2006: Assoc. Prof. R. J. Reeves Gravitation 3.1 P113 Gravitation: Lecture 3 Escape speed from orbit Planets and satellites: Keplers Laws Orbital energy

2006: Assoc. Prof. R. J. Reeves Gravitation 3.2 Escape from Earth’s Gravity We know from everyday experience by throwing stones that they go higher up if we throw them with faster velocity. Can we derive an equation that will give the height as a function of velocity? A mass m starts at the surface of the earth with vertical velocity v. It reaches an altitude h where it turns around and falls down. Conservation of energy gives: Exercise: Complete the algebra to derive the following equation for h.

2006: Assoc. Prof. R. J. Reeves Gravitation 3.3 Escape from Earth’s Gravity - 2 Solving for h gives v (m/s)Altitude h (m) ,00056,000 10, x ,000– 6.7 x 10 6

2006: Assoc. Prof. R. J. Reeves Gravitation 3.4 Escape from Earth’s Gravity - 3 The negative altitude reached when the velocity was 100,000 m/s indicates the mass becomes unbound from the earth’s gravitational field. The escape velocity from the earth’s surface is given by BodyMass (kg)Radius (m)Escape speed (km/s) Earth5.98 x x Earth’s moon7.36 x x Jupiter1.90 x x Sun1.99 x x

2006: Assoc. Prof. R. J. Reeves Gravitation 3.5 Escape from Earth’s Gravity - 4 Escape from earth requires 11.2 km/s = 40,000 km/hr! How do rockets manage this incredible speed? 11.2 km/s is the required speed at the earth’s surface without any additional thrust being applied. We could escape from earth at 40 km/hr providing we applied thrust greater than our weight all the way to the moon! Rockets achieve a balance: –They start out less than 11.2 km/s but continue to apply thrust. –The necessary escape speed decreases as the altitude increases. –Eventually the rocket is travelling faster than the escape speed for its present altitude. The rocket motors can be turned off and then it’s glide to the moon!

2006: Assoc. Prof. R. J. Reeves Gravitation 3.6 Kepler’s Laws applied to the Planets THE LAW OF ORBITS: All planets move in elliptical orbits with the Sun at one focus. The orbit is characterised by its semimajor axis a and its eccentricity e. R p is the perihelion distance. R a is the aphelion distance.

2006: Assoc. Prof. R. J. Reeves Gravitation 3.7 Kepler’s Laws applied to the Planets THE LAW OF AREAS: A line that connects the planet to the Sun sweeps out equal areas in then plane of the planets orbit in equal time intervals. Kepler’s second law is a statement of conservation of angular momentum.

2006: Assoc. Prof. R. J. Reeves Gravitation 3.8 Kepler’s Laws applied to the Planets THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. Planeta (10 10 m)eT (yrs)T2/a3T2/a3 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

2006: Assoc. Prof. R. J. Reeves Gravitation 3.9 Solar system simulation

2006: Assoc. Prof. R. J. Reeves Gravitation 3.10 Orbital Energy - 1 Kepler’s second law stipulates that the speed of a planet in orbit is faster when closer to the sun. Correspondingly, the gravitational potential energy is smaller (more negative) when closer. Assume an exact circular orbit. Newton’s second law gives The kinetic energy is then The total energy of the orbit is

2006: Assoc. Prof. R. J. Reeves Gravitation 3.11 Orbital Energy - 2 If the orbit is elliptical, the total energy is The negative value for the total energy is a general indication of a bound system. Question: Can you think of another bound system with a negative energy?

2006: Assoc. Prof. R. J. Reeves Gravitation 3.12 Problem