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Finally. our two random vectors sj and sy). are indepeudent from one another if aud only if the joined probability distribution Pj.s;44) of both siguals are factorizable into the product of their marginal pdfs. P(sj) aud δεν]: and satisty tle property where g(5;) aud σι) are absolutely integrable funcetious of s; aud δι respectively, | Finally, our two random vectors ${s_{l}}$ and ${s_{l+1}}$ are independent from one another if and only if the joined probability distribution $P({ s_{l}},{ s_{l+1}})$ of both signals are factorizable into the product of their marginal pdfs, $P({ s_{l}})$ and $P({ s_{l+1}})$ : and satisfy the property where $g({s}_{l})$ and $h({s}_{l+1})$ are absolutely integrable functions of ${s}_{l}$ and $s_{l+1}$ respectively. |
Frou the definition of independence in equation ÀA5.. we obtain the definition of uncorrelateduess (equation À3)) in the special case where both s; aud s;;4 are liuear aud are ouly defined by their covariances (Le. no higher order statistical moments) (Ivvarinen&Oja2000:ΠννάμοιOja.2001:Rilevetal.2002). | From the definition of independence in equation \ref{independence2}, we obtain the definition of uncorrelatedness (equation \ref{ortho}) ) in the special case where both ${ s_{l}}$ and ${ s_{l+1}}$ are linear and are only defined by their covariances (i.e. no higher order statistical moments) \citep{hyvarinen00, icabook, riley02}. |
. In other words. uncorrelatedness is a special ease of independence. | In other words, uncorrelatedness is a special case of independence. |
Cucorrelated Gaussian random variables are always also independent aud the definitions of uncorrelateduess. orthogouality (for zero mean) aud statistical independence become identical. | Uncorrelated Gaussian random variables are always also independent and the definitions of uncorrelatedness, orthogonality (for zero mean) and statistical independence become identical. |
The covariance matrix of X. Cy. is eiven by C,=EDET, where E is the matrix of cigcuvectors aud D the diagonal matrix of cigenvalues. D= diag(dy.do.....d,). | The covariance matrix of $\bf{X}$, $\bf{C}_{x}$, is given by $\bf{C}_{x} = \bf{E}\bf{D}\bf{E}^{T}$, where $\bf{E}$ is the matrix of eigenvectors and $\bf{D}$ the diagonal matrix of eigenvalues, $\textbf{D} = diag(d_{1}, d_{2},..., d_{n})$ . |
Using principal compoucut analysis (PCA). we compute aud aud the whitening matrix is hence theinverse square root covariance matrix ο. is then given by equation D2.UIvvirinen&Oja.2001:Jolliffe 2002).. | Using principal component analysis (PCA), we compute and and the whitening matrix is hence theinverse square root covariance matrix $\bf{C_{x}^{-1/2}}$ is then given by equation \ref{Vwhite} \citep{icabook, pcabook}. . |
masses are given by Alcodow ancl Moonign. | masses are given by ${\rm M}_{\rm c0,low}$ and ${\rm M}_{\rm c0,high}$. |
Phe equations are: where M is the ZAAIS mass. | The equations are: where M is the ZAMS mass. |
Vhe coellicients lor the cilferent values of Y and Z are shown in table 4 | The coefficients for the different values of $Y$ and $Z$ are shown in table \ref{tab:mczero}. |
The procedure used is to find the point of intersection between the two fits and then to plug in the relevant. mass. | The procedure used is to find the point of intersection between the two fits and then to plug in the relevant mass. |
Phe most obvious trend is there is an increase in the core mass as the value of 3 increases. | The most obvious trend is there is an increase in the core mass as the value of $Y$ increases. |
This is important since on the AGB a [arger core mass leads to a higher luminosity. | This is important since on the AGB a larger core mass leads to a higher luminosity. |
The luminosity density of the Universe in the range from about to iukl-UV to near-IR. is dominated by stellar cluission. | The luminosity density of the Universe in the range from about to, mid-UV to near-IR, is dominated by stellar emission. |
This wavelength range spans the peak in he spectra (f,) of stars with effective temperatures your about TI, down to 200019]. Therefore. neasurclents of the hununmositv density ia various xoadbands across this rauge provide a powerful constrain ou coste star-formation history (SEIT) and/or a universa stellar initial mass function (IME). | This wavelength range spans the peak in the spectra $f_\nu$ ) of stars with effective temperatures from about K down to K. Therefore, measurements of the luminosity density in various broadbands across this range provide a powerful constraint on cosmic star-formation history (SFH) and/or a universal stellar initial mass function (IMF). |
The stellar IMIF describes the relative probability of stars of different masses foriuius (seeCüluore&Towel1998.forrecentreviewsand analvses). | The stellar IMF describes the relative probability of stars of different masses forming \citep[see][for recent reviews and
analyses]{GH98}. |
Its importance crosses manv fields of astronomy from. for example. star formation (testing theoretical models) to cosmic chenica evolution (heavy metal production from high mass stars). | Its importance crosses many fields of astronomy from, for example, star formation (testing theoretical models) to cosmic chemical evolution (heavy metal production from high mass stars). |
Tt is widely used in the study of the SEIT of galaxies from their iuteerated spectra. | It is widely used in the study of the SFH of galaxies from their integrated spectra. |
Ceucrally, an IMPE is asse and used as an input to evolutionary. stellar population. svuthesis models. aud these models are fitted to inteeratcc spectra. | Generally, an IMF is assumed and used as an input to evolutionary, stellar population, synthesis models, and these models are fitted to integrated spectra. |
The first calculation of au IME. was made by Salpeter(1955) based on the observed Imuuimositv function of solar-ucieghborlood stars. couverting to mass. correcting for mmain-sequence lifetimes aud assuming that the star-formation rate (SER) has been constant for the las 5GGvr. | The first calculation of an IMF was made by \cite{Salpeter55} based on the observed luminosity function of solar-neighborhood stars, converting to mass, correcting for main-sequence lifetimes and assuming that the star-formation rate (SFR) has been constant for the last Gyr. |
Despite the uncertainties m mass-to-lieht ratios. stellar lifetimes and the SFR. this result (a power-law slope of 1.35 measured frou about 0.3 to A[sun)) is still commonly used. Aleasurements of the solu ucighborhood IMF were reviewed by Scalo(19856). produciug an IME with a mass fraction peak around ((Fig. 19). | Despite the uncertainties in mass-to-light ratios, stellar lifetimes and the SFR, this result (a power-law slope of $-1.35$ measured from about 0.3 to ) is still commonly used Measurements of the solar neighborhood IMF were reviewed by \cite{Scalo86} producing an IMF with a mass fraction peak around (Fig. \ref{fig:imfs}) ). |
When applied to galaxy. populations. this IME is unable to reproduce Hào ποτος (I&cuuicutt.Tam-να,&Congdou199 D)... and applied to cosmic evolution. is unable to match observed mean galaxy. colors (Macau.Pozzetti.&Dickinson1998) due to a too low fraction of Hel-mass stars (AL>10 AMsun)). | When applied to galaxy populations, this IMF is unable to reproduce $\alpha$ luminosities \citep*{KTC94}, and applied to cosmic evolution, is unable to match observed mean galaxy colors \citep*{MPD98} due to a too low fraction of high-mass stars $M>10$ ). |
Iu a more recent review bv Scalo(1998).. he coucluded hat the feld star IME was of questionable use for a ΠΠΟΙ of reasons tthe derived IAIF in the rauge ddepends strongly on the assumed solar-neighborhood SPILT). | In a more recent review by \cite{Scalo98}, he concluded that the field star IMF was of questionable use for a number of reasons the derived IMF in the range depends strongly on the assumed solar-neighborhood SFH). |
DIustead. he παπασος the results frou studving star clusters in a tripledudex power-law IMP as au estimate of an average ΙΣΠ (Fig. 13). | Instead, he summarized the results from studying star clusters in a triple-index power-law IMF as an estimate of an average IMF (Fig. \ref{fig:imfs}) ). |
We also noted that --- tho existing enirpirical estimates of the IME are taken at face value. they preseut strong evidence for variations. aud these variatious do not seen to depend systematically ou physical variables such as metallicity or stellar deusitv. | He also noted that “if the existing empirical estimates of the IMF are taken at face value, they present strong evidence for variations, and these variations do not seem to depend systematically on physical variables such as metallicity or stellar density”. |
The observed IME variations from stellar counts could be | The observed IMF variations from stellar counts could be |
the cliscussion iu the previous sectio1. we have also excludec all NLTT clwarls within 10 degrees of the Galactic Plane. | the discussion in the previous section, we have also excluded all NLTT dwarfs within 10 degrees of the Galactic Plane. |
Of the 58815 sowce int ie NLTT catakgue. 23795 (10.156)) have at least one ONLASS source within the search radius: :ipproximately 11OQ have two or more matches. giving a total of 25305 potential uear-infra'ed couerparts to the proper motion stars. | Of the 58845 source in the NLTT catalogue, 23795 ) have at least one 2MASS source within the search radius; approximately 1400 have two or more matches, giving a total of 25305 potential near-infrared counterparts to the proper motion stars. |
This dataset. provides the basis [or coistructing our ximnary NLTT sample of nearby. star Caldidates. | This dataset provides the basis for constructing our primary NLTT sample of nearby star candidates. |
However. it does not itclude al ol the NLTT sta‘s within the area covered by the currently-available 2MLASS data. | However, it does not include all of the NLTT stars within the area covered by the currently-available 2MASS data. |
We identilied those objects by 'emoviug the matched NLTT stars frou the search list. and re-ruuuing the dataowe query. but with a search radius of 60 aresecouds. | We identified those objects by removing the matched NLTT stars from the search list, and re-running the database query, but with a search radius of 60 arcseconds. |
A total of 1875 additional NLTT stars of the catalogue) lave potential 2ALASS counterparts at those larger. | A total of 4875 additional NLTT stars of the catalogue) have potential 2MASS counterparts at those larger. |
. Figure 3 plots the (0.9) distribttion of the two datasets. | Figure 3 plots the $\alpha, \delta$ ) distribution of the two datasets. |
[t is clear that the wide-paired NLTT stars (he [872 stars) are not raneoily clistributecl: there are obvious COLcentratious. uotably near the North Celestial Pole and wear he Sorth Galactic Pole —30°). | It is clear that the wide-paired NLTT stars (the 4875 stars) are not randomly distributed: there are obvious concentrations, notably near the North Celestial Pole and near the Sorth Galactic Pole $\alpha \sim 1^h, \delta \sim -30^o$ ). |
It is likely that these features sten from systematic er‘ors in the NLTT positious iu those regions. | It is likely that these features stem from systematic errors in the NLTT positions in those regions. |
Figure 3 highlights two issues: first. as al'eady discussed. a sizeable subset Zo?°) | Figure 3 highlights two issues: first, as already discussed, a sizeable subset ?) |
of the stars iu the NLTT catalogue have astrometry of ouv modest accuracy: secoucl. even though 23795 NLTT stars have 2ALASS sources within "there is no guarantee that those sources include he NLTT star itself. | of the stars in the NLTT catalogue have astrometry of only modest accuracy; second, even though 23795 NLTT stars have 2MASS sources within, there is no guarantee that those sources include the NLTT star itself. |
Thus. just as the NLTT catalogue includes only an incomplete subset of late-type dwarls with 20 parsecs. our cross-referencing against the 2M1ÀSS database succeeds in identifviug only a subset of the nearby late-type dwarls in the SLTT. | Thus, just as the NLTT catalogue includes only an incomplete subset of late-type dwarfs with 20 parsecs, our cross-referencing against the 2MASS database succeeds in identifying only a subset of the nearby late-type dwarfs in the NLTT. |
We will discuss the 1875 sources in he NLTT wide-matched sample in a later paper in tliis series: for the present. we coucentrate ou tle sample of 23795 NLTT dwarls with 2MASS counterparts within 10ο the predicted J2000 posiIOUS. | We will discuss the 4875 sources in the NLTT wide-matched sample in a later paper in this series; for the present, we concentrate on the sample of 23795 NLTT dwarfs with 2MASS counterparts within of the predicted J2000 positions. |
Clearly it is uureasonable to attempt detailed follow-up observations of all 25000+ potential NLTT/2MASS pairings. | Clearly it is unreasonable to attempt detailed follow-up observations of all 25000+ potential NLTT/2MASS pairings. |
However. we can use Luyten’s im, photometry to pare the sample to a manageable size. | However, we can use Luyten's $m_r$ photometry to pare the sample to a manageable size. |
Dawson's (1986) analysis of data lor over 2000 LHS stars coulirmecl Gliese Jalireib (1980) calibration of my, against standard Ixrou Ry photometry. deriviug The y-Ix;) colour spaus a loug baseline in waveleneth. and ranges [rom ~3.0 at spectral type MOQ to 6.6 at spectral type ΔΙΣ. | Dawson's (1986) analysis of data for over 2000 LHS stars confirmed Gliese ' (1980) calibration of $m_r$ against standard Kron $_K$ photometry, deriving The $_K$ $_s$ ) colour spans a long baseline in wavelength, and ranges from $\sim3.0$ at spectral type M0 to $\sim6.6$ at spectral type M8. |
Thus. even with uucertainties of 40.5 maguituce in 225. the location of a star in the (15... Gn,-1x;)) plaue can discriminate between a relatively distant early-type M dwarl aud an M6 dwarl in the unimediate Solar Neighbourhood. | Thus, even with uncertainties of $\pm0.5$ magnitude in $m_r$, the location of a star in the $m_r$, $m_r$ $_s$ )) plane can discriminate between a relatively distant early-type M dwarf and an M6 dwarf in the immediate Solar Neighbourhood. |
In this appendix we present the folded light-curves of the thirteen low amplitude transiting candidates. | In this appendix we present the folded light-curves of the thirteen low amplitude transiting candidates. |
The left panel shows for each star the complete Hght-curve. while the middle panel zooms in on —-0.15«phase«0.15. | The left panel shows for each star the complete light-curve, while the middle panel zooms in on $-0.15$$<$ $<$ 0.15. |
The right panel is for each star an extract from the Digitized Sky Survey. measuring 2x2 arcminutes. | The right panel is for each star an extract from the Digitized Sky Survey, measuring $\times$ 2 arcminutes. |
In fact, the only way we found to reconcile models and observations within —1σ is to invoke an ad-hoc increase in the atmospheric opacities, by an order of magnitude or more (we used κι=0.4cm?g! and x,=024cm? g!). | In fact, the only way we found to reconcile models and observations within $\sim 1\sigma$ is to invoke an ad-hoc increase in the atmospheric opacities, by an order of magnitude or more (we used $\kappa_{\rm th}=0.4 \rm\,cm^2\,g^{-1}$ and $\kappa_{\rm v}=0.24\rm\,cm^2\,g^{-1}$ ). |
This is in essence similar to what is proposed by Burrows et al. (2007)) | This is in essence similar to what is proposed by Burrows et al. \cite{burrows2007}) ) |
for planets with lower masses. | for planets with lower masses. |
This yields a slower cooling of the object that remains large for a longer time. | This yields a slower cooling of the object that remains large for a longer time. |
However, the comparison between KOI-423b and (Deleuil et al. 2008)) | However, the comparison between KOI-423b and CoRoT-3b (Deleuil et al. \cite{deleuil2008}) ) |
adds to the puzzle: the two companions have different sizes (CoRoT-3b lies where theory predicts it should be, KOI-423b is significantly larger), but they otherwise have very similar characteristics (mass, characteristics of the parent star). | adds to the puzzle: the two companions have different sizes (CoRoT-3b lies where theory predicts it should be, KOI-423b is significantly larger), but they otherwise have very similar characteristics (mass, characteristics of the parent star). |
KOI-423 has vsini=16€2.5kmsl, whereas CoRoT-3 has vsini=17+Ikms!, showing that both are relatively fast rotators. | KOI-423 has $v\sin i=16\pm2.5\rm\,km\,s^{-1}$, whereas CoRoT-3 has $v\sin i=17\pm1\rm\,km\,s^{-1}$, showing that both are relatively fast rotators. |
CoRoT-3 has a metallicity close to the Sun ([M/H]=—0.02+ 0.06) when KOI-423b is metal-poor. | CoRoT-3 has a metallicity close to the Sun $\rm [M/H]=-0.02\pm 0.06$ ) when KOI-423b is metal-poor. |
The smallest massive planet HAT-P-20b (Bakos et al. 2010)) | The smallest massive planet HAT-P-20b (Bakos et al. \cite{bakos2010}) ) |
orbits a host star with the strongest metallicity index [Fe/H] = 0.35. | orbits a host star with the strongest metallicity index [Fe/H] = 0.35. |
On one hand, this follows the trend observed at lower masses that planets around metal-rich stars contain more heavy elements and are therefore on average smaller (see Guillot 2008,, Bouchy et al. 2010)). | On one hand, this follows the trend observed at lower masses that planets around metal-rich stars contain more heavy elements and are therefore on average smaller (see Guillot \cite{guillot2008}, Bouchy et al. \cite{bouchy2010}) ). |
On the other hand, one would expect that the atmosphere of KOI-423b is poorer in heavy elements, hence yielding smaller opacities, and potentially, a smaller size. | On the other hand, one would expect that the atmosphere of KOI-423b is poorer in heavy elements, hence yielding smaller opacities, and potentially, a smaller size. |
One natural possibility would be to invoke age as a way to distinguish between the two objects. | One natural possibility would be to invoke age as a way to distinguish between the two objects. |
However, given the estimated relatively high age, it is difficult to imagine that 423 would be on the pre-main sequence (see Fig. 11)), | However, given the estimated relatively high age, it is difficult to imagine that KOI-423 would be on the pre-main sequence (see Fig. \ref{plotrage}) ), |
whereas CoRoT-3b would be much older. | whereas CoRoT-3b would be much older. |
We cannot pretend that we understand the large radius of KOI-423. | We cannot pretend that we understand the large radius of KOI-423. |
For these systems with a known substellar companion, we plot in Fig. | For these systems with a known substellar companion, we plot in Fig. |
12 the stellar effective temperature as a function of the companion mass for transiting objects or Msini value for nontransiting objects. | \ref{plottemp} the stellar effective temperature as a function of the companion mass for transiting objects or $M\sin i$ value for nontransiting objects. |
The data for low-mass companions (« 13Mjup) was obtained from exoplanets.org (see Wright et al. 2011)) | The data for low-mass companions $<~13\rm M_{Jup}$ ) was obtained from exoplanets.org (see Wright et al. \cite{wright2011}) ) |
and exoplanet.eu. | and exoplanet.eu. |
For objects with higher masses, no publicly available database exists, and we manually extracted parameters for the following transiting substellar systems: OGLE-TR-106 (Pont et al. 2005a)), | For objects with higher masses, no publicly available database exists, and we manually extracted parameters for the following transiting substellar systems: OGLE-TR-106 (Pont et al. \cite{pont2005a}) ), |
OGLE-TR-122 (Pont et al. 20050). | OGLE-TR-122 (Pont et al. \cite{pont2005b}) ), |
OGLE-TR-123 (Pont et al. 2006)), | OGLE-TR-123 (Pont et al. \cite{pont2006}) ), |
HAT-TR-205-013 (Beatty et al. 2007)), | HAT-TR-205-013 (Beatty et al. \cite{beatty2007}) ), |
CoRoT-15 (Bouchy et al. 2011)), | CoRoT-15 (Bouchy et al. \cite{bouchy2011}) ), |
WASP- (Anderson et al. 2011)), | WASP-30 (Anderson et al. \cite{anderson2011}) ), |
and LHS6343C (Johnson et al. 2011)). | and LHS6343C (Johnson et al. \cite{johnson2011}) ). |
And we did it for non-transiting short-period (P«30 days) substellar systems: NLTT41135C (Irwin et al. 2010)), | And we did it for non-transiting short-period $P<30$ days) substellar systems: NLTT41135C (Irwin et al. \cite{irwin2010}) ), |
HD189310 (Sahlmann et al. 2011)), | HD189310 (Sahlmann et al. \cite{sahlmann2011}) ), |
and HD180777 (Galland et al. 2006)). | and HD180777 (Galland et al. \cite{galland2006}) ). |
We numerically integrated the orbits of 500 test particles for different. values of (heir orbital elements. in a region between 30/2, and SOR). | We numerically integrated the orbits of 500 test particles for different values of their orbital elements, in a region between $30{R_J}$ and $80{R_J}$. |
Our integrations indicated. that the long-term stability of these objects is affected by (he values of their initial periastron distances. | Our integrations indicated that the long-term stability of these objects is affected by the values of their initial periastron distances. |
For given values of their semimajor axes. the region of instability of test particles extended to larger distances as the initial values of their orbital eccentricities were increased. | For given values of their semimajor axes, the region of instability of test particles extended to larger distances as the initial values of their orbital eccentricities were increased. |
Our numerical simulations also showed (hat. except al large distances [rom the outer boundaries of the influence zones of Ganvimecle and. Callisto. the lack of irregular satellites between Callisto and Themisto can be attributed to the instability of test particles caused bv their interactions with the two outermost Galilean satellites. | Our numerical simulations also showed that, except at large distances from the outer boundaries of the influence zones of Ganymede and Callisto, the lack of irregular satellites between Callisto and Themisto can be attributed to the instability of test particles caused by their interactions with the two outermost Galilean satellites. |
At larger distances (e.g. between ~404, to 8040; for particles in circular orbits. and between ~6042, to 804, Lor parücles with initial orbital eccentricities of 0.4). however. (he perturbations of Galilean salellites do not seem to be able to account for the instability of small bodies. | At larger distances (e.g., between $\sim 40{R_J}$ to $R_J$ for particles in circular orbits, and between $\sim 60{R_J}$ to $R_J$ for particles with initial orbital eccentricities of 0.4), however, the perturbations of Galilean satellites do not seem to be able to account for the instability of small bodies. |
A possible explanation is (hat their instabilitv is the result of interactions with Jovian satellitesimals and protosatellites during the formation of Jupiters regular moons. | A possible explanation is that their instability is the result of interactions with Jovian satellitesimals and protosatellites during the formation of Jupiter's regular moons. |
Decause the test particles in our simulations were initiallv close to Jupiter. we neglected the elfect of solar perturbations. | Because the test particles in our simulations were initially close to Jupiter, we neglected the effect of solar perturbations. |
As shown by HamiltonandIxrivov.(1997).. for Jupiter. the shortest critical distance bevond which the perturbation from (he Sun cannot be neglected corresponds to prograde orbits and is equal to 3898;. | As shown by \citet{Hamilton97}, for Jupiter, the shortest critical distance beyond which the perturbation from the Sun cannot be neglected corresponds to prograde orbits and is equal to $R_J$. |
In our simulations. the outermost test parücle was placed well insicle this region al 80/2;. | In our simulations, the outermost test particle was placed well inside this region at $R_J$. |
It is important (to note that. although the effects of solar perturbations on our (est particles are small auc will not cause orbital instability. they max. in the long term. create noticeable changes in the orbital evolution of test. particles. | It is important to note that, although the effects of solar perturbations on our test particles are small and will not cause orbital instability, they may, in the long term, create noticeable changes in the orbital evolution of test particles. |
For instance. solar perturbations may enhance the perturbative effects of regular satellites in increasing the orbital eccentricity of test. particles ancl result in (heir capture in Ixozai resonance. | For instance, solar perturbations may enhance the perturbative effects of regular satellites in increasing the orbital eccentricity of test particles and result in their capture in Kozai resonance. |
More nmunerical simulations are needed (o explore these effects. | More numerical simulations are needed to explore these effects. |
As mentioned in 82. different non-zero angular elements may in fact affect the stability of individual test particles. | As mentioned in 2, different non-zero angular elements may in fact affect the stability of individual test particles. |
However. (he analysis of the stability of the svstem. as obtained from our numerical simulations. portrays a picture of the dynamical characteristics of the lest. particles that. in general. is also applicable {ο Jovian irregular satellite svstemis with other initial angular variables. | However, the analysis of the stability of the system, as obtained from our numerical simulations, portrays a picture of the dynamical characteristics of the test particles that, in general, is also applicable to Jovian irregular satellite systems with other initial angular variables. |
The applicabilitw of our results ancl (he extension of our analvsis (ο the satellite-void boundary regions around other giant. planets may be limited due to fact that their satellite svstems are different from that of Jupiter. | The applicability of our results and the extension of our analysis to the satellite-void boundary regions around other giant planets may be limited due to fact that their satellite systems are different from that of Jupiter. |
Although the above-mentioned dvnamicalclearing process can still account lor the instability of many small objects around. these planets. inuuerical simulations. similar to those presented here. are necessary (o understaxd the | Although the above-mentioned dynamical-clearing process can still account for the instability of many small objects around these planets, numerical simulations, similar to those presented here, are necessary to understand the |
contribution increases providing agreement with the COSY data, | contribution increases providing agreement with the COSY data. |
Ou the other hand. more hard NN-models like Paris ancl especially RSC provide (oo large magnitude of (he hieh momentum components of the NN-wave function and. therefore. lead to strong contradiction with the data especially above | GeV (see Ref. [6])). | On the other hand, more hard NN-models like Paris and especially RSC provide too large magnitude of the high momentum components of the NN-wave function and, therefore, lead to strong contradiction with the data especially above 1 GeV (see Ref. \cite{jhuz2003}) ). |
Further analysis [7]. within the OPE model with the subprocess z"d—pn provided an independent confirmation of the dominant contribution of the A(1232)-isohar in this reaction at 0.5Hr—1 GeV. and suggested. sizable acimixture of the ONE mechanism compatible with the CD Bonn moclel. | Further analysis \cite{uzjhcw}
within the OPE model with the subprocess $\pi ^0d\to pn$ provided an independent confirmation of the dominant contribution of the $\Delta(1232)$ -isobar in this reaction at $0.5 - 1$ GeV and suggested sizable admixture of the ONE mechanism compatible with the CD Bonn model. |
The reaction pp—(ppl.s" is the simplest inelastic process in the pp-collision. which can reveal underlving dvnamics of NN interaction. | The reaction $pp\to \{pp\}_s\pi^0$ is the simplest inelastic process in the pp-collision, which can reveal underlying dynamics of NN interaction. |
Restriction to only one pp-partial wave in (he final state considerably simplifies a comparison with theory. | Restriction to only one pp-partial wave (s-wave) in the final state considerably simplifies a comparison with theory. |
The reaction pp>ppl" is very similar kinematically to the reaction pp—dz. but its dynamics can be essentially different. | The reaction $pp\to \{pp\}_s\pi^0$ is very similar kinematically to the reaction $pp\to d\pi^+$, but its dynamics can be essentially different. |
In fact. quantum numbers of the diproton state (/720.f=0.L= 0) diller from these for the deuteron (/=0.0.5LL 0.5]. | In fact, quantum numbers of the diproton state $J^\pi=0^+,\,I=1,\, S=0, \, L=0$ ) differ from these for the deuteron $J^\pi=0^+, I=0,\, S=1, L=0,2$ ). |
Therefore. transition matrix elements for these two reactions are also different. | Therefore, transition matrix elements for these two reactions are also different. |
Due to the eeneralized Pauli principle and angular momentum and P-pariry conservation only negative parity states are allowed in the reaction pp—Ípp]).z". | Due to the generalized Pauli principle and angular momentum and P-pariry conservation only negative parity states are allowed in the reaction $pp\to \{pp\}_s\pi^0$. |
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