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Fout voortplantingsmetodes vir wentelparameters

Fout voortplantingsmetodes vir wentelparameters


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Onlangs het ek 'n artikel in ons plaaslike tydskrif vir astronomiese popularisering teëgekom. Dit gaan oor 'n bekende taak van Sgr A * -swart gat se massaskatting. Die artikel is 'n stap-vir-stap gids vir jong navorsers. Die baanparameters van die S2-ster moet byvoorbeeld gevind word met behulp van 'n tekening op 'n vel papier.

Daarom het ek geglo dat dit nie nodig is nie baie presies berekeninge vir hierdie taak.

Benewens die $ x $, $ y $ koördinate van die S2-ster, kry ek ook die ooreenstemmende onsekerheidswaardes $ Delta x $, $ Delta y $ (19 koördinaatmetings vir die tydperk 1992 tot 2003):

begin {skikking} {| c | c | c | c |} hlyn tyd & x & Delta x & y & Delta y hline 1992.226 & 0.104 & 0.003 & -0.166 & 0.004 1994.321 & 0.097 & 0.003 & -0.189 & 0.004 … &… &… &… & 2003.353 & 0.077 & 0.002 & -0.030 & 0.002 2003.454 & 0.081 & 0.002 & -0.036 & 0.002 hline end {array}

Met behulp van die direkste kleinste kwadraatmetode vir S2-omwenteling deur 'n ellips het ek parameters gevind $ a, b, c, d, e, f $ van die kegelvormige vergelyking: $$ ax ^ 2 + bxy + cy ^ 2 + dx + ey + f = 0 $$ Ek het ook besluit om die simuleer fout voortplanting op 'n Monte Carlo manier om die onsekerheid in die ramings van $ a, b, c, d, e, f $. En ek het daarin geslaag.

Maar hier is Die vraag: is daar enige ander toepaslike, "analitiese" manier (nie 'n Monte Carlo-metode nie) om die onsekerheid in die beramings van $ a, b, c, d, e, f $ wat in die gebruik van die sterrekundiges gebruik kan word?


ProfRob se opmerking is eintlik al die antwoord:

Tensy u die model kan lineêr maak, is daar geen analitiese foutberaming nie.

Ek beskou foutberaming baie belangrik, dus wil ek hier 'n bietjie meer besonderhede toon: 'n manier om die fout van 'n model te skat, is deur die Gaussiese foutverspreiding, sien veral die deel van die Wikipedia-inskrywing oor nie-lineêre kombinasies. Laat ek die idee hierna kort saamvat, aangesien ek die Wikipedia-artikel nie so maklik vind om te volg nie. Ek sal ook verskillende veranderlike name gebruik sodat dit nie oorvleuel met dié in u vraag nie.

Ons begin met 'n funksie $ varphi $ wat afhang van verskillende veranderlikes $ x_1, x_2, ldots $, betekenis $ varphi = varphi (x_1, x_2, ldots) $. Ons neem aan dat ons rondom 'n sekere punt kan lineair $ tilde { bf x} = ( tilde {x} _1, tilde {x} _2, ldots) $, wat hoofsaaklik beteken dat die funksie op daardie stadium as 'n Taylor-reeks geskryf moet word.

Dan kan ons die Gaussiese (maksimum) fout in die omgewing van $ { bf tilde {x}} $.

$$ links. Delta varphi regs | _ { bf tilde {x}} = links. frac { gedeeltelik varphi} { gedeeltelik x_1} regs | _ {( tilde {x} _1, , tilde {x} _2, , ldots)} ! ! ! ! ! cdot Delta x_1 + links. frac { gedeeltelik varphi} { gedeeltelik x_2} regs | _ {( tilde {x} _1, , tilde {x} _2, , ldots)} ! ! ! ! cdot Delta x_2 + cdots $$

In hierdie formule bereken u gedeeltelike afgeleides $ frac { gedeeltelik varphi} { gedeeltelik x_i} $ met betrekking tot elke veranderlike $ x_i $ en steek die waardes van die veranderlikes in die sentrale punt in $ { bf tilde {x}} $. Die $ Delta x_i $ is die foutberaming vir elke individuele veranderlike $ x_i $.

Hoe kan ons dit toepas op u probleem waar u 'n implisiet vergelyking? Aangesien u die Monte-Carlo-benadering (wat die gebruiklike metode in hierdie geval is) uitgesluit het, stel ek die volgende resep voor:

  1. Formuleer eksplisiete funksies $ a (x, y), b (x, y), ldots f (x, y) $, bv. $$ a = - frac {bxy + cy ^ 2 + dx + ey + f} {x ^ 2} $$
  2. Lineariseer die funksies rondom $ x_0, y_0 $. Die $ x_0 $ is genoem $ x $ in u tabel, en onderskeidelik $ y_0 $ is in die $ y $-kolom.
  3. Bepaal $ Delta a, ldots, Delta f $ deur gedeeltelike differensiasie.
  4. Inprop $ Delta x $ en $ Delta y $ soos ook in u tabel gegee.

Foutanalise en maniere om regstelling van die baan van die baan om maanoordrag ☆, ☆☆

Vir 'n terugkeerbare maansonde bestudeer hierdie artikel die eienskappe van die Aarde-Maan-oordragbaan en die terugkeerbaan. Op grond van die fout voortplantingsmatriks word die lineêre vergelyking om die eerste middelweg-regstellingsmanoeuvre (TCM) te skat, uitgewys. Numeriese simulasies word uitgevoer en die kenmerke van die voortplanting van foute in die maanoordragbaan word gegee. Die voordele, nadele en toepassings van twee TCM-strategieë word bespreek, en die berekening van die tweede TCM van die retourbaan word ook gesimuleer onder die omstandighede tydens die hertoetrede.


Die voortplanting en sensitiwiteitsanalise van die baan met geskeide voorstellings

Die meeste benaderings vir stogastiese differensiaalvergelykings met hoë-dimensionele, nie-Gaussiese insette, ly aan 'n vinnige (bv. Eksponensiële) toename in berekeningskoste, 'n probleem wat bekend staan ​​as die vloek van dimensionaliteit. In astrodinamika lei dit tot 'n verminderde akkuraatheid wanneer 'n waarskynlikheidsdigtheidsfunksie van 'n wentelbaanstaat voortplant. Hierdie referaat bespreek die toepassing van geskeide voorstellings vir voortplanting van wentelonsekerheid, waar toekomstige state uitgebrei word tot 'n som van produkte van eenveranderlike funksies van aanvanklike toestande en ander onsekere parameters. 'N Akkurate generasie van geskeide voorstelling vereis 'n aantal toestandsmonsters wat lineêr is in die dimensie van invoeronsekerhede. Die berekeningskoste van 'n geskeide voorstelling skaal lineêr met betrekking tot die steekproefneming, wat die beweegbaarheid verbeter in vergelyking met metodes wat ly aan die vloek van dimensionaliteit. Benewens gedetailleerde besprekings oor die konstruksie en gebruik daarvan in sensitiwiteitsanalise, bied hierdie referaat resultate vir drie toetsgevalle van 'n satelliet wat om die aarde wentel. Die eerste twee gevalle toon dat benadering via geskeide voorstellings 'n oplosbare oplossing lewer vir die voortplanting van die Cartesiese wentel-onsekerheid met tot 20 onseker insette. Die derde geval, wat eerder Equinoctial elemente gebruik, ondersoek 'n scenario wat in die literatuur aangebied word en gebruik die voorgestelde metode vir sensitiwiteitsanalise om die relatiewe effekte van onsekere insette op die vermeerderde toestand deegliker te karakteriseer.

Dit is 'n voorskou van intekenaarinhoud, toegang via u instelling.


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Getalfout by die implementering van universele veranderlikes [gesluit]

Wil u hierdie vraag verbeter? Dateer die vraag op sodat dit onderwerp is vir Physics Stack Exchange.

Ek probeer (in Python voorlopig) voortplanting van lae stuwingsbane vir ruimtetuie met behulp van universele veranderlikes implementeer. Vir 'n gegewe sentrale liggaam met die gravitasieparameter $ mu $ en 'n baan met die semi-hoofas $ a $ en die beginposisie $ vec_0 $ en snelheid $ vec_0 $ teen $ t = t_0 $ word die posisie vir 'n gegewe tyd gegee deur: $ vec = vec_0 f (s) + vec_0 g (s) $ en die snelheid deur $ vec = vec_0 punt(s) + vec_0 punt(s) $

Waar $ f (s) = 1- links ( frac < mu> <| vec_0 |> regs) s ^ 2 c_2 ( alfa s ^ 2) $ $ g (s) = t-t_0- mu s ^ 3c_3 ( alpha s ^ 2) $

Ek bereken $ s $ deur die metode van Newton te gebruik en alles werk goed vir scenario's sonder strekking. Die baan is ellipties en geslote, parabolies vir hoër beginsnelhede, as ek 'n $ vec kies_0 $, $ vec_0 $ en $ t_0 $ en versprei $ t $ vorentoe.

Vir gevalle met stoot waar ek die snelheid in elke iterasie sou verander

dit is nodig om $ vec op te dateer_0 $ en $ vec_0 $ versprei elke keer $ Delta t $ vorentoe en herhaal in plaas van die aanvanklike voorwaardes te kies en net $ t $ vorentoe te propageer. Ter voorbereiding daarop het ek verkies om geen snelheid by te voeg nie, sodat albei benaderings dieselfde resultate moet lewer.

Hier is my probleem: Selfs na die eerste iterasie is die verskille groot en neem die fout vinnig toe

blou: vaste aanvanklike voorwaardes, vermeerder $ t $

rooi: voortplantende aanvanklike voorwaardes, vaste $ Delta t $

Ek het gedink dat ek dit sou kon oplos deur 'n meer gevorderde integrasie-algoritme soos Runge-Kutta te gebruik en het probeer om elke iterasie in so min as moontlik stappe te bereken, maar ek kon die vergelykings nie transformeer sodat ek Runge-Kutta kon gebruik nie (aangesien dit nie 'n ODE nie) en die vermindering van die stappe het niks gehelp nie.

Kan iemand help om dit reg te stel of enige wenke te gee waarom die fout so groot is? Byvoorbaat dankie!


Gebruik van nutsmodules¶

New_tle_kep_state¶

new_tle_kep_state word gebruik om 'n TLE of 'n stel Kepleriaanse elemente in 'n toestandsvektor om te skakel. Om 'n TLE te omskep, maak 'n skikking uit die 2de reël van die TLE. Die skikking moet die volgende vorm hê:

  • tle [0] = helling (in grade)
  • tle [1] = regterklim van stygende knoop (in grade)
  • tle [2] = eksentrisiteit
  • tle [3] = argument van perigee (in grade)
  • tle [4] = gemiddelde anomalie (in grade)
  • tle [5] = gemiddelde beweging (in toere per dag)

Bel nou tle_to_state. Byvoorbeeld:

'N Kepleriaanse versameling kan ook in 'n toestandsvektor omgeskakel word.

Teme_to_ecef¶

teme_to_ecef is used to convert coordinates from TEME frame (inertial frame) to ECEF frame (rotating Earth fixed frame). The module accepts a list of coordinates of the form [t1,x,y,z] and outputs a list of latitudes, longitudes and altitudes in Earth fixed frame. These coordinates can be directly plotted on a map.

The resulting latitudes and longitudes can be directly plotted on an Earth map to visualize the satellite location with respect to the Earth.


6. APPLICATIONS TO REAL SYSTEMS

We now apply our analytical theory to real circumbinary planetary systems. For that purpose, the Kepler-16, Kepler-34, Kepler-35, Kepler-38, Kepler-64 and Kepler-413 systems were selected, as they are currently believed to harbor only one planet in a circumbinary orbit. The systems are assumed to be coplanar, , and , while the rest of the system parameters were taken from the corresponding discovery papers (Doyle et al. 2011 Orosz et al. 2012 Welsh et al. 2012 Schwamb et al. 2013 Kostov et al. 2014). The systems were integrated over one analytical secular period and no other effects than Newtonian gravity were considered, as they were not expected to make a significant contribution to the systems under investigation (e.g., Chavez et al. 2015). Table 1 gives the mass parameters and orbital elements of each system.

Tabel 1. Masses and Orbital Elements for Kepler-16, Kepler-34, Kepler-35, Kepler-38, Kepler-64 and Kepler-413

System a1 (AU) a2 (AU) e1
Kepler-16 0.6897 +0.0035 −0.0034 0.20255 +0.00066 −0.00065 0.333 +0.016 −0.016 0.22431 +0.00035 −0.00034 0.7048 +0.0011 −0.0011 0.15944 +0.00061 −0.00062
Kepler-34 1.0479 +0.0033 −0.0030 1.0208 +0.0022 −0.0022 0.220 +0.011 −0.010 0.22882 +0.00019 −0.00018 1.0896 +0.0009 −0.0009 0.52087 +0.00052 −0.00055
Kepler-35 0.8876 +0.0051 −0.0053 0.8094 +0.0041 −0.0044 0.127 +0.020 −0.021 0.17617 +0.00028 −0.00029 0.60345 +0.00100 −0.00102 0.1421 +0.0014 −0.0014
Kepler-38 0.949 0.249 <0.384 (95% conf.) 0.1469 0.4644 0.1032
Kepler-64 1.384 +0.079 −0.079 0.386 +0.018 −0.018 <0.532 (99.7% conf.) 0.1744 +0.0031 −0.0031 0.634 +0.011 −0.011 0.2117 +0.0051 −0.0051
Kepler-413 0.820 0.5423 0.21 0.10148 0.3553 0.0365

Figures 8 and 9 show the results for the six Kepler systems. Generally, the numerical results are in good agreement with the analytical estimates. Furthermore, one can see that for most planets the current state of eccentricities, indicated by a black horizontal line, is compatible with formation scenarios that predict initial orbits with low eccentricities after the gaseous phase. As in situ planetesimal accretion as well as gravitational collapse have practically been ruled out for most of the circumbinary planets discovered during the Kepler mission (e.g., Pelupessy & Portegies Zwart 2013 Lines et al. 2014), a fast disc driven migration with little time spent near resonances seems to be the most likely formation scenario for Kepler-16, Kepler-35, Kepler-38 and Kepler-64 (e.g., Kley & Haghighipour 2014).

Figure 8. Eccentricity against time for Kepler-16b, Kepler-34b and Kepler-35b. The red curve comes from the numerical integration of the full equations of motion, the green curve is our analytical estimates, the blue curve is the analytical secular solution, while the black line denotes the current value of the planetary eccentricity. The integration time is one planetary period for the left column and one analytical secular period for the right column.

Exceptions are Kepler-34 and Kepler-413, both with a higher planetary eccentricity of e2 = 0.182 and e2 = 0.1181, respectively. Looking at the relevant plots, it is clear that starting the planet on a circular orbit cannot produce a planetary orbit with eccentricities higher than 0.03 for Kepler-34b and 0.04 for Kepler-413b. Moreover, the main eccentricity contribution for both systems comes from short-period activity. This is to be expected, as the stellar masses of Kepler-34 have only around 2.5% difference, and the stellar eccentricity of the Kepler-413 is just 0.0365. As a result, the forced secular eccentricity, which is proportional to the difference between the masses of the stellar components and to the stellar eccentricity is very small. Therefore, either those two planets were formed on a non-circular orbit or, if they were initially circular, some dynamical event may have taken place and pumped up their eccentricity. For instance, an as yet undetected companion as well as an encounter with another star may have injected eccentricity into the planet's orbit. Such an interaction would also explain the slight misalignment of the orbital planes in Kepler-413. Another possible explanation for the elevated eccentricity of Kepler-34b is resonance trapping. If the planet's migration has not been fast enough, the planet may be trapped in a resonance which can cause a significant increase in its orbital eccentricity (Kley & Haghighipour 2014). In the case of Kepler-34b the 10:1 mean motion resonace with the stellar binary may have affected the evolution of the planetary eccentricity to some extent (Chavez et al. 2015).


Opsomming

The simple but often neglected equation for the propagation of statistical errors in functions of correlated variables is tested on a number of linear and nonlinear functions of parameters from linear and nonlinear least-squares (LS) fits, through Monte Carlo calculations on 10 4 −4 × 10 5 equivalent data sets. The test examples include polynomial and exponential representations and a band analysis model. For linear functions of linear LS parameters, the error propagation equation is exact. Nonlinear parameters and functions yield nonnormal distributions, but their dispersion is still well predicted by the propagation-of-error equation. Often the error computation can be bypassed by a redefinition of the least-squares model to include the quantity of interest as an adjustable parameter, in which case its variance is returned directly in the variance-covariance matrix. This approach is shown formally to be equivalent to the error propagation method.


Steven L. Tomsovic

Quantum/wave chaos is an interdisciplinary branch of physics and mathematics which emerged in the second half of the 20th century. It finds application in an incredibly diverse set of research fields, systems, and problems such as: statistical nuclear physics and weak symmetry breaking, quantum dots, disordered electronic conductors, decoherence and fidelity studies, quantum computation, Riemann zeta- and L-functions, optical resonators, ultra-cold atoms in optical lattices, acoustics in crystals, underwater sound propagation, and the Dirac spectrum in non-Abelian gauge field backgrounds. The theoretical underpinnings of quantum/wave chaos are characterized by a number of new statistical and asymptotic methods whose common application in systems such as those cited above leads to strong links in their analysis and understanding despite their seeming to be totally unrelated a priori.

Critical early quantum chaos works include: i) Wigner’s introduction of random matrix theory for modeling slow neutron resonance statistical properties, which are strongly interacting many-body systems ii) Gutzwiller’s derivation of a trace formula, which expresses quantal (or modal) spectra as a sum over periodic classical orbits (or rays) for chaotic systems, iii) Bohigas, Giannoni, and Schmit’s conjecture that random matrix theory applies even to simple, fully chaotic systems (K-systems) iv) Berry and later Voros’ introduction of random plane waves for eigenstates, v) Heller’s scarring of eigenfunctions by short classical periodic orbits, and vi) Wegner and Efetov’s non-linear sigma models of disordered mesoscopic systems.

Over the years the research in our group has included: the random matrix analysis of time reversal and parity violation in strongly interacting nuclear systems analysis of the validity of using chaotic dynamics to construct quantum/wave dynamics (construction of heteroclinic orbit sums) use of transfer matrix methods for disordered, quasi-1D mesoscopic conductors the discovery of chaos-assisted tunneling the derivation of trace formulae valid for systems intermediate between integrability and chaos the application of periodic orbit theory and random matrix theory to Coulomb blockade peak height statistics fidelity, sensitivity-to-perturbation, and irreversibility studies application of semiclassical methods to derive properties of interacting-many electron ground states use of finite-time stability exponents for underwater sound propagation studies (finding branching or clustering behaviors) or locating small islands of regular motion in a dynamical system introducing methods for calculating Kolmogorov-Sinai entropies for interacting, many particle systems introduction of extreme value statistics for understanding eigenstates of chaotic systems studies of the interpretation of scanning gate microscopy experiments and introduction of random matrix theory into long range underwater sound propagation.

Professor of Physics

Office: Webster Physical Sciences 929
Phone: (509) 335-7207
Fax: (509) 335-7816
E-mail: tomsovic at wsu.edu

Research: Chaos, Semiclassical Mechanics, and Symmetry Violation


Kyk die video: 1 planten geslachtelijke voortplanting (Februarie 2023).