gravity and the expanding universe thursday, january 31
TRANSCRIPT
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Gravity and the Expanding Universe
Thursday, January 31
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Isaac Newton (1642-1727)
Discovered 3 Laws of Laws of Motion, Law of Motion, Law of GravityGravity
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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.
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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.
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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.
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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…
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FORCEFORCE = a push or pull acting to accelerate an object.
Examples:
Gravity = pull
Electrostatic attraction = pull
Electrostatic repulsion = push
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Restatement of First Law: In the absence of outside
forces, velocity is constantconstant.
after one
second
after two
seconds
after three
seconds
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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
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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?
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amF F = (0.453 kg) × (9.8 m/sec2)
F = 4.4 kg m / s2
F = 4.4 NewtonsNewtons
F = 1 poundpound
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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.
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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.
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I remove my hand.
Earth pulls on cookies: F = 1 pound, downward.
Cookies pull on Earth:Cookies pull on Earth: F = 1 pound, upward.
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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.
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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.
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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)
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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….
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The universe is full of objects attracting each other: are these attractive forces
enough to stop the expansion?
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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?
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…unless it’s traveling faster than the escape speedescape speed.
What goes up must come down.
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Small initial speed: short distance upward.
Larger initial speed: long distance upward.
Speed > escape speed: to infinity!!
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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
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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.
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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.
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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.)
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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
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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!
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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
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Most of the universe consists of veryvery low density intergalactic voids.
Not immediately obvious that ρ > ρcrit
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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”.
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Distance between two
galaxies
Time
Ω > 1
Ω < 1
Ω = 1
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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)
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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…
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Thursday’s Lecture:
Reading:
Chapter 6
Einstein’s Universe