Kepler’s Laws & Planetary Motion

“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.) Used his analysis to develop “Laws” of Planetary Motion. Kepler’s “Laws” are Laws only in the sense that they agree with observation. They are not true theoretical laws, such as Newton’s Laws of Motion & Newton’s Universal Law of Gravitation.

Newton’s Universal Gravitation Law Kepler studied results of other astronomer’s measurements of portions of the Moon, planets, etc. He found the motion of the Moon and planets could be described by a series of laws Now called Kepler’s Laws of Planetary Motion Kepler’s Laws are mathematical rules inferred from the available information about the motion in the solar system Kepler could not give a scientific explanation or derivation of his laws. Newton’s Laws of Motion & Newton’s Universal Gravitation Law give the explanation

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 planet 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 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 The 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 The Aphelion is the point farthest away from the Sun The distance for the aphelion is a + c For an orbit around the Earth, this point is called the apogee The 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 + 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 First Law of Planetary Motion The Planets Move in Elliptical Orbits The Sun is at one focus This was very different from the previous idea that the planets moved in perfect circles with the Sun at the center

Orbit Examples 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 is located at the second focus.

Kepler’s First Law: Planetary Orbits

Kepler’s 2nd Law The 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 Newton’s Gravitation Law.

Kepler’s 2nd Law of Planetary Motion A line connecting a planet to the sun sweeps out equal areas in equal times as the planet moves around its orbit If the time required for the planet to sweep out area A1 is equal to the time to sweep out A2, the areas will be equal The planet’s speed will be slowest when it is farthest from its sun and fastest when it is closest

axis a of the elliptical orbit 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 Law. This gravitational force supplies a centripetal force for use in Newton’s 2nd Law Ks is a constant Ks is a constant, which is the same for all planets.

Kepler’s 3rd Law It can be shown: This also applies to an elliptical orbit, with replacement of r with a, where a is the semimajor axis. Ks is independent of the planet mass, & is valid for any planet Note: If an object is orbiting another object, the value of K will depend on the mass of the object being orbited. For example, for the Moon’s orbit around the Earth, KSun is replaced with KEarth, where KEarth is obtained by replacing MSun by MEarth in the above equation.

Orbit Examples Low Earth orbit Geosynchronous Orbit International Space Station, for example T ~ 90 minutes r ~ 6.66 x 106 m Geosynchronous Orbit T = 1 day Always above the same position above the Earth r = 4.2 x 107 m

Solar System Data Kepler’s 3rd Law: The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun.