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Register a new userFirst approximation for spacecraft motion relative to (99942) Apophis
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We aim at providing a preliminary approach on the dynamics of a spacecraft in orbit about the asteroid (99942) Apophis during its Earth close approach. The physical properties from the polyhedral shape of the target are derived assigning each tetrahedron to a point mass in its center. That considerably reduces the computation processing time compared to previous methods to evaluate the gravitational potential. The surfaces of section close to Apophis are build considering or not the gravitational perturbations of the Sun, the planets, and the SRP. The Earth is the one that most affects the invisticated region making the vast majority of the orbits to collide or escape from the system. Moreover, from numerical analysis of orbits started on March 1, 2029, the less perturbed region is characterized by the variation of the semimajor axis of 40-days orbits, which do not exceed 2 km very close to the central body ($a < 4$ km, $e < 0.4$). However, no regions investigated could be a possible option for inserting a spacecraft into natural orbits around Apophis during the close approach with our planet. Finally, to solve the stabilization problem in the system, we apply a robust path following control law to control the orbital geometry of a spacecraft. At last, we present an example of successful operation of our orbit control with a total $bigtriangleup v$ of 0.495 m/s for 60 days. All our results are gathered in the CPM-ASTEROID database, which will be regularly updated by considering other asteroids.
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The potentially hazardous asteroid (PHA) (99942) Apophis is one of the most remarkable near-Earth asteroids (NEA) in terms of impact hazard. A good determination of its surface thermal inertia is very important in order to evaluate the Yarkovsky effect on its orbital evolution. We present thermal infrared observations obtained on January 29, 2013, with CanariCam mid-infrared camera/spectrograph attached to the Gran Telescopio CANARIAS (GTC, Roque de los Muchachos Observatory, La Palma, Spain) using the Si2-8.7, Si6-12.5, and Q1-17.65 filters with the aim of deriving Apophis diameter ($D$), geometric albedo ($p_V$), and thermal inertia ($Gamma$). We performed a detailed thermophysical model analysis of the GTC data combined with previously published thermal data obtained using Herschel Space Observatory PACS instrument at 70, 100, and 160 $mu$m.The thermophysical model fit of the data favors low surface roughness solutions (within a range of roughness slope angles $rms$ between 0.1 and 0.5), and constrains the effective diameter, visible geometric albedo, and thermal inertia of Apophis to be $D_{eff} =$~380 -- 393 m, $p_V = $~0.24--0.33 (assuming absolute magnitude $H = 19.09 pm 0.19$) and $Gamma =$~50 -- 500 Jm$^{-2}$ s$^{-0.5}$ K$^{-1}$, respectively.
We assess the risk of an Earth impact for asteroid (99942) Apophis by means of a statistical analysis accounting for the uncertainty of both the orbital solution and the Yarkovsky effect. We select those observations with either rigorous uncertainty information provided by the observer or a high established accuracy. For the Yarkovsky effect we perform a Monte Carlo simulation that fully accounts for the uncertainty in the physical characterization, especially for the unknown spin orientation. By mapping the uncertainty information onto the 2029 b-plane and identifying the keyholes corresponding to subsequent impacts we assess the impact risk for future encounters. In particular, we find an impact probability greater than 10^-6 for an impact in 2068. We analyze the stability of the impact probability with respect to the assumptions on Apophis physical characterization and consider the possible effect of the early 2013 radar apparition.
Tumbling asteroids belong to a small group of objects, whose angular velocity vector is unaligned with any of its principal axes of inertia. This leads to challenging efforts to model the trajectory of any spacecraft designed to orbit these bodies. In this work, we deepen a previous study on this topic, concerning the asteroid (99942) Apophis during its close encounter with the Earth in 2029. We analyze the orbital behaviour of a spacecraft orbiting the asteroid during this event, by including the effects of the changes of orientation of the spin axis of the asteroid, depending on two sets of initial conditions. The global dynamics of the spacecraft around the target are analyzed using three approaches, MEGNO, PMap, and Time-Series prediction. We confirm that no spacecraft with natural orbits could survive the high perturbations caused by the close encounter with our planet.
Avikainen showed that, for any $p,q in [1,infty)$, and any function $f$ of bounded variation in $mathbb{R}$, it holds that $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensiona
The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein W^{s,infty}(mathbb{R})$ for one $sin (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
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