where t is the time of flight, C is a constant having the dimension of velocity cm/s, G is the gravitational constant 6,67.10-8 cm3/G2, M is the mass of the Moon 0,735.1026 g, r – average orbital distance (the semi major axis) at the beginning of the flight, v1 and v2 are the initial and final orbital velocity, calculated at an average orbital distance.
Returning to the question of the effect of aspheric of the moon on the braking of its satellites, note that in equation (3) expressing the time of flight the satellites is no their mass. This confirms the previously made conclusion about the independence of the flight time from the mass of the satellite.
The constant C in equations (1) and (2) if you follow the accepted analogy with electrodynamics, by definition, is the speed of gravitational radiation. Thus, equation (3) can be used to calculate dynamic gravitational constant, i.e. the velocity of propagation of gravitational waves.
The constancy of the constants when calculating for different satellites will confirm the correctness of the methodological approach. Below is data for the calculation of the constants for the evolution of the orbits of the fife satellites of the Moon, including the Soviet satellite Luna-10", American satellite "the lunar Prospector", a satellite of the European space Agency's Smart-1", as well as Japanese and Indian satellites "Kaguya" and "chandrayan-1.
Consider the launch and flight of Sputnik "Luna-10". First, "Luna-10" was put into orbit an artificial satellite of the Earth. Then, using the upper stage, the speed of the station was reduced to 10.9 km/s. At that speed, the duration of the flight to the Moon was slightly less than three and a half days.
Then was the correction of the trajectory, after which the station entered the sphere of gravitational influence of the Moon.
At the final stage of the flight (800 km from the Moon) station has been previously appropriately focused and calculated point remote from the surface of the moon for 1000 km was included braking engine unit and the speed was reduced from 2.1 to 1.25 km/s, which provided the transfer station under the action of the attraction of the Moon with the span of the trajectory on selenocentric orbit with the following parameters: the greatest distance from the surface of the Moon – 1017 km (apocenter 2,755.108 cm); smallest – 350 km (pericenter putting on 2,088.108 cm); the average distance (the semimajor axis) – 2,422.108 cm; average orbital speed – 1,4229.105 cm/s; period of revolution around the moon – 2 hours 58 minutes 15 seconds; the angle of inclination of the satellite's orbit to the plane of the lunar equator – 71° 54. The mass of the spacecraft after separation from the booster was 1582 kg, the mass of the lunar satellite 240 kg
Artificial satellite of the Moon "Luna-10" there were active 56 days (0,0484.108 (s) having 460 revolutions around the Moon. After the batteries have been depleted, the relationship was terminated on May 30, 1966. Orbit at this time had parameters: minimum destruction of 378 km (pericenter 2,116.108 cm), the greatest destruction of 985 km (apocenter 2,723.108 cm and an inclination of 72.2 degrees. The average distance (the semi major axis) – 2,420.108 cm. Average orbital speed – 1,4235.105 cm/s.
Substituting the given data into the formula (3), find the value of the constant C = 3,694.108 cm/s Calculated data are presented in Table 1. Perform similar calculations for other travelers of the Moon.
Table 1. The calculation of the duration of the flight, the constants C and braking force to the satellites of the Moon.
Accordingly, the orbital velocity at the beginning of the highlighted portion of the orbit v1 = 1,665.105 cm/s and at the moment of falling v2 = 1,680.105 cm/s Substituting the above values in the formula 3, we get the value of the constants C = 2,25.108 cm/s, that is close in order of magnitude to the value of the constants calculated for the satellite "Luna-10. The satellite of the moon, Smart-1 (Smart-1: the acronym for Small Mission for Advanced Research in Technology) launched by the European space Agency September 30, 2003 [8]. Initially, it was launched into an elliptical low earth orbit typical of telecommunication satellites with the help of the rocket Ariane-5. Then the output on the lunar orbit was carried out using a low-power (thrust force of 0.07 N) ion propulsion and lasted 16 months.
After moving into the area of the gravity of the Moon and the braking propulsion system on November 11, 2004 "Smart-1" has been translated into lunar orbit. The mass of the satellite 367 kg After number of maneuvers in the period from 28 February to July 18, 2005 the satellite was in free flight, that is, without the inclusion of the propulsion system. The orbital parameters at the beginning of this period: apocenter 4,6182.108 cm and the pericenter 2,2087.108 cm. The average distance (the semi major axis) 3,4134.108 cm After a flight during 0,121.108s apocenter decreased to 4,4957.108 cm, and the pericenter increased to 2,3493.108 cm. The average distance decreased to 3,4025.108 cm.
Orbital speed at the beginning and end of the free flight accordingly was 1,1984.105 and 1,2004.108 cm/s Substituting the obtained values of the average distance in the beginning of the period of free flight and orbital velocity at the beginning and end of the flight in the formula 1, we get the value of the constant C = 1,91.108 cm/s, which is close enough to the values previously given for satellites "Luna-10" (3,69.108 cm/s) and the lunar Prospector" (2,25.108 cm/s).
Japanese satellite of the Moon "Kaguya" was launched on 14 September 2007 with the Japanese Baikonur Tanegasima using booster h-2A (H-2A) [9]. The mass of the satellite 3000 kg. To the orbit of the moon it was only appear on 4 October 2007. After separation of the two auxiliary satellites, test equipment and instruments basic core ("Main orbiter") mass 2 27 1kg December 2007 began their regular observations on polar circular orbit with altitude of 100 km (the distance from the center of 1,838.108 cm, orbital speed 1,6332.105 cm/s).
The time of the flight without the inclusion of the propulsion system lasted until June 11, 2009, that is 0,466.108 s. At the point of activation of the brake motor installation altitude was 27.8 km (the distance from the center of 1,776.108 cm), which corresponds to the orbital velocity 1,6661.105 m/s. Then, after 6 minutes the connection with the satellite was lost. Substituting the values of change of orbital parameters in the formula 1, we get the value of the constant C = 2,34.108 cm/s, very close to the values previously calculated for other satellites.
Indian space research organization (ISRO,) reported [10] about the launch of 22 October 2008 on a circumlunar orbit of his device
"Chandrayan-1 using developed in Indian rocket PSLV–XL (PSLV – Polar Satellite Launch Vehicle from Baikonur Satish Dhawan. Starting weight station was 1380 kg, weight station in lunar orbit – 523 kg.
After a series of maneuvers November 4, the station went on the flight path to the Moon and on 8 November reached the environs of the Moon, where at a distance of 500 km from the surface was included brake motor, resulting in the station moved to a transitional circumlunar orbit resettlement 504 km, aposelene 7502 km and an orbital period of 11 hours. Then on 9 November, after adjustment of the pericenter of the orbit was lowered to 200 km. On November 13, the station was transferred to the circular working circumlunar orbit with altitude of 100 km (1,838.108 cm from the center of the Moon), a cycle time of 120 min, the orbital speed 1,6332.105 cm/s.
On August 29, 2009 ISRO announced that radio contact with the satellite was lost. By the time of the loss of communication with the satellite, it stayed in orbit 312 days (0,27.108 (s) and managed to make a 3400 revolutions around the Moon.
Indian space research organization claims that her device will be in lunar orbit for another 1000 days. The lack of data on the orbital parameters after braking satellite Chandrayaan-1 does not allow the calculation of the constant C. However, determining the average value for other satellites, using equation (3) to confirm or refine the prediction of the lifetime of the satellite "Chandrayan-1.
The average value of the constant C it is advisable to calculate on three.satellites: "the lunar Prospector", "Smart-1" and "Kaguya". It is of 2.16.108 cm/s. The large deviation of the satellite