planetary and satellite motion_lecture

8/8/2019 Planetary and Satellite Motion_lecture

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PlanetaryPlanetaryand Satellite Motionand Satellite Motion


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The EarthThe Earth--Centered UniverseCentered Universe

AA geocentricgeocentric (Earth(Earth--centered) solarcentered) solarsystem is often credited tosystem is often credited to PtolemyPtolemy, an, an

Alexandrian Greek, although the idea isAlexandrian Greek, although the idea is

very old.very old.

Image from: http://abyss.uoregon.edu/~js/ast123/lectures/lec02.html


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Ptolemys Solar SystemPtolemys Solar System

Ptolemys solar system could be made toPtolemys solar system could be made tofit the observational data pretty well, butfit the observational data pretty well, but

only by becomingonly by becoming veryverycomplicated.complicated.



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Copernicus Solar SystemCopernicus Solar System

The Polish clericThe Polish cleric CopernicusCopernicus proposed aproposed aheliocentricheliocentric (Sun centered) solar system in(Sun centered) solar system in

the 1500s.the 1500s.



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Galileo & CopernicusGalileo & Copernicus

GalileoGalileo became convinced thatbecame convinced that

Copernicus was correctCopernicus was correct by observationsby observations ofof

the Sun, Venus, and t

he moons of Jupitert

he Sun, Venus, and t

he moons of Jupiterusing the newlyusing the newly--invented telescope.invented telescope.

Perhaps Galileo was motivated toPerhaps Galileo was motivated to

understandunderstand inertiainertia byh

is desire tobyh

is desire tounderstand and defend Copernicus ideas.understand and defend Copernicus ideas.


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Tycho and KeplerTycho and Kepler

In the late 1500s, a Danish noblemanIn the late 1500s, a Danish nobleman

namednamed Tycho BraheTycho Brahe set out to make theset out to make the

most accurate measurementsmost accurate measurements ofofplanetary motions to date, in order toplanetary motions to date, in order to

validate his own ideas of planetary motion.validate his own ideas of planetary motion.

Tych

os data was successfully interpretedT

ych

os data was successfully interpretedby the German mathematician andby the German mathematician and

scientistscientist Johannes KeplerJohannes Keplerin the earlyin the early

1600s.1600s.


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Keplers LawKeplers Law

Johannes Kepler (1605), developed three laws whichJohannes Kepler (1605), developed three laws whichdescribed the motion of the planets.described the motion of the planets.

1. The1. The Law of OrbitsLaw of Orbits: All planets move in elliptical: All planets move in ellipticalorbits, with the sun at one focus.orbits, with the sun at one focus.

2. The2. The Law of AreasLaw of Areas: A line that connects a planet to: A line that connects a planet tothe sun sweeps out equal areas in equal times.the sun sweeps out equal areas in equal times.

3.3. The Law of PeriodsThe Law of Periods: The square of the period of: The square of the period ofany planet is proportional to the cube of theany planet is proportional to the cube of thesemimajor axis of its orbit.semimajor axis of its orbit.

Kepler's laws were derived for orbits around the sun,Kepler's laws were derived for orbits around the sun,

but they apply to satellite orbits as well.but t

hey apply to satellite orbits as well.


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Law of OrbitsLaw of Orbits

All planets move inAll planets move in

elliptical orbits, withelliptical orbits, with

the sun at onethe sun at one

focusfocus

The elliptical shapeThe elliptical shape

of the orbit is aof the orbit is a

result of theresult of the inverseinversesquare forcesquare force ofof

gravitygravity..


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Eccentricity of ElipseEccentricity of Elipse


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Eccentricity of ElipseEccentricity of Elipse

The eccentricity of anThe eccentricity of an

ellipse can be defined asellipse can be defined as

the ratio of the distancethe ratio of the distance

between the foci to thebetween the foci to the

major axis of the ellipse.major axis of the ellipse.

The eccentricity is zero for a circle. Of the planetaryThe eccentricity is zero for a circle. Of the planetary

orbits, only Pluto has a large eccentricityorbits, only Pluto has a large eccentricity.

a

s

axismajor

distancefocietyeccentrici

!

!,


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Planetary orbit eccentricitiesPlanetary orbit eccentricities

MercuryMercury 0.2060.206

VenusVenus 0.00680.0068

EarthEarth 0.01670.0167

MarsMars 0.09340.0934

JupiterJupiter 0.04850.0485

SaturnSaturn 0.05560.0556

UranusUranus 0.04720.0472

NeptuneNeptune 0.00860.0086

PlutoPluto 0.250.25


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The Law of AreasThe Law of Areas

A line that connects a planet to the sunA line that connects a planet to the sunsweeps out equal areas in equal timessweeps out equal areas in equal times

This empirical law

discovered by Kepler

arises from conservation

of angular momentum.

When the planet is closerto the sun, it moves

faster, sweeping through

a longer path in a given

time.


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The Law of PeriodsThe Law of Periods

The square of the period of any planet isThe square of the period of any planet isproportional to the cube of theproportional to the cube of the

semimajor axis of its orbit.semimajor axis of its orbit.

32

2

32

a

GM

4T

aT

T

E

This law arises from the law of gravitation. NewtonThis law arises from the law of gravitation. Newton

first formulated the law of gravitation from Kepler'sfirst formulated the law of gravitation from Kepler's

3rd law.3rd law.

wherewhereTT PeriodPeriod

aa semi major axissemi major axis

GG Gravitational constantGravitational constant

MM mass of Sunmass of Sun


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GM


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A graph of aA graph of a33 versus Tversus T22

32 aT !a3

T2


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Data: Law of PeriodsData: Law of Periods

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