Calculation and visualization of distortions in morphographic projections

2019 ◽  
Vol 945 (3) ◽  
pp. 26-36
Author(s):  
M.V. Nyrtsov ◽  
A.A. Domatieva
Keyword(s):  
Solar System ◽  
Celestial Body ◽  
Spherical Shape ◽  
Irregular Shape ◽  
Triaxial Ellipsoid ◽  
Reference Surface ◽  
Small Bodies ◽  
The Earth ◽  
Regular Shape

There are different ways of estimating the distortions that help us find the best cartographic projections and recommend them for a specific range of tasks from mapping the Earth to extraterrestrial mapping. Besides the planets and their satellites there are small bodies in the Solar System. For mapping these bodies it is advisable to use morphographic projections to show their non-spherical shape. These projections were developed by Philip Stooke. He also proposed replacement of a regular reference surface with a phoboid and methods for estimating distortions but practically implemented them on the example of a body with a regular shape approximated by test triaxial ellipsoid with a ratio of axes 4:3:2. In this regard we set the task of developing and implementing the method for determining distortions in morphographic projections for a body with an irregular shape. We used the algorithm of comparison cartographic cell on projection and on phoboid in order to estimate distortion. The object of mapping is Phobos, a satellite of Mars. Phobos is a non-spherical celestial body for which morphographic projections were first applied. Comparing our results by the values of distortions with the results obtained by Philip Stooke we can conclude that using morphographic projections makes less distortion at portraying a phoboid than a test triaxial ellipsoid.

Meridian section projections: a new class of the triaxial ellipsoid projections

2021 ◽  
Vol 968 (2) ◽  
pp. 11-22
Author(s):  
M.V. Nyrtsov ◽  
M.E. Fleis ◽  
A.I. Sokolov
Keyword(s):  
Triaxial Ellipsoid ◽  
International Astronomical Union ◽  
Small Bodies ◽  
New Class ◽  
The Earth ◽  
Meridian Section ◽  
International Astronomical ◽  
Global Mapping ◽  
Ellipsoid Of Revolution ◽  
Celestial Bodies

Historically the conformal projections have been used for mapping not only the Earth, but other celestial bodies as well. Their application enables preserving the shape of the relief features on the maps, which is extremely important for various analyses of celestial bodies’ surfaces. For many small bodies of the Solar system the International Astronomical Union recommends to apply a triaxial ellipsoid as a reference surface. But if the conformal projections for the reference surfaces of a sphere and an ellipsoid of revolution already exist, obtaining these projections for a triaxial ellipsoid will be significantly complicated, and the task of preserving the shape of relief features still actual. In general, the article deals with cylindrical and azimuthal projections of the meridian section for global mapping the celestial body surface in accordance with the idea formulated by prof. L. M. Bugaevsky. The projections are implemented for mapping of Phobos, moon of Mars.

Mathematical cartography today

2019 ◽  
Vol 943 (1) ◽  
pp. 52-57
Author(s):  
M.V. Nyrtsov
Keyword(s):  
Solar System ◽  
Initial Data ◽  
Additional Research ◽  
Triaxial Ellipsoid ◽  
International Astronomical Union ◽  
Small Bodies ◽  
Map Projections ◽  
International Astronomical ◽  
A Cell ◽  
Almost All

For most small bodies of the Solar System shapes the International Astronomical Union recommends to use a triaxial ellipsoid. Today almost all major classes of cartographic projections for the triaxial ellipsoid have been developed. They require systematization, classification and additional research. The archives of libraries accumulated a significant list of paper maps of the Soviet times which are compiled in projections developed in our country. It is necessary to implement those projections in the PROJ.4 library used by GIS. Distortions in map projections can be defined as deformation of a cell. If we imagine that the sphere is covered with a net of regular hexagons then it is possible to estimate distortions in the projection by their deformation on the plane. Now the majority of cartographers use online calculators to define the values required for mathematical cartography. Programs are usually written in JavaScript and require the user to only enter the initial data. These are only some of the problems facing modern mathematical cartography.

scholarly journals On secular resonances of small bodies in the planetary systems

2006 ◽  
Vol 2 (S236) ◽  
pp. 77-84
Author(s):  
Jianghui Ji ◽  
L. Liu ◽  
G. Y. Li
Keyword(s):  
Solar System ◽  
Planetary System ◽  
Planetary Systems ◽  
Giant Planets ◽  
Small Bodies ◽  
Secular Resonances ◽  
Mean Motion Resonances ◽  
The Earth ◽  
Habitable Zones ◽  
Earth Like Planets

AbstractWe investigate the secular resonances for massless small bodies and Earth-like planets in several planetary systems. We further compare the results with those of Solar System. For example, in the GJ 876 planetary system, we show that the secular resonances ν1 and ν2 (respectively, resulting from the inner and outer giant planets) can excite the eccentricities of the Earth-like planets with orbits 0.21≤ a <0.50 AU and eject them out of the system in a short timescale. However, in a dynamical sense, the potential zones for the existence of Earth-like planets are in the area 0.50≤ a ≤1.00 AU, and there exist all stable orbits last up to 105 yr with low eccentricities. For other systems, e.g., 47 UMa, we also show that the Habitable Zones for Earth-like planets are related to both secular resonances and mean motion resonances in the systems.

scholarly journals From Discovery to Impact - Near Earth Asteroids

2012 ◽  
Vol 8 ◽  
pp. 73-78
Author(s):  
Miloš Tichý ◽  
Michaela Honková ◽  
Jana Tichá ◽  
Michal Kočer
Keyword(s):  
Solar System ◽  
Current System ◽  
Small Bodies ◽  
Impact Site ◽  
The Earth ◽  
Near Earth Asteroids ◽  
Near Earth Objects

The Near-Earth Objects (NEOs) are the most important of the small bodies of the solar system, having the capability of close approaches to the Earth and the chance to collide with the Earth.  We present here the current system of discovery of these dangerous objects, standards for selecting useful and important targets for NEO follow-up astrometry, system of impact probabilities calculations, and also determination of impact site and evacuation area.

scholarly journals How many interstellar visitors are there in the Solar System?

2021 ◽  
pp. 16-21
Author(s):  
Б.М. Шустов
Keyword(s):  
Particle Size ◽  
Solar System ◽  
Interstellar Dust ◽  
Particle Flux ◽  
Dynamic Evolution ◽  
Small Bodies ◽  
The Earth ◽  
Total Particle ◽  
The Masses ◽  
The Relationship

По результатам краткого обзора наблюдений вошедших в Солнечную систему межзвездных пылинок, метеороидов и более крупных тел и на основе моделей, описывающих спектры масс таких тел, отмечен огромный разброс (много порядков величины) в оценках отношения η потока межзвездных частиц к общему потоку частиц в окрестности Земли и в целом в Солнечной системе. Этот разброс означает, что современные возможности не позволяют определенно ответить на вопрос в заголовке статьи. При анализе результатов наблюдений необходимо учитывать характер зависимости отношения η от размера частиц r, т. е. η(r). Эта зависимость определяется процессами генерации и динамической эволюции населения малых тел в Солнечной системе и за ее пределами. According to the results of a brief review of observations included in the Solar system, interstellar dust, meteoroids and larger bodies, and on the basis of models describing the spectra of the masses of these bodies, we mark a huge spread (many orders of magnitude) in estimates of the relationship η of the flow of interstellar particles to the total particle flux near the Earth and in the whole Solar system. These differences mean that modern capabilities do not allow us to definitely answer the question in the title of the article. When analyzing the results of observations, it is necessary to take into account the nature of the dependence of the ratio η on the particle size r, i.e. η(r). This dependence is determined by the processes of generation and dynamic evolution of the population of small bodies in the Solar system and beyond.

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