Orbits Read Your Textbook: Foundations of Astronomy –Chapter 5 Homework Problems –Review Questions: 3, 4, 5, 9, 10 –Review Problems: 1, 3, 4 –Web Inquiries:

Kepler’s Laws I. Planets orbit the sun in ellipses with the sun located at one focus point.

Ellipses a (1-e) = closest approach e = eccentricity

Ellipses a (1- e) = closest approach (perihelion) a (1+ e) = farthest away (aphelion) Circle e = 0 radius = a Ellipse e > 0

Kepler’s II Law A Planet’s Orbital Vector Sweeps Out Equal Area in Equal Intervals of Time

Equal Area in Equal Time The time required to travel from point A’ to B’ is equal to the time required to travel from A to B.

Angular Momentum Angular Momentum L, is the product of a planet's mass (m), orbital velocity (v) and distance from the Sun (R). The formula is simple: L = m v R, where R is a function of e, the eccentricity.

Kepler’s III Law Period 2 /Distance 3 = constant for any system If the orbital period (P) is measured in years, and the average distance (a) is measured in A.U., the constant for the solar system is P 2 = a 3

Closest Approach For orbits around the Sun, critical points are perihelion and aphelion. The corresponding points for orbits around the Earth are perigee and apogee. The planetary orbital ellipses are very nearly circular. (i.e. the eccentricity is nearly zero, Earth e = 0.03) Circular orbits are a good approximation.

Circular Orbit Approximation The Circular Velocity is the rate at which an object would move in a circular orbit around a given massive body. Examples might be a satellite around the Earth or a planet around the Sun. The equation for circular velocity is: V cir = (GM/R) 1/2

Centripetal Acceleration a = v 2  r Centripetal Force = m a = m v 2  r Since velocity is a vector, even an object with constant speed that does not travel in a straight line, must experience an acceleration since acceleration is a change in velocity. Velocity is a vector which includes speed AND direction.

Two Types of Energy Potential EnergyKinetic Energy

Potential Energy The energy associated with an objects position The object has the potential to do WORK Work was done to give it this potential Examples: Ohh Chop, Chop

Potential Energy The motor pulls the cart up against gravity WORK = Force x distance mg x height

Potential Energy The motor pulls the cart up against gravity WORK = Force x distance mg x height Muscles do work against the tension in the bow string

Potential Energy The motor pulls the cart up against gravity WORK = Force x distance mg x height Muscles do work against the tension in the bow string Muscles do work against gravity to lift the axe above the ground

Potential Energy The roller coaster cart, the bow and the axe were all given potential energy. The change in the potential energy is identical to the work done. These objects now have the potential to do work and convert that stored energy.

Kinetic Energy The energy associated with an objects motion. KE = 1/2 m v 2 m = mass v = velocity Without velocity, there is no KE Chop, Chop

Energy When work is done, there is a change in energy. Energy is a conserved quantity, and remains unchanged in a physical system. TOTAL ENERGY, not Kinetic Energy or Potential Energy by themselves.

The Simple Pendulum The pendulum swings to and fro, where it stops, conservation of energy knows. TOTAL ENERGY = Potential Energy + Kinetic Energy

The Simple Pendulum The total energy of this system is zero.

The Simple Pendulum The total energy of this system is zero. This simple pendulum could be the sway in a grandfather clock, a child on a swing, a hypnotists watch, etc.

The Simple Pendulum Suppose someone does work against gravity to give it some potential energy?

The Simple Pendulum Suppose someone does work against gravity to give it some potential energy? The work done = Force x Distance Force = m g Distance = h The work done = potential energy gained = m g h h

The Simple Pendulum The total energy of the system is now (m g h), reflecting the work done to the system. What happens when we let go of the pendulum? h

To and Fro

h The pendulum swings until it has reached the same height on the other side, before pausing to oscillate back.

Total Energy A projectile has an amount of kinetic energy given by K.E. = 1/2 m V 2 It has a potential energy relative to earth's surface of P.E. = - mGM/R The total energy is conserved so that Total Energy = K.E. + P.E. Total Energy = 1/2 m V 2 - mGM/R In order to escape earth's gravity you have to have Total Energy > 0, so the critical point is when the energy = 0.

Escape Velocity The Escape Velocity is the velocity an object (of ANY mass) must have in order to leave the gravitational field of a massive body. It depends on the total amount of energy an object has. There is an escape velocity from the Earth's surface, from the Sun's surface, even from the solar system, and it depends on how massive that body is and how far you are away from the body.

Escape Velocity 0 = 1/2 m V 2 - mGM/R V 2 /2 = GM/R V esc = (2GM/R) 1/2 The escape velocity at the earth's surface is = 11.2 km/sec.

Orbits Object must have kinetic energy greater than the gravitational potential energy needed to escape the earth. The velocity associated with this kinetic energy is the escape velocity.

Escape Velocity and Orbits Conic Section Velocity Total Energy Orbit Hyperbola V > V esc > 0 Unbound Parabola V = V esc = 0 1 pass Ellipse V < V esc < 0 a(1-e), a(1+e) Circle V = V circ Minimum a = radius e = 0 e > 0 e >> 0