Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations
A new method of determination of planetocentric coordinates of albedo features on the visible disc of a spherical planet is offered under various conditions of its illumination. The method is aimed for processing the images of planets received with ground-based telescopes. The position of a detail o...
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Головна астрономічна обсерваторія НАН України
2005
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| Цитувати: | Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations / V.V. Mikhalchuk // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 557-560. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-797212015-04-04T03:02:41Z Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations Mikhalchuk, V.V. MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics A new method of determination of planetocentric coordinates of albedo features on the visible disc of a spherical planet is offered under various conditions of its illumination. The method is aimed for processing the images of planets received with ground-based telescopes. The position of a detail on the image of a planet is determined not concerning center of a geometric planetary disc, and concerning center of the illuminated part of its visible disc, that allows excluding influence of a phase of a planet. The auxiliary coordinate system connected to equator of intensity is applied for determination of a position of points on the illuminated part of a visible planetary disc. The formulae for transition from auxiliary coordinate system to planetocentric coordinate system are received. The particular cases of determination of planetocentric coordinates of basic points of the illuminated part of the visible disc of a spherical planet from the formulae, which are not requiring of attraction of auxiliary coordinate system, are considered. The method is checked up on an example of processing of the images of a Mercury and Mars. 2005 Article Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations / V.V. Mikhalchuk // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 557-560. — Бібліогр.: 9 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79721 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics |
| spellingShingle |
MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics Mikhalchuk, V.V. Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations Кинематика и физика небесных тел |
| description |
A new method of determination of planetocentric coordinates of albedo features on the visible disc of a spherical planet is offered under various conditions of its illumination. The method is aimed for processing the images of planets received with ground-based telescopes. The position of a detail on the image of a planet is determined not concerning center of a geometric planetary disc, and concerning center of the illuminated part of its visible disc, that allows excluding influence of a phase of a planet. The auxiliary coordinate system connected to equator of intensity is applied for determination of a position of points on the illuminated part of a visible planetary disc. The formulae for transition from auxiliary coordinate system to planetocentric coordinate system are received. The particular cases of determination of planetocentric coordinates of basic points of the illuminated part of the visible disc of a spherical planet from the formulae, which are not requiring of attraction of auxiliary coordinate system, are considered. The method is checked up on an example of processing of the images of a Mercury and Mars. |
| format |
Article |
| author |
Mikhalchuk, V.V. |
| author_facet |
Mikhalchuk, V.V. |
| author_sort |
Mikhalchuk, V.V. |
| title |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| title_short |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| title_full |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| title_fullStr |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| title_full_unstemmed |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| title_sort |
influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations |
| publisher |
Головна астрономічна обсерваторія НАН України |
| publishDate |
2005 |
| topic_facet |
MS6: New Trends, Research Directions, and Perspective Programs in the Field of Astronomy and Astrophysics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79721 |
| citation_txt |
Influence of the phase of a spherical planet on determination of coordinates of albedo features on the planet's surface from ground-based observations / V.V. Mikhalchuk // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 557-560. — Бібліогр.: 9 назв. — англ. |
| series |
Кинематика и физика небесных тел |
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2025-07-06T03:43:38Z |
| last_indexed |
2025-07-06T03:43:38Z |
| _version_ |
1836867564857196544 |
| fulltext |
INFLUENCE OF THE PHASE OF A SPHERICAL PLANET
ON DETERMINATION OF COORDINATES OF ALBEDO FEATURES
ON THE PLANET’S SURFACE FROM GROUND-BASED OBSERVATIONS
V. V. Mikhalchuk
Odesa National Maritime Academy
8 Didrikhsona Str., 65029 Odesa, Ukraine
e-mail: phys@ma.odessa.ua, vmihalchuk@mail.ru
A new method of determination of planetocentric coordinates of albedo features on the visible
disc of a spherical planet is offered under various conditions of its illumination. The method is
aimed for processing the images of planets received with ground-based telescopes. The position of
a detail on the image of a planet is determined not concerning center of a geometric planetary disc,
and concerning center of the illuminated part of its visible disc, that allows excluding influence
of a phase of a planet. The auxiliary coordinate system connected to equator of intensity is
applied for determination of a position of points on the illuminated part of a visible planetary disc.
The formulae for transition from auxiliary coordinate system to planetocentric coordinate system
are received. The particular cases of determination of planetocentric coordinates of basic points
of the illuminated part of the visible disc of a spherical planet from the formulae, which are not
requiring of attraction of auxiliary coordinate system, are considered. The method is checked up
on an example of processing of the images of a Mercury and Mars.
INTRODUCTION
By mapping of albedo details on surfaces of planets of the Solar System and their natural satellites using ground
telescopic observations, there is a problem to determine planetographic coordinates of details on the images of
their visible discs. For the terrestrial planets, it is possible to consider their shape in the first approximation as
spherical. But this problem becomes difficult due to phases causing appreciable damages on the visible disc, that
does not allow one to apply the certain methods for determination of coordinates of observable details [2–4, 9].
The paper [7] is devoted to the finding solution of this problem. In this article a method for determination
of planetocentric coordinates of any point, observable on the illuminated part of the visible disc of a spherical
planet is described.
From ground observations the equatorial planetocentric coordinate system, which determines the position
of each point on the surface of the planet is observed on its visible disc in an orthographic projection. In this
coordinate system, the position of each point on the planet’s surface is specified by the planetocentric latitude b
and longitude l. The basic reference point is the center of the geometric disc of the planet with planetocentric
coordinates (lp, bp), where lp is longitude of the central meridian passing through the center of the disc, bp is
latitude of the center of the geometric disc. The latitude bp is equal to planetocentric declination of the Earth D⊕.
The present contribution is the further development and generalization of the method [5] for any point of
the visible disc of a spherical planet in consideration of a phase by determination of planetographic coordinates
of albedo details on images of spherical planets received from ground-based observations.
INFLUENCE OF THE PHASE ANGLE ON THE CALCULATION
OF THE PLANETOCENTRIC COORDINATES
The methods [2–4, 9] for determination of planetocentric coordinates (l, b) of a detail on the visible planetary
disc are fair only in that specific case, when the disc is completely illuminated, i.e., the phase angle Φ is equal
to zero. If the phase angle Φ differs from zero, the center of the illuminated part of the visible planetary disc
does not coincide the geometric center of the planet’s disc. The limitations of the applicability of those methods
result in the necessity to develop a new method, which allows, without a loss for the accuracy, to exclude
the influence of the phase.
Let us consider the visible disc of a spherical planet illuminated by the Sun at a phase angle Φ (Fig. 1). Let
us introduce O as the geometric center of the planetary disc, E as the subsolar point (pole of illumination),
c© V. V. Mikhalchuk, 2004
557
C as the center of the illuminated part of the planet’s visible disc, T as the visible center of the orthographic
terminator, A and B as orthographic horns of the disc, M as a point on the planet’s surface.
Figure 1. The auxiliary coordinate system (the transverse orthographic projection)
We introduce an auxiliary spherical coordinate system (λ, ϕ) on the planet’s surface, in which the plane of
equator of intensity is considered as the basic plane, and poles are the orthographic horn of the disc [7]. Origin
of the auxiliary system is the point C, which always is on the illuminated part of the visible planetary disc.
Then, the position of each point on the surface of the illuminated part of the visible planetary disc will be
determined by latitude ϕ, which is measured from the equator of intensity, and longitude λ, which is measured
from the mean meridian (it is passing through the point C) westward of the planet. As described in [7], it is
possible to find the auxiliary spherical coordinates (λ, ϕ) of any point on the planet’s surface from its relative
coordinates (ξ, η).
To transform the auxiliary spherical coordinates (λ, ϕ) into the planetocentric coordinates (l, b), we shall
consider a projection of the visible disc of spherical planet on the plane of the sky. Let P be the angle of
position of the rotation axis of the planet on the geocentric celestial sphere, let Q be the angle of position
of point F as the least illuminated on the disc [1]. The planetocentric coordinates of the point O (lp, bp)
are known and bp = D⊕. The planetocentric coordinates of the point C are (l0, b0) and can be determined
using the method [5]. The auxiliary coordinates (λ, ϕ) of the point M are also known. Then, it is possible to
determine the planetocentric coordinates (l, b) of any point of the illuminated part of the visible planetary disc
from the auxiliary spherical coordinates (λ, ϕ) of this point using the equations obtained in [7]:
sin b = ± sinϕ cos D⊕ sin(P − Q) + cosϕ [cosλ0 sin D⊕ ∓ sin λ0 cosD⊕ cos(P − Q)], (1)
tan (l − lp) =
±[cosϕ sin λ0 sin (P − Q) + sin ϕ cos (P − Q)]
cosϕ cosλ0 cosD⊕ ∓ sinD⊕[sin ϕ sin (P − Q) − cosϕ sin λ0 cos (P − Q)]
, (2)
where λ0 = λ + γ, γ is the phase shift of the center of the planetary disc, and the signs of numerator and
denominator in (2) coincide with the signs of sin (P − Q) and cos (P − Q), respectively. The selection of a sign
in expressions (1) and (2) is defined by the rule: the upper sign concerns the case, when sin(P − Q) > 0, and
the lower sign – the case, when sin(P − Q) < 0.
APPLICATION OF THE METHOD OF DETERMINATION OF PLANETOCENTRIC
COORDINATES TO MAPPING ALBEDO DETAILS OF SPHERICAL PLANETS
The planetocentric coordinates of some albedo features on surfaces of Mercury and Mars were determined using
their images received from ground-based observations. Photos and CCD-images of visible discs of planets were
processed (Internet Web site ALPO: the Mars Watch, 1999; the Mercury Watch, 2003). For processing only
those images were used, which were received at considerable phase angles.
The image of Mercury (Fig. 2a) represents drawing, received from visual observations (Internet Web site
ALPO, observer M. Frassati, telescope with diameter of 203 mm and resolution of 0.6′′). The image of Mars
(Fig. 2b) was received using a CCD (Internet Web site ALPO, observers S. Buda and B. Curcic, telescope with
diameter of 254 mm and resolution of 0.5′′).
558
a) Mercury b) Mars
Figure 2. The images of visible discs of the planets
To calculate physical ephemerides of Mercury and Mars, the elements of rotation of planets were used
from [8]. Physical ephemerides of planets at moments of observations, which were calculated using programs
from [6], are given in Table 1.
Table 1. The physical ephemerides of planets
Planet UT r Φ D⊕ P Q lp
Mercury 01/02/2003, 7h00m 3.55′′ 82.45◦ −7.2◦ 356.6◦ 265.6◦ 309.7◦
Mars 03/06/1999, 10h50m 7.03 29.28 +22.8 36.3 111.3 303.9
Using equations from [7] and the measured relative coordinates ξ and η, the auxiliary spherical coordi-
nates λ and ϕ were determined. The coordinates of details on the surfaces of planets, which were found using
the method [7], are listed in Table 2.
Table 2. The coordinates of details on the surfaces of planets
Planet Detail ξ η λ0 ϕ l b Name of region on the map
Mercury 1 0.00 +0.46 +25.7◦ +27.4◦ 333.6◦ +21.3◦ Solitudio Argiphontae
2 −0.56 +0.47 +6.7 +28.0 315.6 +21.0 Solitudio Aphrodites
3 −0.31 +0.27 +15.0 +15.7 324.0 +9.0 Pentas
4 −0.27 −0.05 +16.4 −2.9 326.3 −9.5 Sinus Argiphontae
5 −0.92 −0.11 −4.9 −6.3 304.8 −13.6 Solitudio Alarum
6 −0.63 −0.25 +4.5 −21.3 314.6 −21.6 Pieria
7 −0.66 −0.56 +3.5 −34.1 314.3 −41.2 Solitudio Criophori
8 −0.46 −0.75 +10.0 −48.6 322.8 −55.4 Cyllene
Mars 1 +0.56 +0.31 +27.4 +18.1 332.9 +45.0 Ismenius Lacus
2 +0.86 +0.46 +47.8 +27.4 2.8 +52.9 Mare Acidalium
3 +0.58 +0.57 +28.6 +34.8 333.4 +61.8 Ortygia
4 +0.54 +0.90 +26.2 +64.2 219.2 +87.3 The North polar cap
5 +0.12 +0.41 +2.8 +24.2 298.6 +46.8 Boreosyrtis
6 −0.24 +0.47 −16.8 +28.0 273.6 +43.3 Casius
7 −0.44 +0.36 −28.4 +21.1 265.8 +32.2 Nodus Alcyonius
8 −0.26 +0.10 −17.9 +5.7 283.6 +22.5 Syrtis Major
9 −0.87 −0.16 −61.4 −9.2 250.1 −9.6 Tritonis Sinus
10 −0.70 −0.23 −46.0 −13.3 265.6 −6.3 Syrtis Minor
11 −0.38 −0.14 −24.8 −8.0 282.3 +7.2 Moeris Lacus
12 +0.52 −0.58 +25.0 −35.5 333.1 −8.5 Sinus Sabaeus
13 +0.96 −0.44 +56.6 −26.1 0.8 −1.2 Sinus Meridiani
14 0.00 0.00 −3.7 0.0 300.1 +21.8 Syrtis Major, Aeria
559
The planetocentric coordinates of the main details on images of visible discs of Mercury and Mars, which were
obtained using the above-mentioned method, were compared with coordinates of the same details taken from
maps of albedos of the planets for epoch 2000.0 (Internet Web site ALPO). The results shown a good agreement
the calculated coordinates of albedo details with the coordinates, which have been taken from the map, within
errors of the initial images and the map.
For points located on the equator of intensity, we set ϕ to 0. In this case, it is possible to obtain from expres-
sions (1) and (2) equations for calculation of planetocentric coordinates (l, b) of any point on the illuminated
part of the equator of intensity (the pole of illumination, the pole of phase, the visible center of the orthographic
terminator and the mirror point), in the projection on the plane of the sky. The obtained formulae were also
applied to the determination of planetocentric coordinates of the aforementioned points on illuminated parts of
the visible discs of Mercury and Mars.
CONCLUSION
The main results obtained in this work allow us to draw the following conclusions:
1. The auxiliary spherical coordinate system on the planet’s surface is to be introduced to determine positions
of details on the illuminated part of planet’s visible disc with any values of a phase angle.
2. The method for the determination of the planetocentric coordinates of details on the visible disc of
a spherical planet under various conditions of its illumination, which takes into account the influence of
a phase, is applied to the processing images of Mercury and Mars.
3. The possibility to derive the formulae for the determination of planetocentric coordinates of basic points
on the illuminated part of the visible disc of the spherical planet, which lie on the equator of intensity, is
shown.
Acknowledgements. The author are thankful to anybody who has read this contribution to the end.
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