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3 Celestial Mechanics 3 Celestial Mechanics Assign: Read Chapter 2 of Carrol and Ostlie (2006) 3.1 Kepler’s Laws OJTA: 3. The Copernican Revolution/Kepler – (4) Kepler’s First Law b r' r εa θ a r C r0 D 2a b2 D a2.1 2/ (1) a.1 2/ r D Area D ab (2) 1 C cos Example: For Mars a D 1:5237 AU D 0:0934 At perihelion D 0ı and a.1 2/ .1:5237 AU/.1 0:09342/ r D D D 1:3814 AU 1 C cos 1 C 0:0934 cos.0ı/ At aphelion D 180ı and .1:5237 AU/.1 0:09342/ r D D 1:6660 AU 1 C 0:0934 cos.180ı/ 5 Difference = 19% – (5) Kepler’s Second Law – (6) Kepler’s Third Law P 2.yr/ D a3.AU/ Example: For Venus, a D 0:7233 AU. Then the period is P D a3=2 D .0:7233/3=2 D 0:6151 yr D 224:7 d 3.2 Galileo OJTA: 4. The Modern Synthesis/Galileo – (3) New Telescopic Observations – (4) Inertia 3.3 Mathematical Interlude: Vectors Vectors are objects that have a magnitude (length) and a direction. Direction V h Lengt 6 Therefore, they require more than one number to specify them. In contrast a scalar is specified by only one number. One way to specifiy a vector is to give its components: y Components θ in -1 θ = tan (Vy /V x ) |s V V 2 2 2 |V| = V x + V y + V z = | y θ V x Vx = |V|cos θ The direction of a vector can be specified by orientation angles (one in two dimensions and two in three dimensions). Its length is given by the Pythagorean theorem in terms of its components. For a 3-dimensional vector, q 2 2 2 jV jD Vx C Vy C Vx : The components are related to the angles by basic trigonometry. In the 2-D example above, Â Ã Vy D tan1 Vx Some important vector quantities include the position, velocity, momentum, and accelera- tion of objects. Let us illustrate in 2-D for simplicity. Position: The position of a point can be specified by a vector r 7 y Position θ -1 in θ = tan (r /r ) r y x |s r 2 2 |r| = r x + r y = | y θ r x rx = | r |cos θ where q 2 2 rx D r cos ry D r sin rjrjD rx C ry : Velocity: The velocity vector can be defined in terms of the time derivative of the position vector, dr r r r v ' D 1 2 dt t t1 t2 y Velocity θ -1 in θ = tan (v /v ) v y x |s v 2 2 |v| = v x + v y = | y θ v x vx = |v|cos θ The standard units of velocity in the SI system are m s1. Momentum: The momentum vector is defined to be the mass m times the velocity vector, dr p mv D m : dt Its standard units are kg m s1. 8 Acceleration: The acceleration vector a is defined to be the time derivative of the velocity vector, dv d 2r a D : dt dt2 y Acceleration θ -1 in θ = tan (a /a ) a y x |s a 2 2 |a| = a x + a y = | y θ a x ax = |a|cos θ The standard units of acceleration in the SI system are m s2. Example: Uniform circular motion Consider the case of uniform circular motion v The magnitude of the velocity is constant, but the direction is changing, so there is an acceleration Is this accelerated motion? Yes, because there is a continuous change in the direction of the velocity, even though its magnitude is constant. 9 Addition of vectors: Vectors can be added graphically by a head–to–tail rule: A B C = A + B C Unit vectors: It is convenient to define unit vectors that point along the coordinate system axis and have unit length. For cartesian coordinates, z z x y y x In terms of components, we can write for a vector A A D AxxO C AyyO C AzzO: Scalar product of vectors: There are two kinds of vector products of interest to us. The scalar product of two vectors A and B is a number (a scalar), defined by AB jAjjBj cos D AB cos ; (3) where is the angle between the two vectors. The order does not matter in the scalar product: AB D BA. That is, the scalar product commutes. The scalar product is often called the dot product. Note some special cases of the scalar product: 10 1. If A and B point in the same direction, cos D 1 and AB D AB. 2. If A and B point in the opposite direction, cos D1 and AB DAB. 3. If A and B are perpendicular, cos D 0 and AB D 0. Example: Scalar Product of Two Vectors Consider the following two vectors A 5 | = |A θ = 30o B |B| = 8 Their scalar product is AB D BA D AB cos D .5/.8/ cos.30ı/ D 34:6: Cross product of vectors: The second kind of vector product is called the cross product or the vector product. It differs from the scalar product in that it produces a new vector, not a scalar like the scalar product. The cross product is defined by AB D .AB sin /IO; (4) where is again the angle between the vectors and IO is a unit vector that is perpendicular to the plane containing the vectors A and B, with its direction (up or down) given by the right-hand rule: A B B Rotate A into B with the right hand. The thumb points in the direction (up or down) of A the new vector A x B. Right hand 11 Unlike for the scalar product, the order in the cross product matters: A B DB A (easily seen from the right-hand rule: try to rotate B into A in the preceding diagram and note the direction that your thumb points). Note some special cases of the vector product: 1. If A and B point in the same or opposite directions, sin D 0 and jABjD0. 2. If A and B are perpendicular, sin D 1 and jABjDAB. Example: Angular Momentum Angular momentum L is a vector that measures the tendency of a body in angular motion to remain in that motion. It is conserved (the reason an ice skater spins faster if the arms are drawn in is conservation of angular momentum). Angular momentum with respect to some coordinate system is the cross product of the position vector with the momentum vector: L r p D r .mv/: (5) For example, consider the angular momentum associated with uniform circular motion p L = r p 90o r p r Right hand The magnitude of the angular momentum is L D rp sin 90ı D rp 12 and the direction is out of the paper, as illustrated by the right-hand rule in the figure. The SI units for angular momentum are kg m2s1. 3.4 Conservation Laws The case of angular momentum just considered is an example of a quantity that is conserved by all interactions in Newtonian physics. Conservation of angular momentum is an example of a conservation law. In Newtonian physics, we believe that Energy Momentum Mass Angular momentum are always conserved in isolated systems. Conservation laws are very important. Since they must be obeyed, no matter what, they often can be used to simplify the solution of problems. We will see specific examples shortly. 3.5 Newton’s Three Laws of Motion Newton’s 1st Law: Objects in a state of uniform motion remain in that state of motion unless an external force acts on them (The law of inertia). Newton’s 2nd Law: If an external force F acts on an object, the acceleration a experienced by the object is given by the force divided by the mass, so that: F D ma This permits the change in velocity (acceleration) to be computed. 13 The force F in this case is the vector sum of all forces acting on the object: Xn F D F 1 C F 2 C :::C F n D F i : iD1 Assuming constant mass, Newton’s 2nd law may be written in the equivalent forms dv d.mv/ dp F D ma D m D D : (6) dt dt dt These vector equations are equivalent to three simultaneous equations in the components (in 3-D). For example 8 < Fx D max F D ma ! F D ma : y y (7) Fz D maz: The standard unit of force in the SI system is the Newton: 1 Newton 1 N D 1 kg m s2: Newton’s 3rd Law: For every reaction, there is an equal and opposite reaction. Notice that in Newton’s 3rd law the action and reaction are forces that always act on differ- ent objects (never on the same object): 1 2 F12 F21 If object 1 exerts a force F 21 on object 2, then object 2 exerts a force F 12 DF21 on object 1. These forces are equal in magnitude but opposite in direction. 3.6 Newton’s Universal Law of Gravitation Newton reasoned that gravity was a force, obeying his three laws of motion. 14 3.6.1 The gravitational force From the observed properties of gravity, Newton deduced his Universal Law of Gravita- tion: Universal Law of Gravitation Every mass in the Universe exerts a force on every other mass that is at- tractive and directed along the line of centers for the two masses, with the magnitude of the force given by m1m2 F jF jDG ; (8) r2 F -F m1m2 m1 m2 F = G r2 r where the universal gravitational constant G is measured to be G D 6:673 1011 Nm2kg2 D 6:673 1011 kg1m3s2.
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