Gravity and the Expanding Universe Thursday, January 31

Isaac Newton ( ) Laws of Motion, Law of Gravity Discovered 3 Laws of Motion, Law of Gravity

First 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 SPEED = rate at which an object changes its position. VELOCITY VELOCITY = speed plus direction of travel Example: 65 miles per hour. Example: 65 miles per hour to the north.

ACCELERATION velocity 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 FORCE = a push or pull acting to accelerate an object. Examples: Gravity = pull Electrostatic attraction = pull Electrostatic repulsion = push

constant Restatement of First Law: In the absence of outside forces, velocity is constant. after one second after two seconds after three seconds

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. or

Example: a package of cookies has mass m = kilograms. It experiences a gravitational acceleration a = 9.8 meters/sec 2. How large is the force acting on the cookies?

F = (0.453 kg) × (9.8 m/sec 2 ) F = 4.4 kg m / s 2 Newtons F = 4.4 Newtons pound F = 1 pound

Third Third Law of Motion: For every action, there is an equal and opposite reaction. equal opposite 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: Cookies pull on Earth: F = 1 pound, upward.

Third Law Third Law states: force on Earth = force on cookies. Second Law Second Law states: acceleration = force divided by mass. Mass of Earth = × mass of cookies Mass of Earth = × mass of cookies Therefore, acceleration of cookies = × acceleration of Earth.

Newton’s Law of Gravity attractive all Gravity is an attractive force between all pairs of massive objects. big How big is the force? That’s given by a (fairly) simple formula.

Newton’s Law of Gravity F = force m 1 = mass of one object m 2 = mass of other object r = distance between centers of objects G = “universal constant of gravitation” (G = 6.7 × Newton meter 2 / kg 2 )

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: increasing 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?

escape speed …unless it’s traveling faster than the escape speed. What goes up must come down.

Small initial speed: short distance upward. Larger initial speed: long distance upward. Speed > escape speed: to infinity!!

density (ρ)radius (r) Escape speed from a planet (or star) depends on its density (ρ) & radius (r). Earth Escape speed from Earth: 11 km/sec = 25,000 mph Sun Escape speed from Sun: 620 km/sec = 1,400,000 mph

v v v v r Suppose a sphere of gas (radius = r) is expanding outward at a speed v. If expansion speed is greater than escape speed (v > v esc ), sphere will expand forever.

v v v v r Higher density ρ leads to a higher escape speed v esc. For given values of v and r, there is a critical density ρ crit at which v esc = v.

v v v v r Offered without proof: Offered without proof: critical density below which the sphere expands forever is… (Small, rapidly expanding spheres need a higher density to recollapse them.)

v v v v r Suppose our sphere of gas is part of the expanding universe, so that v = H 0 r

This critical density depends only on the universal constant of gravitation G and on the Hubble constant H 0. We know the values of G and H 0 !

With H 0 = 70 km/sec/Mpc, the critical density for the universe is: ρ crit = 9 × kg/m 3 is Yes, this is a very low density! Water: 1000 kg/m 3 Air: 1 kg/m 3

very Most of the universe consists of very low density intergalactic voids. Not immediately obvious that ρ > ρ crit

density critical density Newton says: fate of the universe depends on the ratio of its density to the critical density. density parameter Omega (Ω) is also called the “density parameter”.

Distance between two galaxies Time Ω > 1 Ω < 1 Ω = 1

The Big Crunch Ω>1 (density greater than critical): The Big Crunch The Big Chill Ω≤1 (density less than or equal to critical): The Big Chill (recollapse, becoming hotter) (perpetual expansion, becoming cooler)

Big Crunch Big Bounce 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: Reading: Chapter 6 Einstein’s Universe