Satellite Motion

Kepler’s Three Laws The Law of Ellipses: The path of the planets around the sun is elliptical in shape. The center of the sun is located at one focus. The Law of Equal Areas: A line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. The Law of Harmonies: The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. Equal areas in equal intervals of time? What do we mean by periods?

The Law of Ellipses All planets orbit the sun in a path resembling an ellipse, where the sun is one foci of the ellipse.

The Law of Equal Areas Describes the speed the planets moves while orbiting the sun. Planets move fastest when closest to the sun and slowest when further away. WHY? Animation: http://www.physicsclassroom.com/mmedia/circmot/ksl.cfm

The Law of Harmonies Compares the orbital period and orbital radius of a planet to other planets. The orbital radius is measured in “au” astronomical unit. 1au=1.4957 x 1011 m the average distance from the earth to the sun. The period is measured in earth years 3.156 x 107 seconds The ratio T2/R3 is nearly the same for all planets. k= 2.97 x 10-19 s2/m3

Newton’s take on it… Fnet=(mplanet x v2) /R Fgrav=Gmplanetmsun/R2 Fnet=Fgrav (mplanet x v2) /R=Gmplanetmsun/R2 v2=(4π2)/T2 (mplanet4π2R2)/(RT2)=Gmplanetmsun/R2 T2/R3=(mplanet4π2)/(Gmplanetmsun) T2/R3=(4π2)/(Gmsun) Newton combined the law of universal gravitation with circular motion principles to show that gravity provides the centripetal force for planets nearly circular orbits. Kepler’s T2/R3 can be predicted from this. Net Centripetal Force Force of Gravity = Replace V2 Substitute Simplified

Practice Problem

Practice Question The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One moon Io has a distance from Jupiter’s center is 4.2 units and it orbits Jupiter in 1.8 Earth-days. Ganymede is 10.7 units from Jupiter’s center. Predict the period of Ganymede using Kepler’s law of harmonies.

A Satellite is a Projectile! The only force acting upon a satellite is the force of gravity. Otherwise it would follow its inertial straight line path. Launched with enough speed a satellite will fall towards the earth at the same rate the earth curves. This speed is 8000 m/s because for every 8000 m along the horizon the earth curves downward 5 m. 5m = .5 (10m/s2) t2 t=1 second.

Based on your knowledge of circular motion: What is the direction of a satellite’s velocity: Perpendicular B. Tangent C. Inward D. Outward What is the direction of a satellite’s acceleration: What is the direction of a satellite’s net force: A. Perpendicular B. Tangent C. Inward D. Outward

Calculating Orbital Speed Fnet=(msat v2)/R Fgrav=(Gmsatmcentral)/R2 Fnet=Fgrav (msat v2)/R=(Gmsatmcentral)/R2 v=√Gmcentral/R

Calculating Orbital Acceleration g= Gm/R2 therefore a= Gmcentral/R2 Orbital Period Equation R radius of orbit T2/R3=(4π2)/(Gmcentral)

What do you notice about these three equations representing the velocity, acceleration, and period of an orbiting satellite? v=√Gmcentral/R a= Gmcentral/R2 T2/R3=(4π2)/(Gmcentral)

Practice Problem A satellite wishes to orbit the earth at a height of 100 km above the surface of the earth. Determine the speed acceleration and orbital period of the satellite. mearth=5.98 x 1024 Rearth+ height=6.47 x 106 V=7.85 x 10^3 A= 9.53 T=5176 seconds or 1.44 hr

Practice Problem The period of the moon is approximately 27.2days (2.35 x 106s). Determine the radius of the moon’s orbit and the orbital speed of the moon. mearth=5.98 x 1024 Rearth=6.37 x 106 Calculate R for the orbit R=3.82 x 10^8 V=1.02 x 103

Weightlessness in Orbit Why do you think astronauts in space feel weightless? Have you ever experienced weightlessness? The feeling of weightlessness is a sensation experienced when contact forces are removed. Contact forces: applied, tension, normal, etc. Requires to objects to make contact and can be felt (provide a sensation of weight) At a distance force: gravity Non contact forces are not felt Imagine the moments of weightless ness while riding on a rollercoaster

Weight Scales measure the upwards force of applied by the scale to balance out the force of gravity acting upon an object. When you are stationary (at equilibrium) this measurement relays your weight. How would taking a scale measurement while jumping or riding an elevator go? How would constant speed or rest compare to an object accelerating?

Weightlessness in Orbit Again: Why do you think astronauts in space feel weightless? They feel a sensation of weightlessness because there are no contact forces acting upon them only the force of gravity. Gravity is the only force acting therefore astronauts can orbit in circular motion, based on the centripetal force allowing inward acceleration.

Practice Problems Will stands on a bathroom scale and rides the elevator. Normally 784N (80kg) Will notices the scale reading depends on what is happening in the elevator. What would the scale read when will accelerates upward at .4 m/s2? 752N b. 816N c. 784N d. 32N What does the scale read when Will travels upwards at a constant velocity of 2.0 m/s? a. 752N b. 816N c. 784N d. 32N As Will reaches the top of the building the elevator slows down at a rate of .4 m/s2. What does the scale read? a.752N b. 816N c. 784N d. 32N

Satellite Energy Relationships Circular Satellite Motion Moves with constant speed. Remains at the same height above the surface of the central object. Tangent velocity is perpendicular to inward movement, therefore Fnet doesn’t impact velocity.

Satellite Energy Relationships Elliptical Satellite Motion Experiences forces in the same and opposing directions of their motion. When the satellite moves away from the earth (or central object) force is in the opposite direction of the satellites motion, resulting in negative work slowing down the satellite. When the satellite towards the earth force is in the same direction of the satellites motion, resulting in positive work speeding up the satellite. Satellites in elliptical motion experience constant changes in speed.

Work-Energy Relationship Remember TMEi + Wext = TMEf Do satellites experience external work? Yes B. No Mechanical energy is conserved in satellite motion. What would PE & KE look like in a satellite moving with circular motion?

What would PE & KE look like in a satellite moving with elliptical motion? Select the Graph that best represents position A of the elliptical orbit A. B. C. D.

What would PE & KE look like in a satellite moving with elliptical motion? Select the Graph that best represents position D of the elliptical orbit A. B. C. D.