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Home , Elliptic orbit

The Elliptic of the

Before proceeding with the subject of dynamics, wc shall briefly describe the remarkable achievement of in establishing the proposition that the planetary orbits are not compounded circles but simple . This great discovery was based almost entirely on the analysis of the motion of a single —Mars. The behavior of Mars had puzzled and exasperated astronomers for a very long time, because the apparent irregularities in its motion were greater than those of any other planet and defied any easy analysis in terms of combinations of uniform circular motions.

To appreciate the development of Kepler's discovery one must constantly keep in mind the fact that the data of observational astronomy (and this was especially true in the days before the telescope) are directions rather than distances. Although it was well understood that variations in the apparent brightness of the planets were linked to variations of distance from the earth, the precise data were only of angular positions. The whole theoretical machinery of superposed circular motions was primarily a means of reproducing the observed angular position of each planet as a function of time.

Kepler began his study of Mars at the direction of the great observational astronomer Tycho Brahe, whom he joined as an assistant in 1600. Kepler's task was to construct the actual path of Mars in space from the accumulation of original observations; it took him 6 years, and many false scents, before he arrived at the picture that is now familiar to us. Kepler fully accepted a heliocentric model of the , and (unlike Copernicus himself) he consistently held to the idea that the path of a planet must be a smooth, continuous curve of some kind around the . His problem was to find this curve on the basis of observations made from a laboratory—the earth— which was itself orbiting the sun in a nonuniform way. A first task was therefore to establish the path of the earth itself. Kepler attacked this problem in several ways. The one most directly based on observation was brilliant.

Kepler published the full story of his labors—the many failures as well as the final successes—in a book, The New Astronomy (Astronomia Nova), published in 1609. It is a classic of scientific discovery.

The law of the inverse square Newton’s law of gravitational attraction states that two particles of masses , at a distance r apart, attract one another with equal and opposite forces of magnitude

, (1) where G is the gravitational constant. Coulomb’s law of electrostatic attraction states that two particles carrying electric charges (in electrostatic units), at a distance r apart, repel one another with equal and opposite forces of magnitude

. (2)

If and have opposite signs, this force is a force of attraction. Here we have two examples of the law of the inverse square. The law (1) governs astronomical phenomena – in particular, the motion of a planet round the sun. The law (2) governs atomic phenomena – in particular, the motion of an electron in an atom about the central nucleus. In this case, of course, the charges have opposite signs, so that the force is one of attraction, as in the gravitational case. It is remarkable that the same form for the law of attraction should hold on such different scales. The expressions (1) and (2), combined with Newton’s law of motion, constitute two hypothesis regarding phenomena in gravitational and electrostatic fields. For a long time, they were accepted as completely valid from a physical point of view, but that is no longer the case. The modern astronomer knows that gravitational attraction should be discussed in terms of the general theory of relativity, and the physicist insists

that problems on the atomic scale belong to quantum mechanics. It would, however, create a completely false impression if we were to say that the law of the inverse square has disappeared from modern science. Nearly all the calculations of astronomers are still bases on (1) and give results in excellent agreement with observation. Moreover, the physicist often falls back on simple atomic picture based on (2) and Newton’s law of motion. In what follows, we shall discuss the motion of a planet attracted by the sun. Obviously, be a mere change of constant, the same reasoning will apply to the motion of an electron in an atom.

Determination of the orbit The sun and a planet are regarded as particles, of masses M and m, respectively. The attraction of the sun on the planet, given by (1), produces an acceleration of the sun and treats it as if it were at rest. We consider then the case of a particle attracted toward a fixed centre by a force per unit mass, where , (3)

being some positive constant. The polar differential equation (the Path Equation (6) of Mechanics-3) for the orbit now becomes

. (4)

The general solution is

, (5) where C and are constants of integration. This is, in polar coordinates, the equation of the most general orbit described under a central force varying as the inverse square of the distance. The potential energy per unit mass (given by equation (1b) of Mechanics-3) is

, (6) the constant of integration being chosen to make V vanish at infinity.

Let us now substitute from (5) in equation (7) of Mechanics-3 i.e. , the equation of energy, in order to express the constant C in terms of E and h (the total energy and angular momentum per unit mass). We get

,

So that

(7)

By rotating the base line , we can make and in (5); this we shall suppose done. Then the equation (5) for the orbit reads

. (8)

From the focus-directrix property of conic, we know that its equation in polar coordinates may be written as , (9)

Where l is the semi-latus-rectum (i.e. half the focal chord parallel to the directrix) and e the eccentricity; is measured from the perpendicular dropped from the focus on the directrix. The conic may be of any of the following types : , , hyperbola . In the case of the hyperbola, (9) gives only the branch adjacent to the focus.

Comparing (8) and (9), we note that it is always possible to bring the equations into complete agreement by choosing for and the values

. (10)

Accordingly, we may say: The orbit described by a particle, attracted to a fixed centre by a force varying as the inverse square of the distance, is a conic having the centre of force for focus. The semi-latus-rectum and the eccentricity are given by (10) in terms of the angular momentum and energy per unit mass. The orbit may be of the following types: ellipse , parabola , hyperbola . The fact that orbits may be classified thus in terms of the total energy is remarkable.

Possible orbits under an inverse square attractive central conservative force

The most important orbits in astronomy (those of the planets) are ellipses. Recurring describe orbits which are elongated ellipses, approximating to . A body with a parabolic or hyperbolic orbit would pass out from the solar system, never to return.

Constants of the elliptical orbit Let us now confine our attention to the elliptical orbit. Since the orbits of the planets are of this type, a great wealth of technical detail has been developed about the elliptical orbit. We shall discuss here in brief. It is evident from (9) that the shape and size of an orbit (but not its orientation in space) are determined by the two constants l, e. These are related to the constants E, h by (10). Thus, of the various constants which appear in our equations, we are to regard (the intensity of the force centre) as given once for all, whereas the constants l, e, E, h take different values for different orbits. On account of (10) only two of these constants are independent. We may use as an independent pair any two which prove convenient. Instead of using (l, e) as fundamental constants, it is better to use (a, e), where a is the semiaxis major of the orbit. Now,

, (11)

B being the semiaxis minor. We shall refer to (a, e) as the geometrical constants of an orbit and (E, h) as its dynamical constants. The formulas of transformation from one set to the other are as follows :

(12)

Velocity at any point There is a simple formula giving the speed at any point of the orbit in terms of the radius vector. By the equation of energy (equation (1) of Mechanics-3) , and (equation (1b) of

Mechanics-3) we have by (3)

. (since , therefore )

Substituting for E from (12), we obtain (13)

The periodic time We now ask : How long does the particle take to describe the elliptical orbit ? This time is called the periodic time ( ). We seek an expression for in terms of the fundamental constants. For obtaining periodic time we refer to the equation (16) of Mechanics-3, which gives for the areal velocity of a particle moving along a plane curve .

Figure : The rate of increase of A is constant

If F is the focus at which the centre of force is situated, it follows at once that the particle describes an arc VP, starting from the vertex V nearer to F, in a time , where is the area of the sector subtended at F by this arc (see figure above). The vertex V is called perihelion – the point closest to the sun – the other vertex being called aphelion – the point away from the sun. The corresponding terms for motion about the earth are called perigee and apogee. In general these positions are called pericenter and apocenter which are the points of and respectively and are the turning points of the orbit. Following the motion of the point P round the orbit, we get the periodic time as , (14) where A is now the total area of ellipse. We might substitute ; and to be systematic, we should express in terms of either the geometrical constants or the dynamical constants . Since we obtain from (14), after using the expressions for and from (12),

,

i.e. (15)

( Remember that for the elliptical orbit )

It is remarkable that the formula involves only one geometrical constant or one dynamical constant. All orbits with the same semi-major axis have the same periodic time; so also have all orbits with the same total energy.

Table: Some Properties of the Principal Objects in the Solar System

aOne (A.U.) is the length of the semimajor axis of earth’s orbit. One A.U. =1.495x1011m or 93x106 miles. bEarth’s mass is approximately 5.976x1024 kg. cProblem : Calculate the missing entries denoted by c in the table.

Apsidal distances of the path The apsidal distances ( and ) as measured from the foci to the orbit , are given by

(16)

Time of description of an arc of an elliptic orbit The equation of the path is , , where is the semi-latus rectum and is the eccentricity. Again, we have or, .

Hence, , where we assume that t is the time taken from the vertex of the ellipse to any point .

Then, . (17)

Now,

.

Therefore,

or,

On integration, we get

Or,

Now, we know that

.

Therefore, .

Then, from (17) we have

Now, as and ,

Therefore, = .

Hence, . (18)

Kepler’s laws Before Newton’s discovery of the laws of motion, Kepler announced the following three laws describing the motion of the planets, deduced from the extensive and accurate observations of planetary motions by Tycho Brahe: (1) The planets move in ellipses with the sun at one focus. (2) Areas swept out by the radius vector from the sun to a planet in equal times are equal. (3) The square of the period of revolution is proportional to the cube of the semimajor axis.

The second law is a consequence of the conservation of angular momentum deduced in earlier notes. It shows that the force acting on the planet is directed toward the sun. The first law follows, as we have shown earlier, from the fact that the force is inversely proportional to the square of the distance. The third law follows from the expression of periodic time deduced in (15) above. Thus, starting from Newton’s law of gravitation, we have shown that all these statements are true. But it is interesting to adopt the historical point of view and face the problem as it presented itself to Newton : Given Kepler’s laws as a statement of fact, what is the law of gravitational attraction ? Law (2) tells us that h – the angular momentum per unit mass – is constant. Hence is constant, so that the cross-radial acceleration of the central orbit which is is zero

Hence the force (acceleration) exerted on the planet must be only radial i.e. directed toward the sun. From Law (1), we know that the equation of an orbit may be written .

Then by the Path Equation (6) of Mechanics-3 , the force per unit mass is

or, (19)

Thus for each planet the force varies inversely as the square of the distance. But it remains to prove that the force is of the form , (20) where m is the mass of the planet and a constant, the same for all the planets. To show this, we appeal to Law (3). We know that for an elliptical orbit, described under a central force directed to a focus, ,

And so,

But, by Law (3), this is a constant, the same for all planets. Thus is the same for all planets; and so, by (19),

, where is the same for all planets. Hence (20) is true, and Newton’s law of gravitation is thus deduced as a consequence of Kepler’s laws.

We have studied the laws of planetary motion. The motion of comets remained an enigma for a long time even after Kepler formulated the three laws. In fact, it was Newton who observed a in 1682 and was the first to explain its trajectory. He could see that the orbit of the comet was governed by the same principles of dynamics that applied to the motion of the planets. He realized that some comets could move past the sun in parabolic and hyperbolic orbits and so would never return. But other comets should move along elliptical path like the planets. Only the eccentricity would be much higher. Newton’s insight revealed that comets are members of the Solar System. We are familiar with the Halley’s Comet which returns every 76 years. It has a with .

If Kepler’s laws were accurately true, we should have to regard the sun as fixed and the planets as attracted only by the sun. More precise measurements show that Kepler’s laws are only an approximation and that the inverse- square law of attraction holds for every pair of bodies. It is fortunate that the observations of Kepler’s time were crude, because otherwise the simplicity of the law of gravitation would have been obscured.