scholarly journals Going back to basics: accelerating exoplanet transit modelling using Taylor-series expansion of the orbital motion

2020 ◽  
Vol 499 (3) ◽  
pp. 3356-3361
Author(s):  
H Parviainen ◽  
J Korth

ABSTRACT A significant fraction of an exoplanet transit model evaluation time is spent calculating projected distances between the planet and its host star. This is a relatively fast operation for a circular orbit, but slower for an eccentric one. However, because the planet’s position and its time derivatives are constant for any specific point in orbital phase, the projected distance can be calculated rapidly and accurately in the vicinity of the transit by expanding the planet’s x and y positions in the sky plane into a Taylor series at mid-transit. Calculating the projected distance for an elliptical orbit using the four first time derivatives of the position vector (velocity, acceleration, jerk, and snap) is ∼100 times faster than calculating it using the Newton’s method, and also significantly faster than calculating z for a circular orbit because the approach does not use numerically expensive trigonometric functions. The speed gain in the projected distance calculation leads to 2–25 times faster transit model evaluation speed, depending on the transit model complexity and orbital eccentricity. Calculation of the four position derivatives using numerical differentiation takes $\sim 1\, \mu$s with a modern laptop and needs to be done only once for a given orbit, and the maximum error the approximation introduces to a transit light curve is below 1 ppm for the major part of the physically plausible orbital parameter space.

2006 ◽  
Vol 2 (S240) ◽  
pp. 531-535
Author(s):  
S. Komonjinda ◽  
J.B. Hearnshaw ◽  
D.J. Ramm

AbstractThe orbital eccentricity of the SB1 system ζ TrA (S.T. F9V, P ∼ 13 d) was found by Skuljan et al. (2004) to be e = 0.01398 ± 0.00019. Lucy (2005) devised a statistical test of the significance of this result, based on the amplitude and phase of the third harmonic in the Fourier analysis of the radial velocity data, and concluded that the non-zero eccentricity measured does not arise from a slightly eccentric Keplerian orbit, but from proximity effects in the binary. He therefore believes a circular orbit should be assigned to this system. In this paper we investigate one possible proximity effect, namely the tidal distortion of the primary star, such that the measured Doppler shift does not accurately indicate the centre of mass radial velocity of the star as a whole. The code of Wilson & Devinney (2003) was used to model the tidal distortion of the measured radial velocities, assuming a range of possible secondary masses, corresponding to M-dwarf companions. The result is that even for the lowest possible mass secondary of 0.09M⊙ with sin i = 1 (this gives the greatest tidal distortion, as it is closest to the primary) there is no significant effect on the radial velocities (the differences are of order 1 m s−1 as a result of the tidal effects). Similar negligible tidal effects arise using a white dwarf companion. We note that the difference between a circular orbit and the observations amounts to as much as 140 m s−1 at some phases, which is essentially the amplitude of the second harmonic in the data. Our conclusion is that this strong and highly significant second harmonic is most probably the result of a small orbital eccentricity as reported by Skuljan et al (2004). We note that the observed third harmonic according to Lucy (2005) has an amplitude of only 5.2 ± 2.0 m s−1, which is just over twice the error bar of its measurement, and that the predicted third harmonic for an eccentric orbit is only 1.6 m s−1.


2008 ◽  
Vol 4 (S253) ◽  
pp. 536-539
Author(s):  
Francis T. O'Donovan ◽  
David Charbonneau ◽  
Joseph Harrington ◽  
Sara Seager ◽  
Drake Deming ◽  
...  

AbstractWe present here results from observations of TrES-2 made using the Infrared Array Camera on the Spitzer Space Telescope. We monitored this transiting system during two secondary eclipses, when the planetary emission is blocked by the star. The resulting decrease in flux is 0.135 ± 0.036%, 0.245 ± 0.027%, 0.162 ± 0.064%, and 0.295 ± 0.066%, at 3.6-μm, 4.5-μm, 5.8-μm, and 8.0-μm, respectively. We find evidence for a temperature inversion in the atmosphere of TrES-2, which is predicted by Fortney and collaborators based on the proposed importance of TiO and VO opacities for this highly irradiated gas giant. We also find the time of the center of the eclipse to be consistent with predictions from transit timing observations of TrES-2. This implies that TrES-2 most likely has a circular orbit, and thus does not obtain additional thermal energy from tidal dissipation of a non-zero orbital eccentricity, a proposed explanation of the large planetary radius.


2016 ◽  
Author(s):  
E. Solazzo ◽  
S. Galmarini

Abstract. In this study, methods are proposed to diagnose the causes of errors in air quality (AQ) modelling systems. We investigate the deviation between modelled and observed time series of surface ozone through a revised formulation for breaking down the mean square error (MSE) into bias, variance, and the minimum achievable MSE (mMSE). The bias measures the accuracy and implies the existence of systematic errors and poor representation of data complexity, the variance measures the precision and provides an estimate of the variability of the modelling results in relation to the observed data, and the mMSE reflects unsystematic errors and provides a measure of the associativity between the modelled and the observed fields through the correlation coefficient. Each of the error components is analysed independently and apportioned to resolved process based on the corresponding timescale (long scale, synoptic, diurnal, and intra-day) and as a function of model complexity. The apportionment of the error is applied to the AQMEII (Air Quality Model Evaluation International Initiative) group of models, which embrace the majority of regional AQ modelling systems currently used in Europe and North America. The proposed technique has proven to be a compact estimator of the operational metrics commonly used for model evaluation (bias, variance, and correlation coefficient), and has the further benefit of apportioning the error to the originating timescale, thus allowing for a clearer diagnosis of the process that caused the error.


2002 ◽  
Vol 17 (20) ◽  
pp. 2756-2756 ◽  
Author(s):  
R. COLISTETE

Starting with an exact and simple geodesic, we generate approximate geodesics1,2 by summing up higher-order geodesic deviations in a fully relativistic scheme. We apply this method to the problem of orbit motion of test particles in Schwarzschild3 and Kerr metrics; from a simple circular orbit as the initial geodesic we obtain finite eccentricity orbits as a Taylor series with respect to the eccentricity. The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme is best adapted for small eccentricities, but arbitrary values of M/R. We also analyse the possible application to the calculation of the emission of gravitational radiation from non-circular orbits around a very massive body3.


2020 ◽  
Author(s):  
Jian Zhou ◽  
Jianyang Xia ◽  
Ning Wei ◽  
Yufu Liu ◽  
Chenyu Bian ◽  
...  

Abstract. The synchronous increase of model complexity and data volume in Earth system science challenges using observations to evaluate Earth system models (ESMs). The challenge mainly stems from the untraceable of model outputs, the lack of automatic algorithms, and the high computational costs. Here, we built up an online Traceability analysis system for Model Evaluation (TraceME), which is traceable, automatic and shareable. The TraceME (v1.0) can trace the structural uncertainty of simulated carbon (C) storage in the state-of-the-art ESMs into gross primary production (GPP), carbon use efficiency (CUE), baseline residence time and environmental scalars (temperature and precipitation). The cloud-based framework used in TraceME provides the scientific workflows and a shareable platform to achieve the automated analysis and distributed data storage to greatly improve the efficiency of model evaluation. Then, we set up a worker node in TraceME (v1.0) to store the data from Coupled Model Intercomparison Project (CMIP6), and submitted tasks through browser to analyze the uncertainties of CMIP6 models in the TraceME system. Overall, this new tool can greatly facilitate model evaluation to identify sources of model uncertainty and provide some new implications for the next generation of model evaluation.


2018 ◽  
Vol 610 ◽  
pp. A55 ◽  
Author(s):  
Emily Garhart ◽  
Drake Deming ◽  
Avi Mandell ◽  
Heather Knutson ◽  
Jonathan J. Fortney

Aims. Previous secondary eclipse observations of the hot Jupiter Qatar-1b in the Ks band suggest that it may have an unusually high day side temperature, indicative of minimal heat redistribution. There have also been indications that the orbit may be slightly eccentric, possibly forced by another planet in the system. We investigate the day side temperature and orbital eccentricity using secondary eclipse observations with Spitzer. Methods. We observed the secondary eclipse with Spitzer/IRAC in subarray mode, in both 3.6 and 4.5 μm wavelengths. We used pixel-level decorrelation to correct for Spitzer’s intra-pixel sensitivity variations and thereby obtain accurate eclipse depths and central phases. Results. Our 3.6 μm eclipse depth is 0.149 ± 0.051% and the 4.5 μm depth is 0.273 ± 0.049%. Fitting a blackbody planet to our data and two recent Ks band eclipse depths indicates a brightness temperature of 1506 ± 71 K. Comparison to model atmospheres for the planet indicates that its degree of longitudinal heat redistribution is intermediate between fully uniform and day-side only. The day side temperature of the planet is unlikely to be as high (1885 K) as indicated by the ground-based eclipses in the Ks band, unless the planet’s emergent spectrum deviates strongly from model atmosphere predictions. The average central phase for our Spitzer eclipses is 0.4984 ± 0.0017, yielding e cos ω = −0.0028 ± 0.0027. Our results are consistent with a circular orbit, and we constrain e cos ω much more strongly than has been possible with previous observations.


2014 ◽  
Vol 14 (19) ◽  
pp. 10845-10895 ◽  
Author(s):  
K. Tsigaridis ◽  
N. Daskalakis ◽  
M. Kanakidou ◽  
P. J. Adams ◽  
P. Artaxo ◽  
...  

Abstract. This paper evaluates the current status of global modeling of the organic aerosol (OA) in the troposphere and analyzes the differences between models as well as between models and observations. Thirty-one global chemistry transport models (CTMs) and general circulation models (GCMs) have participated in this intercomparison, in the framework of AeroCom phase II. The simulation of OA varies greatly between models in terms of the magnitude of primary emissions, secondary OA (SOA) formation, the number of OA species used (2 to 62), the complexity of OA parameterizations (gas-particle partitioning, chemical aging, multiphase chemistry, aerosol microphysics), and the OA physical, chemical and optical properties. The diversity of the global OA simulation results has increased since earlier AeroCom experiments, mainly due to the increasing complexity of the SOA parameterization in models, and the implementation of new, highly uncertain, OA sources. Diversity of over one order of magnitude exists in the modeled vertical distribution of OA concentrations that deserves a dedicated future study. Furthermore, although the OA / OC ratio depends on OA sources and atmospheric processing, and is important for model evaluation against OA and OC observations, it is resolved only by a few global models. The median global primary OA (POA) source strength is 56 Tg a−1 (range 34–144 Tg a−1) and the median SOA source strength (natural and anthropogenic) is 19 Tg a−1 (range 13–121 Tg a−1). Among the models that take into account the semi-volatile SOA nature, the median source is calculated to be 51 Tg a−1 (range 16–121 Tg a−1), much larger than the median value of the models that calculate SOA in a more simplistic way (19 Tg a−1; range 13–20 Tg a−1, with one model at 37 Tg a−1). The median atmospheric burden of OA is 1.4 Tg (24 models in the range of 0.6–2.0 Tg and 4 between 2.0 and 3.8 Tg), with a median OA lifetime of 5.4 days (range 3.8–9.6 days). In models that reported both OA and sulfate burdens, the median value of the OA/sulfate burden ratio is calculated to be 0.77; 13 models calculate a ratio lower than 1, and 9 models higher than 1. For 26 models that reported OA deposition fluxes, the median wet removal is 70 Tg a−1 (range 28–209 Tg a−1), which is on average 85% of the total OA deposition. Fine aerosol organic carbon (OC) and OA observations from continuous monitoring networks and individual field campaigns have been used for model evaluation. At urban locations, the model–observation comparison indicates missing knowledge on anthropogenic OA sources, both strength and seasonality. The combined model–measurements analysis suggests the existence of increased OA levels during summer due to biogenic SOA formation over large areas of the USA that can be of the same order of magnitude as the POA, even at urban locations, and contribute to the measured urban seasonal pattern. Global models are able to simulate the high secondary character of OA observed in the atmosphere as a result of SOA formation and POA aging, although the amount of OA present in the atmosphere remains largely underestimated, with a mean normalized bias (MNB) equal to −0.62 (−0.51) based on the comparison against OC (OA) urban data of all models at the surface, −0.15 (+0.51) when compared with remote measurements, and −0.30 for marine locations with OC data. The mean temporal correlations across all stations are low when compared with OC (OA) measurements: 0.47 (0.52) for urban stations, 0.39 (0.37) for remote stations, and 0.25 for marine stations with OC data. The combination of high (negative) MNB and higher correlation at urban stations when compared with the low MNB and lower correlation at remote sites suggests that knowledge about the processes that govern aerosol processing, transport and removal, on top of their sources, is important at the remote stations. There is no clear change in model skill with increasing model complexity with regard to OC or OA mass concentration. However, the complexity is needed in models in order to distinguish between anthropogenic and natural OA as needed for climate mitigation, and to calculate the impact of OA on climate accurately.


Author(s):  
Wayne Arter

The work extends the linear fields’ solution of compressible nonlinear magnetohydrodynamics (MHD) to the case where the magnetic field depends on superlinear powers of position vector, usually, but not always, expressed in Cartesian components. Implications of the resulting Lie–Taylor series expansion for physical applicability of the Dolzhansky–Kirchhoff (D–K) equations are found to be positive. It is demonstrated how resistivity may be included in the D–K model. Arguments are put forward that the D–K equations may be regarded as illustrating properties of nonlinear MHD in the same sense that the Lorenz equations inform about the onset of convective turbulence. It is suggested that the Lie–Taylor series approach may lead to valuable insights into other fluid models.


2021 ◽  
Author(s):  
Wes Bonifay

Traditional statistical model evaluation typically relies on goodness-of-fit testing and quantifying model complexity by counting parameters. Both of these practices may result in overfitting and have thereby contributed to the generalizability crisis. The information-theoretic principle of minimum description length addresses both of these concerns by filtering noise from the observed data and consequently increasing generalizability to unseen data.


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