# Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

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1 F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F Gm 1 m ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple, so if paticle 1 is attacted by paticles 2 and 3, the total foce on 1 is F 12 F 13. Cental foces ae consevative, we can define gavitational V F d d potential enegy: Gm 1 m 2 1 m 2 2 Define also gavitational potential (aka gavitational potential enegy Φ 1 pe unit mass) Gm (set m m and eplace m 2 with 1). Likewise, gavitational field g as gavitational foce pe unit mass: g ˆ Gm 2 g Field and potential ae elated in the usual way: Φ

2 2 Gavity Fom A Spheical Shell: Diect Calculation Conside a thin spheical m shell of adius a, mass pe unit aea ρ and total mass 2 4πρa Supeposition pinciple leads to element of mass: m dm ρ2πasinθadθ sinθdθ 2 Contibution to the potential fom annulus is: dφ Gdm R R Gm sinθdθ 2 Integate ove θ fom 0 to π. Change the integation 2 2 vaiable 2 fom θ to R, via R a 2a cosθ hence sinθdθr dr a

3 Gma a a If a integation limits ae a and a; if Φ a they ae and : Gm a 2a adr a Gm fo a fo 3 Gavitational g field by diffeentiation: a Gm ˆ 2 fo a 0 fo 1. Outside the shell, the potential is just that of a point mass at the cente. 2. Inside the shell, the potential is constant and so the foce vanishes.

4 4 Now Using Analogy With Coulomb Foce Apply integal fom of Gauss Law to gavitational case: g ds 4πG ρ m dv S V That is: to Suface integal of nomal component of gavitational field ove given suface S is equal 4πG times the mass contained within suface, with mass obtained by integating mass density ρ m ove volume V contained by S. Fom spheical symmety g gavitational field g must be adial: g ˆ Choose a concentic spheical suface with adius a: mass enclosed g is just shell mass m and Gauss Law says 4π 4πGm 2 g giving a. Gm ˆ fo 2 Likewise, choose concentic spheical suface inside shell: mass enclosed is zeo and g vanish.

5 F 5 Obits: Peliminaies Two-body Poblem: Reduced Mass Expess position i as CM location R plus displacement 1 ρ i R elative 1 to it: 2 R 2 ρ ρ F 1 Change vaiables fom 1 and 2 to R and 2 (ecall F 12 1 F 21, intenal 2 F foces only): m 1 m M 1 2 Set m m 2, thus M R m 1 m 2 i.e., CM velocity is constant. Conside elative displacement, F 1 m F m i.e., µ 1 m 1 and e-encounte educed µ mass m 1 1 m 2 2 m m 1 F m m 1 m 2 2

6 µ 6 Fo consevative foce F thee is potential enegy E V, hence total enegy MṘ2 is: µṙ2 V Likewise, when L F is cental, total angula momentum is R Ṙ M µ ṙ Since CM velocity is constant, R choose inetial fame 0 with oigin at R, i.e.: Hence: E L µṙ2 1 ṙ V 2 Rationale: two-body poblem educes to equivalent single body one of mass µ at position elative to fixed cente, acted F V ˆ upon by foce

7 7 2 If m m 1 µ : 1 2 R m 1 m 2 m 1 m m m 1 1 m m m 2 ( fixed Sun and moving Planet appoximation ). Commentay: 1. Appoximation 2 1 valid fo Keple s Laws (m m and m m Sun ). 2. Can ignoe inteactions between Planets in compaison to gavitational attaction Planet-Sun. Planet

8 0 8 Two-body Poblem: Conseved Quantities Gavity is cental foce: gavitational attaction between two bodies acts along line joining them. Gavitational foce on mass µ acts in diection L and no toque is exeted about fixed cente: constant Both magnitude and diection of L ae fixed! Since L p µṙ L is pependicula to plane defined by position and momentum of µ. Convesely: and p must always lie in fixed plane of all diections pependicula to L. Can theefoe descibe motion using plane pola coodinates θ, with oigin at fixed cente! k m 2 Radial and angula equations of motion become: θ 2 θ 2 adial equation 1 d equation dt angula F Gavitational foce is µ m 2 k M kˆ GMm wheein m Planet and m Sun.

9 9 Angula equation expesses L consevation of angula momentum: 2 θ m Othe conseved quantity is total enegy: k E 1 2 mṙ2 1 2 m2 θ 2 wheein gavitational V potential k enegy is Commentay: 1. Gavitational potential enegy will help deducing shape of planetay obits!

10 10 Two-body Poblem: Two Poblems Comet 1. Comet appoaching Sun in plane of Eath s obit (assumed cicula) cosses obit at angle of 60 tavelling at 50kms1. 2. Closest appoach to Sun is 110 of Eath s obital adius ( e ). µ m 3. Ignoe attaction of comet to Eath compaed to Sun (i.e., educed mass m Comet ). 4. Aim: compute comet s speed at point of closest appoach. Solution L p Key: angula momentum consevation mv of comet about Sun At point of closest appoach v comet s velocity must be tangential only: minv max v At cossing point: e vsin30 Equate two expessions: min v max 0 1 e v max ev 1 2

11 11 Finally, v max 5v 250kms1

12 G m 27 5 m 12 Cygnus X1 1. Cygnus X1 is a binay system of a supegiant sta of 25 sola masses and a black hole of 10 sola masses, each in a cicula obit about CM with peiod 5 6 days. 2. Aim: Detemine distance between supegiant and black hole, given sola mass kg. Solution Key: 2-body equation of motion in pola cood.s: Gm 1 m m ω 2 2 ω 2 m m 2 (m mass, distance, ang. velocity). Whee is RHS 2nd tem coming fom? T Intoduce peiod: 2πω Extact distance: 1 3 m 2 T 2 4π m 3 That is, m

13 13 Keple s Laws State Keple s Laws: The obits of the planets ae ellipses with the Sun at one focus. The adius vecto fom the Sun to a planet sweeps out equal aeas in equal times. 3. The squae of the obital peiod of a planet is popotional to the cube of the semimajo axis of the planet s obit (T 2 a 3 ). Next lectue: thei deivation

14 14 Keple s 2nd Law This is statement of angula momentum consevation unde action of cental gavitational foce. Angula equation 2 θ m of motion gives: L const Leads to: 2 θ da 1 L dt 2 2m const

15 15 Obit equation Ellipses ae specific to invese squae law fo foce, hence fist and thid laws ae specific to invese squae law foce. Study adial equation of motion (k GMm) 2 k θ m! 2 (i) Remove θ using angula momentum consevation, θ 2 Lm get 2 k m (ii) Use elation L 2 m 2 3 d θ dt dθ d L d m dθ 2 (diffeential u equation fo in tems of θ). (iii) Substitute 1 to 2 obtain u obit equation: d 2 u mk dθ L 2

16 16 Keple s 1st Law Solution of obit equation 2 1 is 1 mk L ecosθ l 0 e Fist law: fo 1 is an ellipse, with semi latus ectum L 2mk. Keple s 3d Law Stat with 2nd law fo ate of aea: da L dt 2m T A! Integate ove complete obital peiod T: 2mAL πab is aea of ellipse Substituting fo b in 2 tems of a3 a gives thid law: 4π 2 T GM

17 17 Scaling Agument fo Keple s 3d Law Suppose you found a solution to obit equation 2 2 θ km i.e., and θ as functions of t. Scale adial and vaiables by constants α and β: α t βt In tems of and t, LHS of obit equation is: d dt 2 dt 2 β dθ α α α θ θ β 2 β RHS becomes: k 1 m k α m Compae two sides, new 2 solution 3 in tems of and t povided β α That is, T 2 a 3 1. Need solving obit equation fo popotionality constant. 2. Scaling agument makes clea thid law based on invese-squae foce law.

18 18 Poblem Sheet 6 Section B. Eath s speed in cicula motion about Sun!? Obit equation in pola coodinates: 0 l 1! e That is, e l L mk G 2 m 2 v 2 e 2 e m 2 M Sun Inveting, Finally, GM e v 2 GM v 2 e e Sun 2 e Sun e Eveything can be expessed in tems of v e!

19 19 Enegy Consideations: Effective Potential Gavitational foce is consevative, hence total enegy E of obiting body is conseved: V E 1 2 mṙ2 1 2 m2 θ 2 Angula momentum is also conseved (foce is cental), hence use 2 θ Lm to emove θ 2 : E 1 2 mṙ2 L 2 2m 2 V Fomally, enegy equation of paticle in linea motion unde effective U potential 2 V L 2 2m 2 Effective potential contains centifugal tem, L 22m aising because V angula momentum l is conseved. Replace k and use L 2mk: k Fig U kl 2 2

20 mk 20 U 2 Intepet as a function of fo given E By definition ṙ E U 0, implying 2 kl U k 2 Daw a hoizontal line fo E, lies below it! U Cicula Obit At minimum c l E, is constant at E 2mk L hence obit is cicula and total enegy is k2l 2 22L Elliptic Obit If k2l E 0, motion p is a allowed fo peihelion p E 2 and aphelion a k given by oots of kl2 E Paabolic Obit If 0, thee is always minimum value fo but escape to infinity is just possible. E Hypebolic Obit Fo 0, escape to infinity is possible with finite kinetic enegy at infinite sepaation.

21 21 Obits in a Yukawa Potential Conside V Yukawa potential 0 κ 0 αeκ α Descibes, e.g., attactive foce between nucleons in an atomic nuclei. Neglect quantum-mechanics and use classical dynamics. Effective potential U becomes 2 L 2 2m αeκ Tajectoies ae moe complicated: Fig.

22 22 U Intepet E as a function of fo given E 0 but geate than U min Rosette obit, i.e., ellipse with otating oientation, aka pecession of peihelion. Typical of small (κ 0) petubations of planetay obits, e.g., due to othe planets (iegulaities in Uanus motion led to discovey of Neptune, 1846). Lage limit Tem L 22m U 2 dominates exponentially falling Yukawa tem, so becomes positive! If U max E 0, two possible obits classically distinct. In quantum mechanics, tunnelling becomes possible (e.g., alpha decay)!

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