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The only pieces of information that the model requires are the initial conditions of the orbiting galaxy (its gas and dark matter profiles), the density profile of the ICM and the orbit [the latter two are needed to calculate P,am(t)].
The only pieces of information that the model requires are the initial conditions of the orbiting galaxy (its gas and dark matter profiles), the density profile of the ICM and the orbit [the latter two are needed to calculate $P_{\rm ram}(t)$ ].
The model contains two tunable coefficients that are of order unity.
The model contains two tunable coefficients that are of order unity.
Fixing these coefficients to match the stripping in just one of our idealised uniform medium simulations (see §3.2) leads to excellent agreement with all our other simulations.
Fixing these coefficients to match the stripping in just one of our idealised uniform medium simulations (see 3.2) leads to excellent agreement with all our other simulations.
With the exception of cases where the mass of the galaxy is greater than about of the mass of the group and its orbit is highly non-radial, the analytic model reproduces the mass loss in the simulations to &10% accuracy at all times and for all the orbits, galaxy masses, and galaxy concentrations that we have explored.
With the exception of cases where the mass of the galaxy is greater than about of the mass of the group and its orbit is highly non-radial, the analytic model reproduces the mass loss in the simulations to $\approx 10\%$ accuracy at all times and for all the orbits, galaxy masses, and galaxy concentrations that we have explored.
For cases where the mass of the galaxy exceeds of the mass of the group, it will likely be necessary tofactor in the
For cases where the mass of the galaxy exceeds of the mass of the group, it will likely be necessary tofactor in the
JJ0632|057 was observed for 6 hours at GGllz with the Verv Large (VLA) in D-configuration (angular resolution +147) during July September 2008 (program AS9044).
J0632+057 was observed for 6 hours at GHz with the Very Large (VLA) in D-configuration (angular resolution $\approx$ $''$ ) during July – September 2008 (program AS944).
The time was divided into three 2hhour observations separated by ο]. month.
The time was divided into three hour observations separated by $\sim$ 1 month.
The data were calibrated. using the NILAO AIPS (AstronomicalImageProcessingSystem:ο) software package and then loaded into citepDifmap for additional editing and imaging.
The data were calibrated using the NRAO AIPS \citep[Astronomical Image Processing System;][]{AIPS} software package and then loaded into \\citep{Difmap} for additional editing and imaging.
The flux density scale was set using a scan of 1147 at the end of each observation and the phase was monitored. with scans of the calibration source 0632|103.
The flux density scale was set using a scan of 147 at the end of each observation and the phase was monitored with scans of the calibration source 0632+103.
The oll-source rms in each observation (0.03 mmJIv/beam) was estimated from large sourceless boxes. far from the phase centre and was found to be close to the thermal noise limit for a 2 hour observation of z:0.03 mm.v/beam.
The off-source rms in each observation (0.03 – mJy/beam) was estimated from large sourceless boxes, far from the phase centre and was found to be close to the thermal noise limit for a 2 hour observation of $\approx$ mJy/beam.
Five unresolved: sources were found. above So within the primary beam radius of the best fit position of J.J0632|057.
Five unresolved sources were found above $\sigma$ within the $'$ primary beam radius of the best fit position of J0632+057.
Emission is detected from this source in all three observations and the measured [ux varies significantly from observation to observation: from 0.19+0.04 mindy to O.41+40.04 mium] (C clof for a constant Gt = 19.472. chance probability — 10.7). see Fig.
Emission is detected from this source in all three observations and the measured flux varies significantly from observation to observation: from $0.19\pm0.04$ mJy to $0.41\pm0.04$ mJy $\chi^2$ /dof for a constant fit = 19.4/2, chance probability = $\times$ $^{-5}$ ), see Fig.
3.
3.
Additionally. since each 2hhour observation consisted. of four mminute scans of the region surrouncling .JJ0632|057. we searched [or shorter timescale (intrahour) variability.
Additionally, since each hour observation consisted of four minute scans of the region surrounding J0632+057, we searched for shorter timescale (intrahour) variability.
No evidence was found for short-term (~Lhhour) variability of source 4233 in the scan-by-scan light-curves.
No evidence was found for short-term $\sim$ hour) variability of source 3 in the scan-by-scan light-curves.
Source 233 was moclelled with an elliptical 2D Gaussian in the map plane resulting in a size of 7" hy 4" (10). consistent with an unresolved source.
Source 3 was modelled with an elliptical 2D Gaussian in the map plane resulting in a size of $''$ by $''$ $\sigma$ ), consistent with an unresolved source.
A further 3hhour observation at 5 and Ss.5CGCGllIz with the VLA in the high resolution A-configuration (program. ASOGT) was taken in October 2008.
A further hour observation at 5 and GHz with the VLA in the high resolution A-configuration (program AS967) was taken in October 2008.
ILowever. source 333 was not detected. during this observation. presumably due to a low flux state of this object.
However, source 3 was not detected during this observation, presumably due to a low flux state of this object.
| plausible alternative reason for the non-detection is that the source is extended on scales significantly larger. than the 0.47. beam (but smaller than the 2" GGLIz GAIRY beam). reducing the signal/noise achievable in this configuration.
A plausible alternative reason for the non-detection is that the source is extended on scales significantly larger than the $0.4''$ beam (but smaller than the $''$ GHz GMRT beam), reducing the signal/noise achievable in this configuration.
To circumvent this issue. the A configuration data were tapered to match the 27 L28GCOlIz CARP bean and rms point source limits at 5 and SCGCGllIz were measured. and. plotted in Fig.
To circumvent this issue, the A configuration data were tapered to match the $''$ GHz GMRT beam and rms point source limits at 5 and GHz were measured and plotted in Fig.
3.
3.
All four of the field. sources detected. during the D configuration observations were also present in the A configuration observations.
All four of the field sources detected during the D configuration observations were also present in the A configuration observations.
One of the field sources (source #11) shows significant variability both on the ~month separation timescale of these observations and on shorter (~hour) timescales.
One of the field sources (source 1) shows significant variability both on the $\sim$ month separation timescale of these observations and on shorter $\sim$ hour) timescales.
This source is located at 06h33m06.59s + 0.37. 05d53/39"724 + 0.3". and is visible both in the NWSS archive (14Cllz.7) witha flux of ~15mmJv (ef —12mmJv in the GAIRT 1.28Ciz observations) and in the NMM-Neywton image of this field.
This source is located at 06h33m06.59s $\pm$ $''$, $'$ $''$ 24 $\pm$ $''$, and is visible both in the NVSS archive \citep[1.4\,GHz,][]{Condon98} with a flux of $\sim$ mJy (cf $\sim$ mJy in the GMRT GHz observations) and in the XMM-Newton image of this field.
Constant Huxes were recorded for all remaining field sources in all observations.
Constant fluxes were recorded for all remaining field sources in all observations.
We also observed. the field of J.0632|057. using the Ciant Aletrewave Raclio (GMICE) (2). during June September 2008.
We also observed the field of J0632+057, using the Giant Metrewave Radio (GMRT) \citep{Swarup91} during June – September 2008.
Phe CMICE array has 30 antennas. arranged in roughly a Y configuration over 25 kim area.
The GMRT array has 30 antennas, arranged in roughly a `Y' configuration over 25 km area.
The target. field was observed for a total of 36 hours with 616 slots of 3 hours cach at 1280 anc 610/235. MIIz.
The target field was observed for a total of 36 hours with 6+6 slots of 3 hours each at 1280 and 610/235 MHz.
We observed simultaneously. at 235 and 610. MllIz. using synthesised bandwidths of 6 and 16 MlIeZ. respectively.
We observed simultaneously at 235 and 610 MHz, using synthesised bandwidths of 6 and 16 MHz, respectively.
The 1280 Mllz observation was carried out with a bandwidth of 16 Mllz in cach of the two available sidebands.
The 1280 MHz observation was carried out with a bandwidth of 16 MHz in each of the two available sidebands.
Each frequency channel is 125 ΚΙ in width. enabling the removal of narrow-band raclio frequceney interference (REL).
Each frequency channel is 125 kHz in width, enabling the removal of narrow-band radio frequency interference (RFI).
Vhe sources 1147. and 448 were observed at. the beginning and end of the observations and used as amplitude ancl bandpass calibrators to set. the tux density scale.
The sources 147 and 48 were observed at the beginning and end of the observations and used as amplitude and bandpass calibrators to set the flux density scale.
The total area within the conservative boundaries of the spectroscopically surveyed region of the Bootes field is 7.30 deg’.
The total area within the conservative boundaries of the spectroscopically surveyed region of the Bootes field is 7.30 $^{2}$.
The total effective area, taking into account the gaps between the CCDs and degradation of the detection efficiency, is5.90 deg?.
The total effective area, taking into account the gaps between the CCDs and degradation of the detection efficiency, is5.90 $^{2}$.
The model for the source redshift distribution, dN/dz, should reflect both the intrinsic variations of the comoving number density with redshift and all selection effects of the catalog.
The model for the source redshift distribution, $dN/dz$, should reflect both the intrinsic variations of the comoving number density with redshift and all selection effects of the catalog.
A commonly used approach is to model the observed dN/dz distribution with a high-order polynomialexamples).
A commonly used approach is to model the observed $dN/dz$ distribution with a high-order polynomial.
. This approach works well for catalogs with a large number of sources.
This approach works well for catalogs with a large number of sources.
However, for smaller catalogs, like ours, there is a danger that a high-order polynomial fit will follow statistical fluctuations in the observed dN/dz, while a low-order polynomial would be unable to adequately model the strong gradients at low z.
However, for smaller catalogs, like ours, there is a danger that a high-order polynomial fit will follow statistical fluctuations in the observed $dN/dz$, while a low-order polynomial would be unable to adequately model the strong gradients at low $z$.
Therefore, we fit the redshift distribution of AGNs in the Boóttes field with a parametric model based on several physical assumptions.
Therefore, we fit the redshift distribution of AGNs in the Boöttes field with a parametric model based on several physical assumptions.
The first component of the model represents the cosmological comoving volume per unit redshift, dN,/dz« dV/dz.
The first component of the model represents the cosmological comoving volume per unit redshift, $dN_{1}/dz\propto dV/dz$ .
The second component is a power law function of the minimum luminosity which corresponds to the flux limit at redshift z, dN5»/dzοςLee«d?, where d; is the luminosity distance.
The second component is a power law function of the minimum luminosity which corresponds to the flux limit at redshift $z$, $dN_{2}/dz\propto L_{\rm min}^{\alpha/2}\propto d_{L}^{\alpha}$, where $d_{L}$ is the luminosity distance.
This component represents the effect of the low-L, cutoff of the intrinsic luminosity function introduced by the selection which is primarily based on detections.
This component represents the effect of the $L_{x}$ cutoff of the intrinsic luminosity function introduced by the selection which is primarily based on detections.
It also can describe the evolution of the function at high z.
It also can describe the evolution of the luminosity function at high $z$.
The third is a high-z luminositycutoff modeled by a broad Gaussian, componentdN3/dz«
The third component is a $z$ cutoff modeled by a broad Gaussian, $dN_{3}/dz\propto \exp\left(-d_{L}^{2}/C^{2}\right)$ .
This component can represent the high-L, cutoff or (-d?/steepeningc? of the intrinsic AGN luminosity function, and also can describe various observational limits implicitly built into our catalog (e.g., a lower efficiency of optical identifications and redshift measurements for the highest-z ray sources).
This component can represent the $L_{x}$ cutoff or steepening of the intrinsic AGN luminosity function, and also can describe various observational limits implicitly built into our catalog (e.g., a lower efficiency of optical identifications and redshift measurements for the $z$ X-ray sources).
This simple analytic model,which has only two free parameters provides a strikingly good fit to the observed redshift distribution of the Boóttes X-ray selected AGNs (Fig. 5)).
This simple analytic model,which has only two free parameters provides a strikingly good fit to the observed redshift distribution of the Boöttes X-ray selected AGNs (Fig. \ref{fig:dndz}) ).
The best-fit values are a=—1.07 and C=1.50x10?! Mpc.
The best-fit values are $\alpha=-1.07$ and $C=1.50\times10^{3}\,h^{-1}\,$ Mpc.
We use the arguments outlined above only as a motivation for a good analytical description of the dN/dz distribution for our sources.
We use the arguments outlined above only as a motivation for a good analytical description of the $dN/dz$ distribution for our sources.
The functional form and derived parameters are not meant to represent the true ray luminosity function or its evolution.
The functional form and derived parameters are not meant to represent the true X-ray luminosity function or its evolution.
Figure 5 also demonstrates the general characteristics of our sample.
Figure \ref{fig:dndz} also demonstrates the general characteristics of our sample.
The peak in the observed dN/dz distribution is near z~0.6.
The peak in the observed $dN/dz$ distribution is near $z\approx 0.6$.
The median redshift of the is Zmea= 1.04.
The median redshift of the sample is $z_{\rm med}=1.04$ .
The tail in the redshift distribution extends sampleto z~4.5 but the fraction of AGNs with z>3 is very small.
The tail in the redshift distribution extends to $z\approx4.5$ but the fraction of AGNs with $z>3$ is very small.
Overall, the clustering properties of sources in our sample are most sensitive to the distribution of the X-ray AGN population near ze.
Overall, the clustering properties of sources in our sample are most sensitive to the distribution of the X-ray AGN population near $z\approx 1$.
Because the volume is never sampled completely in astronomical surveys, the derivation of the two-point correlation function from the data uses mock catalogs of intrinsically randomly distributed objects, which faithfully reproduces all observational distortions introduced by the survey.
Because the volume is never sampled completely in astronomical surveys, the derivation of the two-point correlation function from the data uses mock catalogs of intrinsically randomly distributed objects, which faithfully reproduces all observational distortions introduced by the survey.
Examples of such distortions are boundaries of the survey region, gaps in the data or spatial variations of the sensitivity, variations of the selection efficiency with redshift, etc.
Examples of such distortions are boundaries of the survey region, gaps in the data or spatial variations of the sensitivity, variations of the selection efficiency with redshift, etc.
Given the catalog of observed sources and the mock random catalog, the two-point correlation function can be estimated as where DD is the number of source pairs in the data for the given distance interval, RR is the number of pairs in the random catalog, DR iscorresponding the number of pairs between the data and random catalog, and N4 and N, are the numbers of objects in the data and random catalogs,
Given the catalog of observed sources and the mock random catalog, the two-point correlation function can be estimated as where $DD$ is the number of source pairs in the data for the given distance interval, $RR$ is the corresponding number of pairs in the random catalog, $DR$ is the number of pairs between the data and random catalog, and $N_{d}$ and $N_{r}$ are the numbers of objects in the data and random catalogs, respectively.
Statistical uncertainties for € can be estimated respectively.as(?);; this equation includes both the Poissonian shot noise and intrinsic variance terms.
Statistical uncertainties for $\xi$ can be estimated as; this equation includes both the Poissonian shot noise and intrinsic variance terms.
To verify the of the error by eq. 4,,
To verify the accuracy of the error by eq. \ref{frm:poisson error},
we used the sample varience ofaccuracy the correlation functions measured in the mock catalogs derived from the Millenium simulation for the survey geometry and object properties similar to those of the survey.
we used the sample varience of the correlation functions measured in the mock catalogs derived from the Millenium simulation for the survey geometry and object properties similar to those of the survey.
This analysis showed that eq.
This analysis showed that eq.
4 is accurate at tsmall scales but may underestimate the uncertainties at large scales.
\ref{frm:poisson error} is accurate at tsmall scales but may underestimate the uncertainties at large scales.
The correction factor can be described by a smooth function which is3%,,23%,, and at separations of 1, 6, and 15! Mpc, respectively.
The correction factor can be described by a smooth function which is, and at separations of $1$, $6$, and $15\,h^{-1}\,$ Mpc, respectively.
This correction is applied to the statistical uncertainties estimated by eq. 4..
This correction is applied to the statistical uncertainties estimated by eq. \ref{frm:poisson error}.
The correlation function in real space is expected to be isotropic, so £ is a function of the 3D separation only.
The correlation function in real space is expected to be isotropic, so $\xi$ is a function of the 3D separation only.
When the object redshifts are used to derive the distances, the correlation function is distorted in the line-of-sight direction because of large-scale flows (the Kaiser effect) and “fingers of God” arising within the virialized dark matter halos.
When the object redshifts are used to derive the distances, the correlation function is distorted in the line-of-sight direction because of large-scale flows (the Kaiser effect) and “fingers of God” arising within the virialized dark matter halos.
The correlation function should then be measured as a function of the projected separation, ry, and the line-of-sight separation, π.
The correlation function should then be measured as a function of the projected separation, $r_{p}$ , and the line-of-sight separation, $\pi$.
Equations [3]] and [4]] still can be used, but the pairs must be counted for each combination (rp,7).
Equations \ref{frm:LSE}] ] and \ref{frm:poisson error}] ] still can be used, but the pairs must be counted for each combination $(r_{p},\pi)$.
Given the angular separation between two objects, 6, and redshifts, z; and z2, the comoving separations r, and π can be computed as follows.
Given the angular separation between two objects, $\theta$ and redshifts, $z_{1}$ and $z_{2}$, the comoving separations $r_{p}$ and $\pi$ can be computed as follows.
First, one computes the radial comoving distances, [ει and D,, corresponding to the object redshifts ?).
First, one computes the radial comoving distances, $D_{c,1}$ and $D_{c,2}$, corresponding to the object redshifts .
. Then, following we have One can also define a formal 3D separation, but it should be kept in mind that s is not equivalent to the true 3D separation, r, because of the redshift space distortions.
Then, following we have One can also define a formal 3D separation, but it should be kept in mind that $s$ is not equivalent to the true 3D separation, $r$ , because of the redshift space distortions.
As onlythe line-of-sight separations, 7, are affected by the object peculiar velocities, it is useful to consider the correlation function projected on the sky plane,
As onlythe line-of-sight separations,$\pi$ are affected by the object peculiar velocities, it is useful to consider the correlation function projected on the sky plane,
are expected (o evolve starüng Irom marginally eravilalionally unstable (Q>1.5) initial conditions (e.g.. Boley 2009).
are expected to evolve starting from marginally gravitationally unstable $Q > 1.5$ ) initial conditions (e.g., Boley 2009).
Such disks (vpically also fragment. but only after a period of dvnamical evolution toward Q~1 in limited regions. such as dense rings (e.g. Boss 2002).
Such disks typically also fragment, but only after a period of dynamical evolution toward $Q \sim 1$ in limited regions, such as dense rings (e.g., Boss 2002).
Large. massive disks have been detected in regions of low-mass star formation. such as the 300-AU-scale. ~LAL. disk wound the class O protostar Serpens FIRS 1 (Enoch et al.
Large, massive disks have been detected in regions of low-mass star formation, such as the 300-AU-scale, $\sim 1 M_\odot$ disk around the class O protostar Serpens FIRS 1 (Enoch et al.
2009).
2009).
Observations of 11 low- and intermedciale-mass pre-main-sequence stars immiplv that (heir circumstellar disks formed with masses in (he range from 0.05 AZ. to 0.4 M... (Isella. Carpenter. Sargent. 2009).
Observations of 11 low- and intermediate-mass pre-main-sequence stars imply that their circumstellar disks formed with masses in the range from 0.05 $M_\odot$ to 0.4 $M_\odot$ (Isella, Carpenter, Sargent 2009).
These and other observations support the choice of the disk Inasses and sizes assumed in the present moclels.
These and other observations support the choice of the disk masses and sizes assumed in the present models.
All of the models clvnamically evolve in much the same wav.
All of the models dynamically evolve in much the same way.
Beginning [rom nearly axisvmmnmetrie configurations (with initial m=1.2.3.4 density perturbations of amplitude 1%)). the disks develop increasingly stronger spiral arm structures.
Beginning from nearly axisymmetric configurations (with initial $m = 1, 2, 3, 4$ density perturbations of amplitude ), the disks develop increasingly stronger spiral arm structures.
Eventually these trailing spiral arms become distinct enough. through sell-gravitational growth and mutual collisions. that reasonably well-defined clamps appear aud maintain (heir identities for some fraction of an orbital period.
Eventually these trailing spiral arms become distinct enough, through self-gravitational growth and mutual collisions, that reasonably well-defined clumps appear and maintain their identities for some fraction of an orbital period.
Ilowever. because the fixed-grid nature of these calculations prevents ihe chumps from contracting to much higher densities. (he clumps are doomed to eventual destruction by a combination of thermal pressure. Uidal forces [rom the protostar. ancl Keplerian shear.
However, because the fixed-grid nature of these calculations prevents the clumps from contracting to much higher densities, the clumps are doomed to eventual destruction by a combination of thermal pressure, tidal forces from the protostar, and Keplerian shear.
ILowever. new clumps continue (o form and orbit the protostar. suggesting that chump formation is inevitable.
However, new clumps continue to form and orbit the protostar, suggesting that clump formation is inevitable.
Previous work (Boss 2005) has shown that as the numerical spatial resolution is increased. the survival of clumps formed by disk instability is enhanced.
Previous work (Boss 2005) has shown that as the numerical spatial resolution is increased, the survival of clumps formed by disk instability is enhanced.
While an adaptive-mesh-refinement code would be desirable for demonstrating that elumps can contract and survive. the present models. combined with the previous work by Boss (2005). are sullicient for a first exploration of this region of disk instability. parameter space.
While an adaptive-mesh-refinement code would be desirable for demonstrating that clumps can contract and survive, the present models, combined with the previous work by Boss (2005), are sufficient for a first exploration of this region of disk instability parameter space.
Figures l through 10 show the midplane density and temperature contours for all [ive models at a (me of ~6/54. where Poy is the Keplerian orbital period at the distance of the inner erid boundary of 20 AU [or à protostar with the given mass.
Figures 1 through 10 show the midplane density and temperature contours for all five models at a time of $\sim 6 P_{20}$, where $P_{20}$ is the Keplerian orbital period at the distance of the inner grid boundary of 20 AU for a protostar with the given mass.
For models 0.1. 0.5. 1.0.
For models 0.1, 0.5, 1.0.
1.5. and 2.0. respectively. P5, is equal to 283 vr. 126 vr. 89.4 vr. 73.0 ve. and 63.2 vr.
1.5, and 2.0, respectively, $P_{20}$ is equal to 283 yr, 126 yr, 89.4 yr, 73.0 yr, and 63.2 yr.
It is clear (hat clamps have formed by this Gime in all five models.
It is clear that clumps have formed by this time in all five models.
Ilowever. in order to become a giant planet. clumps must survive long enough to contract toward planetary densities.
However, in order to become a giant planet, clumps must survive long enough to contract toward planetary densities.
The spherically svnunetric protoplanet models of IIlellel Boclenheimer (2011) suggest contraction time scales ranging from e10% vr to ~10? vr. depending on the metallicity. for protoplanets with masses from 3 to 7 Mj. so these clumps must survive for many orbital periods in order to become planets.
The spherically symmetric protoplanet models of Helled Bodenheimer (2011) suggest contraction time scales ranging from $\sim 10^3$ yr to $\sim 10^5$ yr, depending on the metallicity, for protoplanets with masses from 3 to 7 $M_{Jup}$, so these clumps must survive for many orbital periods in order to become planets.
The virtue of this model is that it allows à. simple estimate of the shape of the correlation function in the Lini in which the subelumps are each a small fraction of the mass of the parent halo. and the total mass in subebunmps is a smal fraction of the total mass.
The virtue of this model is that it allows a simple estimate of the shape of the correlation function in the limit in which the subclumps are each a small fraction of the mass of the parent halo, and the total mass in subclumps is a small fraction of the total mass.
Recall that. in this limit. the tota correlation function is well approximated by the sum of two terms.
Recall that, in this limit, the total correlation function is well approximated by the sum of two terms.
The first term is the contribution from pairs which are [rom the smoothly. distributed component. and the seconc is from the subcelumps.
The first term is the contribution from pairs which are from the smoothly distributed component, and the second is from the subclumps.
The subclump contribution. then. can be determined. by rescaling the contribution from the smooth component at σι.
The subclump contribution, then, can be determined by rescaling the contribution from the smooth component at $z_1$.
If the particles which were strippec [rom the halos present à τι were random particles. then the number of pairs is lower by à factor of f7: since the halos a 2, were a factor of (1|z4* denser. the contribution to the correlation function is dillerent by a factor of /7(1|zi.
If the particles which were stripped from the halos present at $z_1$ were random particles, then the number of pairs is lower by a factor of $f^2$; since the halos at $z_1$ were a factor of $(1+z_1)^3$ denser, the contribution to the correlation function is different by a factor of $f^2 (1+z_1)^3$.
This vields a factor which is probably slightly smaller than unity.
This yields a factor which is probably slightly smaller than unity.
On the other hand. if the mass was stripped [rom the earlier halos in shells. much like lavers olf an onion. as simulations suggest. then a better model of the subelump contribution is to truncate the halos which were present at τι at a fraction f of their virial radii when estimating how the number of pairs changes with scale.
On the other hand, if the mass was stripped from the earlier halos in shells, much like layers off an onion, as simulations suggest, then a better model of the subclump contribution is to truncate the halos which were present at $z_1$ at a fraction $\sim f$ of their virial radii when estimating how the number of pairs changes with scale.
Although this changes the actual shape of the correlation function (e.g.. the subclump pairs are shifted to scales which are a factor of f smaller). the typical factor by which the correlations are allected is fτῇσι), which can be considerably. larger than unity.
Although this changes the actual shape of the correlation function (e.g., the subclump pairs are shifted to scales which are a factor of $f$ smaller), the typical factor by which the correlations are affected is $f^{-1} (1+z_1)^3$, which can be considerably larger than unity.
We have shown how to incorporate the effects. of substructure into. the halo. model. description. of the nonlinear clensitw field.
We have shown how to incorporate the effects of substructure into the halo model description of the nonlinear density field.