Satellite Orbits. Orbital Mechanics. Circular Satellite Orbits

Transcription

1 Obitl Mechnic tellite Obit Let u tt by king the quetion, Wht keep tellite in n obit ound eth?. Why doen t tellite go diectly towd th, nd why doen t it ecpe th? The nwe i tht thee e two min foce tht ct on tellite o ny object tht i in obit ound th (o ny othe plnet o t). Thee foce e clled 1. Centifugl foce: Thi foce ct on the tellite long the line between the tellite nd the cente of m of th in the outwd diection. Thi eult fom the fct tht the object moe nd object in motion tend to tel in tight line.. Centipetl foce: Thi foce ct on the tellite long the line between the tellite nd the cente of m of th in the inwd diection. Thi eult fom the gittionl foce exeted on the tellite by th. In ddition to the boe two foce, thee i le impotnt foce tht my lte the pth of motion of tellite including:. Atmopheic dg: Thi foce my be ignificnt fo tellite tht e t LO obit, epecilly t ltitude le thn 00 km o o. The effect of thi foce eult in lowing the tellite motion nd hence lteing it obitl ltitude. 4. Moon nd ol gittion: Although tellite e uully plced in obit tht e much cloe to th thn to th moon nd knowing tht the gittionl pull of the Moon i itelf much lowe thn the gittionl pull of th, tellite motion neethele i ffected by the Moon (nd un ) gitie jut like e wte i ffected by Moon gity eulting in tide. The effect of thee gitie eult in the tellite wobbling (moing up nd down lowly) while in obit. 5. ol wind nd comic dition: thee foce he mll effect on the motion of tellite. Cicul tellite Obit Conideing the fit two foce only fo the time being, the following figue illutte how thee foce ct on n object in obit ound th:

2 Diection of Motion Centifugl Foce R tellite Centipetl Foce th The Centipetl Foce due to gity of th ( F ) i equl to F M = m whee i clled the Uniel ittionl Contnt, i the m of tellite, th. The Uniel ittionl Contnt i equl to M i the m of th, m i the diu of ottion of the tellite ound the cente of ( N ) m 11 = 6.67*10, kg m but Newton N = kg. Thi mke become m kg m = 6.67*10 kg = 6.67* m kg. The boe contnt ue the unit of (m), but ince ditnce between plnet nd tellite e uully meued in (km), o the we will ue i = 6.67*10 0 km kg. The m of th i equl to

3 M = 5.974*10 4 kg Often, the diu of tellite ottion i gien in tem of tellite ltitude (height boe th ufce) inted of diu. In thi itution, the tellite diu become equl to the ddition of tellite ltitude with th diu ( ): = + whee th diu = km Note tht thi i the ege diu of th. The othe min foce ffecting the tellite (centifugl foce) i gien by FC = m.(cetifegl cceletion) = m whee i the peed of motion of the tellite in obit, i the diu of tellite ottion. m i the m of the tellite, nd Now, uing the fomul fo the boe two foce, we ee tht tellite will emin in fixed cicul obit if the two foce e equl to ech othe, othewie, the tellite will not tel in cicul obit nd my poibly hit th ufce o ecpe th gity. Uing thi, we get the following: M m m = Thi gie fomul fo eluting the needed peed of tellite to mintin cicul obit ound th in tem of the diu of tellite obit o the oppoite: M M = = OR The quntity µ = M i clled Kepple Contnt. Thi quntity i n impotnt quntity tht i found when dicuing object obit th (o plnet in genel). Impotnt Note 1: When uing the boe eqution o ny othe eqution to elute ome quntity, it i ey impotnt to be conitent with unit (i.e., ue the pope unit of the diffeent quntitie uch tht the imil unit in diffeent quntitie e the me (km. km, o kg. kg, nd o on)). o, if one of the quntitie i gien in (km) while the othe quntity i gien in unit of (m), one of them h to be coneted to the othe.

4 Impotnt Note : Becue the diection of the Centipetl nd Centigl foce e oppoite to ech othe nd both of the them ct in pependecul diection to the diection of motion of the tellite, they do not eult in chnging the peed of motion of the tellite (i.e., the peed of tellite emin contnt t ll time). ince tellite in cicul obit ound th with diu tel oe cicle, the totl ditnce telled in complete obit i ( π ). o, the peiod of ottion i T π = = π M econd xmple 1: NAA Hubble pce telecope i plced in cicul obit ound th t n ltitude (height boe th ufce) of 595 km. Detemine: ) The peed of the tellite in obit, b) The tellite obitl peiod. xmple : Detemine: ) The diu of geo-ynchonou o geo-ttiony tellite obit b) The peed of tellite in geo-ynchonou obit. llipticl tellite Obit ee The boe dicuion conide cicul tellite obit, which e only ubcl of tellite obit. The genel fom of tellite obit i n ellipticl obit, which men tht the ditnce fom the tellite to the cente of th (o to the ufce of th) chnge continuouly the tellite moe in obit. An ellipticl obit h two focl point ( comped to ingle focl point in the ce of cicul obit which the cente of the cicle). th i locted in one of the focl point of n ellipticl obit.

5 b 0 φ 0 (1 + e) e P = (1 e) Whee The fit peon to depict the ellipticl motion of plnet ound the un w Keple who cme up with hi lw of plnety motion in the Thee lw pply to ny object of negligible m tht obit much lge m it i the ce fo tellite obiting th. Newton then confimed tht Keple lw e comptible with hi gittionl lw. Keple lw tte the following: 1. The obit of ny mll body bout lge body i lwy n ellipe with the cente of m of the lge body one of the two focl point of the ellipticl obit.. The obit of the mlle body weep equl e of the ellipe in equl time intel.. The que of the peiod of eolution i gien by 4π T = µ whee km 5 µ = M =.986*10. Thee lw llowed the deition of the loction of the tellite t diffeent point duing it obit uing the following et of eqution Ditnce between tellite nd th 0 ( 1 e ) = 1 + e co ( φ ) 0 Reltion between the emi-mjo xi, emi-mino xi, nd eccenticity

6 b e = 1 b e = 1 The elocity of the tellite The obitl peiod µ = 1 0 4π T = µ xmple : A tellite obit th in n ellipticl obit = 15 km nd e = 0.. The plin of tellite obit i the me plne of th equto nd the diection of tellite obit i in the me diection of th obit (i.e., the tellite obit fom W to ). Find the following: ) If the tellite w on longitude 6 when it w in it peigee point, oe which longitude will it be when it eche the next pogee? b) The peed t which the tellite will be teling t the pogee nd peigee point c) The peed of the tellite when it height i height boe th ufce i 4000 km. d) The height of the tellite boe th ufce when it peed i 4.5 km/.