Astronomy 1 - Lecture Notes

Wednesday, September 30, 1998

Homework:
- Problem Solving and Scientific Method 
-- Assignment for next week:
	Chapter #2 Problems #7, #8, and #10
	Chapter #3 Problems #1,#2, and #3
Outline:
--------- Previous Lecture Notes
	Newton's Laws
	Circular Motion
	Law of Gravity
	Centre of Mass
		---- Today's Notes
	Momentum and Energy Concepts
	Application to Orbits
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Chapter #3 Electromagnetic Radiation (Start Reading this chapter for Monday)
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September 28 Lecture on Newton's Laws (Law's and Circular Motion)
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Momentum and Momentum Conservation
	Inertia (as put forward in Newton's first Law)
	is reflected in the quantity, Momenum.

	p = momentum = m v

	It is a vector since it has magnitude and direction.
	
	If there is no force on a group of objects, then their total momentum is
conserved. That means that their total momentum is constant.
			
EXAMPLE:      o--->v0         O   m2
	     m1			oO	(collision or other interaction)
					o-->v1	0-->  v2

Total Momentum before collision = Total Momentum after collision
		m1 v0  =  m1 v1 + m2 v2

Angular Momentum
	Rotational Motion also has a momentum which uses 
	angular velocity, w, rather that linear velocity, v.
	There is also an 'rotational inertia' called moment-of-inertia, I.
	Angular momentum is defined as , L = I w

Example: object in a circular orbit
	Angular velocity of a mass, m, in a circular orbit of radius, r, 
		moving with a velocity, v, is w =  v/r
			units [w] = s-1  (radians/s)
	The moment-of-inertia of the object is I = mr2
			units [I] = km m2
	Angular momentum, L = I w  = mr2 v/r = mrv
			units [L] = kg m2/s = Newton meter

Example: rotating sphere of radius, R, and mass, M,  of uniform density.
		L = I w
	In this case the moment-of-inertia depends on shape and density, but in
	this case I = MR2/2
	
	The angular momentum is thus, L = MR2w/2 
		If we were to write the angular velocity in terms of
		the surface velocity, V, at the equator of the object
		then w = V/R
	
Work, Energy, Power and Conservation of Energy
	Work = Force x distance
	Change in Energy of object (or collection of objects)
			 = work done on object 
		Units of Energy - [E] = Newton meter (N m)= Joule (J)
	Power = time rate of change of Energy 
		Units of Power - [Power] = J/s = Watt (W)

	Types of Energy (of interest to astronomy):

		Kinetic (energy of motion)
		Object of mass, m, and velocity, v.

			KE = mv2/2
		
		Gravitational Potential Energy
		Energy of gravitational attraction a mass, m1,
		to another mass of m2 when their centers of mass
		are a distance, d, apart

			PE = - G m1m2/d2

	Conservation of Energy
		
		Total Energy, E = KE + PE		

		If no energy is going into or out a collection of objects
		then the total energy of the objects is constant.
		Gravitational Potential energy and Kinetic energy can change from
		one type to the other.

EXAMPLE: Escape from the Earth's Surface
		Rocket of mass, m,  at velocity, v, on the Earth's surface
			KE = mv2/2
			PE = - G m ME/RE
	If the velocity of the rocket is just sufficient to get it far 
	from the earth and is slowed to nearly zero velocity, we call that 
	the escape velocity.
	(for large distance from Earth, and zero velocity)
			KE = 0 
			PE = 0	
		So the total energy before launch = total energy after 
			
		mv2/2 - G m ME/RE = 0 
		
		This give a value of the escape velocity from the Earth

		vescape2 = 2 G ME/RE
		
	ME = 6x1024 kg and RE = 6370 km
	gives vescape =  11,200 m/s = 11.2 km/s

EXAMPLE: Escape from the Solar System at Earth's Orbit
		R = 1 AU = 150,000,000 km = 1.5x1011 m
		Msun = 2x1030 km
	vescape2 = 42200 m/s = 42.2 km/s

		Note that vorbit2 = G ME/RE
		So   vescape2 = 2vorbit2
--->			vescape = (2)1/2vorbit
		And since for Earth, vorbit = 29.8 km/s
		then vescape = 1.414 x 29.8 km/s = 42.4 km/s

Energy of Object in Orbit:
	Circular Orbit Example: Period = P,  radius = r
			Central Planet mass = M,  object in orbit mass = m
	Kepler's 3rd Law.
		P2 = 4(pi)2r3/(GMm)
	KE = mv2/2
	PE = - GmM/r
	In the circular orbit the Period is related to the velocity by 
		P = 2(pi)r/ v	and v = 2(pi)r/P
	
	E(circular orbit) = KE + PE = (1/2 - 1) GMm/r = - GMm/r  

Note that the total energy of the circular orbit is less than zero. 
This is a 'bound' system ie. the planet is bound to the Sun. 
Elliptical orbits also have negative energy because they are bound.
	
Energy of Orbits and eccentricity:
	Circular and Elliptical Orbits    	E < 0	eccentricity <1
	Parabolic Orbit				E = 0 	eccentricity = 1
	Hyperbolic Orbit (unbound)		E > 0 	eccentricity > 1

Circular and Elliptical --> Planets, Binary Stars, Short Period Comets
Parabolic --> Long Period Comets
Hyperbolic --> Meteoroids in Earth's System, Space Probe encounting a planet