Apples and PlanetsPTYS206-228 Feb 2008

List of SymbolsF, forcea, acceleration (not semi-major axis in this lecture)v, velocityM, mass of Sunm, mass of planetd, general distancer,radius of circle, semi-major axis of orbitR, radius of Earth

Newtons LawsNewton devised a uniform and systematic method for describing motion, which we today refer to as the Science of Mechanics. It remains the basic description of motion, requiring correction only at very high velocities and very small distances.Newton summarized his theory in 3 laws:An object remains at rest or continues in uniform motion unless acted upon by a force.Force is equal to mass x acceleration (F=ma)For every action there is an equal and opposite reaction.

Newton and GravityLink for animationCambridge was closed because of the Plague. As the story goes, Newton was sitting under the apple tree outside his farmhouse (shown right) and while watching the apples fall he realized that the force that made the apples fall also made the planets orbit the sun. Using his newly invented Calculus, Newton was able to show that Keplers 3 laws of planetary motion followed directly from this hypothesis.

Falling Apples and Orbiting PlanetsSplatWhat do these have in common?

Newtons cannonball From Principia

Apples and PlanetsWe will know analyze the motion of terrestrial falling bodies and orbiting planets in more detail. We will analyze both phenomenon in the same way and show that Newtons theory explains both. The plan is to combine Newtons second law with Newtons law of gravitation to determine the acceleration.

The interesting thing here is that we are applying laws determined for motion on Earth to the motion of heavenly bodies. What an audacious idea!

Gravitational Force: UnitsAccording to Newtons 2nd law, Force=mass x accelerationThe units must also match. Units of mass = kilogramsUnits of acceleration = meters/sec2Unit of force must be kilograms-meters/sec2 = kg m s-2 (shorthand)We define a new unit to make notation more simple. Lets call it a Newton. From the definition we can see that1 Newton = 1 kg m s-2 From now on we measure force in Newtons.

What are the units of G?Newtons law of gravitationF = GMm/d2Lets solve for G (multiply by d2, divide by Mm)G = Fd2/MmExamine the unitsFd2/Mm has units of N m2/kg2 or N m2 kg-2 Or, expressing Newtons in kg, m, and s (1 N = 1 kg m s-2)Fd2/Mm has units of N m2 kg-2 = (kg ms-2)m2 kg-2= m3 s-2 kg-1G has units of m3 s-2 kg-1Numerically, G = 6.6710-11 m3 s-2 kg-1

Newtons Law of GravityAll bodies exert a gravitational force on each other.The force is proportional to the product of their masses and inversely proportional to the square of their separation.F = GMm/d2 where m is mass of one object, M is the mass of the other, and d is their separation.G is known as the constant of universal gravitation.Newtons Second LawForce = mass x accelerationF = ma

Falling Apples: Gravity on Earth F = m a = G m M / R2

F = m a = G m M / R2 (cancel the ms)

a = G M / R2 where: G = 6.67x10-11 m3kg-1s-2 M = 5.97x1024 kg On Earths surface: R = 6371 kmThus: a = G M / R2 = 9.82 m s-210 m s-2 a on Earth is sometimes called g.The separation, d, is the distance between the centers of the objects.

Newton Explains GalileoThe acceleration does not depend on m!Bodies fall at the same rate regardless of mass.

Planetary motion is more complicated, but governed by the same laws.First, we need to consider the acceleration of orbiting bodies

Circular AccelerationAcceleration is any change in speed or direction of motion. Circular motion is accelerated motion because direction is changing. For circular motion: a = v2/r

Real Life ExampleA Circular Race TrackAcceleration

Orbiting Planets ContinuedSo, orbiting planets are accelerating. This must be caused by a force. Lets assume that the force is gravity. We should be able to calculate the force and acceleration using Newtons second law and Newtons law of gravity.

Orbits come in a variety of shapes (eccentricities). In order to keep the math simple, we will consider in this lecture only circular orbits. All of our results also apply to elliptical orbits, but we will not derive them that way.

Step 1: Calculate the VelocityWe take as given that acceleration and velocity in circular motion are related bya = v2/rAccording to Newtons 2nd lawF = ma = mv2/rAccording to Newtons law of gravityF = GMm/r2Equating the expressions for force we havemv2/r = GMm/r2Solving for v2 givesv2 = GM/r

Step 2: The Velocity is related to the semi-major axis and periodThe velocity is related to the semi-major axis and the period in a simple way: velocity = distance/timedistance = 2r, where r=semi-major axis, radius of circletime = Period, Pv = 2r/P = distance/time

Step 3: Relate the Period to the Orbital Radius We have v2 =GM/rAndv = 2r/P So it follows that(2r/P)2 = GM/rOr42r2/P2 = GM/r

How Does This Relate to Keplers Third Law?We have42r2/P2 = GM/rMultiply both sides by r42r3/P2 = GMMultiply both sides by P242r3 = GM P2 Divide both sides by 42r3 = (GM/42) P2

Newtons form of Keplers Third LawWe haver3 = (GM/42) P2 Keplers third law was a3=P2, where a=semi-major axis (not acceleration). Since today we are using r=semi-major axis, this equation is the same as Keplers 3rd if (GM/42) = 1 AU3/year2Lets check

Do Newton and Kepler Agree?We want to know if (GM/42) = 1 AU3/year2Plug in G = 6.710-11 m3 s-2 kg-1, M=2.01030 kg (GM/42) = 3.41018 m3 s-2 Recall 1 AU = 1.51011 m and 1 year = 3.1107 s So1 AU3/year2 = (1.51011 m)3/(3.1107 s)21 AU3/year2 = 3.41018 m3 s-2 Wow!!!

Using Newtons Form of Keplers Third Law: Example 1Planet Gabrielle orbits star Xena. The semi major axis of Gabrielle's orbit is 1 AU. The period of its orbit is 6 months. What is the mass of Xena relative to the Sun?

Using Newtons Form of Keplers Third Law: Example 2Planet Linus orbits star Lucy. The mass of Lucy is twice the mass of the Sun. The semi-major axis of Linus' orbit is 8 AU. How long is 1 year on Linus?

Using Newtons Form of Keplers Third Law: Example 3Jupiter's satellite (moon) Io has an orbital period of 1.8 days and a semi-major axis of 421,700 km. What is the mass of Jupiter?

Using Newtons Form of Keplers Third Law: Example 4The moon has an orbit with a semi-major axis of 384,400 km and a period of 27.32 days. What is the mass of the Earth?