These two years of uninterrupted work were the most productive of his life. He conceived the first two laws of motion, the formula for centripetal acceleration, and, in 1666, when he was 24, the law of universal gravitation. This must have been the time of the famous incident of the falling apple. One of the sources of this story is a biography of Newton written by his friend William Stukeley in 1752; Stukeley records that on one occasion when he was having tea with Newton in a garden under some apple trees, Newton told him that "he was just in the same situation, as when formerly, the notion of gravitation came to mind. It was occasion'd by the fall of an apple, as he sat in a contemplative mood..." This does not mean that he was the first to have the conception of gravity; for this concept had been developing for nearly a century. But it was probably when he saw the connection between the motion of the moon (and the planets) and the falling of the apple; that he saw that gravity was the cause of both celestial and natural terrestrial motion. Newton dates his discovery to 1666;
"In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing the dignity of any Binomial into such a Series. The same year, in May, I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the theory of Colours, and in May following I had entrance into the inverse method of Fluxions. In the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere... from Kepler's Rule of the periodical times of the planets being in a sesquialterate proportion of their distances from of their orbs, I deduced that the forces which keep the planets in their orbs must be reciprocally as the square of their distances from the centers about which they revolve and thereby compared the forces requisite to keep the Moon in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since."
He did not publish any of these ideas for many years, because he could not prove mathematically that the attractive force between two spherical masses could be considered as acting at the geometrical centers of the sphere. He supplied this proof later to his friend, Edmond Halley (1656-1742 A.D.), for whom the comet was named.
In 1667 after the plague, Newton returned to Cambridge, where he remained until 1692. After he successively became a Minor Fellow, a Major Fellow, a Senior Fellow, and following Barrow's resignation in favor of Newton, he was appointed Lucasian Professor of Mathematics at the age of 27. Newton did much of his work on optics then, and at the request of the Royal Society, he sent them in December, 1671, a reflecting telescope that he had designed and constructed; two months later he was elected a Fellow of the Society. He published in 1672 in the Transactions of the Royal Society a paper entitled, "Theory about Light and Color," in which he described and explained the operation of his telescope; but it involved him in such a bitter controversy about the nature of light with Robert Hooke (1635-1703 A.D.) and others, that he resolved never to publish again.
In August of 1684, Halley traveled to Cambridge, to ask Newton's advice on the problem of the nature of the orbit of a planet; that is, what is the shape of its orbit, if its acceleration was always toward the sun and inversely proportional to the square of its distance from the sun. Newton said to his friend's surprise that it was an ellipse. Halley asked him how did he know that. Newton replied that he had calculated it. Halley asked him for the calculation but he could not find it. He had solved the problem of the orbit of a planet around the sun many years before but had misplaced the solution and could not find it at the moment, but he said that he would send it to him later. Three months later Halley received a short treatise with the title On the Motion of Bodies in Orbit, known as De motu, containing nine theorems and seven problems. Halley was so impressed with its importance that he went to Cambridge again to urged Newton to publish it. After some seventeen or eighteen months of labor during 1685 and 1686, the result was the Principia Mathematica Philosophiae Naturalis (Mathematical Principles of Natural Philosophy) or, as it is usually known, the Principia, which was published in 1687, establishing Newton's reputation as one of the all time greatest scientists.
In 1689 and again in 1701 Newton was elected a Member of Parliament by the University. In 1696 he was made Warden of the Mint and he helped materially in re-establishing the currency of the country. In 1699 became Master of the Mint, which position he held until his death. He moved to London, and his home was cared by his niece Catherine Barton. In 1703 he became president of the Royal Society, a position which he held until his death in 1727. In 1704 with the controversy about the nature of light lessened by the death of Hooke, he published his Opticks. This work was written in English instead of the usual Latin, in which the Principia was written. By this time Newton had been become renowned throughout the scientific world and was knighted by Queen Anne in 1705 for his contributions to science. In the last two decades Newton worked on theological subjects and biblical prophecies. From an early period of his life Newton had been much interested in theological studies. Earlier in 1690 he wrote, in the form of a letter to Locke, an Historical Account of Two Notable Corruptions of the Scripture, regarding two passages on the Trinity. He left in manuscript a work entitled Observations on the Prophecies of Daniel and Apocalypse, and other works of biblical exegesis. At the time of his death on March 20, 1727, in his eighty-fifth year, Newton was revered throughout Europe and the honors bestowed upon him at burial in Westminster Abbey, after lying in state in the Jerusalem Chamber, were greater than those accorded most nobility.
The Principia is divided into three books:
Book I, Definitions, Axioms and Procedures;
Book II, Demonstration of propositions of forces in motion;
Book III, System of the World.
In Book I the three laws of motion and their consequences are established. The law of gravitation is set forth as a mathematical hypothesis from which Kepler's three laws of planetary motion is deduced. The orbits of two attracting bodies revolving about their common center of mass is calculated. The more complicated problem of the interaction between three bodies, each attracting the other two (the famous "three body" problem), is analyzed and, although the analysis is not complete, makes a good beginning in developing the modern form of the theory of perturbations. The motion of the simple pendulum and of the cycloidal pendulum is completely discussed.
In Book II the effect of resisting medium upon motion is discussed in detail. The science of fluid dynamics, including streamline effects, is established. The calculus of variations and the mathematical treatment of wave motion is introduced.
In Book III the system of the world is set forth. The motion of the planets, the satellites of Jupiter, and the orbits of the comets are studied in detail and are shown to be explained by the law of gravitation. The complicated motion of the moon, the precession of the equinoxes, and the tides is examined in detail and is systematically explained. The equatorial bulge of the earth is predicted quantitatively, which was later found to be extremely accurate. The density of the earth is estimated as between five and six times that of water, which was later found to be about five and one half times that of water.
The Principia is presented on the model of Euclid's Elements, but does not approach the rigor of the Elements. The book was not written for popular consumption and it is difficult to understand without any background in mathematics, physics or astronomy. The title indicates the level and scope of the book; the words in the title, "Mathematical Principles," sets the level and the words "Natural Philosophy" sets the scope and subject of the book (Natural Philosophy included the sciences of physics and astronomy). Only rarely does the Principia resort to the calculus, preferring to rely on geometrical arguments and demonstrations. And the formulation of the principles of mechanics in the book is not that which is learned today; the present formulation is that of Joseph Louis Lagrange, W. R. Hamilton, and a host of other writers since Newton, rather than of Newton himself. Although the formulation is not that of Newton, the principles are "equivalent" to Newton's, even though covering topics that Newton could not have anticipated.
"Law I. Every body continues in its state of rest, or of uniform of motion in a right line, unless it is compelled to change that state by forces impressed upon it.Expressing Newton's First Law (law of inertia) in symbols:
Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downward by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motion, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time."
This First Law of Motion is sometimes called the Law of Inertia, where inertia is defined as the property of a body by virtue of which a body resists change in its motion. The Dutch scientist, Christian Huygens (1629-1695 A.D.), anticipated Newton's statement here 14 years earlier in his greatest work published in 1673, the Horologium oscillatorium,
"If gravity did not exist, nor atmosphere obstruct motion of bodies, a body would maintain forever a motion once impressed upon it, with uniform velocity in a straight line."
"Law II. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body was moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."
Newton had already defined "quantity of motion" as the
product of mass times velocity, which product is called today
"momentum." In symbols,
p = mv,
where p is momentum of the body, m is its mass and
v is its velocity. Therefore, Newton's phrase "change
of motion" is the same as the change of momentum, that is,
Δp = p2 - p1,
where the Greek letter Δ means
"the change of", p1 is the initial momentum
at the beginning of change of momentum, and p2 is the final
momentum at end of change of momentum. The phrase "motive
force" designates what is now called impulse which
is defined as the product of the impressed force and the elapsed
time the force is applied, that is,
Impulse = FΔt,
where Δt is the elapsed time the force F is applied.
This meaning of the phrase is indicated by Newton's words, "whether
that force be impressed at once, or gradually and successively."
Thus Newton's Second Law may be restated in modern terms: "the
change in momentum is proportional to the impulse applied;"
and in symbols
FΔt = kΔp,
or with units chosen so that the proportionality constant k is one,
FΔt = Δp.
If both sides of this equation is divided by Δt, and canceling,
then
F = Δp / Δt,
where Δp / Δt
is the time rate of change of momentum.
Thus Newton's Second Law may be restated: the rate of change of
momentum is directly proportional to the net impressed force.
This statement of the second law in terms of change of momentum
allows for change of mass as well as change of velocity, that is,
F = Δ(mv) / Δt
F = (vΔm + mΔv) / Δt
F = vΔm / Δt +
mΔv / Δt,
where Δm / Δt is the time rate change of mass
and Δv / Δt is the time rate change of velocity.
The change of mass occurs in a rocket, where some of its mass
(the solid propellant) is burned and expelled, changing its mass,
as the force lifts the rocket. But if the mass of the body involved
remains constant (Δm = 0), the change of momentum will result
from a change of velocity, and the time-rate of change of velocity
is called "acceleration", a. Thus Newton's Second
Law may again be restated: the acceleration of a body is proportional
to the net impressed force and in the direction of that force.
If the unit of force is chosen properly so that the proportionality
constant is one, the second law may be expressed in symbols:
F = ma,
where F is the impressed net force, m is the mass
of the body, and a is the acceleration of the body in the
direction of the force, F.
Mass has been defined as a measure of inertia, that is, the resistance
to change of motion. Newton defined mass as the quantity of matter
that a body possesses and is equal to the product of its density and its
bulk (volume); volume is simply a measure of the physical space
occupied by a body, measured in cubic inches, cubic centimeters,
liters, etc.; it is "size" of the body, the amount of space
that it occupies, and it is not the mass, nor the weight of the body.
Density is a measure of how much mass is contained within a
given volume, that is, "how tightly packed" mass is packed
in space; specifically, it is the ratio of mass to volume:
density = mass / volume, or ρ = m / V.
Density is measured in units of grams per cubic centimeter, etc.
If density is given terms of the density of water, it is called
specific gravity, that is, the relative density of a substance
compared to the density of water, which is 1 gm per cubic centimeter.
For example, the specific gravity of gold is 19.3 or a density
of 19.3 gm per cm3.
Mass is not weight, even though they are related:
the weight of a body is directly proportional
to the mass of the body, where the constant of proportionality
is the acceleration due to the force of gravity; that is, weight
equals mass times the acceleration due to gravity, or
w = mg,
where w is the weight of the body, m is the mass
of the body and g is the acceleration of due to the force
of gravity. This relation is a special case of Newton's Second
Law of Motion were the force is the force of gravity, called the
weight of the body, and the acceleration is the acceleration
due to the force of gravity. The weight of a body is the
force of gravity exerted on the body. On earth the weight of
a body is the force of the gravity of the earth exerted on the
body; on the moon the weight of the body would be the force of
the moon's gravity exerted on the body. Weight is relative to
where it is measured and not the same on every planet or satellite;
but the mass of the body is the same every where.
In order to use Newton's Second Law of motion, a system of
units must be constructed, in which the units are interrelated
such that from a set of primary units all other units are defined.
Today there are three system of units that are currently in use:
foot-pound-second system (FPS), centimeter-gram-second system
(CGS), and meter-kilogram-second system (MKS). The FPS system
is an English system of units in which the primary units are selected
for length, weight, and time: foot for length, pound for weight,
the second for time. The CGS system is the older metric system
of units, in which the primary units are selected for length,
mass, and time: centimeter for length, gram for mass, and the
second for time. The MKS system of units is the newer and recommended
metric system of units, in which the primary units are selected
for length, mass and time: meter for length, kilogram for mass,
and second for time. Other units are defined in terms of these
primary units; for example, the unit of velocity in the FPS system
is defined as the unit of length divided by the unit of time,
that is, foot per second (the word "per" indicates division),
based on the formula for velocity:
v = Δs / Δt,
where Δs is the change in space or distance through
which a body travels in the elasped time Δt.
The unit of force in the CGS system and in the MKS system is defined
as the unit of mass times the unit of acceleration, that is, in
the CGS system, the unit of force is the dyne, and is defined as
gram·cm per sec2 and the unit of force
in the MKS system the newton is defined as
kilogram·meter per sec2;.
These definitions of force units are based on Newton's Second Law:
F = ma.
In the FPS system the unit of mass is the
slug and is defined in terms of the unit of force divided by the
unit of acceleration, that is, the slug is defined as pounds per
feet per sec2 or lb sec2 / ft; this unit of mass
is also based on Newton's Second Law, solved for mass:
m = F / a.
Since w = mg, one slug weighs 32.2 pounds,
one gram weighs 980 dynes, and one kilogram weighs 9.8 newtons.
It may also be shown that 1 newton = 100,000 dynes = 0.22 pounds-force.
| FPS | CGS | MKS | |
|---|---|---|---|
| length | foot (ft) | centimeter (cm) | meter (m) |
| mass | slug | gram (gm) | kilogram (kg) |
| time | second (sec) | second (sec) | second (sec) |
| velocity | ft per sec (fps) | cm per sec (cm/sec) | m per sec (m/sec) |
| acceleration | ft per sec per sec (ft/sec²) | cm per sec per sec (cm/sec²) | m per sec per sec (m/sec²) |
| force | pound (lb) | dyne | newton (nt) |
Note also that Newton was careful to state that the direction of
the forces in the same or opposite direction and are additive or subtractive
respectively. In other words, the forces are vector quantities,
whose direction must be taken into consideration when combining
them with the use of vector addition. There are three special
cases of the direction of the forces and of the motion:
(1) Force applied is in the same direction of the motion of the
body; the speed increases and the body accelerates without change
of direction,
(2) Force applied is in the opposite direction of the motion of
the body; the speed decreases and the body decelerates without
change of direction,
(3) Force applied is at right angles to the direction of the motion
of the body; the direction changes, but not the speed.
As an example of case (3), a body at the end of string moving
in a circle at a constant speed. The string exerts a force on
the body at right angles to the direction of its motion and toward
the center of the circle. This force is called a centripetal
force ("walking toward the center"). The direction
is constantly changing as the body moves about the circle, always
pointing at right angles to the radial line from the body toward
the center of the circle. This is accelerated motion because
the velocity of the body is changing, that is, its direction,
not its speed, is changing. This acceleration is called centripetal
acceleration or radial acceleration; if the path of
a moving body is a circle, then its magnitude may calculated by
the following formula:
centripetal acceleration = square of its constant speed / radius of circle, or
ac = v2 / r,
where ac = centripetal acceleration,
v = constant speed, and r = radius of circular path.
A thorough analysis of centripetal acceleration was first published in 1673 by Christian Huygens, and was probably known to Newton in all its essential details some years earlier.
This formula may derived by the following considerations:
"Law III. To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone;...."
The first two laws were clearly drawn from Galileo's analysis
and statements; this third law is stated by Newton for the first
time. Today this law is stated without the use of the terms "action"
and "reaction" and in terms of forces: for every force
there is a equal and opposite force. Combining these statements:
for every force acting on a body there is equal and opposite reacting
force exerted back upon the body exerting the acting force.
Note that this law says three things about forces:
(1) forces always occurs in pairs, an acting force and reacting force.
(2) these two forces are not exerted on the same body, and, therefore,
cannot cancel each other.
(3) these two forces are equal in magnitude but opposite in direction.
This third law may be expressed in symbols:
F12 = -F21,
where F12 is the force acting on a second body by a first
body and F21 is the force reacting on the first body by the
second body.
For example, the earth is pulled on by the sun by a force of gravity
there is an equal and opposite force of gravity pulling on the
sun by the earth. Since the earth is in motion revolving about
the sun, the force exerted by the sun on the earth is called a
centripetal force (force acting toward the center of motion) and
the equal and opposite force exerted back on the sun by the earth
is called a centrifugal force (force acting away from the
center of motion). As another example, the force exerted on a
whirling body by the string is the centripetal force and the force
exerted back on the hand holding the string is the centrifugal
force and these forces are equal in magnitude and opposite in
direction.
"1. That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.
2. The force of gravity towards the several equal particles of any body is inversely as the square of the distance of places from the particles."
A more recent statement of this law is the following:
"Between every pair of particles in the universe there exists a force of gravitational attraction. This force acts along the line joining the pair of particles and has a magnitude which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."In symbols this may be written:
Let F12 be the gravitational force exerted by mass one
on mass two and F21 be the gravitational force exerted
by mass two on mass one. These forces are equal in magnitude
( |F12| = |F21| )
but act in opposite directions
(F12 = -F21).
The magnitude is calculated by the mathematical formula above.
This gravitational force acts at the center of the body if it is
homogeneous (has same density throughout) and spherical. Since the law
of universal gravitation, as stated above, apply to particles, not to such
bodies, Newton invented the Integral Calculus in order to prove that
the force of attraction exerted on or by a homogeneous spherical body is the
same as that exerted on or by a particle having its mass (a particle is
a mass-point, that is, a geometrical point having the property of inertia).
Newton delayed publication of his law until he was able to prove
mathematically that the law did apply to homogeneous, spherical
bodies as well as to particles. This he was able to do by 1685.
The following summarizes the five steps given in the Principia by which Newton arrived at the law of gravitation:
Newton's proof is a combination of physical intuition and geometry.
The following is a simplified form of that proof. For mathematical
simplification, it will be assumed that the body revolves around
the earth in perfect circle.
(1) As the body moves about the circular path, it may be considered
as "falling" toward the earth at the center of the circle.
The distance h that it falls in unit of time may be calculated
by Galileo's Law of Falling Bodies:
h = ½at2.
(2) At any instant of time at a point P the direction of the body
is tangent to the circle at that point, that is, its direction
is at right angles to the line between the body and the center
of the circle, the radius of the circle.
(3) If at that instant the force on the body moving it in a circle
was stopped, then the body would fly off on the tangent line and would
travel a distance per unit time which may be calculated by the
formula: d = vt; and it then would be at a point A whose
distance from the center of the circle O would be equal to the
radius of the circle plus the distance it would have "fallen,"
that is, r + h, where r is the radius of the circle.
(4) Using the Pythagorean theorem on the right triangle OPA:
(r + h)2
= r2 + d2, or
r2 + 2rh + h2
= r2 + d2, and,
canceling r2,
2rh + h2 = d2.
(5) Since h is very small with respect with d and
r, h2 is very small compared to 2rh. Also
as the time interval t becomes very small, h2 becomes
very small with respect to 2rh. Therefore, a good approximation
is found by neglecting h2: thus,
2rh = d2, and solving for h:
h = d2 / 2r.
(6) From step (3) substitute d = vt for d
in formula (5), and from step (1) substitute
h = ½at2 for h in
formula (5), it becomes
½at2 = [(vt)2] / 2r
and solving for a we get
a = v2 / r,
which is the formula for centripetal acceleration.
(7) Using the Second Law of Motion, F = ma,
the formula for the magnitude of
the centripetal force exerted on a body moving in a circle is
Fc = mv2 / r;
this force is continually changing direction but is
always directed to the center of the circular path and is thus
sometimes called a central force.
The proof for the case of motion in a circle is as follows:
The circumference of the circle C is
C = 2πr,
where r is the radius of the circle.
The speed of the moving body m around the circle is
v = Δs / Δt = C / T =
2πr / T,
where T is the period of the moving body m.
The centripetal force exerted on the moving body is
Fc = mv2 / r =
m[(v)2] / r =
m[(2πr / T)2] / r =
m[4π2r2 / T2] /
r =
4π2mr / T2.
Since T2 = kr3,
which is Kepler's Third Law of Planetary Motion, then
Fc = 4π2mr / kr3 =
4π2m / kr2 =
(4π2 / k)(m / r2),
where 4π2 / k is a constant.
Since Fc = F12, then
F12 is proportional to
m2 / r2.
That is, the force of gravity varies inversely as the square of
distance between the attracting body and the attracted body.
Observations:
(1) Newton's Law of Gravitation is based on the Kepler's Second
and Third Laws of Planetary Motion and Newton's Second and Third
Laws of Motion.
(2) What kind of law is it? What is the character of this law
of gravitation? As a law it is describes how the force of gravity
behaves. But it does more than just describe the force of gravity;
it also explains why bodies fall as they do and why the planets
and the moon move as they do as described by Kepler's first and
second laws of planetary motion. It is a theory which explains
why these laws of planetary motion are as they are. But it is
not an explanation of gravity; it does not explain what gravity
is. And Newton refused to form such a hypothesis to explain the
nature of gravity. At the end of Book III, he says,
"Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power... But hitherto I have not been able to discover the cause of those properties of gravity from phenomena [observations and experimentation], and I frame no hypotheses... And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all motions of the celestial bodies and of our sea."
To verify this value, the actual value of acceleration of the
moon must be found. To obtain the actual value the formula for
centripetal acceleration will be used, assuming that the moon's
orbit is circular and that its centripetal acceleration is the
result of the force of gravity, a = v2 / r.
If the moon traverses a circle of radius r in time T,
called its period, then the speed v of the moon in its orbit
is the circumference (2πr) divided by the time period T,
that is, v = 2πr / T.
The centripetal acceleration of the moon is
ac = 4π2r / T2.
Since the period T of the moon is 27.3 days = 27.3 * 24
* 3600 = 2,358,720 seconds and r = 240000 miles * 5280
ft/mi = 1,267,200,000 ft, then a = 0.00896 ft per sec per
sec. This is very close to the predicted value of 0.00894 ft
per sec per sec. The difference is due to the assumption that
the orbit of the moon is circular instead of slightly elliptical
and to the approximate distance from the moon to the center of
the earth and from the stone to the center of the earth. But
as Newton put it, he had "computed the force requisite to
keep the moon in her orb with the force of gravity at the surface
of the earth, and found them to answer pretty nearly." This
was the first direct independent confirmation of the inverse-square
gravitation law. Some have suggested that Newton withheld for
so long the publication of his law of gravitation because that
when Newton first made his calculation, the ratio of the radius
of moon's orbit to the radius of the earth was not very precisely
known, and his calculation showed only approximate agreement.
It used to be supposed that this decrepancy was reason for him
laying the matter aside, but this view is not generally accepted now.
Newton's invention of the calculus made the calculation of the
elliptical motions of the planets and many other difficult computations
relatively simple. With the use of the calculus he performed
these calculation but we will not repeat them here. Without using
the calculus and making use of the simplifying assumption that
the orbits of the planets are practically circular, we will derive
Kepler's third law. If the centripetal force Fc on any
particular planet with a period of T and a mass of m,
by Newton's second law of motion,
Fc = mac,
where ac is the centripetal acceleration due to the
force Fc.
Fc = mac =
mv2 / R
where R is the radius of the orbit of the planet.
Since v = 2πR / T, then
Fc =
mv2 / R =
4π2mR / T2.
Assuming the centripetal force Fc is due to the force of
gravity Fg of the sun with a mass of M exerted on
the planet with a mass m. That is,
Fg = Fc =
4π2mR / T2,
and by the law of gravitation,
Fg = GmM / R2,
where G is assumed to be a universal constant, then
GmM / R2 =
4π2mR / T2.
Solving this equation for T2, we get
T2 = [4π2 / GM]R3.
Since the quantity in brackets contains the gravitational constant
G, the mass of the sun M, and numerical factors, it
is a constant k; that is,
k = 4π2 / GM.
Thus T2 = kR3.
This is Kepler's Third Law of Planetary Motion.