Download Chapter 3. Sir Isaac Newton

Chapter 3. Sir Isaac Newton
1.
Newton’s Laws of Motion
Kepler was able to describe the motion of the planets, but had no understanding of why
they moved that way. Newton was the first person to suggest an underlying “law” of nature
which could account for the way the planets moved. It was in terms of a force, which has
come to be called “gravity”. It turns out that gravity is one of the four fundamental forces
operating in the Universe today, and an important one for this course, since it controls the
motions of planets, stars and galaxies and a lot about their character. In this Chapter, we
will derive the gravitational force law as Newton did, based on his ideas of force and motion
and the observations of motions of the Moon and planets. It is assumed that all students
in this course have studied Newton’s Laws of mechanics in a physics course, either in high
school or college and are basically familiar with them.
A key test of Newton’s theory of Gravity is that it predicts Kepler’s Laws.....in particular
his Harmonic Law (P2 =a3 ) and provides the basis for determining masses of the planets (and
stars and galaxies) from the motion of their satellites. Because of the important role of this
law in determining masses of all astronomical objects, we emphasize it here.
Newton’s concepts of force and motion are often summarized in terms of three laws,
and can be expressed as follows:
1) A body in motion in a straight line at constant speed continues like that unless acted
on by a force (from outside itself).
2)The amount of acceleration (change in speed or direction) is proportional to the force
applied and inversely proportional to the object’s mass (acceleration=Force/mass)
3. For every action, there is an equal and opposite reaction
A simple equation which expresses these concepts is the famous F = m a where the boldfaced type here reminds the reader that force (F) and acceleration (a) are vector quantities,
while mass (m) is a scalar. Again, if these concepts are not familiar to you you should review
an introductory physics text on force and motion.
A key aspect of these concepts for understanding Newton’s derivation of the law of
gravity is that motion in a circle at constant speed requires a constant force. With a little
calculus (or even without it) it is possible to derive the relationship between the magnitude
of the acceleration that is involved when the circular speed is v and the radius of the circle
–2–
is r. The answer is a = v2 /r. Note that this is written as a scalar relation, since it only
involves the magnitudes of the vectors. Knowing the acceleration of an object moving in
a circle, it was possible for Newton to calculate the amount of force required to keep the
object moving in a circle. The object he had in mind was the Moon, which orbits the Earth
in roughly a circle. This required that there be some force involved and his brilliant concept
was that the force was gravity – the same force that operated within the terrestrial sphere on
falling objects. By this notion of universality of gravity – that it could stretch across the vast
distance to the Moon – into the celestial sphere – he gave birth to modern astronomy. Today
we take it as given that the laws of physics that apply here on Earth also apply across the
cosmos, at least until proven otherwise! Astronomy could not advance as a science without
that presumption and, fortunately, to the extent we can test it (which is considerable) the
principle has held up.
2.
Newton’s Theory of Gravity
Newton knew the distance to the Moon, hence the radius of the circle in which it
orbited. He also knew the sidereal period (P) of the Moon to orbit the Earth, which is 27.3
days. Assuming a circular orbit for the Moon, which is roughly true and appropriate for this
calculation, he could calculate the speed of the Moon in its orbit from the relationship Pv =
2πr, which is simply time x rate = distance traveled. Then, using the relationship between
acceleration, speed and radius of orbit derived above, he could calculate the acceleration of
the Moon in its orbit, which in mks units is about 2.5 × 10−3 m/s/s. This, of course, is
much less than the acceleration of gravity at the surface of the Earth, also known as 1 “gee”,
which is g=9.8 m/s/s. Hence, one might think that the same force that causes things to fall
on Earth could not be the force keeping the Moon in its orbit – one requires a much weaker
force.
Here is where Newton’s genius kicked in and he realized that it could be the same force
if he supposed that gravity actually weakened with distance from its source. The Moon is
much further away from the center of the Earth (the supposed center for the force of gravity
since everything on Earth falls towards its center) than is an object on the surface of the
Earth. If we compare the distance to the Moon (about 250,000 miles) to the radius of the
Earth (which is the distance of an object at the surface from the center), we find an amazing
fact. The ratio of the force of gravity needed to explain the Moon’s acceleration to the force
at the surface of the Earth is precisely the ratio of the distance to the Moon to the radius
of the Earth, quantity squared! In other words, if gravity were inversely proportional to
distance from an object squared, one could account for both the acceleration of objects on
–3–
the Earth and the acceleration of the Moon in its orbit with the same force! In terms of
magnitude F ∝ 1/d2 .
To complete his law, Newton employed the reasoning behind his laws of motion derived
on Earth, including the reciprocity law that for every action there is an equal reaction. That
implied that the Moon must be pulling on the Earth as well and gave rise to the notion
that the force of gravity acted between ANY two masses. What one had to do was to add
up the contributions from all the little bits of matter comprising any object to get the net
resulting force. He had to invent integral calculus to do this kind of calculation, of course.
But, for example, he showed (and you undoubtedly learned in a physics class) that summing
up the gravitational force from all the components of a spherical body leads to the same net
force (outside the object) as if you put all its mass at a single “point mass” located at the
center of mass of the object. And the reciprocity law requires that the force of gravity be
symmetrical. It needs to depend on the masses of both objects (e.g. Moon and Earth) in the
same way. Hence, Newton surmised that gravity was a force acting between any two masses
(call them M and m) separated by a distance (d) and that the magnitude of the force was
proportional to the product of the masses divided by the separation squared: F ∝ mM/d2
while the direction of the force was along the line between them and it was always attractive.
To complete the description as we usually write it today, we can add the “Gravitational
constant”, often called “Big Gee” whose value depends on the unit system used. In the mks
system, where lengths are measured in meters, masses in kilograms and time in seconds, the
value of G is 6.67 × 10−11 . The expression of Newton’s Law of Gravity, then in terms of the
magnitude of the force is:
GMm
F =
.
d2
3.
Testing Newton’s Theory and Deriving Masses in Astronomy
One test of Newton’s theory of gravity was to finally explain Kepler’s mysterious Harmonic Law. The explanatory sheet that is available on the Links page provides the details
of how Newton was able to show, for a circular orbit, that
P2 =
4π 2 3
a
GM
While our version of this law was only derived for a circle (i.e. where a, the semi-major
axis is the same as r, the radius of the circle, or d, the constant distance between the
objects) Newton actually derived this expression for the general case of an elliptical orbit.
He furthermore showed that point masses acting under the influence of his gravitational
–4–
law would orbit in ellipses – explaining Kepler’s first law of planetary motion. He further
showed that the speed in the orbit would change with distance in just the way required by
Kepler’s second law. Hence, Newton’s rather simple formulation of a law of gravity was
highly successful at explaining planetary motion.
Newton’s form of Kepler’s third law is an extremely powerful and useful tool for astronomers since it gives us the ability to determine the masses of any object which has a
satellite orbiting it. In the form given above and derived on the sheet, we assume that there
is a large central mass (M) and a much smaller orbiting mass (m), in which m<<M. (This
assumption is not necessary and a more general form of the law is discussed below). If we
can measure the orbital period of the satellite (e.g. moon of a planet, or planet orbiting a
star) and the radius of its orbit (a) then we can determine M, the mass of the central object.
This is exactly how we do determine the masses of all the planets except Mercury (which
has neither a natural nor artificial satellite). For objects without orbiting satellites we need
a more complex method but one that involves the partial orbit or amount of deflection from
a straight line of a passing object.