Planetary Orbits

General Remarks

An orbit is the path that an object follows around another object. The word orbit is usually associated with celestial objects. There are two types of orbit: closed orbits are those when the object in question can return to the same location after some time, and open orbits are those when the object never returns. The planets in our solar system follow closed orbits. Many of the comets we see are on open orbits, meaning that they pass by the Sun once and then head out to interstellar space.

In our solar system, planetary orbits all lie in the same flat, imaginary surface called the ecliptic plane. The ecliptic plane is defined as the flat surface that contains the Earth's orbit around the Sun. The ecliptic plane therefore always contains the Earth and the Sun. The other planets have their orbits tilted by a few degrees with respect to the plane of the ecliptic.

All planets orbit in the same direction, which is counter-clockwise when viewed from above the plane of the ecliptic (i.e., from the direction of the north ecliptic pole). This similarity of orbit directions is not by chance, but rather is due to the process of formation of the solar system.

Prograde and retrograde motion

We view the Sun and planets from our Earth. Because the Earth is in motion, we perceive odd motions of the planets. The Sun always appears to be moving in the same direction through the stars, at a rate of 360° every 365 days, or about 1 degree per day. The direction of the Sun's apparent drift is defined as east. We call eastward motion "forward", or "prograde motion", or sometimes "progradation".

Planets generally appear to be moving in the same direction as the Sun does, but occasionally reverse their apparent direction and move westwards for a few months. This short-lived, periodic westward motion is called "backward", or "retrograde motion", or sometimes "retrogradation". Retrogradation occurs when a faster planet overtakes a slower planet. For planets outside the Earth's orbit, it is when Earth overtakes each planet that this planet appears to lag behind us, giving the appearance of westward motion. This motion is not real, but is an effect caused by our perspective on a moving Earth. Similarly, planets closer to the Sun than the Earth appear to experience retrograde motion when they pass us on the inside track.

Kepler's Laws

Johannes Kepler formulated his three laws of planetary motion in the early 17th century, publishing his first two laws in 1609, and the third law in 1619. These laws are empirically-based, meaning that they are arithmetical expressions based upon the observations of the planets' paths through the celestial sphere; Kepler did not have any underlying cause in mind when he formulated them. The laws themselves can be shown to be a consequence of Newton's laws of motion and gravitation. Kepler's laws are:

1. Planetary orbits are ellipses. The Sun is located at one focus of the ellipse. The other focus is empty.
2. A planet orbits in such a way that a straight line connecting the Sun with the planet will enclose equal areas over equal times.
3. The period, or time for one complete orbit, is related to the orbit's size. Specifically, the square of a planet's period is equal to the cube of its semi-major axis, if the units are relative to those of the Earth. In other words, the period must be expressed in Earth years and the semi-major axis must be expressed in Earth-Sun distances. We define the average Earth-Sun distance as one astronomical unit (AU). In symbols, we write P²=a³, where "P" represents the orbital period of the planet, and "a" represents the semi-major axis length of the planet's orbit.

The first law tells us that the shapes of closed orbits are oval-like, and note that a circle is a special case of an ellipse. Circular orbits are allowed. The orbits of the major planets in our solar system are very nearly circular. We use the letter "e", called the eccentricity, to describe how much an ellipse has deviated from a circle. The eccentricity of closed orbits is a number between zero and one, where e=0 indicates a perfectly circular orbit, and 0A focus is an imaginary point in an ellipse that defines its shape. There are two focuses in an ellipse. Imagine two lines that extend from the perimeter of the ellipse to each of the two focuses; the sum of the lengths of these two lines is always the same for any point on the circumference of a given ellipse. For a circular orbit, the focuses are both located at the center.

The second law tells us that planets change their speed as they orbit. They speed up when closer and slow down when further from the Sun.

The third law tells us that, the further away a planet is from the Sun, the longer it will take to orbit the Sun. The longer orbit time is because (i) the path length of one orbit travelled by an outer planet is longer than that of an inner planet, and (ii) outer planets travel more slowly than inner planets.

Synodic and sidereal periods

The word "sidereal" means "with reference to the stars". We use the stars as a non-moving reference to measure positions in the solar system. The sidereal period is the time required for a planet to make one complete orbit of 360° around the Sun. The sidereal period of the Earth is one year.

The word "synodic" means "related to an assembly". The assembly in this case is the Earth, the Sun, and a planet. The synodic period is the time between successive occurrences of a given Earth-Sun-planet configuration.

The sidereal and synodic periods of two planets are related by:

   1          1            1
-------- = --------  -  ----------
P(outer)   P(inner)     T(synodic)

where P(outer) represents the sidereal period of the outer planet, P(inner) represents the period of the inner planet, and T(synodic) is the synodic period of the two with respect to the Sun.

For example, every 2.14 years, Earth catches up with Mars and passes it "on the inside track". Therefore, T(synodic)=2.14 years for the Earth-Mars-Sun system. We know that Earth's sidereal period is one year, and we know that Earth's orbit is interior to that of Mars, so P(inner)=1 year. We can calculate Mars's sidereal period:

   1        1        1
-------- = ---  -  ------        # Substitution of values
P(outer)    1       2.14

   1        214       100
-------- = -----  -  -----       # Bring to a common denominator
P(outer)    214       214

   1        114
-------- = -----                 # Perform the subtraction
P(outer)    214

            214
P(outer) = -----                 # Invert the fractions
            114

P(outer) =  1.88                 # Compute the result in decimal notation

Therefore, Mars's sidereal period is 1.88 years. It takes 1.88 Earth-years for Mars to make one complete orbit around the Sun. The answer here is in Earth years, because we used units of Earth years in our substitution. Any length (seconds, days, months) could have been used, as long as it is used consistently throughout.