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1.
Multivariate orthogonal regression in astronomy   总被引：1，自引：0，他引：1  
Total least squares considers the problem of data reduction when error resides in both the data itself and also in the equations of condition. Error may be found in all of the columns of the matrix of the equations of condition, or merely in some; the latter situation is referred to as a mixed total least squares problem. A covariance matrix may be derived for total least squares. Both memory and operation count requirements are more severe than for ordinaty least squares: about four times more memory and, if the problem involves n unknowns, 15n + 4 more arithmetic operations. The method, applicable in any situation where ordinary least squares is relevant, including the estimation of scaled variables, is applied to three examples, one artificial and two taken from astronomy: the estimation of various parameters of Galactic kinematics, and the differential correction of a planetary orbit. In these two examples the results from total least squares are superior to those from ordinary least squares.  相似文献   

2.
Initial orbit determination by least squares of N observations is essentially a linear problem if the coordinates x 0 and x 1 at two standard epochs are used as elements. The orbit of a main belt object is approximated within the observational errors by a third degree polynomial during a month. A 4-observation orbit is useful for the initial linking between two nights. Parallax is treated rigorously and future simultaneous space and Earth based observations determine the critical distance directly. The N-observation method is a great simplification of the classical 3-observation orbit followed by a differential correction by N observations.  相似文献   

3.
Use is made of 93,106 parallaxes from the Hipparcos catalog, with a mixture of spectrum-luminosity classes, to derive the position of the Galactic plane. The reduction technique, mixed total least squares-least squares, takes into account the errors in the parallaxes, and the condition that the direction cosines of the Galactic pole have unit Euclidean norm is rigorously enforced. To obtain an acceptable solution it is necessary to eliminate the stars of classes O and B that belong to the Gould belt. The Sun is found to lie 34.56±0.56 pc above the plane. The coordinates of the Galactic pole, l g , b g, are found to be: l g =0.°004±0.°039; b g =89.°427±0.°035.This agrees well with what radio observations find and demonstrates that the IAU's recommendation in 1960 to use only radio observations to determine the Galactic pole, although correct at the time because of the paucity of optical observations, can no longer be justified given the plethora of observations contained in the Hipparcos catalog and an adequate reduction technique, unavailable in 1960. The reduction technique is also demonstrably superior to others because it involves fewer assumptions and calculates smaller mean errors. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

4.
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5.
We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L 4 when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L 4. We also show that in the case of instability of L 4 the trajectories starting in a vicinity of L 4 remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.  相似文献   

6.
The OB stars are concentrated near the Galactic plane and should permit a determination of the distance to the Galactic center. van Leeuwen’s new reduction of the Hipparcos catalog provides, after 824 Gould belt stars have been excluded, 6288 OB stars out to 1 kpc and Westin’s compilation an additional 112 stars between 1 kpc and 3 kpc. The reduction model involves 14 unknowns: the Oort A and B constants, the distance to the Galactic center R 0, 2 second-order partial derivatives, the 3 components of solar motion, a K term, a first order partial derivative for motion perpendicular to the Galactic plane, a second-order partial for acceleration perpendicular to the plane, two terms for a possible expansion of the OB stars, and a C constant. The model is nonlinear, and the unknowns are calculated by use the simplex algorithm for nonlinear adjustment applied to 14313 equations of condition, 12694 in proper motion and 1619 in radial velocity. Various solutions were tried: an L1 solution, a least squares solution with modest (2.7 %) trim of the data, and two robust least squares solutions (biweight and Welsch weighting) with more extreme trimming. The Welsch solution seems to give the best results and calculates a distance to the Galactic center 6.72±0.39 kpc. Statistical tests show that the data are homogeneous, that the reduction model seems adequate and conforms with the assumptions used in its derivation, and that the post-fit residuals are random. Inclusion of more terms, such as streaming motion induced by Galactic density waves, degrades the solution.  相似文献   

7.
The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL 1. This equation is obtained numerically by a least squares approximation method.  相似文献   

8.
Problems of Galactic kinematics have usually been solved by the method of least squares. As has been known for over forty years, this may lead to biased results because least squares assume that error resides only in the observations, not in the equations of condition. The latter, however, generally incorporate error, at least in some of the columns of the data matrix. Total least squares represents the ideal mathematical tool for just this sort of problem. In this paper the method, or better stated a mixed total-ordinary least squares method, is applied to 3100 stars taken from the Bright star Catalog to calculate eight parameters of Galactic kinematics: two corrections to the precessional constants, the Oort A and B parameters, the components of solar motion, and the K term. Total least squares calculates a reasonable solution, whereas ordinary least squares fails completely.  相似文献   

9.
The problem of stability of the Lagrangian pointL 4 in the circular restricted problem of three bodies is investigated close to the 1 : 2 commensurability of the long and short period libration. By stability we define boundedness of the solution for a given initial finite displacement from the equilibrium point as function of the mass parameter close to the commensurability. A rigorous treatment close to the resonance condition is possible using a transformation that diagonalizes the matrix related to the linear part of the equations of motion. The so obtained equations are further transformed to action angle type variables. Then using an isolated resonance approach, only the slowly varying terms are kept in the equations and two independent isolating first integrals can be found. These integrals finally enable us to solve the stability problem in an exact way. The so obtained results are compared to numeric integration of the equations of motion and are found to be in perfect agreement.  相似文献   

10.
Trojan type orbits in the system of two gravitational centers with variable separation are studied within the framework of the restricted problem of three bodies. The backward numerical integration of the equations of motion of the bodies starting in the triangular libration pointsL 4 andL 5 (reverse problem) finds the breakdown of librations as the separation decreases because of the mass gain of the smaller component and an approach of the body of negligible, mass to the latter up to the distance below its sphere of action with a relative velocity approximately equal to the escape one on this sphere. The breakdown of librations aboutL 5 occurs under the mass gain of the smaller component considerably larger than in the case ofL 4 and implications are made for the asymmetry of the number of librators aboutL 4 andL 5 in the solar system (Greeks and Trojans). Other parameters of the libration motion near 1/1 commensurability are obtained, namely, the asymmetry of the libration amplitudes about the triangular points as well as the values of periods and amplitudes within the limits of those for real Trojan asteroids. Trojans could be supposedly, formed inside the Proto-jupiter and escape during its intensive mass loss.  相似文献   

11.
We present three new categories of exact and spherically symmetric Solutions with finite central parameters of the general relativistic field equations. Two well behaved solutions in curvature coordinates first category are being studied extensively. These solutions describe perfect fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one and causality condition is obeyed at the centre. The outmarch of pressure, density, pressure-density ratio and the adiabatic speed of sound is monotonically decreasing for these solutions. Keeping in view of well behaved nature of these solutions, one of the solution (I1) is studied extensively. The solution (I1) gives us wide range of Schwarzschild parameter u (0.138≤u≤0.263), for which the solution is well behaved hence, suitable for modeling of Neutron star. For this solution the mass of Neutron star is maximized with all degree of suitability and by assuming the surface density ρ b =2×1014 g/cm3. Corresponding to u=0.263, the maximum mass of Neutron star comes out to be 3.369 M Θ with linear dimension 37.77 km and central and surface redshifts are 4.858 and 0.4524 respectively. We also study some well known regular solutions (T-4, D-1, D-2, H, A, P) of Einstein’s field equations in curvature coordinates with the feature of constant adiabatic sound speed. We have chosen those values of Schwarzschild parameter u for which, these solutions describe perfect fluid balls realistic equations of state. However, except (P) solution, all these solutions have monotonically non-decreasing feature of adiabatic sound speed. Hence (P) solution is having a well behaved model for uniform radial motion of sound. Keeping in view of well behaved nature of the solution for this feature and assuming the surface density; ρ b =2×1014 g/cm3, the maximum mass of Neutron star comes out to be 1.34 M Θ with linear dimension 28.74 km. Corresponding central and surface redshifts are 1.002 and 0.1752 respectively.  相似文献   

12.
The near-forward scattering functions of particles in Saturn ring features are related to 3.6 cm radio occultation power spectra by a Fredholm integral equation of the first kind. The equation reduces to an algebraic system of equations whose solution by usual inversion techniques (i.e., least mean squares) is precluded by the near singularity of the forward transformation matrix. The instabilities are reduced by applying a combination of constrained linear inversion and a filtering algorithm based on eigenvector decomposition of the matrix, which yields derived phase functions valid over the range of zero to about 12 mrad. These functions represent the collective forward diffraction lobe of particles greater than about 1 m in radius. Multiple scattering of the signal is a significant effect, and the measured phase functions must be adjusted to obtain the singly scattered component. This single-scattering correction is examined for two physical ring models, (a) the monolayer and (b) the classical discrete random slab, and the fraction of opacity in submeter particles for each model for specific ring features is estimated. Four representative regions of the rings approximately between 1.3 and 1.4Rs, 1.5 and 1.52Rs, 2.0 and 2.02Rs, and 2.08 and 2.16Rs have been studied in detail and single-scatter phase functions produced. Each of these features exhibits effective particle sizes in the range of 3–6 m radius. The approximate fractions of optical thickness due to the submeter particles in each of these regions are 0.58, 0.54, 0.23, and 0.0, respectively, for the many-particle-thick model, and 0.67, 0.67, 0.50, and 0.50, for the monolayer model.  相似文献   

13.
This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function i =X ij x j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes.  相似文献   

14.
We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods, we have found equilibrium points and examined their linear stability. We have also found the zero velocity surface for the present model. In addition to five equilibrium points there exists a new equilibrium point on the line joining the two primaries. It is found that L 1 and L 3 are stable for some values of inner and outer radius of the disk while other collinear points are unstable, but L 4 is conditionally stable for mass ratio less than that of Routh’s critical value. Lastly, we have studied the effects of radiation pressure, oblateness and mass of the disk on the motion and stability of equilibrium points.  相似文献   

15.
Although astronomers have been involved with the development and use of least squares, and alternatives, they have made insufficient use of total least squares: least squares that allow for error in the equations of condition as well as the observations. There exist, however, problems of astronomical data reduction for which total least squares represents the ideal mathematical tool. Among these problems are the differential correction of an orbit and the determination of parameters of Galactic kinematics. Total least squares, although more computationally demanding than ordinary least squares, can be used in any situation where the latter is applicable. However, care must be paid to the proper scaling of the data matrix. The method merits greater attention by the astronomical community.  相似文献   

16.
The relation between the locations of L 1, L 2 Lagrangian points and the boundary to their respective satellite system is brought forth, in that, the Lagrangian points L 1, L 2 are seen to lie just on the boundary to their respective satellite system.  相似文献   

17.
By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω1, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.  相似文献   

18.
In a binary system with both bodies being luminous, the inner collinear equilibrium pointL 1 becomes stable for values of the mass ratio and radiation pressure parameters in a certain region. The kind of periodic motions aroundL 1 is examined in this case. Second-order parametric expansions are given and the families of periodic orbits generated fromL 1 are numerically determined for several sets of values of the parameters. Short- and long-period solutions are identified showing a similarity in the character of periodicity with that aroundL 4. It is also found that the finite periodic solutions in the vicinity ofL 1 are stable.  相似文献   

19.
We have compared the observed distribution of the quantity log(V 2 2 rP) for a sample of 233 pairs of galaxies with Monte-Carlo simulations. From such an analysis we have derived an average mass-to-luminosity ratiosM/L B =18±11. Our result is consistent with a linear increase of the mass with radius at least until distances of about 30 kpc.  相似文献   

20.
Rigozo  N.R.  Echer  E.  Vieira  L.E.A.  Nordemann  D.J.R. 《Solar physics》2001,203(1):179-191
A reconstruction of sunspot numbers for the last 1000 years was obtained using a sum of sine waves derived from spectral analysis of the time series of sunspot number R z for the period 1700–1999. The time series was decomposed in frequency levels using the wavelet transform, and an iterative regression model (ARIST) was used to identify the amplitude and phase of the main periodicities. The 1000-year reconstructed sunspot number reproduces well the great maximums and minimums in solar activity, identified in cosmonuclides variation records, and, specifically, the epochs of the Oort, Wolf, Spörer, Maunder, and Dalton Minimums as well the Medieval and Modern Maximums. The average sunspot number activity in each anomalous period was used in linear equations to obtain estimates of the solar radio flux F 10.7, solar wind velocity, and the southward component of the interplanetary magnetic field.  相似文献   

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