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Physics 320: Astronomy and
Astrophysics – Lecture II
Carsten Denker
Physics Department
Center for Solar–Terrestrial Research
NJIT
Celestial Mechanics
 Elliptical
Orbits
 Newtonian
Mechanics
 Kepler’s Laws
Derived
 The Virial
Theorem
NJIT Center for Solar-Terrestrial Research
September 10, 2003
Elliptical Orbits
Kepler’s 1st Law: A planet orbits the Sun in an
ellipse, with the Sun at on focus of the ellipse.
 Kepler’s 2nd Law: A line connecting a planet to
the Sun sweeps out equal areas in equal time
intervals.
 Kepler’s 3rd Law: The
average orbital distance a
of a planet from the Sun
is related to the planets
sidereal period P by:

P a
2
3
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September 10, 2003
Ellipses
 Focal
points F1 and F2 (sun in principal focus)
 Distance from focal points r1 and r2
 Semimajor axis a
 Semiminor axis b
 Eccentricity 0  e  1
 Ellipse defined:
r1  r2  2a
r1  r2  r  a  r 2  b2  (ae)2
 b2  a 2 (1  e2 )
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a(1  e 2 )
r
1  e cos 
A   ab
September 10, 2003
Conic Sections
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September 10, 2003
Distances in the Planetary System
 Astronomical
unit [AU], average distance
between Earth and Sun:
1 AU = 1.496  108 km
 Light year: 1 ly = 9.461  1012 km
 Light minute: 1.800  107 km
(1 AU = 8.3 light minutes)
 Parsec: 1 pc = 3.0857  1013 km = 3.262 ly
NJIT Center for Solar-Terrestrial Research
September 10, 2003
Newtonian Physics

Galileo Galilei (1564–1642)






Heliocentric planetary model
Milky Way consists of a multitude of stars
Moon contains craters  not a perfect sphere
Venus is illuminated by the Sun and shows phases
Sun is blemished possessing sunspots
Isaac Newton (1642–1727)


1687 Philosophiae Naturalis Principia Mathematica
 mechanics, gravitation, calculus
1704 Optiks  nature of light and optical experiments
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September 10, 2003
Laws of Motion
Newton’s 1st Law: The law of inertia. An object at rest will
remain at rest and an object in motion will remain in
motion in a straight line at a constant speed unless acted
upon by an unbalanced force.
n
 Newton’s 2nd Law: The net force
Fnet   Fi  ma
(the sum of all forces) acting on
i 1
an object is proportional to the
dv d (mv ) dp
Fnet  m


object’s mass and it’s resultant
dt
dt
dt
acceleration.
 Newton’s 3rd Law: For every
F12   F21
action there is an equal and
opposite reaction.

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September 10, 2003
Gravitational Force
P 2  kr 3 (Kepler’s 3rd law, circular orbital motion, M >> m)
2 r
(constant velocity)
P
v
4 2 r 2
v 2 4 2 m
3

 kr  m 
(centripetal force)
2
2
v
r
kr
4 2 m
Mm
F
 G 2 (law of universal gravitation)
2
kr
r
Universal gravitational constant: 6.67  10–11 Nm2 / kg2
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September 10, 2003
Gravity Near Earth’s Surface
M m
F G
2
( R  h)
M m
G 2
R
(h
R )
M
F  ma  mg  g  G 2
R
M   5.974 10 kg 
m
  g  9.799 2
3
s
R  6.378 10 km 
24
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September 10, 2003
Potential Energy
rf
U f  U i  U    F  dr
ri
 1 1
Mm
U   G 2 dr  GMm   
 rf ri 
r
ri


rf
Mm
U  G
r
( F  dr   Fdr )
(U f  0 if rf  )
U ˆ U ˆ U ˆ
F  U  
i
j
k
x
y
z
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September 10, 2003
Work–Kinetic Energy Theorem
rf
ti
dp
W  U   F  dr  
 (vdt )
dt
ri
tf
ti
ti
dv
 dv 
  m  (vdt )   m  v  dt
dt
 dt 
tf
tf
ti
vi
d (v / 2)
1 2
 m
dt   md  v 
dt
2 
tf
vf
2
1 2 1 2
 mv f  mvi  K
2
2
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September 10, 2003
Escape Velocity
Total mechanical energy:
1 2
Mm
E  mv  G
2
r
Conservation of mechanical energy:
1 2
Mm
mv  G
 vesc  2GM / r  11.2 km/s
2
r
Minimal launch speed:
v2
g   vmin  Rg  7.9 km/s
r
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September 10, 2003
Group Problem
What is the minimum launch speed required to
put a satellite into a circular orbit?
 How many times higher is the energy required
to to launch a satellite into a polar orbit than that
necessary to put it into an equatorial orbit?
 What initial speed must a space probe have if it
is to leave the gravitational field of the Earth?
 Which requires a a higher initial energy for the
space probe – leaving the solar system or hitting
the Sun?

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September 10, 2003
Center of Mass
mr
m1r1  m2 r2

i 1 i i
r  r2  r1  R 
R
n
m1  m2
 mi
n
i 1
n
n
n
i 1
i 1
i 1
  mi R   mi ri  MR  mi ri
n
dri
dR n
M
  mi
 MV   mi vi
dt i 1
dt
i 1
dP n dpi
dP
d 2R

 Fnet 
M 2 0
dt i 1 dt
dt
dt
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September 10, 2003
Binary Star System in COM
Reference Frame
R 0
m1r1  m2 r2
0
m1  m2
m2

r1   m  m r

1
2
r2  r1  r  
 r  m1 r
 2 m1  m2


r1   m r
m1m2

1


m1  m2
r   r
 2 m2
Reduced mass
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September 10, 2003
Energy and Angular Momentum
m1m2
1
1
2
E  m1 v1  m2 v2  G
2
2
r2  r1
1 2
M
E  v  G
2
r
dr


, and r  r2  r1 
v  v , v 
dt


L  m1r1  v1  m2r  v2
L  r  v  r  p
In general, the two–body
problem may be treated as
and equivalent one–body problem
the reduce mass moving about a fixed mass M at a distance r.
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with
September 10, 2003
Kepler’s 2nd Law
dL d
dr
dp
 r  p   p  r 
 v  p  r  F  0!
dt dt
dt
dt
1 2
dA 1 2 d
dA  dr  r d   r dr d  r d 
 r
2
dt 2 dt
dr ˆ
d ˆ
dA 1
v  vr  v  r  r

 rv
dt
dt
dt 2
L
L
dA 1 L
rv  r  v 
 

 
dt 2 
The time rate of change of the area swept out by a line
connecting a planet to the focus of an ellipse is a constant.
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September 10, 2003
Kepler’s 3rd Law
rp  a(1  e) (perihelion)
v r 
 L   rv   rp v p   ra va
 ra  a(1  e) (aphelion)
1 e
1 2
M
1 2
M


and  v p  G
  va  G
va 1  e
2
a(1  e) 2
a(1  e)
vp
GM
v 
a
2
p
GM  1  e 
 1 e 
2

 and va 


1

e
a
1

e




 L   rp v p   GMa(1  e2 )
Virial Theorem
m1m2 1
1 2
M
M
 E  v p  G
 G
 G
 U
2
rp
2a
2a
2
NJIT Center for Solar-Terrestrial Research
September 10, 2003
Kepler’s 3rd Law (cont.)
M 1 2
M
E  G
 v  G
2a
2
r
2 1
 v  G (m1  m2 )   
r a
2
P
P
Virial Theorem:
For gravitationally bound
systems in equilibrium, it can
be shown that the total energy
is always one–half of the time
averaged potential energy.
P
dA
1L
1L
1L
A
dt  
dt 
dt

P

dt
2
20
2
0
0
2 2 2 2
4 2 a 2b2  2
 2 A  4 a b 
2
P 

 
2
L
 L 
 2 GMa 1  e2
2

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

4 2 3

a
GM
September 10, 2003
Class Project
Exhibition
Science
Audience
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September 10, 2003
Homework Class Project
 Read
the Storyline hand–out
 Prepare a one–page document with
suggestions on how to improve the
storyline
 Choose one of the five topics that you
would like to prepare in more detail
during the course of the class
 Homework is due Wednesday September
23rd, 2003 at the beginning of the lecture!
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September 10, 2003
Homework Solutions
Problem 1.5 (a)
(b)
 90  42  23.5  71.5
 90  42  23.5  24.5
Problem 1.6 (a) 90  l    90
(b)
(c)
l  66.5
l  90
Problem 1.7 (a)  =9.9m  2.48,   10  0.167,   1.23°
(b)
s=d =8.56 1011 km = 5720 AU
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September 10, 2003
Homework
 Homework
is due Wednesday September
16th, 2003 at the beginning of the lecture!
 Homework assignment: Problems 2.3, 2.9,
and 2.11
 Late homework receives only half the
credit!
 The homework is group homework!
 Homework should be handed in as a text
document!
NJIT Center for Solar-Terrestrial Research
September 10, 2003