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Gravity and the
Expanding Universe
Thursday, January 31
Isaac Newton
(1642-1727)
Discovered 3
Laws of Motion,
Law of Gravity
Newton’s 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.
SPEED = rate at which an
object changes its position.
Example: 65 miles per hour.
VELOCITY = speed plus
direction of travel
Example: 65 miles per hour to the north.
ACCELERATION = rate at which
an object changes its velocity.
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…
FORCE = 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 constant.
after
three
seconds
after
two
seconds
after
one
second
Second Law of Motion:
The acceleration of an object is directly
proportional to the force acting on it,
and inversely proportional to its mass.
a  F /m
or
F  m a
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?
F  m a
F = (0.453 kg) × (9.8 m/sec2)
F = 4.4 kg m / s2
F = 4.4 Newtons
F = 1 pound
Third 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 equal in magnitude
and opposite 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:
F = 1 pound, upward.
Third Law states:
force on Earth = force on cookies.
Second Law states:
acceleration = force divided by mass.
Mass of Earth = 1025 × mass of cookies
Therefore, acceleration of cookies =
1025 × acceleration of Earth.
Newton’s Law of Gravity
Gravity is an attractive force between
all pairs of massive objects.
How big is the force? That’s given
by a (fairly) simple formula.
Newton’s Law of Gravity
 m1m2 
F  G 2 
 r 
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 increasing.
Is the attractive force
between apple & Earth
enough to stop the
apple from rising?
What goes up must come down.
…unless it’s traveling faster
than the escape speed.
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 (ρ) & radius (r).
Escape speed from Earth:
11 km/sec = 25,000 mph
Escape speed from Sun:
620 km/sec = 1,400,000 mph
v
r
v
Suppose a sphere of
gas (radius = r) is
v expanding outward
at a speed v.
v
If expansion speed is greater than
escape speed (v > vesc), sphere
will expand forever.
v
r
v
v
Higher density ρ
leads to a higher
escape speed vesc.
v
For given values of v and r, there is a
critical density ρcrit at which vesc = v.
v
r
v
v
v
 crit
Offered without proof:
critical density below
which the sphere
expands forever is…
2
3 v

2
8 G r
(Small, rapidly expanding spheres need
a higher density to recollapse them.)
v
r
v
v
v
 crit
2
3 v

2
8 G r
Suppose our sphere of gas is part of
the expanding universe, so that v = H0r
 crit
3
2

(H 0 )
8 G
 crit
2
3 H0

8 G
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 ×
-27
3
10 kg/m
Yes, this is a very low density!
3
Water: 1000 kg/m
Air: 1 kg/m3
Most of the universe consists of very
low density intergalactic voids.
Not immediately obvious that ρ > ρcrit
Newton says: fate of the universe
depends on the ratio of its density
to the critical density.


 crit
Omega (Ω) is also called the
“density parameter”.
Ω<1
Ω=1
Distance
between two
galaxies
Ω>1
Time
Ω>1 (density greater than critical):
The Big Crunch
(recollapse, becoming hotter)
Ω≤1 (density less than or equal to critical):
The Big Chill
(perpetual expansion, becoming cooler)
Amusing speculation of the day:
perhaps a Big Crunch would lead to
a Big Bounce.
You are here
Or maybe here
Or here…
Thursday’s Lecture:
Einstein’s Universe
Reading:
Chapter 6