Calculating Ephemerides

Calculating an Object's Ephemeris from its Elements

This tool calculates an ephemeris for an objects whose orbital elements are supplied on a range of dates starting at a specified time and advancing by a specified interval. It works for unperturbed objects in elliptical orbits (planets, asteroids, most comets and non-powered spacecraft). Due to limitations of the method employed (oscillating elements) and single precision arithmetic of most browsers, the accuracy will be no better than a few minutes of time or degree. Adequate for pointing telescopes but not accurate enough for more demanding purposes.

Acknowledgments: While none of the code is copied from or directly derived from either source, many of the concepts come from either Peter Duffett-Smith's Practical Astronomy with your Calculator [1979 Cambridge University Press] or Jean Meeus' Astronomical Algorithms [Second Edition 1998 Willmann-Bell, Inc.]. Any errors are my own.

Step 1: Provide Orbital Elements
These calculations require certain attributes of the objects' orbit collectively called the orbital elements:

(date at which time these values are correct)

(half the length of the orbit's longest axis in AUs)

(shape defining parameter (0<=e<1.1) for object's orbit)

(tilt of object's orbit to Earth's orbit)

(orbital plane intersection for object and Earth)

(closest approach of the body to the Sun)

(position of object at epoch)

* NOTE: For highly eccentric orbits the semimajor axis must be replaced by the perihelion distance (indicated by a negative number). The epoch must be replaced the time of perihelion passage. The longitude at epoch must be zero.

Before you panic, these values usually appears in articles about newly discovered comets and asteroids. Astronomy books tend to have them in tables in the appendices. If they aren't there, try IAU Ephemerides and Orbital Elements which is kept up to date frequently. You can also go to their software update pages as a link from this page. The default values describe the planet Mars. You can also get the information for the major planets from FDO Solar System Reference.

Step 2: Select Outputs
Decide which outputs interest you. Check off boxes for the outputs you desire. The report get long if everything is checked off. Display the date for each sighting. Display the position of the object in its orbit. Display heliocentric ecliptic coordinates. Display geocentric ecliptic coordinates. Display Right Ascension and declination. Display heliocentric Cartesian coordinates.
Step 3 Initial Observation Date and Interval
Put in the first (or only) date and time you are interested in viewing. Specify an interval between viewing times. You may specify intervals as a integer or decimal number for days and as a sexagesimal number for hours and minutes.
Step 4 Display an Ephemerides
Repeat this step as often as you like. The program will increment the date by the interval each time. You can avoid moving the mouse each time by simply pressing the space bar twice, once to remove the current information panel and a second time to request the next panel.

Orbital elements are often not reported in the same parameters as shown here. However they are always shown with enough information so that the parameters can be determined (or else they simply are not orbital elements). Here are some of the more common modifications to the list I've chosen above.

  • a-semi major axis Particularly for comets the semi major axis is not provided. In these cases q - perihelion distance is usually specified. It is related to a by q=1/(1-e). If the perihelion distance is provided rather than the semi major axis enter q, the perihelion distance as negative in the box labeled a. The tool will create the semi major axis for you. If very rare cases, a value labeled z (reciprocal of the semi-major axis) is provided. Compute a=1/z and enter in it in the box labeled a. The semi-major axis will be converted to AUs if it is specified in any other units. A semi-major axis with no unit is assumed to be AUs.
  • p-period The period is often not included. It can be determined from the relationship p2=a3. The FDO tool always computes it from the semi-major axis.
  • T-epoch and L - longitude at epoch Comets are often not reduced to a common epoch. Generally what is reported is the perihelion passage (the time where the comet approaches the Sun most closely). Use the perihelion passage as the epoch and set the longitude at epoch to zero.
  • ε This is an alternate symbol for L longitude at epoch in some sources.
  • π This is an alternate symbol for ω longitude of perihelion in some sources.
  • n - degrees per day This value is sometimes reported. It is not required by the FDO tools. It is related to the period by n=360°/365.25p.
  • P and Q These values provide an alternate way to specify the inclination i and the longitude of the ascending node ω. The FDO tools do not use P or Q.
  • J19yy.0 or J20yy.0 If the epoch or time of perihelion passage is specified as a J followed by a year between 1900 and 2099, you may enter it as such. For example J2000.0 is valid as a date input.

This program supports circular, elliptical, parabolic and hyperbolic orbits. These are all the possible orbital shapes that an object can assume when affected solely by the Sun's gravity. The program will give spurious results if an object is too close to a large planet. Notice that this includes all moons and spacecraft in orbits about a planet.

For closed orbits (circles and ellipses) you have the option of supplying either the semi major axis or the perihelion distance. The epoch may or may not be the same as the perihelion passage time. Circular orbits have eccentricities e=0 . Elliptical orbits have eccentricities 0<e<1 . However when eccentricities exceed the program treats the elliptical orbit as nearly parabolic because round off errors make an elliptical solution less accurate than a similar parabolic solution. When you have an elliptical orbit with very high eccentricities, you should supply the perihelion distance and the time of perihelion passage.

The semi major axis of an open orbit (parabolas and hyperbolas) is effectively infinite and cannot be supplied. You must supply the perihelion distance. The epoch date box is assumed to be the perihelion passage time. Every parabola will give results for any number of days from perihelion passage. Hyperbolas will only give reasonable answers for a relatively short period around perihelion passage and for eccentricities not too much greater than 1. The program will warn you when a hyperbolic solution does not converge. It will switch to a parabolic solution but the results will be poor for long time periods or high eccentricities.

This utility was authored by Les Coleman and is subject to Copyrights belonging to Les Coleman. This material may be referenced and reproduced as long as proper attribution is given as specified in Proper Usage Guidelines for Frosty Drew and Related Materials.