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phys documents (dragged) 11 - T 2 a 3 = 4 2 GM tot 1.3.5...

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4 Physics Formulary by ir. J.C.A. Wevers Kepler’s orbital equations In a force Feld F = kr - 2 , the orbits are conic sections with the origin of the force in one of the foci (Kepler’s 1st law). The equation of the orbit is: r ( θ )= ± 1+ ε cos( θ - θ 0 ) , or: x 2 + y 2 =( ± - ε x ) 2 with ± = L 2 2 M tot ; ε 2 =1+ 2 WL 2 G 2 μ 3 M 2 tot =1 - ± a ; a = ± 1 - ε 2 = k 2 W a is half the length of the long axis of the elliptical orbit in case the orbit is closed. Half the length of the short axis is b = a ± . ε is the excentricity of the orbit. Orbits with an equal ε are of equal shape. Now, 5 types of orbits are possible: 1. k< 0 and ε =0 : a circle. 2. k< 0 and 0 < ε < 1 : an ellipse. 3. k< 0 and ε =1 : a parabole. 4. k< 0 and ε > 1 : a hyperbole, curved towards the centre of force. 5. k> 0 and ε > 1 : a hyperbole, curved away from the centre of force. Other combinations are not possible: the total energy in a repulsive force Feld is always positive so ε > 1 . If the surface between the orbit covered between t 1 and t 2 and the focus C around which the planet moves is A ( t 1 ,t 2 ) , Kepler’s 2nd law is A ( t 1 ,t 2 )= L C 2 m ( t 2 - t 1 ) Kepler’s 3rd law is, with T the period and M tot the total mass of the system:
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Unformatted text preview: T 2 a 3 = 4 2 GM tot 1.3.5 The virial theorem The virial theorem for one particle is: m v r = 0 T =-1 2 F r = 1 2 r dU dr = 1 2 n U if U =-k r n The virial theorem for a collection of particles is: T =-1 2 particles F i r i + pairs F ij r ij These propositions can also be written as: 2 E kin + E pot = 0 . 1.4 Point dynamics in a moving coordinate system 1.4.1 Apparent forces The total force in a moving coordinate system can be found by subtracting the apparent forces from the forces working in the reference frame: F = F- F app . The different apparent forces are given by: 1. Transformation of the origin: F or =-m a a 2. Rotation: F =-m r 3. Coriolis force: F cor =-2 m v 4. Centrifugal force: F cf = m 2 r n =- F cp ; F cp =-mv 2 r e r...
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