Mechanics of the universe (or at least our part of it)




Early Astronomical Thought

The Development of "Modern" Astronomy

Newton and the Classical Laws of Motion

Relativity and Space-time

Angular Momentum and Gravity

Orbital Mechanics



This is where some of the heavy-duty laws of nature come into effect

Lots of math and physics

This should be the toughest two weeks of the course

Survive this and you'll probably make it!

In large part first clearly defined by Sir Issac Newton (1642-1727)

"Principia" in 1686

We'll get to him soon enough

But first some background...


Matter: The stuff of reality

Mass: How much stuff there is in an object

Does not change with a change of location

Volume: how much space a mass occupies

DIGRESS TO: Nature loves spheres

Density: a very important concept (so pay attention)

In my opinion: one of the fundamental driving forces of nature

Defined as: Density = mass / volume (units: g/cm3)

EXAMPLES: See table 3:1 pg. 50

Panning for gold: gold in a placer deposit

Oil, water, antifreeze, and mercury

Hot air vs. cold air


Pepsi cans

Candle in the wind

Some extremes:

Comet's tail: 10-16 g/cm3

Neutron star: 1015 g/cm3

Density differences are responsible for internal structure of the earth

To sum it up:

Mass = How much

Volume = How big

Density = how tightly packed at atomic level

Some basic definitions:

Star vs. planet vs. moon (flashlights vs. mirrors)

Click here for a summary of some other physics and chemistry concepts


Early Astronomical Thought

The Ancients - far more advanced than Europe before the Renaissance

The Chinese - for 5000 years or more...

Europe - Stonehenge (pg. 12) in England

Where did this come from?

The Original Americans (Maya, Aztec, Inca)

These cultures clearly understood part of the reality of celestial motions

They had great calendars

Polynesia (Michenor's "Hawaii")

Driven out for religeous reasons

Sail north - navigation always a problem

Polaris (fig. 1.2, pg. 13)

The Greeks (a.k.a. the Ionians)

Very sharp culture

First postulated "atoms"

Root for the term "ion"

Pythagoras - we still use formulas which bear his name

Aristotle (384 - 322 B.C.)

Earth and Moon both spheres

Orbits in perfect, uniform circles (fig. 1.13, pg. 22)

Needed "epicycles" to explain subtle variations

Phases of the moon

Aristarchus (310 - 230 B.C.)

Credited with first heliocentric model (sun in the center)

Hipparchus (worked from 160 - 127 B.C.)

Excellent and subtle observations

Stellar magnitudes - we still use his original scale

Identified the precession of the earth's axis (approx. 22,000 year cycle)

DEMO: gyroscope

Ptolemy (140 A.D.)

Supported geocentric model

Published "Almagest" - accepted in Europe for over 1000 years

Also began formal study of Astrology

Not much more happened during the Dark Ages

If anything, European cultures retreated into ignorance

But there were some who kept a thread of knowledge concerning the sky

Belief in Astrology definitely survived

Requires some rudimentary understanding of celestial events

Magicians & eclipses ("Listen up, King...")


The Development of "Modern" astronomy (the European version)

Copernicus (1473-1543)

Agreed with Aristotle - uniform circular motion (with epicycles)

Also agreed with Aristarchus and resurrected the heliocentric model

Said that earth (and the other planets) were in orbit around sun

Described basic layout of solar system (fig. 2.3, pg. 31)

Superior vs. inferior planet

Opposition vs. conjunction



Calculated relative distances for the planets

Was remarkably accurate (table 2.1, pg. 32)

Redshift - Tours ("Solar System" 5/20)

Tycho Brahe (1546-1601)

Was the court mathematician in Prague

"Arrogant and extravagant"

Did not accept heliocentric model

Was, however, and excellent observer and took great notes

Proof of the value of note taking: you can be a bozo and still get a crater on the moon named after you if you take good notes

His work was later used as support of the Copernican heliocentric model

Johannes Kepler (1571-1630)

Succeeded Tycho as the court mathematician in Prague

Used Tycho's observations to come up with the 3 fundamental "laws" of planetary motion that are named after him

Published the first two of his laws in 1609 and the third law nearly a decade later, in 1618.

Kepler's first law

A planet describes an ellipse in its orbit around the Sun, with the Sun at one focus

Some appropriate terms:

Ellipse: basically a flattened circle (fig. 2.9, pg. 35)

Major axis - the maximum diameter of the ellipse (through the foci)

Semimajor axis -half the major axis

Commonly used as the "distance from the planet to the sun"

Eccentricity - the shape of the ellipse (fig. 2.10, pg. 36)

A circle has an eccentricity of zero (0)

Kepler's second law (law of equal areas)

A ray directed from the Sun to a planet sweeps out equal areas in equal times

See fig. 2.11, pg. 36

This means that a planet's speed changes as it moves around the sun

Faster as it gets closer, slower as it moves away

Kepler's third law (the Harmony of the Worlds)

Keppler tried to come up with an underlying harmony to nature which could be defined mathematically

He published The Harmony of the Worlds in 1619

The square of the period of a planet's orbit is proportional to the cube of its semimajor axis (p2 = a3)

This proportionality is the same for all planets (table 2.2, pg. 37)

How about artificial satellites?

Galileo Galilei (1564-1642)

Experimental physics and astronomy



The property of matter which resists any change in motion - either while moving or at rest

The lack of motion is no more natural than motion


Different sized bodies fall at the same rate (fig. 2.14, pg. 38)


Supported heliocentric model

Cost him dearly later in life

In 1616 the church stated the heliocentric model was "false and absurd"

Galileo was forced to recant on pain of torture and excommunication

Invented the telescope

Defined the Milky Way

Discovered lots of other new stuff

Four of Jupiter's moons

The phases of Venus

The "seas" of the moon

Sunspots (and evidence of the sun's rotation)


Newton and the Classical Laws of Motion

Sir Issac Newton (1642-1727)

VIDEO: Inertia video clip

Newton's 1st Law of Motion: Inertia

"A body will continue in a state of rest, or in uniform motion in a straight line, unless it is acted upon by a net external force"

Inertia: a property of matter which requires a force to cause acceleration

NOTE: acceleration is a vector: both magnitude and direction

Can you hold an inertia in your hand?

Momentum: a measure of the inertia or state of motion of a body

Momentum = mass X velocity


Changing mass: hit by a VW vs. a dump truck

Changing velocity: speed up the VW

Both together: bullet vs. a medicine ball

DIGRESS TO: a sniper uses small caliber rounds

Low mass and resistance, but very high velocity

Newton's 2nd Law of Motion: changes in momentum

"When an unbalanced force acts upon a body, the body will be accelerated in the direction of the greater force"

EXAMPLE: Tug of war (use vector addition)

Defines force: Force = mass X acceleration

Refer back to video clip

Newton's 3rd Law of Motion: Law of action and reaction

"For every action there is an equal and opposite reaction"

A single force cannot exist or there would never be balance

Every force MUST be accompanied by an equal and opposite force


When I push on the wall, the wall has to be pushing back

Recoil of a rifle: the shooter's mass is far greater than the bullet

Rockets: work best in a vacuum

Therefore, it is clear that the force need not be pushing against something


Relativity and Space-time

The classical laws of physics work well for most of what we have to deal with

Start to break down when we get too big or small

Or overcome inertia and start moving fast

DIGRESS TO: Relativistic speeds

With respects to Professors Einstein and Hawking...

I'm not qualified to discuss their work, but I'll give it a shot

Albert Einstein: Early 1900's

Spatial reality and time - we think we handle them pretty well

We define our reality relative to what is around us

All points in space defined relative to another point

EXPAND TO: Where is the fixed reference point?

How about time?

The concepts of past, present, and future are all relative

Is there a fixed reference point for time?

For most of what we do, this doesn't matter at all

In our daily lives we don't notice anything weird

But there are some weirdities...

FOR EXAMPLE: How do we measure speed, distance, and time

We all think that these can be defined as absolute and specific quantities

We even have formulas (with equal signs) to relate them to each other

Rate = Distance / Time (re-arrange and solve for each)

They are all inter-related and inter-dependent

If any of these 3 are not absolute, then none can be absolute

Let's start with distance and shoot some free throws in the gym

(or play catch, or kick a soccer goal, or throw a touchdown pass, or toss a pom-pom into the air and catch it)

We know the exact distance to the basket, and if we can control the trajectory of the ball we can make them all day

But what if Shaq's nightmare comes true and the NBA introduces the "moving basket" in 2003

Suddenly our "absolute distance" is gone and we've got a problem

How about on a ship at sea

Isn't the basket in motion now?

Yes, but so are we

So, we are motionless relative to each other

And, thanks to inertia, can still make the shot

But is the basket in the gym motionless?

The gym is attached to the earth, and the earth is in motion

(REFER BACK TO: "How fast are we moving" discussion)

So, if distance is measured between 2 points, and both are in motion...

How can we obtain an absolute value for distance

There can't be any absolute distances

The important thing is the relative motion of the 2 points

We can only define relative distances between two points (which are in motion)

The distance constant only if they are in uniform motion

Therefore motionless relative to each other

How about time?

VIDEO: "Back to the Future" scene with 2 synchronized clocks

They did this with accurate clocks and jet planes

Time changes as you speed up! (oops)

So if distance and time are both variable, so must be rate

How can we attempt to define absolute spatial reality?

We can only define a relative sense of mechanical position and motion

How about electro-magnetics? Does relativity also apply here?

Radios and computers both work at high rates of speed

So the principle of relativity must also work for electromagnetic waves

Speed of light: here's where it gets seriously weird

We like the speed of light

We base quite a bit of scientific "truth" on how fast it goes

And that it is a constant (absolute) value (2.998 X 108 meters per second)

But relativity states that it, too, must be dependent upon the motion of the observer

But, here comes the weird part...

Speed of light / bullet train example (DESCRIBE)

East-west across Kansas at sunrise and sunset

No measurable change in the speed of light

If speed of light is an absolute value then how can this be?

It must be different for each observer

Or the train changes in length and messes up our measurement

Time and distance are both relative and change as you speed up! (oops)2

So how can we successfully define our reality?

Everything we see is only what it is relative to everything else

And if everything we see is relative to everything we see, we are probably relative, too

I'm getting a headache (but it's only relative)

Clearly there are things here that we still don't fully understand

Einstein also said that at sufficient speed matter and energy are interchangeable (E = MC2)

See Sec. 8.3g; pg. 137

Stephen Hawking: severely handicapped (physically) but still alive and working

Hopes to unify classical mechanics, relativity, and quantum mechanics

Way too many formulas

Very confusing for modern astrophysicists

Not to mention us normal mortals

Remember Kepler and The Harmony of the Worlds?

Hawking is convinced that there is a GUT

Combines everything into a single mathematical expression


Angular Momentum and Gravity

Angular Momentum

A measure of the momentum of a body as it rotates around a fixed point

Angular Momentum = mass X velocity X radius

Angular momentum is conserved

EXAMPLE: a spinning ice skater

DEMONSTRATION: swivel chair, a student, and 2 books

This is an important concept which we will refer to later

For example: formation of the solar system

As nebulas condense they must speed up their rotation to conserve angular momentum

Causes them to flatten and bulge out in the center

Leads to planetary formation

Newton's "Universal" Law of Gravitation: G=M1M2/D2

There has to be something which attracts two bodies

Several observations indicate this

1) The orbits of the planets:

1st law says they will go in a straight line unless acted upon by a net external force

EXAMPLE: ball on a string

Spin it around your head and let go

Flies off in a straight line

Ask Goliath about this one!

Since the planets are not orbiting in a straight line, there has to be an external force which "attaches" them to the star

2) Back to definition of mass: does not change with a change of location

Astronauts on moon: bounced around like Roger Rabbit

Why: mass is same, what was different

Weight: mass under the influence of some external (and invisible) force

This one gets even weirder

Astronauts in space: weightless

It seems as though an object only has "weight" if it is associated with a second object

We now know that there is a force of mutual attraction between all objects

Newton's "Universal" Law of Gravitation

Defined as: Fg = G X M1 X M2 / D2


M1 = mass of an object (in kilograms)

M2 = mass of an object (in kilograms)

D = distance between the objects (in meters)

G = gravitational constant (6.67 X 10-11 N m2/kg2 on earth)

Give some examples:

Any 2 people are bound by this force

Even people you don't like!

Close to earth, its greater mass results in sufficient force to mask the others forces of attraction

In space, however...

Powerful implications in this

Pulls planets out of their straight-line trajectory and into orbit

Called Centripetal Force and results in Centripetal Acceleration

The gravitational force is proportional to the amount of mass

With a larger mass exerting more gravitational force

Therefore, since the moon has less mass than the earth, we can all play Roger Rabbit there

Also, the world's quickest diet

The gravitational force is inversely proportion to the square of the distance between the bodies

As they get farther apart, the force lessens by the square of the distance

DIGRESS TO: What is the net result of the gravitational constant

This makes gravity a VERY weak force

Even the smallest and weakest baby can defeat gravity

The problem is that gravity never stops

How to apply this to a spherical object (such as the sun or a planet)

Pretty complex math - Newton invented calculus to solve it

Beyond the scope of this class (fortunately!)

What Newton found (see fig. 3.6; pg. 52)

"A spherical mass acts gravitationally as though all its mass were concentrated at a point at its center"

Called the "Center of Mass"

Allows us to consider all celestial bodies as "points" with regard to their gravitational forces

Therefore, the weight of an object is defined as the gravitational force between the object and the earth

Weight of an object on earth = G X M1 X M2 / r2 where

r = radius of the earth in meters (6.4 X 106 meters)

M1 = mass of the earth in kilograms (6 X 1024 kg)

M2 = mass of any object (like an apple, or me)

Let's try one: the attraction between the earth and a 65 kg ball of toe jam

g = 6.67 X 10-11 N m2/kg2 X ((6 X 1024 kg) X 65 kg) / (6.4 X 106 meters)2

= 635 Newtons

One step farther...

From this we can calculate the gravitational acceleration

On earth this is 9.8 m/s2

An object will fall towards the earth at this rate

Section 3.2(d) pg. 52 details the mathematical proof of this


Orbital Mechanics

The ancients thought that the natural path of an object was a perfect circle

Newton's 1st law says it's a straight line

Laws of motion allow us to predict the motions of 2 bodies under the influence of their mutual gravitational attraction

Review center of mass

Newton says "A spherical mass acts gravitationally as though all its mass were concentrated at a point at its center"

How about 2 spherical bodies?

They also have a common center of mass

Called the "barycenter" (the true center of mass in a two body system)

Must lie on the plane between the centers of mass of the 2 bodies

The distance of each body from the barycenter will be inversely proportional to its mass.

Means that the barycenter will be closer to the more massive object

Like 2 kids on a seesaw

How does this impact the orbits of planets?

Both objects move about the barycenter

EXAMPLE: earth and moon

Does the moon orbit the earth?

Yes, but the center of the earth is not the center of the orbit

Both actually orbit each other about the barycenter

VIDEODISC: Earth-Moon System (Barycenter clip)

Putting all this together, we can explain the orbit of the moon

See 3.4 pg. 54 and figure 3.9 pg. 55

An object dropped on the earth will fall at 9.8 m/s2

Average speed during first second is 4.9 m/s, so it drops that far

How about the orbit of the moon

In 1 second the moon travels 1 km horizontally, and wants to continue in a straight line (Law of Inertia)

Which would carry it out into space and away from the earth

However, the earth exerts a gravitational attraction on the moon

But the moon is 400,000 km from the earth, so the force is less

The earth's attraction on the moon is 1/3600 as strong, so it "falls" only 1.4mm in the same 1 second (not 4.9 meters)

Fortunately, due to the curvature of the earth, the ground drops away from the moon at the same rate, so it doesn't get any closer

The moon actually "falls" around the earth without ever getting any closer

The orbits of the planets around the sun can be explained in the same way

Also explains any satellite in orbit around any space body