Apples and PlanetsApples and Planets

PTYS206-228 Feb 2008

List of SymbolsList of Symbols

• F, force• a, acceleration (not semi-major axis in this

lecture)• v, velocity• M, mass of Sun• m, mass of planet• d, general distance• r,radius of circle, semi-major axis of orbit• R, radius of Earth

Newton’s LawsNewton devised a uniform and systematic method for

describing motion, which we today refer to as theScience of Mechanics. It remains the basic descriptionof motion, requiring correction only at very highvelocities and very small distances.

Newton summarized his theory in 3 laws:

1. An object remains at rest or continues in uniform motionunless acted upon by a force.

2. Force is equal to mass x acceleration (F=ma)

3. For every action there is an equal and opposite reaction.

Newton and Gravity

Link for animation

Cambridge was closed becauseof the Plague. As the story goes,Newton was sitting under theapple tree outside his farmhouse(shown right) and while watchingthe apples fall he realized that theforce that made the apples fallalso made the planets orbit thesun. Using his newly inventedCalculus, Newton was able toshow that Kepler’s 3 laws ofplanetary motion followed directlyfrom this hypothesis.

Falling Apples and Orbiting PlanetsFalling Apples and Orbiting Planets

Splat

What do these have in common?

Newton’s cannonball

From Principia

Apples and PlanetsApples and PlanetsWe will know analyze the motion of terrestrial fallingbodies and orbiting planets in more detail. We willanalyze both phenomenon in the same way and showthat Newton’s theory explains both. The plan is tocombine Newton’s second law with Newton’s law ofgravitation to determine the acceleration.

The interesting thing here is that we are applying lawsdetermined for motion on Earth to the motion ofheavenly bodies. What an audacious idea!

Gravitational Force: UnitsGravitational Force: Units

According to Newton’s 2nd law, Force=mass x acceleration

The units must also match.

Units of mass = kilograms

Units of acceleration = meters/sec2

Unit of force must be kilograms-meters/sec2 = kg m s-2 (shorthand)

We define a new unit to make notation more simple. Let’s call it aNewton. From the definition we can see that

1 Newton = 1 kg m s-2

From now on we measure force in Newtons.

What are the units of G?What are the units of G?Newton’s law of gravitation

F = GMm/d2

Let’s solve for G (multiply by d2, divide by Mm)

G = Fd2/Mm

Examine the units

Fd2/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-1

G has units of m3 s-2 kg-1

Numerically, G = 6.67×10-11 m3 s-2 kg-1

NewtonNewton’’s Law of Gravitys Law of Gravity• All 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 theirseparation.

F = GMm/d2

where m is mass of one object, M is the mass of theother, and d is their separation.

• G is known as the constant of universal gravitation.

NewtonNewton’’s Second Laws Second LawForce = mass x acceleration

F = ma

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

F = m a = G m M / R2 (cancel the m’s)

a = G M / R2

where: G = 6.67x10-11 m3kg-1s-2

M = 5.97x1024 kg On Earth’s 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, isthe distance betweenthe centers of theobjects.

Newton Explains Galileo

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

F = GMm/d2Newton’s law of gravity:

a = GM/R2Cancel m on both sides ofthe equation

ma = GMm/R2Set forces equal

F = GMm/R2

The separation d is thedistance between thefalling body and the centerof the Earth d=R

F = maNewton’s 2nd Law:

Planetary motion is morecomplicated, but governed by

the same laws.First, we need to consider theacceleration of orbiting bodies

Circular AccelerationCircular Acceleration

Acceleration is any change in speed or directionof motion.Circular motion isaccelerated motionbecause direction ischanging. For circularmotion:

a = v2/r

Real Life ExampleReal Life ExampleA Circular Race TrackA Circular Race Track

Acceleration

Orbiting Planets ContinuedOrbiting Planets Continued

So, orbiting planets areaccelerating. Thismust be caused by aforce. Let’s assumethat the force is gravity.We should be able tocalculate the force andacceleration usingNewton’s second lawand Newton’s law ofgravity.

Orbits come in a varietyof shapes (eccentricities).In order to keep the mathsimple, we will considerin this lecture only circularorbits. All of our results also apply to ellipticalorbits, but we will not derive them that way.

Step 1: Calculate the VelocityStep 1: Calculate the VelocityWe take as given that acceleration and velocity in circular motion arerelated by

a = v2/r

According to Newton’s 2nd law

F = ma = mv2/r

According to Newton’s law of gravity

F = GMm/r2

Equating the expressions for force we have

mv2/r = GMm/r2

Solving for v2 gives

v2 = GM/r

Step 2: The Velocity is related to theStep 2: The Velocity is related to thesemi-major axis and periodsemi-major axis and period

The velocity is related to the semi-major axis and theperiod in a simple way: velocity = distance/time

distance = 2πr,

where r=semi-major axis, radius of circle

time = Period, P

v = 2πr/P = distance/time

Step 3: Relate the Period to theStep 3: Relate the Period to theOrbital RadiusOrbital Radius

We have

v2 =GM/r

And

v = 2πr/P

So it follows that

(2πr/P)2 = GM/r

Or

4π2r2/P2 = GM/r

How Does This Relate to How Does This Relate to KeplerKepler’’ssThird Law?Third Law?

We have

4π2r2/P2 = GM/r

Multiply both sides by r

4π2r3/P2 = GM

Multiply both sides by P2

4π2r3 = GM P2

Divide both sides by 4π2

r3 = (GM/4π2) P2

NewtonNewton’’s form of s form of KeplerKepler’’s s Third LawThird Law

We have

r3 = (GM/4π2) P2

Kepler’s 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 Kepler’s 3rd if

(GM/4π2) = 1 AU3/year2

Let’s check

Do Newton and Do Newton and Kepler Kepler Agree?Agree?

We want to know if

(GM/4π2) = 1 AU3/year2

Plug in G = 6.7×10-11 m3 s-2 kg-1, M=2.0×1030 kg

(GM/4π2) = 3.4×1018 m3 s-2

Recall 1 AU = 1.5×1011 m and 1 year = 3.1×107 s

So

1 AU3/year2 = (1.5×1011 m)3/(3.1×107 s)2

1 AU3/year2 = 3.4×1018 m3 s-2 Wow!!!

Using NewtonUsing Newton’’s Form of s Form of KeplerKepler’’ssThird Law: Example 1Third Law: Example 1

Planet Gabrielle orbitsstar Xena. The semimajor axis of Gabrielle'sorbit is 1 AU. Theperiod of its orbit is 6months. What is themass of Xena relative tothe Sun?

Using NewtonUsing Newton’’s Form of s Form of KeplerKepler’’ssThird Law: ExampleThird Law: Example 22

Planet Linus orbits starLucy. The mass of Lucy istwice the mass of the Sun.The semi-major axis ofLinus' orbit is 8 AU. Howlong is 1 year on Linus?

Using NewtonUsing Newton’’s Form of s Form of KeplerKepler’’ssThird Law: ExampleThird Law: Example 33

Jupiter's satellite (moon)Io has an orbital period of1.8 days and a semi-majoraxis of 421,700 km. Whatis the mass of Jupiter?

Using NewtonUsing Newton’’s Form of s Form of KeplerKepler’’ssThird Law: ExampleThird Law: Example 44

The moon has an orbit witha semi-major axis of384,400 km and a period of27.32 days. What is themass of the Earth?