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In Figures 4. and ll. the faintest galaxies and the most deviant data points are based on our DEIMOS observations.
In Figures \ref{fig:fphotp} and \ref{fig:deviation}, , the faintest galaxies and the most deviant data points are based on our DEIMOS observations.
One could argue that our σ΄ measurements might be overestimated.
One could argue that our $\sigma$ measurements might be overestimated.
In paper L. we have considered all source of uncertainties in measuring the velocity
In paper I, we have considered all source of uncertainties in measuring the velocity
The combinations of the baseline CSP model and the helium-enhanced (Y = 0.33 and 0.38) SSP models would probably explain the UV spectra of the strongest UVX (FUV—yz6) galaxies under the ad/roc assumption that all the UV flux originates from the heltum-enhanced HB stars only.
The combinations of the baseline CSP model and the helium-enhanced (Y = 0.33 and 0.38) SSP models would probably explain the UV spectra of the strongest UVX $FUV-r\approx6$ ) galaxies under the $ad~hoc$ assumption that all the UV flux originates from the helium-enhanced HB stars only.
An even higher helium abundance (e.g.. Y = 0.43) seems unlikely to contribute in this color-eolor diagram as the HB stars of such high helium abundance are too hot and faint in the GALEX FUV bandpass.
An even higher helium abundance (e.g., Y = 0.43) seems unlikely to contribute in this color–color diagram as the HB stars of such high helium abundance are too hot and faint in the GALEX FUV bandpass.
The UV spectral shapes of the helium-enhanced stars are clearly disentangled from those of the young stars which have a much stronger UV flux both in FUV and NUV at the same optical luminosity.
The UV spectral shapes of the helium-enhanced stars are clearly disentangled from those of the young stars which have a much stronger UV flux both in FUV and NUV at the same optical luminosity.
In summary. we made two main findings with the latest GALEX UV data of the ETGs in the nearby universe.
In summary, we made two main findings with the latest GALEX UV data of the ETGs in the nearby universe.
1) The "old and dead" ETGs consist of a well-defined sequence in UV colors in which the stronger UVX galaxies should have a harder UV spectral shape systematically.
1) The “old and dead” ETGs consist of a well-defined sequence in UV colors in which the stronger UVX galaxies should have a harder UV spectral shape systematically.
The UV color distributions of ETGs at 0.05<z«0.12 are very similar to those observed for the local RC3 galaxies.
The UV color distributions of ETGs at $0.05<z<0.12$ are very similar to those observed for the local RC3 galaxies.
The comparison with the population synthesis models indicates that the hot HB stars are responsible for the observed "UV red sequence". while the contamination of recently-formed young stars can be effectively discriminated in the UV color-color diagram.
The comparison with the population synthesis models indicates that the hot HB stars are responsible for the observed “UV red sequence”, while the contamination of recently-formed young stars can be effectively discriminated in the UV color–color diagram.
2) The observed slope of the UV color-color relation. of quiescent ETGs ts too steep to be reproduced by the canonical stellar population models in which the UV flux is mainly controlled by age or metallicity parameters.
2) The observed slope of the UV color-color relation of quiescent ETGs is too steep to be reproduced by the canonical stellar population models in which the UV flux is mainly controlled by age or metallicity parameters.
Moreover. 2 mag of color spreads both in FUV—NUV and FUV—r appear to be ubiquitous among any subsets in distance or luminosity. which can hardly be explained by the canonical models with even large spreads in age (7 +2 Gyrs) and mean metallicity (> +0.3 dex) assumed.
Moreover, 2 mag of color spreads both in $FUV-NUV$ and $FUV-r$ appear to be ubiquitous among any subsets in distance or luminosity, which can hardly be explained by the canonical models with even large spreads in age $>$ $\pm$ 2 Gyrs) and mean metallicity $>$ $\pm$ 0.3 dex) assumed.
This implies that the UVX in ΕΤΟΣ could be driven by yet another parameter which might be even more influential than age or metallicity.
This implies that the UVX in ETGs could be driven by yet another parameter which might be even more influential than age or metallicity.
The recent debate on the helium-enhanced populations has an interesting implication for this issue.
The recent debate on the helium-enhanced populations has an interesting implication for this issue.
Observations suggest that there are some heltum-enhanced subpopulations m the extended HB globular clusters (D'Antona&Calor2004:Nor-al.2006;Leeet2007) and in the massive globular clusters in M87 (Sohnetal.2006).
Observations suggest that there are some helium-enhanced subpopulations in the extended HB globular clusters \citep{dan04,nor04,lee05,pio05,rec06,lee07} and in the massive globular clusters in M87 \citep{soh06}.
. If present. such subpopulations would steepen the UV spectral slope systematically by increasing the mean temperature of helium burning stars. as demonstrated in Fig.
If present, such subpopulations would steepen the UV spectral slope systematically by increasing the mean temperature of helium burning stars, as demonstrated in Fig.
3.
3.
They would also enhance the UV color spreads among the ETG stellar populations with even small variations in the number fractions or in the amount of helium enhancement (e.g..Kavirayetal.2007c;Chung2011).
They would also enhance the UV color spreads among the ETG stellar populations with even small variations in the number fractions or in the amount of helium enhancement \citep[e.g.,][]{kav07c,chu11}.
Whether the postulated helium enhancement in old stellar systems would occur also in galactic scales or in the intracluster medium of galaxy clusters (Peng is in question.
Whether the postulated helium enhancement in old stellar systems would occur also in galactic scales or in the intracluster medium of galaxy clusters \citep{pen09} is in question.
Observational tests by Loubser&Sánchez-Blázquez(2011) and Yietal.(2011) do not support the helium sedimentation hypothesis. while Carteretal.(2011) argue that the helium abundance is a plausible candidate to explain the UV radial gradient.
Observational tests by \citet{lou11} and \citet{yi11} do not support the helium sedimentation hypothesis, while \citet{car11} argue that the helium abundance is a plausible candidate to explain the UV radial gradient.
By the comparison between the UV color distribution and some toy models presented in this Letter. it is suggested that the observed steep slope of the UV color-color relation of ETGs would be better explained with the helirum-enhanced subpopulations.
By the comparison between the UV color distribution and some toy models presented in this $Letter$, it is suggested that the observed steep slope of the UV color–color relation of ETGs would be better explained with the helium-enhanced subpopulations.
The excessive spreads in the UV colors observed among the ETGs. besides the variations expected by the age/metallicity spreads. would be attributed to small variations in helium abundance or to the low-level star formation that is hard to be identified with the current survey depths and resolutions of the GALEX and SDSS data.
The excessive spreads in the UV colors observed among the ETGs, besides the variations expected by the age/metallicity spreads, would be attributed to small variations in helium abundance or to the low-level star formation that is hard to be identified with the current survey depths and resolutions of the GALEX and SDSS data.
ὃν inverting equation 2.. cach point in the (V-h. tx.) CAID ean then be assigned an estimate of its metallicity.
By inverting equation \ref{eq:fidiso}, each point in the $_s$ $_s$ ) CMD can then be assigned an estimate of its metallicity.
‘Taking into account only the uncertainty of the accuracy on the polvnomical regression of equations 2. and 3. and the photometric errors. the tvpical uncertainty in the resulting measurement of the metallicity ofa individual star is smaller than 0.04 dex.
Taking into account only the uncertainty of the accuracy on the polynomical regression of equations \ref{eq:fidiso} and \ref{eq:fidmet} and the photometric errors, the typical uncertainty in the resulting measurement of the metallicity of a individual star is smaller than 0.04 dex.
The resulting metallicity distribution function is represented in figure S.
The resulting metallicity distribution function is represented in figure \ref{fig:mdf}.
The secondary peak at 2.2 isan artefact duc to AGB stars and should be ignored.
The secondary peak at $= -2.2$ is an artefact due to AGB stars and should be ignored.
The mean metallicity obtained is Fe/l]2142 with a dispersion of 0.2 dex.
The mean metallicity obtained is $ = -1.42$ with a dispersion of 0.2 dex.
Fhis mean metallicity is in agreement with the value obtained from the RGB bump.
This mean metallicity is in agreement with the value obtained from the RGB bump.
A comparison of those metallicity estimates with the literature is given in the discussion section.
A comparison of those metallicity estimates with the literature is given in the discussion section.
Since the RGB in our photometric system provides a good indicator of metallicity. we studied its variation with radius as atest of a possible metallicity gradient in Sculptor.
Since the RGB in our photometric system provides a good indicator of metallicity, we studied its variation with radius as a test of a possible metallicity gradient in Sculptor.
The metallicity indicators. derived. in. the previous section have been studied as a function of radius.
The metallicity indicators derived in the previous section have been studied as a function of radius.
Phe central 10 arcminutes of Sculptor has zero ellipticity. (Irwin.&Latzicimitriou 1995)). so as our data co not extend bevond a radius of 15 arcminutes. we use a simple circular annulus.
The central 10 arcminutes of Sculptor has zero ellipticity \citealt{IRW95}) ), so as our data do not extend beyond a radius of 15 arcminutes, we use a simple circular annulus.
ligure 9 show that no metallicity gradient is detected within our data.
Figure \ref{fig:metrad} show that no metallicity gradient is detected within our data.
This gives an upper limit of 0.03 dex for the metallicity gradient within twice the core radius of Sculptor.
This gives an upper limit of 0.03 dex for the metallicity gradient within twice the core radius of Sculptor.
The horizontal branch of Sculptor can be clearly divided into a blue (19) ancl a red (10) part. lving on either side of the instability strip (V).
The horizontal branch of Sculptor can be clearly divided into a blue (B) and a red (R) part, lying on either side of the instability strip (V).
The ratio of the number counts of those dillerent. parts. quantified by the HD. index. (B-10/051VIR). is dependent on the metallicitv.
The ratio of the number counts of those different parts, quantified by the HB index (B-R)/(B+V+R), is dependent on the metallicity.
But there is à well known ‘second parameter problem. which could be age (e.g. Leeetal. 19949).
But there is a well known `second parameter problem', which could be age (e.g. \citealt{LEE94}) ).
The LLB index was computed from the (V-L1.V) €MD. as illustratec in figure 7.. for the same radial annuli as used for he RGB studs.
The HB index was computed from the (V-I,V) CMD, as illustrated in figure \ref{hbindex}, for the same radial annuli as used for the RGB study.
Both the full ESO WEI field of view data and the sub-area in common with the CLRSL field of view are presented in figure. LO.
Both the full ESO WFI field of view data and the sub-area in common with the CIRSI field of view are presented in figure \ref{hbrad}.
A Ix-S test for the hypothesis hat the red and blue horizontal branch stars have the same racial clistribution gives a significance level of 4...
A K-S test for the hypothesis that the red and blue horizontal branch stars have the same radial distribution gives a significance level of $^{-9}$.
‘This confirms the LIB gradient. detected by Llurley-Kellerctαἱ.(1999) and Majewskietal.(1999).
This confirms the HB gradient detected by \cite{HUR99} and \cite{MAJ99}.
The Leeetal.(1994) models give theoretical isochrones hat show. for an WB index between -0.5 anc 0.5 and a eiven age. a linear relation between the HD. index and. the metallicity: The observed gradient in HEBindex of about 0.5 then corresponds to a gradient in. metallicity of 0.17. dex.
The \cite{LEE94} models give theoretical isochrones that show, for an HB index between -0.5 and 0.5 and a given age, a linear relation between the HB index and the metallicity: The observed gradient in HBindex of about 0.5 then corresponds to a gradient in metallicity of 0.17 dex.
Considering the upper limit of 0.04 dex for a metallicity eracticnt derived previously from the RGB morphology. an age gradient is required to explain the observed LLB eracticnt.
Considering the upper limit of 0.04 dex for a metallicity gradient derived previously from the RGB morphology, an age gradient is required to explain the observed HB gradient.
As always when discussing LB morphology however. one must recall that the "second. parameter problemi is not vet solved. and that another parameter may. inlluence the LB morphology.
As always when discussing HB morphology however, one must recall that the `second parameter problem' is not yet solved and that another parameter may influence the HB morphology.
The simplest conclusion is that a small gradient in mean
The simplest conclusion is that a small gradient in mean
same generic parameter X.
same generic parameter $\widetilde{X}$.
For sake of simplicity in this paper. let us assume that these functions are power laws: with Ay. A» and Ax being constant.
For sake of simplicity in this paper, let us assume that these functions are power laws: with $A_1$ , $A_2$ and $A_3$ being constant.
Any linear correlation of the form where a. b and c are constant. is thus expressed as This expression should be valid for all X. implying that This set of two linear equationsgenerally vields solutions for « and P.
Any linear correlation of the form where $a$, $b$ and $c$ are constant, is thus expressed as This expression should be valid for all $\widetilde{X}$, implying that This set of two linear equations generally yields solutions for $a$ and $b$.
The resultis thus a plane in the3-D space (O4.Q».Ον).
The result is thus a plane in the 3-D space $\left( \Omega_1, \Omega_2, \Omega_3 \right)$.
The generic parameter X can be a multivariate component. making the parametric dependence ofO,.Ον and QO; mutlivariate.
The generic parameter $\widetilde{X}$ can be a multivariate component, making the parametric dependence of $\Omega_1, \Omega_2$ and $\Omega_3$ mutlivariate.
For instance. consider power laws with two parameters. X, and X»: Then it is easy to show that a correlation like equation (20) holds if: This set of equations is more constraining than equation GE.
For instance, consider power laws with two parameters, $X_1$ and $X_2$: Then it is easy to show that a correlation like equation \ref{eq:fundplanegen}) ) holds if: This set of equations is more constraining than equation \ref{eq:condition1gen}) ).
But if for instance the second relation in equation (6)) is not exactly fulfilled. then the parameter X» can be considered as noise adding a dispersion to the correlation defined by the two other relations.
But if for instance the second relation in equation \ref{eq:condition1mult}) ) is not exactly fulfilled, then the parameter $X_2$ can be considered as noise adding a dispersion to the correlation defined by the two other relations.
In other words. X» could generate a thickness to the plane defined by equation (29).
In other words, $X_2$ could generate a thickness to the plane defined by equation \ref{eq:fundplanegen}) ).
To be complete. we must diseuss the physies in the parameter X.
To be complete, we must discuss the physics in the parameter $\widetilde{X}$.
This parameter is supposed to influence all the three variables.
This parameter is supposed to influence all the three variables.
We distinguish two possbilities: causality or evolution.
We distinguish two possbilities: causality or evolution.
In the first case. the relation between the variables and X is driven by direct causality. that is some parameter influences directly each of the three variables Oj.Ον,O; through physical laws.
In the first case, the relation between the variables and $\widetilde{X}$ is driven by direct causality, that is some parameter influences directly each of the three variables $\Omega_1, \Omega_2, \Omega_3$ through physical laws.
This is illustrated in Sect.3.2..
This is illustrated in Sect. \ref{virial}.
In the second case. the relation is statistical in the sense that each variable is bound to evolve so that even totally unrelated variables can show an apparent correlation.
In the second case, the relation is statistical in the sense that each variable is bound to evolve so that even totally unrelated variables can show an apparent correlation.
This is discussed in Sect.
This is discussed in Sect.
3.3 and Sect. 4.
\ref{novirial} and Sect. \ref{evolutionary}.
Let us putO,=r, the effective radius. Ων=o the velocity dispersion. andQO;=L the luminosity.
Let us put $\Omega_1 = r_e$ the effective radius, $\Omega_2 = \sigma$ the velocity dispersion, and $\Omega_3 = L$ the luminosity.
equation ¢1)) becomes: The surface brightness ccan be expressed as where nr is a constant of normalisation.
equation \ref{eq:powerlawsgen}) ) becomes: The surface brightness can be expressed as where $m$ is a constant of normalisation.
Any linear correlation of the form translates to This implies If a solution can be found for & and from equation (11). then the equation of the fundamental plane equation (9)) is obtained.
Any linear correlation of the form translates to This implies If a solution can be found for $a$ and $b$ from equation \ref{eq:condition1}) ), then the equation of the fundamental plane equation \ref{eq:fundplane}) ) is obtained.
There is no need of any further assumption to explain the fundamental plane.
There is no need of any further assumption to explain the fundamental plane.
This demonstration is made here with a simple power-law assumption in equation (73).
This demonstration is made here with a simple power-law assumption in equation \ref{eq:powerlaws}) ).
But this result is also true for more complex functions. with equation (119) being replaced by more complicated conditions.
But this result is also true for more complex functions, with equation \ref{eq:condition1}) ) being replaced by more complicated conditions.
To understand the origin equation (9). we need to solve equation (LI.
To understand the origin equation \ref{eq:fundplane}) ), we need to solve equation \ref{eq:condition1}) ).
Since there are too many unknowns. additional input is required.
Since there are too many unknowns, additional input is required.
Two approaches are possible: either input some a priori knowledge to determine the functions of X and derive coetficients a and P (Sect.3.2)). or conversely use the observations to determine « and P and derive constraints on the functions of X (Sect.3.3)).
Two approaches are possible: either input some a priori knowledge to determine the functions of $\widetilde{X}$ and derive coefficients $a$ and $b$ \ref{virial}) ), or conversely use the observations to determine $a$ and $b$ and derive constraints on the functions of $\widetilde{X}$ \ref{novirial}) ).
In this section. we consider the input of a priori knowledge believed to be relevant for the physics of galaxies.
In this section, we consider the input of a priori knowledge believed to be relevant for the physics of galaxies.
At first glance. it is quite logical to consider that mass. either dynamical. true or stellar. is somehow related to the radius. the velocity dispersion and the surface brightness. essentially because it influences the density of stars and their kinematies.
At first glance, it is quite logical to consider that mass, either dynamical, true or stellar, is somehow related to the radius, the velocity dispersion and the surface brightness, essentially because it influences the density of stars and their kinematics.
So we consider X=M in equation (7)).
So we consider $\widetilde{X}=M$ in equation \ref{eq:powerlaws}) ).
One obvious way to link the kinematics to the mass is through the virial theorem.
One obvious way to link the kinematics to the mass is through the virial theorem.
Hence. let us assume that the virial equation holds and that the ratio between the average squared velocity and c7. and the ratio between r, and the gravitational radius. are constant.
Hence, let us assume that the virial equation holds and that the ratio between the average squared velocity and $\sigma^2$, and the ratio between $r_e$ and the gravitational radius, are constant.
This gives Using equation €11)) and equation (123) we obtain Unfortunately. the brightness has no direct relationto mass. but it might be assumed that the mass is essentially due to the stars that are responsible for the luminosity.
This gives Using equation \ref{eq:condition1}) ) and equation \ref{eq:psvirial}) ) we obtain Unfortunately, the brightness has no direct relationto mass, but it might be assumed that the mass is essentially due to the stars that are responsible for the luminosity.
If we also assume that the ratio M/L is constant for à given population of galaxies. we must have: Replacing this value of / in equation ¢13)). one obtains two solutions:
If we also assume that the ratio $M/L$ is constant for a given population of galaxies, we must have: Replacing this value of $t$ in equation \ref{eq:psvirial2}) ), one obtains two solutions:
»ossibility is accretion of small fragments.
possibility is accretion of small fragments.
S1uall fragments are produced by disruptive collisious of jxanetesimals in the vicinity of a large core (Iuaba.Wether
Small fragments are produced by disruptive collisions of planetesimals in the vicinity of a large core \citep{IWI03}.
ill.&Uksoma2003).. Since crag is stroug for the lragments. they are accreted outo the core (Inaba&Ikoma2003).
Since atmospheric-gas drag is strong for the fragments, they are accreted onto the core \citep{II03}.
. Because hese studies are limited to the suberitical core case. we need to exteud the calculation to the case oL supercritical cores. in which the planet mass rapidly Increases aud uou-tuiformity of clisk gas is »ououncecd.
Because these studies are limited to the subcritical core case, we need to extend the calculation to the case of supercritical cores, in which the planet mass rapidly increases and non-uniformity of disk gas is pronounced.
We uext explore the possibility of enrichment of heavy elements in HDI19026b (i.e.. bombarclinent of another planet or plauetesimals) substantial accretion of the envelope.
We next explore the possibility of enrichment of heavy elements in HD149026b (i.e., bombardment of another planet or planetesimals) substantial accretion of the envelope.
If the late bomibardimeut happened. it is likely to have occurred alter the planet migrated to its current location: The close-in planet collided with another plauet(s) or planetesiimals that had been eravitationally scattered by an outer giant. planet(s).
If the late bombardment happened, it is likely to have occurred after the planet migrated to its current location: The close-in planet collided with another planet(s) or planetesimals that had been gravitationally scattered by an outer giant planet(s).
At ~ 0.05 AU the planet’s Hill radius is only several times as large as its plivsical radius. while the former is much larger (by 100-1000). than he latter in outer regionse (=LAN1. AU).
At $\sim$ 0.05 AU the planet's Hill radius is only several times as large as its physical radius, while the former is much larger (by 100–1000) than the latter in outer regions $\ga 1$ AU).
That means the ratio of collision to scatteringOm cross sections is much larger at 0.05 AU relative to in outer regions.
That means the ratio of collision to scattering cross sections is much larger at $\sim$ 0.05 AU relative to in outer regions.
The scattering cross section is further reduced by high speed encounter.
The scattering cross section is further reduced by high speed encounter.
The impact. velocities (timp) of the scattered bodies in nearly yarabolic orbits are as large as the local Iweplerian velocity at 0.05 AU. which is a few times arger than surface escape velocity of the inner Saturu-mass planet (tee).
The impact velocities $v_{\rm imp}$ ) of the scattered bodies in nearly parabolic orbits are as large as the local Keplerian velocity at $\sim$ 0.05 AU, which is a few times larger than surface escape velocity of the inner Saturn-mass planet $v_{\rm esc}$ ).
The collision cross section is. in general. larger than the scattering cross section [or timp>Uese.
The collision cross section is, in general, larger than the scattering cross section for $v_{\rm imp} > v_{\rm esc}$.
Note that the shephercing does not work for highly eccentric orbits.
Note that the shepherding does not work for highly eccentric orbits.
To know how efficieutly such scattered bodies collide with the close-in giant planet. we perform the following numerical simulation.
To know how efficiently such scattered bodies collide with the close-in giant planet, we perform the following numerical simulation.
We cousicder an inmer planet of 0.5AZj in a circular orbit a 0.05 AU and a planetesimal (test particle) or another giant planet of 0.5244; in a nearly parabolic orbit ofe1 and ¢=0.01 with initial semimajor axis of «=1 AU.
We consider an inner planet of $0.5M_{\rm J}$ in a circular orbit at 0.05 AU and a planetesimal (test particle) or another giant planet of $0.5M_{\rm J}$ in a nearly parabolic orbit of $e \simeq 1$ and $i = 0.01$ with initial semimajor axis of $a = 1$ AU.
For each initial pericenter distauce (q). LOO cases with random augular distributions are nunerically integrated by Ith order Hermite integrator for 100 Ixeplerian periods at 1 AU.
For each initial pericenter distance $q$ ), 100 cases with random angular distributions are numerically integrated by 4th order Hermite integrator for 100 Keplerian periods at 1 AU.
Then. collision probability with the inner dlanet (4,4). that with the parent star (P2). and ejection prability (je) are counted.
Then, collision probability with the inner planet $P_{\rm col}$ ), that with the parent star $P_{\rm col}^*$ ), and ejection probability $P_{\rm ejc}$ ) are counted.
The 'esidual fraction. 1—(Peg)+Pdge). corresponds to the cases in which the plauetesimal/plane isstill orbiting after 100 Ixeplerian periods.
The residual fraction, $1 - (P_{\rm col} + P_{\rm col}^* + P_{\rm ejc})$, corresponds to the cases in which the planetesimal/planet isstill orbiting after 100 Keplerian periods.
We also did longercalculations aud fouud that PootPe is similar. although individual absolute values of {οι and Lj. increases.
We also did longercalculations and found that $P_{\rm col}/P_{\rm ejc}$ is similar, although individual absolute values of $P_{\rm col}$ and $P_{\rm ejc}$ increases.
The results are insensitive o «d as long as aὃν0.05 AU. because velocity aud specilic angular momentum of the incomiug dlanetesimal/planet are given by e2726M,/q aud L226M,q that are indepeudent of initial setajor axis « of incoming bodies.
The results are insensitive to $a$ as long as $a \gg 0.05$ AU, because velocity and specific angular momentum of the incoming planetesimal/planet are given by $v \simeq \sqrt{2GM_*/q}$ and $L \simeq \sqrt{2 GM_* q}$ that are independent of initial semimajor axis $a$ of incoming bodies.
Figure 10. shows the probabilities of the three outcomes of encounters between the inuer planet and a planetesimal (panel a) or another giant planet (panel b).
Figure \ref{fig:scat} shows the probabilities of the three outcomes of encounters between the inner planet and a planetesimal (panel a) or another giant planet (panel b).
As shown in Fie. 10..
As shown in Fig. \ref{fig:scat}, ,
when O.OLAL<q O.OGAL. the incoming bodies closely approach the inner planet.
when $0.01 {\rm AU} < q < 0.06 {\rm AU}$ , the incoming bodies closely approach the inner planet.
Although some
Although some