ORBITS

Key Concepts

Newton modified and expanded Kepler's three laws of planetary motion.
Kepler described how planets move; Newton explained why they move the way they do.

(1) Newton modified and expanded Kepler's three laws of planetary motion.

Kepler's laws of planetary motion, in their original form, describe the motion of planets around the Sun. However, not only do planets orbit the Sun, but the Moon orbits the Earth, the Galilean moons orbit Jupiter, and so forth. Since gravity, according to Newton, is universal, orbits must be a universal phenomenon. Newton was able to use his laws of motion and laws of gravity to make minor corrections to Kepler's laws of planetary motion, and to expand their scope, making them universally applicable.

Consider Kepler's laws one by one.

FIRST LAW of planetary motion:
Kepler's version: The orbits of planets around the Sun are ellipses with the Sun at one focus.
Newton's revised version: The orbits of any pair of objects are conic sections with the center of mass at one focus.

Strictly speaking, the Earth doesn't orbit the Sun. As the Sun pulls on the Earth, the Earth pulls on the Sun with an equal but opposite force. Thus, both the Earth and the Sun are being accelerated, and both the Earth and the Sun are orbiting the center of mass of the Sun-Earth system. Imagine placing the Sun and Earth on a gigantic seesaw in a uniform gravitational field. In order for the seesaw to remain level, the balance point couldn't be midway between them; it would have to be much closer to the Sun, which is much more massive than the Earth. The center of mass is the point at which the seesaw would balance. The center of mass of a pair of objects is located on the line connecting the objects, and is closer to the more massive object. In the Sun-Earth system, the Sun is 330,000 times more massive than the Earth, and thus the center of mass is 330,000 times closer to the center of the Sun than to the center of the Earth; that's only 450 kilometers from the center of the Sun, buried deep within the Sun's interior. In the Earth-Moon system, the Earth is 81 times the mass of the Moon, and the center of mass is 4700 kilometers from the center of the Earth; it is inside the Earth, but a distant observer would be able to see the Earth going around on its tiny orbit at the same time the Moon goes around on its much larger orbit.

Newton was aware of the concept of an artificial satellite. To put an object (a cannonball or a weather satellite or a Space Shuttle) into orbit, you can launch it sideways (parallel to the surface of the Earth) with a speed v. The shape of the orbit depends on v; in every case, however, the orbit is a conic section. A conic section is one of a family of curves obtained by slicing a cone with a plane. A circle is a conic section, and so is an ellipse, a parabola, and a hyperbola.

Low initial speed for the satellite: closed orbit (circle or ellipse)
High initial speed for the satellite: open orbit (parabola or hyperbola)

So, not all orbits, as Newton recognized, are closed loops (an ellipse or circle). If an artificial satellite is launched at a high enough speed, it escapes the Earth altogether, and the path it traces is an open curve (a hyperbola or parabola, technically speaking).

Most artificial satellites are on nearly circular orbits. We can compute the speed at which a satellite must travel if it is on a circular orbit. This speed is called the circular speed, and is designated by the symbol v_c. The gravitational acceleration felt by the satellite will be, from Newton's Law of Gravity and Second Law of Motion,
a = G m / r²
where m is the mass of the planet that the satellite is orbiting, and r is its distance of the satellite from the center of mass. (Since artificial satellites are always much much less massive than the Earth, in the case of a satellite orbiting Earth, it is safe to say that the center of mass is indistinguishable from the center of the Earth.) The acceleration required to keep a satellite with speed v_c on a circular orbit of radius r is
a = v_c² / r
Set the two accelerations equal, and you find
v_c² / r = G m / r²
or, with a bit of algebra,
v_c = (Gm/r)^1/2 .

To take an example, the International Space Station is in a circular orbit about 370 kilometers above the Earth's surface (only 230 miles). Since the radius of the Earth is 6380 kilometers, that means the distance of the Space Station from the center of the Earth is r = 370 km + 6380 km = 6750 km = 6,750,000 meters. Using the mass of the Earth and G, you can compute the circular speed for the International Space Station:

v_c = 7680 meters/second = 7.68 kilometers/second = 17,000 mph.

At this speed, it takes the Space Station only 92 minutes to complete one orbit.

SECOND LAW of planetary motion:
Kepler's version: A line from a planet to the Sun sweeps out equal areas in equal time intervals.
Newton's revised version: Angular momentum is conserved.

Newton's revised version doesn't sound anything like Kepler's initial version! Nevertheless, Kepler's second law of planetary motion is the inevitable consequence of the conservation of angular momentum.

The angular momentum of an orbiting object is given by a simple formula:

L = m v r

where

L = angular momentum
m = mass of orbiting object
v = orbital speed of object
r = distance of object from center of mass

The statement ``angular momentum is conserved'' simply means that L remains constant as an object moves along its orbit. Since the mass m is also constant, this means that the product of the orbital speed v and the distance r must remain constant. As r gets smaller, the speed v must get bigger. As r gets larger, the speed v gets smaller. [The customary analogy is a spinning ice skater, who spins faster and faster as she pulls in her arms closer to her body.] Thus, conservation of angular momentum implies that PLANETS MOVE FASTEST WHEN THEY ARE CLOSEST TO THE SUN. Moreover, a careful mathematical analysis reveals that r and v vary in such a way that a line between an orbiting object and the center of mass sweeps out equal areas in equal times.

THIRD LAW of planetary motion:
Kepler's version: P² = a³ (when P is measured in years, and a in A.U.)
Newton's revised version: P² = { 4 pi² / [ G (m₁+m₂) ] } a³

In the above equations,

P = orbital period
a = semimajor axis of orbit
G = gravitational constant (6.7 x 10^-11 in metric units)
m₂ = mass of orbiting body
m₁ = mass of other body in system
pi = 3.14159265...

The original version of Kepler's third law applies ONLY to objects orbiting the Sun. Newton's revision applies to ALL pairs of objects orbiting their center of mass. For planets orbiting the Sun, the mass of the planet (m₂) is negligibly small compared to the mass of the Sun (m₁). Even Jupiter, the biggest planet, has a mass which is only 1/1000 of the Sun's mass. Thus, the factor inside the curly brackets in Newton's revised formula reduces to
4 pi² / ( G m₁ )
where m₁ is the mass of the Sun. This number is the same for all objects orbiting the Sun, and when you measure time in years and distances in astronomical units, it is equal to ONE. Thus, Kepler's third law in its original form is merely a special case of Newton's more general formula.

Note that Newton's form of Kepler's third law can be used to determine the mass of distant objects. For the Galilean moons of Jupiter, for example, we can measure their orbital periods (P) and the sizes of their orbits (a). Assuming that the masses of the Galilean moons are much smaller than the mass of Jupiter (which is quite correct), we can use Newton's form of the third law to find the mass m₁ of Jupiter.

(2) Kepler described HOW planets move; Newton explained WHY they move the way they do.

Kepler's laws are purely descriptive; they tell us how the planets move, but do not provide an explanation of the forces which affect their motion. On the other hand, Newton's laws of motion and law of gravity explain why the planets move the way they do.

Newton, using his UNIVERSAL laws of motion and law of gravity, was able to modify Kepler's laws of planetary motion so that they too are UNIVERSAL. They're not just for planets any more; in the revised form, they apply to any pair of objects moving freely under their mutual gravitational attraction.

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