chapter 3.3 newton's universal law of gravitation & 3.4 orbits in the solar system

gravity

the force that attracts a body towards the centre of the earth, or towards any other physical body having mass

• newton believed that Earth’s gravity might extend as far as the Moon and produce the force required to curve the Moon’s path from a straight line and keep it in its orbit

• newton hypothesised that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies

• whenever there are masses in the universe, they will interact via the force of gravitational attraction

• never becomes 0 (implied by newton’s law)

universal law of gravitation

newton hypothesised that there is a universal attraction among all bodies everywhere in space

• he had to find a precise mathematical description of that gravitational force that had to dictate that the planets move exactly as Kepler had described them to

• also, that gravitational force had to predict the correct behavior of falling bodies on Earth, as observed by Galileo

• he concluded that the magnitude of the force of gravity must decrease with increasing distance between the Sun and a planet (or any two objects) and that gravitational attraction between two bodies must be proportional to their masses

• F_gravity= gravitational force between two objects

• M₁ and M₂= masses of the two objects

• R= their separation

• G= ‘universal gravitational constant’

weight

depends on the local force of gravity -

• a body’s relative mass or the quantity of matter contained by it, giving rise to a downward force

acceleration on Earth vs Moon

• the Moon is 60 Earth radii away from the centre of Earth

• if gravity (thus acceleration) gets weaker with distance squared, the acceleration the Moon experience should be a lot less for the object compared to Earth

• The acceleration should be (1/60)² = 1/3600 (or 3600 times less—about 0.00272 m/s)

Kepler’s relationship between the orbital period of a planet’s revolution and its distance from the Sun reformulated by newton

eccentricity of planets

most eccentric orbit; Mercury (0.21)

rest of planets; less than 0.1

orbital period lengths

planets closest to the Sun have the shortest orbital period, increasing as distance from the Sun increases

• Mercury (88 Earth days), thus it has the highest orbital speed (48km/s)

• Neptune (165 years), average orbital speed of 5km/s

planetary orbits

confined to a common plane (near the plane of Earth’s orbit (ecliptic)

asteroid

small rocky body orbiting the sun

• have smaller semimajor axes than comets

• the majority of asteroids lie between 2.2 and 3.3 AU (asteroid belt) —> middle gap between the orbits of Mars and Jupiter

comet

a celestial object consisting of a nucleus of ice and dust and, when near the sun, a ‘tail’ of gas and dust particles pointing away from the sun

• tend to have larger orbits with greater eccentricity than asteroids

• eccentricity of their orbits = 0.8 or greater

• according to Kepler’s second law, they spend most their time far from the Sun, moving slowly

• as they approach perihelion (point of an orbit which is closest to the Sun) , the comets speed up and whip through the inner parts of their orbits more rapidly