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Orbital Mechanics:

Orbital Mechanics:. Kepler’s Third Law:. P 2 = a 3 x M -1. Where M = Mass of the System in SOLAR MASSES. Where a = Semi-Major Axis of Orbit in AU. Where P = Orbital Period in YEARS.

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Orbital Mechanics:

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  1. Orbital Mechanics:

  2. Kepler’s Third Law: P2= a3 x M-1 • Where M = Mass of the System in SOLAR MASSES • Where a = Semi-Major Axis of Orbit in AU • Where P = Orbital Period in YEARS • This law is General, which means that it works for ANY orbiting body (that is much smaller than the object it orbits).

  3. What’s a Moon Good For? • This new equation means that we can find out exactly how big all the planets are relative to the Sun! • How? Using their moons! • Galileo had discovered that some planets had moons that appeared to orbit them, just like our moon. • Thus, we could use the period of these orbits to get the masses of the planets! C = angle on sky B = planet-moon dist. (AU) B = A x TAN(C) A = sun-Earth dist. (AU)

  4. Jupiter’s Mass from its Moon Ganymede: • Let’s use Jupiter’s Moon Ganymede as an example. Ganymede Jupiter C • To get the semi-major axis we look to see how far away the moon gets from Jupiter! On the sky, that distance is measured as an angle. • Using a telescope we can measure the angle to be C=0.1º. B = A x TAN(C) B = 4.2 x TAN(0.1) B = 0.007 AU C = angle on sky B = planet-moon dist. (AU) A = Jupiter-Earth dist. = 4.2 AU

  5. Jupiter’s Mass from its Moon Ganymede: • Let’s use Jupiter’s Moon Ganymede as an example. Ganymede Jupiter C • Now, to get the Mass of Jupiter we use Kepler’s Law. P2= a3 x M-1 M = a3/P2 • We just learned that a = 0.007 AU. • Galileo measured the time for Ganymede’s orbit to be P = 7.5 days or P = 0.02 years. M = (0.007)3/(0.02)2 = 9x10-4Solar Masses • Therefore, Jupiter is about 0.09% as big as the Sun!

  6. What’s a Moon Good For? • This new equation means that we can find out exactly how big all the planets are relative to the Sun! • Notice that we used a different moon of Jupiter in this table, but we got the same answer!

  7. What is an Astronomical Unit? • Kepler’s laws are all you need to determine the scales and masses that describe orbits. The one thing that is missing what an AU is! • One way is to put a satellite in a known orbit and measure its period. • The other is to use radar and to measure the time it takes to reflect back. 2xD = c/time D • Where c is the speed of light c = 3x105 km/sec • One Astronomical Unit = 1.5x108 km

  8. a Kepler’s Third Law (Revised Edition) • With a direct measurement of what an AU is, we can develop a more comprehensive form for Kepler’s Law (with gravity) • Where G=Gravitational Constant • G=6.7x10-11 (M in kg, P in sec, and a in m). • In the above equation M = M1+M2, the total mass of the orbiting system • Most of the time the mass of the orbiting body is small enough that we can ignore it.

  9. 86400 seconds = 24 hours! • We call that orbit geosynchronous. An Example: • Suppose we have a satellite orbiting Earth with a = 4.2x107 m. G=6.67x10-11 a • Plug into the equation and we get that P = 86400 seconds.

  10. Orbital Velocity: • Note that the speed of an object in its orbit depends on its distance. • The Speed of a satellite is given simply by…. Vorbit = 2a/P Vorbit a • For the EARTH • P = 3.1 x 107 sec • a = 1.5 x 108 km Earth Sun VEarth = 29.8 km/sec

  11. Orbital Velocity: • Note that the speed of an object in its orbit depends on its distance. • The Speed of a satellite is given simply by…. Vorbit = 2a/P Vorbit a • For MARS: • P = 5.8 x 107 sec • a = 2.3 x 108 km Mars Sun VMars = 24.2 km/sec

  12. Hitting the Road: • So what does any of this mean to space travel? • Kepler’s laws are the roadmap for roaming the solar system! • Suppose we want to go to Mars. Mars Earth Sun • The orbit at right will always go from the Earth’s orbit to Mars. • This is called a transfer orbit.

  13. Setting things up: • Some rules to keep in mind. • For two objects in an orbit, the one going faster will get further away. 1.5 AU 1.0 AU • So…a spaceship in our transfer orbit must be going faster than the Earth at 1 AU and slower than Mars at 1.5! Mars Earth Sun How much faster and slower????

  14. V1 = 30.7 km/sec V2 = 21.8 km/sec Setting things up: • We can calculate this. 1.0 AU 1.5 AU • Recall that the Earth and Mars are moving at 29.9 km/sec and 24.2 km/sec. V2 Mars Earth Sun V1 • Our satellite must leave going 0.8 km/sec fasterthan Earthand arrive at Mars going 2.4 km/sec slower than Mars.

  15. Escape Velocity: • It’s not enough to give a spacecraft 0.8 km seconds, we must also get it off of the Earth. • To leave a planet requires overcoming gravity, which acts like an elastic band attached to our spacecraft. We have to snap it. • To escape, our spacecraft must have energy of motion equal (or greater than) the gravitational energy holding it down. spacecraft mass 1/2 MV2 = gMr starting altitude acceleration of gravity V2 = 2gR • This reduces to which is 11.8 km/sec for Earth and 5.2 km/sec for Mars.

  16. Escape Velocity: • So to go to Mars we have to give our spacecraft an initial boost of 12.6 km/sec! • Recall we will be moving 2.4 km/sec slower than Mars when we will arrive. 1.0 AU 1.5 AU V2 • Is this ok? Will Mars Capture us? Mars Earth Sun V1 • If we are close enough to Mars when we arrive, then yes, because 2.4 < 5.2!

  17. Mission Planning: • Does the configuration here work for the launch of our spacecraft? • Remember that it will take some time to get there. 1.0 AU 1.5 AU • How long? V2 Mars Earth Sun P2= a3 V1 • In this case 2a = 2.5 AU • Therefore P = 1.4 years • We arrive 8.5 months after launch!

  18. Mission Planning: • We have to plan our flight so that Mars is there when we get there! Earth (arrive) • The transfer orbit is the lowest energy orbit for getting from place to place. 1.0 AU 1.5 AU Earth (Launch) Mars (arrive) Sun Mars (launch) • What if we want to arrive sooner? What if we want to send people and bring them home?

  19. Mars Option 1: • Slow trip to Mars….. Mars (arrive) • Instead of 0.8 km/sec, we give our spacecraft 7 km/sec. Earth (arrive) • We can arrive within three months now. Sun • But how long do we have to stay? Mars (launch) Earth (Launch)

  20. Mars Option 1: • We have to wait until the planets are in the right alignment. Mars (arrive) • Then we launch to a net slower velocity than Mars, and ‘fall’ back toward Earth. Earth (arrive) Sun Earth (Departure) • The total trip takes 2.6 years! Earth (homecoming) • Can we do it faster? Mars (Departure) Mars (homecoming)

  21. Mars Option 1 (return): • Slow trip to Mars….. Mars (arrive) • Instead of 0.8 km/sec, we give our spacecraft 7 km/sec. Earth (arrive) • We can arrive within three months now. Sun • But how long do we have to stay? Mars (launch) Earth (Launch)

  22. Mars Option 2: • We have to wait until the planets are in the right alignment. Mars (arrive) • Then we launch to a net slower velocity than Mars, and ‘fall’ back toward Earth. Earth (arrive) Sun Earth (Departure) • The total trip takes 2.6 years! Earth (homecoming) • Can we do it faster? Mars (Departure) Mars (homecoming)

  23. Pluto Neptune Uranus Saturn Jupiter Mars Earth Venus Mercury

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