Gravity and the Expanding Universe

Thursday, January 31

Isaac Newton (1642-1727)

Discovered 3 Laws of Laws of Motion, Law of Motion, Law of GravityGravity

Newton’s First First Law of Motion: An object remains at rest, or moves in a straight line at constant speed, unless acted on

by an outside force.

Mathematical laws require precise definitions of terms.

SPEEDSPEED = rate at which an object changes its position.

VELOCITYVELOCITY = speed plus direction of travel

Example: 65 miles per hour.

Example: 65 miles per hour to the north.

ACCELERATION ACCELERATION = rate at which an object changes its velocityvelocity.

Acceleration can involve:

1) increase in speed

2) decrease in speed

3) change in direction.

Example of acceleration: an apple falls from a tree.

Acceleration = 9.8 meters/second/second.

After 1 sec, speed = 9.8 meters/sec, After 2 sec, speed = 19.6 meters/sec, etc…

FORCEFORCE = a push or pull acting to accelerate an object.

Examples:

Gravity = pull

Electrostatic attraction = pull

Electrostatic repulsion = push

Restatement of First Law: In the absence of outside

forces, velocity is constantconstant.

after one

second

after two

seconds

after three

seconds

SecondSecond Law of Motion: The acceleration of an object is

directly proportional to the force acting on it, and inversely proportional to its

mass.

mFa /

amF or

Example: a package of cookies has mass m = 0.453 kilograms.

It experiences a gravitational acceleration a = 9.8 meters/sec2.

How large is the force acting on the cookies?

amF F = (0.453 kg) × (9.8 m/sec2)

F = 4.4 kg m / s2

F = 4.4 NewtonsNewtons

F = 1 poundpound

ThirdThird Law of Motion: For every action, there is an

equal and opposite reaction.

If A exerts a force on B, then B exerts a force on A that’s equalequal in magnitude and oppositeopposite in direction.

Example: I balance a package of cookies

on my hand.

Cookies push on hand: F = 1 pound, downward.

Hand pushes on cookies: F = 1 pound, upward.

I remove my hand.

Earth pulls on cookies: F = 1 pound, downward.

Cookies pull on Earth:Cookies pull on Earth: F = 1 pound, upward.

Third LawThird Law states: force on Earth = force on cookies.

Second LawSecond Law states: acceleration = force divided by mass.

Mass of Earth = 10Mass of Earth = 1025 25 ×× mass of cookies mass of cookies Therefore, acceleration of cookies =

1025 × acceleration of Earth.

Newton’s Law of Gravity

Gravity is an attractiveattractive force between allall pairs of massive objects.

How bigbig is the force? That’s given by a (fairly) simple formula.

Newton’s Law of Gravity

221

r

mmGF

F = force m1 = mass of one object m2 = mass of other object r = distance between centers of objects G = “universal constant of gravitation” (G = 6.7 × 10-11 Newton meter2 / kg2)

Gravity makes apples fall; it also keeps the Moon on its orbit around the Earth, the Earth on its orbit around the Sun, the Sun on its orbit around the Galactic center….

The universe is full of objects attracting each other: are these attractive forces

enough to stop the expansion?

Let’s start with a related problem:

A boy standing on the Earth throws an apple upward: initially, the distance

between apple & Earth is increasingincreasing.

Is the attractive force between apple & Earth

enough to stop the apple from rising?

…unless it’s traveling faster than the escape speedescape speed.

What goes up must come down.

Small initial speed: short distance upward.

Larger initial speed: long distance upward.

Speed > escape speed: to infinity!!

Escape speed from a planet (or star) depends on its

density (density (ρρ)) & radius (r)radius (r).

Escape speed from EarthEarth: 11 km/sec = 25,000

mph

Escape speed from SunSun: 620 km/sec = 1,400,000

mph

vv

vv

vv

vv

rr

Suppose a sphere of gas (radius = r) is

expanding outward at a speed v.

If expansion speed is greater than escape speed (v > vesc), sphere

will expand forever.

vv

vv

vv

vv

rr

Higher density ρ leads to a higher

escape speed vesc.

For given values of v and r, there is a critical density ρcrit at which vesc = v.

vv

vv

vv

vv

rrOffered without proof:Offered without proof:

critical density below which the sphere

expands forever is…

2

2

crit r

v

G 8

3

(Small, rapidly expanding spheres need a higher density to recollapse them.)

vv

vv

vv

vv

rr

2

2

crit r

v

G 8

3

Suppose our sphere of gas is part of the expanding universe, so that v = H0r

20crit )(H

G 8

3

G 8

H 3 20

crit

This critical density depends only on

the universal constant of gravitation G

and on the Hubble constant H0.

We know the values of G and H0!

With H0 = 70 km/sec/Mpc, the critical density for the universe is:

ρcrit = 9 × 10-27 kg/m3

Yes, this is is a very low density! Water: 1000 kg/m3

Air: 1 kg/m3

Most of the universe consists of veryvery low density intergalactic voids.

Not immediately obvious that ρ > ρcrit

Newton says: fate of the universe depends on the ratio of its densitydensity

to the critical densitycritical density.

crit

Omega (Ω) is also called the “density parameterdensity parameter”.

Distance between two

galaxies

Time

Ω > 1

Ω < 1

Ω = 1

Ω>1 (density greater than critical):

The Big CrunchThe Big Crunch

Ω≤1 (density less than or equal to critical):

The Big ChillThe Big Chill

(recollapse, becoming hotter)

(perpetual expansion, becoming cooler)

Amusing speculation of the day: perhaps a Big CrunchBig Crunch would lead to

a Big BounceBig Bounce.

You are here Or maybe here Or here…

Thursday’s Lecture:

Reading:

Chapter 6

Einstein’s Universe