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A change in the EOS changes the shape and magnitude of the dip in Py at the Ile II ionization zone for the same helium abundance. | A change in the EOS changes the shape and magnitude of the dip in $\Gamma_1$ at the He II ionization zone for the same helium abundance. |
Figure 5. shows the relative difference in Py between two solar envelope models. one constructed with the so-called MIID equation of state (IIunmer Milalas 1988: \lihalas. Dapppen IIummer 1933: Dápppen οἱ al. | Figure \ref{fig:gamdif} shows the relative difference in $\Gamma_1$ between two solar envelope models, one constructed with the so-called MHD equation of state (Hummer Mihalas 1988; Mihalas, Däpppen Hummer 1988; Däpppen et al. |
1983). and the second with the OPAL equation of state (Rogers. Swenson Iglesias 1996). | 1988), and the second with the OPAL equation of state (Rogers, Swenson Iglesias 1996). |
Both models were constructed. with the same opacilies. have identical helium ancl heavy element abundances in the CZ (0.242 and 0.018 respectivelv) and were constructed (to have the same CZ depth (0.28711. ). | Both models were constructed with the same opacities, have identical helium and heavy element abundances in the CZ (0.242 and 0.018 respectively) and were constructed to have the same CZ depth $_\odot$ ). |
We can see that ihe models have substantial differences in Dy in the region of the Ie II ionization zone and higher. | We can see that the models have substantial differences in $\Gamma_1$ in the region of the He II ionization zone and higher. |
The amplitude of the He II signal for the MIID model is 1.1645/dIz and that of the OPAL model is 1.027,/0HIz. | The amplitude of the He II signal for the MHD model is $\mu$ Hz and that of the OPAL model is $\mu$ Hz. |
The difference between the He II amplitudes of the two models (0.13740HIz) is larger than the total range of change in amplitude seen in Fig. | The difference between the He II amplitudes of the two models $\mu$ Hz) is larger than the total range of change in amplitude seen in Fig. |
4. (which is only 0.09954]Iz according to (he linear fits to the results). | \ref{fig:high} (which is only $0.099\mu$ Hz according to the linear fits to the results). |
Thus as a first approximation we can sav that the change in [Py near the solar helium ionization zone between solar minimum and maxinnun is less (han that | Thus as a first approximation we can say that the change in $\Gamma_1$ near the solar helium ionization zone between solar minimum and maximum is less than that |
The results of DEMO2 have been previously preseuted in in Bernstein(1998.1999a.b.2001).. and distributed iu a preprint form for some time (Berustein.Freediuan.&Madore1999. 2000): in the following they are collectively referred to as BFALD98-O1. | The results of BFM02 have been previously presented in in \citet[]{ber98,bea99,beb99,ber01}, and distributed in a preprint form for some time \citep[]{bfm99,bfm00}; in the following they are collectively referred to as BFM98-01. |
These results have been widely cited in the literature as the “EBL standard reference values”. see eg. Barecretal.(2001):Hauser(2001):Wright (2001). | These results have been widely cited in the literature as the “EBL standard reference values”, see e.g. \citet[]{bar01, hau00, jim99,
lon01, mad00, pag01,pee01, poz01, pri99, ren01, wri01}. |
. It is therefore important to poiut out the between the BEM98-01 and DEMO2 results: Iu BFM98-01. the reduction method for the Zociacal Light measurements differed in a fundamental wav frou that iu DEMQO2b. | It is therefore important to point out the between the BFM98-01 and BFM02 results: In BFM98-01, the reduction method for the Zodiacal Light measurements differed in a fundamental way from that in BFM02b. |
Instead of Eq.(2). the following formula was applied: i.c. the atinospheric scattered light term Loca(A.t.XN) was completely omitted without explanation. thus neelectiug this fundamental aspect of the diffise might skv photometry. | Instead of Eq.(2), the following formula was applied: i.e. the atmospheric scattered light term $I_{\rm sca}(\lambda,t,X)$ was completely omitted without explanation, thus neglecting this fundamental aspect of the diffuse night sky photometry. |
Ouly in BFA02) has an Appeudix on scattered light model calculations now been added. | Only in BFM02b has an Appendix on scattered light model calculations now been added. |
Using the results of these caleulatious a lower limit to ια at L650 cai be estimated by adopting VY=1.1 for the ainmass. ic. a value at the lower eud of the BFALO2b airiass ranee. | Using the results of these calculations a lower limit to $I_{\rm sca}$ at 4650 can be estimated by adopting $X=1.1$ for the airmass, i.e. a value at the lower end of the BFM02b airmass range. |
L4 amounts to —18 of Izy. | $I_{\rm sca}$ amounts to $\sim$ of $I_{\rm ZL}$ . |
It. consists of the Ravleigh (1054 of Iz.) aud Mie (5 of zt) components of scatteredZodiacal Light. as well as of a scattered ISL component with a "ZL-like spectrmm (~ of £z,) (see Sect. | It consists of the Rayleigh $\sim$ of $I_{\rm ZL}$ ) and Mie $\sim$ of $I_{\rm ZL}$ ) components of scatteredZodiacal Light, as well as of a scattered ISL component with a “ZL-like” spectrum $\sim$ of $I_{\rm ZL}$ ) (see Sect. |
Land 5. aud. Fies.20. 21. aud 30 of BFAL0O2b). | 4 and 5, and Figs.20, 21, and 30 of BFM02b). |
This value translates. for the airumass of 1.1. to au outsice-the- value of ~22% of Ig, which. for the BFAIO2) value of Iz,=109.1«10. tat 1650. corresponds to ~21L\101. | This value translates, for the airmass of 1.1, to an outside-the-atmosphere value of $\sim$ of $I_{\rm ZL}$ which, for the BFM02b value of $I_{\rm ZL} = 109.4\times$ at 4650, corresponds to $\sim$ $\times$. |
2 Since not subtracted. Z4, Was erroneously iucluded iuto the Zodiacal Light. νε. iu the BFAI9s-01 analysis. | Since not subtracted, $I_{\rm sca}$ was erroneously included into the Zodiacal Light, $I_{\rm ZL}$, in the BFM98-01 analysis. |
Thus. the difference in the reduction methods necessarily should have resulted ina ~22% larger ZL value in DEMO8-01 than in BFALO2). | Thus, the difference in the reduction methods necessarily should have resulted in a $\sim$ larger ZL value in BFM98-01 than in BFM02b. |
This ~22% difference in Zzt is at least 7 times ax large as the EBL valueof BFAIN2 at 5500 and 5000A. | This $\sim$ difference in $I_{\rm ZL}$ is at least 7 times as large as the EBL valueof BFM02 at 5500 and 8000. |
. However. iu BFALI98-01 the EBL intensities at 3000. 5500. and 5000 are identical to the values in DEMO2 of LO. 2.7. aud 224.10 sto within €0.14101. | However, in BFM98-01 the EBL intensities at 3000, 5500, and 8000 are identical to the values in BFM02 of 4.0, 2.7, and $\times$ to within $\leqq 0.1\times$. |
Tt is unclear how this puzzling situation should be uuderstood. | It is unclear how this puzzling situation should be understood. |
Iu an Appendix of DEMO2b. model calculations for the atmospheric scattered light are now presented. | In an Appendix of BFM02b model calculations for the atmospheric scattered light are now presented. |
However. the nature of the problem dictates that it is hardly possible to achieve an absolute accuracy of - σαν] as required iu the DEMO2 method. | However, the nature of the problem dictates that it is hardly possible to achieve an absolute accuracy of $\lesssim$ $\times$ as required in the BFM02 method. |
Major problems are caused by e.g. the varving properties of the atimospheric acrosols and ground reflectance. as well as bw the insuffiicicutly known iutensity distributious of the main ight sources. the ZL. ISL. and DOL over the sky. | Major problems are caused by e.g. the varying properties of the atmospheric aerosols and ground reflectance, as well as by the insufficiently known intensity distributions of the main light sources, the ZL, ISL, and DGL over the sky. |
BFALO2b have adopted for the albedo of the aerosol particles a valueof ¢=0.59. | BFM02b have adopted for the albedo of the aerosol particles a valueof $a =0.59$. |
This value. given in Staude(1975).. was calculated for an ad hioc particle composition with a refractive index ofa7=το0.14. | This value, given in \citet[]{sta75}, was calculated for an ad hoc particle composition with a refractive index of$m = 1.5-0.1i$ . |
For a realistic acrosol composition. according to Carstane(1991) aud MeClatcheyetal. (1978).. the albedo is —0.91. | For a realistic aerosol composition, according to \citet[]{gar91} and \citet[]{mcc78}, , the albedo is $\sim$ 0.94. |
and for a varicty of differcut measured aerosol populations and conditions. | and for a variety of different measured aerosol populations and conditions. |
necessarily imply a dissipative pulse (due to the presence of pulse broadening) and may result in a misinterpretation of the true nature of the disturbance. | necessarily imply a dissipative pulse (due to the presence of pulse broadening) and may result in a misinterpretation of the true nature of the disturbance. |
This event shows inconclusive variation in the PA-averaged integrated pulse intensity with distance from both observed passbands. | This event shows inconclusive variation in the PA-averaged integrated pulse intensity with distance from both observed passbands. |
The higher cadence 171 ddata exhibits a generally increasing trend with distance. while the lower cadence 195 oobservations show negligible variation on average. but strong point-to-point variation. | The higher cadence 171 data exhibits a generally increasing trend with distance, while the lower cadence 195 observations show negligible variation on average, but strong point-to-point variation. |
The multi-passband. high-cadence observations afforded by the (SDO)) will allow the true variation (if any) to be determined to a high degree of accuracy. | The multi-passband, high-cadence observations afforded by the ) will allow the true variation (if any) to be determined to a high degree of accuracy. |
Three additional CBF events from 2007 December 12. 2009 February 12. and 2009 February 13 were also studied using the intensity profile technique (see Figures + to 7)). | Three additional CBF events from 2007 December 12, 2009 February 12, and 2009 February 13 were also studied using the intensity profile technique (see Figures \ref{fig:profile_20071207_A} to \ref{fig:profile_20090213_B}) ). |
The relationships discussed in Section 3. were plotted for these additional events. with approximately similar results observed for each event. | The relationships discussed in Section \ref{sect:methods} were plotted for these additional events, with approximately similar results observed for each event. |
Table | shows the kinematics of all the pulses studied and indicates that they displayed similar initial propagation velocities ( 240—450 km s! y. | Table \ref{tbl:events_characteristics} shows the kinematics of all the pulses studied and indicates that they displayed similar initial propagation velocities $\sim$ 240–450 km $^{-1}$ ). |
The 2007 May 19 event appears to have been a relatively fast event. with an initial velocity of «450 km s! and a statistically significant non-zero acceleration. | The 2007 May 19 event appears to have been a relatively fast event, with an initial velocity of $\sim$ 450 km $^{-1}$ and a statistically significant non-zero acceleration. |
The event of 2007 December 07 was much slower. with an initial velocity of ~260 km s! and a statistically significant negative acceleration as observed by both spacecraft. | The event of 2007 December 07 was much slower, with an initial velocity of $\sim$ 260 km $^{-1}$ and a statistically significant negative acceleration as observed by both spacecraft. |
The quadrature events of 2009 February 12 and 13 were different from each other despite originating from the same region. | The quadrature events of 2009 February 12 and 13 were different from each other despite originating from the same region. |
The 2009 February 12 event showed a faster initial velocity (~405 km s!) and stronger deceleration (~ -29|] m s). while the 2009 February 13 event had a slower initial velocity (~274 km s') and a much weaker negative acceleration ( —49 m 877). | The 2009 February 12 event showed a faster initial velocity $\sim$ 405 km $^{-1}$ ) and stronger deceleration $\sim$ $-$ 291 m $^{-2}$ ), while the 2009 February 13 event had a slower initial velocity $\sim$ 274 km $^{-1}$ ) and a much weaker negative acceleration $\sim$ $-$ 49 m $^{-2}$ ). |
The large errors associated with the acceleration terms given here indicate the difficulties associated with accurately determining. the kinematics of CBFs from low cadence observations. despite the minimization of errors through the use of both the intensity profile technique and bootstrapping analysis. | The large errors associated with the acceleration terms given here indicate the difficulties associated with accurately determining the kinematics of CBFs from low cadence observations, despite the minimization of errors through the use of both the intensity profile technique and bootstrapping analysis. |
The distance-time plots for each event are given in the online Figures 10 and 11. for comparison. | The distance-time plots for each event are given in the online Figures \ref{fig:app_kinematics_20071207} and \ref{fig:app_kinematics_200902} for comparison. |
Table | also shows the rate of change of spatial and temporal pulse width Ar and Ar with distance and time respectively. as well as the rate of change of the PA-averaged integrated pulse intensity. AZ,,;. associated with each event tthe slope of the lines shown in Figures 12 and 16)). | Table \ref{tbl:events_characteristics} also shows the rate of change of spatial and temporal pulse width $\Delta r$ and $\Delta \tau$ with distance and time respectively, as well as the rate of change of the PA-averaged integrated pulse intensity, $\Delta I_{tot}$, associated with each event the slope of the lines shown in Figures \ref{fig:broadening} and \ref{fig:intensity}) ). |
The results indicate that all of the observed events exhibited clear pulse broadening in both the spatial and temporal domains. with the d(Ar)/dr and d(AT)/dt parameters positive for both the 171 aand 195 ppassbands. | The results indicate that all of the observed events exhibited clear pulse broadening in both the spatial and temporal domains, with the $d(\Delta r)/dr$ and $d(\Delta \tau)/dt$ parameters positive for both the 171 and 195 passbands. |
This suggests that pulse broadening is a general characteristic of CBPFs and must be accounted for by all theories that seek to explain this phenomenon. | This suggests that pulse broadening is a general characteristic of CBFs and must be accounted for by all theories that seek to explain this phenomenon. |
The plots showing the variation in both spatial and temporal pulse width with distance and time respectively for each event are given in the online Figures 13 to 15.. | The plots showing the variation in both spatial and temporal pulse width with distance and time respectively for each event are given in the online Figures \ref{fig:app_broadening_20071207} to \ref{fig:app_broadening_20090213}. |
The variation 1n. PA-averaged integrated pulse intensity with distance is more difficult to interpret. | The variation in PA-averaged integrated pulse intensity with distance is more difficult to interpret. |
Both of the online Figures 17 and 18 show the variation in peak intensity (top). FWHM (middle) and PA-averaged integrated pulse intensity (bottom) with distance. | Both of the online Figures \ref{fig:app_intensity_20071207} and \ref{fig:app_intensity_200902} show the variation in peak intensity (top), FWHM (middle) and PA-averaged integrated pulse intensity (bottom) with distance. |
In each case. the 195 ppeak intensity drops with distance while the 171 ppeak intensity variation ts inconclusive. although the FWHM of each passband tends to increase. | In each case, the 195 peak intensity drops with distance while the 171 peak intensity variation is inconclusive, although the FWHM of each passband tends to increase. |
For the 2007 December 07 event. the resulting PA-averaged integrated pulse intensity shows a slight decrease with distance for the 195 ddata. while showing an apparent increase with distance for the 171 ddata. with a large separation between the 171 aand 195 ppassbands. | For the 2007 December 07 event, the resulting PA-averaged integrated pulse intensity shows a slight decrease with distance for the 195 data, while showing an apparent increase with distance for the 171 data, with a large separation between the 171 and 195 passbands. |
In contrast. both the 171 aanc 195 oobservations tend to drop with distance for the 2009 February 12 event. while the 2009 February 13 event shows an increase and decrease with distance for the 171 aanc 195 ddata respectively. | In contrast, both the 171 and 195 observations tend to drop with distance for the 2009 February 12 event, while the 2009 February 13 event shows an increase and decrease with distance for the 171 and 195 data respectively. |
These observations suggest that the variation in PA-averaged integrated pulse intensity with distance is not well-defined and requires further investigation. | These observations suggest that the variation in PA-averaged integrated pulse intensity with distance is not well-defined and requires further investigation. |
Gaussian constant of gravitational k=I. | Gaussian constant of gravitational $\mathbf{ k}^{2}=1$. |
Then perturbed mean motion » of (he primaries is⋅⋅given by n?=14"nd-Apr,3 τ. Where =a4b. a.b are flatness anc core parameters| respectively1 which1 determine: the density: profilet of2 the belt. | Then perturbed mean motion $n$ of the primaries isgiven by $n^{2}=1+\frac{3A_{2}}{2}+\frac{2M_b
r_c}{\left(r_c^2+T^2\right)^{3/2}}$ , where $T=\mathbf{a}+\mathbf{b}$, $\mathbf{a,b}$ are flatness and core parameters respectively which determine the density profile of the belt. |
r2Q=(1—409,2/43»7446.rz ==a"n is the oblateness coellicient of ma: r;. rj are the equatorial and polar radii of mie respectively. | $r_c^2=(1-\mu)q_1^{2/3}+\mu^2$ , $A_{2}=\frac{r^{2}_{e}-r^{2}_{p}}{5r^{2}}$ is the oblateness coefficient of $m_{2}$; $r_{e}$ , $r_{p}$ are the equatorial and polar radii of $m_{2}$ respectively. |
reντ is the distance between primaries and. =fi(/).yfo(/) ave the functions of lime / ie. / is only independent variable. | $r=\sqrt{x^2+y^2}$ is the distance between primaries and $x=f_1(t), y=f_2(t)$ are the functions of time $t$ i.e. $t$ is only independent variable. |
The mass parameter is po="ETT (9.537x10.! for the Sun-Jupiter and 3.00348x10." for the Sun-Earth mass distributions respectively | The mass parameter is $\mu=\frac{m_{2}}{m_{1}+m_{2}}$ $9.537 \times
10^{-4}$ for the Sun-Jupiter and $3.00348 \times 10^{-6}$ for the Sun-Earth mass distributions respectively ). |
loquid ds a mass reduction factor. where PF, is the solar radiation pressure force which is exactly apposite to the gravitational attraction force Fy. | $q_1=1-\frac{F_p}{F_g}$ is a mass reduction factor, where $F_{p}$ is the solar radiation pressure force which is exactly apposite to the gravitational attraction force $F_g$. |
The coordinates of mq. mo are {μιο) (1—yt.0) respectively. | The coordinates of $m_1$, $m_2$ are $(-\mu,0)$, $(1-\mu,0)$ respectively. |
In the above mentionedreference svstem and Mivamoto(1975). model. the equations of motion of the infinitesimal massparticle in the vy-plane formulated as[please see Ixushvah.(2008)..IXushivah. (2009a)]]: where From equations (1)) and (2)). the Jacobian integral is givenby: isrelated to the Jacobian constant Co =—2E. | In the above mentionedreference system and \citet*{MiyamotoNagai1975PASJ} model, the equations of motion of the infinitesimal massparticle in the $x
y$ -plane formulated as[please see \citet{Kushvah2008Ap&SS318}, \cite{Kushvah2009Ap&SS}] ]: where From equations \ref{eq:Omegax}) ) and \ref{eq:Omegay}) ), the Jacobian integral is givenby: whichis related to the Jacobian constant $C=-2E$ . |
The location ofthree collinear equilibrium points and twotriangular equilibrium points is computed by dividing the orbital plane into three parts Ly. Lus; ppc«(1— qu). Lo: (1—p)κc and La: c«-μι | The locationof three collinearequilibrium points and twotriangular equilibrium points is computed by dividing the orbital plane into three parts $L_1,L_{4(5)}$ : $\mu<x<(1-\mu)$ , $L_2$ : $(1-\mu)<x$ and $L_3$ : $x<-\mu$. |
For the | For the |
the cooling efficiency declines again. | the cooling efficiency declines again. |
This implies that a softening value should exist which maximizes the amount of cooled gas. while smaller and larger values lead to less cold gas as a result of spurious gas heating or lack of resolution. respectively. | This implies that a softening value should exist which maximizes the amount of cooled gas, while smaller and larger values lead to less cold gas as a result of spurious gas heating or lack of resolution, respectively. |
To check for this effect. we show in Figure 8. the dependence of the fraction of cooled baryons. fi. and of the number of identified galaxies 7,4 within the virial radius on the adopted gravitatiobal softening length. | To check for this effect, we show in Figure \ref{fig:fstar_soft} the dependence of the fraction of cooled baryons, $f_c$, and of the number of identified galaxies $n_{\rm gal}$ within the virial radius on the adopted gravitatiobal softening length. |
We show results both for the low—resolution (LR) and the high-resolution (HR) versions of the CL? and CLA clusters. | We show results both for the low--resolution (LR) and the high–resolution (HR) versions of the CL2 and CL4 clusters. |
In order to avoid mixing the effects of numerica heating with that of efficient feedback. we have performed these simulations by switching off feedback by galactic winds (NW). | In order to avoid mixing the effects of numerical heating with that of efficient feedback, we have performed these simulations by switching off feedback by galactic winds (NW). |
In the plot of Figure 8.. the softening is given in units of that adopted for our reference simulations. as reported in Table 2.. | In the plot of Figure \ref{fig:fstar_soft}, the softening is given in units of that adopted for our reference simulations, as reported in Table \ref{tab:res}. |
The results clearly contirm our expectation: both a too large and a too smal softening lead to a decrease of the amount of cooled gas. | The results clearly confirm our expectation: both a too large and a too small softening lead to a decrease of the amount of cooled gas. |
Quite interestingly. the softening (in units of the reference value) at which the cooled baryon fraction is maximised is the same at low and | Quite interestingly, the softening (in units of the reference value) at which the cooled baryon fraction is maximised is the same at low and |
Figure 4 presents the derived orbit of IID 164427. | Figure 4 presents the derived orbit of HD 164427. |
Traditionally. an astrometric solution is presented. eraphically by a diagram of the derived orbit on the plane of the sky. usually together with the individual txvo-dimensional measurements. | Traditionally, an astrometric solution is presented graphically by a diagram of the derived orbit on the plane of the sky, usually together with the individual two-dimensional measurements. |
This is impossible for the Hipparcos data. because these measurements are only one-dimensional. observed along the instantaneous relerence ereal circle at the time of the measurement. | This is impossible for the Hipparcos data, because these measurements are only one-dimensional, observed along the instantaneous reference great circle at the time of the measurement. |
However. (wo LHipparcos measurements observed αἱ very close ming will (wo different great-circle directions allow us. in principle. to derive a (6wo-dimensional stellar position. | However, two Hipparcos measurements observed at very close timing with two different great-circle directions allow us, in principle, to derive a two-dimensional stellar position. |
Because of the intrinsic uncertainty of the nmeasurenienis. more (han (wo measurements are desired for such an exercise. | Because of the intrinsic uncertainty of the measurements, more than two measurements are desired for such an exercise. |
Such a averaging of the Llipparcos data was used by IHalbwachs et ((2000) as a eraphic representation of (he astrometric orbit for the long-period binaries they have studie. | Such a ``two-dimensional averaging” of the Hipparcos data was used by Halbwachs et (2000) as a graphic representation of the astrometric orbit for the long-period binaries they have studied. |
The graphical presentation of WD 164427 was done in a similar way. | The graphical presentation of HD 164427 was done in a similar way. |
We folded the data with the orbital period and looked [or small groups of Hipparcos data points that. cluster around (he same orbital phase. | We folded the data with the orbital period and looked for small groups of Hipparcos data points that cluster around the same orbital phase. |
Such small clusters. with at least thiree points. were averaged. | Such small clusters, with at least three points, were ``two-dimensionally averaged”. |
The resulting points are presented in Figure ta. | The resulting points are presented in Figure 4a. |
The figure shows that the points are at about 2.3 meas away from the center. aid (therefore indicates. although without any quantitative measure. (hat (he orbit is real. | The figure shows that the points are at about 2–3 $mas$ away from the center, and therefore indicates, although without any quantitative measure, that the orbit is real. |
Figure da does not show the temporal dependence of the points. | Figure 4a does not show the temporal dependence of the points. |
To show this dependence we derived (he mean anomaly of each point from the correspondingobserved (rue anomaly. together with the other orbital elements. | To show this dependence we derived the mean anomaly of each point from the corresponding true anomaly, together with the other orbital elements. |
In Figure 4b we plot the mean anomalies of the ὃ points as a function of their orbital phase. | In Figure 4b we plot the mean anomalies of the 8 points as a function of their orbital phase. |
Without noise. ancl assuming (here is a genuine orbital motion. we would expect the points to lie along a straight line. which is also plotted. | Without noise, and assuming there is a genuine orbital motion, we would expect the points to lie along a straight line, which is also plotted. |
The figure clearly demonstrates the nature of the stellar orbital revolution. | The figure clearly demonstrates the nature of the stellar orbital revolution. |
We stress again that these 8 points are used only for graphic presentation and no real conclusion is drawn from them. | We stress again that these 8 points are used only for graphic presentation and no real conclusion is drawn from them. |
Any quantitative statement is based upon the full set of 28 one-dimensional measurements. | Any quantitative statement is based upon the full set of 28 one-dimensional measurements. |
The analvsis presented here shows that among (he planet candidates no orbit was detected with a significance higher than9956. | The analysis presented here shows that among the planet candidates no orbit was detected with a significance higher than. |
.. Out of 47 svstems. six orbits were derived with significance higher than90%. | Out of 47 systems, six orbits were derived with significance higher than. |
. These orbits are probably all false. as we expect to derive [rom the whole sample 4.7£2.1 false orbits with (his sienilicance or higher. | These orbits are probably all false, as we expect to derive from the whole sample $4.7 \pm 2.1$ false orbits with this significance or higher. |
Three of these orbits were derived with significance higher than9556.. while we expect 2.3+1.5 false orbits. | Three of these orbits were derived with significance higher than, while we expect $2.3 \pm 1.5$ false orbits. |
Although we ean not rule out the possibility that one or two of the six orbits are real. apparently the Hipparcos precision is not enough to vield detections of reflex motion induced by extrasolar planets with a high enough statistical sienilicance. | Although we can not rule out the possibility that one or two of the six orbits are real, apparently the Hipparcos precision is not enough to yield detections of reflex motion induced by extrasolar planets with a high enough statistical significance. |
This conclusion is contrary | This conclusion is contrary |
Moreover. since mereing or tidal capture requires (hat the planet-planet periapse clistance be no larger than a lew times Zi. The final expressions for AL/E and II/H follow from the assumption that the planet and star have similar densities. | Moreover, since merging or tidal capture requires that the planet-planet periapse distance be no larger than a few times $R_p$, The final expressions for $\Delta E/E$ and $\Delta H/H$ follow from the assumption that the planet and star have similar densities. |
In what follows we discard |.NI/II|<.NE/E since 1. | In what follows we discard $|\Delta H/ H|\ll \Delta E/E$ since $R_*/a\ll 1$ . |
Eccentricity is related to orbital energy and angular momentum by for Z and // using equations (1) (2) 30 the leading contribution in |rq —rs|/(rq+re)and writing AE A SI.we deduce that the eccentricity is bounded by EE: Maximal\laximal eccentricityniricitv iis achievedhi| if the initial orbits |have uthe same1 semimajornmiünmjor axis. | Eccentricity is related to orbital energy and angular momentum by Substituting for $E$ and $H$ using equations and yields Taking the leading contribution in $|r_1-r_2|/(r_1+r_2)$ and writing where $A\lesssim 1$, we deduce that the eccentricity is bounded by Maximal eccentricity is achieved if the initial orbits have the same semimajor axis. |
axi Alergers or captures are not possible for initial separations much larger than (he Ill radius. | Mergers or captures are not possible for initial separations much larger than the Hill radius. |
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