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Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator
S. Valk N. Delsate and A. Lemaître and T. Carletti University of Namur (FUNDP), Département de Mathématique, Unité de Systèmes Dynamiques, 8, Rempart de la Vierge, B-5000, Namur, Belgium Corresponding author
In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta and Simó (2000), Mean Exponential Growth factor of Nearby Orbits (MEGNO), which provides an efficient tool to investigate both regular and chaotic components of the phase space.
We compute a stability atlas, for a large set of near-geosynchronous space debris, by numerically computing the MEGNO indicator, to provide an accurate understanding of the location of stable and unstable orbits as well as the timescale of their exponential divergence in case of chaotic motion. The results improve the analysis presented in Breiter et al. (2005) notably by considering the particular case of high-area-to-mass ratios space debris. The results indicate that chaotic orbits regions can be highly relevant, especially for very high area-to-mass ratios.
We then provide some numerical investigations and an analytical theory that lead to a detailed understanding of the resonance structures appearing in the phase space. These analyses bring to the fore a relevant class of secondary resonances on both sides of the well-known pendulum-like pattern of geostationary objects, leading to a complex dynamics.
Solar radiation pressure – Space debris – MEGNO – Detection of chaos – Long-term evolution – Geosynchronous orbit – High area-to-mass ratios – Secondary resonances
Recent optical surveys in high-altitude orbits, performed by the European Space Agency 1m telescope on Tenerife (Canary islands), have discovered a new unexpected population of 10 cm sized space debris in near geosynchronous orbits (GEO). These objects sometimes present highly eccentric orbits with eccentricities as high as 0.55 (Schildknecht et al., 2004 and 2005). Following the initial guess of Liou and Weaver (2004) who suggested that this new population may be constitued by GEO objects with high area-to-mass ratios, recent numerical and analytical investigations were performed to support this assumption (Anselmo and Pardini, 2005; Liou and Weaver, 2005) . In addition, these authors and others, such as Chao (2006) and later Valk et al. (2008a) , presented some detailed results concerning the short- and long-term evolution of high area-to-mass ratios geosynchronous space debris subjected to direct solar radiation pressure. More specifically, these latter authors mainly focused their attention on the long-term variation of both the eccentricity and the inclination vector. Moreover, some studies concerning the effects of the Earth's shadowing effects on the motion of such space debris were given in Valk and Lemaître (2008b) . However, nobody ever dealt with the question to know whether these orbits are really predictable or not on the time scales of their investigations.
The objective of this paper is twofold. The first goal is the investigation of the long-term stability of high area-to-mass ratio space debris subjected to direct solar radiation pressure, by means of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) criterion. Second, still considering high area-to-mass ratios, we bring to the fore a relevant class of additional secondary structures appearing in the phase space.
The paper is organized as follows. In Section 2, we focus our attention to the specification of the underlying model and we give some details about the numerical aspects of the method. In Section 3, for the sake of completeness, we dwell upon the detailed definition of the MEGNO indicator, also providing a review of its main properties, in order to understand the behavior of the chaos indicator. Then in Section 4, in the framework of the validation of our implementation, we retrieve the results obtained by Breiter et al. (2005) . We also discuss the significance of the time of integration, recently reported by Barrio et al. (2007) . In Section 5, we apply the MEGNO technique in order to give a insightful understanding of the stability of high area-to-mass ratio space debris. More specifically, we show that the orbits of such peculiar space debris are extremely sensitive to initial conditions, especially with respect to the mean longitude and the semi-major axis. Second, we perform extended numerical analyses, showing that the related 2-dimensional phase space is dominated by chaotic regions, in particular when the area-to-mass ratio is large. In addition, we also provide some results presenting the importance of the initial eccentricity value in the appearance of chaotic regions. Finally, in Section 6, we present extensive numerical and analytical investigations of the additional patterns which will be identified as secondary resonances.
For the purpose of our study, we consider the modeling of a space debris subjected to the influence of the Earth's gravity field, to both the gravitational perturbations of the Sun and the Moon as well as to the direct solar radiation pressure. As a consequence the differential system of equations governing the dynamics is given by
¨r=apot+a\leftmoon+a⊙+arp,
where apot is the acceleration induced by the Earth's gravity field, which can be expressed as the gradient of the following potential
U(r,λ,ϕ)=−μr∞∑n=0n∑m=0(Rer)nPmn(sinϕ)(Cnmcosmλ+Snmsinmλ), (1)
where the quantities Cnm and Snm are the spherical harmonics coefficients of the geopotential. The Earth's gravity field adopted is the EGM96 model (Lemoine et al., 1987) . In Eq. (1), μ is the gravitational constant of the Earth, Re is the Earth's equatorial radius and the quantities (r,λ,ϕ) are the geocentric spherical coordinates of the space debris. Pmn are the well-known Legendre functions. It is worth noting that the potential of Eq. (1) is subsequently expressed in Cartesian coordinates by means of the Cunningham algorithm (Cunningham, 1970) .
Both the accelerations a\leftmoon and a⊙ result from the gravity interaction with a third body of mass m∗, where ∗=\leftmoon and ∗=⊙, and can be expressed with respect to the Earth's center of mass as
a∗=−μ∗(r−r∗∥r−r∗∥3+r∗∥r∗∥3),
where r and r∗ are the geocentric coordinates of the space debris and of the mass m∗, respectively. The quantity μ∗ is the gravitational constant of the third-body. In our implementation, we chose the high accurate solar system ephemeris given by the Jet Propulsion Laboratory (JPL) to provide the positions of both the Sun and the Moon (Standish, 1998) .
Regarding direct solar radiation pressure, we assume an hypothetically spherical space debris. The albedo of the Earth is ignored and the Earth's shadowing effects are not taken into account either. The acceleration induced by direct solar radiation pressure is given by
arp=CrPr[a⊙∥r−r⊙∥]2Amr−r⊙∥r−r⊙∥,
where Cr is the adimensional reflectivity coefficient (fixed to 1 further on in this paper) which depends on the optical properties of the space debris surface; Pr=4.56×10−6 N/m2 is the radiation pressure for an object located at the distance of 1 AU; a⊙=1 AU is a constant parameter equal to the mean distance between the Sun and the Earth and r⊙ is the geocentric position of the Sun. Finally, the coefficient A/m is the so-called area-to-mass ratio where A and m are the effective cross-section and mass of the space debris, respectively.
3 The Mean Exponential Growth factor of Nearby Orbits
For the sake of clarity we present in this section the definition and some properties of the MEGNO criterion.
Let H(p,q), with p∈Rn, q∈Tn, be a n-degree of freedom Hamiltonian system and let us introduce the compact notation x=(p,q)∈R2n as well as f=(−∂H/∂q,∂H/∂p)∈R2n, then the dynamical system is described by the following set of ordinary differential equations
ddtx(t)=f(x(t),α),x∈R2n, (2)
where α is a vector of parameters entirely defined by the model. Let ϕ(t)=ϕ(t;x0,t0) be a solution of the flow defined in Eq. (2) with initial conditions (t0,x0), then it has associated the Lyapunov Characteristic Number (hereafter LCN), defined by (Benettin et al., 1980)
λ=limt→∞1tln∥δϕ(t)∥∥δϕ(t0)∥, (3)
where δϕ(t), the so-called tangent vector, measures the evolution of an initial infinitesimal deviation δϕ(t0)≡δ0 between ϕ(t) and a nearby orbit, and whose evolution is given by the variational equations (terms of order O(δ2) are omitted)
˙δϕ=ddtδϕ(t)=J(ϕ(t))δϕ(t),withJ(ϕ(t))=∂f∂x(ϕ(t)), (4)
where J(ϕ(t)) is the Jacobian matrix of the differential system of equations, evaluated on the solution ϕ(t). Let us note that the definition of LCN, given by Eq. (3), can also be written in an integral form
λ=limt→∞1t∫t0˙δϕ(s)δϕ(s)ds,
where δϕ=∥δϕ∥, ˙δϕ=˙δϕ⋅δϕ/δϕ.
The Mean Exponential Growth factor of Nearby Orbits Yϕ(t) is based on a modified time-weighted version of the integral form of LCN (Cincotta and Simó, 2000) . More precisely
Yϕ(t)=2t∫t0˙δϕ(s)δϕ(s)sds,
as well as its corresponding mean value, to get rid of the quasi-periodic oscillation possibly existing in Yϕ(t)
¯¯¯¯Yϕ(t)=1t∫t0Yϕ(s)ds.
In the following we will omit the explicit dependence of Y and ¯¯¯¯Y on the specific orbit ϕ, when this will be clear from the context.
Actually, ¯¯¯¯Y(t) allows to study the dynamics for long time scales, where generically Y(t) does not converge, while limt→∞¯¯¯¯Y(t) is well defined (Cincotta et al., 2003) . Consequently, the time evolution of ¯¯¯¯Y(t) allows to derive the possible divergence of the norm of the tangent vector δ(t), giving a clear indication of the character of the different orbits. Indeed, for quasi-periodic (regular) orbits, Y(t) oscillates around the value 2 with a linear growth of the separation between nearby orbits. On the other hand, for chaotic (irregular) motion, the norm of δ grows exponentially with time, and Y(t) oscillates around a linear divergence line. Cincotta et al. (2003) showed that, for the quasi-periodic orbits, ¯¯¯¯Y(t) always converges to 2, that is a fixed constant. Moreover, it has been shown that ordered motions with harmonic oscillations, i.e. orbits very close to a stable periodic orbit, tend asymptotically to ¯¯¯¯Y(t)=0.
These latter properties can also be used to compute efficiently a good estimation of LCN, or similarly the Lyapunov time Tλ=1/λ, by means of a linear least square fit of ¯¯¯¯Y(t). Indeed, in the case of an irregular orbit, the time evolution of ¯¯¯¯Y(t) may be easily written as
¯¯¯¯Y(t)≃a⋆t+d,t→∞,
where a⋆ is simply related to LCN by the relation a⋆=λ/2 and d is small for irregular and chaotic motion. Bur for regular orbits, after a transitory time, d is not necessarily close to zero. Thus, the value of d may be considered as the measure of how long the orbits sticks to a regular torus before getting chaotic (Cincotta and Simó, 2000) .
Regarding the numerical computation of the MEGNO indicator, we adopt the same strategy as in Goździewski et al. (2001) . To be specific, in addition to the numerical integrations of both the equations of motion and the first order variation equations, we consider the two additional differential equations
ddty=˙δ⋅δδ⋅δ,ddtw=2yt, (5)
which allow to derive the MEGNO indicators as
Y(t)=2y(t)/t,¯¯¯¯Y(t)=w(t)/t.
The MEGNO criterion, unlike the common Lyapunov variational methods, takes advantage of the whole dynamical information for the orbits and the evolution of its tangent vector, which results in shorter times of integration to achieve comparable results. Moreover, a couple of applications found in the literature (e.g. Goździewski et al. 2001, 2008 ; Cincotta and Simó 2000; Breiter et al. 2005 ) justify and confirm that MEGNO is relevant, reliable and provides an efficient way for the investigation of the dynamics by detecting regular as well as stochastic regimes.
3.1 Megno and numerical integrations
As previously mentioned, in order to evaluate the MEGNO indicator, we have to integrate the differential system of Eq. (2), the linear first order variational system of Eq. (4), as well as the two additional differential Eq. (5). We choose to write both the expressions of the perturbing forces and the variational system, i.e. the Jacobian matrix, in rectangular coordinates positions-velocities. In such a way we can overcome both the null eccentricity and the null inclination singularity present in the dynamics of space debris (Valk et al., 2008c) . Moreover, the explicit analytical expressions of the vector fields allow us to avoid the difficulties inherent in the classical method of neighboring trajectories (two particles method).
In order to integrate numerically the two differential systems of equations, we adopted the variable step size Bulirsh-Stoer algorithm (see e.g. Bulirsh and Stoer, 1966, and Stoer and Bulirsch, 1980). Let us note that, for the purpose of validation, the numerical integrations were also made with a couple of other numerical integrators. However, the Bulirsh-Stoer algorithm seems to be the best compromise between accuracy and efficiency. Moreover, as quoted by Wisdom (1983) : What is more important for this study, Benettin et al. (1980) found that the maximum LCE111Lyapunov Characteristic Exponent. did not depend on the precision of their calculation. It appears likely that as long as a certain minimum precision is kept, maximum LCE's may be accurately computed, even though it is not possible to precisely follow a specified trajectory for the required length of time.
Although this latter observation was formulated in the framework of both Lyapunov variational method and Hamiltonian systems, it seems that it remains relevant in the computation of the MEGNO criterion, at least in the particular case of our analysis.
3.2 Influence of the initial tangent vector δ0
By construction MEGNO depends on the initial value of the tangent vector δ0 as the LCE (Benettin et al., 1980) . That's way we preferred to adopt the strategy of initialize randomly the initial tangent vectors in order to avoid some parts of the artificially created zones of low MEGNO due to the proximity of δ0 to the minimum Lyapunov exponent direction (Breiter et al., 2005) . Moreover, as pointed out by Goździewski et al. (2001) , the random sampling of δ0 is relevant in the sense that different initial tangent vectors can lead to different behaviors of the MEGNO time evolution while considering the same orbit. This observation has been reported in the framework of extra-solar planetary systems and seems to remain similar in the case of Earth orbiting objects and more generally for high-dimensional dynamical systems (having more than 3 degrees of freedom).
Regarding the impact of the choice of the initial tangent vector δ0, we performed a set of exhaustive numerical investigations of regular orbits. More specifically, we compared the time-evolution of MEGNO using different initial tangent vectors and identical generic initial conditions. The results confirm that the random choice of the initial tangent vector induces a significant random behavior in the way MEGNO approaches the limit value 2, hence preventing this information from being useful to check the stability/instability character of regular orbits. Actually, when considering a slightly perturbed two-body problem (such as the central attraction disturbed by the oblateness of the Earth), the way MEGNO converges to 2 is completely unpredictable , leading to more or less 50% of convergence of ¯¯¯¯Y(t) to 2 from above and the other remaining 50% from below. This result is formally discussed in the following subsections. However, when the order of magnitude of the perturbation is larger, the result does not completely hold anymore. In particular, when considering the perturbing effects induced by the 1:1 resonance, the MEGNO evolution no longer depends on the random choice of the initial tangent vector. In this latter case, the intrinsic stability of the chosen orbits seems also to dictate the evolution of MEGNO as reported in Cincotta et al. (2003) . More specifically, the stability of the orbit seems to influence the time evolution of MEGNO the more the orbit is closer to a stable or unstable equilibrium point. For instance, regarding the orbits extremely close to a stable equilibrium point, MEGNO generally approaches slowly the limit value 2 from below, even though some infrequent orbits present a MEGNO convergence from above. Conversely, the orbits initially close to the separatrices generally present a MEGNO approaching the value 2 from above.
3.3 Megno for integrable systems
In this section we will study the MEGNO indicator for integrable Hamiltonian systems and we will show that generically (if the system is not isochronous) it always converges to 2, moreover the way Y(t) reaches this limit value, say from higher or lower values, depends only on the choice of the initial tangent vector and not on the orbit itself.
So let us consider an integrable Hamiltonian system and suppose to write it in action-angle variables, H=H(p), where p∈B⊂Rn denotes the action variables and q∈Tn denotes the angle variables. Then the Hamiltonian equations are
˙p=0,˙q=∂H∂p=ω(p).
The tangent space (to a given orbit) can be split into the action and angle direction, namely δ=(δp,δq), thus the variational system can be written as
\setstretch1.4˙δp=0,˙δq=∂2H∂p2δp=M(p)δp.
If the system is isochronous then M≡0, thus δp and δq are constant and Y(t)=0 for all t. On the other hand, if the system is non-isochronous we get δp(t)=δp(0) and δq(t)=δq(0)+M(p(0))δp(0)t. To simplify the notations, let us introduce
M(p(0))=M0,δp(0)=ξ0 and δq(0)=η0.
Using the definition of MEGNO, we get
Y(t)=1t∫t0(M0ξ0)2s+M0ξ0⋅η0(ξ0)2+(η0)2+2M0ξ0⋅η0s+(M0ξ0)2s2sds,
and this integral can be explicitly computed, obtaining
\setstretch2Y(t)=2−M0ξ0⋅η0t(M0ξ0)2log[1+2M0ξ0⋅η0t+(M0ξ0)2t2]+−2t√(M0ξ0)2−(M0ξ0⋅η0)2(M0ξ0)2[arctanM0ξ0⋅η0+(M0ξ0)2t2√(M0ξ0)2−(M0ξ0⋅η0)2−arctanM0ξ0⋅η0√(M0ξ0)2−(M0ξ0⋅η0)2]. (6)
One can check that the square root is well defined, i.e. positive, and thus one can cast Eq. (6) into
Y(t)=2−M0ξ0⋅η0tF1(t)−1tF2(t),
where F1 and F2 are positive functions and F2 is bounded. We can then conclude that (see Fig. 1)
if M0ξ0⋅η0>0 then Y(t) approaches 2 from below;
if M0ξ0⋅η0<0 then Y(t) approaches 2 from above, in fact for large t the first contribution dominates the bounded term F2.
In this last part we will consider if and under which assumptions the previous results concerning the convergence Y→2 are still valid, for a quasi–integrable Hamiltonian system of the form H(p,q,ϵ)=H0(p)+ϵV(p,q). The main idea is the following: fix ϵ>0, but small, and consider a "non–chaotic"orbit ϕϵ, namely an orbit without a positive Lyapunov exponent (or "if you prefer" with a bounded MEGNO), then if ϵ is sufficiently small this orbit is a perturbation of an orbit existing also for ϵ=0, ϕ0, and we can check that Yϕϵ=Yϕ0+O(ϵ), hence the smallness of such ϵ–correction cannot change "the way Y goes to 2". More precisely, the Hamilton equations are now
\setstretch1.6˙p=−∂H∂q=−ϵ∂V∂q˙q=∂H∂p=ω(p)+ϵ∂V∂p,
and a similar decomposition can be done for the variational system
\setstretch1.6˙δp=−ϵ∂2V∂p∂qδp−ϵ∂2V∂q2δq˙δq=(∂2H∂p2+ϵ∂2V∂p2)δp+ϵ∂2V∂p∂qδq.
Figure 1: MEGNO for quasi–integrable adimensional Hamiltonian system. We consider the evolution of Yϕϵ for the system H=p21/2+p2+ϵcosq1+ϵcos(q1−q2). On the left panel ϵ=10−4, while on the right panel ϵ=10−3. In both cases ϵ is small enough to confirm the theoretical predictions; let observe that in this case the matrix M is given by (1000) and thus the sign condition reads Mδp,0⋅δq,0=δ1p,0δ1q,0. The unit of time corresponding to 1/10 of period of the orbit.
Looking for δp and δq as ϵ–power series, i.e. δp=δp,0+ϵδp,1+… and δq=δq,0+ϵδq,1+…, and collecting together, in the definition of MEGNO, terms contributing to the same power of ϵ, we can thus get
\setstretch1.6Yϕϵ(t)=1t∫t0(M0δp,0)2s+M0δp,0⋅δq,0(δp,0)2+(δp,0)2+2M0δp,0⋅δq,0s+(M0δq,0)2s2sds+O(ϵ)=Yϕ0(t)+O(ϵ).
4 Validation of the method
To validate our method we first apply the technique on a simplified model, containing only the Earth's gravity field expanded up to the second degree and order harmonics, namely, J2=−C20,C22 and S22. For the purpose of the analysis, we followed a set of 12 600 orbits, propagated over a 30 years time span, that is the order of 104 fundamental periods (1 day) empirically required by the method (Goździewski et al., 2001) . As reported in Breiter et al. (2005) , a 30 years time span seems to be relatively small for long-term investigations of geosynchronous space debris. However, the numerical integration of variational equations in addition to the extrapolation of the orbit is quite time consuming. Indeed, the simulation with an entry level step size of 400 seconds takes approximately 20 seconds per orbit when including only the Earth's gravity field, whereas it takes 42 seconds with a complete model. Thus, the examination of large sets of initial conditions can take a lot of time (typically 5 days for 104 orbits). On the other hand, the analysis of the following section will bring to the fore some indications about the Lyapunov times (smaller than 30 years). As a consequence, the integration time can be considered as sufficiently large in the particular case of our study.
For the purpose of this validation study, we consider a set of initial conditions defined by a mean longitude λ grid of 1∘, spanning 90∘ on both sides of the first stable equilibrium point and a semi-major axis a grid of 1 km, spanning the 42164±35 km range. The other fixed initial conditions are e0=0.002 for the eccentricity, i0=0.004 rad for the inclination, Ω0=ω0=0 rad for the longitude of the ascending node and the argument of perigee, respectively. These values have been fixed to compare our results for the nearly-geosynchronous orbits with the ones of Breiter et al. (2005) . As pointed out by Breiter et al. (2005) , due to the 1:1 resonance, good variables to present our results will be (a0,σ0), where a0 is the osculating initial semi-major axis and σ is the so-called resonant angle, i.e. σ=λ−θ, where θ is the sidereal time.
Figure 2: The MEGNO computed as a function of initial mean longitude λ0 and osculating semi-major axis a0. The equations of motion include the central body attraction as well as the second degree and order harmonics J2,C22 and S22. The mean longitude grid is 1∘ and the semi-major axis grid is 1 km, spanning the 42164±35 km range. The initial conditions are e0=0.002, i0=0.004 rad, Ω0=ω0=0 rad. Time at epoch is 25 January 1991. The patterns have been obtained using two different times of integration, tf=30 years [left] and tf=300 years [right].
Figure 2 (left panel) shows the MEGNO values computed using 30 years of integration time. We identify clearly a blow-up of the typical double pendulum-like pattern related to the 1:1 resonance. Here, we plot only over a horizontal range of 180∘, i.e. only one eye. The existence of both the stable and the two unstable equilibrium points can be easily inferred. We observe that the phase space seems to be essentially filled in with MEGNO values ¯¯¯¯Y(t)≃2, that is plenty of regular orbits. Moreover, the two separatrices are also identifiable and are associated with neighboring MEGNO values 2<¯¯¯¯Y(t)≤4. Therefore, following the properties defined in Section 3, one could consider that these orbits are chaotic. However, we will show that this conclusion is false. Indeed, a careful identification of the MEGNO time evolution shows that the latter always approach slowly the limit 2 from above. The closer to the separatrice, the slower the convergence. More precisely, orbits close to the separatrix integrated over long time span present a bounded MEGNO evolution. Hence they should be considered as non chaotic.
To clarify this point, we performed a similar study, but using a significantly longer time-span, namely 300 years. The results are shown in Figure 2 (right panel). For the sake of comparison, the color bars have been taken identical on both plots. Let us observe that the maximum value reached by the MEGNO is 4 in the left panel and 2.5 in the right one. In the 300 years simulation (Figure 2, right), the MEGNO values, associated with orbits close to the separatrices, turn out to be, on average, smaller than in Figure 2 (left panel), reaching almost the limit ¯¯¯¯Y(t)→2, due to the longer time of integration. Similarly, the dark zone in the neighborhood of the stable equilibrium point, corresponding to MEGNO values close to zero, is strongly shrunk, supporting the result that, in the limit of infinitely large t, only the orbit originating from the exact stable equilibrium point leads to ¯¯¯¯Y=0, whereas the neighboring trajectories converge slowly to ¯¯¯¯Y(t)=2.
Let us note that the importance of the integration time has been recently reported by Barrio et al. (2007) in the framework of applications of the MEGNO method. And, we confirm that a too short time of integration can give wrong conclusions about the dynamical behavior. Moreover, the latter paper also underlines some spurious structures appearing in the maps of the variational chaos indicators, explaining the presence of the sine wave of lower MEGNO with a bulge at the center of Figures 2, "suggesting that the same periodic orbit is more or less regular depending on the initial conditions choice". Actually, accordly to the latters authors and to our analysis, this conclusion is wrong because this spurious structure is related to numerical artifacts.
5 High area-to-mass ratios analysis
The study of the long-term stability of near-geosynchronous objects has recently prompted an increasing interest of the scientific community. In the particular case of classical near-geosynchronous objects, the problem has been solved by computing the MEGNO indicator for a family of simulated geostationary, geosynchronous and super-geosynchronous orbits. A classical near-geosynchronous object has a period close to one sidereal day and is subjected to the main gravitational effects of the Earth, including the 1:1 resonance, luni-solar perturbing effects, as well as solar radiation pressure associated to a small area-to-mass ratio (A/m≪1 m2/kg). According to Breiter et al. (2005) and Wytrzyszczak et al. (2007) , the near-geostationary region presents chaotic orbits only very close to the separatrices, due to the irregular transits between the libration and the circulation regimes. Regarding the super-geostationary orbits, all of them seem to be entirely regular on the time scale of the investigations, that is a few decades.
The aim of this section is to provide a more extensive analysis of the dynamics of near-geosynchronous space debris with high area-to-mass ratios (A/m≫1 m2/kg), subjected to direct solar radiation pressure. Our main objective is to study the effects of high area-to-mass ratios on the stability of the principal periodic orbits and on the chaotic components. This analysis is divided into three parts. First, in section 5.1, we focus our attention on the sensitivity to initial conditions; then, in section 5.2, we report the results of dedicated numerical analyses which emphasize the importance of the area-to-mass ratio value. Finally, in section 5.3, we study the influence of both the initial eccentricity and time at epoch.
Let us recall that for large area-to-mass ratios (A/m≥10 m2/kg), the solar radiation pressure may become the major perturbation, by far larger than the dominant zonal gravity term J2 (Valk et al., 2008a) . In this particular case, the larger the area-to-mass ratio, the more affected the dynamics of the near-geosynchronous space debris, leading to daily high-amplitude oscillations of the semi-major axis, yearly oscillations of the eccentricity as well as long-term variations of the inclination.
Figure 3: Time-evolution of high area-to-mass ratio space debris. Orbital elements over 210 years for a A/m=10 m2/kg; initial conditions are: a0=42166.473 km, e0=0.002,i0=0.004 rad, Ω0=ω0=0 rad and M0=4.928 rad. Time at epoch is 25 January 1991.
As an illustration, Figure 3 shows the orbital elements histories of the first 210 years of a geosynchronous high area-to-mass ratio space debris (A/m=10 m2/kg). The yearly variation of the eccentricity reaches 0.2, which confirms the expected values predicted (e.g. Anselmo and Pardini, 2005, and Liou and Weaver, 2005). The inclination evolution presents a well known long-term variation whose period is directly related to the area-to-mass ratio value. Regarding the longitude of ascending node as well as the argument of perigee, they both present a libration due to the chosen set of initial conditions. For further details, we refer to Valk et al. (2008a) as well as Chao (2006) , where a full description of the long-term motion of high area-to-mass ratios space debris is given.
5.1 Sensitivity to initial conditions
To start with, we follow the evolution of two high area-to-mass ratio space debris (A/m=10 m2/kg) defined by two sets of very close initial conditions, differing only in the 10th digits in mean longitude. Figure 3 shows the first one and Figure 4 shows the second near orbit. We observe that the most difference (in the behavior) take place in the semi-major axis and resonant angle panels. We notice that there are some differences in the dynamics of the semi-major axis already after 20 years and at the end of the integration. This is the same for resonant angle, confirming the hypothesis that the sensitivity to initial conditions is especially relevant for the semi-major axis and resonant angle whereas the difference between the other orbital elements remains small. We first focus our attention on the time evolution of the semi-major axis and resonant angle. As a complement to Figure 3, we numerically computed two orbits for two space debris with different area-to-mass ratios, A/m=1 m2/kg and A/m=10 m2/kg, whose initial conditions have been chosen near the separatrices, to emphasize their chaotic behaviors.
Figure 5 shows a blow-up of the evolution of the semi-major axis (top panels) and resonant angle (middle panels) over the time span of 250 years. It is clear that the semi-major axis presents some irregular components over its evolution, related to some transitions between different regimes of motion, clearly identifiable in the resonant angle plots. In addition, we also computed the corresponding MEGNO time evolution. The bottom panel in each graph shows the time evolution of the MEGNO indicator as well as its corresponding mean value. First, we see that the time evolution of ¯¯¯¯Y(t) presents a quasi-linear growth almost since the beginning of the integration process, leading to the conclusion that these orbits are clearly chaotic over that time scale.
Figure 4: Time-evolution of high area-to-mass ratio space debris. Orbital elements over 210 years for a A/m=10 m2/kg; the initial conditions are the same of Figure 3 but differ in the 10th digits in mean longitude.
Therefore, we also computed the linear fit ¯¯¯¯Y(t)≃a⋆t+d in order to evaluate the Lyapunov time Tλ. Tλ is the inverse of the LCN (λ) calculated by the linear regression coefficients a⋆=λ/2. Let us remark that to avoid the initial transient state, the least square fits were performed on the last 85% of the time interval. This latter analysis brings to the fore the fact that larger area-to-mass ratios lead to smaller Lyapunov times, i.e. larger Lyapunov characteristic numbers. Indeed, for A/m=1 m2/kg, the Lyapunov time turns out to be on the order of 11 years, whereas it reaches the value Tλ≃3.7 years for A/m=10 m2/kg.
Figure 5: For each graph, we show the orbital evolution of the semi-major axes a (solid line) superimposed with the evolution of the inclinations (dashed line) [top panels]. The time evolution of the resonant angles [middle panels] and the time evolution of the MEGNO indicator (Y and ¯¯¯¯Y=<Y(t)>), as well as the corresponding linear fit ¯¯¯¯Y(t)≃a⋆t+d [bottom panels]. The area-to-mass ratios are A/m=1 m2/kg in the upper panel and A/m=10 m2/kg in the lower one. The initial conditions are chosen near the separatrices. The computed linear regression coefficients are given by a⋆=0.043 (for A/m=1 m2/kg) and a⋆=0.134 (for A/m=10 m2/kg).
Second, let us also remark that the behavior of the MEGNO indicator is of particular interest in these cases. A careful analysis of Y(t) underlines some irregular patterns directly related to the evolution of σ, in particular when the orbits seem to transit across the separatrices. Finally, we can also highlight the fact that the sudden changes between libration and circulation regimes occur mainly when the inclination changes its sign of variation, especially at the maximum value for A/m>>1 m2/kg and at the minimum for A/m≤1 m2/kg (Figure 5, top panels, dashed line), with an empirical long-term periodicity of TΩ, that is the long-term periodicity of the longitude of the ascending node, which is all the more smaller when A/m is large (Valk et al., 2008a) .
5.2 Extended numerical analyses
We considered a set of 12600 simulated orbits with various initial semi-major axes and mean longitudes. We took into account the following perturbing effects: second degree and order harmonics (J2,C22 and S22), the luni-solar interaction as well as the perturbing effects of the solar radiation pressure with four values of the area-to-mass ratio (A/m=1,5,10,20 m2/kg). The results are reported in Figure 6.
In the case with A/m=1 m2/kg (top left panel) we recognize the same pendulum-like pattern as in Figure 2. Considering the same integration time (30 years), we notice that the MEGNO values tend to be slightly larger than in Figure 2 (left). Moreover, some irregularly distributed MEGNO values are clearly visible close to the two saddle unstable stationary points.
Figure 6: The MEGNO computed as a function of initial mean longitude λ0 and initial (osculating) semi-major axis a0. The equations of motion include the central body attraction, the second degree and order harmonics J2,C22 and S22, the luni-solar interaction as well as the perturbing effects of the solar radiation pressure. The mean longitude grid is 1∘ and the semi-major axis grid is 1 km, spanning the 42164±35 km range. The initial conditions are e0=0.002,i0=0.004 rad and Ω0=ω0=0 rad. The integration time is 30 years from epoch fixed at 25 January 1991. The patterns have been obtained using four different area-to-mass ratios, A/m=1,5,10,20 m2/kg, represented, respectively in the top left, top right, bottom left and bottom right panel.
These results completely agree with those presented by Breiter et al. (2005) , where the solar radiation pressure was taken into account, but only for very small area-to-mass ratios (typically 0.005 m2/kg). Indeed, our latter analysis shows that in addition to the luni-solar perturbations, solar radiation pressure with small to moderate area-to-mass ratios, that is 0≤A/m≤1 m2/kg, do not change considerably the phase space pattern.
On the other hand, the remaining panels of Figure 6 show that the phase portrait becomes significantly more intricate with increasing area-to-mass ratios. Indeed, the width of the stochastic zone in the neighbourhood of the separatrices becomes relevant, with a large displacement of the separatrices on the phase plane. The larger chaotic region can readily be explained by the osculating motion of the separatrices due to the before-mentioned daily variations of the semi-major axis with respect to some mean value as well as by the increasing amplitudes of the eccentricities. These variations lead inevitably to transits through both the regions separating libration and circulation motion for orbits initially close to the separatrices.
Figure 7: Cartoon to illustrate the difference between mean and osculating initial conditions with respect to the semi-major axis (s.m.a.) evolution. For the sake of simplicity, the mean semi-major axis does not present any long-term variation, whereas the osculating semi-major axis presents daily oscillations related to direct solar radiation pressure (the implicit underlying model is radiation pressure only). It is clear that even if the osculating initial conditions aosc1 and aosc2 are identical, the corresponding mean initial conditions amean1 and amean2 can be significantly different, due to different initial mean longitudes (or similarly different initial resonant angle values).
Moreover, it is also clear that the usual double pendulum-like phase space shows a tendency to be distorted with an apparent displacement of the unstable equilibrium points, whereas the stable equilibrium points remain almost fixed. This last result is however quite awkward insofar as there is no physical interpretation to this phenomenon. Indeed, direct solar radiation pressure does not depend explicitly on the (mean) resonant angle with respect to the long-term investigations after averaging over short perioding terms. Therefore, it can not induce a displacement of the equilibrium points in the phase space. Actually, a clever explanation can be found regarding the way the sampling is considered in the elaboration of the graphics. More specifically, it is worth noting that, at first, the sampling is carried out with respect to osculating initial conditions.
Second, within the framework of mean motion theory, it is well-known that, due to the short-period oscillations, the mean and the osculating initial conditions can not be considered to be equal. In other words, for the same fixed value of the initial osculating semi-major axis and for various initial mean longitude, we obtain different values for the mean semi-major axis; as explained with Figure 7. Actually, the different initial mean longitudes induce a phase difference in the corresponding evolution of the semi-major axis, leading to different mean initial semi-major axes. Let us remark that the maximum difference between both the mean semi-major axes is directly related to the order of magnitude of the short-period variations, and, as a consequence, is also directly related to the area-to-mas ratio.
Figure 8: Relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis. The first osculating semi-major axis is taken above the libration region, the second is related to an osculating semi-major axis sampling which crosses the libration region and, finally, the third sampling is taken below this region.
More rigorously, the difference between osculating and mean initial conditions is a well-defined transformation, depending on the generating function used within the averaging process allowing to change from mean to osculating dynamics. For further details concerning this explicit transformation, we refer to the Lie algorithm discussed in Deprit (1969) and Henrard (1970) . However, because we bound our analysis mainly to numerical simulations, we cannot access such generating function; we can nevertheless overcome this problem by numerically computing, for each semi-major axis osculating initial condition, the related mean initial semi-major axis, by considering the average over a short time span of 10 days. As an illustration, in Figure 8, we give the relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis (A/m=10 m2/kg). The first difference is related to a semi-major axis sampling taken above the libration region, the second is related to a semi-major axis sampling which crosses the libration region and finally, the third sampling is taken below this region. In conclusion, we clearly see that the order of magnitude of the differences is, as previously mentioned, the order of the amplitudes of the daily variations observed in the semi-major axis dynamics. Let us note that in the latter case, i.e. A/m=10 m2/kg, the differences reach at most 27 km, which correspond exactly to the difference between the stable and unstable equilibrium points, as shown in Figure 6 (bottom, left).
We can thus apply numerically the transformation as a post-treatment process, that is considering the MEGNO values not in the osculating initial conditions phase space, but in the mean initial conditions phase space. For the sake of comparison with Figure 6, we show the results once such a transformation has been applied (Figure 9): it is clear that now the vertical gaps between both the stable and unstable equilibrium points are almost completely eliminated, hence these points have almost the same mean semi-major axis, getting rid of the what we called the "short-period artefact". The thin light waves crossing the Figure 9 are due to gaps in the set of initial conditions and have no dynamical significance (also valid for the only one light wave crossing the Figure 13 in section 6). Let us also remark that, from now on, all the results will be shown in the mean initial conditions phase space.
Figure 9: The MEGNO computed as a function of initial mean longitudes λ0 and initial mean semi-major axis a0. The model is the same as in Figure 6. The area-to-mass ratio is A/m=5 and 10 m2/kg for the left and for the right graph, respectively.
5.3 Initial time at epoch and importance of the mean eccentricity
One should also recall that solar radiation pressure leads to a theoretical equilibrium defined both in eccentricity e0 and longitude of perigee ϖ0. The conditions leading to such an equilibrium can be written as
⎧⎪⎨⎪⎩e0=32CrPrAm1nan⊙cos2ϵ2≃0.01CrAm,ϖ0=λ⊙(0).
where n and n⊙ are the angular motions of both the space debris and the Sun respectively, ϵ is the obliquity of the Earth with respect to the ecliptic and λ⊙(0) the initial ecliptic longitude of the Sun. If these conditions are fulfilled, it has been shown (Chao, 2006, and later Valk et al., 2008), that the eccentricity vector (ecosϖ,esinϖ) remains constant, leading to a fixed value of both the eccentricity and longitude of perigee.
As an illustration, Figure 10 shows the mid-term variations of the eccentricity for a fixed value of the area-to-mas ratio (A/m=10 m2/kg) and fixed initial conditions, namely, a0=42164 km, e0=0.1, i0=0 rad, Ω0=ω0=λ0=0 rad. It is clear that, apart from a phase difference, the amplitudes of the variations of the eccentricities are qualitatively the same, except when adopting an initial time at epoch equal to 21 March. In this latter case, the eccentricity remains almost constant, as expected by the theory.
Figure 10: Mid-term variations (yearly oscillations) of the eccentricity for fixed initial conditions a0=42164 km, e0=0.1, i0=0 rad, Ω0=ω0=λ0=0 rad and A/m=10 m2/kg. Various initial times at epoch t0, namely different initial ecliptic longitudes of the Sun λ⊙(0), were used for the numerical propagations. Only J2,C22,S22 and direct solar radiation pressure taken into account.
Figure 11 shows the phase space in mean semi-major axis and longitude for A/m = 10 m2/kg and fixed values of the initial conditions, namely e0=0.1, i0=0.004 rad, Ω0=ω0=0 rad. The differences between the two graphs only depends on the initial time at epoch parameter t0. We could actually expect that different initial times at epoch, namely, different initial ecliptic longitudes of the Sun λ⊙(0), will reveal a quite rich collection of behaviors, depending on the different states with respect to the before-mentioned eccentricity equilibrium. Actually, assuming an initial time at epoch of 21 December 2001, we see clearly that the phase space is filled by a large number of chaotic orbits (Figure 11, left). On the contrary, starting with an initial time at epoch of 21 March 2000, that is adopting a Sun pointing longitude of perigee (λ⊙(0)=0 rad), the MEGNO values tend to be smaller and associated with significantly narrower chaotic regions, always located close to the separatrices (Figure 11, right). In the latter case, the eccentricity presents only small yearly variations due to the proximity of the theoretical equilibrium.
Figure 11: MEGNO computed as a function of initial mean longitude λ0 and semi-major axis a0. The equations of motion include the central body attraction, the second degree and order harmonics J2,C22 and S22, the luni-solar interaction as well as the perturbing effects of solar radiation pressure. The mean longitude grid is 1∘ and the semi-major axis grid is 500 m spanning the 42164±35 km range. The initial conditions are e0=0.1,i0=0.004 rad, Ω0=ω0=0 rad with A/m=10 m2/kg. The patterns have been obtained using two different initial times at epoch, namely 21 December 2000 (left) and 21 March 2000 (right), respectively.
Therefore, these results seem to suggest that high amplitude variations of the eccentricity increase considerably the extension of chaotic regions close to the separatrices and, conversely, small eccentricity variations seem to minimize considerably the extent of chaotic regions. To justify this assumption, we performed a dedicated numerical simulation with the same set of parameters used in the one reported in Figure 11, but considering higher values of the initial eccentricity. The results are reported in Figure 12, the chosen time at epoch is 21 December 2000 and the initial eccentricities are, e0=0.2 (left panel) and e0=0.4 (right panel). In the latter case, the huge variations of the perigee altitude, induced by the large variations of the eccentricity as well as by the variations of the semi-major axis, leads to even more complicated dynamics. These results thus confirm the importance of the initial eccentricity in the appearance of chaos.
Figure 12: MEGNO computed as a function of initial mean longitude λ0 and semi-major axis a0. The equations of motion include the central body attraction, the second degree and order harmonics J2,C22 and S22, the luni-solar interaction as well as the perturbing effects of solar radiation pressure. The mean longitude grid is 1∘ and the semi-major axis grid is 500 m spanning the 42164±35 km range. The initial conditions are i0=0.004 rad, Ω0=ω0=0 rad with A/m=10 m2/kg. Time at epoch is 21 December 2000. The patterns have been obtained using two initial eccentricities, e0=0.2 (left) and e0=0.4 (right).
6 Secondary resonances
It is worth noting that inspecting Figures 9, 11 and 12 we clearly note the presence of some additional patterns located on both sides of the separatrices in the phase space. These never seen before regions, hence unexplained so far, are actually characterized by very low MEGNO values. Indeed, this observation underlines the fact that the dynamics of high area-to-mass ratio space debris is even more intricate than expected. In the following two sections we will provide some numerical results and an analytical theory, based on a simplified model, to better understand such zones.
6.1 Numerical investigations
We followed a large set of near-geosynchronous space debris, related to an extremely large set of initial conditions taken on both sides of the pendulum-like pattern, and for each one of the 72000 orbits we computed the related MEGNO indicator. The initial conditions have been fixed by a mean longitude grid of 1∘, spanning 360∘, and a semi-major axis grid of 1 km, spanning the 42164±100 km range, while the remaining orbit parameters and time at epoch are the same as in Figure 6. Moreover, as in the previous extended analyses, the model of forces also includes the central body attraction, the second degree and order harmonics J2,C22 and S22 as well as the combined attractions of the Sun and the Moon. The perturbing effects of direct solar radiation pressure are also taken into account for a high area-to-mass ratio fixed at 10 m2/kg.
The results are reported in Figure 13, which is nothing but an extensive enlargement of the phase space presented in Figure 6 (bottom, left). This phase space widening clearly underlines the before-mentioned additional structures located at ± 40 km on each side of the resonant area. Furthermore, besides these patterns, what is of special interest is that this Figure also brings to the light supplementary structures located at approximately 80 km on both sides of the main resonance, suggesting that the phase space is actually foliated by a larger set of secondary structures. Moreover, the width of these additional patterns and the numerical values of the MEGNO both seem to be directly related to the inverse of the distance with respect to the resonant area.
Figure 13: MEGNO computed as a function of initial mean longitude λ0 and semi-major axis a0. The equations of motion include the central body attraction, the second degree and order harmonics J2,C22 and S22 as well as the luni-solar perturbations. The mean longitude grid is 1∘ and the semi-major axis grid is 1 km, spanning the 42164±100 km range. The initial conditions are e0=0.002, i0=0.004 rad and Ω0=ω0=0 rad. The area-to-mass ratio is 10 m2/kg. Time at epoch is 25 January 1991.
In addition, we also performed a set of similar numerical investigations, in order to distinguish qualitatively the relative relevance of some parameters such as the initial mean eccentricity, the value of the area-to-mass ratio, as well as the importance of the 1:1 resonance and of the third-body perturbations in the occurrence of such secondary structures. Even though these results are not presented here in detail, we can draw the following preliminary conclusions: the second order harmonic J2, as well as the third-body perturbations, do not seem to be really relevant and crucial in the appearance of these additional patterns. In other words, the unexpected patterns occur only when taking into account the combined effects of both the second order and degree harmonic and direct radiation pressure. As a matter of fact, the extended numerical investigations performed in Figure 6 (top, left), or similarly those shown in Breiter et al. (2005) , also present these structures, even though they are difficult to perceive. Actually, the extension and chaoticity indicator of the secondary patterns seem to be directly proportional to the area-to-mass ratio value or, equivalently directly proportional to the mean value of the eccentricity.
Figure 14: Blow-up of the phase space with the specification of a resonant angle section (horizontal black solid line), that is the set of orbits having the same (osculating) initial resonant angle value, near the first stable equilibrium point, namely σsection0=81.67∘ (top panel). Evolution of MEGNO with respect to the initial semi-major axis a0 for the specified section (middle panel). The fundamental period of σ with respect to the initial semi-major axis a0, computed by means of frequency analysis for the specified section (bottom panel). The estimation of the periods are made over a 20 years period of time.
To get even more concluding results, we considered a blow-up of the phase space (dashed line rectangle in Figure 13) with really high resolution sampling (150 meters in the semi-major axis a and 0.3∘ in the resonant angle σ). Figure 14 (top) shows this phase space widening wherein we defined a so-called resonant angle section (horizontal black solid line), that is the subset of orbits having the same initial resonant angle value. This resonant angle section spans the complete range in semi-major axis and passes close to the stable equilibrium point. For each orbit defined on this section, we computed the MEGNO indicator and in Figure 14 (middle) we report this value at the end of the simulation as a function of the semi-major axis.
To double check our results, we performed a frequency analysis investigation (see Laskar, 1990 and 1995, and Noyelles et al., 2008) aimed to study the behavior of the proper frequency of the resonant angle σ, whose results are reported in Figure 14 (bottom). Here one can clearly notice the distinctive characteristics regarding the well-known 1:1 resonance between the mean longitude and the sidereal time. Indeed, both MEGNO and the fundamental period show distinctively a minimum close to the stable equilibrium point. In this case, as previously mentioned in Section 4, MEGNO should slowly converge to ¯¯¯¯Y(t)=2 everywhere, except at the equilibrium point where the limit value is ¯¯¯¯Y(t)=0, that's why, using a finite integration time, we obtain such V-shaped curve, close to 0 in the center of the resonance and to 2 on the borders. It is also worth noting that the fundamental period of σ is reported to be close to 2.25 years, which is in good agreement with the well-known 818 days libration period of a typical uncontrolled near-geosynchronous object. Near the separatrices, MEGNO clearly presents some obvious high values which confirms the presence of chaotic orbits. Here, the fundamental period reaches significant values and, as a matter of fact, is not well determined, once again supporting the result of the existence of a chaotic zone.
Figure 15: Time evolution of the angles σ, 2σ+λ⊙ and 2σ−λ⊙ (in radians) for several semi-major axes. In the lower major secondary resonance, a0=42122 km. In the eye of the principal resonance, a0=42188 km. Between the primary resonance and the upper secondary resonance, a0=42204 km. Inside the upper major secondary resonance, a0=42212 km. Outside the upper major secondary resonance, a0=42230 km.
Moreover, the use of frequency analysis allows us to support strongly the hypothesis that the additional patterns are actually related to secondary resonances. Indeed, if we look at the evolution of the fundamental period with respect to the semi-major axis, it is clear that the so-called secondary resonances are associated, regarding the angle σ, with periods which are commensurate with 1 year. More precisely, the major secondary resonances, located at approximately 40 km on both sides of the pendulum-like pattern, are related to a 2 years fundamental period of σ. Concerning the farther patterns located at ±80 km, the fundamental period of σ turns out to be very close to 1 year. As a consequence, we can presumably assume that these secondary resonances are actually related to a commensurability between σ and the 1 year period angle λ⊙, that is the ecliptic longitude of the Sun.
To justify this assumption, we focused our attention to the major secondary resonances located at ±40 km on both sides of the pendulum-like pattern, considering the time evolution of various linear combinations of σ and λ⊙. For this purpose, we considered various initial semi-major axes in the phase space. The results are shown in Figure 15. At first glance, it is apparent that three propagations stand apart from the others. In the first row of Figure 15, that is regarding the evolution of the resonant angle σ, we clearly identify the well-known characteristics related to the primary resonance. In particular, in Figure 15a, that is when considering an initial semi-major axis inside the primary resonant (a0=42 188 km), σ shows the well-known long-period libration (2.25 years), whereas σ circulates outside this region. Furthermore, what is of special interest is the time evolution of both 2σ+λ⊙ and 2σ−λ⊙, shown in the second and third row, respectively. It is clear that most of the time these angles show a circulation regime. However, when considering an initial semi-major axis inside the major lower secondary resonance for 2σ−λ⊙ or, similarly inside the major upper secondary resonance for 2σ+λ⊙, both these angles show a significant long-term evolution (Figure 15b and 15c).
6.2 Analytical investigation – simplified model
The presence and the location of these secondary resonances can be studied using an appropriate simplified model. Hence we model the averaged geostationary motion by a pendulum-like system, given by its Hamiltonian formulation (Valk et al., 2008c) up to order e2 in the series expansion
H=−μ22L2−˙θL+3μ4L6R2e(1−52e2)S2200(Ω,ω,M,θ),
L=√μaandS2200(Ω,ω,M,θ)=C22cos2σ+S22sin2σ.
In the context of direct solar radiation pressure, we can introduce the factor Z proportional to A/m through the eccentricity e (for further details, we refer to the averaged simplified analytical model developed in Valk et al., 2008). As a first approximation, the time evolution of both the eccentricity e and the longitude of perigee ϖ were found to be (neglecting the obliquity of the Earth with respect to the ecliptic)
\setstretch1.4ecosϖ=ZLn⊙cosλ⊙+α0,esinϖ=ZLn⊙sinλ⊙−β0,
which introduces λ⊙ in the Hamiltonian. The quantity n⊙ is the mean motion of the Sun and both α0 and β0 are related to the initial conditions with respect to the eccentricity and the longitude of perigee). The resulting Hamiltonian takes the generic form
H=−μ22L2−˙θL+FL6cos(2σ−2σ0)−GL62cos(2σ−2σ0)cos(λ⊙+δ),
where δ, F, G, σ0 are constants. A suitable transformation is then necessary to introduce action-angle variables (ψ,J) in the libration and in the circulation region of the double pendulum, in such a way any trajectory of the double pendulum is characterized by a constant action J and a corresponding constant frequency ˙ψ. Rewriting the perturbed system (because of λ⊙ terms) by means of these new variables and then using the expansions in Bessel functions, we could isolate any resonance of the type kψ±λ⊙ in the circulation region, for any |k|, and in the libration region, for |k|≥3, which corresponds to our frequency analysis. This analysis is surely promising, but it is outside the goals of this paper. Further investigations will be detailed in a forthcoming publication (Lemaître et al., submitted for publication) .
The predictability of the trajectory high area-to-mass ratio space debris located near the geosynchronous region was investigated by means of a recent variational chaos indicator called MEGNO. Thanks to this technique, we clearly identified the regular (stable) and irregular (chaotic) orbits. This efficient method allowed us to obtain a clear picture of the phase space, hence showing that chaotic regions can be particularly relevant, especially for very high area-to-mass ratio objects. Moreover, we discussed the importance of both the initial eccentricity and time at epoch in the appearance of chaos.
Finally, we brought to the fore a relevant class of additional unexpected patterns which were identified as secondary resonances, that were numerically studied by means of both the MEGNO criterion and frequency map analysis, to eventually conclude that they involve commensurabilities between the primary resonant angle and the ecliptic longitude of the Sun. We also presented an analytical scheme that could explain their existence. It will be the subject of further work.
The authors thank S. Breiter for helpful discussions about both the MEGNO criterion as well as numerical issues which led to substantial improvement of the present paper, also providing some useful references. Finally, we are grateful for the opportunity to use the frequency analysis tools developed by B. Noyelles and A. Vienne. The authors also warmly thank the two referees for their suggestion that allowed us to strongly improved the paper.
Anselmo and Pardini (2005) Anselmo, L., Pardini, C., Orbital evolution of geosynchronous objects with high area-to-mass ratios. In: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, (ESA SP-587). ESA Publications Division, Noordwijk, The Netherlands, pp. 279–284, 2005.
Barrio et al. (2007) Barrio, R., Borczyk, W., Breiter, S., Spurious structures in chaos indicators maps. In press, Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2007.09.084, 2007.
Benettin et al. (1980) Benettin, G., Galgani, L., Giorgilli, A., et al., Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica 15, 9–20, 1980
Breiter et al. (2005) Breiter, S., Wytrzyszczak, I., Melendo, B., Long-term predictability of orbits around the geosynchronous altitude. Advances in Space Research 35, 1313–1317, 2005.
Bulirsh and Stoer (1966) Bulirsh, R., Stoer, J., Numerical treatment of ordinary differential equations by extrapolation methods. Numerische Mathematik 8, 1–13, , March 1966.
Chao (2006) Chao, C. C., Analytical investigation of GEO debris with high area-to-mass ratio, AIAA Paper No. AIAA-2006-6514, Presented at the 2006 AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, August 2006.
Cincotta et al. (2003) Cincotta, P. M., Giordano, C. M., Simó, C., Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182, 151–178, 2003.
Cincotta and Simó (2000) Cincotta, P. M., Simó, C., Simple tools to study global dynamics in non-axisymmetric galactic potentials - I. Astronomy and Astrophysics, Supplement 147, 205–228, 2000.
Cunningham (1970) Cunningham, L. E., On the computation of the spherical harmonics terms needed during the numerical integration of the orbital motion of an artificial satellite. Celestial Mechanics 2, 207–216, 1970.
Deprit (1969) Deprit, A., Canonical transformations depending on a small parameter. Celestial Mechanics 1, 12–30, 1969.
Goździewski et al. (2001) Goździewski, K., Bois, E., Maciejewski, A. J., et al., Global dynamics of planetary systems with the MEGNO criterion. Astronomy and Astrophysics 378, 569–586, 2001.
Goździewski et al. (2008) Goździewski, K., Breiter, S., Borczyk, W., The long-term stability of extrasolar system HD37154. numerical study of resonance effects. Monthly Notices of the RAS 383, 989–999, 2008.
Henrard (1970) Henrard, J., On a perturbation theory using lie transforms. Celestial Mechanics 3, 107–120, 1970.
Laskar (1990) Laskar, J., The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291, 1990.
Laskar (1995) Laskar, J., Introduction to frequency map analysis. In: Proceedings of 3DHAM95 NATO Advanced Institute. Vol. 533. S'Agaro, 134–150, June 1995.
Lemaître et al. (submitted for publication) Lemaître, A., Delsate, N., Valk, S., A web of secondary resonances for large A/m geostationary debris. Submitted to Celestial Mechanics and Dynamical Astronomy, 2009.
Lemoine et al. (1987) Lemoine, F. G., Kenyon, S. C., Factor, J. K., et al., The development of the joint nasa gsfc and nima geopotential model EGM96. Tech. rep., NASA, TP-1998-206861, 1987.
Liou and Weaver (2004) Liou, J.-C., Weaver, J. K., Orbital evolution of GEO debris with very high area-to-mass ratios. The Orbital Quarterly News 8 issue 3, The NASA Orbital Debris Program Office, 2004.
Liou and Weaver (2005) Liou, J.-C., Weaver, J. K., Orbital dynamics of high area-to-mass ratio debris and their distribution in the geosynchronous region. In: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris (ESA SP-589). ESA Publications Division, Noordwijk, The Netherlands, pp. 285–290, 2005.
Noyelles et al. (2008) Noyelles, B., Lemaître, A., Vienne, A., Titan's rotation. A 3-dimensional theory. Astronomy and Astrophysics 475, 959–970, 2008.
Schildknecht et al. (2005) Schildknecht, T., Musci, R., Flury, W., et al., Optical observations of space debris in high-altitude orbits. In: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587. ESA Publications Division, Noordwijk, The Netherlands, pp. 113–118, 2005.
Schildknecht et al. (2004) Schildknecht, T., Musci, R., Ploner, M., et al., Optical observations of space debris in GEO and in highly-eccentric orbits. Advances in Space Research 34, 901–911, 2004.
Standish (1998) Standish, E. M., JPL planetary and lunar ephemeris, DE405/LE405. JPL Interoffice Memorandum IOM 312.D-98-048, August 1998.
Stoer and Bulirsch (1980) Stoer, J., Bulirsch, R., Introduction to numerical analysis. Springer-Verlag, New York, 1980.
Valk et al. (2008a) Valk, S., Lemaître, A., Anselmo, L., Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Advances in Space Research 41, 1077–1090, 2008a.
Valk and Lemaître (2008b) Valk, S., Lemaître, A., Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including earth's shadowing effects. Advances in Space Research 42, Issue 8, 1429–1443, 2008b.
Valk et al. (2008c) Valk, S., Lemaître, A., Deleflie, F., Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. In press, Advances in Space Research, doi: 10.1016/j.asr.2008.12.015, 2008c.
Wisdom (1983) Wisdom, J., Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74, 1983.
Wytrzyszczak et al. (2007) Wytrzyszczak, I., Breiter, S., Borczyk, W., Regular and chaotic motion of high altitude satellites. Advances in Space Research 40, 134–142, 2007.


text:
Opeth's Mikael Akerfeldt on His Gutturals: 'I've Moved on and So Should You'
Graham Hartmann
Anne C. Swallow
It's been over a decade since Opeth frontman Mikael Akerfeldt recorded guttural vocals in the studio. Despite the time passed, Akerfeldt still receives questions on why he stopped using his demonic style. Instead of getting annoyed, however, Mikael appreciates the question, though he advises fans to "move on" just like he did.
Akerfeldt's best gutturals were arguably released in 2008, the final year fans heard fresh screams from the musician. Opeth released the vicious and beautiful Watershed in June 2008, while Bloodbath issued their Unblessing the Purity EP in March, followed by the full-length Fathomless Mastery in September. Akerfeldt seemed to further perfect his "sing you to sleep / scare you to death" style with each album, which is why fans were so shocked once Heritage came around in 2011.
During a recent Q&A session for Opeth's newest album, In Cauda Venenum, Akerfeldt commented on his gutturals remaining a major talking point amongst metalheads.
"I get that question a lot from fans and journalists," Akerfeldt shares. "Journalists are a bit more careful, diplomatic. They don't tell me, like, 'You suck now. You should do that.' But contrary to what people think, I'm happy to get that question, because that means that that stuff meant a lot to people … The last death metal record I bought with my own money and enjoyed was Domination by Morbid Angel, and that's, like, '95."
"Our first album came out in '95, and at that point, I was still into that death metal - whatever scene we were part of - finding our feet, and I wanted us to be that band and have both the screaming vocals and the clean vocals," he continues. "But even then, I wanted to add more clean vocals; it was just that I didn't dare to - I didn't have the confidence to sing more clean. And with more record, there was more and more and more and more, and then you end up where there's no screams, because of - I guess - confidence."
"I stopped after Watershed because Watershed was the pinnacle to that style for me. And it's kind of well-known fact that I tried to make a Watershed 2.0 or whatever, that was then deleted. And Heritage was born and with that a new path for us. And I wanted to see what was around the corner there. And it's because I'm a musician and I love music. We're not entrepreneurs or businessmen," Akerfeldt concludes. "Not catering to your previous success - that's not interesting to me. Unless that's what I wanna do in music, and it's not right now. I don't wanna do Watershed 2.0 … We still play those songs live. It's not that I'm ashamed - I love it. But yeah, I've moved on. And so should you." [via Ultimate Guitar]
In Cauda Venenum is now available for fans to hear. Opeth also just announced a 2020 North American tour with Graveyard, so check out those dates here.
2019's Best Metal Songs… So Far
Source: Opeth's Mikael Akerfeldt on His Gutturals: 'I've Moved on and So Should You'
Filed Under: mikael akerfeldt, Opeth

redpajama_set_name: RedPajamaCommonCrawl


text:
Third Sector Policy at the Crossroads
An International Non-profit Analysis
Jeremy Kendall
Published October 2, 2012 by Routledge
Third Sector Policy at the Crossroads: An International Non-profit Analysis
Book Description Author(s) Support Material
The nonprofit sector occupies an ever more central role in economic and social policies, from the redesign of welfare system in the U.S. or UK, to fostering democracy in Central and Eastern Europe. At the same time, non-profit organizations face increased public scrutiny, and calls for more 'efficiency' and greater 'accountability'. Against this background, this revealing book explore keys policy issues such as:
* Is the nonprofit sector in crisis?
* What are the common themes and patterns in current policy debates concerning the future of the nonprofit sector?
* What policy models are being discussed, and what are their implications?
* How can nonprofit organiztions negotiate a course between commercialization and tighter government regulation?
As this sector is confronted with often conflicting demands of new public management, tight budgeting, and greater competition, it arrives at a crucial crossroad in its development. Through the detailed investigations presented in this key book, postgraduate students of business and politics can fully explore this significant sector and analyze it's position in today's society.
Routledge Studies in the Management of Voluntary and Non-Profit Organizations


text:
Personal Data Processing and Protection Policy (JSCo "RZD")
The history of railways, XIX century
The history of railways, XX century
The history of railways, XXI Century
History of Russian rail lines
History of the development of rolling stock
The history of the development of the railway track
History of terminals and stations
Railways during the war (1941 - 1945)
Transsib during the war
Belorussky station
Kazansky station
Kievsky station
Kursky station
Leningradsky station
Paveletsky station
Rizhskaya station
Savelovsky Station
Yaroslavsky Station
Moscowsky station
East Siberian Railways
Gorky Railways
Far Eastern Railways
Trans-Baikal Railways
West Siberian Railways
Kaliningrad Railways
Krasnoyarsk Railway
October Railways
Volga Railways
Sverdlovsk Railways
Northern Railways
North Caucasus Railways
South-East Main Line
South Urals Railways
Stations-museums
Savyolovsky Station, Moscow
In 2002, Moscow's "newest" station celebrated its 100th anniversary. Savyolovsky is Moscow's only railway station whose name is derived not from a town, but a village.
The construction of the Savyolovskaya line was initiated by Savva Mamontov, the famous industrialist and philanthropist who was also Chairman of the Board at the Moscow-Yaroslavskaya Railway Company. Largely due to his energy, the concession for the construction of the line, which had originally been awarded to another private company, the Second Sidings Company, was transferred to "the Yaroslavskaya".
After received Imperial Permission, the Moscow-Yaroslavl-Arkhangelsk Railway began surveying the route in 1897 and then started laying the new line from Moscow to the village of Savyolovo, which stood on the bank of the Volga opposite Kimry. The new branch line was not very long, just 130 km, but promising. The trading village of Kimry was famous at that time for its master cobblers and the ancient town of Kashin was nearby. In the future, it was planned to extend the line to Kalyazin, Yglich and Rybinsk.
For the construction of the Savyolovsky line, a special department was created "under the management of the Chief Engineer K. A. Savitsky." The line was envisaged as single-track with a capacity of two pairs of passenger trains and five freight trains per day and an average speed of 20 versts per hour. A verst is a Russian unit of distance equal to 1.067 kilometres 0.6629 miles.
The line was built from both ends - from Moscow and Savyolovo. Only rails made by domestic factories in Putilovsky, South Dneprovsky and Bryansk were used. Construction began with the laying of a branch line from the 10th verst on the Moscow-Yaroslavl railway, from the marshalling yards at Losinoostrovskaya station to Beskudnikovo station, from where, in fact, the Savyolovskaya line was to begin.
A site for the station was chosen at Butyrskaya Zastava on the outskirts, where the price of land was low. The Savyolovskaya branch line was extended from the Beskudnikovo station to the Kamer-Kollezhsky Val.
After numerous delays, the Moscow City Duma gave its permission for the line, and sand, stone and other construction materials were brought to Butyrskaya Zastava. The building was scheduled to be completed by winter 1899.
However, work was halted suddenly, as the Vindavo-Rybinsk Railway Company proposed to the Management Board of the Moscow-Yaroslavl-Arkhangelsk line to buy the section of the Savyolovsky line from Beskudnikovo station to Savyolovo. The station's prospective new owners were going to build the passenger station elsewhere.
Meanwhile, by the beginning of 1900, the main work on the Savyolovsky branch line had been completed and temporary services had begun. Trains to Savyolovo departed from Yaroslavsky station, which greatly inconvenienced passengers, who, after reaching the "milestone" at the 10th verst on the Yaroslavl line, were forced to change to carriages owned by the Savyolovskaya line.
In summer 1900, the Moscow-Yaroslavl-Arkhangelsk line was transferred to public ownership, so the sale of the Moscow section of the Savyolovsky line to the Vindavo-Rybinsk Railway Company never took place.
In September 1900, construction work on the station resumed under the direction of the engineer A. S. Sumarokov, and there is speculation that he also developed the project. The station building was quite modest, and did not have even a front entrance. It consisted mostly of just one storey, and only in the centre did it boast two storeys to accommodate the service apartments. Separately from the passenger station, a so-called military barracks was built, which far exceeds the size of the station building. It was suggested that it house a temporary passenger station. In any event, the freight yard and its tracks were some distance away.
Construction work was completed by spring 1902. On Sunday, 10 March (Old Style), the station, which had received the name Butyrsky, was blessed and the first train departed. "In the morning, the new station building and the whole station yard," wrote the daily newspaper the Moscow Listok, "were decorated with flags and green garlands which covered the main entrance. At about 12 o'clock in the afternoon, a service train arrived from Yaroslavl station carrying commanders and the invited representatives from other railway lines. The ceremony began in the 3rd-class hall with a litany before shrines brought from the local church. After the litany and the sprinkling of holy water on the building, everyone was invited to the 1st-class hall, where champagne was served."
Regular services began. Initially, two pairs of trains ran per day, with a passenger train departing at 10:35 am and a mail train at 7:30 pm.
The railway line and station transformed the life of this quiet corner of Moscow, from Novoslobodskaya Street to Marina Roshcha from the one direction, and to Butyrsky Khytor and Petrovsko-Razymovsky, where formerly mostly horse-cab drivers, artisans and farmers used to live, from the other.
The industrialist Gustav Liszt built a new factory near the station, and counted on the labour force from the suburbs to man the plant. In anticipation of an influx of visitors, Moscow homeowners built about 30 new homes in the neighbourhood, and land prices rose sharply.
The station was built behind the city zastava or gates, that is, outside Moscow. However, in view of the prospects which would open up for the region, in mid-1899 the Moscow City Duma issued documents for a new delineation of the city and county, and from 1900, parts of the suburban land were incorporated into Moscow. As a result, residents of the Butyrka suburban settlement became Muscovites thanks to the railway line and the train station.
For many years Butyrskaya Station, which was later renamed Savyolovskaya, successfully fulfilled its function, but with the growth of traffic, especially in the suburban areas, it began to lag behind the times and fell into disrepair. In the 1980s, it was decided it was in need of major repairs and restoration and plans were drawn up by the Institute of Moszheldorproekt under the direction of Y. V. Shamrai. The work took several years, but train services continued and the ticket offices worked from temporary premises.
On 1 September 1992, 90 years after its original construction, the updated and rejuvenated station reopened its doors. It was now a two-storey structure, but the building still retains the original architectural appearance. Today, Savyolovskaya Station is a modern passenger terminal, offering travellers a wide range of railway services.


"The house for sale has three bedrooms, plus a den and playroom."
"I live with my family, plus two dogs and a cat."
"I added two plus two."
"I'm paying the principal amount plus the interest."
"How much is the total plus tax."
"I like vegetables, plus they're good for you."
"I like concerts in the park, plus they are free."
"I enjoy going to school, plus I get to see my friends."
"I enjoy bike riding, plus it's good exercise."
"The apartment is small, but having a swimming pool is a plus."
"The house is old, but being close to school is a plus."
"Since I have a dog, having a big yard is a plus."
"Being able to work part-time while my children are in school is a plus."


Today's episode reveals the hidden aspects of our nature that determine how we respond to the expectations of ourselves and others.
These 4 Tendencies give us a framework for how we react to the internal and external demands in our lives. When we better understand these patterns in our behavior, we gain immense control over our actions and decisions.
Although our personality type may be deeply ingrained, we can benefit from knowing these predictable behavioral patterns and set ourselves up for success.
Listen to Wellness Force Radio 129 as Gretchen shares her insight from decades of research in happiness, habits, and so much more.
These personality traits may be deeply ingrained, but luckily we don't need to try and change who we are. Just by knowing our tendencies, we can set ourselves up for success.
If you fall into the large category of Obligers, remember to be mindful of Obliger Rebellion. This is when the demands and expectations from others begin to overwhelm us to the point of burnout and resentment. Typically, this happens when the Obliger feels used or taken advantage of, which they usually are. We get to set healthy boundaries for our external accountability.
It's important to know that not everyone thinks like we do. Sometimes this gets us upset when people don't live up to our expectations. There are also times when it's hard to comprehend why people do the things they do. If we understand the personality traits and nature of others, we can remove judgment and enhance our impact, interactions, and relationships.
During her multi-book investigation into understanding human nature, Gretchen Rubin realized that by asking the seemingly dry question "How do I respond to expectations?" we gain explosive self-knowledge. She discovered that based on their answer, people fit into Four Tendencies: Upholders, Questioners, Obligers, and Rebels. Our Tendency shapes every aspect of our behavior, so using this framework allows us to make better decisions, meet deadlines, suffer less stress, and engage more effectively.
Gretchen's co-host and guinea pig is her younger sister, Elizabeth Craft, a TV writer and producer living in Los Angeles, who (lovingly) refers to Gretchen as her happiness bully. Part of the Panoply Network.
Gretchen Rubin is the author of several books, including the blockbuster New York Times bestsellers, Better Than Before, The Happiness Project and Happier at Home. She has an enormous readership, both in print and online, and her books have sold almost three million copies worldwide, in more than thirty languages. She makes frequent TV appearances and is in much demand as a speaker.
On her weekly podcast Happier with Gretchen Rubin, she discusses good habits and happiness with her sister Elizabeth Craft. Rubin started her career in law and was clerking for Supreme Court Justice Sandra Day O'Connor when she realized she wanted to be a writer. She lives in New York City with her husband and two daughters.
Read more about Gretchen on her About page.
To Cheat Day or Not To Cheat Day?


Discover the glamorous Principality of Monaco at your own pace on this unguided tour. With a map and booklet to guide you, perhaps visit attractions in Monaco, such as the Prince's Palace, the Palace Square, the Oceanographic Museum, Aquarium or Monaco Cathedral. There'll be a complimentary bus ride to Monte Carlo to shop along the Place du Casino, admire the fabulous yachts in the harbor, have a flutter at the Casino or visit highlights like the Japanese Garden and the National Museum.


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Beat the Heat: Public safety officers are redirecting street racers to drag strips by racing with them. Read more....
Beat The Heat: Public Safety Officers Engage with Youth to Combat Street Racing
by Ziva Allen
We love to drive fast. That's why we seek out tracks to bring our cars to. Check out our TrackDrivers article. And we also know that keeping it on the track is the right thing to do, not only for us, but for the general public. However, we are also aware that, on any given day, many are engaging in the risky and irresponsible choice to participate in illegal street racing. And for that irresponsible choice, we all are at risk. For this reason I hope that through our monthly Keeping It On The Track column, we can bring to you not only stories illustrating reasons why to keep it on the track, but uplifting stories depicting those who devote their lives to that end.
If you talk to police officers about their experiences in having the unfortunate duty to work a crash fatality, no matter how long ago that fatality occurred, you cannot help but come away with the realization that officers lose a piece of themselves with each and every fatality. When an illegal street racer not only loses a race but their life as well, someone has to make a dreadful house call. Someone has to ring a doorbell and then witness utter hell begin emerging on a parent's face upon realizing why there is a police officer standing on their front porch in the middle of the night. In speaking with Retired Lieutenant Tom Brown, President of Beat the Heat, I had the opportunity to get an intimate sense of the emotional toll it takes to be charged with the duty of having to ring a doorbell in the middle of the night and deliver the ultimate in horrible news. As Brown, who has had to make such house calls, described to me, "The worst job I've ever had as a police officer is going and giving a death sentence where I have to go knock on a parent's door and when they come to the door and they see a police officer standing there, they know why you're there. You've got to tell them that their child's not coming home. And then you have to stay there with them and it's the most uncomfortable thing you've ever done. There's no set program where you read a script. You just go there and you make it up…once you've worked a fatality, it stays with you. I retired several years ago and every fatality I've ever worked, they come back and visit me in the night. I mean I still have dreams about it."
Time to Take Action
And it was in delivering too many of these messages that Brown decided he had had enough and chose to do something positive. Brown is now President of Beat the Heat, a non-profit organization based out of Texas that educates young people in the dangers of illegal street racing. As Brown states, "We want to be the alternative." Beat the Heat is comprised of police officers and firefighters and those in their respective communities who volunteer their time to, as cliché as it sounds, truly make a difference in the communities they serve and in the lives that they save. As Brown explained, "I love drag racing. And I like young people and I like to talk to them and this is just an ideal program for me. Everything this organization does, if we save just one young person and save one innocent that's killed by an illegal street race, everything that we've done is worth it."
Beat the Heat involves the young people in their communities by enticing them off of the streets and into organized drag racing events against none other than the police officers themselves! Yes, the police officers literally race against the kids. But, as opposed to illegal middle of the night street racing amongst 40 to 50 cars in uncontrolled environments on narrow, rural, dark roads, lined by spectators standing within inches of the cars racing by them at 100 miles an hour, these Beat the Heat challenges are conducted in an organized format on properly designed drag strips. The events consist of volunteer police officers, firefighters, paramedics, and the like to ensure safety for all participants, including spectators who don't have to risk losing a toe and more but can properly view from bleachers at a respectable and safe distance. "We're giving them an alternative," Brown says. To illustrate the dangers in illegal street racing, Brown makes the point that even the road surface is a known quantity. "Out on the street you don't know what kind of surface you've got. You've got asphalt which is really slick. You've got concrete which is polished concrete. You've got new concrete, the coefficient comes into effect. On a drag strip, it's a prepared surface."
Beat the Heat began in 1984 by Sgt. Don Robertson of the Jacksonville, Florida Sheriff's Department. A second chapter sprung up in 1990 by Senior Corporal Mac Sibley in Texas. By 1993, the program expanded to 15 police agencies around the State of Texas. In 1994, the program grew to over 25 cars and the Heat Teams participated in over 50 events engaging almost 100,000 young people. Currently, there are over 400 Beat the Heat members spanning across 36 states. Additionally, Beat the Heat has members in other countries now, such as Canada, Puerto Rico, Australia, and England. Over 1.5 million kids are reached each year! In working with young people about the dangers of illegal street racing, Beat the Heat volunteers additionally incorporate into their message the dangers of distracted and drug and alcohol impaired driving. Encouraging kids to stay in school and be achievers is also a top priority.
In writing this article, I started out with the intent of bringing to you the story of Beat the Heat. But after having the good fortune to speak with Brown, it is difficult to separate the good work that the organization does from the heart that Brown possesses. Brown educates kids. But the kids consider Brown a mentor. It is his goal to not lecture or intimidate young people into making good choices but instead to engage kids on an equal level and to earn their respect. As a tool in reaching young people, Brown also understands the importance of educating other police officers in how to interact with kids. One of Beat the Heat's main goals is to foster a link or understanding between police officers and the kids they come in contact with. Brown wants young people to view police officers as on their side, not their enemy. As Brown states, "When the kids are out there racing and the police get there, they look at us like we're not human. Well we've got families too and we are human" and Brown wants kids to understand this. When a kid sees a police car drive by, he doesn't want the kid to look in the window and say to himself, "there's a cop." He wants the kid to say to himself, "Hey, I know that cop."
But how does this organization entice young people to come to one of their drag races? Brown understands that kids could be quite intimidated by the notion of drag racing against police officers in front of not only their peers but the police officers themselves. In understanding this, Brown does not ever want a kid to come to an event and be embarrassed. "When we challenge them to come to the track, we don't want them to come out and just beat up on them," says Brown. "A problem with a lot of these street racers is they don't want to come to the track because they've never been to the track. They're afraid of the unknown. So we have classes for them before the track ever opens." Because some of these young people have never been to the track, Brown shows them the ropes early so "that way when their friends get there, they've already been down the track a couple of times and they're not going to get embarrassed. I don't want to embarrass anybody because if I do, he won't come back." In challenging young people to drag race against the police and engaging with them in a sport they all have in common, the kids then view the police officers differently, as human and with more respect. "All of a sudden they see that we love the sport of drag racing too and they'll sit and talk to us. They'll come over to my racecar trailer and we'll sit under an awning when it's hot outside and I'll give them something to drink and we'll sit and we'll just talk about all kind of things."
Unfortunately, there is very little data regarding illegal street racing. Sure we all hear about the Paul Walkers. But on any given night, all across this country, on dark, rural back roads, kids are organizing illegal street races. And these races are often quite sophisticated in their attempts to dodge the police, such as using radio relay systems. Gathering data and statistics of the number of injuries and deaths due to illegal street racing is very difficult. As Brown explains, "You may have an illegal street race and somebody will pull off the road and hit a tree and get killed or the car he was racing [and the spectators] they all jump up and they leave and so when the police get there it's just a one car accident." Therefore, because gathering data is so difficult, curbing this illegal activity is very challenging. But Beat the Heat knows they make a difference because they are on the ground engaging one-on-one with these young people and so they witness for themselves the difference they are making. Communities who have brought in Beat the Heat's program have seen improvements. One city Brown talks about that had brought in a taskforce to try to counter the illegal street racing occurring in their city, was able to disband that taskforce within six months of Beat the Heat's program being instituted! If you are interested in reading further about Beat the Heat or interested in bringing a program or chapter to your school, church or community, please visit their website at beattheheatinc.org or by clicking here.
Check out their segment on Top Gear
S5_DISQUS_COMMENTS_VIEW_THE_DISCUSSION_THREAD


Why were undercover Israeli agents creeping inside a Palestinian village?
Undercover Israeli agents shoot three men in the middle of the night, killing one. Al-Haq's independent investigation exposes the chilling story. Read more about Why were undercover Israeli agents creeping inside a Palestinian village?


text:
Q: Openpyxl and Binary Search The problem: I have two spreadsheets. Spreadsheet 1 has about 20,000 rows. Spreadsheet 2 has near 1 million rows. When a value from a row in spreadsheet 1 matches a value from a row in spreadsheet 2, the entire row from spreadsheet 2 is written to excel. The problem isn't too difficult, but with such a large number of rows, the run time is incredibly long.
Book 1 Example:
|Key   |Value             |
|------|------------------|
|397241|587727227839578000|
An example of book 2:

ID
a
b
c

587727227839578000
393
24
0.43

My current solution is:

g1 = openpyxl.load_workbook('path/to/sheet/sheet1.xlsx',read_only=True)
grid1 = g1.active
grid1_rows = list(grid1.rows)

g2 = openpyxl.load_workbook('path/to/sheet2/sheet2.xlsx',read_only=True)
grid2 = g2.active
grid2_rows = list(grid2.rows)

for row in grid1_rows:
        value1 = int(row[1].value)
        print(value1)
        for row2 in grid2_rows:
            value2 = int(row2[0].value)
            if value1 == value2:
                new_Name = int(row[0].value)
                print("match")
                output_file.write(str(new_Name))
                output_file.write(",")
                output_file.write(",".join(str(c.value) for c in row2[1:]))
                output_file.write("\n")

This solution works, but again the runtime is absurd. Ideally I'd like to take value1 (which comes from the first sheet,) then perform a binary search for that value on the other sheet, then just like my current solution, if it matches, copy the entire row to a new file. then just
If there's an even faster method to do this I'm all ears. I'm not the greatest at python so any help is appreciated.
Thanks.

A: You are getting your butt kicked here because you are using an inappropriate data structure, which requires you to use the nested loop.
The below example uses sets to match indices from first sheet to those in the second sheet.  This assumes there are no duplicates on either sheet, which would seem weird given your problem description.  Once we make sets of the indices from both sheets, all we need to do is intersect the 2 sets to find the ones that are on sheet 2.
Then we have the matches, but we can do better.  If we put the second sheet row data into dictionary with the indices as the keys, then we can hold onto the row data while we do the match, rather than have to go hunting for the matching indices after intersecting the sets.
I've also put in an enumeration, which may or may not be needed to identify which rows in the spreadsheet are the ones of interest.  Probably not needed.
This should execute in the blink of an eye after things are loaded.  If you start to have memory issues, you may want to just construct the dictionary at the start rather than the list and the dictionary.
Book 1:

Book 2:

Code:
import openpyxl

g1 = openpyxl.load_workbook('Book1.xlsx',read_only=True)
grid1 = g1.active
grid1_rows = list(grid1.rows)[1:]  # exclude the header

g2 = openpyxl.load_workbook('Book2.xlsx',read_only=True)
grid2 = g2.active
grid2_rows = list(grid2.rows)[1:]  # exclude the header

# make a set of the values in Book 1 that we want to search for...

search_items = {int(t[0].value) for t in grid1_rows}
#print(search_items)

# make a dictionary (key-value paring) for the items in the 2nd book, and
# include an enumeration so we can capture the row number

lookup_dict = {int(t[0].value) : (idx, t) for idx,t in enumerate(grid2_rows, start=1)}

#print(lookup_dict)

# now let's intersect the set of search items and key values to get the keys of the matches...

keys = search_items & lookup_dict.keys()

#print(keys)

for key in keys:
    idx = lookup_dict.get(key)[0]       # the row index, if needed
    row_data = lookup_dict.get(key)[1]  # the row data
    print(f'row {idx} matched value {key} and has data:')
    print(f'  name: {row_data[1].value:10s} \t qty: {int(row_data[2].value)}')

Output:
row 3 matched value 202 and has data:
  name: steak        qty: 3
row 1 matched value 455 and has data:
  name: dogfood      qty: 10

redpajama_set_name: RedPajamaStackExchange


There are two basic steps to server-side configuration: creating an oauth client in Google and configuring authentication in npm Enterprise.
These steps are based on Google's documentation.
First, log in to the Google Developers API console at https://console.developers.google.com/apis/credentials.
Click the Create credentials button and choose OAuth client ID.
Choose Web application as the "Application type".
Enter a name like npm Enterprise.
Under Restrictions, enter a callback URL for your npm Enterprise instance under Authorized redirect URIs. The value should be the Full URL of npm Enterprise website (e.g. http://npm.mycompany.com:8081) from the npm Enterprise admin console settings, plus the /auth/oauth2/callback route. A full example would be http://npm.mycompany.com:8081/auth/oauth2/callback. Note that you must use a domain name (IP address not allowed). We will enter this same URL as the Redirect URI in npm Enterprise's admin console when configuring authentication.
Click the Create button. On the next page, you will be shown the client ID and client secret that we will plug into npm Enterprise's admin console.


text:
Q: Unit root Test on binary Variables I am working on idea of how Unit roots test will perform on binary variables. But I got some unexpected results:
library(quantmod)
library(tseries)
library (urca)
library (forecast)

getSymbols("MSFT")

 names(MSFT)<-c("Open", "High", "Low", "Close", "Volume", "Adjusted.Close")
 p<-data.frame(MSFT)

 n=length(p$Adjusted.Close)
# d; third column indicates direction of market.
 d<-cbind(tdy=p$Adjusted.Close[1:n-1],twm=p$Adjusted.Close[2:n],
      direct=sign( p$Adjusted.Close[2:n]-p$Adjusted.Close[1:n-1]))

plot(d[1:100,3],type="l",col = "blue")

table (d[,3])

d<-ts(d)
Ar.model <- auto.arima(d[,3], stationary = TRUE, seasonal = FALSE, ic="aic")

print(Ar.model)

    Series: d[, 3]
    ARIMA(2,0,2) with zero mean

    Coefficients:
     ar1      ar2      ma1     ma2
    0.0213  -0.8571  -0.0489  0.8692
     s.e.     NaN      NaN      NaN     NaN

 sigma^2 estimated as 0.9859:  log likelihood=-3843.91
   AIC=7697.82   AICc=7697.84   BIC=7727.37
   Warning message:
   In sqrt(diag(x$var.coef)) : NaNs produced

As you can see, the S.E. are NaN. I was wondering may be I am facing a unit root process. So I run ADF test:
  ho_adf <- ur.df(d[,3])
  summary(ho_adf)

###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################

Test regression none

Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
    Min      1Q  Median      3Q     Max
-1.0376 -0.9786  0.9624  1.0081  1.0376

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
z.lag.1    -1.021448   0.027505 -37.136   <2e-16 ***
z.diff.lag -0.008069   0.019173  -0.421    0.674
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9938 on 2720 degrees of freedom
Multiple R-squared:  0.5149,    Adjusted R-squared:  0.5146
F-statistic:  1444 on 2 and 2720 DF,  p-value: < 2.2e-16

Value of test-statistic is: -37.1364

Critical values for test statistics:
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62

We know that H0 of ADF-test declares unit root in data. The p-value and test-statistics above rejects H0. Meaning that we can reject unit root existence. Now, my questions:

*

*Why I get NaN is S.E in absence of unit root of data?

*How can I fix the problem of S.E.? ( i don't want to use other ADF.test functions of R, I am aware of KPSS and PP tests already;
unless it is related to just Binary Times series data. Meaning that
your recommending specific Unit root test for binary time series.
like BARMA. In this case, the question arises why ADF test fails?)

A: I ran your codes and the results are:
> print(Ar.model)
Series: d[, 3]
ARIMA(2,0,2) with non-zero mean

Coefficients:
          ar1      ar2     ma1     ma2    mean
      -0.0566  -0.8144  0.0208  0.8187  0.0262
s.e.   0.1570   0.1050  0.1506  0.1122  0.0183

sigma^2 estimated as 0.9852:  log likelihood=-3997.69
AIC=8007.39   AICc=8007.42   BIC=8043.08

perhaps you can retry it.