Solution of the basic equation of stellar statistics

Monthly Notices of the Royal Astronomical Society

Hammersley

Garzón

et al. 2000

A method based on Lucy’s iterative algorithm is developed to invert the equation of stellar statistics for the Galactic bulge and is then applied to the K‐band star counts from the Two‐Micron Galactic Survey in a number of off‐plane regions (10°>|b|>2°, |l|<15°). The top end of the K‐band luminosity function is derived and the morphology of the stellar density function is fitted to triaxial ellipsoids, assuming a non‐variable luminosity function within the bulge. The results, which have already been outlined by López‐Corredoira et al., are shown in this paper with a full explanation of the steps of the inversion: the luminosity function shows a sharp decrease brighter than MK=−8.0 mag when compared with the disc population; the bulge fits triaxial ellipsoids with the major axis in the Galactic plane at an angle with the line of sight to the Galactic centre of 12° in the first quadrant; the axial ratios are 1:0.54:0.33, and the distance of the Sun from the centre of the triaxial ellipsoid is 7860 pc. The major–minor axial ratio of the ellipsoids is found not to be constant, the best fit to the gradient being Kz=(8.4±1.7)×exp(−t/(2000±920) pc), where t is the distance along the major axis of the ellipsoid in parsecs. However, the interpretation of this is controversial. An eccentricity of the true density‐ellipsoid gradient and a population gradient are two possible explanations. The best fit for the stellar density, for 1300 pc

Section: Inversion Of the Stellar Statistics Equationsupporting

confidence: 83%

“…The inversion of integral equations such as is ill‐conditioned. Typical analytical methods for solving these equations (see Balázs 1995) cannot achieve a good solution because of the sensitivity of the kernel to the noise of the counts (see, for instance, Craig & Brown 1986, chapter 5).…”

Section: Inversion Of the Stellar Statistics Equationmentioning

confidence: 99%

Inversion of stellar statistics equation for the Galactic bulge

Monthly Notices of the Royal Astronomical Society

Hammersley

Garzón

et al. 2000

“…Typical analytical methods for solving these equations (see Balázs 1995) cannot achieve a good solution because of the sensitivity of the kernel to the noise of the star counts (see, e.g., Craig & Brown 1986, chapter 5). Since the functions in these equations have a stochastic rather than analytical interpretation, it is to be expected that statistical-inversion algorithms will be more robust (Turchin et al 1971;Jupp & Vozoff 1975;Balázs 1995). Among these statistical methods, the iterative method of Lucy's algorithm (Lucy 1974;Turchin et al 1971;Balázs 1995;López-Corredoira et al 2000) is an appropriate one.…”

Section: Appendix A: Lucy's Methods For the Inversion Of Fredholm Intementioning

confidence: 99%

Gaia-DR2 extended kinematical maps

Labini

2019

A&A

Context. The Gaia Collaboration has used Gaia-DR2 sources with six-dimensional (6D) phase space information to derive kinematical maps within 5 kpc of the Sun, which is a reachable range for stars with relative error in distance lower than 20%. Aims. Here we aim to extend the range of distances by a factor of two to three, thus adding the range of Galactocentric distances between 13 kpc and 20 kpc to the previous maps, with their corresponding error and root mean square values. Methods. We make use of the whole sample of stars of Gaia-DR2 including radial velocity measurements, which consists in more than seven million sources, and we apply a statistical deconvolution of the parallax errors based on the Lucy’s inversion method of the Fredholm integral equations of the first kind, without assuming any prior. Results. The new extended maps provide lots of new and corroborated information about the disk kinematics: significant departures of circularity in the mean orbits with radial Galactocentric velocities between −20 and +20 km s−1 and vertical velocities between −10 and +10 km s−1; variations of the azimuthal velocity with position; asymmetries between the northern and the southern Galactic hemispheres, especially towards the anticenter that includes a larger azimuthal velocity in the south; and others. Conclusions. These extended kinematical maps can be used to investigate the different dynamical models of our Galaxy, and we will present our own analyses in the forthcoming second part of this paper. At present, it is evident that the Milky Way is far from a simple stationary configuration in rotational equilibrium, but is characterized by streaming motions in all velocity components with conspicuous velocity gradients.

“…Typical analytical methods for solving these equations (Balázs 1995) cannot achieve a good solution because the kernel is sensitive to the noise of the star counts (Craig & Brown 1986, chapter 5). Because the functions in these equations have a stochastic rather than analytical interpretation, it is to be expected that statistical inversion algorithms are more robust (Turchin et al 1971;Vozoff & Jupp 1975;Balázs 1995). These statistical methods include the iterative method of Lucy's algorithm (Lucy 1974;Turchin et al 1971;Balázs 1995;López-Corredoira et al 2000), which is appropriate here.…”

mentioning

confidence: 99%

“…Because the functions in these equations have a stochastic rather than analytical interpretation, it is to be expected that statistical inversion algorithms are more robust (Turchin et al 1971;Vozoff & Jupp 1975;Balázs 1995). These statistical methods include the iterative method of Lucy's algorithm (Lucy 1974;Turchin et al 1971;Balázs 1995;López-Corredoira et al 2000), which is appropriate here. Its key feature is the interpretation of the kernel as a conditioned probability and the application of Bayes' theorem.…”

mentioning

confidence: 99%

Structure of the outer Galactic disc with Gaia DR2

Chrobáková

Nagy

2020

A&A

Context. The structure of outer disc of our Galaxy is still not well described, and many features need to be better understood. The second Gaia data release (DR2) provides data in unprecedented quality that can be analysed to shed some light on the outermost parts of the Milky Way. Aims. We calculate the stellar density using star counts obtained from Gaia DR2 up to a Galactocentric distance R=20 kpc with a deconvolution technique for the parallax errors. Then we analyse the density in order to study the structure of the outer Galactic disc, mainly the warp. Methods. In order to carry out the deconvolution, we used the Lucy inversion technique for recovering the corrected star counts. We also used the Gaia luminosity function of stars with MG < 10 to extract the stellar density from the star counts. Results. The stellar density maps can be fitted by an exponential disc in the radial direction hr = 2.07 ± 0.07 kpc, with a weak dependence on the azimuth, extended up to 20 kpc without any cut-off. The flare and warp are clearly visible. The best fit of a symmetrical S-shaped warp gives zw ≈ z + (37 ± 4.2(stat.) − 0.91(syst.))pc · (R/R ) 2.42±0.76(stat.)+0.129(syst.) sin(φ + 9.3 • ± 7.37 • (stat.) + 4.48 • (syst.)) for the whole population. When we analyse the northern and southern warps separately, we obtain an asymmetry of an ∼ 25% larger amplitude in the north. This result may be influenced by extinction because the Gaia G band is quite prone to extinction biases. However, we tested the accuracy of the extinction map we used, which shows that the extinction is determined very well in the outer disc. Nevertheless, we recall that we do not know the full extinction error, and neither do we know the systematic error of the map, which may influence the final result. The analysis was also carried out for very luminous stars alone (MG < −2), which on average represents a younger population. We obtain similar scale-length values, while the maximum amplitude of the warp is 20 − 30% larger than with the whole population. The north-south asymmetry is maintained.