The surface of the Moon is covered with millions of impact craters due to the fact that there is no atmosphere on the Moon to protect it from collisions by asteroids, comets, or meteorites. These celestial objects hit the Moon at a wide range of speeds, about 8 to 20 km/s. Also, there is no erosion phenomena and little geologic activity to wear away these craters, so they remain unchanged until another new impact changes it. These facts and others led the celestial dynamists and geophysicists, over the years, to conclude that the Moon's gravity is stronger in some regions than others, creating a lumpy gravitational field. In particular, impact basins exhibit unexpectedly strong gravitational pull. Scientists have suspected that the explanation has to do with an excess distribution of mass below the lunar surface, and have dubbed these regions mass concentrations, or “mascons”. These collisions send material flying out, creating a dense band of debris around the crater’s perimeter. The impacts send a shockwave through the Moon’s interior, reverberating within the crust and producing a counter wave that draws dense material from the lunar mantle toward the surface, creating a dense centers within the craters. See the URL, http://newsoffice.mit.edu/2013/an-answer-to-why-lunar-gravity-is-so-uneven-0530 and for more details see Melosh, et al. (2013).
The physical exploration of the Moon began when Luna 2, a space probe launched by the Soviet Union, made an impact on the surface of the Moon on September 14, 1959. In 1969, NASA's Project Apollo first successfully landed humans on the Moon. They landed scientific instruments on the Moon and returned lunar samples to the Earth. In 1966, Luna 9 became the first spacecraft to achieve a controlled soft landing, and Luna 10 became the first mission to enter orbit.
Between 1968 and 1972, manned missions to the Moon were conducted by the United States as part of the Apollo program. Apollo 8 was the first manned mission to enter orbit in December 1968, and was followed by Apollo 10 in May 1969. Six missions landed men on the Moon, beginning with Apollo 11 in July 1969.
Also in last decade, on September 14, 2007 Japan launched an artificial satellite around the Moon. China also launched a lunar satellite on October 24, 2007. Both countries have participated in many missions to the Moon in this decade. Also, India launched a lunar orbiter (Chandrayaan 1) on October 22, 2008, for two-year mission aimed to map the Moon, Carvalho et al. (2010). This competition motivate the researchers from all countries to focus on the orbits of lunar orbiters.
The motion of an artificial satellite around the Moon is quite different from the one around the Earth on several aspects. The most important difference is that the Moon is a slowly rotating body. Secondly the Moon's atmosphere has nearly negligible effect on the motion of the Moon's satellite in contrast to that of the Earth. It is also well known that, the lunar gravity field is far from being central, nor does it exhibit any strong symmetry of revolution; the order of magnitude for the Earth spherical harmonics can be found in Kaula (1966), see e.g. Konopliv et al. (2001) for a recent model in spherical harmonics, and for the Moon, Bills & Ferrari (1980).
The Moon is much less flattened than the Earth, which makes the
In this paper, we will focus on the combined effect of the lunar perturbing terms factored by
The value of inclination that enforces the rate of argument of periapsis change of an orbit under the effect of considered perturbations to be equal zero is called critical inclination
In the Earth artificial satellite theory, the critical inclination under
Delhaise & Morbidelli (1993) investigated the luni-solar effects on a geosynchronous satellite in near the critical inclination. Yi & Choi (1993) studied the characteristics and perturbation effects on the artificial satellite orbit with critical inclination. Abd El-Salam & Ahmed (2005a,b) discussed Post-Newtonian (PN) effects on the critical inclination angle in the Earth artificial satellite theory including the zonal harmonics up to
On the other hand, the dynamics of lunar orbiters is quite different from that of an artificial Earth satellite in which the Moon is less flattened than the Earth, that cause
The flatness of the Moon is less than the Earth which makes the
In this study, the effect of
Our results can be applied to a special type of orbit, a Sun-synchronous orbit for Moon’s artificial satellites. The Sun-synchronous orbit is a particular case of an almost polar orbit. The satellite travels from the North Pole to the South Pole and vice versa, but its orbital plane is always fixed for an observer that is posted in the Sun. Thus the satellite always passes approximately on the same point of the surface of the Moon every day in the same hour. In such a way the satellite can transmit all the data collected for a lunar fixed antenna, during its orbits. An analysis of Sun synchronous orbits considering the non-uniform distribution of mass of the Moon is done for the longitude of the ascending node with an approach based on Park & Junkins (1995).
To achieve our goal, this paper is organized as follows. In Section 2, the potential of the Moon is presented and the Hamiltonian of the system is constructed using Delaunay canonical variables. In Section 3 and Section 4, we eliminated the short periodic terms from the Hamiltonian and we obtained the new families of critical inclinations using a single-average procedure. In Section 5 and Section 6, we gave numerical simulations and discussions on the results. Finally, in Section 7 a conclusion is presented to draw attention about the importance of
In the main problem, the following approximations are made by Giacaglia et al. (1970): i) the Moon's orbit around the Earth lies on the lunar equatorial plane; ii) the orbit of the Moon around the Earth is circular; iii) the longitude of lunar longest meridian λ22 is equal to the longitude of the Earth λ⊕. The classical Keplerian orbital elements are used to define the orbit of a lunar orbiter; i.e. semi-major axis
Using the lunar equatorial plane as the fundamental plane, then lunar disturbing potential per unit mass can be written as (d’Avanzo et al. 1997, Radwan 2002)
The leading harmonics of lunar potential are almost of the same order, this complicates the choice of the harmonics at which the potential may be truncated and makes the choice somewhat arbitrary. In terms of the orbital elements the Legendre terms will now be (El-saftawy, 1991)
The Delaunay canonical variables are usually defined as
3. SHORT PERIODIC TERMS ELIMINATION
In order to study the motion of an orbiter a single-average is taken over the mean anomaly,
To calculate the effects of
The critical inclination
which can be written as
4. SOLUTION OF THE CUBIC EQUATION (7)
Let equation (7) be rewritten as
The three roots of this cubic equation are given by
Depending on Equation (8); note that Equation (9) is neglected hence since it contain imaginary part, different simulations will be introduced here to distinguish the effect of
In all figures the black curves shows the critical inclination considering the effcts of
In the first case labeled by Figs. 1(a), 2(a), 3(a), we set the semi-major axis to the range 2000≤
[Fig. 1.] Critical inclination as a function of, 1- semi-major axis (Fig. a, Fig. d) with eccentricity =0.1,0.5, 2-eccentricity (Fig. b, Fig. e) with semi-major axis =4000km,10000km, 3- both semi-major axis and eccentricity (Fig. c, Fig. f), argument of perigee ω=90°, and longitude of the ascending node Ω=270°.
[Fig. 2.] Critical inclination as a function of, 1- semi-major axis (Fig. a, Fig. d) with eccentricity =0.1,0.5, 2-eccentricity(Fig. b, Fig. e) with semi-major axis =4000km,10000km, 3- both semi-major axis and eccentricity (Fig. c, Fig. f), argument of perigee ω=270°, and longitude of the ascending node Ω=270°.
[Fig. 3.] Critical inclination as a function of, 1- semi-major axis (Fig. a, Fig. d) with eccentricity =0.1,0.5, 2-eccentricity(Fig. b, Fig. e) with semi-major axis =4000km,10000km, 3- both semi-major axis and eccentricity (Fig. c, Fig. f), argument of perigee ω=60°, and longitude of the ascending node Ω=18°.
In the second case labeled by Figs. 1(d), 2(d), 3(d), we set the semi-major axis to the range 4000≤
In the third case labeled by Figs. 1(b), 2(b), 3(b), we set the critical inclination versus the eccentricity 0.1≤
In the fourth case labeled by Figs. 1(e), 2(e), 3(e), we set the critical inclination versus the eccentricity 0.4≤
We repeat these case with different argument of perigee and longitude of the ascending node.
In Fig. 1:
taking account of the dynamical shape parameters
In Fig. 1 (c) and Fig. 1 (f), we plotted three dimensional figures of the critical inclinations versus the semi-major axis and the eccentricity for the values mentioned in Fig. 1 (a), Fig. 1 (b), Fig. 1(d) and Fig. 1 (e) respectively.
In Fig. 2:
taking account of the dynamical shape parameters
In the Fig. 2 (c) and Fig. 2 (f), we plotted three dimensional figures of the critical inclination versus the semi-major axis and the eccentricity for the values mentioned in Fig. 2 (a) and Fig. 2 (b), and Fig. 2 (d) and Fig. 2 (e), respectively.
In Fig. 3:
taking account of the dynamical shape parameters
In Fig. 3 (c) and Fig. 3 (f), we plotted three dimensional figures of the critical inclination versus the semi-major axis and the eccentricity for the values mentioned in Fig. 3 (a), Fig. 3 (b), Fig. 3 (d) and Fig. 3 (e), respectively.
We can observe the following from the figures:
1) The orbital parameters especially the eccentricity and the semi-major axis do not affect so much the location of the critical inclination when we have considered only J2+C22, but when we consider J2+C22+C31, we have a number of families of critical inclinations as analyzed before. 2) Setting the argument of pericentre ω=90°, and the longitude of the ascending node Ω=270° yields one family of increasing critical inclinations. And we two families of decreasing critical inclinations when we set ω=270°, Ω=270°or ω=60°, Ω=18°, respectively. 3) The remarkable effect is that C31 is the responsible parameter for all these families of critical inclination.4) The reversed nature among some figures may be interpreted due to different selections of argument of pericentre. 5) For the same values of semi-major axes or the same eccentricities, we have different families of critical inclinations. These may be attributed to the passage of a satellite over different regions on the peculiar surface of the Moon with different mass distribution. It is quite clear from the result for the different choices of argument of pericentre and the longitude of the ascending node. 6) The slopes of all curves are not same which means the effects are of nonlinear nature.
Finally, we can collect our in the following Tables 1-2.
[Table 1.] Critical inclinations as seen from Fig. 1？ Fig. 3. Where 2000 km≤a≤4000 km, e=0.1 and 4000 km≤e≤10000 km, e=0.5.
Critical inclinations as seen from Fig. 1？ Fig. 3. Where 2000 km≤a≤4000 km, e=0.1 and 4000 km≤e≤10000 km, e=0.5.
[Table 2.] Critical inclinations as seen from Fig. 1 - Fig. 3. Where 0.1≤e≤0.4, a=4000 km and 0.4≤e≤0.6, a=10000 km.
Critical inclinations as seen from Fig. 1 - Fig. 3. Where 0.1≤e≤0.4, a=4000 km and 0.4≤e≤0.6, a=10000 km.
This study clearly shows the change in critical inclination values based on the investigation of the effect of the first tesseral harmonic coefficient,