There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the ...basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed.
We present a local convergence analysis for general multi-point-Chebyshev–Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space ...setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study.
18 mm-sized Orionid meteoroids were captured in 2019 and 2020 by the Canadian Automated Observatory’s mirror tracking system. Meteor position measurements were made to an accuracy of ∼1m and the ...meteors were tracked to a limiting magnitude of about +7.5 at the faintest point. The trajectory estimation shows the intrinsic physical dispersion of the Orionid radiant is 0.400°±0.062°. An erosion-based entry model was fit to the observations to reproduce ablation and fragmentation for each meteor, simultaneously reproducing the light curve, the dynamics, and the wake. Wake observations were found to directly inform the grain mass distribution released in the modelled erosion. A new luminous efficiency model was derived from simultaneous radar and optical observations and applied in the modelling to improve its accuracy. The results show that the apparent strength of Orionids varies with radiant location and time of appearance during the period of shower activity. The average differential grain mass distribution index was 2.15, higher than found from in-situ estimates, possibly due to the evolution of the physical properties of meteoroids since ejection. All Orionids showed leading fragment morphology which was best explained by stopping the erosion at the peak of the light curve, leaving a non-fragmenting meteoroid with ∼10% of the original mass. The inverted Orionid meteoroid average bulk density of ∼300kgm−3, corresponding to porosities of ∼90%, is consistent with in-situ measurements of larger dust particles by Vega-2 at 1P/Halley and Rosetta at 67P.
•Captured 18 Orionid meteoroids using the Canadian Automated Observatory’s mirror-tracking system.•Determined Orionid radiant intrinsic physical dispersion: 0.400°±0.062°.•Introduced a new luminous efficiency model derived from radar and optical observations.•Utilized an erosion-based entry model to simulate meteor ablation, fragmentation, and light curve.•Orionid meteoroid bulk density: ∼ 300 kg/m3, aligning with in-situ measurements from Vega-2.
In this paper, the dynamics of the Chebyshev–Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This set ...has interesting similarities with the Mandelbrot set. The parameter space has allowed us to find different elements of the family which have bad convergence properties, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.
We numerically study the dynamical evolution of observed samples of active and inactive Centaurs and clones that reach the Jupiter-Saturn region. Our aim is to compare the evolution between active ...and inactive Centaurs, their end states and their transfer to Jupiter family comets and Halley-type comets. We find that the median lifetime of inactive Centaurs is about twice longer than that for active Centaurs, suggesting that activity is related to the residence time in the region. This view is strengthened by the observation that high-inclination and retrograde Centaurs (Tisserand parameters with respect to Jupiter TJ<2) which have the longest median dynamical lifetime (=1.37×106 yr) are all inactive. We also find that the perihelion distances of some active, comet-like Centaurs have experienced drastic drops of a few au in the recent past (∼102−103 yr), while such drops are not found among inactive Centaurs. Inactive Centaurs with TJ≲2.5 usually evolve to Halley-type comets, whereas inactive Centaurs with TJ≳2.5 and active Centaurs (that also have TJ≳2.5) evolve almost always to Jupiter family comets and very seldom to Halley type comets. Inactive Centaurs are also more prone to end up as sungrazers, and both inactive and active Centaurs transit through different mean motion resonances (generally with Jupiter) during their evolution.
•We analyze the orbital evolution of active and inactive Centaurs with perihelia in the Jupiter-Saturn zone.•We analyze their end states as Jupiter family comets and Halley-type comets.•Inactive Centaurs lead to more sungrazers than active Centaurs.•The dynamical half-life of inactive Centaurs is found to be longer than that of active Centaurs.
We study the local convergence of Chebyshev–Halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation. In Sharma (2015) (see Theorem 1, ...p. 121) the convergence of the method was shown under hypotheses reaching up to the third derivative. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamics of these methods are also studied. Finally, numerical examples examining dynamical planes are also provided in this study to solve equations in cases where earlier studies cannot apply.
The two-parameter Chebyshev–Halley-like family of optimal two-point fourth-order methods proposed by Babajee (2015), is further extended to solve systems of nonlinear equations. This two-step ...fourth-order family is further extended to obtain a two-parameter family of sixth-order methods which requires only one extra function evaluation. The performance of some special members of the proposed families using only single inverse per iteration have been tested through numerical examples and the results show that these are effective and comparable to existing methods both in order and efficiency.
Origin and Evolution of Short-period Comets Nesvorný, David; Vokrouhlický, David; Dones, Luke ...
Astrophysical journal/The Astrophysical journal,
08/2017, Volume:
845, Issue:
1
Journal Article
Peer reviewed
Open access
Comets are icy objects that orbitally evolve from the trans-Neptunian region into the inner solar system, where they are heated by solar radiation and become active due to the sublimation of water ...ice. Here we perform simulations in which cometary reservoirs are formed in the early solar system and evolved over 4.5 Gyr. The gravitational effects of Planet 9 (P9) are included in some simulations. Different models are considered for comets to be active, including a simple assumption that comets remain active for perihelion passages with perihelion distance . The orbital distribution and number of active comets produced in our model is compared to observations. The orbital distribution of ecliptic comets (ECs) is well reproduced in models with and without P9. With P9, the inclination distribution of model ECs is wider than the observed one. We find that the known Halley-type comets (HTCs) have a nearly isotropic inclination distribution. The HTCs appear to be an extension of the population of returning Oort-cloud comets (OCCs) to shorter orbital periods. The inclination distribution of model HTCs becomes broader with increasing , but the existing data are not good enough to constrain from orbital fits. is required to obtain a steady-state population of large active HTCs that is consistent with observations. To fit the ratio of the returning-to-new OCCs, by contrast, our model implies that , possibly because the detected long-period comets are smaller and much easier to disrupt than observed HTCs.
Context.
Detection of abundant O
2
at 1–10% relative to H
2
O ice in the comae of comets 1P/Halley and 67P/Churyumov-Gerasimenko has motivated attempts to explain the origin of the high O
2
ice ...abundance. Recent chemical modelling of the outer, colder regions of a protoplanetary disk midplane has shown production of O
2
ice at the same abundance as that measured in the comet.
Aims.
We aim to carry out a thorough investigation to constrain the conditions under which O
2
ice could have been produced through kinetic chemistry in the pre-solar nebula midplane.
Methods.
We have utilised an updated chemical kinetics code to evolve chemistry under pre-solar nebula midplane conditions. Four different chemical starting conditions and the effects of various chemical parameters have been tested.
Results.
Using the fiducial network, and for either reset conditions (atomic initial abundances) or atomic oxygen only conditions, the abundance level of O
2
ice measured in the comets can be reproduced at an intermediate time, after 0.1–2 Myr of evolution, depending on ionisation level. When including O
3
chemistry, the abundance of O
2
ice is much lower than the cometary abundance (by several orders of magnitude). We find that H
2
O
2
and O
3
ices are abundantly produced (at around the level of O
2
ice) in disagreement with their respective abundances or upper limits from observations of comet 67P. Upon closer investigation of the parameter space, and varying parameters for grain–surface chemistry, it is found that for temperatures 15–25 K, densities of 10
9
−10
10
cm
−3
, and a barrier for quantum tunnelling set to 2 Å, the measured level of O
2
ice can be reproduced with the new chemical network, including an updated binding energy for atomic oxygen (1660 K). However, the abundances of H
2
O
2
and O
3
ices still disagree with the observations. A larger activation energy for the O + O
2
→ O
3
reaction (
E
act
> 1000 K) helps to reproduce the non-detection of O
3
ice in the comet, as well as reproducing the observed abundances of H
2
O
2
and O
2
ices. The only other case in which the O
2
ice matches the observed abundance, and O
3
and H
2
O
2
ice are lower, is the case when starting with an appreciable amount of oxygen locked in O
2
.
Conclusions.
The parameter space investigation revealed a sweet spot for production of O
2
ice at an abundance matching those in 67P and 1P, and O
3
and H
2
O
2
ice abundances matching those in 67P. This means that there is a radial region in the pre-solar nebula from 120–150 AU, within which O
2
could have been produced in situ via ice chemistry on grain surfaces. However, it is apparent that there is a high degree of sensitivity of the chemistry to the assumed chemical parameters (e.g. binding energy, activation barrier width, and quantum tunnelling barrier). Hence, because the more likely scenario starting with a percentage of elemental oxygen locked in O
2
also reproduces the O
2
ice abundance in 67P at early stages, this supports previous suggestions that the cometary O
2
ice could have a primordial origin.
In this paper, a family of modified Chebyshev–Halley’s methods free from second derivative is presented. Per iteration the new methods require three evaluations of the function and one of its first ...derivatives. A detailed convergence analysis of the new methods shows that the new methods are at least fifth-order convergent and especially, the modified super-Halley’s method is sixth-order convergent. Numerical examples are given to illustrate the efficiency and performance of the new methods.
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