# Dynamical properties of the Solar System. Second Kepler s Law. Dynamics of planetary orbits. ν: true anomaly

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1 First Kepler s Law The secondary body moves in an elliptical orbit, with the primary body at the focus Valid for bound orbits with E < 0 The conservation of the total energy E yields a constant semi-major axis: a = G (m 1 +m 2 )/(2E) Dynamical properties of the Solar System Suggested reading: Physics of the Earth and the Solar System B. Bertotti and P. Farinella, Kluwer Academic Publishers Planets and Astrobiology ( ) G. Vladilo apocentre pericentre ν: true anomaly 1 3 Dynamics of planetary orbits In the limit of non-relativistic motion, Newton s laws can be applied Relativistic effects should be taken into account for orbits very close to the star Even using the newtonian approximation, the treatment of N-body systems, such as the Solar System, is extremely complex However, over a short period of time, the motion of a planet around its host star can be treated as a 2-body problem, neglecting perturbations due to other bodies in the system In this limit, two objects can be considered to be isolated in space and to move around each other according to Keplerian laws Kepler s laws are easily shown to be the result of the inverse-square law of gravity, with the Sun as the central body, and the conservation of angular momentum and energy These laws are fundamental to understand the dynamics of the Solar System and extrasolar planetary systems Second Kepler s Law The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time 2 4

2 Second Kepler s Law da: infinitesimal area swept by an infinitesimal displacement dθ of the radial vector r Area swept per unit time, where ω is the angular velocity Third Kepler s Law The meaning of the third Kepler s law can be easily understood in the case of a circular orbit of radius a By equating the centripetal force of the planet with the gravitational force exerted by the central body, we have From this we obtain the circular velocity L: angular momentum Kepler s 2 nd Law follows from the conservation of the angular momentum and the period This last expression yields the third Kepler s law for the case M p << M * 5 7 Third Kepler s Law The square of a planet s orbital period P about the Sun (in years) equals the cube of its semimajor axis a (in AU) By solving the two-body problem one can derive the more general expression where M * and M p are the mass of the central star and planet, respectively. Since M p << M *, in practice we have Orbital elements In order to locate in 3D space the positions of a body moving in an elliptical orbit, a set of 6 orbital parameters (or elements) is required The orbital elements specify the size, shape, and orientation of the orbit in space and the position of the body at a particular instant in time (the epoch ) Commonly used parameters are a = the semi-major axis of the ellipse e = the eccentricity of the ellipse i = the inclination of the orbital plane to the reference plane Ω = the longitude of the node of the orbital plane on the reference plane ω = the argument of the pericenter T = the epoch at which the body is at the pericenter 6 8

3 Orbital elements Rotation periods and tilt of rotation axis Rotation is generally prograde with the rotation spin of the Sun Obliquity with respect to the ecliptic generally < 30 degrees Exceptions: Venus, Uranus 9 11 Orbital parameters of the Solar System planets Gravitational perturbations When more than 2 bodies are present, the gravitational perturbations induced by the other bodies will alter the orbital and rotational parameters of the system Example: precession of the pericenter which, in the case of the Earth, yields the well-known precession of the equinox Period and semimajor axis obey to 3 rd Kepler s law Eccentricity is generally low Orbits are approximately coplanar, with small inclinations I relative to the ecliptic Orbital spins are generally aligned with the spin of solar rotation 10 Different types of perturbations exist, characterized by different intensities and time scales - To study perturbations we introduce the concept of resonance 12

4 Resonances When the ratio of dynamical periodicities of two bodies can be expressed as the ratio of two small integers, gravitational perturbations tend to cumulate, leading to a resonance P 1 /P 2 = n/m Orbital resonances Examples of orbital stabilization Concentrations of minor bodies stabilized by dynamical effects Hilda family: orbital resonance 3:2 with Jupiter Trojans: resonace 1:1 with Jupiter Resonances may stabilize or destabilize orbits Since the orbital period P and semimajor axis a are linked though the 3 rd Kepler s law, resonances play a key role in shaping the architecture of planetary systems 13 Resonances P Orbital resonance 1,orb /P 2,orb = n/m When the orbital periods are in resonance It is called MEAN MOTION ORBITAL RESONANCE Orbital resonance can generate a dynamical instability In the most extreme case, it may lead to the ejection of one of the two bodies from the system in dynamical (i.e., fast) time scales Orbital resonances Examples of orbital destabilization Distribution of semimajor axis in the main asteroid belt Gaps result from the ejection of asteroids from regions in resonance with Jupiter s orbital period Secular resonance P 1,prec /P 2,prec = n/m When the periods of precession of the periapsides are in resonance Secular resonances may lead to gradual changes of the orbital eccentricity and inclination on longer time scales, of the order of millon years 14 16

5 Other examples of resonances The architecture of rings around giant planets results from resonance effects between debris material that surrounds the planets and satellites orbiting the same planets Such resonance effects generate gaps in the debris material, leading to the formation of the characteristic ring pattern Evolution of the obliquity of the Earth s rotation axis during the past 5 millon years, as predicted by numerical simulations Effects of gravitational perturbations on the tilt of the planet rotation axis Evolution of the obliquity of the Mars rotation axis during the past 10 millon years, as predicted by numerical simulations The tilt of the rotation axis influences the planetary climate and habitability Gravitational torques between different planets tend to alter the tilt of the rotation axis in a chaotic fashion For instance, the axis tilt of Mars is believed to evolve significantly, leading to extreme inclinations in some epochs This is not the case for the Earth, since the axist tilt of our planet is stabilized by the presence of the Moon Therefore, the Earth s axis experiences small oscillations of its obliquity around its typical value of 23.5 degrees 18 20

6 Relativistic effects Surface tides Relativistic effects play a small but detectable role In the Solar System they are most evident in the precession of the perihelion of the orbit of Mercury, the planet deepest in the Sun s gravitational potential well General relativity adds 43 arcsec/century to the precession rate of Mercury s orbit, which is 574 arcsec/century Prior to Eintein s theory of General Relativity (1916) it was thought that the excess in the precession rate of Mercury was due to an hypothetical planet ( Vulcan ) orbiting interior to it, which has never been detected Surface tides can be understood by calculating the vectorial sum of the centrifugal force and the gravitational force (with its gradient inside the planet or satellite) Tidal effects Tidal effects are ubiquitous in the universe An example is provided by two bodies interacting gravitationally, the size of at least one of them being not negligible with respect to their mutual distance R ~ r In this case gravity gradients give rise to forces and torques that cause deformations and changes in the rotational state of affected bodies Tidal forces tend to vanish very rapidly with distance From differentiation of Newton s law in distance, r f T ~ r 3 Tidal locking Mechanism Tidal forces create a bulge If P rot > P orb, the body rotates through its tidal bulge, generating internal friction Friction slows downs the rotation period The friction stops when P rot = P orb We say that a body is tidally locked when tidal effects have slowed down the rotation period to P rot = P orb Examples Satellites around planets: Moon-Earth and Titan-Saturn systems Important for extrasolar planets close to their central star 22 24

7 Tidal heating The dissipation of tidal forces leads to internal heating of satellites The best known example is the satellite Io, orbiting Jupiter Dynamical stability of the Solar System How stable is the Solar System? We are not able to provide a simple analytical description of the dynamical evolution of a set of N > 3 bodies under the effects of recyprocal gravitational interaction The long-term dynamical stability of the Solar System can be tested with N-body gravitational simulations N-body simulations indicate that small variations of the orbital parameters of the solar system bodies lead to unstable configurations, with possible ejection of a body from the system This type of investigation confirm that dynamical interactions are essential in shaping the architecture of planetary systems Combined effects of tidal locking and resonances The case of the Jupiter satellites Io is tidally locked Europa and Ganymede are in resonance 26

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