6.1-6.3 Planetary Motion

1. 6.1-6.3 Planetary Motion

2. Force fields • Not the kind you find in Star Trek (the coolest show ever) • Force fields are used to describe the amount of a given type of force generated by an object on other objects that are near it per a unit measurement

3. For example… • A gravitational field affects objects that have mass • Therefore, any object that possesses mass that is within a gravitational field will experience a gravitational force acting on it • How much force is acting on it will depend on where this object is – and how much mass it has

4. Why use force fields? What happens if objects of the same mass move further or closer to the object creating the field? Is the amount of gravity acting on two objects of different mass the same, if their distances from the earth are the same?

5. Force fields are like prices • What force fields allow you to do is to calculate the value of a force acting on an object depending on a set of conditions • It’s like pricing objects in a store • Is it easier to give the price of one, two or three objects… • Or give a price per object?

6. Price • A price gives you the cost per item – so you can predict the cost of a purchase based on how many items you purchase • Mathematically, a simple way of viewing a field is looking at the measurement as a price • The strength of a field at one point tells you how much the total value will be based on the amount of a particular quality that the field affects

7. Take another look at an equation you know…. How does: F = Gm1m2 d2 Define a gravitational field? See notebook file for derivation of gravitational field

8. Planetary orbits • The orbits of planets are not circular; they are actually ellipses: • http://mistupid.com/astronomy/orbits.htm • However, in order to derive the velocity of orbits based on the gravitational pull between two bodies can be dealt with by assuming that the orbit is circular

9. Circular motion and orbits • Recall that when looking at circular motion, an object maintains a constant velocity in a circular path if there is a constant force that pulls the circulating object towards the center of its rotation • This situation is similar to how orbits are formed

10. Circular orbit and planetary motion • Compare the relationship between an orbiting planet and the motion of an object on a string • http://www.physclips.unsw.edu.au/jw/circular.htm • Therefore, equations related to circular motion can be used to approximate the velocity of objects in orbit • See notebook file for the derivation of orbital velocity

11. Change in orbit • What happens if Fg was to change? • What happens if v changes?

12. Kepler’s Laws • Johannes Kepler a German mathematician, astrologer and astronomer determined 3 laws that govern planetary motion • Though Kepler finally deduced the real motion of the planets, he did so by analyzing data gathered by another scientist by the name of Tycho Brae

13. Kepler’s First Law • Planetary orbits are elliptical, with the sun at one focus of the ellipse

14. Kepler’s Second Law • The straight line connecting the planet and the sun sweeps out equal areas in the same amount of time Kepler's Second Law Interactive

15. Kepler’s Third Law • The cube of the average radius , r, of a planet’s orbit is directly proportional to the square of its period, T • Namely: r3α T2 • Therefore: r3= CsT2 • Where: Cs = constant of proportionality for the sun • Note that: Cs is based on the object that is creating the gravitational field • Kepler's Third Law Interactive • See derivation of Cs in notebook file

16. Understanding Escape Energy • Objects on planets are “bound” to the planet in a situation very similar to the following • Imagine being tied to a bungee cord to another object, and the only method of escape that you have is to run as fast as you can in order to try to “break” the cord

17. Discuss the energy • What must you do in order break free? Discuss your energy expenditure • What is the relationship between your energy expenditure and the distance between you and the object?

18. You’re in an energy debt • You can view the energy of this system in terms of how much you “owe” the bungee cord in order to get free • All the effort that you put in in order to break free of the bungee cord doesn’t increase your speed – it’s all put into stretching or trying to break the cord • If the cord wasn’t there – you would be going at a much faster speed for the same distance travelled!

19. Escape energy E Radius of planet r

20. How do you get away from the bungee cord? • An object exiting a gravitational field must do so by paying the energy “debt” with kinetic energy • In order to escape the pull of the planet, the total kinetic energy of the rocket must EQUAL OR EXCEED the energy debt owed to the planet • ESCAPE VELOCITY refers to the minimum velocity required for an object to just escape the gravitational pull of the planet, • ESCAPE ENERGY refers to the energy associated with the kinetic energy required to repay the energy “debt” • See notebook file for derivation

21. Tied up in debt • Any object that is “bound” to earth remains so because its total energy does not exceed the energy debt owed to the planet • Think about an orbiting satellite: • It remains in orbit around the earth (therefore, it is still “tied” to the earth – how do we know? What happens if it stops moving?) • But it also has a kinetic energy • See notebook file for derivation

22. The debt in orbit E r r Etotal Ek Eg • Total energy for objects in orbit will equal to HALF of the potential energy owed at that particular radius

23. Total energy of objects in a gravitational field • Therefore, total energy of any object in a gravitational field is therefore equal to the sum of its gravitational potential energy and the object’s kinetic energy • Therefore, there are 3 cases that can be set up due to this

24. Case 1 – Object just escapes: ET = 0 E • In this situation, the kinetic energy is just enough for the object to escape • All the kinetic energy is used to pay the energy debt • What will the object’s motion be like when it escapes the field? Ek r Eg

25. Case 2 – Object escapes with v > 0: ET > 0 E • In this situation, there is enough Ek to pay the debt and provide the object with enough Ek to continue onwards with a constant velocity as r ∞ Ek r Eg

26. Case 3 – Bound object: ET < 0 E • In this situation, there is enough Ek is not great enough, so total energy is still negative • Since there is still an energy debt, the object remains bound to the planet and cannot escape Ek r Eg