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.

Ellipsesa (1-e) = closest approach e = eccentricity

Ellipsesa (1- e) = closest approach (perihelion)

a (1+ e) = farthest away (aphelion)

Circle e = 0

radius = a

Ellipse e > 0

Kepler’s II LawA Planet’s Orbital Vector Sweeps Out Equal Area

in Equal Intervals of Time

Equal Area in Equal TimeThe time required to travel from point A’ to B’ is

equal to the time required to travel from A to B.

Angular MomentumAngular 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 LawPeriod2/Distance3 = 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 1.000.

P2 = a3

Closest ApproachFor 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 ApproximationThe 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:

Vcir= (GM/R)1/2

Centripetal Accelerationa = v2 r

Centripetal Force = m a

= m v2 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 Energy Kinetic 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 v2

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

To and Fro

To and Fro

To and Fro

To and Fro

To and Fro

To and Fro

h

The pendulum swingsuntil it has reached thesame height on the otherside, before pausing tooscillate back.

Total EnergyA projectile has an amount of kinetic energy given by

K.E. = 1/2 m V2

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 V2 - 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 VelocityThe 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 Velocity0 = 1/2 m V2 - mGM/R

V2/2 = GM/R

Vesc = (2GM/R)1/2

The escape velocity at the earth's

surface is = 11.2 km/sec.

OrbitsObject 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 > Vesc > 0 Unbound Parabola V = Vesc = 0 1 pass Ellipse V < Vesc < 0 a(1-e), a(1+e) Circle V = Vcirc Minimum a = radius

e = 0

e > 0

e >> 0