Orbits and Kepler's and Newton's Laws: Understanding Voyager's Journey, Study notes of Astronomy

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Lecture 18 – Orbits, Applying Kepler’s and

Newton’s Laws

Voyager’s Journey

Voyager’s Orbits:

~ 100 AU (Sept. 2005)

Kepler’s First Law

Major axis Minor axis An ellipse has two foci b c a e=c/a

P

2

~ a

3

P

2

= Ka

3

(K is a constant of proportionality; K

= 1 if a is in AUs and P is in years.)

Kepler’s Third Law

Newton

Physical Basis

Generalization of possible orbits

P

2 = 4 π 2

a

3 G(m 1 +m 2

NP

Not to scale!!

A

B

Getting to Orbit

A

B

Getting to Orbit – > V

B

< V

circ NP Perigee here Apogee here Path is ellipse with Earth at one focus

A

B

Getting to Orbit – > V

B

= V

circ NP Path is circle with Earth at center V circ ~ 8 km/s = 28 800 km/hr LEO ~ 90 min.

A

B

Getting to Orbit – > V

B

= V

esc NP Path is parabolic V esc ~ 11 km/s = 39 600 km/hr

How fast does a satellite go in low Earth orbit? First, we equate gravitational and centripetal forces: F c = F g mv 2 /r = GMm/r 2 (approximate version) v = (GM/r) 1/ (notice that v is independent of sat. mass) Inserting numbers: v = [(6.67 x 10

• )(5.98 x 10 24 )/(65001000)] 1/ v = 7.8 km/sec = 28,000 km/hr

Rocket Science

A

B

Getting to Orbit – > V

B

> V

esc NP Path is hyperbolic As V B the spacecraft will have a higher velocity once it is far from Earth

A

B

Modifying Orbits

NP

Apogee here Perigee here Initial orbit has low perigee Rocket Burn adds energy