1

Class #22

Celestial engineering Central Forces DVD The power of Equivalent 1-D problem and

Pseudopotential Kepler’s 3rd law

Orbits and Energy The earth-moon flywheel

Energy and Eccentricity

:02

2

Two particles with central forces

2 2 2

1 2

2 2 2 1 2

21 22

1 2 1( ) ( )

2

1( )

1

2

1( )

2

0:

:

;

2

CM CM

r r U r

mmr r G

r

m mr r G

r

m mr G r

r

M R R

m m

M

MR R

r

rel

CM rel

rel

L

LL L

L

+

r

CMR

1m

2m

1 2mm MG G

r r

3

The power of

:20

21 2

2 2 2 2 21 2

2 21 2 1 2

1 2 2 2

1 1 1 1

2 2 2 2

i

i

Two real masses One reduced mass

I I I r

T T I I T I r

m v m m mmvr r r G G

r r r r

4

The power of - part 2

:25

2 2

1 2 1 1 2 2

2 2

2 11 2

2 222 1

1 22 2

2 22 1 1 21 2 2

2 1

2

I I m r m r

m mm r m r

M M

m mm m r

M M

m m mmmm r r

M m m

r

1r

2r

moonm

Earthm

21 2

2 2 2 2 21 2

2 21 2 1 22 2

1 1 1 1

2 2 2 2

i

i

Two real masses One reduced mass

I I I r

T I I T I r

m v m m mmvG G

r r r r

5

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

6

Pseudopotential and Energy

:35

21 2

22pseudo

mmU G

r r

1 2mmG

r

2

22 r

pseudoU

2 2 2 1 2

2 2 2 1 2

22 1 2

2

2

2

1( )

21( )

2

1

2 2

1

21

2

pseudo

mmT U r r G

rm m

T U r r Gr

m mE r G

r r

E r U

E r

rel

rel

rel

L

H

H

7

Pseudopotential and Energy

:37

21 2

22pseudo

mmU G

r r

2

2

1

21

2

pseudoE r U

E r

0E bounded orbit

0E unbounded orbit

8

Earth and Moon

211

26.67 10

28

2,419,200sec

?

NmG

kg

Lunar month days

T

24

22

6.0 10

7.3 10

Earth

moon

m kg

m kg

1 2 1

11 2

2

1 11

2 2

1

1; (1 )

mm mmm mm

m mIf m

m m

1. What is the reduced mass for

the earth-moon system (in kg)?2. How many percent different is

it than the lunar mass3. What is theta-dot?4. What is the radius of a circular

orbit?5. How would this change if the

earth were fixed in space by the hand of God or a Borg tractor beam?

9

Orbital Energy

:47

1. Last class we derived values for omega, mu and r for the earth-moon system

2. Total energy consumption on Earth is 1000 Terajoules/day

3. If we could power human activites by stealing the angular kinetic energy from the earth-moon system, how much should omega change to give us 100 years of power?

24

22

22

6

8

6.0 10

7.3 10

7.2 10

2.6 10 /

3.9 10

Earth

moon

m kg

m kg

kg

rad s

r m

10

Converting .

:30

21 22 3

2

1 22

2

1 2

1;

( ) ( )

( )(1 cos )

mmr G

r r

d d d u dr

u dt dt d d

Gm mu u

rGm m

( ) ( )r t to r

11

Properties of ellipses

:30

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

12

E, L and Eccentricity

:30

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).

13

Energy and Eccentricity

:30

2

1 2 22

2

1 2

12

( )(1 cos )

GmmE

rGmm

1

1

0

0 1 Eccentricity

Energy Orbit

E<0 Circle

E<0 Ellipse

E=0 Parabola

E>0 hyperbola

0

1 1

0 1

14

Kepler’s 1st, 2nd and 3rd laws (1610)

:37

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

15

E, L and Eccentricity

:30

2

2

1 2

21E

Gmm

1

1

0

0 1

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

16

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

17

Predicting collisions

:37

21 2

22pseudo

mmU G

r r

impactrr pp

Case Need

0

Eminr

min 6400r km

v

impactr

18

Predicting collisions

:37

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

19

Predicting collisions

:37

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

20

Planetary Scattering Angle

:37

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