High order normal form stability estimates for co-orbital motion

We present estimates of the size of the analytic domain of stability for co-orbital motions obtained by a high order normal form in the framework of the elliptic restricted three body problem. As a demonstration example, we consider the motion of a Trojan body in an extrasolar planetary system with a giant planet of mass parameter $\mu =0.005$ μ = 0.005 and eccentricity $e^{\prime }=0.1$ e ′ = 0.1 . The analysis contains three basic steps: (i) derivation of an accurate expansion of the Hamiltonian, (ii) computation of the normal form up to an optimal order (in the Nekhoroshev sense), and (iii) computation of the optimal size of the remainder at various values of the action integrals (proper elements) of motion. We explain our choice of variables as well as the method used to expand the Hamiltonian so as to ensure a precise model. We then compute the normal form up to the normalisation order $r=50$ r = 50 by use of a computer-algebraic program. We finally estimate the size $||R||$ | | R | | of the remainder as a function of the normalization order, and compute the optimal normalization order at which the remainder becomes minimum. It is found that the optimal value $\log (||R_{opt}||)$ log ( | | R o p t | | ) can serve in order to construct a stability map for the domain of co-orbital motion using only series. This is compared to the stability map found by a purely numerical approach based on chaotic indicators.     

Computerized series solution of relativistic equations of motion

A method of solution of the equations of planetary motion is described. It consists of the use of numerical general perturbations in orbital elements and in rectangular coordinates. The solution is expanded in Fourier series in the mean anomaly with the aid of harmonic analysis and computerized series manipulation techniques. A detailed application to the relativistic motion of the planet Mercury is described both for Schwarzschild and isotropic coordinates.Receipt delayed by the postal strike in Great Britain.     

Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound _* ^r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe ^–(*₁/) ^a , with positive constantsA,a and ₁ ^* exponential estimate of Nekhoroshev's type.     

On optimal estimates for the solutions of linear difference equations on the circle

A linear difference equation arising in the proof of Moser's twist mapping theorem is solved and optimal estimates for the solution are established.     

Asymptotic series for planetary motion in periodic terms in three dimensions

For the planetary case of the gravitationaln-body problem in three dimensions, a sequence of Lie series contact transformations is used to construct asymptotic series representations for the canonical parameters of the instantaneous orbits in a Jacobi formulation. The series contain only periodic terms, the frequencies being linear combinations of those of the planetary orbits and those of the secular variations of the apses and nodes, and the series are in powers of the masses of the planets in terms of that of the primary, and of a quantity of the order of the excursions of the eccentricities and inclinations of the orbits. The treatment avoids singularities for circular and coplanar orbits. It follows that the major axes are given by series of periodic terms only, to all orders in the planetary masses.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

Areas of Polar Coronal Holes from 1996 Through 2010

Polar coronal holes (PCHs) trace the magnetic variability of the Sun throughout the solar cycle. Their size and evolution have been studied as proxies for the global magnetic field. We present measurements of the PCH areas from 1996 through 2010, derived from an updated perimeter-tracing method and two synoptic-map methods. The perimeter-tracing method detects PCH boundaries along the solar limb, using full-disk images from the SOlar and Heliospheric Observatory/Extreme ultraviolet Imaging Telescope (SOHO/EIT). One synoptic-map method uses the line-of-sight magnetic field from the SOHO/Michelson Doppler Imager (MDI) to determine the unipolarity boundaries near the poles. The other method applies thresholding techniques to synoptic maps created from EUV image data from EIT. The results from all three methods suggest that the solar maxima and minima of the two hemispheres are out of phase. The maximum PCH area, averaged over the methods in each hemisphere, is approximately 6 % during both solar minima spanned by the data (between Solar Cycles 22/23 and 23/24). The northern PCH area began a declining trend in 2010, suggesting a downturn toward the maximum of Solar Cycle 24 in that hemisphere, while the southern hole remained large throughout 2010.     

Long-Wavelength Observations of Jets from Polar Regions of the Sun

We report radio observations of enhanced emission associated with the extreme-ultraviolet (EUV) jets from polar coronal hole regions of the Sun, with the Gauribidanur radioheliograph (GRH). We have estimated the brightness temperature, electron density and mass of the ejected material. These jets were not accompanied by nonthermal radio bursts, particularly Type III events.     

On the exactness of estimates for irregularly structured bodies of the general term of Laplace series

The main form of the representation of a gravitational potential V for a celestial body T in outer space is the Laplace series in solid spherical harmonics \((R/r)^{n+1}Y_n(\theta ,\lambda )\) with R being the radius of the enveloping T sphere. The surface harmonic \(Y_n\) satisfies the inequality

$$\begin{aligned} \langle Y_n\rangle < Cn^{-\sigma }. \end{aligned}$$

The angular brackets mark the maximum of a function’s modulus over a unit sphere. For bodies with an irregular structure \(\sigma = 5/2\), and this value cannot be increased generally. However, a class of irregular bodies (smooth bodies with peaked mountains) has been found recently in which \(\sigma = 3\). In this paper, we will prove the exactness of this estimate, showing that a body belonging to the above class does exist and

$$\begin{aligned} 0<\varlimsup n^3\langle Y_n\rangle <\infty \end{aligned}$$

for it.

     

Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion

We present new important results about the intermediate-type Seyfert galaxy Mrk 315, recently observed through optical imaging and integral-field spectroscopy. Broad-band images were used to study the morphology of the host galaxy, narrow-band Hα images to trace the star-forming regions, and middle-band [O  iii ] images to evidence the distribution of the highly ionized gas. Some extended emission regions were isolated and their physical properties studied by means of flux-calibrated spectra. High-resolution spectroscopy was used to separate different kinematic components in the velocity fields of gas and stars. Some peculiar features characterize this apparently undisturbed and moderately isolated active galaxy. Such features, already investigated by other authors, are re-analysed and discussed in the light of these new observations. The most relevant results we obtained are: the multitiers structure of the disc; the presence of a quasi-ring of regions with star formation much higher than previous claims; a secondary nucleus confirmed by a stellar component kinematically decoupled by the main galaxy; a new hypothesis about the controversial nature of the long filament, initially described as hook shaped, and more likely made of two independent filaments caused by interaction events between the main galaxy and two dwarf companions.     

Polar wander of Mars: Evidence from giant impact basins

We investigate the polar wander of Mars in the last ∼4.2 Ga. We identify two sets of basins from the 20 giant impact basins reported by Frey [Frey, H., 2008. Geophys. Res. Lett. 35, L13203] which trace great circles on Mars, and propose that the great circles were the prevailing equators of Mars at the impact times. Monte Carlo tests are conducted to demonstrate that the two sets of basins are most likely not created by random impacts. Also, fitting 63,771 planes to randomly selected sets of 5, 6, or 7 basins indicated that the identified two sets are unique. We propose three different positions for the rotation pole of Mars, besides the present one. Accordingly, Tharsis bulge was initially formed at ∼50 N and moved toward the equator while rotating counterclockwise due to the influence of the two newly forming volcanic constructs, Alba Patera and Elysium Rise. The formation of the giant impact basins, subsequent mass concentrations (mascons) in Argyre, Isidis, and Utopia basins, and surface masses of volcanic mountains such as Ascraeus, Pavonis, Arsia and Olympus, caused further polar wander which rotated Tharsis bulge clockwise to arrive at its present location. The extensive polar motion of Mars during 4.2-3.9 Ga implies a weak lithosphere on a global scale, deduced from a total of 72,000 polar wander models driven by Tharsis bulge, Alba Patera and Elysium Rise as the major mass perturbations. Different compensation states, 0-100%, are examined for each of the surface loads, and nine different thicknesses are considered for an elastic lithosphere. The lithosphere must have been very weak, with an elastic thickness of less than 5 km, if the polar wander was driven by these mass perturbations.     

Equations of motion for interconnected rigid and elastic bodies: A derivation independent of angular momentum

Equations of motion are derived for systems of rotationally interconnected bodies in which the terminal bodies may be flexible and the remaining bodies are rigid. The bodies may have an arbitrary topological tree arrangement; that is, there are no closed loops of bodies. This derivation extends earlier results for systems of interconnected rigid bodies only, and is much simpler than several other recent works on terminal flexible bodies. The model for a flexible body assumes that the elastic deformation is representable as a time-varying linear combination of given mode shapes.The paper also derives the appropriate form for gravitational terms, so that the equations can be used for flexible satellites. Also included are expressions for kinetic energy and angular momentum so that in case these are theoretically constant, they can be used to monitor the accuracy of the numerical integration. The paper concludes with a section showing how interbody constraint forces and torques (which do not appear in the equations of motion) can be recovered from quantities available in this formulation, and also how to treat state variables which are prescribed functions of time.A digital computer program based on the equations derived here has been used to simulate a spinning Skylab (with flexible booms) and also the interplanetary Viking (with flexible solar panels and thrust vector control).We announce with regret that Bill Hooker died in an avalanche while on a mountain-climbing expedition in Peru, July 1974.     

The Taylor series method for the problem of the rotational motion of a rigid satellite

This paper presents the Taylor series method for integration of differential equations describing the rotational motion of a rigid satellite. We compared the presented algorithm with other methods, and we show that it gives the most accurate results with reasonable efficiency.     

Spectro-Polarimetry of Polar Faculae

This contribution deals with the properties of small-scale magnetic elements at the polar caps of the Sun. Spectro-polarimetric observations, obtained with high spatial resolution with the Gregory Coudé Telescope at the Observatorio del Teide on Tenerife, were analysed. We find, though with limited data sets, that polar faculae differ in two aspects from faculae of the network in non-active regions near the equator (equatorial faculae): (1) Polar faculae appear to have the same magnetic polarity as the general polar magnetic field. Presumably, the latter is rooted in the small-scale faculae. The equatorial faculae show both magnetic polarities. (2) Polar faculae, with a size of 3.5 ± 1.3, are larger than equatorial faculae with 2.1 ± 0.4. Yet as for equatorial faculae, polar faculae possess kilogauss magnetic fields.     

Transformations to extend the range of application of power series solutions of differential equations of motion

Let the solution of a differential equation, expanded in powers of the independent variablet, have radius of convergenceT. let , wheret=t(), be a new independent variable, and let the corresponding power series in have radius of convergenceS. Thent(S) will not in general be equal toT. Ift(S)>T, then the series in powers of may have advantages over those in powers oft. Mathematical consequences of this distinction have been appreciated since the time of Poincaré. In this note the practical applications of some transformations are investigated.     

A linear solution of the equations of motion of an earth-orbiting satellite based on a Lie-series

Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method.     

Estimates of Chandler's Component of Polar Motion as Derived from Various Data Sets

Data sets consisting of several combinations of classical and modern techniques (Doppler and laser ranging to artificial sattellites) have been analysed by autoregressive (AR), sinusoidal and autoregressive moving averag (ARMA) methods in order to obtain estimates of the frequency and quality factor of the Chandler component of polar motion. The ARMA method shows advantages over the other methods and the results obtained do not furnish evidence of temporal variation in these polar motion parameters. The disadvantages associated with smoothing the data sets and the difficulties as sociated with short time intervals for the data are cleary demonstrated. There is a great need for regular and long observational series obtained by the modern techniques.     

Unbiased estimates of galaxy scaling relations from photometric redshift surveys

Many physical properties of galaxies correlate with one another, and these correlations are often used to constrain galaxy formation models. Such correlations include the colour–magnitude relation, the luminosity–size relation, the fundamental plane, etc. However, the transformation from observable (e.g. angular size, apparent brightness) to physical quantity (physical size, luminosity) is often distance dependent. Noise in the distance estimate will lead to biased estimates of these correlations, thus compromising the ability of photometric redshift surveys to constrain galaxy formation models. We describe two methods which can remove this bias. One is a generalization of the V _max method, and the other is a maximum-likelihood approach. We illustrate their effectiveness by studying the size–luminosity relation in a mock catalogue, although both methods can be applied to other scaling relations as well. We show that if one simply uses photometric redshifts one obtains a biased relation; our methods correct for this bias and recover the true relation.     

Latitudinal Distribution of Polar Faculae

The distribution of polar faculae with respect to latitude is investigated, using data obtained at the Ussuriysk Observatory during the years 1963–1994. To correct the data for the effect of visibility, a visibility function of polar faculae is derived. Corrected surface density of polar faculae is calculated as a function of latitude and time. During most part of each solar cycle, polar faculae exhibit pronounced concentrations at high latitudes with maxima of the surface density located near the poles. Such concentrations of polar faculae (below referred to as `polar condensations') are formed after a lapse of 1–2 years from the polar magnetic field reversals, and then they persist for 7–9 years, until the high-latitude magnetic fields again start to reverse. During several years after the sunspot minima, the polar condensations co-exist with the new latitudinal belts of polar faculae which appear at middle latitudes and then migrate toward the poles. To describe the evolution of the polar condensations quantitatively, the polar faculae density n at latitudes above 60° has been approximated by means of the power law nn ₀ cos^m where is polar angle. The parameters n ₀ and m both are found to vary during the course of the solar cycle, reaching maximum values near or shortly after the minimum of sunspot activity. At the minimum phase of the solar cycle, on average, the surface density of polar faculae varies as cos¹⁴. In addition to the 11-yr variation, the latitude–time distribution of polar faculae exhibits short-term variations occurring on the time scale of 2–3 years.     

Implications from the motion of outer belt asteroids

A principal feature of the asteroidal distribution is the rapid truncation of its population outward from ～3.4 AU. This paper presents further evidence, based on the motion and distribution of certain minor planets with large semimajor axes, that this truncation cannot be strictly the result of gravitational perturbations of the major planets even acting over times of ～10⁹ years. The motion of other outer asteroids sets a probable upper limit of 0.081 on Jupiter's eccentricity.     

New age estimates of M31 globular clusters from multicolour photometry

The large majority of extragalactic star cluster studies performed to date essentially use multicolour photometry, combined with theoretical stellar synthesis models, to derive ages, masses, extinction estimates and metallicities. M31 offers a unique laboratory for studies of globular cluster (GC) systems. In this paper, we obtain new age estimates for 91 M31 GCs, based on improved photometric data, updated theoretical stellar synthesis models and sophisticated new fitting methods. In particular, we used photometric measurements from the Two Micron All Sky Survey (2MASS), which, in combination with optical photometry, can partially break the well-known age–metallicity degeneracy operating at ages in excess of a few Gyr. We show robustly that previous age determinations based on photometric data were affected significantly by this age–metallicity degeneracy. Except for one cluster, the ages of our other sample GCs are all older than 1 Gyr. Their age distribution shows populations of young- and intermediate-age GCs, peaking at ∼3 and 8 Gyr, respectively, as well as the 'usual' complement of well-known old GCs, i.e. GCs of similar age as the majority of the Galactic GCs. Our results also show that although there is significant scatter in metallicity at any age, there is a notable lack of young metal-poor and old metal-rich GCs, which might be indicative of an underlying age–metallicity relationship among the M31 GC population.