Apples and Planets PTYS Feb 2008

List 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 Laws Newton 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 Planets Splat What do these have in common?

Newton’s cannonball From Principia

Apples and Planets We 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: Units According to Newton’s 2nd law, Force=mass x acceleration The units must also match. Units of mass = kilograms Units of acceleration = meters/sec 2 Unit of force must be kilograms-meters/sec 2 = 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? Newton’s law of gravitation F = GMm/d 2 Let’s solve for G (multiply by d 2, divide by Mm) G = Fd 2 /Mm Examine the units Fd 2 /Mm has units of N m 2 /kg 2 or N m 2 kg -2 Or, expressing Newtons in kg, m, and s (1 N = 1 kg m s -2 ) Fd 2 /Mm has units of N m 2 kg -2 = (kg ms -2 )m 2 kg -2 = m 3 s -2 kg -1 G has units of m 3 s -2 kg -1 Numerically, G = 6.67  m 3 s -2 kg -1

Newton’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/d 2 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 Law Force = mass x acceleration F = ma

Falling Apples: Gravity on Earth F = m a = G m M / R 2 F = m a = G m M / R 2 (cancel the m’s) a = G M / R 2 where: G = 6.67x m 3 kg -1 s -2 M = 5.97x10 24 kg On Earth’s surface: R = 6371 km Thus: a = G M / R 2 = 9.82 m s -2  10 m s -2 a on Earth is sometimes called g. 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 2 nd Law: F = ma Newton’s law of gravity:F = GMm/d 2 The separation d is the distance between the falling body and the center of the Earth d=R F = GMm/R 2 Set forces equal ma = GMm/R 2 Cancel m on both sides of the equation a = GM/R 2

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

Circular Acceleration Acceleration is any change in speed or direction of motion. Circular motion is accelerated motion because direction is changing. For circular motion: a = v 2 /r

Real Life Example A Circular Race Track Acceleration

Orbiting 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.

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 Velocity We take as given that acceleration and velocity in circular motion are related by a = v 2 /r According to Newton’s 2nd law F = ma = mv 2 /r According to Newton’s law of gravity F = GMm/r 2 Equating the expressions for force we have mv 2 /r = GMm/r 2 Solving for v 2 gives v 2 = GM/r

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

Step 3: Relate the Period to the Orbital Radius We have v 2 =GM/r And v = 2  r/P So it follows that (2  r/P) 2 = GM/r Or 4  2 r 2 /P 2 = GM/r

How Does This Relate to Kepler’s Third Law? We have 4  2 r 2 /P 2 = GM/r Multiply both sides by r 4  2 r 3 /P 2 = GM Multiply both sides by P 2 4  2 r 3 = GM P 2 Divide both sides by 4  2 r 3 = (GM/4  2 ) P 2

Newton’s form of Kepler’s Third Law We have r 3 = (GM/4  2 ) P 2 Kepler’s third law was a 3 =P 2, 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/4  2 ) = 1 AU 3 /year 2 Let’s check

Do Newton and Kepler Agree? We want to know if (GM/4  2 ) = 1 AU 3 /year 2 Plug in G = 6.7  m 3 s -2 kg -1, M=2.0  kg (GM/4  2 ) = 3.4  m 3 s -2 Recall 1 AU = 1.5  m and 1 year = 3.1  10 7 s So 1 AU 3 /year 2 = (1.5  m) 3 /(3.1  10 7 s) 2 1 AU 3 /year 2 = 3.4  m 3 s -2 Wow!!!

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