreased bv | In the test simulation, the location of the disk outer edge was shifted farther out by $\sim 3 \%$ and the total disk mass increased by $\sim 5\%$. |
The changes are a result of a greater Traction of he iufaling mass being traisported outward ratjor thaji inward. | The changes are a result of a greater fraction of the infalling mass being transported outward rather than inward. |
This test indicates hat it is nuyortant to ideutifv the exact location at which the iufalΠιο παν» accretes onto the circunplaueary clisk. | This test indicates that it is important to identify the exact location at which the infalling mass accretes onto the circumplanetary disk. |
Ilowever. the current 3-D ivdrodyvuauuical παπαΊος used to mocel imfall rou the solar ueula onto cireiunplauetarv disks have insufficient resolutio1 to ideutifv the location exactly. | However, the current 3-D hydrodynamical simulations used to model infall from the solar nebula onto circumplanetary disks have insufficient resolution to identify the location exactly. |
We are satisfiec that our treatment is πιaficient for this stidy. vet we plan on inchiding more precise results as they become available. | We are satisfied that our treatment is sufficient for this study, yet we plan on including more precise results as they become available. |
While the streugho | fhe viscosity. plavs uo 1e iun the location of: v clisks outer bouudary. it does play a siguificau role in the toal niss COtained iu a given disk. | While the strength of the viscosity plays no role in the location of a disks outer boundary, it does play a significant role in the total mass contained in a given disk. |
In a steady sate. the mass accretion rate. ALXEM. 1 coustaut. | In a steady state, the mass accretion rate, $\dot{M}\propto \nu \Sigma$, is constant. |
A Cistant mass accretion rate implies tlat he nass surface density mst be proportional to the inverse of the viscosity. | A constant mass accretion rate implies that the mass surface density must be proportional to the inverse of the viscosity. |
Iu our models. this niens that the surface density is nwerscly proportional to the viscosity. paraieteor. o. | In our models, this means that the surface density is inversely proportional to the viscosity parameter, $\alpha$. |
Oue would naivelv assume that a arecr surface deusitv would result in larger satellites. but in actuality he opposite is true due to he increased rate of mügration. | One would naively assume that a larger surface density would result in larger satellites, but in actuality the opposite is true due to the increased rate of migration. |
A MOLE Massive. ower viscosity disk results in a less luassive satellite system. | A more massive, lower viscosity disk results in a less massive satellite system. |
thud that satellites will ouly survive against type I uueration for values of a>107. | \citet{canup02}f find that satellites will only survive against type I migration for values of $\alpha \geq 10^{-3}$ |
parameters simultaneously. which would probably become prohibitive first.) | parameters simultaneously, which would probably become prohibitive first.) |
The DoF issue can be addressed if the DDE in question can be represented by a parametrized model for E. | The DoF issue can be addressed if the DDE in question can be represented by a parametrized model for $\jones{E}{p}$. |
We can then solve for the parameters of that model (presumably. few in number). and then correct for the resulting £, estimate using one of the methods of Sect. 2.3.. | We can then solve for the parameters of that model (presumably, few in number), and then correct for the resulting $\jones{E}{p}$ estimate using one of the methods of Sect. \ref{sec:dde-correction}. |
A number of approaches have shown that this is feasible. | A number of approaches have shown that this is feasible. |
For the tonosphere. the (FBC) method of ? uses the position offsets of sources (1n individual snapshot images) to fit a global phase screen over the array. | For the ionosphere, the (FBC) method of \citet{Cotton:FBC} uses the position offsets of sources (in individual snapshot images) to fit a global phase screen over the array. |
The (SPAM) algorithm of ? does a similar fit to phase solutions obtained via peeling (in AIPS). | The (SPAM) algorithm of \citet{Intema:SPAM} does a similar fit to phase solutions obtained via peeling (in AIPS). |
Both methods show how to work around the limitations of 2GC packages: since direct fits to visibilities are impossible in the framework of the latter. especially without a fully-fledged RIME. they rely on standard calibration methods (including peeling). and fit a model to the of calibration. | Both methods show how to work around the limitations of 2GC packages: since direct fits to visibilities are impossible in the framework of the latter, especially without a fully-fledged RIME, they rely on standard calibration methods (including peeling), and fit a model to the of calibration. |
? have demonstrated a similar approach for E-Jones. using source fluxes to fit the FWHM parameter of the ATA beam. | \citet{Hull:ata-beam-fitting} have demonstrated a similar approach for $E$ -Jones, using source fluxes to fit the FWHM parameter of the ATA beam. |
Given an explicit RIME. it should be possible to fit parametrized models directly to the observed visibilities. | Given an explicit RIME, it should be possible to fit parametrized models directly to the observed visibilities. |
The (MIM) approach proposed by Noordam is similar to FBC and SPAM. in that it purports to fit a smooth model for ionospheric phase. but is different in that it uses visibilities (but also other sources of data. such as GPS measurements). | The (MIM) approach proposed by Noordam is similar to FBC and SPAM, in that it purports to fit a smooth model for ionospheric phase, but is different in that it uses visibilities (but also other sources of data, such as GPS measurements). |
This requires a software system where explicit RIMEs may be implemented. and so cannot be adapted to 2GC packages. but it has been demonstrated in the LOFAR BBS system. using a simple linear-slope MIM. | This requires a software system where explicit RIMEs may be implemented, and so cannot be adapted to 2GC packages, but it has been demonstrated in the LOFAR BBS system, using a simple linear-slope MIM. |
The pointing selfcal method (?) already mentioned above is an application of the same approach to pointing errors, | The pointing selfcal method \citep{SB:pointing} already mentioned above is an application of the same approach to pointing errors. |
All these methods have the common feature of relying onsources.. that 1s. having enough sources in the field to constrain the solutions. | All these methods have the common feature of relying on, that is, having enough sources in the field to constrain the solutions. |
The availability of a sufficient number of beacons ts a crucial question for the calibratability of future instruments. | The availability of a sufficient number of beacons is a crucial question for the calibratability of future instruments. |
[ will return to this in the conclusion to Paper III (?).. after the results presented therein have been considered. | I will return to this in the conclusion to Paper III \citep{RRIME3}, after the results presented therein have been considered. |
Note that. just as in the DFT-vs.-FFT debate discussed in Sect. 2.3.3... | Note that, just as in the DFT-vs.-FFT debate discussed in Sect. \ref{sec:subtraction-uv-plane}, |
there is a related dichotomy between the parametrized model approach. and methods based on direction-dependent solutions (peeling. differential gains). | there is a related dichotomy between the parametrized model approach, and methods based on direction-dependent solutions (peeling, differential gains). |
The latter methods the use of DFTs at the predict stage. since the FFT approach (AW-projection) cannot be applied without a model of £,,(/) for the entire field. | The latter methods the use of DFTs at the predict stage, since the FFT approach (AW-projection) cannot be applied without a model of $\jones{E}{p}(\vec l)$ for the entire field. |
Parametrized models. on the other hand. may be applied both via DFT and FFT. | Parametrized models, on the other hand, may be applied both via DFT and FFT. |
Once again. [ suggest that the two approaches should be treated as complementary. | Once again, I suggest that the two approaches should be treated as complementary. |
Looking ahead. the results of Paper HI (2). will show that brighter off-axis sources exhibit all sorts of complicated structure in their AZ, solutions. even in the relatively uncomplicated (1.6. low-DDE) case of WSRT 21 em observations. | Looking ahead, the results of Paper III \citep{RRIME3} will show that brighter off-axis sources exhibit all sorts of complicated structure in their $\Delta\jones{E}{p}$ solutions, even in the relatively uncomplicated (i.e. low-DDE) case of WSRT 21 cm observations. |
It is hard to see how this can be captured by a parametrized DDE model to a precision sufficient for error-free subtraction of such sources. | It is hard to see how this can be captured by a parametrized DDE model to a precision sufficient for error-free subtraction of such sources. |
This suggests a similar trade-off in accuracy vs. computing cost as that deseribed in Sect. 2.3.3.. | This suggests a similar trade-off in accuracy vs. computing cost as that described in Sect. \ref{sec:subtraction-uv-plane}, |
leading to the following hybrid approach for dealing with DDEs: Note that the sets of sources involved at steps 2. 3 and 4 are conceptually similar to "Cat I and "Cat II sources proposed for LOFAR calibration (?).. but here I suggest three sets rather than two. | leading to the following hybrid approach for dealing with DDEs: Note that the sets of sources involved at steps 2, 3 and 4 are conceptually similar to “Cat I” and “Cat II” sources proposed for LOFAR calibration \citep{JEN:LOFAR3}, but here I suggest three sets rather than two. |
The exact partitioning of sources into sets determines the accuracy vs. computing cost trade-off. | The exact partitioning of sources into sets determines the accuracy vs. computing cost trade-off. |
It may be interesting to compare the different approaches to a particular class of DDE. for instance pointing error. | It may be interesting to compare the different approaches to a particular class of DDE, for instance pointing error. |
Pointing errors introduce an E-Jones às given by Eq. (9)). | Pointing errors introduce an $E$ -Jones as given by Eq. \ref{eq:mispointing}) ). |
To date. three relevant approaches have been proposed: pointing selfcal (?).. peeling (Sect. 2.4.2)) | To date, three relevant approaches have been proposed: pointing selfcal \citep{SB:pointing}, peeling (Sect. \ref{sec:peeling}) ) |
and differential gains (Sect. 2.4.3)). | and differential gains (Sect. \ref{sec:dEs}) ). |
Of these. peeling is by far the best tested. since it Is available with all 2GC software packages. | Of these, peeling is by far the best tested, since it is available with all 2GC software packages. |
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