njit physics 320: astronomy and astrophysics – lecture ii carsten denker physics department center...
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NJIT
Physics 320: Astronomy and Astrophysics – Lecture II
Carsten Denker
Physics DepartmentCenter for Solar–Terrestrial Research
September 10, 2003NJIT Center for Solar-Terrestrial Research
Celestial Mechanics
Elliptical OrbitsNewtonian
MechanicsKepler’s Laws
DerivedThe Virial
Theorem
September 10, 2003NJIT Center for Solar-Terrestrial Research
Elliptical OrbitsKepler’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:2 3P a
September 10, 2003NJIT Center for Solar-Terrestrial Research
Ellipses
Focal points F1 and F2 (sun in principal focus)
Distance from focal points r1 and r2
Semimajor axis aSemiminor axis bEccentricity 0 e 1Ellipse defined:1 2 2r r a
2 2 21 2
2 2 2
( )
(1 )
r r r a r b ae
b a e
2(1 )
1 cos
a er
eA ab
September 10, 2003NJIT Center for Solar-Terrestrial Research
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 kmLight minute: 1.800 107 km
(1 AU = 8.3 light minutes)Parsec: 1 pc = 3.0857 1013 km =
3.262 ly
September 10, 2003NJIT Center for Solar-Terrestrial Research
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
September 10, 2003NJIT Center for Solar-Terrestrial Research
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.
Newton’s 2nd Law: The net force (the sum of all forces) acting on an object is proportional to the object’s mass and it’s resultant acceleration.
Newton’s 3rd Law: For every action there is an equal and opposite reaction.
net1
net
( )
n
ii
F F ma
dv d mv dpF m
dt dt dt
12 21F F
September 10, 2003NJIT Center for Solar-Terrestrial Research
Gravitational Force
2 3P kr (Kepler’s 3rd law, circular orbital motion, M >> m)
2 rP
v
(constant velocity)
2 2 2 23
2 2
4 4r v mkr m
v r kr
(centripetal force)
2
2 2
4 m MmF G
kr r
(law of universal
gravitation)
Universal gravitational constant: 6.67 10–11 Nm2 / kg2
September 10, 2003NJIT Center for Solar-Terrestrial Research
Gravity Near Earth’s Surface
2 2( )
M m M mF G G
R h R
( )h R
2
MF ma mg g G
R
24
23
5.974 10 kg m9.799
s6.378 10 km
Mg
R
September 10, 2003NJIT Center for Solar-Terrestrial Research
Potential Energy
f
i
r
f i
r
U U U F dr
2
1 1f
i
r
f ir
MmU G dr GMm
r r r
( )F dr Fdr
MmU G
r ( 0 if )f fU r
ˆˆ ˆU U UF U i j k
x y z
September 10, 2003NJIT Center for Solar-Terrestrial Research
Work–Kinetic Energy Theorem
22
2 2
( )
( )
( / 2) 1
2
1 1
2 2
f i
i f
i i
f f
i i
f f
r t
r t
t t
t t
t v
t v
f i
dpW U F dr vdt
dt
dv dvm vdt m v dt
dt dt
d vm dt md v
dt
mv mv K
September 10, 2003NJIT Center for Solar-Terrestrial Research
Escape Velocity
21
2
MmE mv G
r
Total mechanical energy:
2esc
12 / 11.2 km/s
2
Mmmv G v GM r
r
Conservation of mechanical energy:
Minimal launch speed:2
min 7.9 km/sv
g v Rgr
September 10, 2003NJIT Center for Solar-Terrestrial Research
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?
September 10, 2003NJIT Center for Solar-Terrestrial Research
Center of Mass
11 1 2 22 1
1 21
n
i iin
ii
m rm r m rr r r R R
m m m
1 1 1
n n n
i i i i ii i i
m R m r MR m r
1 1
n ni
i i ii i
drdRM m MV m v
dt dt
2
net 21
0n
i
i
dpdP dP d RF M
dt dt dt dt
September 10, 2003NJIT Center for Solar-Terrestrial Research
Binary Star System in COM Reference Frame
1 1 2 2
1 2
0 0m r m r
Rm m
2
11 2
2 11
21 2
mr r
m mr r r
mr r
m m
111 2
1 22
2
r rmm m
m mr r
m
Reduced mass
September 10, 2003NJIT Center for Solar-Terrestrial Research
Energy and Angular Momentum
2 1 21 1 2 2
2 1
1 1
2 2
m mE m v m v G
r r
21
2
ME v G
r
2 1, , and dr
v v v r r rdt
1 1 1 2 2L m r v m r v
L r v r p
In general, the two–body problem may be treated as and equivalent one–body problem with the reduce mass moving about a fixed mass M at a distance r.
September 10, 2003NJIT Center for Solar-Terrestrial Research
Kepler’s 2nd Law
0!dL d dr dp
r p p r v p r Fdt dt dt dt
2 21 1
2 2
dA ddA dr r d r dr d r d r
dt dt
1ˆ ˆ2r
dr d dAv v v r r rv
dt dt dt
1
2
L L dA Lrv r v
dt
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.
September 10, 2003NJIT Center for Solar-Terrestrial Research
Kepler’s 3rd Law
(1 ) (perihelion)
(1 ) (aphelion)p
p p a aa
r a ev r L rv r v r v
r a e
2 21 1 1 and
1 2 (1 ) 2 (1 )p
p aa
v e M Mv G v G
v e a e a e
2 21 1 and
1 1p a
GM e GM ev v
a e a e
2(1 )p pL r v GMa e
2 1 21 1
2 2 2 2pp
m mM ME v G G G U
r a a
Virial Theorem
September 10, 2003NJIT Center for Solar-Terrestrial Research
Kepler’s 3rd Law (cont.)
21
2 2
M ME G v G
a r
21 2
2 1( )v G m m
r a
0 0 0
1 1 1
2 2 2
P P PdA L L LA dt dt dt P
dt
2 2 2 2 2 2 2 2 2 2
2 32 2 2
2 4 4 4
1
A a b a bP a
L L GMGMa e
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.
September 10, 2003NJIT Center for Solar-Terrestrial Research
Class Project
Exhibition
Science
Audience
September 10, 2003NJIT Center for Solar-Terrestrial Research
Homework Class Project
Read the Storyline hand–outPrepare 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!
September 10, 2003NJIT Center for Solar-Terrestrial Research
Homework Solutions
(a) 90 42 23.5 71.5Problem 1.5
(b) 90 42 23.5 24.5
Problem 1.6 (a) 90 90
(b) 66.5
(c) 90
l
l
l
m
11
Problem 1.7 (a) =9.9 2.48 , 10 0.167 , 1.23°
(b) s=d =8.56 10 km = 5720 AU
September 10, 2003NJIT Center for Solar-Terrestrial Research
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!