Kepler Orbits Last time we saw that this equation describes an ellipse. r min = perihelion (for solar orbits) or perigee r max = aphelion (for solar orbits)

Kepler Orbits

Last time we saw that this equation describes an ellipse.

rmin = perihelion (for solar orbits) or perigeermax = aphelion (for solar orbits) or apogee

Kepler Orbits

• We can rewrite this in a the more familiar equation for an ellipse:

Kepler’s First Law: the planets follow elliptical orbits with the Sun at one focus

© 2005 Pearson Education Inc., publishing as Addison-Wesley

Brief History of Astronomy

Copernicus (1473-1543):• proposed Sun-centered model (published 1543)• used model to determine layout of solar system (planetary distances in AU)

But . . .

• model was no more accurate than Ptolemaic model in predicting planetary positions, because still used perfect circles.


Tycho Brahe (1546-1601)

• Compiled the most accurate (one arcminute) naked eye measurements ever made of planetary positions.

• Still could not detect stellar parallax, and thus still thought Earth must be at center of solar system (but recognized that other planets go around Sun)

• Hired Kepler, who used his observations to discover the truth about planetary motion.


Johannes Kepler(1571-1630)

• Kepler first tried to match Tycho’s observations with circular orbits

• But an 8 arcminute discrepancy led him eventually to ellipses…

If I had believed that we could ignore these eight minutes [of arc], I wouldhave patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.


An ellipse looks like an elongated circle

What is an Ellipse?


Kepler’s First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus.


Kepler’s Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal times.

means that a planet travels faster when it is nearer to the Sun and slower when it is farther from the Sun.

Whiteboards: Derive Kepler’s 2nd Law from conservation of angular momentum.


More distant planets orbit the Sun at slower average speeds, obeying the relationship

p2 = a3

p = orbital period in years a = avg. distance from Sun in AU

Whiteboards: Derive Kepler’s 3rd Law starting with F = m a.

Kepler’s Third Law


Graphical version of Kepler’s Third Law


Question

An asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun?

A. 4 years

B. 8 years

C. 16 years

D. 64 years


Galileo Galilei

Galileo (1564-1642) overcame major objections to Copernican view. Threekey objections rooted in Aristotelian view were:

1. Earth could not be moving because objects in air would be left behind.

2. Non-circular orbits are not “perfect” as heavens should be.

3. If Earth were really orbiting Sun,we’d detect stellar parallax.


Galileo’s experiments showed that objects in air would stay with a moving Earth.

Overcoming the first objection (nature of motion):

• Aristotle thought that all objects naturally come to rest.• Galileo showed that objects will stay in motion unlessa force acts to slow them down (Newton’s first law of motion).


Overcoming the second objection (heavenly perfection):

• Tycho’s observations of comet and supernova already challenged this idea.

• Using his telescope, Galileo saw:

• sunspots on Sun (“imperfections”)

• mountains and valleys on the Moon (proving it is not a perfect sphere)


• Tycho thought he had measured stellar distances, so lack of parallax seemed to rule out an orbiting Earth.

• Galileo showed stars must be much farther than Tycho thought — in part by using his telescope to see the Milky Way is countless individual stars.

• If stars were much farther away, then lack of detectable parallax was no longer so troubling.

Overcoming the third objection (parallax):


Galileo also saw four moons orbiting Jupiter, proving that not all objects orbit the Earth…


… and his observations of phases of Venus proved that it orbits the Sun and not Earth.


Galileo Galilei

The Catholic Church ordered Galileo to recant his claim that Earth orbits the Sun in 1633

His book on the subject was removed from the Church’s index of banned books in 1824

Galileo was formally vindicated by the Church in 1992

Historical Overview

Isaac Newton (1643 - 1727)

• Building on the results of Galileo and Kepler

Major achievements:1. Invented calculus as a necessary tool to solve mathematical

problems related to motion

• Adding physics interpretations to the mathematical descriptions of astronomy by Copernicus, Galileo and Kepler

2. Discovered the three laws of motion

3. Discovered the universal law of mutual gravitation

The Universal Law of Gravity

• Any two bodies are attracting each other through gravitation, with a force proportional to the product of their masses and inversely proportional to the square of their distance:

F = - G Mm

r2

(G is the gravitational constant.)

Orbital Motion (II)

In order to stay on a closed orbit, an object has to be within a certain range of

velocities:

Too slow : Object falls back down to Earth

Too fast : Object escapes the Earth’s gravity

Relating Energy to Eccentricity

• Use the relation that E = Ueff(rmin) and rmin = c/(1+ε), you can show

Summary of Orbits

eccentricity

energy orbit

e = 0 E < 0 circle

0 < e < 1 E < 0 ellipse

e = 1 E = 0 parabola

e > 1 E > 0 hyperbola

Changes in Orbit

• One way to change an objects orbit is using a tangential thrust at perigee.

• Let λ= thrust factor • (ratio of speed after/ speed before)

• Angular momentum changes by the same factor

• The eccentricity of the new orbit is:

Changes in Orbit

The Slingshot Effect

• A close encounter with a planet can dramatically increase a spacecraft’s speed

• Elastic Collision• Momentum is conserved• Kinetic energy is conserved


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The probe’s speed after the encounteris exceeds 2x the planet’s initial speed!

Eyes on the Solar System

• Go to http://solarsystem.nasa.gov/eyes/

• Watch the 4 tutorials

• Follow the Voyager 2 spacecraft on its journey from Earth to the outer solar system. Identify significant energy boosts that the probe received and explain qualitatively how it received them (through what mechanism).

• Repeat for a mission to Mars (you decide on the mission).

Documents

Kepler Orbits Last time we saw that this equation describes an ellipse. r min = perihelion (for solar orbits) or perigee r max = aphelion (for solar orbits)