Lecture 15 - Newton and Gravitation (2/16/96)

Seeds: Chapter 4

Isaac Newton
- born Lincolnshire, England (1643-1727), year after Galileo's death
- in 1669 appointed Professor of Mathematics 8 years after entering Cambridge as an undergraduate in 1661
- In 1665 and 1666 Newton worked out most of his ideas on mechanics and gravitation, but did not write it up for publication or presentation.
- Newton concentrated for almost two decades on optics (developing the "Newtonian" telescope), and on the invention and development of calculus
- At the urging of Edmund Halley, the discoverer of Halley's comet, Newton presented his work in 1865 to the Royal Society.
- In 1686, he published the landmark The Mathematical Principles of Natural Philosophy, usually known at the Principia.
- In the Principia, Newton presented his three Laws of Motion.
- Newton eventually acheived fame for his scientific acheivements, as was elected as President of the Royal Society.
Newton's Laws of Motion
1. Every body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. (This is Galileo's Law of Inertia)
2. Any change of motion is proportional to the force that acts, and it is made in the direction of the straight line in which that force is acting.
3. To every action there is always an equal and opposite reaction; or, the mutual actions of two bodies upon each other are always equal and act in opposite directions.
Understanding Newton's Mechanics
- Units - it is standard to use SI (Standard International) units for the three principle quantities of distance, mass, and time: meter, kilogram, second (also abbreviated MKS)
- A vector is a quantity with a magnitude and a direction, like an arrow. We will denote vectors as boldface variables.
- Speed and Velocity - the velocity (v) is a vector expressing the motion of a body by its speed and direction in space. The units of velocity are km/s (plus direction).
- Acceleration (a) - is the vector change in velocity per unit time, or (change in velocity)/(change in time). The units are velocity/time = km/s/s = km/s^2.
- Force (F)- from Second Law F = ma. Units are kg m/s^2 (= 1 Newton)
- Momentum (p) - is the measure of inertia in a body. Force = (change in momentum)/(change in time), so p = mv
- Angular Momentum (L) - is the inertia contained in rotation. For an object of mass m orbiting in a circle of radius r, the angular momentum is L = mvr. Conservation of angular momentum causes the speed to increase as the radius decreases, like spinning skater speeding up as they fold in their outstretched arms.
- Reaction - Newton's 3rd. F_21 = -F_12. But the accelerations are not equal: m_1 a_1 = - m_2 a_2 and the more massive body suffers less acceleration
- The Law of Universal Gravitation - says F = G m_1 m_2 / r^2. G is Newton's gravitational constant (6.67 x 10^-11 N m^2 / kg). The mutual force on the two masses is equal, but the accelerations differ by the ratio of the masses.
- Centripetal Force - we can express the force holding a body in its orbit as the centripetal force F = m v^2 / r, and corresponding acceleration a = v^2 / r
- Kepler's 3rd Revisited: By equating the centripetal and gravitational forces, we can see why P^2 = A^3!
- Center of Mass: because of Newton's third Law of reaction, two bodies to each orbit a point on the line between them, but closer to the more massive body, with m_1 r_1 = m_2 r_2 giving the distances from the center of mass of the two bodies.
- The center of mass of the Earth-Sun orbit lies very close to the Sun: r_s / r_e = m_e / m_s = 3 x 10^-6, or r_s = 450 km
- For two orbiting bodies, the center of mass is called the barycenter. The Sun and Earth mutually revolve around the Sun-Earth barycenter.
- Kepler's 3rd (Newton's Version) - Taking into account of the barycenter, Kepler's 3rd law becomes: (m_1 + m_2)/M_sun (P / 1 yr)^2 = (a / 1 AU)^3. Remember to use masses in units of the Sun's mass, periods in units of years, and distances in units of AU, do not measure masses in kilograms for this equation!

Isaac Newton was born in Lincolnshire, England in the year 1643, a year after Galileo's death. He was appointed as the Lucasian Professor of Mathematics at Cambridge University 8 years after entering Cambridge as an undergraduate and 2 years after beginning graduate studies. Stephen Hawking, the theoretical astrophysicist and cosmologist, currently holds the Lucasian professorship that Newton once held.

As a young man Newton was interested in "Natural Philosophy", as Science was called then, and this continued to hold his interest along with mathematics, optics and a number of other topics. Newton was a slow publisher, and usually needed some instigation to get him to complete his work. For example, in 1666 and 1666, just after completing his bachelor of arts degree and while Cambridge was closed for the plague, Newton worked out most of his ideas on mechanics and gravitation, but did not write it up for publication or presentation. In fact, Newton concentrated for almost two decades on optics, on which he wrote some very important treatises, and mathematics. He designed a type of reflecting telescope, now called the Newtonian telescope. In the area of mathematics, he invented differential and integral Calculus, which he needed to do his gravitational calculations.

In 1684, Edmund Halley, the discoverer of Halley's comet, consulted Newton on the problem of orbital mechanics. He was astounded to find that Newton had worked out the solution 20 years earlier! There were several scientists, like Halley and the physicist Robert Hooke, who were working on the problem at this time. At Halley's urging, in the following year Newton presented his work to the Royal Society. In 1686, he published the landmark The Mathematical Principles of Natural Philosophy, usually known at the Principia. It was in the Principia that Newton presented his three Laws of Motion, placing the mechanics of gravtitation on a physical basis.

Newton's Laws of Motion

In the Principia, Newton stated his three Laws of Motion:

Every body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
Any change of motion is proportional to the force that acts, and it is made in the direction of the straight line in which that force is acting.
To every action there is always an equal and opposite reaction; or, the mutual actions of two bodies upon each other are always equal and act in opposite directions.

Newton's First Law is just a restatement of Galileo's Law of Inertia. Application of a force causes the acceleration needed to change the velocity of a body. The momentum of a body is a measure of the inertia. Newton's Second Law states that the acceleration induced by application of a force is proportional to the force, with the constant of proportionality given by the mass (F=ma). Note that this means that the momentum is given by the mass x velocity. Finally, the Third Law of action and reaction is something entirely new. He noticed that forces between bodies alway come in pairs, and that when you push on something it pushes back with equal force.

Understanding Newton's Mechanics

Newton's work was the first real mathematical formulation of mechanics that allowed true calculation of observable quantities based upon a few relatively simple principles.

To quantify Newton's mechanics, we first need to understand the quantities involved. First, we need to decide what units we use to measure things in. In physics, it is standard to use SI (Standard International) units for the three principle quantities of distance, mass, and time: meter, kilogram, second (also abbreviated MKS). Sometimes, especially in astronomy, CGS (centimeter, gram, second) units are used. English units (foot, pound, second) are not used for serious calculations.

There are also different types of quantities. A scalar quantity is one that behaves just like a pure number (think of a "scale factor"). The mass of a body is a scalar. On the other hand, some quantities have not only a scale or magnitude, but a direction. These are called vectors, and can be thought of as arrows with a length given by thier magnitude, and pointing in some direction. In this section, we will denote vectors as boldface variables.

The velocity (v) of a body is a vector quantity expressing the motion of a body by its speed (v) and direction in space. The units of velocity are km/s (plus a direction).Acceleration (a) is the vector change in velocity per unit time. The units are velocity/time = km/s/s = km/s^2.

The Second Law gives acceleration proportional to force, with the mass as the constant of proportionality. Thus

F = ma

The units are thus kg m/s^2 which is defined a 1 Newton of force.

As mentioned previously, the momentum (p) of a body is the measure of the inertia it posesses. Force can be expressed as a change in momentum per change in time. From definition of acceleration and F = ma, we see that the momentum is just

p = mv.

The angular momentum (L) of a body or system of bodies expresses the inertia contained in rotation. We will not discuss this in detail, but for those interested

L = mv x r where r is the distance vector from an arbitrary point in space (usually the center of rotation) to the body, and the "x" operator denotes the vector cross-product, which means to take distance r times the velocity perpendicular to r. Thus, an object of mass m orbiting in a circle of radius r, the angular momentum is L = mvr. The angular momentum is an interesting quantity because it is conserved in the orbit, and it is what causes the speed to increase as the radius of the orbit decreases - its like a spinning skater speeding up as they fold in outstretched arms.

Newtons 3rd law of action and reaction tells us that if if we denote the force vector on body 2 caused by body 1 as F_12, then the force on 1 by 2 is

F_21 = -F_12

But equal forces do not mean equal accelerations! The two accelerations are related by

m_1 a_1 = - m_2 a_2

so the accelerations are different by the ratio of the masses. The recoil or back reaction of the more massive body is less than that of the lighter one. (Watch a small child run down a hallway and run into a large adult! See who goes flying!)

Newton found that all motion qcould be explained by a force that decreased with the inverse square of the distance between the masses, and proportional to the product of the masses. In other words, the magnitude of the gravitational force F is given by:

F = G m_1 m_2 / r^2

where G is Newton's gravitational constant (6.67 x 10^-11 N m^2 / kg). Note that the strength of the force is the same on the interchange of m_1 and m_2 (the vector direction will reverse though). Thus, the gravitational accelerations are given by

a_1 = G m_2 / r^2

and

a_2 = G m_1 / r^2

Even before Newton's formulation of gravity, Robert Hooke (1635-1703) understood that there must be some central force holding planets in orbit or inertia would send them flying out in straight lines. He postulated a centripetal force and corresponding centripetal acceleration holding it on its orbit. By construction , we can show that the centripetal acceleration must be

a = v^2 / r Thus, the centripetal force on a mass m is: F = m v^2 / r

The centripetal force is a useful concept, and we will use it frequently.

From the centripetal acceleration and gravitational force law, we can see why P^2 = A^3! We ignore the mass m of the Earth, and consider only the mass M of the sun. Then, if we equate

F = G M m / r^2 = m v^2 / r

then we find that

v^2 = G M / r

For a circular orbit of period P, the velocity is given by

v = 2 Pi r/ P

so substitution gives

v^2 = ( 2 Pi r/ P )^2 = G M / r

( G M / 4 Pi^2 ) P^2 = r^3

which is Kepler's third law!

Because of Newton's third Law of reaction, if the Earth is made to orbit because of the gravitational force between the Earth and Sun, then the Sun must also orbit by the equal and opposite force. Because the forces are equal, then the accelerations are inversely proportional to the masses, and we would expect the two bodies to each orbit a point on the line between them, but closer to the more massive body (the Sun).

We will consider circular orbits as usual.

To see this, equate the centripetal forces on two masses m_1 and m_2 orbiting at distances r_1 and r_2 respectively from the orbital center:

m_1 v_1^2 / r_1 = m_2 v_2^2 / r_2

To remain on opposite sides of the orbital center, which they must do to for force symmetry reasons, then the periods of the orbits must be the same:

P = 2 Pi r_1 / v_1 = 2 Pi r_2 / v_2

and thus

r_1 / v_1 = r_2 / v_2.

Combining with the previous equation for v_1^2 / v_2^2, we get

m_1 r_1 = m_2 r_2

which is the equation for the center of mass of two bodies. The relative distances from the center of mass are inversely proportional to the respective masses.

Because the mass of the Sun (2 x 10^30 kg) is so much greater than that of the Earth (6 x 10^24 kg), the center of mass of the Earth-Sun orbit lies very close to the Sun:

r_s / r_e = m_e / m_s = (6 x 10^24 kg)/(3 x 10^30 kg) = 3 x 10^-6

Since the sum of the distances r_s + r_e = 1 AU = 1.5 x 10^8 km, then r_s = 450 km, which is still very close to the center of the Sun! Q: What is the distance of the center-of-mass of the Earth-Moon system from the center of the Earth?

For two orbiting bodies, the center of mass is called the barycenter. The Sun and Earth mutually revolve around the Sun-Earth barycenter.

Because two bodies mutually revolve about their barycenter, Newton knew he had to modify Kepler's third law to take this into account. If we equate the centripetal force about the barycenter with the gravitational force between the bodies, then we find

G m_1 m_2 / (r_1 + r_2)^2 = m_1 v_1^2 / r_1 = m_1 (2 Pi r_1 / P)^2 / r_1

Let the semimajor axis a = r_1 + r_2. The equation for the center of mass m_1 r_1 = m_2 r_2 tells us

a = r_1 + r_1 = ( 1 + m_1/m_2 ) r_1

r_1 = a m_2 / (m_1 + m_2)

and upon substitution, cancellation, and rearrangement

( G / 4 Pi^2 ) (m_1 + m_2) P^2 = a^3

which is Kepler's 3rd law but now with both masses included.

We can get an easier equation by using the Earth's orbit to scale things:

(m_1 + m_2)/M_sun (P / 1 yr)^2 = (a / 1 AU)^3

Remember to use masses in units of the Sun's mass, periods in units of years, and distances in units of AU, and you can simply write:

(m_1 + m_2) P^2 = a^3

Do not measure masses in kilograms for this equation!

We will use this for finding masses of binary stars, as well as masses of galaxies. Remember this formula!

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Steven T. Myers - Last revised 14May96