LawsofPlanetaryMotion

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Kepler’sLawsofPlanetaryMotion

Sun

Planet

Kepler’sLawsofPlanetaryMotion

“ReminderoftheGravity”

F1 =m1v12 / r1 =4p2m1r1/P2(ß v1 =2pr1/P)

F2 =m2v22 / r2 =4p2m2r2/P2

r1 / r2 =m2 /m1 &a = r1 + r2

® r1 =m2a /(m1 +m2)orr2 =m1a /(m1 +m2)

Fgrav = F1 = F2 = Gm1m2 / a2

GravityGeneral

Newton’sformofKepler’s3rd law

YoucandriveKepler’s3rd lawdirectlyfromthegravity.

Foraboundsystem:

Describablebyrμ &vμ

L&Econservation

Equationofaconicsection.

Threesolutionsforrfordifferente.

Ellipse!

Eccentricity

Parabola!

Hyperbola!

“Reduced-massSolution”

Let’schangethe2-bodyproblemto1-bodyproblemaroundthecenterofmass!

GravityGeneral

● Threedifferentorbitsarepossible:elliptical,parabolic,&hyperbolic;● Ellipticalforboundsystems(E<0;e.g.,solarsystemplanets);parabolicforE=0;hyperbolicforopen(E>0;unbound)systems;● “Kepler’slawsrepresentthecaseforboundsystems(motionsaroundCM)”

Whycouldn’tKeplerknowtheexistenceoftheparabolicand/orhyperbolicorbits?

WhereistheCMoftheSolarsystem?

Ellipse!

Parabola!

Hyperbola!

Newton’sformofKepler’s3rd law(seenextslides)

Generalsolutionofbound2-bodyorbit

“Reduced-massSolution”

Let’schangethe2-bodyproblemto1-bodyproblemaroundthecenterofmass!

GravityGeneral

2-BodyProblem

ConicSections

Parabola: Thesetofallpointsintheplanewhosedistancesfromafixedpoint,calledthefocus,andafixedline,calledthedirectrix,arealwaysequal.

Hyperbola: Thesetofallpointsintheplane,thedifferenceofwhosedistancesfromtwofixedpoints,calledthefoci,remainsconstant.

Ellipse:Acurveonaplanesurroundingtwofocalpointssuchthatastraightlinedrawnfromoneofthefocalpointstoanypointonthecurveandthenbacktotheotherfocalpointhasthesamelengthforeverypointonthecurve.

ConicSections

“ConicSections”2-bodygravitysystempredictsorbitsofconicsections

GravityGeneral

Hyperbola!

Orbitsof2-bodygravityproblemCircularVelocity EscapeVelocity

• Circularorbitisthespecialcaseoftheellipticorbits.

• v<vE →Ellipse

• v=vE →Parabola

• v>vE →Hyperbola

GravityGeneral

Example:2-bodyorbit“Cometshaveellipticalorparabolic/hyperbolicorbits”

Orbits of Comet Kohoutek (red) and Earth (blue), illustrating the high eccentricity of the orbit and more rapid motion when closer to the Sun.

CometsofhyperbolicorbitswillleavetheSolarsystemattheend(cf:non-periodiccomets)

GravityGeneral

SolarSystemPlanets(inellipticalorbits)

Planetaryorbitsareinclined.

X

X

OrbitalElementsSohowmanyparameters(=orbitalelements)doweneedtodescribeaplanetaryorbit?

OrbitalElementsSohowmanyparameters(=orbitalelements)doweneedtodescribeaplanetaryorbit?

Referenceplane(e.g.,ecliptic,equatorial)

Planetaryorbitalplane

PlanetaryorbitalplaneReferenceplane(e.g.,ecliptic,equatorial)

OrbitalElementsSohowmanyparameters(=orbitalelements)doweneedtodescribeaplanetaryorbit?

Referenceplane(e.g.,ecliptic,equatorial)

Planetaryorbitalplane

PlanetaryorbitalplaneReferenceplane(e.g.,ecliptic,equatorial)

1.Weneedtoknowthesizeandshapeoftheellipticalorbititself.

2.Weneedtoknowtheorientationsoftheorbit.

3.Weneedareferencetime.

OrbitalElements

Vernalpoint Ascendingnode

1.Semi-majoraxis(a)andeccentricity(e).

2.Inclination(i),longitudeofascendingnode(W),argumentofperihelion(w):ß 3angles;W andw aremeasuredinthecounter-clockwisedirectionandshowthelocationofthenodeandrotationoftheplane,respectively.

3.Epochofperihelion(t).

From Wikipedia

Motion of an asteroid w.r.t. background stars

TheSolarSystemTodayandThePossibleDiscoveryofaNinthPlanet