Apples and PlanetsApples and Planets

PTYS206-2

28 Feb 2008

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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 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:

1. An object remains at rest or continues in uniform motion unless 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 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 Kepler’s 3 laws of planetary motion followed directly from this hypothesis.

Falling Apples and Orbiting PlanetsFalling Apples and Orbiting Planets

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Splat

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What do these have in common?

Newton’s cannonball

From Principia

Apples and PlanetsApples 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 Newton’s theory explains both. The plan is to combine Newton’s second law with Newton’s 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: 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 a Newton. 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 = 6.6710-11 m3 s-2 kg-1

Newton’s Law of GravityNewton’s 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 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.

Newton’s Second LawNewton’s 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.

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The separation, d, is the distance between the centers of the objects.

Newton Explains Galileo

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

Newton’s 2nd Law: F = ma

Newton’s law of gravity: F = GMm/d2

The separation d is the distance between the falling body and the center of the Earth d=R

F = GMm/R2

Set forces equal ma = GMm/R2

Cancel m on both sides of the equation

a = GM/R2

Planetary motion is more complicated, but governed by

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

Circular AccelerationCircular Acceleration

Acceleration 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 ExampleReal Life ExampleA Circular Race TrackA Circular Race Track

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Acceleration

Orbiting Planets ContinuedOrbiting Planets Continued

So, orbiting planets are accelerating. This must be caused by a force. Let’s assume that the force is gravity. We should be able to calculate the force and acceleration using Newton’s second law and Newton’s law of gravity.

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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 are related 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 the Step 2: The Velocity is related to the semi-major axis and periodsemi-major axis and period

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

distance = 2r,

where r=semi-major axis, radius of circle

time = Period, P

v = 2r/P = distance/time

Step 3: Relate the Period to the Step 3: Relate the Period to the

Orbital RadiusOrbital Radius We have

v2 =GM/r

And

v = 2r/P

So it follows that

(2r/P)2 = GM/r

Or

42r2/P2 = GM/r

How Does This Relate to Kepler’s How Does This Relate to Kepler’s Third Law?Third Law?

We have

42r2/P2 = GM/r

Multiply both sides by r

42r3/P2 = GM

Multiply both sides by P2

42r3 = GM P2

Divide both sides by 42

r3 = (GM/42) P2

Newton’s form of Kepler’s Third LawNewton’s form of Kepler’s Third Law

We have

r3 = (GM/42) 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/42) = 1 AU3/year2

Let’s check

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

We want to know if

(GM/42) = 1 AU3/year2

Plug 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

So

1 AU3/year2 = (1.51011 m)3/(3.1107 s)2

1 AU3/year2 = 3.41018 m3 s-2 Wow!!!

Using Newton’s Form of Kepler’s Using Newton’s Form of Kepler’s Third Law: Example 1Third Law: Example 1

Planet 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 Newton’s Form of Kepler’s Using Newton’s Form of Kepler’s Third Law: Example 2Third Law: Example 2

Planet 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 Newton’s Form of Kepler’s Using Newton’s Form of Kepler’s Third Law: Example 3Third Law: Example 3

Jupiter'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 Newton’s Form of Kepler’s Using Newton’s Form of Kepler’s Third Law: Example 4Third Law: Example 4

The 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?