Astronomy Principles and Practice Fourth Edition.pdf

170 Celestial mechanics: the two-body problem
13.3 Newton’s laws of motion
Kepler’s three laws provide a convenient and highly accurate way of describing the orbits of planets
and the way in which the planets pursue such orbits. They do not, however, give any physical reason
whatsoever why planetary motions obey these laws. Newton’s three laws of motion, coupled with his
law of gravitation, provided the reason. We consider the three laws of motion before looking at the law
of gravitation.
Newton’s three laws of motion laid the foundations of the science of dynamics. Though some, if
not all of them, were implicit in the scientific thought of his time, his explicit formulation of these laws
and exploration of the consequences in conjunction with his law of universal gravitation did more to
bring into being our modern scientific age than any of his contemporaries’ work.
They may be stated in the following form
(1) Every body continues in its state of rest or of uniform motion in a straight line except insofar
as it is compelled to change that state by an external impressed force.
In other words, in the absence of any force (including friction), an object will remain stationary
or, if moving, will continue to move with the same speed in the same direction forever.
(2) The rate of change of momentum of the body is proportional to the impressed force and
takes place in the direction in which the force acts.
Momentum is mass multiplied by velocity. The second law states that the rate at which the
momentum changes will depend upon the size of the force acting on the object and naturally enough
also depends upon the direction in which the force acts.
In calculus, d/dt denotes a rate of change of some quantity.
Both velocity v and force F are directed quantities or vectors, i.e. they define specific directions
and such directed quantities are usually underlined or printed in bold type. Mass m is not a directed
quantity. We may, therefore, summarize laws (1) and (2) by writing
d(mv)
= F (13.5)
dt
where we have chosen the unit of force so that the constant of proportionality is unity. Since in most
dynamical problems the mass is constant, we may rewrite equation (13.5) as
or
m dv
dt = F (13.6)
mass × acceleration = impressed force. (13.7)
(3) To every action there is an equal and opposite reaction.
The rocket working in the vacuum of space is an excellent example of this law. The action of
ejecting gas at high velocity from the rocket engine in one direction results in the acquiring of velocity
by the rocket in the opposite direction (the reaction). It is also found that the momentum given to the
gas is equal to the momentum acquired by the rocket in the opposite direction.
13.4 Newton’s law of gravitation
One of the most far-reaching scientific laws ever formulated, Newton’s law of universal gravitation
is the basis of celestial mechanics—that branch of astronomy dealing with the orbits of planets and
satellites—and astrodynamics—the branch of dynamics that deals with the orbits of space probes and