Newton s Law of Gravity and Orbits of Planets & Satellites

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Bathsheba McCormick
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1 he Uniesal Law of Gaitation Newton s Law of Gaity and Obits of Planets & Satellites Newton s Uniesal Law of Gaitation states that any two point asses attact each othe with a foce popotional to the poduct of thei asses and inesely popotional to the squae of the distance between the. he foce of uniesal gaitation is a ey weak foce and only becoes noticeable when at least one of the objects is exteely assie. he foce acts equally on both asses. G whee gaitational foce on each ass (N) Whee G N kg - is uniesal gaitational constant,, ae the point asses, is the distance between the asses. he diection of this foce is always along the line between the two asses. If one object is spheically syetic, the foce exeted on anothe object outside the sphee is exactly as if the whole spheical ass wee concentated at the cente. Gaitational ield A gaitational field is the foce field ceated aound an object that causes gaitational attaction of othe objects. he gaitational field stength at a point in a gaitational field is the foce pe unit of ass of an object placed at that point. o Newton s Second Law of Motion, we can egad "foce pe unit ass" as being equialent to acceleation. heefoe, gaitational field stength is anothe nae fo acceleation due to gaity. M G M g G, whee is the distance of the test ass fo the cente of the ass M ceating gaitational field. he unit of gaitational field stength is the sae as that of acceleation, i.e. N/kg, o s -.

2 o the gaitational field ceated by ath, when <, the gaitational field g is popotional to the distance of the test ass and the cente of ath. When ~, the gaitational field g 9.8s -. When >, the cue is an inese squae law. Shell heoe Shell theoe states the following:. A unifo spheical shell of atte attacts a paticle that is outside as if all the shell s ass wee concentated at its cente.. A unifo shell of atte exets no net gaitational foce on a paticle located inside the shell. Applying the shell theoe to the gaitational field ceated by ath, we can see that fo < (points inside the ath), the gaitational field stength is diectly popotional to the distance fo the cente of ath. By the shell theoe, the asses outside the test ass gie no foce to the test ass, and the gaitational foce is solely esulted fo the eaining ass 4 M V ρ π ρ (eebe that the olue of a sphee is 4 π ), whee ρ is the aeage density of the ath. Howee, the ass M can be egaded as a point ass located at the cente of ath, hence can expess the gaitational field as g 4 G( π ρ) 4 4 M Gπ ρ Gπ ( 4 π ) As shown in the gaph below, when <, the gaitational field g is popotional to the distance. g ath's Suface distance fo the cente of ath

3 of the test ass and the cente of ath. When >, the cue is an inese squae law. Gaitational Potential negy he gaitational potential enegy of a syste consisting of asses and M with a distance between thei centes is M U G he negatie sign eflects on the fact that the gaitational foce is attactie. U is equal to the wok done of the gaitational foce to oe a ass fo a point easued fo the cente of the ass M to infinity. Note that U appoaches zeo as appoaches infinity. Satellite Obits he centipetal foce fo the satellite in its obit is the gaitational attaction. Since the gaity can be consideed the only foce acting on the satellite, by Newton s Second Law: g a c, whee a c and g. hus, Multiplying both the nueato and denoinato of this expession by and eaanging, we get g Satellite ath Obit o exaple, if the obit of the satellite is 00 k fo the ath s suface, and consideing that the ath s adius is 6400 k, the speed of the satellite can be found as

4 6 (6.4 0 ) 9.8( ) (6.4 0 ) / s he tie fo the satellite to ake one coplete obit of the eath is π ; 86 inutes Mechanical negy of a Satellite Conside a satellite oing along a cicula obit with tangential elocity at a distance fo the cente of eath. Satellite ath t he potential enegy of the satellite is U ) (. he total echanical enegy K.. + P.. K.. + U() Hence, we hae.

But the centipetal foce is poided by the gaitational attaction which gies the elocity, i.e.. inally, we obtain the total echanical enegy ( ) Launching a Satellite he satellite can escape fo the ath whenee it has enough enegy.

5 But the centipetal foce is poided by the gaitational attaction which gies the elocity, i.e.. inally, we obtain the total echanical enegy ( ) Launching a Satellite he satellite can escape fo the ath whenee it has enough enegy. scape in this context eans that the satellite gets to the point whee the gaitational foce is appoaching zeo. he potential enegy of the satellite at this point is zeo as well. P.. 0 Satellite ath P.. On the suface of ath, the potential enegy is k P.. and its kinetic enegy is. hen, the iniu speed fo the satellite to escape can be found fo the Law of Conseation of negy: 0. (We assue that the kinetic enegy of the satellite at the distant point is zeo as well.)

6 ( ) g / s / k s Keple s Laws In 609, Keple, a Gean astonoe, published his thee laws on the otion of planets:. he planets oe in elliptical obits with the sun at one focus.. he line connecting a planet to the sun sweeps out equal aeas in equal ties.. o eey planet, the atio of the cube of the aeage obital adius to the squae of the peiod of eolution is the sae, i.e. constant. peihelion A planet B M aphelion S Sun L As aea A aea B, then the tie it takes to oe fo to S is the sae as to oe fo L to M. he constant fo the thid law coes fo 4π 4π hat is, fo a gien cental body of ass M, the atio satellites. is constant and equal to fo all its 4π

Determining solar characteristics using planetary data

Determining solar characteristics using planetary data Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation