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Class #19 of 30

Celestial engineering - II Reduced 2-body problem Kepler 1st, 2nd and 3rd laws Cometary collision prediction A whiff of scattering theory DVD – Gravitational slingshot

:02

2

Reduced two-body problem

:15

1r

2r

relativer

moonm

Earthm

3

Equivalent 1-D problem

:30

2 2 2 1 2

21 22

21 22 3

1( )

2

:

mmr r G

rm m

r r G rr

m mr G

r r

relL

Relative Lagrangian

Radial equation

Total Radial ForcetotalF

total pseudoF dr U

2r

2

1 222pseudo

mmU G

r r

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Kepler’s 1st, 2nd and 3rd laws (1610)

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1st Law – Planets move in ellipses with sun at one focusThird law demonstrated previously relates period to semi-minor radius2nd law is direct consequence of momentum conservation

“Equal areas are swept out in equal times”True for ALL central forces

2 32 4 r

GM

2

2 '

"

.

.

1 1( )( )

2 2

.

r const

So r const

dAbh r r

dt

const

1b 2b1h 2h

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Predicting collisions

:37

21 2

22pseudo

mmU G

r r

2

2

1

21

2

pseudoE r U

E r

1 2mmG

r

2

22 r

pseudoU

r p Case All objects hit the earth,

regardless of initial distance.

0

At turning point 0r

6

Predicting collisions

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21 2

22pseudo

mmU G

r r

impactrr pp

Case Need

0

Eminr

min 6400r km

v

impactr

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Predicting collisions

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22

2

2

2

22

min min

1

2 2

1;

2

( )1

2 2

impact

impact

ME r G

r r

E v v r

v rMv G

r r

impactrr pp

2 2 2 2min min

2 2 4 2

min 2

min min

2 0

2 4 4

2

; 0

impact

impact

impact

v r GMr v r

GM G M v rr

v

If GM is weak r r If GM is strong r

22

min 21 1impactr vGM

rv GM

8

Predicting collisions

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For v-infinity comparable to v-escape, R-impact is approx same as R-earth. For v-infinity < v-escape, R-impact is many times R-earth

112.5

37.3

11.3

3.9 1.5 1.10.0

20.0

40.0

60.0

80.0

100.0

120.0

0 0.5 1 1.5 2 2.5 3

V-infinity / Escape Velocity

R-i

mp

act

/ R-E

arth

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Properties of ellipses

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2

1 2

min max

2

2

1 2 22

2

1 2

;1 1

1

12

( )(1 cos )

cGm m

c cr r

b

a

Gm mE

rGmm

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E, L and Eccentricity

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2

1 2 22

2

2

1 2

2

1 2

12

21

( )(1 cos )

GmmE

EGmm

rGmm

1

1

0

0 1

The physics is in E and L. Epsilon is purely a geometrical factor.Epsilon equation applies to ALL conic sections (hyperbolae, ellipses, parabolas).

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E, L and Eccentricity

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2

2

1 2

21E

Gmm

1

1

0

0 1

The physics is in E and L, and it transfers to quantum mechanics.

12

Planetary Scattering Angle

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Epsilon -1/eps 2*arccos Omega1 -1.00 360.0 0.0

1.001 -1.00 354.9 5.11.02 -0.98 337.3 22.71.05 -0.95 324.5 35.51.1 -0.91 310.8 49.21.3 -0.77 280.6 79.42 -0.50 240.0 120.05 -0.20 203.1 156.920 -0.05 185.7 174.350 -0.02 182.3 177.7100 -0.01 181.2 178.8

2

1 2

( )(1 cos )

rGmm

( )r

( )r 1180 arccos

impactr

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Strong Force Scattering Angle

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Quantum mechanics is needed, but…Particle physicists probe strong interactions by scattering particles just as we can probe gravity by scattering meteors

14

Problem L19-1

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2

1 2

( )(1 cos )

rGmm

Two rocks are scattering off the Earth’s gravitational fieldOne has orbit with Epsilon=1.0015Other has orbit with Epsilon=3.0How much are their paths deflected by the Earth?Sketch the two orbits.

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Problem L19-2

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2

2

1 2

2

1 2

21

( )(1 cos )

EGmm

rGmm

A 1000 kg rock is aimed in the vicinity of EarthOut beyond the orbit of pluto, it has a kinetic energy of 50 billion joules and an angular momentum of 96 trillion kg-m^2/s. Calculate epsilonWill the rock hit earth?If not, what is the minimum radius?

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Class #19 Windup

Epsilon is function of L and E and GMm

Office hours today 3-5, tomorrow 4-5:30

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2

2

1 2

2

1 2

21

( )(1 cos )

EGmm

rGmm

KEPLER1st Law – Planets move in ellipses 2nd law dA/dt=const3rd law Period goes as semi-minor radius to 3/2 power