Orbits and gravity

• The Earth lay at the centre of nested “crystal spheres” each rotating at different rates.
• Retrograde motion: when a planet temporarily moves westward in the sky over the course of several weeks or months (instead of eastward, as it typically does)
• Ptolemy devised a system of circular epicycles to explain retrograde motion.

  • It is the movement of small circles upon larger circles explaining retrograde motion.
• Copernicus -Earth-centered model of the universe, which proposed that the sun was the center of the universe and that the planets revolved around it.
• • Tycho Brahe was a Danish astronomer who was the first to use a telescope to observe the sky.
  • Compiled the most accurate (one arcminute) naked eye measurements ever made of planetary positions.
  • Thought Earth was the centre of the solar system but recognized that other planets go around the sun.
• Kepler first tried to match Tycho’s observations with circular orbits• But an 8-arcminute discrepancy led him eventually to adopt elliptical orbits for the planets around the Sun.
• An ellipse looks like an elongated circle.

  
• Kepler's 3 laws of planetary motion:

  • 1st law: The orbit of each planet around the sun is an ellipse with the sun at one focus.
  • 2nd law suggests that a planet moves faster when its closer to the sun and slower when further away. (As a planet moves around its orbit, it sweeps out equal areas in equal times).
  • 3rd law: More distant planets orbit the sun at a slower speed.
• Why did Kepler's 2nd law suggest that?

  Because the angular momentum is conserved.

  mass x velocity x distance
• Kepler's third law obeys a relationship of:

  p^2=a^3

  P = orbital period in years

  a = semi-major axis of orbit in AU
• 1 Astronomical unit is the average distance between Earth and the sun.
• Galileo pointed a telescope at the sky for the first time.
• Galileo saw 4 moons orbiting Jupiter, proving that not all objects orbit Earth.
• The Catholic Church ordered Galileo to recant his claim that Earth orbits the Sun in 1633. His book on the subject was removed from the Church’s index of banned books in 1824. Galileo was formally vindicated by the Church in 1992.
• The universal law of gravitation:

  1. Every mass attracts every other mass.
  2. Attraction is directly proportional to the product of their masses.
  3. Attraction is inversely proportional to the square of the distance between their centres.
• Orbits can be: Bound ( ellipses ) & unbound -

  ( Parabola & hyperbola )
• Newton and Kepler’s Third Law: Newton’s laws of gravity and motion showed that the relationship between the orbital period and average orbital distance of a system tells us the total mass of the system.
• Newton's version of Kepler's law:

  P 2 = (4π 2/G(M1 + M2) ) x a 3

  p = Orbital period

  a = Average orbital distance between centres

  (M1+M2) = masses
• Why does Kepler’s Third "Law" work?

  P 2 = (4π 2/G(M1 + M2)) x a 3

  • M1 + M2 = sum of object masses

  • But in our Solar System, the mass of the Sun M1 ≫ M2 (mass of planet). So we can ignore M2.
• Two objects (Sun and Earth, binary star system) don’t orbit the more massive object:

  • They both orbit around their centre of mass
• Because of momentum conservation, objects orbit around their centre of mass.