Announcements Labs open this week to finish. You may go to any lab

section this week (most people done). Lab exercise 2 starts Oct 2. It's the long one!! Midterm exam likely Oct 23 or 25, in class. Assignment 2 handed out Oct. 2, due Oct 10.

Supplementary text problems (should be straightforward)

3-51, 3-54, 4-53, 4-57a,b,c Answers will be on web site

Chapter 4. Chapter 4.

Motion and Motion and

gravitygravity

Objects in Motion : A reviewObjects in Motion : A review

speed – rate at which an object moves, i.e. the

distance traveled per unit time [m/s; mi/hr]

velocity – an object’s speed in a certain

direction, e.g. “10 m/s moving east”

acceleration – a change in an object’s velocity,

i.e. a change in either speed or direction is an

acceleration [m/s2]

Newton’s Laws of MotionNewton’s Laws of Motion

Examples of the 3 laws of motion, paraphrased:

1. Move in straight lines unless outside force.

2. Forces cause accelerations.

3. Equal and opposite reactions.

Forces : A reviewForces : A review

Forces change the motion of objects.

momentum – the (mass x velocity) of an object

force – anything that can cause a change in an

object’s momentum

As long as the object’s mass does not change, the

force causes a change in velocity, or an…

Falling objectsFalling objects

When you drop a ball:

It starts with vo= 0

It accelerates down

a = g = 10 m/s2

This value of g only true near the Earth's surface!

After 1 sec, v=10 m/s

After 2 sec, v=20 m/s. v = vo + a t

The acceleration is of course caused by gravity

We can calculate the mass of the Earth!

Universal Law of GravitationUniversal Law of Gravitation Between every two objects there is an

attractive force, the magnitude of which is

directly proportional to the mass of each object

and inversely proportional to the square of the

distance between the centers of the objects.

Mass of the EarthMass of the Earth

A mass m feels a downward force and accelerates

F = m a , but here a=g , so F = m g

This downward force is of course gravity due to the Earth of

mass M. Since we are the Earth's radius R above the Earth's

center, gravity's force is GMm/R2

Setting these two equal : mg = GMm/R2

But the m's cancel, so for ANY object, g=GM/R2

So M = gR2/G

= (10 m/s2)(6.4x106 m)2/(6.67x10-11Nm2/kg2)

= 6 x 1024 kg (confirm the units!)

Acceleration due to gravityAcceleration due to gravity

Galileo experimentally showed

that the acceleration an object

experiences under gravity is

independent of its mass.

This means that 'inertial mass'

(the tendency of objects to keep

their motion) is the same as

'gravitational mass' (which

appears in the law of gravity)

Newton was puzzled by this...

Einstein explained it.

(see ASTR 102...)

Central forces explain orbitsCentral forces explain orbitsThe central (or centripital) force provided by

gravity causes objects to orbit the origin of the

force, continually changing the direction of the

object's velocity

If the force were to stop, linear motion resumes

In some sense, objects in orbit are always In some sense, objects in orbit are always

`falling' towards the gravitational center`falling' towards the gravitational center

Newton performed this

thought experiment with

a cannon on a mountain

top.

A cannon ball shot fast

enough will 'fall around

the Earth' in an orbit.

Something shot very fast

can escape.

Orbital Paths : Orbital Paths :

Conic sectionsConic sectionsExtending Kepler’s First

Law, Newton found that

ellipses were not the only

orbital paths.

possible orbital paths

circle (bound) e=0

ellipse (bound) 0<e<1

parabola (unbound)

hyperbola (unbound)

Newton’s Version of Kepler’s 3rd Law Using calculus, Newton derived Kepler’s three

Laws from his own Law of Gravity.

In the 3rd Law's most general form:

P2 = 4 2 a3 / G (m1 + m2) SO : If you can measure the orbital period of two

objects (P) and the distance between them (a), then

you can calculate the sum of the masses of both

objects (m1 + m2).

TEST : Ensure you can recover Kepler's 3rd law for

objects orbiting the Sun! (i.e., prove Earth's orbit

takes 1 year given Msun

, aEarth

in kg, km)

Celestial Mechanics...Celestial Mechanics...

...is the study of

orbital motion

Has been a driving

force behind the

development of

mathematics and

physics

An object's distance varies as it An object's distance varies as it

moves around its orbit (if e>0)moves around its orbit (if e>0)

'helios' = around Sun, so perihelion

'gee' = around Earth, so perigee

AphelionAphelionPerihelionPerihelion

Note period is independent of 'e' !?!Note period is independent of 'e' !?!

Kepler's Third Law (and

Newton's version of it

also) depends ONLY on

the semimajor axis 'a'

While the speeds at

various places around

the orbit may differ, the

period is the same if a is

identical.

The next step in the scaleThe next step in the scaleSo far in the course we have seen

how to measure (in absolute units)

The radius of the Earth

The radius of the Moon

The distance to the Moon

But how does one figure out:

The distance to the Sun?

the scale of the planetary orbits

(even in a relative sense)?

Kepler managed the latter

Geometry establishes relative scale

Kepler used geometry of some planetary configurations to

calculate the size of other planetary orbits.

Interior planet (left) is easy: use greatest elongation.

Exterior planet (right) needs more advanced trigonometry

1 AU defined to be semimajor axis of Earth; the

astronomical unit becomes the yardstick

sin sin = r/(1AU) = r/(1AU)

so rso rAUAU = sin = sin . .

eg: Venus: eg: Venus: =44°=44°

so r = 0.72 AUso r = 0.72 AU

But...but...but...what is an AU???But...but...but...what is an AU???Astronomers didn't know the

AU in terms of anything else

(like kilometers).

Estimates ranged from

8 million km (Copernicus)

to

111 million km (Halley)

Measuring the AU became a

major goal of astronomy

Halley realized that 'transits'

could do the trick.

Transit: Mercury or Venus pass in front Transit: Mercury or Venus pass in front

of the Sun's diskof the Sun's disk

Doesn't happen every

orbit. (Why?)

Because Mercury is so

small and close to the

Sun, this is difficult to

observe precisely.

There was one in

November 1999, and

one in May 2003

Venus transits are better, but rareVenus transits are better, but rare

Halley realized that the transits of 1761 and 1769

could be used to measure the AU

How?

Page 218 of text.Page 218 of text.

1 Astronomical Unit1 Astronomical Unit

= 1.50 x 10 = 1.50 x 1088 km km

Venus transits are better, but rareVenus transits are better, but rareIf observed from different latitudes and longitudes on Earth, the

parallax of Venus on the solar disk and the timing of transit yields

the AU.

Worked in 1761, and 1% accuracy was obtained.

Page 218 of text.Page 218 of text.

1 Astronomical Unit1 Astronomical Unit

= 1.50 x 10 = 1.50 x 1088 km km

Some more orbital mechanicsObjects circle around on their orbits forever

until something `perturbs them' and changes

their orbit.

Examples:

Passing close to another large object

A spacecraft burning fuel

Friction with a thin upper atmosphere

The last example is why satellites in Earth orbit

sometimes slowly spiral down to Earth.

The first two examples are important for

spacecraft mechanics.

How do I get to Mars???Several spacecraft did this recently. Two

arrived in January 2004.

Surely the easiest spot to do the transfer is

when the planets are closest, right?

How do I get to Mars???Several spacecraft did this recently. Two

arrived in January 2004.

Surely the easiest spot to do the transfer is

when the planets are closest, right?

Wrong.

Because things just don't

`go straight' between the

planets. You have to orbit

the Sun.

If you orbit the Sun, how do you

increase the size of your orbit? You DON'T accelerate

towards Mars.

You DO accelerate in

the direction you are

already moving.

What does this do to the

semimajor axis and

eccentricity?

The Hohman transfer ellipseThe thrust ('delta v') applied

in the direction of motion

increases the a and produces

an e, so one is at perihelion

P of an ellipse.

With the right delta v you

can get the aphelion A at

Mars.

This is an 'economical '

orbit; less fuel but not the

fastest.

The Hohman transfer ellipseWhat is the Earth-Mars orbit's

parameters?

r2 = 1 AU = a(1-e)

r1 = 1.52 AU = a(1+e)

Add the equations:

2.52 AU=a(1+e)+a(1-e)

2.52 AU=2a, so a=1.26 AU

The use 1AU=(1.26)(1-e) to

calculate e=0.206

P = a3/2 = 1.41 yr for whole

orbit, so 0.70 yr for half.

Getting the timing right

Need to launch when

Mars slightly ahead of

Earth so that Mars is at

the right place for arrival

Exercise: Show this is

when Mars is about 136

degrees before the

opposition point of the

launch date.

Orbit changes can also be natural

Flying near a planet will cause a small object to

'change its orbit'

The gravitation pull of the planet changes the orbit

to a new one.

Here, a comet --->>

Pre-encouter:

Large parabola

Post-encounter

Small ellipse

Remains on ellipse

Until another Jupiter encounter.

One can arrange that spacecraft

`benefit' from the flybys.

Free fuel!