Kepler --- | ---
Orbits
Kepler --- | ---
Orbits
Reading:
Chapter 1-3, P1-2, P1-3, P1-4 (ZG4)
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Key Question: | What keeps a planet or satellite in orbit? |
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Key Principle: | Newton's Laws of Motion |
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Key Problem: | What is the acceleration of the Moon in its orbit? |
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Key Quote: | "Nature and nature's laws lay hid in night. God said, Let Newton be!, and all was light." - A. Pope |
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.
In the Principia, Newton stated his three Laws of Motion:
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.
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.
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. An example of a vectory is the 3-dimensional position of a body in space x.
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). The velocity is the time derivative of the position
The units are thus kg m/s^2 which is defined a 1 Newton of force.
From definition of acceleration and F = ma, we see that the momentum is just
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.
But equal forces do not mean equal accelerations! The two accelerations are related by
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!)
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
and
Thus, the centripetal force on a mass m is
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
then we find that
For a circular orbit of period P, the velocity is given by
so substitution gives
or
which is Kepler's third law! Note that the relation now involves the central mass M that the body is orbiting, and thus the proportionality of Kepler's law can be used to measure the mass of the body that the satellite or planet is orbiting. The above formula was derived neglecting the mass of the satellite. We will derive the general result in the next lecture.
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smyers@nrao.edu Steven T. Myers