1 SATELLITES AND GRAVITATION John Parkinson ©

2 SATELLITES

3 Newton’s Law of Gravitation M1M1 M2M2 r F F

4 CIRCULAR MOTION r F CENTRIPETAL FORCE

5 SATELLITES r m v Equation of Motion M

6 SATELLITE VELOCITY v r m M

7 v R m For an orbit CLOSE to the surface F = mg v = √ r g v = √ 6.4x10 6 x 10 = 8000 ms -1 v = 8 km/s

8 SATELLITE PERIOD r m ω Equation of Motion M

9 GE0SYNCHRONOUS COMMUNICATIONS SATELLITE TO REMAIN OVER ONE PLACE ON THE EARTH’S SURFACE, THE PERIOD HAS TO BE THE SAME AS THE EARTH’S DAY.

10 COMMUNICATIONS SATELLITE THIS GIVES A RADIUS OF km r

11 VGVG Definition The gravitational potential at a point is the work done in moving unit mass (1kg) from infinity to that point. But potential energy at infinity is zero – no attraction. Hence the gravitational potential at the point really measures the absolute potential energy that 1 kilogram has at that point. M 1 kg r FOR A RADIAL FIELD UNITSJ kg -1

12 The gravitational potential at a point measures the potential energy per kilogram at that point. M m kg r Joules The potential energy of a mass of m kilograms is given by: E p = mV G

13 g Relationship between Field Strength and Potential At any point Field Strength = POTENTIAL GRADIENT

14 V G = -40 MJkg -1 V G = -10 MJkg -1 A How much energy is needed to move a 200 tonne spaceship from A to B? B POTENTIAL DIFFERENCE, ΔV G = - 10 – ( - 40 ) = 30 MJ kg -1 E P = m ΔV G = kg x 30 MJ kg -1 ENERGY NEEDED = 6 TJ

15 TOTAL ENERGY OF A SATELLITE m v r M

16 ORBITAL DECAY m v r M When a satellite enters the atmosphere, drag heats it up. It falls to a lower orbit and SPEEDS UP. The decrease in total energy is only half of the decrease in potential energy. As r decreases, the potential energy becomes numerically larger but decreases overall as it is more negative. The kinetic energy is numerically equal to the total energy, but is positive, so increases.