Outline - Jan. 28, 2010

• Correct description of planetary orbits (Kepler; pgs. 72-74)

• Death of “Aristotelian” view of the heavens (Galileo; pgs. 75-78)

• Properties of astronomical objects (mass, temperature, energy)

• Newton’s laws of motion (radical departure from “Aristotelian” physics); pgs. 121-129

• Newton’s law of gravity (134-136)

• Types of Energy (pgs. 130-134)

Correct Model of the Solar SystemJohannes Kepler (1571-1630)

using Tycho Brahe’s lifetime’s worth of planetary motion data

First Law: The orbit of each planet around the sun is an ellipse with the sun at one focus.

Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal time.

Third Law: The relationship between a planet’s period of orbit about the sun (P) and the semi-major axis of its orbit (a) is

P2 = a3

where P is measured in (earth!) years, and a is measured in AU

Purely empirical laws; they only tell you “what”, not “why”

Kepler’s First Lawplanets move on elliptical orbits

An ellipse is a geometric figure with two symmetry axes: major axis (= long axis) and minor axis (= short axis). Half the length of the major axis is called the “semi-major axis”.

The farther apart are the two foci, the “flatter” is the ellipse. The closer together are the two foci, the “rounder” is the ellipse.

A circle is a special case of an ellipse in which the two foci are on top of each other (in the center of the circle).

Orbital Shapes and Distances

a = semi-major axis

b = semi-minor axis

c = distance from center of ellipse to one focus

Shape of ellipse described by “eccentricity”:

e = c / a

Closest approach of planet to sun:

perihelion distance = a(1-e)

Farthest distance of planet from sun:

aphelion distance = a(1+e)

Kepler’s Second Lawequal areas in equal time

The areas (“square footage”) of each of the blue triangles are identical. The time (t) is also identical for each triangle.

Consequence of Kepler’s second law: planets move faster when they are close to the sun than when they are far from the sun.

speed = distance / time

Kepler’s Third LawP2 = a3

P = the time it takes a planet to orbit once around the sun, measured in EARTH years

a = half the length of the long axis of the planet’s orbit, measured in AU

Galileo Galilei(1564-1642)

First person known to use a telescope to make detailed study of the sky; a contemporary of Kepler (8 years older)

Observations provided strong proof that the Aristotelian (= ancient Greek) philosophy of the heavens was wrong

Observations provided direct proof that not all objects orbited the earth and at least one (Venus) orbited the sun

Sun, Moon and Jupiter are not “Flawless Orbs”

Galileo observed that sun has black spotches (sunspots), moon has craters and mountains (very “earth-like” terrain), and Jupiter has big red spot

Galilean Moons of Jupiter(not all objects orbit the earth)

Galileo discovered the 4 largest moons of Jupiter, and showed that they orbit Jupiter, not the earth

Venus Orbits the SUN!Venus could not show “full” phase in Ptolemy’s model

According to Ptolemy, Venus would never show the “full” phase because it could never be on the opposite side of the sky as the sun (as seen from earth). Note: Venus is often (mistakenly) called the “morning star” or the “evening star”.

Phases of planets can only be seen using a telescope. Venus shows the full range of phases (full, gibbous, crescent, new) just like the moon.

Venus is smallest (and dimmest) in its full phase.

Phases of Venus in the “Keplerian” Model (Venus orbits the sun, and is always closer to the sun than Earth)

Venus is in the “full” phase when it is at its farthest distance from the earth, therefore it appears very small in the sky.

Venus is in the “new” phase when it is at its closest distance to the earth, so the “narrow crescent” Venus looks very large (and very bright)!

Where do we go from here?Real Physical Insight about the Universe

Want to deduce information about astronomical objects:

• Mass

• Temperature

• Luminosity and Energy

• Motion

Need a toolbox to do this:

• Fundamentals of motion and gravity (gives MASS)

• Properties of light (gives TEMPERATURE, LUMINOSITY and MOTION towards/away from observer)

• Properties of matter (gives ENERGY production in stars)

Descriptions of Motion(and related quantities)

Speed, s

Velocity, v

Acceleration, a

(Linear) Momentum, p = mv

Angular Momentum, l = mvr (on a curve of radius r)

“p” and “l” are examples of “conserved quantities”

When you step on the bathroom scale are you measuring your MASS?

Change in Velocity (= Acceleration) without a Change in Speed?

Any object moving with a constant speed on a circular orbit is being constantly accelerated because the direction of motion is

constantly changing!

Cause of accelerations: forces!

The “why’s” of orbital motion (planets, moons) come from combining the idea of a generic force that causes accelerations with the specific force of gravity.

Isaac Newton1642-1727

Three Laws of Motion

N1: An object moves at a constant velocity if there is no net force acting upon it. (“Law of Inertia”) a truly radical idea!!!!

N2: A force on an object of mass “m” produces an acceleration, where F = ma (or, equivalently, a = F / m).

N3: For any force, there is always an equal and opposite reaction force.

Units of force: “Newton” (N), 1 N = 1 kg x m/s2

Newton’s Explanation of Orbital Motion

Originally formulated for motion of the moon around the earth, but works equally well for planets orbiting the sun

Law of Gravity - All objects that have mass attract each other with a very specific relationship for the force:

FG = (G m1 m2)/d2

where m1 and m2 are the masses of two objects (in kg), d is the distance between them (in km), and G is “Newton’s Constant”, G = 6.67x10-11 m3/(kg x s2)

Planets orbit (instead of traveling in straight lines at a constant speed) because a force is applied to them (by the sun), and that force is the force of gravity.

What a minute!! What about N3??

N3: For any force, there is always an equal and opposite reaction force.

So, doesn’t that mean that the earth exerts an identical force of gravity on the sun as the sun exerts on the earth?

Why does the earth “orbit the sun”?

Shouldn’t the sun also move in reaction to the force from the earth?

Believe it or not, this is one of the ways that astronomers have found planets orbiting stars other than the sun!!

Newton’s Version of Kepler’s Third Law

Any two objects with masses M1 and M2 orbit about each other according to:

Here G is Newton’s gravitational constant, P is the orbital period and a is the distance between the two objects. If you use G = 6.67x10-11 m3/(kg x s2), then you want P in units of seconds and a in units of meters.

If M2 is MUCH SMALLER than M1, this formula gives you a way to measure M1 (you just need to know “P” and “a”)

See Mathematical Insight 4.3 on page 137 where the mass of the sun is calculated using the earth’s orbit.

Warning! Book Typo! The period of the earth’s orbit is 1 year = 3.15x107 seconds.

Types of EnergyUnits: Joules (J), 1 Joule = 1 kg x m2/s2

Kinetic Energy - energy of motion

Potential Energy - stored energy that can be converted into another kind of energy at some later time (gravitational and chemical are common)

Radiative Energy - energy of light

Mass-Energy - energy contained within matter itself (E = mc2); stars power themselves by converting mass into light!!

Thermal Energy - kinetic energy of many particles (higher the temperature, the faster the speed, and the greater the kinetic energy)

TemperatureUnits: Kelvin (K)

Celsius scale is defined such that water freezes at 0oC and water boils at 100oC

Kelvin scale is defined by how cold it needs to be for molecules to be unable to move/vibrate (“Absolute Zero”)

Minute Paper

A few sentences on one of the following:

* something you found particularly interesting today

* something you found particularly confusing today

* questions on things from today that you would like to know more about

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