Geometrical properties of the conic sections

  We have already seen how the bodies of our system follow conic orbits. In this section we shall see what are exactly these orbits and discuss briefly some of its properties.

Figure 2.3

  Conic sections have known since ancient times and some of its more interesting characteristics were widely known to the Greeks. Their name is a consequence of the fact that any of the basic 4 curves can be generated by the intersection of a plane (orbital plane !) and a right circular cone, as depicted in figure 2.3.

  If said plane cuts only one of the sections of the cone, then the generated curve will be an ellipse which is the shape of all the orbits of the planets of the solar system. In fact, the circle is a special form of the ellipse (with eccentricity zero) and occurs when the intersection is parallel to the base of the cone. One of the characteristics of these curves is that they a re closed and that is why all bodies which follow elliptic orbits are periodic and therefore remain orbiting the central body which governs their motion.

  If the plane is parallel to the surface of the cone but only cuts one section, then the curve receives the name of a parabola. This curve usually represents the orbits of many Comets with the characteristic that being open such comets only come once through the solar system. Although such comets can be captured by the Sun or giant planets and become periodic as a consequence of perturbational gravitational forces, but this a different problem altogether.

  If the plane cuts both sections of  the cone, then the curve will be a Hyperbola with two branches. Although much less common, some comets are know to follow this type of orbits which are also of open shape.

  From a mathematical point of view, the conic sections can be defined as a curve in which the ratio between the distance of a point, on the curve, to a fixed point, called focus, and the absolute distance to a line called directix, is a constant known as eccentricity.

  Numerically, we have that if the eccentricity of a conic is:

-         equal to 0, then the orbit/curve is a circle

-         greater than 0, but less than 1, then the orbit/curve is an ellipse

-         equal to 1, then the orbit/curve is a parabola

-         greater than 1, then the orbit/curve is a hyperbola

  As mentioned above, all the planets of the solar system follow elliptic orbits, being Venus that of the least elongated orbit (near to zero) and Mercury that with the most elongated orbit (highest eccentricity).

Figure 2.4

  Figure 2.4 depicts all the family of conic curves that can follow the celestial bodies under the effects of the action of the force of gravity. Due to their symmetry, conics have two foci, one of which is called the central focus and contains the central body which governs the movement. The secondary focus is empty and has no physical relevance in the study of the two body problem.

  For the specific case of the parabola, which marks the transition between the closed and open conics, we can consider that said curve has the secondary focus located at an infinite distance to the left of the central or main focus.

  Additionally, we have that the magnitude of the "span" of the curve, passing through the focus, is twice the parameter of the conic (known as "latus rectum") and is represented by 2p in the above figure.

  The chord that passes though the foci is called the major axis and is denoted by 2a in the figure. In the circular orbits the major axis is nothing but the diameter of the orbit. In the elliptic orbit is customary to use half of this dimension and is then is know as the semi-major axis of the orbit. In the parabolic orbits, the major axis is infinite as well as in the hyperbolic orbits, although is considered as negative in the latter.

  The distance between the foci is depicted in figure 2.4 as 2c, and is zero for the circular orbits (for having the foci coincident in the center of the orbit). For the parabolic and hyperbolic orbits, this distance is considered to be infinite.

  For all conics, with the exception of the parabola, we have:

                                                                                                               2.13

                                                                                                   2.14

  Relationship of the energy and angular momentum with the geometry of the orbits

  By studying equation 2.12 we can conclude that the parameter of the conic depends solely on the angular momentum and thus an increment in the angular momentum must have the consequence of increasing the size of the orbit.

                                                                     Figure 2.5

  With the well-known analogy (again due to Newton) depicted in figure 2.5, let us imagine a cannon located in the summit of a mountain. If the angle of shot is zero (ie. parallel to the horizon) then h=rv, because is zero. By increasing the speed of the bullet, thus increasing its energy, we will equivalently be increasing the angular momentum.

  In this figure we can observe the family of curves that are generated by gradually increasing the "angular momentum" and such curves are all of conic nature. Conversely, we can also conclude that by increasing the angular momentum we are effectively also increasing the parameter of the orbit, such as stated by equation 2.12.

  With the help this example we can also deduce that at the pericenter (nearest point) or at the apocenter (farthest point) of any orbit, the velocity vector is always tangential to the orbit itself and therefore has a horizontal direction normal (ie. perpendicular) to the position vector and thus as a corollary of equation 2a we have:

                                                                                                2.15

  If we calculate the energy of the orbit by means of equation 2.6 for the pericenter and using equation 2.15, we obtain:

  but we also know that:

  therefore:

  also, we have that:

  then, as a conclusion we can state that:

                                                                                                         2.16

  This quite simple relation, valid for all conics, indicates us that the semi-major axis of an orbit depends only on the energy of the orbiting body, which in turn depends on the velocity and position of the body at a given point of the orbit. Going back to figure 2.5, we can have a clearer picture of this statement as we can see that by increasing the energy we are indeed increasing the semi-major of its orbit.

  Now, remembering figure 2.4 we must bear in mind that the circular and elliptic orbits have their semi-major axes defined as a positive quantity whereas it is considered as infinite in the parabolic orbits and negative in the hyperbolic orbits. This means, on the light of equation 2.16, that the specific mechanic energy or a body in circular of elliptic orbits is negative, zero for parabolic orbits and positive for hyperbolic orbits.

Additionally, as , we have that

but, 

  Therefore, we can conclude that for any conic:

                                                                                               2.17

  Studying this equation we can see that if the energy is negative then the eccentricity is positive and numerically less than one, ie. and ellipse. If the energy is zero, then the eccentricity is numerically equal to one, ie. a parabola. Finally, if the energy is positive then the eccentricity will also be positive but numerically greater than one and therefore the orbit will be hyperbolic.

  It is also important to note that in case that the angular momentum h is zero, independently of the energy of the orbit, then the eccentricity will be exactly one, but not really a parabola but rather a so-called degenerate conic represented by a straight line or a point is space.

Conclusions:

Once all the above mathematical analyses have been done, we can state with some rigor the following important conclusions:

1.      The family of curves called "conic sections" (circle, ellipse, parabola and hyperbola) represent the only possible curves that a celestial body can follow while orbiting another one under the force of gravity.

2.      The focus of an orbit is locates in the center of mass of the central body which in turn governs the movement.

3.      The mechanic energy, ie. the sum of the kinetic and potential energy, does not suffer modifications during the movement of the bodies. However, there exists an exchange between the kinetic and potential energies which means that the orbiting body must decrease its velocity as its distance increases and that its velocity increases as its distance to the central body decreases.

4.      The orbital movement is confined to a fixed plane

5.      The angular momentum of a body in orbit around a central body remains constant

6.      The energy of the orbit of a body, regardless or its geometry, allows us to determine the type and shape of the conic orbit over which the secondary body orbits the central body

7.      The angular momentum is directly related to the geometry of the orbit