Class 34: The Orbits Class 34: The Orbits Keplers Laws 1. The Sun is - - PowerPoint PPT Presentation

Class 34: The Orbits Class 34: The Orbits

Kepler’s Laws

• 1. The Sun is at the focus of the orbits (conic sections).
• 2. The line joining the Sun and a planet sweeps out equal areas in equal times. This

is conservation of angular momentum, true for any central force field. g , y

Move faster Move slower

constant 2 L r 2 1 dt dA

2

     

3.

2 2

G G L ab 2 2 L ab dt dA ellipse

• f

Area T       

For bound orbit (E<0), the period of revolution is proportional to (semi‐major

2 2 p a 2 2 2 2

L L | E | 2 GM 2E GM 2 r r a E 2 L

• r

E GM r r 2 L r GM

• E

                

proportional to (semi major axis)3/2. This is true only for inverse square central force.

2 2 2 p a

2 b 2 b 2 a GM b | E | 2 b L and 2a GM | E | | E | 2 L E 2 L r r b             

3/2 2

a GM 2 a GM b ab 2 L ab 2 T          

Hohmann Transfer (Tangential Thrust at extreme radii)

Thrust Thrust

| E | 2 GM a  

Thrust

| | d d

| E | 2 L b  

| | d d |E| decrease and L increase |E| increase and L decrease Hohmann transfer is the most fuel

Thrust Thrust

Hohmann transfer is the most fuel efficient path, but not in time.

Radial Thrust at extreme radii

|E| decrease and L constant

Thrust Thrust

|E| decrease and L constant

Ueff Energy E r

E E

“Slingslot” Vf’

f

Vf’ Vf Vf Vplanet Vplanet Vi Vplanet Vi’

i

Vi’

planet

To the moving planet (prime frame), |vi’|=|vf’| To an inertial frame where the planet is moving , |vf’|> vi’|