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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