physics 320: astronomy and astrophysics – lecture ii

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


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


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


Conic Sections


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


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


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


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


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


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


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


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


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?


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


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


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.


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.


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


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.


Class Project

Exhibition

Science

Audience


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!


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


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!

physics 320: astronomy and astrophysics – lecture ii

Documents

solar system

ly center

law of inertia

average distance

constant speed

initial speed

higher initial energy

equal time intervals