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In the ultra-relativistic limit {51.52 4/3). equation (5)) gives ο2=e{91 and in the non-relativistic limit (5~1.52 5/3). we simply get ez25 η, | In the ultra-relativistic limit $\gamma \gg 1, \hat{\gamma} \approx 4/3$ ), equation \ref{cs5}) ) gives $c_{\rm s}^2 = c^2/3$; and in the non-relativistic limit $\gamma \sim 1, \hat{\gamma} \approx 5/3$ ), we simply get $c_{\rm s}^2 = 5 \beta^2 c^2/9$ . |
As usual.we assume that the magnetic energy. density in the co-moving frame is a fraction € of the total thermal energy density (Dai. Luang Lu 1999). BCssw=, £pc. and that the shock accelerated electrons. carry a fraction | As usual,we assume that the magnetic energy density in the co-moving frame is a fraction $\xi_{\rm B}^2$ of the total thermal energy density (Dai, Huang Lu 1999), $B'^2 / 8 \pi = \xi_{\rm B}^2 e'$ , and that the shock accelerated electrons carry a fraction |
When estimating a photometric redshift, we assume that there is a mapping between a galaxy's true redshift z, and photometry vector p such that z=Fp). | When estimating a photometric redshift, we assume that there is a mapping between a galaxy's true redshift $z$, and photometry vector $\mathbf{p}$ such that $z = F({\mathbf p})$. |
If such a mapping exists, then the information to find F should be latent in a large galaxy catalogue where both photometry and spectroscopic redshifts are available. | If such a mapping exists, then the information to find $F$ should be latent in a large galaxy catalogue where both photometry and spectroscopic redshifts are available. |
Self-organisation of such training set will naturally encode P, and therefore a SOM acan be used to predict the redshifts (or indeed any other parameter that was involved in the training) of new galaxies where, for example, only a subset of photometry is known. | Self-organisation of such a training set will naturally encode $F$, and therefore a SOM can be used to predict the redshifts (or indeed any other parameter that was involved in the training) of new galaxies where, for example, only a subset of photometry is known. |
This technique could be easily applied to a large imaging survey that contains a smaller spectroscopic component in order to robustly estimate redshifts for those galaxies lacking spectroscopic coverage. | This technique could be easily applied to a large imaging survey that contains a smaller spectroscopic component in order to robustly estimate redshifts for those galaxies lacking spectroscopic coverage. |
The advantage of using a SOM for photometric redshift estimation is that it is completely empirical, requires no assumptions about the spectral properties of the galaxies and involves no user intervention to guide the learning tthe learning is unsupervised). | The advantage of using a SOM for photometric redshift estimation is that it is completely empirical, requires no assumptions about the spectral properties of the galaxies and involves no user intervention to guide the learning the learning is unsupervised). |
However, there are two fundamental limitations to the method: | However, there are two fundamental limitations to the method: |
constrain the photon cussion spectrum of the excess. shedding light ou its origin. | constrain the photon emission spectrum of the excess, shedding light on its origin. |
We perform a joiut analvsis of the CBI results aud the new ACBAR data at(>2000. assunune the coutributious frou primary auisotropy are known. | We perform a joint analysis of the CBI results and the new ACBAR data at$\ell>2000$, assuming the contributions from primary anisotropy are known. |
In cach experiucut. the theoretical baud-powoers for priuary anisotropy are calculated from the product of the ACDME power spectrun and baud window fictions. which are then subtracted from the observed baud-powers. | In each experiment, the theoretical band-powers for primary anisotropy are calculated from the product of the $\Lambda$ CDM power spectrum and band window functions, which are then subtracted from the observed band-powers. |
A two-dimensional likelihood fuuctiou is calculated from these excess baud-powers aud thei uncertainties. where the two paraneters are the ratio of the 230CIIz and 150CIIz excess. C. and the power at 30CIIz. o3, (Gu | A two-dimensional likelihood function is calculated from these excess band-powers and their uncertainties, where the two parameters are the ratio of the $30\,$ GHz and $150\,$ GHz excess, $\zeta$ and the power at $30\,$ GHz, $\sigma^2_{30}$ (in $\mu K_{CMB}^2$ ). |
We thon marginalize over the oy parameter and plot ανα].the likelihood fuuctiou for the power ratio ¢ in Figure L. | We then marginalize over the $\sigma^2_{30}$ parameter and plot the likelihood function for the power ratio $\zeta$ in Figure \ref{fig:excess}. |
Since ACBAR iicasures siguificautlv less power at 150CIIz. the data disfavor sources that result in a blackbody spectrin(56. ¢= 1). | Since ACBAR measures significantly less power at $150\,$ GHz, the data disfavor sources that result in a blackbody spectrum, $\zeta=1$ ). |
Using the ACDAR and CBI frequency response and equation (6)). we calculate the power ratio ¢=L3 for the thermal SZ effect. | Using the ACBAR and CBI frequency response and equation \ref{szs}) ), we calculate the power ratio $\zeta=4.3$ for the thermal SZ effect. |
From the likehhood plot. we couclude that it is 1.5 times more likely that the excess seen by CBI aud ACBAR is the result of the thermal SZ effect (¢= L3) than a primordial process (ὁ= l1). | From the likelihood plot, we conclude that it is 4.5 times more likely that the excess seen by CBI and ACBAR is the result of the thermal SZ effect $\zeta=4.3$ ) than a primordial process $\zeta=1$ ). |
Since the expected ratio of flux at 30 IIz to 150 GIIz from radio sources is expected to be «0.1. such sources are disfavored as being responsible for the excess power seen by CDI at about the same significance (~ 1.20) with which ACBAR detects excess power. | Since the expected ratio of flux at $30\,$ GHz to $150\,$ GHz from radio sources is expected to be $<$ 0.1, such sources are disfavored as being responsible for the excess power seen by CBI at about the same significance $\sim 1.2 \sigma$ ) with which ACBAR detects excess power. |
Additional data from ACBAR or other higher frequency instruments will be required to make a cefinitive statement about the ore of the excess power seen by CDI and DIMÁ. | Additional data from ACBAR or other higher frequency instruments will be required to make a definitive statement about the origin of the excess power seen by CBI and BIMA. |
Iu this section. we estimate cosimological parameters or a nunmal iuflation-based. spatially-flat.— tilted. eravitationally leused. ACDAM model characterized by. six xwanieters, and then investigate models including extra xuwanueters to test extensions of the theory. | In this section, we estimate cosmological parameters for a minimal inflation-based, spatially-flat, tilted, gravitationally lensed, $\Lambda$ CDM model characterized by six parameters, and then investigate models including extra parameters to test extensions of the theory. |
For our base uodel. the six parameters are: the plysical deusitv of xuvonie and dark matter. Oh? and 0,77: à coustaut spectral iudex ον and amplitude Iud, of the primordial vower spectu. the optical depth to last scatterine. Tr: and the ratio of the sound horizon at last scattering o the angular diameter distauce. 0. | For our base model, the six parameters are: the physical density of baryonic and dark matter, $\Omega_bh^2$ and $\Omega_ch^2$; a constant spectral index $n_s$ and amplitude $\ln A_s$ of the primordial power spectrum, the optical depth to last scattering, $\tau$ ; and the ratio of the sound horizon at last scattering to the angular diameter distance, $\theta$. |
The primordial comoving scalar curvature power spectrums is expressed as Paik)=Ah)Us where the normalization (pivot-»)ut) wavenunber is chosen to be fk,=0.05MpeB | The primordial comoving scalar curvature power spectrum is expressed as ${\cal P}_s(k) = A_s (k/k_n)^{(n_s-1)}$, where the normalization (pivot-point) wavenumber is chosen to be $k_n
= 0.05\, {\rm Mpc}^{-1}$. |
The parameter @ maps angles observed at our location ο comoving spatial scales at recombination: changing 0 shifts the cutive acoustic peak/valley aud damping patte of the CAMB power spectra. | The parameter $\theta$ maps angles observed at our location to comoving spatial scales at recombination; changing $\theta$ shifts the entire acoustic peak/valley and damping pattern of the CMB power spectra. |
Additional paramcters are erived from the basic set. | Additional parameters are derived from the basic set. |
These include: the energy chsityv of a cosmological coustaut in uuits of the critical density. O4: the age of universe: the cuerey density of non-relativistic matter. O,,: the (near) matter fluctuation ish !IMpe spheres. ox: the redshift to reionization. L4: aoeud the value of the present dav IIubble coustaut. fy. iu units of Mpc +. | These include: the energy density of a cosmological constant in units of the critical density, $\Omega_\Lambda$; the age of universe; the energy density of non-relativistic matter, $\Omega_m$; the (linear) matter fluctuation in $8h^{-1}$ Mpc spheres, $\sigma_8$; the redshift to reionization, $z_{re}$; and the value of the present day Hubble constant, $H_0$, in units of $^{-1}$ $^{-1}$. |
Tilted primordial spectra indicate the presence of a tensor-incduced anisotropv componcut. however. we do not iuchide this potential contribution due to its uncertain amplitude. | Tilted primordial spectra indicate the presence of a tensor-induced anisotropy component, however, we do not include this potential contribution due to its uncertain amplitude. |
The influence of the tensor COL)sent woulL only be significant at low-f. not iu the regine which ACBAR probes. | The influence of the tensor component would only be significant at $\ell$, not in the regime which ACBAR probes. |
We also restrict this work to flat ACDAL models. inotivated by the observed. curvature being so close to zero. | We also restrict this work to flat $\Lambda$ CDM models, motivated by the observed curvature being so close to zero. |
However. we have run models with non-zero curvature Oy. and find that they reproduce the standard ecometrical degeneracy associated with 0 aud Ox, | However, we have run models with non-zero curvature $\Omega_K$, and find that they reproduce the standard geometrical degeneracy associated with $\Omega_K$ and $\Omega_\Lambda$. |
We have also considered two exteusious to the basic model which coul potentially inipact the interpretation of the ACBAR hancelpowers. | We have also considered two extensions to the basic model which could potentially impact the interpretation of the ACBAR bandpowers. |
These extended models include flat AC'DAL inodccds with a runndus scalar spectral dex characterized by the derivative οΠα(κ). aud flat ACDAL aodels with a Sunvaev-Zeldovicl1 contribution to the angular power spectrum with amplitude parametrized by 077, | These extended models include flat $\Lambda$ CDM models with a running scalar spectral index characterized by the derivative $dn_s/d\ln k (k_n)$, and flat $\Lambda$ CDM models with a Sunyaev-Zel'dovich contribution to the angular power spectrum with amplitude parametrized by $\alpha^{\rm SZ}$. |
We also investigate a model where both a runing spectral iudex aud a SZ contribution are considered. stuultanecously. | We also investigate a model where both a running spectral index and a SZ contribution are considered simultaneously. |
The parameter constraints are obtained using a Alonte Carlo Mawkov Chain (CMCMC) sampling of the uultidiueusional likelihood as a function of model xuauneters, | The parameter constraints are obtained using a Monte Carlo Markov Chain (MCMC) sampling of the multi-dimensional likelihood as a function of model parameters. |
Our software is based ou tle ]mblicly available package λεν | Our software is based on the publicly available package \citep{Lewis:2002ah}. |
CAIB angular power spectra aud matter power spectra are computes using the code (?).. | CMB angular power spectra and matter power spectra are computed using the code \citep{lewis00}. |
We approximate the ful non-Coussiun baudpower Usclihoods with an offset lognormal distribution (7) ound by explicit fits (see NOL for a detailed discussion of he calculation). | We approximate the full non-Gaussian bandpower likelihoods with an offset lognormal distribution \citep{bond2000} found by explicit fits (see K04 for a detailed discussion of the calculation). |
Our παπάαχ results include he effects of weal eravitational lenxiug ou the CAIB (?7).. | Our standard results include the effects of weak gravitational lensing on the CMB \citep{seljak96,lewis00}. |
Leusine effects in the telmpcrature spectruni are expected to become significant at scales (>1000. hence it is müportant to include this effect when interpreting the ACBAR results. | Lensing effects in the temperature spectrum are expected to become significant at scales $\ell > 1000$, hence it is important to include this effect when interpreting the ACBAR results. |
The major effect of leusing ix a scale-depeudent smoothing of the angular power spectrun which diminishes the peaks aud vallevs of the spectrm. | The major effect of lensing is a scale-dependent smoothing of the angular power spectrum which diminishes the peaks and valleys of the spectrum. |
Tuclusion of lensing iuin the model improves the fit to the data for all experiment combinations. | Inclusion of lensing in the model improves the fit to the data for all experiment combinations. |
However. we fiud that the parameter mean values and 1:mcertaitics are largely unaffectec w the inclusion of lensine with some exceptions. in particular the iutroductiou of leuxing tends to increase the value of os. | However, we find that the parameter mean values and uncertainties are largely unaffected by the inclusion of lensing with some exceptions, in particular the introduction of lensing tends to increase the value of $\sigma_8$ . |
The typical computation consists of 5 separate chains. each having different initial random parameter choices. | The typical computation consists of $8$ separate chains, each having different initial random parameter choices. |
The chains are run until the largest eigenvalue of the Cehnan-Rubin test is sumaller than 0.1 after accounting for burudn. | The chains are run until the largest eigenvalue of the Gelman-Rubin test is smaller than 0.1 after accounting for burn-in. |
ΕΠΟΠΠ priors with very broac cüstributions are assuned or the basic parameters. | Uniform priors with very broad distributions are assumed for the basic parameters. |
The standard run also includes a weak prior on the ITubble coustaut (15«90 + 1) and on the age ofthe uulverse (>LO Cars). | The standard run also includes a weak prior on the Hubble constant $45 < H_0 < 90$ $^{-1}$ $^{-1}$ ) and on the age of the universe $>10$ Gyrs). |
We also investigate the influeuce of adding Large Seale Structure (LSS) data from the 2 deoree Field Cilaxy Redshitt Survey (2dFGRS) (2) aud 1ο Sloan Digital Sky Survev (SDSS) (?).. | We also investigate the influence of adding Large Scale Structure (LSS) data from the 2 degree Field Galaxy Redshift Survey (2dFGRS) \citep{cole05} and the Sloan Digital Sky Survey (SDSS) \citep{tegmark04}. . |
When iucludiuue the LSS data.we use only the baud powers for leugth scales larger than koeOAPApe ! to avoid non-linear clusering and scale- galaxybiasine effects. | When including the LSS data,we use only the band powers for length scales larger than $k \sim 0.1 h$ $^{-1}$ to avoid non-linear clustering and scale-dependent galaxybiasing effects. |
We nuwelnalize over a paranieter 55JD whichB describesH the (linear). biasing+ of. the ealaxy-galaxy power spectrum for L, σαaxies relative to the underlyingmass density power specrun. | We marginalize over a parameter $b^2_g$ which describes the (linear) biasing of the galaxy-galaxy power spectrum for $L_\star$ galaxies relative to the underlyingmass density power spectrum. |
We adopt a Cassial prior on centered. around b,=10 witha conservative width equivalentbo to ób,=0: 3: all parameters | We adopt a Gaussian prior on $b^2_g$ centered around $b_g=1.0$ with a conservative width equivalent to $\delta b_g = 0.3$ ; all parameters |
(7).. | \citep{Soltan1982}. |
We have only included solutions in both this and the following sections (Ligures 11- 16)) that describe. quasars which would lie in the SDSS catalog with Aeg=Aly and redshift corresponding to fy. | We have only included solutions in both this and the following sections (Figures \ref{fig:allpl4panel}- \ref{fig:allintbothMgoverlap}) ) that describe quasars which would lie in the SDSS catalog with $M_{BH} = M_0$ and redshift corresponding to $t_0$. |
We can also restrict the above solutions to lie at. least 0.2 dex from cach boundary to account for scatter. due to intrinsic variability in the observed quasar population. | We can also restrict the above solutions to lie at least $0.2$ dex from each boundary to account for scatter due to intrinsic variability in the observed quasar population. |
However. there are no non-trivial solutions lving at least 0.2 dex away from cach boundary with fy=4.5 Gyr. | However, there are no non-trivial solutions lying at least $0.2$ dex away from each boundary with $t_0 = 4.5$ Gyr. |
This may be because the boundaries change discontinuously when the quasar redshift moves across a bin boundary in Table L.. and some bouncdaries in this table match poorly. | This may be because the boundaries change discontinuously when the quasar redshift moves across a bin boundary in Table \ref{table:boundaries}, and some boundaries in this table match poorly. |
The greatest mismatches occur when virial masses transition from using to broad emission lines at z=0.8. | The greatest mismatches occur when virial masses transition from using to $\beta$ broad emission lines at $z = 0.8$. |
Phere are many wavs one might. correct. [or this binning problem. | There are many ways one might correct for this binning problem. |
As here we are simply developing models in an effort to establish rough guidelines for what properties are required of quasar evolution. we will just give the results of using a simple interpolation. | As here we are simply developing models in an effort to establish rough guidelines for what properties are required of quasar evolution, we will just give the results of using a simple interpolation. |
In Figures 13. and 14. we take the boundaries in Table 1. to be correct at the midpoint (in time) of each in. and interpolate in both slope and intercept to find the roundaries used at other redshifts. | In Figures \ref{fig:plint4panel} and \ref{fig:expint4panel}, we take the boundaries in Table \ref{table:boundaries} to be correct at the midpoint (in time) of each bin, and interpolate in both slope and intercept to find the boundaries used at other redshifts. |
Using these interpolated »»undaries. Figures 13. and 14 include parameters lor all racks of the sort included. in Figures 11. and 12.. with Πο<Ay1077. departing from Eddington at the same our fy. | Using these interpolated boundaries, Figures \ref{fig:plint4panel} and \ref{fig:expint4panel} include parameters for all tracks of the sort included in Figures \ref{fig:allpl4panel} and \ref{fig:allexp4panel}, with $10^5 < M_0 < 10^{10}$, departing from Eddington at the same four $t_0$. |
Η quasar activity at all times is controlled by the identical physical constraints. all quasars might lie on a zunilv of tracks with identical shape (A and &) but cillerent Ao and fo. | If quasar activity at all times is controlled by the identical physical constraints, all quasars might lie on a family of tracks with identical shape $k$ and $\kappa$ ) but different $M_0$ and $t_0$. |
There are no quasar tracks with either à power- or exponential decline in Exldington ratio with & and & lving in the allowed range in all panels of either Figure. | There are no quasar tracks with either a power-law or exponential decline in Eddington ratio with $k$ and $\kappa$ lying in the allowed range in all panels of either Figure. |
‘Thus. Families of tracks with the same fen at all times are disallowed with this interpolation. | Thus, families of tracks with the same $k,\kappa$ at all times are disallowed with this interpolation. |
Finally. recent stuclies Comparing 1.1 ancl Megll-based. virial masses have proposed that a substantial correction might be required to bring the two into agreement (??).. although the correction might not be needed for the brightest quasars (?).. | Finally, recent studies comparing $\beta$ - and -based virial masses have proposed that a substantial correction might be required to bring the two into agreement \citep{Onken2008,Risaliti2009}, although the correction might not be needed for the brightest quasars \citep{Steinhardt2010c}. |
The allowed parameters in the three panels with fy<6.0 Cyr of Figures Ll and 12) are. primarily determined using masses. while the fy=7.5 Civr (2= 0.68) panel is determined using primarily 11 masses. | The allowed parameters in the three panels with $t_0 < 6.0$ Gyr of Figures \ref{fig:allpl4panel} and \ref{fig:allexp4panel} are primarily determined using masses, while the $t_0 = 7.5$ Gyr $z = 0.68$ ) panel is determined using primarily $\beta$ masses. |
Requiring that parameters appear only in all threeMgll panels. procuces the allowed. parameters in Figures 15. and. for interpolated boundary conditions. the power-Iaw. parameters. indicated in Fig. 16.. | Requiring that parameters appear only in all three panels produces the allowed parameters in Figures \ref{fig:allbothMgoverlap} and, for interpolated boundary conditions, the power-law parameters indicated in Fig. \ref{fig:allintbothMgoverlap}. |
Phere are still no parameters for an exponential decay that appear in all three panels of Figure 12. (for the interpolated boundary. conditions) | There are still no parameters for an exponential decay that appear in all three panels of Figure \ref{fig:allexp4panel} (for the interpolated boundary conditions). |
We have shown that the quasar mass-Iuminosity. plane can o used to restrict possible tracks for the evolution of incliviclual quasars. | We have shown that the quasar mass-luminosity plane can be used to restrict possible tracks for the evolution of individual quasars. |
Phe allowed parameters presented in this oper correspond. to simple miocels anc analysis methods. | The allowed parameters presented in this paper correspond to simple models and analysis methods. |
In 5.. we discuss some of the properties that appear o be common to all allowed tracks. | In \ref{sec:trdiscussion}, we discuss some of the properties that appear to be common to all allowed tracks. |
Additional tuning of he quasar track shape. e.g.. requiring a different minimum. distance from boundaries on the quasar distribution. or the use of interpolation methods produces altered sets of allowed xwameters. | Additional tuning of the quasar track shape, e.g., requiring a different minimum distance from boundaries on the quasar distribution, or the use of interpolation methods produces altered sets of allowed parameters. |
“These figures are intended. to serve only as a rough guide to the types of changes that result from. using dilferent. assumptions. | These figures are intended to serve only as a rough guide to the types of changes that result from using different assumptions. |
The production of these tracks was motivated. by a | The production of these tracks was motivated by a |
In the next subsections we give four examples of 3D chaotic orbits lying near and along some main unstable periodic orbits, having initial conditions inside corotation, near corotation and outside corotation. | In the next subsections we give four examples of 3D chaotic orbits lying near and along some main unstable periodic orbits, having initial conditions inside corotation, near corotation and outside corotation. |
Figure 12 gives an example of the density distribution of orbits around an unstable periodic orbit of the —1:1 family with E;——1252560 (within the range of the top-right panel of Fig. | Figure 12 gives an example of the density distribution of orbits around an unstable periodic orbit of the $-1:1$ family with $E_j=-1252560$ (within the range of the top-right panel of Fig. |
11) where the motion is limited outside corotation. | 11) where the motion is limited outside corotation. |
We have taken 20000 initial conditions all along this 3D unstable periodic orbit (and its symmetric with respect to the origin), with small deviations from it, on the 6-dimensional space (z,y,z,Uz,vy, Vz). | We have taken 20000 initial conditions all along this 3D unstable periodic orbit (and its symmetric with respect to the origin), with small deviations from it, on the 6-dimensional space ${x,y,z,v_x,v_y,v_z}$ ). |
Then we integrated all these orbits for a time corresponding to many rotations of the bar. | Then we integrated all these orbits for a time corresponding to many rotations of the bar. |
In Fig. | In Fig. |
12a we plot the density distribution of these orbits on the rotation plane. | 12a we plot the density distribution of these orbits on the rotation plane. |
In this figure we have considered a time interval within 5-6 bar rotations. | In this figure we have considered a time interval within 5-6 bar rotations. |
In the same figure we have superimposed the high density contours of the corresponding Jacobi constant (black curves) and we see that the orbits close to the —1:1 periodic orbit reproduce well the density distribution giving the outer parts of the spiral structure. | In the same figure we have superimposed the high density contours of the corresponding Jacobi constant (black curves) and we see that the orbits close to the $-1:1$ periodic orbit reproduce well the density distribution giving the outer parts of the spiral structure. |
In Fig. | In Fig. |
12b(12c) we plot the density distribution of the pericentres (apocentres) of the orbits of Fig. | 12b(12c) we plot the density distribution of the pericentres (apocentres) of the orbits of Fig. |
12a together with the high density contours (black curves) corresponding to this E; value. | 12a together with the high density contours (black curves) corresponding to this $E_j$ value. |
On the one hand the pericentric density maxima seem to follow a part of the spiral structure but at a slightly smaller radius. | On the one hand the pericentric density maxima seem to follow a part of the spiral structure but at a slightly smaller radius. |
On the other hand the apocentric distribution does not fit well the real density maxima. | On the other hand the apocentric distribution does not fit well the real density maxima. |
Finally we remark that the loci of the velocity minima practically coincide with the ones of the pericentres. | Finally we remark that the loci of the velocity minima practically coincide with the ones of the pericentres. |
Figure 13 gives a similar information as Fig. | Figure 13 gives a similar information as Fig. |
12, but for an unstable periodic orbit of the 3:1 family, which is located inside corotation for E;=—1090000 (within the range of the middle-right panel of Fig. | 12, but for an unstable periodic orbit of the 3:1 family, which is located inside corotation for $E_j=-1090000$ (within the range of the middle-right panel of Fig. |
11). | 11). |
This Jacobi constant value, is close to E;(Li), and the areas inside and outside corotation can communicate. | This Jacobi constant value, is close to $E_j(L_1)$, and the areas inside and outside corotation can communicate. |
We considered again 20000 initial conditions all along this 3D (3:1) unstable periodic orbit (and its symmetric with respect to the origin), with small deviations from it, in the 6-dimensional space (x,y,z,Us,vy,Uz). | We considered again 20000 initial conditions all along this 3D $(3:1)$ unstable periodic orbit (and its symmetric with respect to the origin), with small deviations from it, in the 6-dimensional space ${x,y,z,v_x,v_y,v_z}$ ). |
These orbits follow, in the phase space, the unstable directions of the asymptotic manifolds originating from the periodic orbit. | These orbits follow, in the phase space, the unstable directions of the asymptotic manifolds originating from the periodic orbit. |
For a short time (the shortness depends on how large is the initial divergence from the periodic orbit) these orbits stay very close to the periodic orbit, but later they start deviating considerably from it. | For a short time (the shortness depends on how large is the initial divergence from the periodic orbit) these orbits stay very close to the periodic orbit, but later they start deviating considerably from it. |
In Fig. | In Fig. |
13a we see the density distribution (in color scale) of these orbits after within a time interval of 3-4 bar rotations. | 13a we see the density distribution (in color scale) of these orbits after within a time interval of 3-4 bar rotations. |
The black curves indicate the high density contours of the real particles for the same Jacobi constant value. | The black curves indicate the high density contours of the real particles for the same Jacobi constant value. |
It is evident that these two distributions fit one another very well. | It is evident that these two distributions fit one another very well. |
A similar behavior has been found by Patsis(2006) for chaotic orbits near the 4:1 resonance. | A similar behavior has been found by \cite{b18}
for chaotic orbits near the 4:1 resonance. |
This type of orbits was considered responsible for the inner parts of the spiral arms in an analytical model representing the spiral galaxy NGC 4314. | This type of orbits was considered responsible for the inner parts of the spiral arms in an analytical model representing the spiral galaxy NGC 4314. |
In Fig. | In Fig. |
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