Paul Wiegert

Paul Wiegert
The University of Western Ontario | UWO · Department of Physics and Astronomy

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191
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Publications

Publications (191)
Preprint
Full-text available
The Taurid Meteoroid Complex (TMC) is a broad stream of meteoroids that produces several annual meteor showers on Earth. If the linkage between these showers and 2P/Encke is at the centre of most TMC models, the small size and low activity of the comet suggest that 2P/Encke is not the unique parent body of the Taurids. Here we simulate the formatio...
Article
The Taurid Meteoroid Complex (TMC) is a broad stream of meteoroids that produces several annual meteor showers on Earth. If the linkage between these showers and 2P/Encke is at the centre of most TMC models, the small size and low activity of the comet suggest that 2P/Encke is not the unique parent body of the Taurids. Here, we simulate the formati...
Article
Full-text available
We provide an overview of the observational properties of the four major Taurid showers, namely the Northern and Southern Taurids (#017 NTA and #002 STA), the β Taurids (#173 BTA) and the ζ Perseids (#172 ZPE). Analysing more than two decades of meteor observations from visual, optical and radar measurements we present the Taurids average activity,...
Preprint
Full-text available
We provide an overview of the observational properties of the four major Taurid showers, namely the Northern and Southern Taurids (#017 NTA and #002 STA), the Beta Taurids (#173 BTA) and the Zeta Perseids (#172 ZPE). Analysing more than two decades of meteor observations from visual, optical and radar measurements we present the Taurids average act...
Article
The goal of this work is to determine if the dynamics of individual Taurid Complex (TC) objects are consistent with the formation of the complex via fragmentation of a larger body, or if the current orbital affinities between the TC members result from other dynamical processes. To this end, the orbital similarity through time of comet 2P/Encke, fi...
Preprint
Full-text available
The goal of this work is to determine if the dynamics of individual Taurid Complex (TC) objects are consistent with the formation of the complex via fragmentation of a larger body, or if the current orbital affinities between the TC members result from other dynamical processes. To this end, the orbital similarity through time of comet 2P/Encke, fi...
Article
If a transiting exoplanet has a moon, that moon could be detected directly from the transit it produces itself, or indirectly via the transit timing variations it produces in its parent planet. There is a range of parameter space where the Kepler Space Telescope is sensitive to the TTVs exomoons might produce, though the moons themselves would be t...
Article
Full-text available
Context. We present a new numerical model of the η -Aquariid and Orionid meteor showers. Aims. The model investigates the origin, variability, and age of the η -Aquariid and Orionid apparitions from 1985 to the present day in order to forecast their activity over the next several decades. Methods. Through the numerical integration of millions of si...
Preprint
We present a new numerical model of the eta-Aquariid and Orionid meteor showers. The model investigates the origin, variability and age of the eta-Aquariid and Orionid apparitions from 1985 to the present day, in order to forecast their activity over the next several decades. Through the numerical integration of millions of simulated meteoroids and...
Article
Full-text available
This review discusses the unsolved problem of the detection of interstellar particles in the Earth’s atmosphere and the presence of interstellar meteors in meteor databases. Owing to the difficulties in obtaining accurate meteor measurements and, consequently, the meteoroids’ orbital parameters, the identification of interstellar meteors based on t...
Preprint
If a transiting exoplanet has a moon, that moon could be detected directly from the transit itproduces itself, or indirectly via the transit timing variations it produces in its parent planet. There is a range of parameter space where the Kepler Space Telescope is sensitive to the TTVs exomoons might produce, though the moons themselves would be to...
Article
Full-text available
Aims. We present a multi-instrumental, multidecadal analysis of the activity of the Eta-Aquariid and Orionid meteor showers for the purpose of constraining models of 1P/Halley’s meteoroid streams. Methods. The interannual variability of the showers’ peak activity and period of duration is investigated through the compilation of published visual and...
Preprint
Full-text available
We present a multi-instrumental, multidecadal analysis of the activity of the Eta-Aquariid and Orionid meteor showers for the purpose of constraining models of 1P/Halley's meteoroid streams. The interannual variability of the showers' peak activity and period of duration is investigated through the compilation of published visual and radar observat...
Article
A strong and unexpected meteor shower outburst was observed by the Southern Argentina Agile MEteor Radar Orbital System (SAAMER-OS) at high southern ecliptic latitude within the South Toroidal region. The outburst, which was active throughout solar longitudes 351° and 352°, peaked at 09:30 UT on 2020 March 12, has a mean Sun-centered ecliptic radia...
Article
Full-text available
The Double Asteroid Redirection Test (DART) spacecraft is planned to impact the secondary of the binary asteroid (65803) Didymos in 2022 to assess deflection strategies for planetary defense. The impact will create a crater and release asteroidal material, some of which will escape the Didymos system. Because the closest point of approach of Didymo...
Article
Granvik et al. reported an absence of asteroids on orbits with perihelia near the Sun that they attribute to the “supercatastrophic disruption” of these bodies. Here we investigate whether there is evidence for this process among other bodies with similarly low perihelia: near-Earth asteroids, Solar and Heliospheric Observatory ( SOHO ) comets, and...
Article
The low velocity of interstellar asteroid 1I/’Oumuamua with respect to our galaxy’s local standard of rest implies that it is young. Adopting the young age hypothesis, we assess possible origin systems for this interstellar asteroid and for 2I/Borisov, though the latter’s higher speed means it is unlikely to be young. First, their past trajectories...
Preprint
Granvik et al. (2016) report an absence of asteroids on orbits with perihelia near the Sun that they attribute to the 'supercatastrophic disruption' of these bodies. Here we investigate whether there is evidence for this process among other bodies with similarly low perihelia: near-Earth asteroids, SOHO comets, as well as meter-sized and millimeter...
Article
In the first paper of this series, we examined existing methods of optical meteor trajectory estimation and developed a novel method which simultaneously uses both the geometry and the dynamics of meteors to constrain their trajectories. We also developed a simulator which uses an ablation model to generate realistic synthetic meteor trajectories w...
Preprint
The DART spacecraft is planned to impact the secondary of the binary asteroid (65803) Didymos in 2022, to assess deflection strategies for planetary defense. The impact will create a crater and release asteroidal material, some of which will escape the Didymos system. Because the closest point of approach of Didymos to Earth's orbit is only 6 milli...
Preprint
Full-text available
In the first paper of this series we examined existing methods of optical meteor trajectory estimation and developed a novel method which simultaneously uses both the geometry and the dynamics of meteors to constrain their trajectories. We also developed a simulator which uses an ablation model to generate realistic synthetic meteor trajectories wh...
Article
It has recently been shown by Egal et al. (2017) that some types of existing meteor in-atmosphere trajectory estimation methods may be less accurate than others, particularly when applied to high precision optical measurements. The comparative performance of trajectory solution methods has previously only been examined for a small number of cases....
Preprint
Full-text available
It has recently been shown by Egal et al. (2017) that some types of existing meteor in-atmosphere trajectory estimation methods may be less accurate than others, particularly when applied to high precision optical measurements. The comparative performance of trajectory solution methods has previously only been examined for a small number of cases....
Preprint
The speed of interstellar asteroid `Oumuamua with respect to the Local Standard of Rest of our Galaxy is low, implying it is young. With this as our hypothesis, we assess possible origin systems for this interstellar asteroid. First,`Oumuamua's past trajectory is modelled under the influence of 'disk heating' (ie. gravitational scattering) to asses...
Preprint
Full-text available
There is a vibrant and effective planetary science community in Canada. We do research in the areas of meteoritics, asteroid and trans-Neptunian object orbits and compositions, and space weather, and are involved in space probe missions to study planetary surfaces and interiors. For Canadian planetary scientists to deliver the highest scientific im...
Article
In June 2019 the Earth will approach within 5° mean anomaly of the centre of the Taurid resonant swarm, its closest post-perihelion encounter with Earth since 1975. This will be the best viewing geometry to detect and place limits on the number of NEOs proposed to reside at the swarm centre until the early 2030s. We present an analysis of the optim...
Article
We report on a multiobservatory campaign to examine asteroid 3200 Phaethon during its 2017 December close approach to Earth, in order to improve our measurements of its fundamental parameters, and to search for surface variations, cometary activity, and fragmentation. The mean colors of Phaethon are B – V = 0.702 ± 0.004, V – R = 0.309 ± 0.003, and...
Preprint
We introduce a new technique to estimate the comet nuclear size frequency distribution (SFD) that combines a cometary activity model with a survey simulation and apply it to 150 long period comets (LPC) detected by the Pan-STARRS1 near-Earth object survey. The debiased LPC size-frequency distribution is in agreement with previous estimates for larg...
Preprint
We report on a multi-observatory campaign to examine asteroid 3200 Phaethon during its December 2017 close approach to Earth, in order to improve our measurements of its fundamental parameters, and to search for surface variations, cometary activity and fragmentation. The mean colors of Phaethon are B-V = 0.702 +/- 0.004, V-R = 0.309 +/- 0.003, R-I...
Preprint
In June 2019 the Earth will approach within 5{\deg} mean anomaly of the centre of the Taurid resonant swarm, its closest post-perihelion encounter with Earth since 1975. This will be the best viewing geometry to detect and place limits on the number of NEOs proposed to reside at the swarm centre until the early 2030s. We present an analysis of the...
Article
This work presents numerical simulations of meteoroid streams released by comet 21P/Giacobini-Zinner over the period 1850–2030. The initial methodology, based on Vaubaillon et al. (2005), has been updated and modified to account for the evolution of the comet's dust production along its orbit. The peak time, intensity, and duration of the shower we...
Article
We introduce a new technique to estimate the comet nuclear size frequency distribution (SFD) that combines a cometary activity model with a survey simulation and apply it to 150 long period comets (LPC) detected by the Pan-STARRS1 near-Earth object survey. The debiased LPC size-frequency distribution is in agreement with previous estimates for larg...
Preprint
This work presents numerical simulations of meteoroid streams released by comet 21P/Giacobini-Zinner over the period 1850-2030. The initial methodology, based on Vaubaillon et al. (2005), has been updated and modified to account for the evolution of the comet's dust production along its orbit. The peak time, intensity, and duration of the shower we...
Article
The Kepler Space Telescope has discovered thousands of planets via the transit method. The transit timing variations of these planets allows us not only to infer the existence of other planets, transiting or not, but to characterize a number of parameters of the system. Using the transit timing variations of the planets Kepler-159b and 159c, the tr...
Preprint
The Kepler Space Telescope has discovered thousands of planets via the transit method. The transit timing variations of these planets allows us not only to infer the existence of other planets, transiting or not, but to characterize a number of parameters of the system. Using the transit timing variations of the planets Kepler-159b and 159c, the tr...
Article
Predictions of the 2018 Draconid activity at the Earth and the Sun-Earth L1 and L2 Lagrange points are presented. Numerical simulations of the meteoroids' ejection and evolution from comet 21P/Giacobini-Zinner are performed with a careful implementation of the results analysis and weighting. Model meteoroid fluxes at Earth are derived using as cali...
Preprint
Predictions of the 2018 Draconid activity at the Earth and the Sun-Earth L1 and L2 Lagrange points are presented. Numerical simulations of the meteoroids' ejection and evolution from comet 21P/Giacobini-Zinner are performed with a careful implementation of the results analysis and weighting. Model meteoroid fluxes at Earth are derived using as cali...
Article
Near-Earth asteroid (3200) Phaethon is notable for its association with a strong annual meteor shower, the Geminids, indicative of one or more episodes of mass ejection in the past. The mechanism of Phaethon's past activity is not yet understood. Here, we present a Hubble Space Telescope (HST) search of meter-sized fragments in the vicinity of Phae...
Preprint
Near-Earth asteroid (3200) Phaethon is notable for its association to a strong annual meteor shower, the Geminids, indicative of one or more episodes of mass ejection in the past. The mechanism of Phaethon's past activity is not yet understood. Here we present a Hubble Space Telescope (HST) search of meter-sized fragments in the vicinity of Phaetho...
Article
Full-text available
Observations have revealed rich structures in protoplanetary disks, offering clues about their embedded planets. Due to the complexities introduced by the abundance of gas in these disks, modeling their structure in detail is computationally intensive, requiring complex hydrodynamic codes and substantial computing power. It would be advantageous if...
Article
Jupiter-family Comet D/1770 L1 (Lexell) was the first discovered Near-Earth Object (NEO), and passed the Earth on 1770 Jul 1 at a recorded distance of 0.015 au. The comet was subsequently lost due to unfavorable observing circumstances during its next apparition followed by a close encounter with Jupiter in 1779. Since then, the fate of D/Lexell ha...
Article
Asteroid 2007 VW266 is among the rare objects with a heliocentric retrograde orbit, and its semimajor axis is within a Hill sphere radius of that of Jupiter. This raised the interesting possibility that it could be in co-orbital retrograde resonance with Jupiter, a second "counter-orbital" object in addition to recently discovered 2015 BZ509. We fi...
Article
Full-text available
Structures observed in debris disks may be caused by gravitational interaction with planetary or stellar companions. These perturbed disks are often thought to indicate the presence of planets and offer insights into the properties of both the disk and the perturbing planets. Gaps in debris disks may indicate a planet physically present within the...
Article
Full-text available
In this work we model the dynamical evolution of meteoroid streams of comet 96P/Machholz, and the largest member of the Marsden sunskirters, comet P/1999 J6. We simultaneously fit the characteristics of eight meteor showers which have been proposed to be linked to the complex, using observations from a range of techniques - visual, video, TV and ra...
Article
Full-text available
We describe how meteors recorded at the Earth can be used to partly reconstruct the meteoroid environments of the planets if a large sample (i.e. millions of orbits at a minimum) is available. The process involves selecting from the Earth-based sample those meteors which passed near a planet's orbit prior to arriving at the Earth, and so carry info...
Article
Full-text available
We examine the conditions under which material from the martian moons Phobos and Deimos could reach our planet in the form of meteorites. We find that the necessary ejection speeds from these moons (900 and 600 m/s for Phobos and Deimos respectively) are much smaller than from Mars' surface (5000 m/s). These speeds are below typical impact speeds f...
Article
Recent theoretical work in celestial mechanics has revealed that an asteroid may orbit stably in the same region as a planet, despite revolving around the Sun in the sense opposite to that of the planet itself. Asteroid 2015 BZ509 was discovered in 2015, but with too much uncertainty in its measured orbit to establish whether it was such a retrogra...
Article
Full-text available
Centaurs are objects whose orbits are found between those of the giant planets. They are supposed to originate mainly from the TransNeptunian objects, and they are among the sources of NearEarth Objects.TransNeptunian Objects (TNOs) cross Neptune's orbit and produce the Centaurs. We investigate their interactions with main belt asteroids to determi...
Article
Full-text available
Encke-type comet 332P/Ikeya-Murakami is experiencing cascading fragmentation events during its 2016 apparition. It is likely the first splitting Encke-type comet ever being observed. A nongravitational solution to the astrometry reveals a statistical detection of the radial and transverse nongravitational parameters, $A_{1} = \left(1.54 \pm 0.39\ri...
Article
The daytime Arietid meteor shower is active from mid-May to late June and is amongst the strongest of the annual meteor showers, comparable in activity and duration to the Perseids and the Geminids. Due to the daytime nature of the shower, the Arietids have mostly been constrained by radar studies. The Arietids exhibit a long-debated discrepancy in...
Conference Paper
Full-text available
67P/Churyumov-Gerasimenko is a bi-lobed comet connected by a thick neck. The aim of our work is to find the possible origins of this comet, dynamically and physically.
Article
Previous studies have revealed Jupiter-family comet 252P/LINEAR as a comet that was recently transported into the near-Earth object (NEO) region in $\sim1800$~AD yet only being weakly active. In this Letter, we examine the "formed (almost) dead" hypothesis for 252P/LINEAR using both dynamical and observational approaches. By statistically examining...
Article
Full-text available
Temporarily Captured Orbiters (TCOs) are Near-Earth Objects (NEOs) which make a few orbits of Earth before returning to heliocentric orbits. Only one TCO has been observed to date, 2006 RH120, captured by Earth for one year before escaping. Detailed modeling predicts capture should occur from the NEO population predominantly through the Sun-Earth L...
Article
We have analysed the orbits and ablation characteristics in the atmosphere of 59 earth-impacting fireballs, produced by meteoroids one meter in diameter or larger, described here as meter-scale. Using heights at peak luminosity as a proxy for strength, we determine that there is roughly an order of magnitude spread in strengths of the population of...
Article
Full-text available
It is speculated that some weakly active comets may be transitional objects between active and dormant comets. These objects are at a unique stage of the evolution of cometary nuclei, as they are still identifiable as active comets, in contrast to inactive comets that are observationally indistinguishable from low albedo asteroids. In this paper, w...
Article
Full-text available
The Quadrantid meteor shower is among the strongest annual meteor showers, and has drawn the attention of scientists for several decades. The stream is unusual, among others, for several reasons: its very short duration around maximum activity (~12 - 14 hours) as detected by visual, photographic and radar observations, its recent onset (around 1835...
Article
Full-text available
The gravitational influence of a planet on a nearby disk provides a powerful tool for detecting and studying extrasolar planetary systems. Here we demonstrate that gaps can be opened in dynamically cold debris disks at the mean-motion resonances of an orbiting planet. The gaps are opened away from the orbit of the planet itself, revealing that not...
Article
Full-text available
NASA's Asteroid Redirect Mission (ARM) has been proposed with the aim to capture a small asteroid a few meters in size and redirect it into an orbit around the Moon. There it can be investigated at leisure by astronauts aboard an Orion or other spacecraft. The target for the mission has not yet been selected, and there are very few potential target...
Article
Full-text available
The Kreutz family of sungrazing comets contains over 2~000 known members, many of which are believed to be under $\sim 100$~m sizes (\textit{mini~comets}) and have only been studied at small heliocentric distances ($r_\mathrm{H}$) with space-based SOHO/STEREO spacecraft. To understand the brightening process of mini Kreutz comets, we conducted a su...
Article
The impact of a meteoroid onto an asteroid transfers linear and angular momentum to the larger body, which may affect its orbit and its rotational state. Here we show that the meteoroid environment of our Solar System can have an effect on small asteroids that is comparable to the Yarkovsky and Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effects u...
Article
The arrival of solid particles from outside our solar system would present us with an invaluable source of scientific information. Attempts to detect such interstellar particles among the meteors observed in Earth's atmosphere have almost exclusively assumed that those particles moving above the Solar System's escape speed -- particles on orbits hy...
Article
Long-period Comet C/2013 A1 (Siding Spring) will experience a close encounter with Mars on 2014 October 19. As of 2013 October 21, the distance of closest approach between the two is projected to be between 89,000 km and 173,000 km, with a nominal value of 131,000 km. Thus, a collision between the comet and the planet has been ruled out, but the co...
Article
The direct detection of Kuiper Belt Objects (KBOs) by telescopic imaging is not currently practical for objects much less than 100 km in diameter. However, indirect methods such as serendipitous stellar occultations might still be employed to detect these bodies. The method of serendipitous stellar occultations has been previously used with some su...
Article
Full-text available
An unexpected intense outburst of the Draconid meteor shower was detected by the Canadian Meteor Orbit Radar (CMOR) on October 8, 2012. The peak flux occurred at ~16:40 UT on October 8 with a maximum of 2.4 +/- 0.3 hr-1 km-2 (appropriate to meteoroid mass larger than 10-7 kg), equivalent to a ZHRmax = 9000 +/- 1000 using 5-minute intervals, using a...
Article
Earth is continuously colliding with fragments of asteroids and comets of various sizes. The largest encounter in historical times occurred over the Tunguska river in Siberia in 1908, producing an airburst of energy equivalent to 5-15 megatons of trinitrotoluene (1 kiloton of trinitrotoluene represents an energy of 4.185 × 10(12) joules). Until rec...
Article
Full-text available
Previous studies have suggested that Comet 209P/LINEAR may produce strong me- teor activity on Earth on 2014 May 24; however, exact timing and activity level is difficult to estimate due to the limited physical observations of the comet. Here we reanalyze the optical observations of 209P/LINEAR obtained during its 2009 appari- tion. We find that th...
Article
The Kreutz sungrazer family is one of the best-known comet families and contains some of the most spectacular comets in history. However, due to its extreme orbit geometry, most members in this family are either too faint or too close to the sun to be observed from the ground. Space-based coronagraphs launched in recent decades, notably the Solar a...
Article
We describe the hardware and software for the Canadian Automated Meteor Observatory (CAMO), an automated two-station video meteor system designed to facilitate simultaneous radar-video meteor detections, to help constrain numerical ablation models with higher precision meteor data, and to measure the meteoroid mass influx at the Earth. A guided sys...
Article
Full-text available
The size distribution of the asteroid belt is examined with 16956 main belt asteroids detected in data taken from the Canada-France-Hawaii Telescope Legacy Survey in two filters (g' and r'). The cumulative H (absolute magnitude) distribution is examined in both filters, and both match well to simple power laws down to H = 17, with slopes in rough a...
Article
Full-text available
We acquired Johnson BV photometry of the binary Be disk system delta~Scorpii during its 2009, 2010, 2011, and 2012 observing seasons and used it to probe the innermost regions of the disk. We found that several disk building events have occurred during this time, resulting in an overall brightening in the V-band and reddening of the system. In addi...
Article
Continuing improvements in observational technologies are providing a wealth of new observations of disk structures that surround B-emission (Be) stars. For example, high-resolution interferometry in the optical and the infrared now routinely provides information at sub-milliarcsecond scales advancing our understanding of the structure and the phys...
Article
Accurate simulations of meteoroid streams permit the prediction of stream interaction with Earth, and provide a measure of risk to Earth satellites and interplanetary spacecraft. Current cometary ejecta and meteoroid stream models have been somewhat successful in predicting some stream observations, but have required questionable assumptions and si...
Article
The Andromedid meteor shower, originating from comet 3D/Biela (which disintegrated in the mid-1800's) has been weak or absent since the late 19th Century. Here we report the return of this shower in Dec 2011 with a zenith hourly rate of approximately 50, the strongest return of the shower in a century. The outburst was detected by the Canadian Mete...
Article
The Andromedid meteor shower underwent spectacular outbursts in 1872 and 1885, producing thousands of visual meteors per hour and described as `stars fell like rain' in Chinese records of the time. The shower originates from comet 3D/Biela whose disintegration in the mid-1800's is linked to the outbursts, but the shower has been weak or absent sinc...
Article
The logarithmic potential is of great interest and relevance in the study of the dynamics of galaxies. Some small corrections to the work of Contopoulos & Seimenis (1990) who used the method of Prendergast (1982) to find periodic orbits and bifurcations within such a potential are presented. The solution of the orbital radial equation for the purel...
Article
Full-text available
The NEOSSat spacecraft will efficiently discover Atira and Aten orbital class near-Earth asteroids by searching arcs of sky across the ecliptic plane to within 45 degrees of the Sun. Approximately 10 Atiras are expected to be discovered per year.
Article
Near-Earth asteroid 2010 TK7 was discovered by the Widefield Infrared Survey Explorer (WISE) satellite on October 1, 2010. With followup observations from the Canada-France-Hawaii telescope, it has been shown to be the first known Trojan companion of Earth, in an orbit with large librational and epicyclic range about the leading L4 Lagrange point....
Article
We report high-resolution multi-station observations of meteors by the Canadian Automated Meteor Observatory recorded from 2009 June to 2010 August. Our survey has a limiting detection magnitude of +5 mag in R band, equivalent to a limiting meteoroid mass of ~2 × 10–7 kg. The high metric trajectory accuracy (of the order of 30 m perpendicular to th...
Article
We have carried out a search for Main Belt Asteroids (MBAs) co-orbiting with the large MBA Vesta and the dwarf planet Ceres. Through improving the search criteria used in (Christou, 2000b) and numerical integrations of candidate coorbitals, we have identified approximately 51 (44) objects currently in co-orbital libration with Ceres (Vesta). We sho...
Article
We show that many MBAs are presently in the 1:1 resonance ('co-orbitals') with dward planet Ceres and large MBA Vesta. They most likely represent a transient population; their dynamical evolution is similar to that of objects known to co-orbit with Earth and Venus. We discuss this population in terms of the longevity of co-orbital capture, its rela...
Article
Full-text available
The Jupiter family comet 169P/NEAT (previously known as asteroid 2002 EX 12) has a dynamical associ-ation with the α-Capriconid meteoroid stream. In this paper, we present photometric observations of comet 169P/NEAT to further investigate the physical char-acters of its disintegration state related to the stream. The comet shows a point-like surfac...
Article
Analytic methods by Ceplecha have long been used for the determination of meteoroid heliocentric orbits. These methods include both the derivation of an initial atmospheric contact position and velocity state, and the calculation of an orbit at infinity based on zenithal attraction assumptions. Herein, we describe a numerical integration-based veri...
Article
It was realized in 1772 that small bodies can stably share the same orbit as a planet if they remain near 'triangular points' 60° ahead of or behind it in the orbit. Such 'Trojan asteroids' have been found co-orbiting with Jupiter, Mars and Neptune. They have not hitherto been found associated with Earth, where the viewing geometry poses difficulti...