L3GH09_Orbits

L3GH09_Orbits - Orbits etc. Lecture topics: orbital energy,...

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January 13, 2009 orbits Orbits etc. •Lecture topics: orbital energy, Vis Viva equation, tides, orbital resonance etc. •Text reading: Chapter 3
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January 13, 2009 orbits 1. Gravitating objects orbit one another in an ellipse of eccentricity e and semimajor axis length a. 2. Conservation of angular momentum means that objects move faster when they are closest to one focus (e.g. the Sun) 3. The orbital period P increases with the size of the semimajor axis, a (a=a M +a m ). Review: Generalized Kepler’s Laws ( ) θ cos 1 1 2 e e a r + = π mrv P e a m L = = 2 2 1 2 ) ( 4 3 2 2 m M G a P + = rP e a v ) 1 ( 2 2 2 = Note: angular momentum equation can be rearranged to give an expression for velocity in terms of r, a, e, P
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January 13, 2009 orbits Circular Velocity •A body in circular motion will have a constant velocity determined by the force it must “balance” to stay in orbit. •By equating the circular acceleration and the acceleration of a mass due to gravity: r GM v circ = where M is the mass of the central body and r is the separation between the orbiting body and the central mass. •T h i s r e l a t i o n i s c o n v e n i e n t a n d useful for our purposes because most planet and satellite orbits are close to circular. 2 2 2 r GMm r m r a circ = = υυ
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January 13, 2009 orbits Escape velocity •Escape velocity is the velocity a mass must have to escape the gravitational pull of the mass to which it is “attracted”. •We define a mass as being able to escape if it can move to an infinite distance just when its velocity reaches zero. At this point its net energy is zero (i.e. PE grav +KE = 0) and so we have: 2 2 1 esc mv r GMm = r GM v esc 2 = Note: circ esc v v 2 =
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January 13, 2009 orbits Orbital Energy • In the solar system we observe bodies of all orbital types: ¾ planets etc. = elliptical, some nearly circular; ¾ comets = elliptical, parabolic, hyperbolic; ¾ some like comets or miscellaneous debris have low energy orbits and we see them plunging into the Sun or other bodies orbit type v E tot e circular v=v circ E<0 e=0 elliptical v circ <<v esc E<0 0<e<1 parabolic v=v esc E=0 e=1 hyperbolic v>v esc E>0 e>1 r GMm v m a GMm E = = 2 2 2
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January 13, 2009 orbits Vis-Viva Equation • to derive the Vis Viva equation we begin with: • then substitute relations for perihelion and aphelion distance (from definition of ellipse) and velocity (from conservation of orbital angular momentum) •and also show that: potential kinetic orbit E E E + = = a r GM r v 2 1 1 2 ) ( 2 • we can find a more general formula which relates velocity, distance and semimajor axis, but does not involve eccentricity •t h i s i s t h e Vis Vivaequation a GMm E orbit 2 =
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This note was uploaded on 04/15/2009 for the course PHYS 275 taught by Professor Harris during the Winter '09 term at Waterloo.

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L3GH09_Orbits - Orbits etc. Lecture topics: orbital energy,...

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