The orbit of a comet can be defined with six numbers, called the orbital elements, and by entering these numbers into your favorite planetarium software, you can view the location of the comet at any given time reasonably near the epoch date. The epoch date is a particular date for which the orbital elements are calculated and therefore the most accurate around that time.
Different sets of six parameters can be used, but the most common are shown below. Example values are given for Comet Holmes (17P), which exhibited a remarkable outburst in October 2007, now almost 12 years ago.
Perihelion distance, q
This is the center-to-center distance from the comet to the Sun when the comet is at perihelion, its closest point to the Sun. For Comet Holmes, this is 2.05338 AU, well beyond the orbits of both the Earth and Mars.
Orbital eccentricity, e
This is a unitless number that is the measure of the amount of ellipticity an orbit has. For a circular orbit, e = 0. A parabolic orbit, e = 1. A hyperbolic orbit, e > 1. Many comets have highly elliptical orbits, often with e > 0.9. Short-period comets, such as Comet Holmes (17P), have more modest eccentricities. Comet Holmes has an orbital eccentricity of 0.432876. This means that at perihelion, Comet Holmes is 43.3% closer to the Sun than its midpoint distance, and at aphelion Comet Holmes is 43.3% further away from the Sun than its midpoint distance.
Date of perihelion, T
This is a date (converted to decimal Julian date) that the comet reached perihelion, or will next reach perihelion. For example, Comet Holmes reached perihelion on 2007 May 5.0284.
Inclination to the Ecliptic Plane, i
Longitude of the ascending node, Ω
The intersection between the comet’s orbital plane and the Earth’s orbital plane forms a line, called the line of nodes. The places where this line intersects the comet’s orbit forms two points. One point defines the location where the comet crosses the ecliptic plane heading from south to north. This is called the ascending node. The other point defines the location where the comet crosses the ecliptic plane heading from north to south. This is called the descending node. 0° longitude is arbitrarily defined to be the direction of the vernal equinox, the point in the sky where the Sun in its apparent path relative to the background stars crosses the celestial equator heading north. The longitude of the ascending node (capital Omega, Ω) is the angle, measured eastward (in the direction of the Earth’s orbital motion) from the vernal equinox to the ascending node of the comet’s orbit. For Comet Holmes, that angle is 326.8532°.
Argument of perihelion, ω
The angle along the comet’s orbit in the direction of the comet’s motion between its perihelion point and its ascending node (relative to the ecliptic plane) is called the argument of perihelion (small omega, ω). For Comet Holmes, this angle is 24.352°.
If all the mass of the Sun and the comet were concentrated at a geometric point, and if they were the only two objects in the universe, these six orbital elements would be fixed for all time. But these two objects have physical size, and are affected by the gravitational pull of other objects in our solar system and beyond. Moreover, nongravitational forces can act on the comet’s nucleus, such as jets of material spewing out into space, exerting a tiny but non-negligible thrust on the comet, thus altering its orbit. Because of these effects, in practice it is a good idea to define a set of osculating orbital elements which will give the best positions for the comet around a particular date. These osculating orbital elements change gradually with time (due to gravitational perturbations and non-gravitational forces acting on the comet) and give the best approximation to the orbit at a given point in time. The further one strays from the epoch date for the osculating elements, the less accurate the predicted position of the comet will be.
For example, the IAU Minor Planet Center gives a set of orbital elements for Comet Holmes that has a more recent epoch date than the one given by the JPL Small-Body Database Browser. The MPC gives an epoch date of 2015 Jun 27.0, reasonably near the date of the most recent perihelion passage of this P = 6.89y comet (2014 Mar 27.5736). JPL, on the other hand, provides a default epoch date of 2010 Jan 17.0, nearer the date of the 2007 May 5.0284 perihelion and the spectacular October 2007 apparition. For the most accurate current position of Comet Holmes in your planetarium software, you’ll probably want to use the MPC orbital elements, since they are for an epoch nearest to the date when you’ll be making your observations.