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A Study on Phases of Moon,Chandrayaan-Ii ,Sidereal Month and Synodic Month The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 A STUDY ON PHASES OF MOON,CHANDRAYAAN-II ,SIDEREAL MONTH AND SYNODIC MONTH Dr.K.Kalaiarasi ¹ and D.Haritha² 1.Pg and Research Department of Mathematics,Cauvery college for women,Trichy-18,Tamilnadu,India. [email protected] 2.Pg and Research Department of Mathematics,Cauvery college for women ,Trichy-18,Tamilnadu,India [email protected] ABSTRACT: In this resaearchpaper,a study on phase of moon,introduction of chandrayaan,siderealmonthsynodic month and their relationship are discussed. KEYWORDS : Phases of moon,siderealmonth,synodicmonth,history of chandrayaan ,composite of chandrayaan-II. INTRODUCTION: Chandrayaan-2 (candra-yāna, transl. "mooncraft";[15][16] pronunciation (help·info)) is the second lunar exploration mission developed by the Indian Space Research Organisation (ISRO),[17][18] after Chandrayaan-1.[19][20] It consisted of a lunar orbiter, the Vikram lander, and the Pragyan lunar rover, all of which were developed in India.[21] The main scientific objective is to map and study the variations in lunar surface composition, as well as the location and abundance of lunar water.[22][23] The mission was launched on its course to the Moon from the second launch pad at SatishDhawan Space Centre on 22 July 2019 at 2.43 PM IST (09:13 UTC) by a Geosynchronous Satellite Launch Vehicle Mark III (GSLV Mk III).[9][10][24] The craft reached the Moon's orbit on 20 August 2019 and began orbital positioning manoeuvres for the landing of the Vikram lander.[25]Vikram and the rover were scheduled to land on the near side of the Moon, in the south polar region[26] at a latitude of about 70° south at approximately 20:23 UTC on 6 September 2019 and conduct scientific experiments for one lunar day, which approximates two Earth weeks. However, the lander deviated from its intended trajectory starting at 2.1 kilometres (1.3 mi) altitude,[27] and had lost communication when touchdown confirmation was expected.[28][29] Initial reports suggesting a crash[30][31] have been confirmed by ISRO chairman K. Sivan, stating that the lander location had been found, and "it must have been a hard landing".[32] Volume XI, Issue X, October/2019 Page No:380 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 Both ISRO and NASA tried unsuccessfully to communicate with the lander for two weeks before the lunar night set in. The orbiter, part of the mission with eight scientific instruments, remains operational and is expected to continue its seven-year mission to study the Moon. Chandrayaan-2 Chandrayaan-2 composite Mission type Lunar orbiter, lander, rover Operator Indian Space Research Organisation (ISRO) SATCAT no. 2019-042A Website www.isro.gov.in/chandrayaan2-home-0 Mission duration Orbiter: ~ 7 years Elapsed: 23 days Vikram lander ≤ 14 days[1][2] Volume XI, Issue X, October/2019 Page No:381 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 Pragyan rover: ≤ 14 days[2] Spacecraft properties Manufacturer Indian Space Research Organisation (ISRO) Launch mass Combined (wet): 3,850 kg (8,490 lb)[3][4][5] Combined (dry): 1,308 kg (2,884 lb)[6] Orbiter (wet): 2,379 kg (5,245 lb)[4][5] Orbiter (dry): 682 kg (1,504 lb)[6] Vikram lander (wet): 1,471 kg (3,243 lb)[4][5] Vikram lander (dry): 626 kg (1,380 lb)[6] Pragyan rover: 27 kg (60 lb)[4][5] Power Orbiter: 1 kW[7] Vikram lander: 650 W Pragyan rover: 50 W Start of mission Launch date 22 July 2019, 14:43:12 IST (09:13:12 UTC)[8] Rocket GSLV Mk III[9][10] Launch site SatishDhawan Space Centre Second Launch Pad Contractor Indian Space Research Organisation (ISRO) Moon orbiter Orbital insertion 20 August 2019, 09:02 IST (03:32 UTC) [11][12] Orbital parameters Periapsis altitude 100 km (62 mi)[13] Apoapsis altitude 100 km (62 mi)[13] Inclination 90° (polar orbit) Moon lander Volume XI, Issue X, October/2019 Page No:382 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 Spacecraft component Rover Landing date 7 September 2019, 01:53 IST (6 September 2019, 20:23 UTC)[14][12] Chandrayaanprogramme ← Chandrayaan-1 Chandrayaan-3 → History : On 12 November 2007, representatives of the Russian Federal Space Agency (Roscosmos) and ISRO signed an agreement for the two agencies to work together on the Chandrayaan-2 project.[33] ISRO would have the prime responsibility for the orbiter and rover, while Roscosmos was to provide the lander. The Indian government approved the mission in a meeting of the Union Cabinet, held on 18 September 2008 and chaired by Prime Minister Manmohan Singh.[34] The design of the spacecraft was completed in August 2009, with scientists of both countries conducting a joint review.[35][36] Although ISRO finalised the payload for Chandrayaan-2 per schedule,[37] the mission was postponed in January 2013[38] and rescheduled to 2016 because Russia was unable to develop the lander on time.[39][40] Roscosmos later withdrew in wake of the failure of the Fobos-Grunt mission to Mars, since the technical aspects connected with the Fobos-Grunt mission were also used in the lunar projects, which needed to be reviewed.[39] When Russia cited its inability to provide the lander even by 2015, India decided to develop the lunar mission independently.[38][41] The spacecraft's launch had been scheduled for March 2018, but was first delayed to April and then to October to conduct further tests on the vehicle.[42][43] On 19 June 2018, after the program's fourth Comprehensive Technical Review meeting, a number of changes in configuration and landing sequence were planned for implementation, pushing the launch to the first half of 2019.[44] Two of the lander's legs got minor damage during one of the tests in February 2019.[45] Chandrayaan-2 launch was initially scheduled for 14 July 2019, 21:21 UTC (15 July 2019 at 02:51 IST local time), with the landing expected on 6 September 2019.[20] However, the launch was [46][8][47] aborted due to a technical glitch and was rescheduled. The launch occurred on 22 July 2019 at 09:13 UTC (14:43 IST) on the first operational flight of a GSLV MK III M1.[48 THE MOON: THE Moon is at a distance of about 2,40,000 miles (384403 km) from the earth and as such it is the nearest celestial neighbor to the earth. The linear diameter of the moon is about 2163 miles (3476km) and its mass is about 1 times that of the earth, the 81 angular diameter of the moon is about 30’ and its mean horizontal parallax is about 57’ Volume XI, Issue X, October/2019 Page No:383 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 The moon is not a self-luminous body. It shines in the light it receives from the sun. The moon is a satellite of the earth. It moves around the earth following the laws of Kepler. 1 The lunar orbit is at an angle of about 5 ̊8’ to the ecliptic and is eccentricity is . The two points of 18 intersection of the lunar orbit and ecliptic are called the nodes of lunar orbit. The point where the moon crosses the ecliptic in going north is called the ascending node* and the other point where the moon crosses the ecliptic in going south is called the descending node*. The line joining the nodes of the lunar orbit is called the nodal line and it is the line of intersection of the planes of the orbit of the moon and the ecliptic. PHASES OF THE MOON : The phases of the moon are different forms in which the moon is seen by terrestrial observer. The phase of the moon is mathematically defined as the ratio of visible of moon’s disc to the whole illuminated disc. Let S be the sun E the earth and M the Centre of Moon.Moon receives light form the sun in the direction SM and its hemisphere ACB turned towards the sun is illuminated.The moon is observed from the earth in the direction EM so that the visible portion is the projection of the illuminated portion on the plane LN perpendicular to the line sight of sight EM. Let D be the foot of the Perpendincular from A on this plane sothat the visible portion is the area XLYDX. =semicircle XLY-semi ellipse XDY Volume XI, Issue X, October/2019 Page No:384 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367 휋푎² 휋푎푏 = - 2 2 Where a is the radius of moon and b=MD, Since the distance between the earth and the moon is very small compared to the distance of moon or earth from the sun,the earth also receives light from the sun in the same direction as the moon does. Let 휃 be the elongation of moon Clearly we get ∠퐴푀퐷 From ∆ AMDMD=AMcos휃 i.e. b=a cos 휃 휋푎² 휋푎2 cos 휃 ∴ visible portion = - 2 2 Whole illuminated disc = πa² 풗풊풔풊풃풍풆 풑풐풓풕풊풐풏 Phase of the moon = 풘풉풐풍풆 풊풍풍풖풎풊풏풂풕풆풅 풅풊풔풄 흅풂² 흅풂ퟐ 퐜퐨퐬 휽 − ퟐ ퟐ = 흅풂² i.e phase=ퟏ−퐜퐨퐬 휽 ퟐ successive phases of moon : As the moon moves along its orbit around the earth its hemisphere turned towards the sun is illuminated. For an observer on the earth only that part of the hemisphere turned towardsthe earth is visible. Therefore the whole illuminated portion of the moon is not visible and it is seen in different shapes.
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