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We plan to perform additional HIFI observations in the THz region towards LI157-B1 to observe more species and transitions. thus to be able to derive reliable abundances and study of the different gas components associated with the bow structure. | We plan to perform additional HIFI observations in the THz region towards L1157-B1 to observe more species and transitions, thus to be able to derive reliable abundances and study of the different gas components associated with the bow structure. |
io 5% (Albrecht Weller 2000). | to 5 (Albrecht Weller 2000). |
This raises svstematic effects as (he main source of concern (Iloflichetal.1998). | This raises systematic effects as the main source of concern \citep{Hoeflich:1998}. |
. In this context. it is important (to understand observationally whether or not SN Ia are spherically svimnietric since any deviations [rom sphericity could potentially affect the accuracy of distance estimates. | In this context, it is important to understand observationally whether or not SN Ia are spherically symmetric since any deviations from sphericity could potentially affect the accuracy of distance estimates. |
Aside from the new focus on SN Ia as cosmological tools. there are long-standing problems associated with determining (he progenitor evolution aud the physical processes involved in the explosion bv thermonuclear combustion. | Aside from the new focus on SN Ia as cosmological tools, there are long-standing problems associated with determining the progenitor evolution and the physical processes involved in the explosion by thermonuclear combustion. |
It is nearly universally assumed (hat SN Ia result [rom some form of binary evolution. but no direct observational evidence [or (his conjecture has ever been presented. | It is nearly universally assumed that SN Ia result from some form of binary evolution, but no direct observational evidence for this conjecture has ever been presented. |
There is also general agreement (hal SNe Ia result [rom some process involving the combustion of a degenerate white clwarl (WD) 1960). | There is also general agreement that SNe Ia result from some process involving the combustion of a degenerate white dwarf (WD) \citep{Hoyle:1960}. |
. Within this general picture. three classes of models have been considered: (1) an explosion of a carbou/oxveen (CO) WD. with mass close to the Chandrasekhar limit. (hat accretes mass through Roche-lobe overflow from an evolved companion star 19173): (2) an explosion of a rotating configuration formed from the merging of (vo low-mass WDs. caused by the loss of angular momentum through gravitational radiation (WebbinkIben&Tutukov1984:Paczviski 1955): and (3) explosion of a low mass CO-WD triggered by the detonation of a helium laver (Weaver.Axelrod.&Woosley1980:WallaceWooslev1981:Nomoto1982:Woosley&Weaver 1936).. | Within this general picture, three classes of models have been considered: (1) an explosion of a carbon/oxygen (CO) WD, with mass close to the Chandrasekhar limit, that accretes mass through Roche-lobe overflow from an evolved companion star \citep{Whelan:1973}; (2) an explosion of a rotating configuration formed from the merging of two low-mass WDs, caused by the loss of angular momentum through gravitational radiation \citep{Webbink:1984,Iben:1984,Paczynski:1985}; and (3) explosion of a low mass CO-WD triggered by the detonation of a helium layer \citep{Weaver:1980,Wallace:1981,Nomoto:1982,WoosWeav:1986}. |
Only the first two models appear to be viable for observed SN Ia (IHóflich&Khokhlov1996:Nugentοἱal.1997). | Only the first two models appear to be viable for observed SN Ia \citep{HofKho:1996, Nugent:1997}. |
. Although not [avored [or most SN Ia. the merging of two WDs max contribute to the SNe Ia population particularly. perhaps. to subluminous events (ILowelletal.2001).. | Although not favored for most SN Ia, the merging of two WDs may contribute to the SNe Ia population particularly, perhaps, to subluminous events \citep{Howell:99by}. |
All three scenarios might leave some (race in polarization data. | All three scenarios might leave some trace in polarization data. |
There are at least three wavs in which the [act that SN Ia evolve and explode in binary svstems could impart a dominant axis to the explosion that could be reflected in (he polarization. | There are at least three ways in which the fact that SN Ia evolve and explode in binary systems could impart a dominant axis to the explosion that could be reflected in the polarization. |
Perhaps the most general is that the WD could be rotating. | Perhaps the most general is that the WD could be rotating. |
Rotation could allect the shape of the WD. or it could affect the propagation of the thermonuclear burning (see below) and hence the distribution of either (he elements produced or the density or both (Ilowelletal.2001).. | Rotation could affect the shape of the WD, or it could affect the propagation of the thermonuclear burning (see below) and hence the distribution of either the elements produced or the density or both \citep{Howell:99by}. |
Another generic asvuunetric feature is an accretion disk. | Another generic asymmetric feature is an accretion disk. |
Accretion disks resulting [from mass transfer are not expected (o be very massive and hence it is difficult to see how the swept up disk matter could directly alfect the asymmetry. | Accretion disks resulting from mass transfer are not expected to be very massive and hence it is difficult to see how the swept up disk matter could directly affect the asymmetry. |
The exception might be if the disk comes from the disruption of a binary WD companion. in which case ihe remnant disk at the time of the explosion could be thick and dense. | The exception might be if the disk comes from the disruption of a binary WD companion, in which case the remnant disk at the time of the explosion could be thick and dense. |
A third asvanmetry eeneric to binary evolution models for SN Ia 15 a binary companion. | A third asymmetry generic to binary evolution models for SN Ia is a binary companion. |
In (wo of the scenarios. the Ave, models and (he helium detonators. the binary companion should still exist. | In two of the scenarios, the $M_{Ch}$ models and the helium detonators, the binary companion should still exist. |
The collision of the ejecta with this companion must induce some level of asymmetry in the ejecta (Livneetal.1992:Marietta 2000).. | The collision of the ejecta with this companion must induce some level of asymmetry in the ejecta \citep{Livne:1992,Marietta:2000}. . |
Ever since the first extrasolar planet around a solar-type star was detected (?).. questions about the composition and origin of extrasolar planetary objects (exo-planets) have been of major interest. | Ever since the first extrasolar planet around a solar-type star was detected \citep{MayorQueloz95}, questions about the composition and origin of extrasolar planetary objects (exo-planets) have been of major interest. |
Models of exoplanets are often little constrained based on the observable parameters mass. radius. and effective temperature. in particular metal-rich planets (2)... | Models of exoplanets are often little constrained based on the observable parameters mass, radius, and effective temperature, in particular metal-rich planets \citep{Adamsetal08}. |
For solar planets additional constraints are provided by the gravitational moments which have been measured by spacecraft or Earth-based observations of the motion of satellites and hence are not accessible for extrasolar planets. | For solar planets additional constraints are provided by the gravitational moments which have been measured by spacecraft or Earth-based observations of the motion of satellites and hence are not accessible for extrasolar planets. |
However. a similar quantity does exist: the tidal Love number ‘>. | However, a similar quantity does exist: the tidal Love number $k_2$. |
To first order in. the dimensionless number that describes the effect of rigid rotation or degree 2 tidal distortion A> is equivalent to J+ (seee.g.?).. | To first order in the dimensionless number that describes the effect of rigid rotation or degree 2 tidal distortion $k_2$ is equivalent to $J_2$ \citep[see e.g.][]{Hubbard84}. |
The tidal Love number &» is a potentially observable parameter. | The tidal Love number $k_2$ is a potentially observable parameter. |
?.— showed that the dominant source of apsidal precession of Hot Jupiters ts the tidal interaction between the planet and its star. | \citet{RagozzineWolf09} showed that the dominant source of apsidal precession of Hot Jupiters is the tidal interaction between the planet and its star. |
This tidally induced apsidal precession creates a unique variation in the transit light curve which is detectable by space-based missions likeKepler. | This tidally induced apsidal precession creates a unique variation in the transit light curve which is detectable by space-based missions like. |
.. Another possibility of determining A> is the measurement of the orbital parameters of a two-planet system in apsidal alignment (?).. | Another possibility of determining $k_2$ is the measurement of the orbital parameters of a two-planet system in apsidal alignment \citep{Batyginetal09}. |
Due to tidal dissipation a coplanar two-planet system can evolve into a tidal fixed point which is characterized by the alignment of the apsidal lines (?) and both orbits precess with the same rate. | Due to tidal dissipation a coplanar two-planet system can evolve into a tidal fixed point which is characterized by the alignment of the apsidal lines \citep{Mardling07} and both orbits precess with the same rate. |
? showed that in this state the Love number K is à function of the inner planets eccentricity. | \citet{Batyginetal09} showed that in this state the Love number $k_2$ is a function of the inner planet's eccentricity. |
Like Js for the solar system planets. Ko. if known. can be used to further constrain the models of extrasolar planets as It Is sensitive to the internal density distribution of the planet. | Like $J_2$ for the solar system planets, $k_2$, if known, can be used to further constrain the models of extrasolar planets as it is sensitive to the internal density distribution of the planet. |
Understanding the planetary interior ts important for determining not only physical processes but also the formation history. | Understanding the planetary interior is important for determining not only physical processes but also the formation history. |
Hence. it is crucial to analyze what information can be extracted from a measured &» and its implications on the planetary interior. | Hence, it is crucial to analyze what information can be extracted from a measured $k_2$ and its implications on the planetary interior. |
First. we will deseribe the definition and calculation of the Love numbers in Sect. 2.. | First, we will describe the definition and calculation of the Love numbers in Sect. \ref{sec:DefCalc2L}. |
We also confirm the correlation between the central condensation of a planet and its Love number Kk» within a simple two-layer model. | We also confirm the correlation between the central condensation of a planet and its Love number $k_2$ within a simple two-layer model. |
In Sect. | In Sect. |
3. we introduce à more sophisticated three-layer planetary model and demonstrate the degeneracy of &» with respect to the density discontinuity in the envelope. | \ref{sec:3L-model} we introduce a more sophisticated three-layer planetary model and demonstrate the degeneracy of $k_2$ with respect to the density discontinuity in the envelope. |
We apply these results to Saturn and to the Hot Neptune 4436b in Sect. 4.. | We apply these results to Saturn and to the Hot Neptune 436b in Sect. \ref{sec:planets}. |
The main results of this paper are summarized in Sect. 5.. | The main results of this paper are summarized in Sect. \ref{sec:sum}. |
Love Numbers quantify the deformation of the gravity field of a planet in response to an external perturbing body of mass M. which can be the parent star. another planet or a satellite. M. | Love Numbers quantify the deformation of the gravity field of a planet in response to an external perturbing body of mass $M$, which can be the parent star, another planet or a satellite. $M$, |
moving in a circular orbit of radius ¢ around a planet. causes a tide-raising potential (?) where s is the radial coordinate of the point under consideration inside the planet. 6 the angle between the planetary mass element at s» and the center of mass of M at a. and P, are Legendre polynomials. | moving in a circular orbit of radius $a$ around a planet, causes a tide-raising potential \citep{ZharkovTrubitsyn78}
where $s$ is the radial coordinate of the point under consideration inside the planet, $\theta'$ the angle between the planetary mass element at $s$ and the center of mass of $M$ at $a$, and $P_n$ are Legendre polynomials. |
Due to the tidally induced mass shift the planet’s potential changes by VCs) K,GOW,CG). where Λι) 1s the Love function (2).. | Due to the tidally induced mass shift the planet's potential changes by $V^{\mathrm{ind}}_n(s)=K_n(s)W_n(s)$ , where $K_n(s)$ is the Love function \citep{Love11}. |
Thus. at the planet's surfacethe definition of the Love numbers &, reads As we are interested in low eccentricity synchronous orbits. we concentrate on the purely hydrostatic tides. | Thus, at the planet's surfacethe definition of the Love numbers $k_n$ reads As we are interested in low eccentricity synchronous orbits, we concentrate on the purely hydrostatic tides. |
For the calculation of the Love numbers we follow the approach by ?.. see also ? and ?.. | For the calculation of the Love numbers we follow the approach by \citet{ZharkovTrubitsyn78}, , see also \citet{GavrilovZharkov77} and \citet{Gavrilovetal75}. |
A Love number of degree 7 is obtained from where ΤΑ) is the value of the function Τη) at the planet's surface. Δρ is the radius of the planet and go the surface gravity for the unperturbed planet. | A Love number of degree $n$ is obtained from where $T_n(R_\mathrm{p})$ is the value of the function $T_n(s)$ at the planet's surface, $R_\mathrm{p}$ is the radius of the planet and $g_0$ the surface gravity for the unperturbed planet. |
The function 7,,(s) satisfies the following second order differential equation: The radial coordinate ts represented by s» and p(s) and V(s) give the unperturbed density distribution and.potential of the planet. respectively. | The function $T_n(s)$ satisfies the following second order differential equation: The radial coordinate is represented by $s$ and $\rho(s)$ and $V(s)$ give the unperturbed density distribution andpotential of the planet, respectively. |
The primes denote first and second differentiation with respect to the radius s. | The primes denote first and second differentiation with respect to the radius $s$ . |
If the planet has an | If the planet has an |
A closer look al the members of the spectroscopically confirmed. GRB/SN eroup (on which the GRB/SNe association hinges) reveals (hat four out of the six detected bursts: GhRDB9850425 (1005) GRB031203 (8N20031w). GRD060218 (SN2006a7) and GRD100316D (SN2010bh). are (uite different (han “normal” LGRDs (see 82): Thev are less Iuminous: have a smooth non-variable lighteurve. and show no evidence for à high enereyv power-law (ail. | A closer look at the members of the spectroscopically confirmed GRB/SN group (on which the GRB/SNe association hinges) reveals that four out of the six detected bursts: GRB980425 (SN1998bw), GRB031203 (SN2003lw), GRB060218 (SN2006aj) and GRB100316D (SN2010bh), are quite different than “normal" LGRBs (see 2): They are less luminous; have a smooth non-variable lightcurve, and show no evidence for a high energy power-law tail. |
Although only a handful of such bursts were observed. the small observable volume set by (heir low luminosity implies an event rate much hieher than the rate of LGRBs pointing towards Earth (Coward2005:Cobbetal.2006:PianSoderberg20062:Liangetal.2007:Guetta&DellaValleFan 2011). | Although only a handful of such bursts were observed, the small observable volume set by their low luminosity implies an event rate much higher than the rate of LGRBs pointing towards Earth \citep{Coward05,Cobb06,Pian06,Soderberg06,Liang07,Guetta07, Fan11}. |
. The unique characteristics of these low luminosity GRBs (denoted hereafter /-GRBs) suggest that thev may be generated bv a totally different process than most LGRBs. | The unique characteristics of these low luminosity GRBs (denoted hereafter ) suggest that they may be generated by a totally different process than most LGRBs. |
As such it is of great. interest to. check whether can arise rom Collapsars. | As such it is of great interest to check whether can arise from Collapsars. |
Specilically we ask (he question: can be eenerated by relativistic jets that break out of their progenitor stars. | Specifically we ask the question: can be generated by relativistic jets that break out of their progenitor stars. |
To answer this question we study. in 33. (Iollowing Bromberg et al. | To answer this question we study, in 3, (following Bromberg et al. |
2011: hereafter D11) the propagation of a relativistic jet in a stellar envelope. | 2011; hereafter B11) the propagation of a relativistic jet in a stellar envelope. |
We obtain the minimal conditions required [or the jet to break out of the star. | We obtain the minimal conditions required for the jet to break out of the star. |
Specifically. we estimate the minimal tme (hat (he central engine must power the jet [or a successful crossing of (he star. | Specifically, we estimate the minimal time that the central engine must power the jet for a successful crossing of the star. |
Using this minimal breakout time we examine the expected duration distributions of LGDs. SGRBs and (84). | Using this minimal breakout time we examine the expected duration distributions of LGRBs, SGRBs and 4). |
We discuss the implications of this distribution on the origin of in 35 and we summarize our results in $6. | We discuss the implications of this distribution on the origin of in 5 and we summarize our results in 6. |
are characterized bv isotropic equivalent luminosities. 10!—107 erg/s. that are much lower than twpical. 10?!—10?* eres/s. emitted bv LGRDs. | are characterized by isotropic equivalent luminosities, $10^{46}-10^{48}$ erg/s, that are much lower than typical, $10^{51}-10^{53}$ ergs/s, emitted by LGRBs. |
The durations range between ~10 sec to an hour (in an extreme case of GRD 060218). and the corresponding isotropic equivalentenergies of E,=107— a [ew times LO! eres. are two to three orders of magnitude lower than those of typical LGRDs. | The durations range between $\sim10$ sec to an hour (in an extreme case of GRB 060218), and the corresponding isotropic equivalentenergies of $E_{\gamma}=10^{48}-$ a few times $10^{49}$ ergs, are two to three orders of magnitude lower than those of typical LGRBs. |
Apart from the Iow-Iuminositv which defines (his sub-eroup. have a softer spectrum with tvpical peak energies sienilicantly below the average of LGRBs and with no evidence of high energy tail. | Apart from the low-luminosity which defines this sub-group, have a softer spectrum with typical peak energies significantly below the average of LGRBs and with no evidence of high energy tail. |
Finally. 4-GIJDs. lighteurves are smooth. each contsüning only a single pulse. | Finally, ' lightcurves are smooth, each containing only a single pulse. |
Most are accompanied by energetic broad line type Ic SNe with a strong radio emission. | Most are accompanied by energetic broad line type Ic SNe with a strong radio emission. |
Radiation models ascribe the radio emission to a mildly relativistic shock moving ahead of the non-relativistic SN material (e.g.al. 1998). | Radiation models ascribe the radio emission to a mildly relativistic shock moving ahead of the non-relativistic SN material \citep[e.g.][]{Kulkarni98}. |
. Rebrightening episodes in the radio emission are commonly associated wilh additional supply of energy that refresh the shock. indicating the presence of an internal engine (hat can operate for long times (Li&Chevalier 1999).. | Rebrightening episodes in the radio emission are commonly associated with additional supply of energy that refresh the shock, indicating the presence of an internal engine that can operate for long times \citep{Li99}. . |
Finally. the alterelow of some | Finally, the afterglow of some |
Greenberg.Singh.&deAlmeida(1993) proposed that three effects could combine linearly to inhibit the detection of CN and Cs in Comet Yanaka. | \citet{G+93} proposed that three effects could combine non-linearly to inhibit the detection of CN and $_2$ in Comet Yanaka. |
First. cosmic rav exposure could "glue together” or "carbonize" the comet (deplete II. N. and O relative to C). with two ellects: trapping carbon-bearing eas molecules between dust grains. and producing larger dust grains more resistant to heating and fragmentation. | First, cosmic ray exposure could “glue together” or “carbonize” the comet (deplete H, N, and O relative to C), with two effects: trapping carbon-bearing gas molecules between dust grains, and producing larger dust grains more resistant to heating and fragmentation. |
Lastly. since Comet Yanaka was observed over a smaller spatial dimension (660 x 16.470 kin) than the other comets in Fink (1992: 3420 x 34.200 km and 2400 x 145.600 km). there may be less time for molecule production to occur within the spectrograph slit aperture. | Lastly, since Comet Yanaka was observed over a smaller spatial dimension (660 $\times$ 16,470 km) than the other comets in Fink (1992; 3420 $\times$ 84,200 km and 2400 $\times$ 145,600 km), there may be less time for molecule production to occur within the spectrograph slit aperture. |
Our observation of 96P covers a spatial area of about 820 x 70.570 km. | Our observation of 96P covers a spatial area of about 820 $\times$ 70,570 km. |
Our slit length is comparable to the larger spatial observations in Fink(1992). | Our slit length is comparable to the larger spatial observations in \citet{F92}. |
. Cochran(1985) reviews 3.5x10! km). and Cy (1.02.5x 10° km). which are all shorter than our slit length. | \citet{C85} reviews Haser-model decay scale lengths for the parent molecules of CN $\times$ $^4$ km), $_2$ $\times$ $^4$ km), and $_3$ $\times$ $^3$ km), which are all shorter than our slit length. |
If cosmic rav gluing increased (he parent decav scale length of Cy bv an order of magnitude. we still expect to see Cy production. | If cosmic ray gluing increased the parent decay scale length of $_3$ by an order of magnitude, we still expect to see $_3$ production. |
Brightness and polarization measurements of dust in the coma οἱ 96P (Grvnko.Jockers.&Schwenn2004). show grain sizes consistent with in-situ measurenientis of Halley dust. also pointing to limited cosmic ray. eluine. | Brightness and polarization measurements of dust in the coma of 96P \citep{G+04} show grain sizes consistent with in-situ measurements of Halley dust, also pointing to limited cosmic ray gluing. |
Greenberg.Singh.&deAlmeida(1993) admit that other dvaamically new comets do not display extreme carbon depletions. | \citet{G+93} admit that other dynamically new comets do not display extreme carbon depletions. |
Likewise. other short-period comets (2?< 7 vears). presumably with surfaces similarly processed by solar radiation as 96P. displav measurable carbon production: 2P/Encke and 6P/d'Arrest (AHearn.Millis.&Bireh1979).. 9P/Tempel 1 (Laraetal.2006).. L9P/Borrelly (lamaneetal...2002).. 26P/Grige-Skjellerup (Jockerselal. 1993).. 46D/Wirtanen (Jockers.Credner.&Boney1998).. and 67P/Churvumov- (Schulz.Stiwe.&Boehnhardt2004). | Likewise, other short-period comets $P <$ 7 years), presumably with surfaces similarly processed by solar radiation as 96P, display measurable carbon production: 2P/Encke and 6P/d'Arrest \citep{A+79}, 9P/Tempel 1 \citep{L+06}, 19P/Borrelly \citep{H+02}, 26P/Grigg-Skjellerup \citep{J+93}, 46P/Wirtanen \citep{J+98}, and 67P/Churyumov-Gerasimenko \citep{S+04}. |
. In old “dusty” comets. or comets moving outbound from perihelion. the CN and CS emission bands are the last to be observable as elobal gas production turns off (ie. ΑHearn et al. | In old “dusty” comets, or comets moving outbound from perihelion, the CN and $_2$ emission bands are the last to be observable as global gas production turns off (i.e. A'Hearn et al. |
1995. Fink Hicks 1996). such that CN is used as a gas-to-dust indicator (i.e. Storrs et al. | 1995, Fink Hicks 1996), such that CN is used as a gas-to-dust indicator (i.e. Storrs et al. |
1992). | 1992). |
IIowever. CN is not detected in the Nast spectra of 96D. Other dormant comets may perhaps show mostly ices. and nol carbon-bearing gas. when (μον outburst. | However, CN is not detected in the Kast spectra of 96P. Other dormant comets may perhaps show mostly ices, and not carbon-bearing gas, when they outburst. |
However. CN was detected in 95P/Chiron bx Busetal.(1991). | However, CN was detected in 95P/Chiron by \citet{B+91}. |
Comet 42/Fave was observed in outburst by Grothues(1996).. ancl has measurable CN. Co. and Cy production rates (Gil-DIntton&Licancdro1994). | Comet 4P/Faye was observed in outburst by \citet{G96}, and has measurable CN, $_2$, and $_3$ production rates \citep{GL94}. |
. Chamberlinetal.(1996) detected no CN in three candidate comet-asteroid (transition objects. although their signal-to-noise deteriorates in the region of the CN band. | \citet{C+96} detected no CN in three candidate comet-asteroid transition objects, although their signal-to-noise deteriorates in the region of the CN band. |
Unfortunately. none of these observations were programmed to look for the NIT 3360 [ρα]. | Unfortunately, none of these observations were programmed to look for the NH 3360 band. |
llaving exhausted several explanations for the odd spectrum of 962. we can compare il | Having exhausted several explanations for the odd spectrum of 96P, we can compare it |
demonstrates (hat in (he separator case a magnetic null is also kev to Che reconnection with field lines passing through the null at the moment their global connectivity changes. again when the flux velocity is infinite. | demonstrates that in the separator case a magnetic null is also key to the reconnection with field lines passing through the null at the moment their global connectivity changes, again when the flux velocity is infinite. |
However. (his is a non-local process with the null itself [ar removed [rom the reconnection site (indeed the null may lie in an ideal environment). | However, this is a non-local process with the null itself far removed from the reconnection site (indeed the null may lie in an ideal environment). |
The magnetic separator itself has only an indirect role in (he process. | The magnetic separator itself has only an indirect role in the process. |
We suggest that. physically. the locations of the singularity in the flux. velocity may be associated with regions of strong particle acceleration in real separator reconnection events. | We suggest that, physically, the locations of the singularity in the flux velocity may be associated with regions of strong particle acceleration in real separator reconnection events. |
llaving considered the wav in which field line reconnection occurs in the separator configuration we proceed next (o a euantitative analvsis where we ask how to measure and interpret the rate of reconnection in the configuration wilh one or more separators. | Having considered the way in which field line reconnection occurs in the separator configuration we proceed next to a quantitative analysis where we ask how to measure and interpret the rate of reconnection in the configuration with one or more separators. |
The question here is whether il is necessary to know the elobal field topology (including the location and number of magnelicc» separators) in order to determine the reconnection rate. | The question here is whether it is necessary to know the global field topology (including the location and number of magnetic separators) in order to determine the reconnection rate. |
In a (wo-dimensional conlieuration where reconnection takes place at an A-type null point of the field the reconnection rate is given by the value of the electric field at the null point ancl measures (he rate at which magnetic [lux is transferred between the four topologically distinct [lux domains. | In a two-dimensional configuration where reconnection takes place at an X-type null point of the field the reconnection rate is given by the value of the electric field at the null point and measures the rate at which magnetic flux is transferred between the four topologically distinct flux domains. |
In order to express the rate as a dimensionless quantity that electric field is normalised to a characteristic convective electric field and so the reconnection rate measured in terms (fractions) of the Alfvénn Mach number. | In order to express the rate as a dimensionless quantity that electric field is normalised to a characteristic convective electric field and so the reconnection rate measured in terms (fractions) of the Alfvénn Mach number. |
In three-dimensions we also have a measure for the rate of reconnection. | In three-dimensions we also have a measure for the rate of reconnection. |
This is given bv the maximum integrated parallel electric field over all field lines (hat thread the non-ideal region (Schindler 1938): | This is given by the maximum integrated parallel electric field over all field lines that thread the non-ideal region (Schindler 1988): |
and subsampline where blocks of data. instead of individual cata points. are resampled. | and subsampling where blocks of data, instead of individual data points, are resampled. |
We introduce these methods in Section 2 and deseribe their shortcomings. | We introduce these methods in Section \ref{sect:bootspatial} and describe their shortcomings. |
li Section we describe (he marked. point bootstrap (Loh&Stein2004). as a wav to address these shortcomings. | In Section \ref{sect:improve} we describe the marked point bootstrap \citep{loh02a} as a way to address these shortcomings. |
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