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Planetary Motions • Planets are observed to move relative to the background stars ¾ Motion is generally regular, but sometimes shows retrograde motion that was very difficult to explain in geocentric theories Led to use of epicycles ¾ In a heliocentric theory retrograde motion is a natural consequence of the inner planets orbiting more quickly than the outer planets • Distances to planets can be determined using simple geometric calculations ¾ Interior: measure angle between planet and the Sun at greatest elongation (when they are farthest apart) ¾ Exterior: measure same angle, separated in time by planet’s orbital period. ¾ Copernicus had excellent values for all of the known planets, but they were less precise for the outermost ones (Jupiter and Saturn) because their periods are long enough that fewer oppositions could be observed Kepler’s Laws • Based on very accurate orbital data obtained by Tycho Brahe, Kepler derived three laws of orbital motion: 1. A planet orbits the Sun in an ellipse, with the Sun at one focus 2. A line connecting a planet to the Sun sweeps out equal areas in equal time intervals 3. For planets orbiting the Sun, P2=a3, where P is the period (in years) and a is the average distance from the Sun (in AU). Kepler’s First Law • Planets move around the Sun in an ellipse, with the Sun at one focus ¾ The distance r between the Sun and the planet is related to the semimajor axis length, a, and the ( ) a 1 − e2 eccentricity, e, by: r = 1 + e cos θ , where θ is the angle between r and the semimajor axis. ¾ The eccentricity relates the length of the semimajor axis (a) and semiminor axis (b) : a 2 = a 2e 2 + b 2 b = 1 − e2 a • The aphelion is the point in the orbit when the planet is farthest from the Sun. Perihelion is the point of closest approach. Kepler’s Second Law • Conservation of angular momentum means the quantity r L = mrvθ does not change with time. Here, vθ is the component of the planet’s velocity perpendicular to the vector r. ¾ Thus the planet moves fastest at perihelion, and slowest at aphelion: va vp = rp ra = (1 − e) (1 + e) ¾ The angular momentum is given by L = 2m πa 2 1 − e 2 P Kepler’s Third Law • This law relates the Period (P) and semimajor axis length (a) for an object of mass m, orbiting a more massive object of mass M. 4π 2 a 3 2 P = G ( M + m) where, G is the gravitational constant. • For the solar system, where the mass of the Sun is much greater than that of any planet, a very good 4π 2 a 3 2 approximation is P = GM Sun ¾ Since the constants are the same for all the objects in the solar system, we can write P 2 = a 3 , if P is measured in years, and a is measured in AU. Exercises 1. Calculate the aphelion and perihelion distances for Halley’s comet, which has a semi-major axis of 17.9 AU and an eccentricity of 0.967. From the equation for an ellipse: ( ) a 1 − e2 r= 1 + e cos θ we have a=17.9 AU and e=0.967. At closest approach (perihelion), θ=0 degrees, so: ( ) a 1 − e2 = a(1 − e) rp = 1+ e = 17.9 × 0.033 = 0.5907 AU While at aphelion, θ=180 degrees: ( ) a 1 − e2 = a(1 + e) ra = 1− e = 17.9 × 1.967 = 35.21AU which is beyond the orbit of Neptune. 2. How much faster does Earth move at perihelion, compared with its velocity at aphelion? The eccentricity of Earth’s orbit is e=0.0167. The ratio of the velocities at perihelion and aphelion are given by: vp va = (1 + e ) (1 − e ) 1.0167 0.9833 = 1.034 = So the Earth moves 3.4% faster at perihelion. 3. The dwarf planet Eris has a small moon, Dysnomia. This moon orbits at a distance of about 30,000 km, with a period of about 14 days. What is the combined mass of the Eris/Dysnomia system? 4π 2 a 3 From Kepler’s third law, P = G ( M + m) , we can solve 4π 2 a 3 22 for ( M + m) = GP 2 = 1.09 × 10 kg , about 1/7th the mass 2 of the Earth’s moon.