Gravity

Planetary Configurations:

The planets outside of the Earth's orbit (Mars, Jupiter, Saturn, Uranus, Neptune, Pluto) are called superior planets

Likewise, the planets inside of the Earth's orbit (Mercury, Venus) are called inferior planets.

Other configurations are:

Object at greatest western elongation = "morning star"
Object at greatest eastern elongation = "evening star"
only inferior planets have phases
transit = passage of an inferior planet across the Sun

Galileo's laws of Motion:

Aside from his numerous inventions, Galileo also laid down the first accurate laws of motion for masses. Galileo realized that all bodies accelerate at the same rate regardless of their size or mass. Everyday experience tells you differently because a feather falls slower than a cannonball. Galileo's genius lay in spotting that the differences that occur in the everyday world are in incidental complication (in this case, air friction) and are irrelevant to the real underlying properties (that is, gravity). He was able to abstract from the complexity of real-life situations the simplicity of an idealized law of gravity.

Key among his investigations are:

developed the concept of motion in terms of velocity (speed and direction) through the use of inclined planes.
developed the idea of force, as a cause for motion.
determined that the natural state of an object is rest or uniform motion, i.e. objects always have a velocity, sometimes that velocity has a magnitude of zero = rest.
objects resist change in motion, which is called inertia.

Galileo also showed that objects fall with the same speed regardless of their mass. The fact that a feather falls slowly than a steel ball is due to amount of air resistence that a feather experiences (alot) versus the steel ball (very little).

Hammer and Feather on Moon

Kepler's laws of Planetary Motion:

Kepler developed, using Tycho Brahe's observations, the first kinematic description of orbits, Newton will develop a dynamic description that involves the underlying influence (gravity)

1st law (law of elliptic orbits): Each planet moves in an elliptical orbit with the Sun at one focus.

Ellipses

2nd law (law of equal areas): a line connection the Sun and a planet (called the radius vector) sweeps out equal areas in equal times

Objects travel fastest at the low point of their orbit, and travel slowest at the high point of their orbit.
3rd law (law of harmonics): The square of a planet's orbital period is proportional to its mean distance from the Sun cubed.
The mathematical way to describe Kepler's 3rd law is:

P² α R³
where the α symbol means `proportional to'. Proportions are expressions that imply there exists some constant, k, that relates the period, P, and the radius, R, such that

P² = kR³
We can determine k by expressing the formula in units of the Earth and its orbit around the Sun, such that

(1 yr)² = k (1 A.U.)³
so k is equal to one, as long as we use units of years and A.U.'s (the Astronomical Unit, i.e. the distance from the Earth from the Sun). With k=1, then kepler's 3rd law becomes

P² = R³
The 3rd law is used to develop a ``yardstick'' for the Solar System, expressing the distance to all the planets relative to Earth's orbit by just knowing their period (timing how long it takes for them to go around the Sun).

click here to see the inner SS orbits

click here to see the outer SS orbits

click here to see orbits of equal or near equal mass objects

Orbits:

Many years after Kepler, it was shown that orbits actually come in many flavors, ellipses, circles, parabolas and hyperbolas; a family of curves called conic sections. There are five basic types of simple orbits: radial, ballistic, stable, polar and geosynchronous.

For an escape orbit, the velocity sufficient to escape gravitation pull of the planet, i.e. the major axis is infinite, such as the Voyager spacecraft

The direction a body travels in orbit can be direct, or prograde, in which the spacecraft moves in the same direction as the planet rotates, or retrograde, going in a direction opposite the planet's rotation.

The semi-major axis of an orbit is determined by the kinetic energy acquired by the rocket at burnout. This is equilvent to the burnout velocity. For low burnout velocities (below 25,000 ft/sec) the orbit is ballistic, meaning it does not escape the surface of the Earth. Burnout velocities above 25,000 ft/sec achieve stable orbit. At 35,000 ft/sec, the orbit reaches the distance of the Moon.

The amount of burnout velocity also determines the orbit type, an ellipse, a parabola or a hyperbolic path.

Newton:

Newton expanded on the work of Galileo to better define the relationship between energy and motion. In particular, he developed the following concepts:

change in velocity = acceleration -> caused by force
inertia = resistance to change in velocity and is proportional to the mass of the object
momentum = quantity of motion energy and is equal to mass times velocity
law of conservation of momentum = total momentum (mass x velocity) of an interaction is conserved -> is the same before and after

Example: Cars and Trucks on Ice!

A corollary to Newton's ideas was the so called Clockwork Universe model. A concept that states that the total momentum of the Universe is conserved, interactions redistribute the momentum, but the total never changes. In this model, God only starts the clock (initial cause), then it runs by itself for the rest of time.

Newton's laws of motion:

1st law: a body remains at rest or moves in a straight line of constant velocity as long as no external forces acts on it
2nd law: a body acted on by a force will accelerate such that force equals mass times acceleration (F=ma)
3rd law: for every action there is an equal and opposite reaction

Example: from Newton's 1st law we know that an object travels in a straight line unless acted upon by an external force. A circular orbit is clearly not a straight line, what is the force? Newton showed that the planets are acted on by the force of gravity arising from the Sun. Each orbit is a constantly changing velocity where gravity adds a small ``delta-vee'' at each moment. This ``delta-vee'' is what produces the elliptical curvature that is the orbit.

Example: from Newton's 2nd law when a baseball player hits a ball he applies a force, F, to the ball of mass m. Let's say he hits a tennis ball of mass of one-tenth the mass of a regular baseball (1/10m). What is the resulting acceleration? Ten times the acceleration of a regular baseball and, therefore, ten times the final velocity and ten times the distance hit.

Example: You are trapped on a lake of ice with a sandbag. Remembering Newton's 3rd law, how do you escape?

Newton's Law of Universal Gravitation:

Galileo was the first to notice that objects are ``pulled'' towards the center of the Earth, but Newton showed that this same force (gravity) was responsible for the orbits of the planets in the Solar System.

Objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances

All masses, regardless of size, attract other masses with gravity. You don't notice the force from nearby objects because their mass is so small compared to the mass of the Earth. Consider the following example:

With vector calculus, Newton was able to develop a cosmology which included the underlying cause of planetary motion, gravity, completed the solar system model begun by the Babylonians and early Greeks. The mathematical formulation of Newton's dynamic model of the solar system became the science of celestial mechanics, the greatest of the deterministic sciences.

Although Newtonian mechanics was the grand achievement of the 1700's, it was by no means the final answer. For example, the equations of orbits could be solved for two bodies, but could not be solved for three or more bodies. The three body problem puzzled astronomers for years until it was learned that some mathematical problems suffer from deterministic chaos, where dynamical systems have apparently random or unpredictable behavior (see below).

Differential Gravitational Forces (Tides):

Tides are caused by the interaction of a body's motion around a planet or the Sun and the internal force of gravity.

Water Tides:

Water tides are caused by the fact that the water on the Earth's surface is more easily deformed by tidal forces than the rocky crust. And the strength of tides are dependent on three factors:

location on the Earth's surface
orientation of the Sun and the Moon (both has approximately equal tidal influence on the surface of Earth)
geographic features (shape of bay, inlets, etc.)

Roche Limit:

What happens when the tidal forces become greater than the internal gravity of an object? The object is torn apart. This occurs when a moon approaches too close to its primary, a point called the Roche limit. The tidal forces increase as R, the distance between the planet and the moon, becomes smaller until the moon is disrupted into numerous small bodies. This is the origin to the rings around Saturn and other Jovian worlds.