Lecture 22: Gravitational Orbits

Transcription

1 Lecture : Gravitational Orbits Astronomers were observing the motion of planets long before Newton s time Some even developed heliocentric models, in which the planets moved around the sun Analysis of their measurements showed that the orbital shapes were ellipses We can now see that Newton s form of the force of gravity eplains these observations Start from the epression for θ(r): θ ( r) = ± l / r µ ( E U ( r) ) l r dr

2 Plugging in the potential for gravity gives: l / r θ r = dr; k = GMm k l µ E + r r To take this integral, we change variables: 1 1 u = ; du = dr r r l / r du dr = ± l k l µ E + µ ku l u µ E + r r 1 µ k ul l ( ) + 8l µ E 1 = l sin + C Equation E.8c of the tet

3 This can be simplified to: θ sin ( θ C) = 1 ul k = sin + 1 = sin + ul 1 1+ l E + 8l µ E ul 1+ 1 l E µ C C

4 To save some writing, let s define a couple of constants: Latus rectum With this, we have: l El ; = 1+ sin ( θ C) = u 1 We can choose C in any way we find convenient We ll pick the one that makes θ = 0 at the minimum radius (or maimum u): 1+ sin ( θ C) u = du cos( θ C) = = 0 when θ = 0 dθ Eccentricity

5 So, we want C = -π/, which means the orbit is given by: u π 1+ sinθ + = π π 1+ sin θ cos + sin cosθ = = 1+ cosθ Putting this in terms of r gives: = 1+ cosθ r

6 What Do These Orbits Look Like? We can gain some intuition into the shape of the orbits by considering various values of : = 0 : = 1 r r = A circle! y = 1: r = 1 + cosθ + y = 1+ + y

7 1 = y + + This is a parabola y + y = + y + = = y Both of these are special cases Let s consider the energy associated with = 0 or 1: El = 0 = 1 + ; E = l Minumum of V(r) El = 1 = 1 + ; E = 0 Lowest-energy unbound orbit

8 For other values of, we have: r = 1 + cos θ If is greater than one, r is undefined for some values of θ (since a negative r is not physically meaningful) This corresponds to a hyperbolic orbit: Real orbit y Unphysical solution Asymptotes

9 Bounded Gravitational Orbits If < 1, / r = 1+ cosθ is the equation of an ellipse with one focus at the origin This is the general case for a bounded orbit due to gravity We can gain some insight into the properties of the ellipse by writing the orbital equation in Cartesian coordinates: r = 1 + cosθ + y = 1+ + y 1 = y + + y + = + y = +

10 + + y = Doesn t look familiar yet, but let s add a constant to both sides: y = + y + + = + + y = 1 Recall from geometry that the general form for the equation of an ellipse is: ( ) ( y y ) a c c + = 1 b 1 a and b are the semi-major and semi-minor aes

11 Just a reminder about ellipses: Semi-minor ais Semi-major ais Latus rectum Focus

12 So not only have we proven that the orbit is an ellipse, we ve also found the lengths of the aes: a b µ µ = = = = El 1 + l / k l / k k 1 El l / l = = = El µ E E Note that the total energy must be negative since these are bound orbits so the minus signs in the above epressions make sense! Also, we see that the major ais is determined by E alone, while the minor ais depends on both E and l

13 Other Properties of the Orbit We can also readily determine the closest and furthest distances from the origin for any orbit: r = 1 + cosθ dr sinθ = = 0 when θ = 0 or π dθ ( 1+ cosθ ) r min = = a Pericenter r ma = = a 1+ Apocenter 1+ For orbits abound the Earth, the terms apogee and perigee are used For orbits around the sun, it s aphelion and perhelion

14 Orbital Period From conservation of angular momentum that the areal velocity is constant, and given by: da l = dt µ Therefore the time it takes to travel all the way around an elliptical orbit is: A π ab πµ ab τ = = = da l l dt µ In terms of E, l, µ, and k, this is: k l πµ E µ E µ τ = = π k l E 3/

15 We can also write the period as: ab a a τ = = = l l l 1/ πµ πµ π µ l 4π µ 4π µ 3 3/ µ 3 τ = a = a = 4π a l l k If the planet s mass is small compared to the Sun, we have: m 3 4π 3 τ 4π a = a GM sunm GM sun With this, we ve now proven all three of Kepler s Laws of planetary motion: 1. Planets move in elliptical orbits with the sun at one focus. The areal velocity is constant 3. The square of the period is proportional to the cube of the major ais of the orbit a 3/