Planetary orbits

The planets of the Solar System move around the Sun following elliptical paths, also called orbits. All planets move in the same direction but with different velocities, so they sweep out equal areas in equal times; due to this, the farther the planets are from the Sun, the slower they move.
The paths of celestial bodies are curves of the conical family: ellipses, parabolas and hyperbolas. The equations of these curves in Cartesian coordinates are:

Ellipses:

The elliptical orbits have the Sun at one focus; the point of the ellipse nearest to the Sun is called perihelion and the farthest from it is called aphelion. Most of the bodies of the Solar System (planets, comets, asteroids, ...) follow elliptical orbits albeit with very different eccentricities.

If the distance from the planet at the perihelion to the Sun is similar to the distance to the aphelion, the ellipse has a small eccentricity: the orbit is almost circular. Planets belong to this group.

On the other hand, if the distance from the planet at the perihelion to the Sun is much smaller than the distance at the aphelion, the ellipse has a very small eccentricity: the orbit is almost a parabola. Comets, and some minor bodies of the Solar System, have orbits of this type.

The bodies with hyperbolic paths are not trapped by the Sun's gravity and are able to escape from the Solar System. Since the Solar System is quite old, the number of bodies with hyperbolic paths is very low: some loose fragments of rock from collisions between bodies of the Solar System (e.g. in meteoroid currents or in the belt of asteroids) and the space probes designed to this end (for example NASA's Pioneer o Voyager probes).

In the following table there is a summary of expressions of some interesting data from the conical figures:

	Circle	Ellipse	Parabola	Hyperbola
Equation	x2 + y2 = r2		y2 = 4px
Parameters	r = circle radius	a = major radius (= 1/2 the length of the major axis) b = minor radius (= 1/2 the length of the minor axis) c = distance from the center to the focus	p = distance from the vertex to the focus (or the directrix)	a = 1/2 the length of the major axis b = 1/2 the length of the minor axis c = distance from the center to the focus
Excentricity (c/a)	0	(0,1)	1	>1
Relation to the focus	p = 0	a2 - b2 = c2	p = p	a2 + b2 = c2
Geometric definition	Geometric layout of points equidistant from the origin	Geometric layout of points whose sum of distances from the focuses is constant	Geometric layout of points whose distance to the focus is the same as the distance to the directrix	Geometric layout of points whose difference of distances to each focus is constant

Position of the planetary orbits in space

Although the orbit of all the planets is elliptical, not all of them are aligned nor are on the same plane. In general, most of the planets' orbit on planes close to that of the Earth's orbit, also called ecliptic. The exception is Pluto, whose orbital plane is inclined more than 17º with respect to the ecliptic.

Some useful formulae for the exercises

To solve these exercises you need to know how to draw up Cartesian equations for an ellipse whose symmetry axes are rotated according to your Cartesian axis. If you pay attention to the drawing and the following expressions, you will see that it is not that difficult... We are going to develop some properties of the points and elements of an ellipse, observing the figure.

The relation between a point P of the ellipse and its focus is:

a2 = b2 + c2

There is a relation between the two distances to the focus, R1 y R2:

R1 - R2 = 2c
R1 = a + c

The following relations are useful in determining the different elements of an ellipse:

R1 = a(1-e)
R2 = a(1+e)
R1 + R2 = 2a
b2 = a2(1-e2)

Therefore, the cartesian equation of the ellipse will be:

If the ellipse is rotated an angle "ω" from the X axis, the formulation becomes a bit complicated:

in this case the formula of the ellipse in the X`Y` system will be the usual one:

yet to write it in the XY system, one should apply the following conversion:

x' = x·cosα - y·sinα
y' = x·sinα + y·cosα

and we get:

and from here we derive the expression (and the coefficients) that you need to fill in and calculate in the exercise:

x2(b2cos2α + a2sin2α) + y2(b2sin2α + a2cos2α) + 2xysinαcosα(a2 - b2) - a2b2 = 0

It`s known, on the other hand, that the general equation of a conic is the following:

Ax2 + By2 + Cxy + D = 0

and comparing the two equations, you will clearly see that:

b2cos2α + a2sin2α = A
b2sin2α + a2cos2α = B
2sinαcosα(a2 - b2) = C
-a2b2 = D