Astronomie si astrofizica_problema Kepler.ppt

7/29/2019 Astronomie si astrofizica_problema Kepler.ppt

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Kepler laws


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The two-body problem

What defines the problem? A large planet and a smaller satellite

Different views on the solar system Nicolaus Copernicus

Tycho Brahe Johannes Kepler

Keplers laws on orbit motions Elliptical orbits within an orbital plane

Equal area law

Scale vs orbital period law

Equations of Motion


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Keplerian Laws

Law I

Each planet revolves around the Sun in anelliptical path, with the Sun occupying one ofthe foci of the ellipse.

Law II

The straight line joining the Sun and aplanet sweeps out equal areas in equalintervals of time.

Law III The squares of the planets' orbital periods

are proportional to the cubes of the semi-major axes of their orbits.


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Keplers first law

)cos(1

)1()(

2

e

ear

rThere are 4 cases:

e=0, circle

0


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In-plane Kepler parameters

)1(ear )1( ear

ea


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Keplers second law

D

B

O

C

A

ABO CDO


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Keplers Third law

nT 2)(32 mMGan

The variable n represents the mean motion in radians persecond, a is the semi-major axis, G is the gravitational

constant, Mis the mass of the Sun, m is the mass of the

satellite (m


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How to obtain r()

A particle moves in a central force field The motion takes place within an orbital plane

The solution of the equation of motion isrepresented in the orbital plane

Substitution 1: polar coordinates in the orbitalplane

Substitution 2: replace r by 1/u

Analogy with a mathematical pendulum

Solve this an substitute elliptical configuration Final step: transformation orbital plane to 3D

(this gives us the set of 6 Keplerian elements)


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Mathematics on r():

Substitute:

Characteristic:

(h = length ang mom vector)


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Velocity and Position

(aka vis-viva equations)

arGMv

Eea

e

ear

12

)cos.1(

cos1

)1( 2

radius r

velocity v

Note: in this case only , or E or M depend on time.


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Total Energy

The total energy inthe system is the

sum of kinetic and

potential energy

For the Kepler

problem one can

show that the total

energy is half thatof the potential

energy

ca

a

GMc

mc

r

GMmmv

EEE totpotkin

2atEnergyPotential

2

.

2

1 2

Astronomie si astrofizica_problema Kepler.ppt

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