m 2 r 1 = m 1 + m 2 M r 2 = m 1 = m 1m 2

Transcription

1 Celestial Mechanics - A.A Calo Nipoti, Dipatimento di Fisica e Astonomia, Univesità di Bologna 26/3/ The gavitational two-body poblem 2.1 The educed mass [LL] Two-body poblem: two inteacting paticles. Lagangian L = 1 2 m 1 ṙ m 2 ṙ 2 2 V ( 1 2 ) Now take the oigin in the cente of mass, 1 m m 2 = 0, and define 1 2, 1 = m 2 m 1 + m 2 Substituting these in the Lagangian, we get 2 = m 1 m 1 + m 2 L = 1 2 µ ṙ 2 V (), whee µ m 1m 2 = m 1m 2 m 1 + m 2 M is the educed mass, whee M m 1 + m 2 is the total mass. So the two-body poblem is educed to the poblem of the motion of one paticle of mass µ in a cental field with potential enegy V ().

2 2 Lauea in Astonomia - Univesità di Bologna 2.2 Keple s poblem: integation of the equations of motion [LL; R05] In the case of gavitational potential V ( 1 2 ) = Gm 1m 2 1 2, V () = Gm 1m 2 = Gµ M A widely used notation in celestial mechanics is µ GM = G(m 1 +m 2 ), whee µ is called the gavitational mass. Note that µ does not have units of mass, while µ and M have units of mass. The poblem is educed to the motion of a paticle of mass m = µ in a cental field with potential enegy 1/: this is known as Keple s poblem. Newtonian gavity: attactive. Coulomb electostatic inteaction: attactive o epulsive. Let s focus on the attactive case = V = α/ with constant α > 0. We ae descibing the motion of a paticle m moving in a cental potential V = α/. In the case of the gavitational two-body poblem m = µ (educed mass) and α = G(m 1 + m 2 )µ = GMµ = µµ We have seen that fo motion in a cental field, the adial motion is like 1-D motion with effective potential enegy V eff () = α + L2 2m 2, See plot of V eff and enegy levels (See fig 10 of LL; FIG CM2.1) Minimum of V eff at = L 2 /mα. V eff,min = α 2 m/2l 2. Motion is possible only when E > V eff. If E < 0 motion is finite. If E > 0 motion is infinite. Path: φ = φ(). Take and substitute V = α/. dφ = Ld 2 2m[E V eff ()] = Ld 2 2m[E V ()] L2 2 We get which can be integated analytically to obtain: φ = accos Ld dφ = 2m [ 2 E + α (L/) (mα/l) 2mE + m2 α 2 L 2 ] L 2 2, + φ 0 = accos L 2 mα EL2 mα 2 + φ 0,

3 Celestial Mechanics - A.A with φ 0 constant (veified by diffeentiation). Note that d dx accos x = 1 1 x 2 Defining l L 2 /mα and e 1 + (2EL 2 /mα 2 ) we get whee f = φ φ 0 is called the tue anomaly. l = 1 + e cos f, This is the equation of a conic section whee l is the semi-latus ectum and e is the eccenticity. is the distance fom one focus. φ 0 is such that φ = φ 0 at the peicente (peihelion). In the two-body poblem each obit is a conic section with one focus in the cente of mass (see plot of conic sections: fig. 2.4 of MD; FIG CM2.2). We have seen that fo motion in a cental field the time dependence of the coodinates is given by dt = md, 2[E V ()] L2 m 2 which fo a Keple potential can be integated analytically (see below, fo instance fo elliptic obits). Depending on the sign of E (and theefoe on the value of e) we distinguish: - E < 0 (e < 1): elliptic obits - E = 0 (e = 1): paabolic obits - E > 0 (e > 1): hypebolic obits Elliptic obits [LL; R05; MD] E < 0 = e < 1 = elliptic obit. Bief summay of popeties of the ellipse (see fig. 4.3 of R05; FIG CM2.3). S focus, S othe focus C cente, P any point on ellipse, CA = a, CB = b, CS = ae, SQ = l, SP/P M = e < 1 (eccenticity), a = l/(1 e 2 ) SP + P S = 2a, x 2 a 2 + y2 b 2 = 1 (Semi majo axis) b = l/ 1 e 2 (Semi mino axis), b = a 1 e 2 (I)

4 4 Lauea in Astonomia - Univesità di Bologna Fom the elations among e, l, L and E, we get a = l/(1 e 2 ) = α/2 E b = l/ 1 e 2 = L/ 2m E (II) (III) Peicente and apocente. We ecall that the equation fo the distance fom one of the focus is = l 1 + e cos φ = a(1 e2 ) 1 + e cos φ, whee we have assumed φ 0 = 0, f = φ. Theefoe the apocente (cos φ = 1) is apo = a(1 + e) and the peicente (cos φ = 1) is pei = a(1 e). We have seen that fo motion in a cental field the sectoial velocity da/dt is constant (Keple s second law). Keple s thid law. Using consevation of angula momentum we get peiod T fo elliptic obit: whee A = πab is the aea of the ellipse. L = m 2 φ = 2m da dt = const Ldt = 2mdA = T L = 2mA = 2mabπ, = T = 2πa 3/2 m/α = πα m/2 E 3, which is Keple s thid law T a 2/3. Note that peiod depends on enegy only. We have used definitions of a, b and L as functions of l (semi-latus ectum): a = l/(1 e 2 ) = α/2 E, b = l/ 1 e 2 = L/ 2m E, L = mαl. In the case of the gavitational two-body poblem we have m = µ and α = GMµ, T 2 GM = 4π 2 a 3 o GM = n 2 a 3, with n 2π/T is the mean motion (i.e. the mean angula velocity), e = E = GMµ 2a 1 + 2EL2 G 2 M 2 µ 3 = 1 L 2 GMa whee L L/µ = 2 φ is the modulus of the angula momentum pe unit mass. In the limit m 2 m 1 (whee m 1 and m 2 ae the masses of the two bodies, fo instance Sun and planet), then µ m 2, M m 1, 1 0, 2, we have Keple s fist law: the obit of each planet is an ellipse with the Sun in one of its foci.

5 Celestial Mechanics - A.A Using the expession fo the obital enegy we can elate the velocity modulus v to and a as follows: E = T + V = 1 2 µ v 2 GMµ = GMµ 2a, o v 2 = GM a = ( 2 1 ), a ( ) 1 2 v2. GM Keple s equation Fom the time dependence of adial coodinate (see above) we have, in the case of elliptic obit: dt = d 2 E /m 2 + α/ E L 2 /2m E, Note that and L 2 2m E = b2 = a 2 (1 e 2 ) = a 2 a 2 e 2 α/ E = 2a, because α = 2a E dt = d 2 E /m a 2 e 2 ( a) 2. Let us intoduce the angula vaiable ξ, known as the eccentic anomaly. We substitute = a(1 e cos ξ), dt = a 2 m/2 E (1 e cos ξ)dξ t = ma 3 /α(ξ e sin ξ), whee we have used α = 2 E a. Note that 0 < ξ < 2π: we do the calculation fo [0, π] ( sin ξ = 1 cos 2 ξ. The calculation fo [π, 2π] is simila, with sin ξ = 1 cos 2 ξ. So, fo an elliptic obit = a(1 e cos ξ), t τ = ma 3 /α(ξ e sin ξ), whee τ is the time of peicentic passage (because when t = τ ξ = 0, = a(1 e) = pei ). The latte equation is known as Keple s equation. Hee 0 ξ 2π fo one peiod. Note that ma 3 /α = T/2π. To obtain ξ (then ) as a function of t Keple s equation must be lved numeically. Note that ma 3 /α = T/2π. Note that often the eccentic anomaly is indicated with E, instead of ξ (see, e.g., R05, MD).

6 6 Lauea in Astonomia - Univesità di Bologna Mean anomaly, tue anomaly and eccentic anomaly. In Keple s equation ξ is the eccentic anomaly. Keple s equation can be witten as M = ξ e sin ξ, whee M = n(t τ) is the mean anomaly, with n = 2π/T mean motion and T = 2πa 3/2 m/α is the peiod. The geometic intepetation of the eccentic anomaly ξ is given in fig. 2.7b of MD (FIG CM2.4). So: x = a cos ξ y 2 = b 2 b2 a 2 x2 = b 2 (1 cos 2 ξ) = b 2 sin 2 ξ = a 2 (1 e 2 ) sin 2 ξ 2 = (x ae) 2 + y 2 = x 2 2aex + a 2 e 2 + y 2 = a 2 cos 2 ξ 2a 2 e cos ξ + a 2 e 2 + a 2 sin 2 ξ a 2 e 2 sin 2 ξ = = a 2 2a 2 e cos ξ + a 2 e 2 cos 2 ξ = a 2 (1 e cos ξ) 2 = a(1 e cos ξ) f as a function of ξ. It is useful to deive elations between the eccentic anomaly ξ and the tue anomaly f (see, e.g., VK 3.8). We know that = a(1 e2 ) 1 + e cos f 1 e cos ξ = 1 e2 1 + e cos f e cos f = 1 e2 1 e cos ξ 1 cos f = cos ξ e 1 e cos ξ, and, using sin f = 1 cos 2 f, sin f = 1 e 2 sin ξ 1 e cos ξ. In summay, we ecall that thee ae thee diffeent anomalies : tue anomaly f (o φ φ 0 = φ ω), mean anomaly M = n(t τ) and eccentic anomaly ξ (see definitions above). 2.3 Systems of coodinates and obital elements [R05, chapt. 2]

7 Celestial Mechanics - A.A Celestial systems of coodinates See fig. 2.5 of R05 (FIG CM2.5). Celestial sphee: fictitious sphee of abitay adius suounding the Eath, on which the celestial bodies ae pojected. Celestial equato: geat cicle obtained intesecting the plane of the Eath equato and the celestial sphee. Celestial poles: intesections of Eath axis with celestial sphee Celestial meidians: geat cicles on the celestial sphee joining the poles Plane of the ecliptic: plane of the Eath obit aound the Sun Ecliptic: geat cicle obtained intesecting the plane of the ecliptic with the celestial sphee Venal and autumnal equinoctial points: intesections between ecliptic and celestial equato. Al known as fist point of Aies and Liba. Venal equinox o fist point of Aies (Υ) efeence point on the ecliptic and on the celestial equato Angle between ecliptic and celestial equato (i.e. inclination of Eath axis w..t. Eath obit) is Equatoial coodinates Right ascension α in hous (0-24) o degees (0-360) fom Υ eastwads. Declination δ: angle along the meidian (in degees) 0 at the equato, 90 at the noth pole, -90 at the uth pole. Ecliptic coodinates Ecliptic longitude λ: in degees (0-360) o hous (0-24) fom Υ eastwads. Al known as celestial longitude. Ecliptic latitude β: in degees fom 0 (ecliptic) to 90 (at the Noth pole of the ecliptic K). Al known as celestial latitude Obital elements Fo simplicity it is convenient to efe to a body obiting the Sola System, but the same fomalism applies to any body obiting anothe body, when a efeence plane is fixed. See fig. 2.6 of R05 (FIG CM2.6). Position and obit of a celestial body (e.g. planet) defined by 6 quantities called elements. Let us specialize to the case of the elliptic obit.

8 8 Lauea in Astonomia - Univesità di Bologna 3 elements define the oientation of the obit (Ω, i, ω) 2 elements define size and shape of the obit (a, e) 1 element defines position of the body at a given time (τ) Line of nodes: intesection between body obital plane and ecliptic plane Nodes: intesections between ecliptic and line of nodes Ascending node N: when in this node the body goes fom the uth ecliptic hemisphee to the noth ecliptic hemisphee (Ω) Longitude of the ascending node Ω: angle fom Υ to N measued eastwad on the ecliptic (in degees fom 0 to 360). (i) Inclination i: angle between body obital plane and ecliptic plane (in degees) Line of apses: line joining the apocente (aphelion) and peicente (peihelion), intecepting the celestial sphee in B (poj. of peicente) and B (poj. of apocente) (ω) Agument of peicente ω: angle between N and B (a) Semi-majo axis of the elliptic obit a = GMµ /2 E = GM m planet /2 E (size of obit) (e) Eccenticity of the obit e: distance between focus and cente is ae (shape of the obit) (τ) Time of peicente passage τ: epoch at which the body was at peicente. Obital elements fo a body obiting in the Sola System: longitude of the ascending node (Ω), inclination (i), agument of peihelion (ω), semi-majo axis (a), eccenticity (e), time of peihelion passage (τ) When the 6 elements ae given, position of body known at any time. Simila obital elements ae taken fo satellites of the Eath (taking the equatoial plane as efeence) o fo satellites of othe planets (with planet s equato as efeence) Al used as element the longitude of peicente ϖ = ω +Ω (dog-leg angle, because Ω and ω lie in diffeent planes if i 0) Al used χ = nτ (mean anomaly at epoch, metimes indicated with M 0 ), instead of τ (see R05 page 211). The element χ is elated to the value of the mean anomaly M n(t τ) = nt + χ at t = 0. Al used mean longitude at epoch ɛ = ϖ nτ = ϖ + χ (see R05 211; MD 6.8) ɛ is the value of the mean longitude λ M + ϖ = n(t τ) + ϖ = nt + ɛ at t = 0. R05 uses the pai (ω, χ). MD use the pai (ϖ, ɛ).

9 Celestial Mechanics - A.A Keple s poblem in Hamiltonian mechanics Keple s poblem in two dimensions [VK 4.10] Let us conside fo simplicity the plana poblem: φ and ae pola coodinates in the plane of the obit. In othe wods, we ae assuming that the efeence plane of ou coodinate system coincides with the plane of the obit (inclination i = 0). The kinetic enegy is The potential enegy is with µ = GM = G(m 1 + m 2 ) The Lagangian is The genealized momenta ae The Hamiltonian is T = 1 2 µ ( ṙ φ2 ) V = µµ L = 1 2 µ ( ṙ φ2 ) V (). p = L ṙ = µ ṙ, p φ = L φ = µ 2 φ. H = 1 2 µ (ṙ φ2 ) µµ, ( ) H = 1 2µ p 2 + p2 φ 2 µµ, which does not depend on φ (i.e., φ is a cyclic coodinate) We want to tansfom q = (, φ) and p = (p, p φ ) into a new set of canonical coodinates (Q, P), fo which the Hamiltonian is zeo. So we take the action S as the geneating function in the fom S = S(q, P, t) = S(, φ, P 1, P 2, t), with P 1 and P 2 new momenta, which must be constants (because the new Hamiltonian H = 0). Let us wite the H-J equation. We have p i = S/ q i, p = S/, p φ = S/ φ, and the H-J equation H + S/ t = 0 eads 1 2µ [ ( S ) 2 + ( 1 ) ] S 2 µµ + S φ t = 0.

10 10 Lauea in Astonomia - Univesità di Bologna Using the method of sepaation of vaiables, we look fo a lution in the fom S(, φ, t) = S () + S φ (φ) + S t (t) We get 1 2µ [ (ds ) 2 + d ( 1 ) ] ds 2 φ µµ dφ = ds t dt. Fo this to be satisfied we must have ds t dt = α 1 = const [ (ds ) 1 2 ( ) ] 1 ds 2 φ 2µ + µµ = α 1, d dφ which can be witten as ( dsφ dφ ) 2 ) = [2µ (α µµ ( ) ] 2 ds. d Fo this to be tue we must have ds d = ds φ dφ = α 2 ( ) 2µ α 1 + µµ α Thus, the geneating function is S = α 1 t + α 2 φ + d ( ) 2µ α 1 + µµ α We take the new genealized momenta as P 1 = α 1, P 2 = α 2 the new genealized coodinates ae Q 1 = β 1 = S P 1 = S α 1 Q 2 = β 2 = S P 2 = S α 2 The new Hamiltonian is the integal of motion is the total enegy. H = H + S t = H α 1 = 0, α 1 = H = µ Ẽ = E = µ µ 2a

11 Celestial Mechanics - A.A We have intoduced the total enegy pe unit mass Ẽ = E/µ = 1 2 v2 µ. We focus on the elliptic case, Ẽ = µ 2a We al have S φ = p φ, α 2 = p φ = µ 2 φ = L = µ L = µ aµ(1 e 2 ), which is the modulus of the angula momentum. We have intoduced the angula momentum pe unit mass L 2 φ = aµ(1 e 2 ), whee we have used the definition of eccenticity: e 1 + (2EL 2 /µ 3 µ 2 ) = 1 L 2 /aµ 2 µ). So We now deive Q 1 and Q 2 : whee and whee = I 1 = P 1 = µ µ 2a P 2 = µ aµ(1 e 2 ) Q 1 = S α 1 = t + I 1 2µ ( µ µ 2a = 1 µ I 2 = µ d 2µ ( α 1 + µµ µ d + µµ ) α ) µ 2 aµ(1 e 2 ) 2 d, 2 a + 2 a(1 e2 ) Q 2 = S α 2 = φ α 2 µ I 2, µ d 2 2µ ( ) α 1 + µµ α 2 2 2

12 12 Lauea in Astonomia - Univesità di Bologna = 2 2µ ( µ µ 2a = 1 µ µ d + µµ ) µ 2 aµ(1 e 2 ) 2 d 2 a + 2 a(1 e2 ) The integals I 1 and I 2 can be lved analytically with the change of vaiable = a(1 e cos ξ), d = ae sin ξdξ, whee ξ is the eccentic anomaly. We have = 1 µ I 1 = 1 µ d 2 a + 2 a(1 e2 ) a(1 e cos ξ)ae sin ξdξ a(1 e cos ξ) 2 + 2a(1 e cos ξ) a(1 e 2 ) = a3/2 µ (1 e cos ξ)e sin ξdξ 1 + 2e cos ξ e 2 cos 2 ξ + 2 2e cos ξ 1 + e 2 ) = a3/2 µ = a3/2 µ (1 e cos ξ)e sin ξdξ e 2 sin 2 ξ (1 e cos ξ)dξ = a3/2 µ (ξ e sin ξ), Q 1 = t + I 1 = t + a3/2 (ξ e sin ξ) = t + 1 M = t + (t τ) = τ. µ n We have = 1 µ = 1 aµ I 2 = 1 µ d 2 /a + 2 a(1 e 2 ) ae sin ξdξ a(1 e cos ξ) a(1 e cos ξ) 2 + 2a(1 e cos ξ) a(1 e 2 ) e sin ξdξ (1 e cos ξ) 1 + 2e cos ξ e 2 cos 2 ξ + 2 2e cos ξ 1 + e 2 ) = 1 e sin ξdξ aµ (1 e cos ξ) e 2 sin 2 ξ = 1 dξ aµ 1 e cos ξ = 1 1 e 2 dξ aµ(1 e 2 ) 1 e cos ξ

13 Celestial Mechanics - A.A We ecall that the elation between tue anomaly f and eccentic anomaly ξ is sin f = 1 e 2 sin ξ 1 e cos ξ, cos fdf = 1 e 2 cos ξ(1 e cos ξ) e sin2 ξ (1 e cos ξ) 2 dξ, cos fdf = 1 e 2 cos ξ e (1 e cos ξ) 2 dξ, cos fdf = 1 e 2 cos f (1 e cos ξ) dξ, because cos f = cos ξ e 1 e cos ξ, df = 1 e 2 (1 e cos ξ) dξ, thus I 2 = 1 aµ(1 e 2 ) df = f aµ(1 e 2 ) So Q 2 = φ α 2 µ I 2 = φ α 2 µ f aµ(1 e 2 ) = φ aµ(1 e 2 f ) aµ(1 e 2 ) = φ f = φ (φ ω) = ω whee we have used α 2 = µ aµ(1 e 2 ). In summay, the new genealized coodinates ae Q 1 = β 1 = τ (minus the time of peicentic passage) and Q 2 = β 2 = ω (agument of peicente). The new genealized momenta ae P 1 = α 1 = µ Ẽ (total enegy) and P 2 = α 2 = µ aµ(1 e 2 ) (angula momentum modulus). All of these ae constants. The new Hamiltonian is H = 0. With the above canonical tansfomation we have obtained 4 constant canonical coodinates, i.e. 4 integals of motion. These integals of motions fully constain the obit. The lution of the equations of motion, i.e. = (t) and φ = φ(), is given by the equations β 1 = S α 1, β 2 = S α 2.

14 14 Lauea in Astonomia - Univesità di Bologna Keple s poblem in thee dimensions [VK 4.11] We now conside Keple s poblem in 3D: in pactice we take a system of coodinates in which the efeence plane does not coincide with the obital plane. We ecall that the motion is plana: howeve it is often convenient to descibe it in 3D (fo instance, when descibing the obit of a body in the Sola System, it is useful to take as efeence plane the ecliptic: see al Chapte on petubation theoy). The Lagangian in spheical coodinates is L = 1 2 µ ( ṙ θ2 + 2 sin 2 θ φ 2) + µ µ. Note that when sin θ = 1 and θ = 1 we educe the poblem to the 2D case. The genealized momenta ae p = L ṙ = µ ṙ, p θ = L θ = µ 2 θ, p φ = L φ = µ 2 sin 2 θ φ. The Hamiltonian of the Keple poblem in spheical coodinates is The H-J equation eads 1 2µ [ ( S ) 2 + H = 1 2 µ (ṙ θ2 + 2 sin 2 θ φ 2 ) µ µ. ( ) H = 1 2µ p 2 + p2 θ 2 + p2 φ 2 sin 2 µ µ θ. ( 1 ) S 2 ( 1 + θ sin θ ) ] S 2 µ µ + S φ t = 0, The action as geneating function is in the fom S = S(q, P, t) = S(, θ, φ, P 1, P 2, P 3, t) with P i = α i = const. Using the method of sepaation of vaiables, we look fo a lution in the fom S(, θ, φ, t) = S () + S θ (θ) + S φ (φ) + S t (t) We have S t t = α 1 S φ φ = α 3 ( ) 2 Sθ + α2 3 θ sin 2 θ = α2 2 ( ) 2 S + α2 2 2 = 2µ ( α 1 + µ µ )

15 Celestial Mechanics - A.A Pefoming calculations analogous to the 2-D case we get the (constant) genealized coodinates Q 1 = β 1 = τ, (time of peicentic passage) Q 2 = ω, (agument of peicente) Q 3 = Ω, (longitude of the ascending node), and the (constant) genealized momenta P 1 = µ Ẽ = µ µ 2a, (total enegy), P 2 = µ aµ(1 e 2 ), (angula momentum modulus), P 3 = µ aµ(1 e 2 ) cos i, (z-component of angula momentum). The coesponding Hamiltonian is H = 0. With the above canonical tansfomation we have obtained 6 constant canonical coodinates, i.e. 6 integals of motion. These integals of motions fully constain the obit. The lution of the equations of motion is given by the equations β i = S α i, i = 1, 2, 3. It is al possible to obtain anothe set of canonical coodinates, maintaining the genealized coodinates as they ae, but mass-nomalizing the genealized momenta and the Hamiltonian. So, we stat fom Q i, P i and define a new set of vaiables q i = Q i, p i = P i /µ. The equations of motion keep the canonical fom with the Hamiltonian H = H /µ (i.e. this is a canonical tansfomation; see G09). This can be seen al by noting that P i = H Q = P i µ = /µ ) (H = p Q i = H i q i and Q i = H = (H /µ ) P i P i /µ = q i = H p i. The (constant) mass-nomalized canonical coodinates ae β 1 = q 1 = τ, β2 = q 2 = ω, β3 = q 3 = Ω α 1 = p 1 = µ 2a, α 2 = p 2 = aµ(1 e 2 ), α 3 = p 3 = aµ(1 e 2 ) cos i. The mass-nomalized Hamiltonian is H = H /µ = 0.

16 16 Lauea in Astonomia - Univesità di Bologna Delaunay s vaiables [VK 4.12] We can pefom a futhe canonical tansfomation to tansfom the above set in a new set of angle-action coodinates, known as Delaunay s vaiables (coodinates l D, g D, h D, and momenta L D, G D, H D ). While q 2 and q 3 ae angles, q 1 = τ is not. So we eplace τ with the mean anomaly M = n(t τ) = n(t+ q 1 ), which is an angle. We want to find the tansfomation to have l D = n(t + q 1 ) g D = q 2 = ω h D = q 3 = Ω L D =? (to be detemined) G D = p 2 = aµ(1 e 2 ) H D = p 3 = aµ(1 e 2 ) cos i whee L D must be found consistently with the choice of l D. A geneating function that does the job is F ( q 1, q 2, q 3, L D, G D, H D, t) = ( nl D 3µ ) (t + q 1 ) + q 2 G D + q 3 H D, 2a whee the tem 3µ/2a appeas to simplify the fom of L D and of the Hamiltonian. We have L D = 1 n l D = F L D = n(t + q 1 ) g D = F G D = q 2 h D = F H D = q 3 p 1 = F = nl D 3µ q 1 2a p 2 = F q 2 = G D, p 3 = F q 3 = H D, ( p 1 + 3µ ) ( = a3/2 2a µ 1/2 µ 2a + 3µ ) = a3/2 µ 2a µ 1/2 a = aµ

17 Celestial Mechanics - A.A The new (Delaunay s) Hamiltonian is K D = 0 + F t = nl D 3µ 2a = µa 3/2 aµ 3µ 2a = µ 2a = µ2 2aµ = µ2 2L 2 D In summay the Delaunay s vaiables ae l D = n(t τ) = M g D = q 3 = ω h D = q 3 = Ω L D = aµ G D = p 2 = aµ(1 e 2 ) H D = p 3 = aµ(1 e 2 ) cos i and the coesponding Hamiltonian is K D = µ2 2L 2. D Five of the Delaunay s vaiables ae constants, but l D (mean anomaly) is not constant and vaies linealy with time. Bibliogaphy Giogilli A., 2009, Appunti di Meccanica celeste (G09) Landau L.D., Lifshitz E.M., 1982, Mechanics, Buttewoth-Heinemann (LL) Muay C.D: & Demott S.F:, 1999, Sola system dynamics, Cambidge Univesity Pess, Cambidge (MD) Roy A.E. 2005, Obital motion, 2005 Taylo & Fancis (R05) Valtonen M., Kattunen H., 2006 The thee body poblem, Cambidge Univesity Pess, Cambidge (VK)