# HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES IN CELESTIAL MECHANICS. 1. Introduction

Save this PDF as:

Size: px
Start display at page:

Download "HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES IN CELESTIAL MECHANICS. 1. Introduction"

## Transcription

1 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES IN CELESTIAL MECHANICS ALESSANDRA CELLETTI 1 and LUIGI CHIERCHIA 2 1 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I Roma, Italy, 2 Dipartimento di Matematica, Università Roma Tre, Largo San Leonardo Murialdo 1, I Roma, Italy, (Received: 15 June 1999; accepted: 17 July 2000) Abstract. The behaviour of resonances in the spin orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. Key words: resonances, spin orbit problem, periodic orbits, Hamiltonian dynamics 1. Introduction This note is an attempt to justify from a purely conservative point of view why certain resonances in the solar system are more common than others. By resonances here we mean resonances in the spin orbit problem, namely commensurabilities within the period of rotation and revolution of an oblate satellite orbiting around a central planet (see Goldreich and Peale, 1966, 1970; Peale, 1973; Murdock, 1978; Henrard, 1985; Wisdom, 1987; Celletti, 1990; Celletti and Falcolini, 1992; Celletti, 1994; Celletti and Chierchia, 1998, and references therein for related papers on this subject). As examples, we consider the Moon Earth and Mercury Sun systems. We then consider an extremely simple mathematical model for the spin orbit problem, described by a time-periodic forced pendulum. In particular, we assume that an oblate satellite moves on a Keplerian orbit around a central planet and rotates about an internal spin-axis. We define a spin orbit resonance of order p : q whenever the ratio between the revolutional and rotational periods is rational, say T rev /T rot = p/q for some positive integers p, q. The perturbing parameter is the equatorial oblateness of the satellite (the eccentricity of the Celestial Mechanics and Dynamical Astronomy 76: , c 2000 Kluwer Academic Publishers. Printed in the Netherlands.

2 230 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA Keplerian orbit is considered as a given parameter). For a fixed frequency p/q, we compute the critical value ε (p/q) at which the elliptic periodic orbit with frequency p/q becomes unstable, while it is stable for 0 <ε<ε (p/q). Such value is obtained by studying the trace of the linearized matrix over the full cycle of the periodic orbit. We then introduce the characteristic stability indicator σ(p/q), which provides a measure of the stability of the different periodic orbits. The plot of σ(p/q)vs. p/q shows that for low values of the eccentricity there is only a marked peak corresponding to the 1:1 commensurability (occurring when the periods of revolution and rotation are the same). Increasing the eccentricity, other resonances appear. Indeed, the 3:2 resonance can be observed at eccentricities larger than Astronomical observations show that most of the evolved satellites or planets of the solar system are trapped in a 1:1 or synchronous spin orbit resonance. The most familiar example is provided by the Moon; due to the equality of the periods of rotation and revolution, the Moon always points the same face to the Earth. The only known exception is provided by Mercury, which moves in a 3:2 resonance. The mathematical model for the spin orbit motion (see Section 2) is described by an equation of the form ẍ εf x (x, t) = 0, (1.1) for a suitable function f depending also on the orbital eccentricity e of the Keplerian orbit; the parameter ε is proportional to the equatorial oblateness coefficient of the satellite. The above equation can be rewritten as ẋ = y, ẏ = εf x (x, t), (1.2) which can be viewed as Hamilton s equations associated to the non autonomous, one-dimensional Hamiltonian H(y,x,t) = y2 εf (x, t ), 2 y R, x T R/2π Z being conjugate coordinates and t T. The variable x represents the angle between the longest axis of the ellipsoidal satellite and the periapsis line (see Figure 1) and ẋ dx/dt represents its velocity. In Section 3 we introduce a stability criterion for periodic orbits. Such criterion involves the computation of trajectories associated to periodic orbits. This is achieved by a method introduced by (Greene, 1979) for the standard map, which we generalize to our specific case in Appendix A. Results and conclusions are presented in Section The Spin Orbit Model We briefly recall a mathematical model introduced in (Celletti, 1990) to describe the spin orbit interaction in celestial mechanics. Let S be a triaxial ellipsoidal

3 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES 231 Figure 1. The spin-orbit geometry. satellite orbiting around a central planet P.LetT rev and T rot be the periods of revolution of the satellite around P and the period of rotation about an internal spin-axis. A p : q spin orbit resonance occurs whenever T rev T rot = p q, for p, q N, q = 0. In particular, when p = q = 1 we speak of 1:1 or synchronous spin orbit resonance, which implies that the satellite always points the same face towards the host planet. In the solar system most of the evolved satellites or planets (e.g., the Moon) are trapped in a 1:1 resonance (Astronomical Almanac, 1997). The only exception is provided by Mercury which is observed in a nearly 3:2 resonance. We introduce a mathematical model describing the spin orbit interaction. In particular we assume that: (i) the center of mass of the satellite moves on a Keplerian orbit around P with semimajor axis a and eccentricity e; (ii) the spin-axis is perpendicular to the orbit plane; (iii) the spin-axis coincides with the shortest physical axis (i.e., the axis whose moment of inertia is largest); (iv) dissipative effects as well as perturbations due to other planets or satellites are neglected. Remark. Assumptions (i) and (ii) imply that we neglect secular perturbations on the orbital parameters as well as the obliquity of the spin axis. As shown in (Laskar and Robutel, 1993) a more relevant model from the physical point of view needs to consider a non-keplerian orbit as well as a non-zero obliquity. However, our model may be considered as a starting point to study the spin orbit dynamics at the most elementary level. Results could be generalized so as to include secular perturbations and the spin axis obliquity. In particular, as is well known in the case

4 232 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA of the Mercury Sun system it is definitely important to include the effect of secular variations of the elliptic elements. We introduce now a mathematical model (compare with Celletti, 1990) apt to describe the spin orbit interaction under the above assumptions. Let A<B<C be the principal moments of inertia of the satellite. We denote by r and ν, respectively, the instantaneous orbital radius and the true anomaly of the Keplerian orbit. Finally, let x be the angle between the longest axis of the ellipsoid and the periapsis line (see Figure 1). The equation of motion may be derived from the standard Euler s equations for rigid body. In normalized units (i.e. assuming that the mean motion is one, 2π/T rev = 1) we obtain ẍ + ε ( ) 1 3 sin(2x 2ν) = 0, (2.1) r where ε 3/2(B A)/C is proportional to the equatorial oblateness coefficient (B A)/C (and the dot denotes time differentiation). Notice that (2.1) is trivially integrated when A = B or in the case of zero orbital eccentricity. Due to assumption (i), the quantities r and ν are known (periodic) Keplerian functions of the time; therefore we can expand (2.1) in Fourier series as ẍ + ε ( m ) W 2,e m =0,m= sin(2x mt) = 0, (2.2) where the coefficients W(m/2,e) decay as powers of the orbital eccentricity as W(m/2,e) e m 2 (see Cayley 1859, for explicit expressions). A further simplification of the model is performed as follows. According to assumption (iv), we neglected dissipative forces and any gravitational attraction beside that of the central planet. The most important contribution comes from the non-rigidity of the satellite, which provokes a tidal torque due to internal friction. Since the magnitude of the dissipative effects is small compared to the gravitational term, we simplify (2.2) retaining only those terms which are of the same order or bigger than the average effect of the tidal torque. Thus we are led to consider an equation of the form ẍ + ε N 2 ( m ) W 2,e sin(2x mt) = 0, (2.3) m =0,m=N 1 where N 1 and N 2 are suitable integers (depending on the structural and orbital properties of the satellite), while W(m/2,e) are truncations of the coefficients W(m/2,e)(which are power series in e). In this paper we are mainly concerned with the Moon Earth and the Mercury Sun systems. In the case of the Moon Earth system N 1 and N 2 are, respectively, 1 and 7. For such system we obtain the equation

5 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES 233 ) e2 e3 ẍ + ε [( + sin(2x t) + 16 ( e ) ( 7 16 e4 sin(2x 2t) + 2 e 123 ) 16 e3 sin(2x 3t) + ( e2 115 ) ( e4 sin(2x 4t) + 48 e ) 768 e5 sin(2x 5t) e4 sin(2x 6t) ] 3840 e5 sin(2x 7t) = 0. (2.4) Notice that the largest coefficient (of order unity) corresponds to the synchronous resonance (i.e., the term related to 2x 2t). For e = 0 Equation (2.4) reduces to ẍ + ε sin(2x 2t) = 0 which corresponds to a pendulum (after the substitution x x t). Other examples, such as the Mercury Sun system, would be described by a different Fourier series truncation (for the Mercury Sun case, it is N 1 = 17, N 2 = 6in(2.3)and with slightly different truncations of the W s); however, we checked that results do not change significantly as new terms are added to Equation (2.4). Precisely, more terms alter the outputs for any initial condition, provoking a global rescaling of the results, without conditioning the qualitative conclusions. The (non-integrable) dynamics of (2.4) is represented in Figure 2 by the Poincaré (or stroboscopic) map (at t = 2π) around the synchronous resonance Figure 2. Poincaré map associated to Equation (2.4) around the synchronous resonance in the (x, ẋ) plane for e = 0.004, ε = 0.2. The initial conditions are (x, y) = (0, 0.8), (0, 1.5), (0, 1), (0,1.1), (1.8,1), (2.1,1), (1.57, 1). Notice that Equation (2.4) is π periodic in x. Librational and rotational regions are divided by the chaotic separatrix.

6 234 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA for e = 0.004, ε = 0.2: the resonance is surrounded by librational surfaces, whose amplitude increases as the chaotic separatrix is approached. Outside this region rotational surfaces can be found. 3. Stability Criterion For Periodic Orbits In this section we investigate numerically the stability of the periodic orbits. We define a p : q periodic orbit (or Birkhoff periodic orbit with rotation number p/q) as a solution of (1.1), x : R R, such that x(t + 2πq) = x(t) + 2πp, (3.1) that is, after q orbital revolutions around the central body the satellite makes p rotations about the spin-axis. By applying a symplectic Euler s method with step size h to the differential equations (1.2), we are led to study the mapping y n+1 = y n + εf x (x n,t n )h, x n+1 =x n +y n+1 h, t n+1 = t n + h. (3.2) Notice that the determinant of the jacobian of the mapping (3.2) is identically one. Let M be the product of the jacobian of (3.2) over the full cycle of the periodic orbit with frequency p/q, thatis, M= q i=1 ( ) 1 εfxx (x i,t i )h h 1 + εf xx (x i,t i )h 2, where the points (x i,t i ), i = 1,...,q, are computed along the periodic orbit. Following (Greene, 1979) the three-dimensional search (in the variables y, x, t) for periodic orbits can be reduced to a one-dimensional problem, using the symmetry property of the mapping (3.2). In particular, use is made of the fact that the mapping (3.2) can be decomposed as the product of two involutions. The initial value of a periodic orbit can be found along the lines of fixed points of one of the above involutions. More details about this method can be found in Appendix A (see also Celletti, 1990). The eigenvalues of M are the associated Floquet multipliers; due to the area preserving property of (3.2), one has det M = 1. Therefore, the product of the eigenvalues λ 1, λ 2 of M is unity, while their sum equals the trace T of M: λ 1 + λ 2 = T.Sinceλ 1,λ 2 must satisfy the equation λ 2 Tλ+1=0, one finds λ 1,2 = T ( ) T ± i. 2 According to the value of T, one has different solutions for the eigenvalues providing the stability character of the periodic orbit as follows:

7 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES 235 (i) if T < 2, then λ 1, λ 2 are complex conjugates on the unit circle and the motion is stable; (ii) if T > 2, λ 1 and λ 2 are real numbers with λ 2 = λ 1 1 and one of the eigenvalues is in modulus greater than 1, providing the instability of the periodic orbit. Notice that in the integrable case ε = 0, T = 2andλ 1,2 =1. As is well known, for any value of the frequency p/q there exist, in general, periodic orbits of elliptic and hyperbolic type. We are interested in investigating the stability of the elliptic periodic orbits as the perturbing parameter ε is increased. In particular, for a fixed eccentricity we look at the value of ε at which the trace T, which originates from the value T = 2asε=0, decreases down to the transition value T = 2becom- ing, immediately after, T< 2. We denote by ε (p/q) the value of the perturbing parameter at which the transition occurs. We take the behaviour of ε (p/q) versus the frequency p as an index of stability. The graph of the analog of the critical q function ε (ω) versus ω (diophantine) for invariant tori (i.e. the so-called breakdown threshold ) of the spin-orbit problem has been investigated in (Celletti and Falcolini, 1992). Next, we want to give a weight to peaks of the function ε ; this weight should take into account, besides the absolute magnitude of the peak, also its relative value compared to nearby (eventually smaller) peaks, or its degree of isolation (an isolated peak is more stable than a peak with other closeby peaks of comparable magnitude), etc. We proceed as follows: assume that ε has a peak (a relative maximum) at ω 0 = p/q;letδ 0 be a positive real number such that ε (ω 0 +δ 0 ) = ε (ω 0 ) or ε (ω 0 δ 0 ) = ε (ω 0 ),andε (ω )<ε (ω 0 ) for any ω I 0 (δ 0 ) (ω 0 δ 0,ω 0 + δ 0 )\{ω 0 }. Then, the quantity α(ω 0,δ 0 ) ε (ω 0 ) ε (ω ) max ω I 0 (δ 0 ) ε (ω 0 ) ω 0 ω provides the relative distance of the transition values in a neighbourhood of size δ 0 around the frequency ω 0. In order to have an indicator of stability, we multiply α(ω 0,δ 0 )by the half size of the domain I 0 (δ 0 ), that is we define the characteristic stability indicator (hereafter, CSI) as σ(ω 0 ) δ 0 α(ω 0,δ 0 ). We will see in the next section that the plot of σ(ω 0 ) versus ω 0 provides useful informations about stability properties of periodic orbits. 4. Results and Conclusions According to standard evolutionary theories, satellites and planets were rotating fast in the past; a constant decrease of the period of rotation about the spin-axis was provoked by tidal friction due to the internal non-rigidity. Therefore, a common scenario suggests that celestial bodies were slowed down until they reached their

8 236 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA actual dynamical configuration, being typically trapped in a 1:1 (with the 3:2 Mercury exception) resonance. This hypothesis implies that higher order resonances (i.e., 2:1, 5:4, 7:3, etc.) were bypassed during the slowing process. Beside the tidal effect, a parameter which might direct the motion toward a resonance is the orbital eccentricity. In this section we investigate the stability of the resonant motions corresponding to periodic orbits as the orbital eccentricity is varied. More precisely, we compute the critical value ε (p/q) and the CSI σ(p/q) for different periodic orbits of frequency p/q. We let the eccentricity vary in a reasonable (astronomical) range of values. In particular, we consider the following set of periodic orbits with frequencies p/q where and q = 1,...,14, p = q +1,...,15, p = 1,...,14, q = p+1,...,15. In order to have a better precision around the main resonances we consider also the periodic orbits with frequencies 1 2 ± 1 10 k, 1 ± 1 10 k, 5 4 ± 1 10 k, 4 3 ± 1 10 k, 3 2 ± 1 10 k, 5 3 ± 1 10 k, 7 4 ± 1 10 k, 2 ± 1 10 k, where k = 1,...,10. For a fixed value of the eccentricity we compute ε (p/q) and σ(p/q) corresponding to the set of rational numbers p/q listed above. Figures 3 and 4 show the graphs of ε (p/q) and σ(p/q) versus p/q for, respectively, the Moon s eccentricity (i.e., e = ) and Mercury s eccentricity (e = ). In order to have a better understanding of the behaviour around the resonances of astronomical interest, we report in panel (a) the results around the 1:2 resonance, while panel (b) corresponds to the synchronous resonance, panel (c) to the 3:2 resonance and panel (d) to the 2:1 resonance. A comparison between the graphs of σ(p/q) vs. p/q reported in Figures 3 and 4 indicates that the most stable resonances are the 1:1, 3:2 and 2:1. However, the stability indicator changes as the eccentricity is varied. In particular, the 1:1 resonance is the most stable one for the Moon s eccentricity (see Figure 3). When the eccentricity is increased up to Mercury s value (i.e. e = ), the 3:2 resonance gains stability and its CSI becomes comparable to that of the synchronous

9 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES 237 Figure 3. The transition value ε (p/q) vs. p/q for e = (i.e. the eccentricity of the Moon). The dots correspond to the actual computations associated to the frequencies listed in the text. Lines are due to interpolation performed by the graphic program. In abscissa are reported the rotation numbers p/q (the 1:1 resonance corresponds to 1, the 1:2 to 0.5 and so on). (a) 1:2 resonance, (b) 1:1 resonance, (c) 3:2 resonance, (d) 2:1 resonance. (e) The CSI σ(p/q)vs. p/q for e = resonance (see Figure 4). This remark suggests that the higher value of the eccentricity might be responsible of the actual dynamical configuration of Mercury in the 3:2 resonance. Indeed, it is worth noticing that the 1:1 and 3:2 resonances attain a comparable stability level for a value of the eccentricity of the order of e = 0.2. At lower values of e, for example e = 0.1 as in Figure 5(c), the motion associated to the synchronous resonance is still the most stable one. We report in Figure 5 the behaviour of ε (p/q) and of σ(p/q) for e = (Figure 5(a)), e = 0.01 (Figure 5(b)), e = 0.1 (Figure 5(c)). A comparison among the different graphs suggests that resonances different from the synchronous one are born as the eccentricity is increased, while at low values of the eccentricity, say e = 0.001, the 1:1 resonance is rather isolated. This result is supported by the astronomical observations, since most of the satellites observed to move in the synchronous resonance have eccentricities in the range [0, 0.01] (see Astronomical Almanac, 1997).

10 238 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA Figure 4. The transition value ε (p/q) vs. p/q for e = (i.e., the eccentricity of Mercury). The dots correspond to the actual computations associated to the frequencies listed in the text. Lines are due to interpolation performed by the graphic program. In abscissa are reported the rotation numbers p/q (the 1:1 resonance corresponds to 1, the 1:2 to 0.5 and so on). (a) 1:2 resonance, (b) 1:1 resonance, (c) 3:2 resonance, (d) 2:1 resonance. (e) The CSI σ(p/q)vs. p/q for e = We finally mention that for a further development of this study it would be clearly necessary to include the spin-axis obliquity as well as the effect of dissipative terms. In the latter case one is led to the question of the existence of attracting tori for a dissipative system (i.e., including tidal forces) corresponding to the most stable resonances. We plan to address this problem in a later study. Appendix A: Computation of Periodic Orbits Let S be the mapping (3.2) which we rewrite as y = y + εf x (x, t) h, x = x + y h, t = t + h, where y,x,t R. Finding periodic orbits associated to S can be considerably simplified exploiting the symmetry property of S. Indeed, this fact was already observed in (Greene, 1979) for the standard mapping y = y + ε sin x, x = x + y.

11 HAMILTONIAN STABILITY OF SPIN ORBIT RESONANCES 239 Figure 5. Left panel: plot of ε (p/q) vs. p/q for the frequencies listed in the text; right panel: plot of the CSI σ(p/q)vs. p/q for the same frequencies as in the left panel. (a) e = 0.001, (b) e = 0.01, (c) e = 0.1. (y R, x R/2πZ) and can be generalized to our case as follows. The basic remark is that S can be decomposed as the product of two involutions: S = I 2 I 1, I 2 1 = I2 2 = 1, I 2 = SI 1, where I 1 is given by y = y + εf x (x, t)h, x = x, t = t, and I 2 is defined as y = y, x = x+hy, t = t+h. The periodic orbits correspond to fixed points of I 1 or I 2 (Greene, 1979, Appendix A). Since the function f x (x, t) is π-periodic in x and 2π periodic in the time,

12 240 ALESSANDRA CELLETTI AND LUIGI CHIERCHIA the periodic orbits can be found at (x, t) = (0, 0), (π/2, 0), (0,π), (π/2,π). This reduces the search for periodic orbits to a one-dimensional problem in the y variable. References Cayley, A.: 1859, Tables of the developments of functions in the theory of elliptic motion, Mem. Roy. Astron. Soc. 29, 191. Celletti, A.: 1990, Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance, (Part I) J. Appl. Math. Phys. (ZAMP) 41, 174; Part II: ZAMP 41, 453. Celletti, A.: 1994, Construction of librational invariant tori in the spin-orbit problem, J. Applied Math. Physics (ZAMP) 45, 61. Celletti, A. and Chierchia, L.: 1998, Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics, Reg. Chaotic Dyn. 3, 107. Celletti, A. and Falcolini, C.: 1992, Construction of invariant tori for the spin-orbit problem in the Mercury Sun system, Celest. Mech. & Dyn. Astr. 53, 113. Greene, J. M.: 1979, A method for determining a stochastic transition, J. Math. Phys. 20, Goldreich, P. and Peale, S.: 1966, Spin orbit coupling in the solar system, Astron J. 71, 425. Goldreich, P. and Peale, S.: 1970, The dynamics of planetary rotations, Ann. Rev. Astron. Astroph. 6, 287. Henrard, J.: Spin-orbit resonance and the adiabatic invariant, in: S. Ferraz Mello and W. Sessin (eds), Resonances in the Motion of Planets, Satellites and Asteroids, Sao Paulo, 19. Laskar, J., Robutel, P.: 1993, The chaotic obliquity of the planets, Nature 361, 608. Murdock, J. A.: 1978, Some mathematical aspects of spin-orbit resonance I, Celest. Mech. & Dyn. Astr. 18, 237. Peale, S. J.: 1973, Rotation of solid bodies in the solar system, Rev. Geoph. Space Phys. 11, 767. Wisdom, J.: 1987, Rotational dynamics of irregularly shaped satellites, Astron. J. 94, (No author listed) 1997, The Astronomical Almanac. Washington: U.S. Government Printing Office.

### Abstract. 1 Introduction

Regions of non existence of invariant tori for spin orbit models WARNING: THE FIGURES OF THIS PAPER CAN BE DOWNLOADED FROM http : //www.mat.uniroma.it/celletti/cm Kf igures.pdf. Alessandra Celletti Dipartimento

### DYNAMICS OF THE LUNAR SPIN AXIS

The Astronomical Journal, 131:1864 1871, 2006 March # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A. DYNAMICS OF THE LUNAR SPIN AXIS Jack Wisdom Massachusetts Institute

### A qualitative analysis of bifurcations to halo orbits

1/28 spazio A qualitative analysis of bifurcations to halo orbits Dr. Ceccaroni Marta ceccaron@mat.uniroma2.it University of Roma Tor Vergata Work in collaboration with S. Bucciarelli, A. Celletti, G.

### Research Article Variation of the Equator due to a Highly Inclined and Eccentric Disturber

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 009, Article ID 467865, 10 pages doi:10.1155/009/467865 Research Article Variation of the Equator due to a Highly Inclined and

### Resonance Capture. Alice Quillen. University of Rochester. Isaac Newton Institute Dec 2009

Resonance Capture Alice Quillen University of Rochester Isaac Newton Institute Dec 2009 This Talk Resonance Capture extension to nonadiabatic regime Astrophysical settings: Dust spiraling inward via radiation

### = 0. = q i., q i = E

Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations

### arxiv: v1 [astro-ph.ep] 18 Mar 2010

On the dynamics of Extrasolar Planetary Systems under dissipation. Migration of planets arxiv:1005.3745v1 [astro-ph.ep] 18 Mar 2010 John D. Hadjidemetriou and George Voyatzis Department of Physics, Aristotle

### DIFFUSION OF ASTEROIDS IN MEAN MOTION RESONANCES

DIFFUSION OF ASTEROIDS IN MEAN MOTION RESONANCES KLEOMENIS TSIGANIS and HARRY VARVOGLIS Section of Astrophysics, Astronomy and Mechanics, Department of Physics, University of Thessaloniki, 540 06 Thessaloniki,

### Chaos Indicators. C. Froeschlé, U. Parlitz, E. Lega, M. Guzzo, R. Barrio, P.M. Cincotta, C.M. Giordano, C. Skokos, T. Manos, Z. Sándor, N.

C. Froeschlé, U. Parlitz, E. Lega, M. Guzzo, R. Barrio, P.M. Cincotta, C.M. Giordano, C. Skokos, T. Manos, Z. Sándor, N. Maffione November 17 th 2016 Wolfgang Sakuler Introduction Major question in celestial

### Resonance and chaos ASTRONOMY AND ASTROPHYSICS. II. Exterior resonances and asymmetric libration

Astron. Astrophys. 328, 399 48 (1997) ASTRONOMY AND ASTROPHYSICS Resonance and chaos II. Exterior resonances and asymmetric libration O.C. Winter 1,2 and C.D. Murray 2 1 Grupo de Dinâmica Orbital e Planetologia,

### arxiv: v1 [math.ds] 29 Dec 2013

arxiv:1312.7544v1 [math.ds] 29 Dec 2013 The spin orbit resonances of the Solar system: A mathematical treatment matching physical data Francesco Antognini Departement Mathematik ETH-Zürich CH-8092 Zürich,

### Astronomy Section 2 Solar System Test

is really cool! 1. The diagram below shows one model of a portion of the universe. Astronomy Section 2 Solar System Test 4. Which arrangement of the Sun, the Moon, and Earth results in the highest high

### Why is the Moon synchronously rotating?

MNRASL 434, L21 L25 (2013) Advance Access publication 2013 June 19 doi:10.1093/mnrasl/slt068 Why is the Moon synchronously rotating? Valeri V. Makarov US Naval Observatory, 3450 Massachusetts Ave NW, Washington,

### ORBITS WRITTEN Q.E. (June 2012) Each of the five problems is valued at 20 points. (Total for exam: 100 points)

ORBITS WRITTEN Q.E. (June 2012) Each of the five problems is valued at 20 points. (Total for exam: 100 points) PROBLEM 1 A) Summarize the content of the three Kepler s Laws. B) Derive any two of the Kepler

### THE exploration of planetary satellites is currently an active area

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 9, No. 5, September October 6 Design of Science Orbits About Planetary Satellites: Application to Europa Marci E. Paskowitz and Daniel J. Scheeres University

### EXTENDING NACOZY S APPROACH TO CORRECT ALL ORBITAL ELEMENTS FOR EACH OF MULTIPLE BODIES

The Astrophysical Journal, 687:1294Y1302, 2008 November 10 # 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A. EXTENDING NACOZY S APPROACH TO CORRECT ALL ORBITAL ELEMENTS

### The steep Nekhoroshev s Theorem and optimal stability exponents (an announcement)

The steep Nekhoroshev s Theorem and optimal stability exponents (an announcement) Nota di Massimiliano Guzzo, Luigi Chierchia e Giancarlo Benettin Scientific chapter: Mathematical analysis. M. Guzzo Dipartimento

### Research Article Dynamics of Artificial Satellites around Europa

Mathematical Problems in Engineering Volume 2013, Article ID 182079, 7 pages http://dx.doi.org/10.1155/2013/182079 Research Article Dynamics of Artificial Satellites around Europa Jean Paulo dos Santos

### expression that describes these corrections with the accuracy of the order of 4. frame usually connected with extragalactic objects.

RUSSIAN JOURNAL OF EARTH SCIENCES, English Translation, VOL, NO, DECEMBER 998 Russian Edition: JULY 998 On the eects of the inertia ellipsoid triaxiality in the theory of nutation S. M. Molodensky Joint

### Formation and evolution of the two 4/3 resonant giants planets in HD

DOI: 10.1051/0004-6361/201424820 c ESO 2014 Astronomy & Astrophysics Formation and evolution of the two 4/3 resonant giants planets in HD 200964 M. Tadeu dos Santos, J. A. Correa-Otto, T. A. Michtchenko,

### arxiv: v1 [astro-ph.ep] 13 May 2011

arxiv:1105.2713v1 [astro-ph.ep] 13 May 2011 The 1/1 resonance in Extrasolar Systems: Migration from planetary to satellite orbits John D. Hadjidemetriou and George Voyatzis Derpartment of Physics, University

### arxiv: v4 [nlin.cd] 12 Sep 2011

International Journal of Bifurcation and Chaos c World Scientific Publishing Company Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential

### arxiv: v1 [astro-ph.ep] 8 Feb 2017

Dynamics and evolution of planets in mean-motion resonances Alexandre C. M. Correia, Jean-Baptiste Delisle, and Jacques Laskar arxiv:1702.02494v1 [astro-ph.ep] 8 Feb 2017 Abstract In some planetary systems

### Ay 1 Lecture 2. Starting the Exploration

Ay 1 Lecture 2 Starting the Exploration 2.1 Distances and Scales Some Commonly Used Units Distance: Astronomical unit: the distance from the Earth to the Sun, 1 au = 1.496 10 13 cm ~ 1.5 10 13 cm Light

### October 19, NOTES Solar System Data Table.notebook. Which page in the ESRT???? million km million. average.

Celestial Object: Naturally occurring object that exists in space. NOT spacecraft or man-made satellites Which page in the ESRT???? Mean = average Units = million km How can we find this using the Solar

### Dynamical behaviour of the primitive asteroid belt

Mon. Not. R. Astron. Soc. 293, 405 410 (1998) Dynamical behaviour of the primitive asteroid belt Adrián Brunini Observatorio Astronómico de La Plata, Profoeg, Paseo del Bosque, (1900) La Plata, Argentina

### Name Period Date Earth and Space Science. Solar System Review

Name Period Date Earth and Space Science Solar System Review 1. is the spinning a planetary object on its axis. 2. is the backward motion of planets. 3. The is a unit less number between 0 and 1 that describes

### Two-Body Problem. Central Potential. 1D Motion

Two-Body Problem. Central Potential. D Motion The simplest non-trivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of

### Poincaré Map, Floquet Theory, and Stability of Periodic Orbits

Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct

### Lecture 13. Gravity in the Solar System

Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws

### Publ. Astron. Obs. Belgrade No. 90 (2010), DYNAMICAL CHARACTERISTICS OF HUNGARIA ASTEROIDS 1. INTRODUCTION

Publ. Astron. Obs. Belgrade No. 9 (21), 11-18 Invited Lecture DYNAMICAL CHARACTERISTICS OF HUNGARIA ASTEROIDS Z. KNEŽEVIĆ1, B. NOVAKOVIĆ2, A. MILANI 3 1 Astronomical Observatory, Volgina 7, 116 Belgrade,

### Resonance In the Solar System

Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012 Goal numerically investigate the dynamics of the asteroid belt

### An introduction to Birkhoff normal form

An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

### FROM ORDERED TO CHAOTIC MOTION IN CELESTIAL MECHANICS

From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com FROM ORDERED TO CHAOTIC MOTION IN CELESTIAL MECHANICS From Ordered to Chaotic Motion in Celestial Mechanics

### Astronomy 6570 Physics of the Planets

Astronomy 6570 Physics of the Planets Planetary Rotation, Figures, and Gravity Fields Topics to be covered: 1. Rotational distortion & oblateness 2. Gravity field of an oblate planet 3. Free & forced planetary

### Basics of Kepler and Newton. Orbits of the planets, moons,

Basics of Kepler and Newton Orbits of the planets, moons, Kepler s Laws, as derived by Newton. Kepler s Laws Universal Law of Gravity Three Laws of Motion Deriving Kepler s Laws Recall: The Copernican

### Stability of the Lagrange Points, L 4 and L 5

Stability of the Lagrange Points, L 4 and L 5 Thomas Greenspan January 7, 014 Abstract A proof of the stability of the non collinear Lagrange Points, L 4 and L 5. We will start by covering the basics of

### Geometric Mechanics and the Dynamics of Asteroid Pairs

Geometric Mechanics and the Dynamics of Asteroid Pairs WANG-SANG KOON, a JERROLD E. MARSDEN, a SHANE D. ROSS, a MARTIN LO, b AND DANIEL J. SCHEERES c a Control and Dynamical Systems, Caltech, Pasadena,

### The observed Trojans and the global dynamics around the Lagrangian points of the Sun Jupiter system.

The observed Trojans and the global dynamics around the Lagrangian points of the Sun Jupiter system. Philippe Robutel, Frederic Gabern and Àngel Jorba Abstract In this paper, we make a systematic study

### Tides and Lagrange Points

Ast111, Lecture 3a Tides and Lagrange Points Arial view of Tidal surge in Alaska (S. Sharnoff) Tides Tidal disruption Lagrange Points Tadpole Orbits and Trojans Tidal Bulges Tides Tidal Force The force

### On Sun-Synchronous Orbits and Associated Constellations

On Sun-Synchronous Orbits and Associated Constellations Daniele Mortari, Matthew P. Wilkins, and Christian Bruccoleri Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843,

### PHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015

PHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015 When you do a calculation, show all your steps. Do not just give an answer. You may work with others, but the work you submit should be your own.

### Lectures on Periodic Orbits

Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete

### On the complex analytic structure of the golden invariant curve for the standard map

Nonlinearity 3 (1990) 39-44. Printed in the UK On the complex analytic structure of the golden invariant curve for the standard map Albert0 BerrettitSO and Luigi ChierchiaSII t Institut de Physique ThBorique,

### 8.1 Bifurcations of Equilibria

1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations

### Observational Astronomy - Lecture 5 The Motion of the Earth and Moon Time, Precession, Eclipses, Tides

Observational Astronomy - Lecture 5 The Motion of the Earth and Moon Time, Precession, Eclipses, Tides Craig Lage New York University - Department of Physics craig.lage@nyu.edu March 2, 2014 1 / 29 Geosynchronous

### Investigation of evolution of orbits similar to that of (4179) Toutatis

Investigation of evolution of orbits similar to that of (4179) Toutatis Wm. Robert Johnston May 1994 Abstract This project compared the orbital evolution of asteroid (4179) Toutatis to that of several

### On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems

Celestial Mechanics and Dynamical Astronomy DOI 1007/s10569-013-9501-z On the extension of the Laplace-Lagrange secular theory to order two in the masses for extrasolar systems Anne-Sophie Libert Marco

### PH 120 Project # 2: Pendulum and chaos

PH 120 Project # 2: Pendulum and chaos Due: Friday, January 16, 2004 In PH109, you studied a simple pendulum, which is an effectively massless rod of length l that is fixed at one end with a small mass

### Orbits, Integrals, and Chaos

Chapter 7 Orbits, Integrals, and Chaos In n space dimensions, some orbits can be formally decomposed into n independent periodic motions. These are the regular orbits; they may be represented as winding

### INVARIANT TORI IN THE LUNAR PROBLEM. Kenneth R. Meyer, Jesús F. Palacián, and Patricia Yanguas. Dedicated to Jaume Llibre on his 60th birthday

Publ. Mat. (2014), 353 394 Proceedings of New Trends in Dynamical Systems. Salou, 2012. DOI: 10.5565/PUBLMAT Extra14 19 INVARIANT TORI IN THE LUNAR PROBLEM Kenneth R. Meyer, Jesús F. Palacián, and Patricia

### A SYMPLECTIC KEPLERIAN MAP FOR PERTURBED TWO-BODY DYNAMICS. Oier Peñagaricano Muñoa 1, Daniel J. Scheeres 2

AIAA/AAS Astrodynamics Specialist Conference and Exhibit 8 - August 008, Honolulu, Hawaii AIAA 008-7068 AIAA 008-7068 A SYMPLECTIC KEPLERIAN MAP FOR PERTURBED TWO-BODY DYNAMICS Oier Peñagaricano Muñoa,

### From the Earth to the Moon: the weak stability boundary and invariant manifolds -

From the Earth to the Moon: the weak stability boundary and invariant manifolds - Priscilla A. Sousa Silva MAiA-UB - - - Seminari Informal de Matemàtiques de Barcelona 05-06-2012 P.A. Sousa Silva (MAiA-UB)

### Nonlinear Control Lecture 2:Phase Plane Analysis

Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53

### Schilder, F. (2005). Algorithms for Arnol'd tongues and quasi-periodic tori : a case study.

Schilder, F. (25). Algorithms for Arnol'd tongues and quasi-periodic tori : a case study. Early version, also known as pre-print Link to publication record in Explore ristol Research PDF-document University

### Control of chaos in Hamiltonian systems

Control of chaos in Hamiltonian systems G. Ciraolo, C. Chandre, R. Lima, M. Vittot Centre de Physique Théorique CNRS, Marseille M. Pettini Osservatorio Astrofisico di Arcetri, Università di Firenze Ph.

### 1. Solar System Overview

Astronomy 241: Foundations of Astrophysics I 1. Solar System Overview 0. Units and Precision 1. Constituents of the Solar System 2. Motions: Rotation and Revolution 3. Formation Scenario Units Text uses

### Lecture Notes for PHY 405 Classical Mechanics

Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter

### Controlling chaotic transport in Hamiltonian systems

Controlling chaotic transport in Hamiltonian systems Guido Ciraolo Facoltà di Ingegneria, Università di Firenze via S. Marta, I-50129 Firenze, Italy Cristel Chandre, Ricardo Lima, Michel Vittot CPT-CNRS,

### Research Paper. Trojans in Habitable Zones ABSTRACT

ASTROBIOLOGY Volume 5, Number 5, 2005 Mary Ann Liebert, Inc. Research Paper Trojans in Habitable Zones RICHARD SCHWARZ, 1 ELKE PILAT-LOHINGER, 1 RUDOLF DVORAK, 1 BALINT ÉRDI, 2 and ZSOLT SÁNDOR 2 ABSTRACT

### Phys 270 Final Exam. Figure 1: Question 1

Phys 270 Final Exam Time limit: 120 minutes Each question worths 10 points. Constants: g = 9.8m/s 2, G = 6.67 10 11 Nm 2 kg 2. 1. (a) Figure 1 shows an object with moment of inertia I and mass m oscillating

### One Dimensional Dynamical Systems

16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar

### Introduction to Accelerator Physics Old Dominion University. Nonlinear Dynamics Examples in Accelerator Physics

Introduction to Accelerator Physics Old Dominion University Nonlinear Dynamics Examples in Accelerator Physics Todd Satogata (Jefferson Lab) email satogata@jlab.org http://www.toddsatogata.net/2011-odu

### SWING-BY MANEUVERS COMBINED WITH AN IMPULSE

Copyright 2013 by ABCM SWING-BY MANEUVERS COMBINED WITH AN IMPULSE Alessandra Ferraz da Silva Antonio F. Bertachini de A. Prado Instituto Nacional de Pesquisas Espaiciais, Av. dos Astronautas 1758, São

### The MATROS project: Stability of Uranus and Neptune Trojans. The case of 2001 QR322

A&A 41, 725 734 (23) DOI: 1.151/4-6361:231275 c ESO 23 Astronomy & Astrophysics The MATROS project: Stability of Uranus and Neptune Trojans. The case of 21 QR322 F. Marzari 1, P. Tricarico 1, and H. Scholl

### Transverse wobbling. F. Dönau 1 and S. Frauendorf 2 1 XXX 2 Department of Physics, University of Notre Dame, South Bend, Indiana 46556

Transverse wobbling F. Dönau and S. Frauendorf XXX Department of Physics, University of Notre Dame, South Bend, Indiana 46556 PACS numbers:..re, 3..Lv, 7.7.+q II. I. INTRODUCTION TRANSVERSE AND LONGITUDINAL

### 1 General Definitions and Numerical Standards

This chapter provides general definitions for some topics and the values of numerical standards that are used in the document. Those are based on the most recent reports of the appropriate working groups

### On the reliability of computation of maximum Lyapunov Characteristic Exponents for asteroids

Dynamics of Populations of Planetary Systems Proceedings IAU Colloquium No. 197, 25 Z. Knežević and A. Milani, eds. c 25 International Astronomical Union DOI: 1.117/S1743921348646 On the reliability of

### Chapter 5 - Part 1. Orbit Perturbations. D.Mortari - AERO-423

Chapter 5 - Part 1 Orbit Perturbations D.Mortari - AERO-43 Orbital Elements Orbit normal i North Orbit plane Equatorial plane ϕ P O ω Ω i Vernal equinox Ascending node D. Mortari - AERO-43 Introduction

### Radial Acceleration. recall, the direction of the instantaneous velocity vector is tangential to the trajectory

Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential to the trajectory 1 Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential

### arxiv:chao-dyn/ v1 20 May 1994

TNT 94-4 Stochastic Resonance in Deterministic Chaotic Systems A. Crisanti, M. Falcioni arxiv:chao-dyn/9405012v1 20 May 1994 Dipartimento di Fisica, Università La Sapienza, I-00185 Roma, Italy G. Paladin

### PHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements.

PHYSICS Course Structure Unit Topics Marks I Physical World and Measurement 1 Physical World 2 Units and Measurements II Kinematics 3 Motion in a Straight Line 23 4 Motion in a Plane III Laws of Motion

### How Kepler discovered his laws

How Kepler discovered his laws A. Eremenko December 17, 2016 This question is frequently asked on History of Science and Math, by people who know little about astronomy and its history, so I decided to

### Dynamical Stability of Terrestrial and Giant Planets in the HD Planetary System

Dynamical Stability of Terrestrial and Giant Planets in the HD 155358 Planetary System James Haynes Advisor: Nader Haghighipour ABSTRACT The results of a study of the dynamical evolution and the habitability

### Patterns in the Solar System (Chapter 18)

GEOLOGY 306 Laboratory Instructor: TERRY J. BOROUGHS NAME: Patterns in the Solar System (Chapter 18) For this assignment you will require: a calculator, colored pencils, a metric ruler, and meter stick.

### Binary star formation

Binary star formation So far we have ignored binary stars. But, most stars are part of binary systems: Solar mass stars: about 2 / 3 are part of binaries Separations from: < 0.1 au > 10 3 au Wide range

### Orbital Motion in Schwarzschild Geometry

Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation

### Kinematics of fluid motion

Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx

### D. most intense and of longest duration C. D.

Astronomy Take Home Test Answer on a separate sheet of paper In complete sentences justify your answer Name: 1. The Moon s cycle of phases can be observed from Earth because the Moon 4. The accompanying

### Orbital resonances in the inner neptunian system I. The 2:1 Proteus Larissa mean-motion resonance

Icarus 188 (2007) 386 399 www.elsevier.com/locate/icarus Orbital resonances in the inner neptunian system I. The 2:1 Proteus arissa mean-motion resonance Ke Zhang, Douglas P. Hamilton Department of Astronomy,

### arxiv: v1 [astro-ph.ep] 10 Sep 2009

Astronomy & Astrophysics manuscript no. secular c ESO 29 September 1, 29 Constructing the secular architecture of the solar system I: The giant planets A. Morbidelli 1, R. Brasser 1, K. Tsiganis 2, R.

### Study of the decay time of a CubeSat type satellite considering perturbations due to the Earth's oblateness and atmospheric drag

Journal of Physics: Conference Series PAPER OPEN ACCESS Study of the decay time of a CubeSat type satellite considering perturbations due to the Earth's oblateness and atmospheric drag To cite this article:

### On fully developed mixed convection with viscous dissipation in a vertical channel and its stability

ZAMM Z. Angew. Math. Mech. 96, No. 12, 1457 1466 (2016) / DOI 10.1002/zamm.201500266 On fully developed mixed convection with viscous dissipation in a vertical channel and its stability A. Barletta 1,

### 1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point

Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point

### Trajectories of the planets - impact of parameters on the shape of orbit (Translated from Polish into English by Maksymilian Miklasz)

Trajectories of the planets - impact of parameters on the shape of orbit (Translated from Polish into English by Maksymilian Miklasz) Abstract: This article contains presentation of mathematical function,

### SPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODIC FORCING. splitting, which now seems to be the main cause of the stochastic behavior in

SPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODIC FORCING A. DELSHAMS, V. GELFREICH, A. JORBA AND T.M. SEARA At the end of the last century, H. Poincare [7] discovered the phenomenon of separatrices splitting,

### THE FOUCAULT PENDULUM (WITH A TWIST)

THE FOUCAULT PENDULUM (WITH A TWIST) RICHARD MOECKEL The Foucault pendulum is often given as proof of the rotation of the earth. As the pendulum swings back and forth, the positions of maximum amplitude

### Can we see evidence of post-glacial geoidal adjustment in the current slowing rate of rotation of the Earth?

Can we see evidence of post-glacial geoidal adjustment in the current slowing rate of rotation of the Earth? BARRETO L., FORTIN M.-A., IREDALE A. In this simple analysis, we compare the historical record

### Astronomy 111 Exam Review Problems (Real exam will be Tuesday Oct 25, 2016)

Astronomy 111 Exam Review Problems (Real exam will be Tuesday Oct 25, 2016) Actual Exam rules: you may consult only one page of formulas and constants and a calculator while taking this test. You may not

### Spiral Structure Formed in a Pair of Interacting Galaxies

J. Astrophys. Astr. (1993) 14, 19 35 Spiral Structure Formed in a Pair of Interacting Galaxies Ch. L. Vozikis & Ν. D.Caranicolas Department of Physics, Section of Astrophysics, Astronomy and Mechanics,

### I ve Got a Three-Body Problem

I ve Got a Three-Body Problem Gareth E. Roberts Department of Mathematics and Computer Science College of the Holy Cross Mathematics Colloquium Fitchburg State College November 13, 2008 Roberts (Holy Cross)

### SUN INFLUENCE ON TWO-IMPULSIVE EARTH-TO-MOON TRANSFERS. Sandro da Silva Fernandes. Cleverson Maranhão Porto Marinho

SUN INFLUENCE ON TWO-IMPULSIVE EARTH-TO-MOON TRANSFERS Sandro da Silva Fernandes Instituto Tecnológico de Aeronáutica, São José dos Campos - 12228-900 - SP-Brazil, (+55) (12) 3947-5953 sandro@ita.br Cleverson

### APS 1030 Astronomy Lab 79 Kepler's Laws KEPLER'S LAWS

APS 1030 Astronomy Lab 79 Kepler's Laws KEPLER'S LAWS SYNOPSIS: Johannes Kepler formulated three laws that described how the planets orbit around the Sun. His work paved the way for Isaac Newton, who derived

### Invariant Manifolds of Dynamical Systems and an application to Space Exploration

Invariant Manifolds of Dynamical Systems and an application to Space Exploration Mateo Wirth January 13, 2014 1 Abstract In this paper we go over the basics of stable and unstable manifolds associated

### Mean-Motion Resonance and Formation of Kirkwood Gaps

Yan Wang Project 1 PHYS 527 October 13, 2008 Mean-Motion Resonance and Formation of Kirkwood Gaps Introduction A histogram of the number of asteroids versus their distance from the Sun shows some distinct

### The Restricted 3-Body Problem

The Restricted 3-Body Problem John Bremseth and John Grasel 12/10/2010 Abstract Though the 3-body problem is difficult to solve, it can be modeled if one mass is so small that its effect on the other two

### Continuous Wave Data Analysis: Fully Coherent Methods

Continuous Wave Data Analysis: Fully Coherent Methods John T. Whelan School of Gravitational Waves, Warsaw, 3 July 5 Contents Signal Model. GWs from rotating neutron star............ Exercise: JKS decomposition............