Sect. 6-5: Kepler’s Laws & Newton’s Synthesis

“Laws” of Planetary Motion. Johannes Kepler German astronomer (1571 – 1630) Spent most of his career tediously analyzing huge amounts of observational data (most compiled by Tycho Brahe) on planetary motion (orbit periods, orbit radii, etc.) He used his analysis to develop “Laws” of Planetary Motion. “Laws” in the sense that they agree with observation, but not true theoretical laws, such as Newton’s Laws of Motion & Newton’s Universal Law of Gravitation.

Kepler’s “Laws” Kepler’s First Law Kepler’s Second Law Kepler’s “Laws” are consistent with & are obtainable from Newton’s Laws Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus Kepler’s Second Law The radius vector drawn from the Sun to a plane sweeps out equal areas in equal time intervals Kepler’s Third Law The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit

Math Review: Ellipses  Typical Ellipse The points F1 & F2 are each a focus of the ellipse Located a distance c from the center Sum of r1 and r2 is constant Longest distance through center is the major axis, 2a a is called the semimajor axis Shortest distance through center is the minor axis, 2b b is called the semiminor axis  Typical Ellipse The eccentricity is defined as e = (c/a) For a circle, e = 0 The range of values of the eccentricity for ellipses is 0 < e < 1 The higher the value of e, the longer and thinner the ellipse

Ellipses & Planet Orbits The Sun is at one focus Nothing is located at the other focus Aphelion is the point farthest away from the Sun The distance for aphelion is a + c For an orbit around the Earth, this point is called the apogee Perihelion is the point nearest the Sun The distance for perihelion is a – c For an orbit around the Earth, this point is called the perigee

All planets move in elliptical orbits with the Sun at one focus Kepler’s 1st Law All planets move in elliptical orbits with the Sun at one focus A circular orbit is a special case of an elliptical orbit The eccentricity of a circle is e = 0. Kepler’s 1st Law can be shown (& was by Newton) to be a direct result of the inverse square nature of the gravitational force. Comes out of N’s 2nd Law + N’s Gravitation Law + Calculus Elliptic (and circular) orbits are allowed for bound objects A bound object repeatedly orbits the center An unbound object would pass by and not return These objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)

Kepler’s 1st Law Each planet’s orbit is an ellipse, with the Sun at one focus. Figure 6-16. Caption: Kepler’s first law. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1 and F2) remains constant. That is, the sum of the distances, F1P + F2P, is the same for all points on the curve. A circle is a special case of an ellipse in which the two foci coincide, at the center of the circle. The semimajor axis is s (that is, the long axis is 2s) and the semiminor axis is b, as shown. The eccentricity, e, is defined as the ratio of the distance from either focus to the center divided by the semimajor axis a. Thus es is the distance from the center to either focus, as shown. For a circle, e = 0. The Earth and most of the other planets have nearly circular orbits. For Earth e = 0.017.

Orbit Examples eHalley’s comet = 0.97 eMercury = 0.21 Fig. (a): Mercury’s orbit has the largest eccentricity of the planets. eMercury = 0.21 Note: Pluto’s eccentricity is ePluto = 0.25, but, as of 2006, it is officially no longer classified as a planet! Fig. (b): Halley’s Comet’s orbit has high eccentricity eHalley’s comet = 0.97 Remember that nothing physical is located at the second focus The small dot

Kepler’s 2nd Law The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals  Kepler’s 2nd Law can be shown (& was by Newton) to be a direct result of the fact that N’s Gravitation Law gives Conservation of Angular Momentum for each planet. The Gravitational force produces no Torque (it is  to the motion) so that Angular Momentum is conserved. (Neither torque nor angular momentum have been discussed yet.)

Kepler’s 2nd Law An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times. Figure 6-17. Caption: Kepler’s second law. The two shaded regions have equal areas. The planet moves from point 1 to point 2 in the same time as it takes to move from point 3 to point 4. Planets move fastest in that part of their orbit where they are closest to the Sun. Exaggerated scale.

Kepler’s 2nd Law Geometrically, in a time dt, the radius vector r sweeps out the area dA = half the area of the parallelogram The displacement is dr = v dt Mathematically, this means That is: the radius vector from the Sun to any planet sweeps out equal areas in equal times

Kepler’s 3rd Law The square of the orbital period T of any planet is proportional to the cube of the semimajor axis a of the elliptical orbit If the orbit is circular & of radius r, this follows from Newton’s Universal Gravitation. This gravitational force supplies a centripetal force for user in Newton’s 2nd Law    Ks is a constant Ks is a constant, which is the same for all planets.