1. Goal: To understand orbits
Objectives:
To learn about Projectiles of different velocities
To understand how Orbits work
To learn about Escape Velocity
To learn about Kepler’s Laws
To understand Our Solar System

2. Slow projectile
I toss a ball, what is the shape of the motion like and why?

3. Cannon ball!
Goes a lot faster, what is its motion like and why?

4. Super fast cannon ball
Now we examine a cannonball traveling at a velocity of 8 km per second.
What will its motion be like?

5. Orbital Velocity
The velocity needed for a circular orbit is given by the following:
Orbital Velocity = square root of (G * Mass you are orbiting / orbital radius)

6. What happens if you speed up while in orbit?

7. Elliptical Orbits Typically orbits are elliptical (egg shaped).
The more you speed up instantly, the more egg shaped your orbit would become.
If you did it slowly, you would just have a bigger orbit.
What happens though if you make the velocity too big?

8. Escape velocity You escape! But how much do you need to escape?
The orbital velocity around the earth is 8 km/s. What do you think the escape velocity is?

9. 40% rule
If you multiply the circular orbit velocity by 40% that is the escape velocity.
There is not a big range of velocities.

10. Our solar system The earliest model made an earth centered model.
While this seems nuts, it was in fact very accurate.
This model took 1500 years before the wheels fell off of it.

11. Kepler
Using the very accurate observations by his mentor Tycho Brahe, Kepler tried to make a circular model for the solar system with the sun in the center.
For most of the planets this worked very well.
Mars was a problem. During most of Mars’s orbit, it fit very well.
However, at a few spots, Mars’s position would differ by up to 8 arc minutes from where his model would predict.

12. Kepler’s dilemma:
A) I have the right model for the solar system at long last! Those 8 arc min are just errors by Tycho.
B) I am a little concerned with the deviations, but my model is very good, much better than anyone else. It should work for 1000 years before anyone else needs to tweak it further.
C) No, no, this won’t do. I am going to scrap this and go another direction completely!

13. Kepler’s decision:
Kepler decided his mentor had the correct observations.
The error lay in his model.
The planets must NOT have circular orbits around the sun.
There must be another explanation.

14. Solution: Orbits have another shape.
After testing many shapes, Kepler found one that worked!
From that Astronomy was changed forever, and his method of using observations combined with mathematical models became the Scientific Method!

15. Law 1 Orbits are not circular. They are elliptical.
Ellipses are egg shapes which have 2 foci.
The separation of the foci is determined by the “eccentricity”.
There is a constant distance such that for any point on the ellipse the sum of the distances to the foci is a constant.

16. Law 2 In an orbit, the area covered per unit time is a constant.
Therefore, when an object is closer to the object it orbits, it travels faster, and similarly, slower when it is further away.
Laws 1 and 2 published in 1609

17. Law 3 Published 1619 Period2 = constant * a3
a = semi-major axis = half the longest length of the ellipse.

18. Law 3 question Period2 = constant * a3
What is the value of the constant for any object orbiting the sun (denoted usually as k)? Hint, think about the earth sun system to try to calculate it using as easy of #s as possible.

19. Another question
You discover an asteroid which has a semi-major axis (a) of 4 AU.
What is its orbital period in years?
A) 2 years
B) 4 years
C) 8 years
D) 16 years
Hint: Kepler’s 3rd law.

20. Our solar system
Period2 = a3, just as long as your period is in years and value for a (semi major axis) is in AU.
Orbital velocities:
Velocity = 30 km/s divided by square root of distance from sun (in AU)
Saturn orbits at 10 AU. What is the orbital velocity of Saturn?

21. Conclusion
Orbits can be fun, and are not as complicated as you might expect.