Mid-Kent Astronomical Society - Calculation of planetary positions

Calculation of planetary positions

Written by Noel
Monday, 24 March 2008
In order to observe the planets, it is helpful to know exactly where they will appear in the sky. The quickest and simplest way to do this is to look them up on a system or almanac. However, if you are interested in the process of calculating orbital positions (or are simply a glutton for punishment) Here are the steps. Find out where we were with respect to the Sun (from an almanac) at a point in time Find out where the planet was with respect to the Sun (at that time) Work out how far it's moved since (mean anomoly) Correct for eliptical orbits (calculate difference due to eccentricity of orbits) Calculate the resultant position with respect to Earth now Format the results (with respect to a coordinate system) The method shown here is based on an article by Keith Burnett, see references at end of this document. There is a worked example attached to this document in Microsoft Exel (but you need Exel to run it). The method described below is an approximation to the actual position of the planets. In trials, it was accurate to no more than 0.4 hrs (4°) of Right Ascension on the inner planets (the outer ones were better than 1°), but better than a degree in declination. Arriving at a resolved position, requires several stages in calculation. Starting points: Our point of reference (obviously) will be a point on the Earth's surface. This however gives us a few issues when calculating the position of planets that are orbiting the Sun: The Earth rotates once every 23:56 hrs (a sidereal day) The Earth orbits the Sun once every 365.25 days (a year) The Earth's axis of rotation is inclined (obliquity) at an angle of 23.5° to the plane of it's orbit around the Sun (the ecliptic). The Earth's axis of rotation moves in a circle (precession) once every 26,000 years The Earth's axis of rotation is nodding up and down (nutation) once every 19 years Light is travelling at a finite speed from the planets to us (aberration) This article doesn't deal with any of these problems, instead I will declare the positions with reference to the celestial coordinates (Right Ascension and Declination) with reference to our start position, leaving you to calculate where in the sky this will appear to you. Calculating the position of the planets requires a starting position (as we are unable to go back 4.5Bn Years to the begining!) we will therefore need to take a zero position or starting point from an observation or almanac. For this example I used the 'Osculating Elements for 2008' published by the U.S. Naval Observatory. Specification of the orbits of the planets needs a number of elements (since orbits are elliptical, not circular), here are the elements we will use: Inclination (i) - the angle between the plane of the Ecliptic and the plane of the orbit. Longitude of the Ascending Node (o) - the position in the orbit where the elliptical path of the planet passes through the plane of the ecliptic, from below the plane to above the plane. Longitude of Perihelion (p) - the position in the orbit where the planet is closest to the Sun. Mean distance (a) - the value of the semi-major axis of the orbit - measured in Astronomical Units for the major planets. Daily motion (n) - how far in degrees the planet moves in one (mean solar) day. This figure can be used to find the mean anomaly of the planet for a given number of days either side of the date of the elements. The figures quoted in the Astronomical Almanac do not tally with the period of the planet as calculated by applying Kepler's 3rd Law to the semi-major axis. Eccentricity (e) - eccentricity of the ellipse which describes the orbit Mean Longitude (L) -Position of the planet in the orbit on the date of the elements. In the image above, the orbits are shown going clockwise (contrary to the way the planets actually orbit!). Mean Anomaly (M) in degrees The Mean Anomaly of the planet (how far it would have moved) from start position is given by the formula; M = \|(n * d_e + L – p)\|₃₆₀ Where d_e(referred to as de) was the number of days since the reference date of the elements. True Anomaly (v) in degrees To correct for eliptical orbits, we will be using ‘Equation of Centre’ approximation: Radius Vector (r) in Au The radius vector r would be in the same units as a, Au. in this case. Conversion to rectangular heliocentric co-ordinates Calculation of Earths heliocentric coordinates Calculation of the Earths position was carried out using the above method (using the calculation spreadsheet itself), the planet for calculation was set to ‘Earth – Moon’ and the elements published for the Earth – Moon Barycentre were used. The calculated planetary positions (X, Y &Z heliocentric position) were then copied and pasted into the adjacent cells for Earths position (i.e. the table of heliocentric positional information for the Earths orbit for the year was set as static data). Geocentric elliptical coordinates Geocentric equatorial coordinates Using an eccentricity constant for Earth (J2000 eccentricity figure)⁴:The Published figures were in arc seconds; this figure used was the converted equivalent in Radians.The previously calculated values were converted to equatorial coordinates using: Convert to RA and Dec Overflow correction is required (to bring the planet back onto the chart): v If Xq <0 (add 180°)v If Yq<0 and Xq>0 (add 360°) Microsoft Excel Example The spreadsheet programme; Microsoft Excel was chosen because of its inbuilt charting functions, which could be used to plot the data directly. The method of implementation of the above calculations should be viewed in the workbook. The Equation of Centre calculation (step 2 of the calculations above) was implemented in Excel as: =H7+((2e-POWER(e,3)/4)SIN(H7)+1.25POWER(e,2)SIN(2H7)+13/12POWER(e,3)SIN(3H7)) Example calculation taken from cell I7 of the Calculations sheet of the Solar System workbook. Note that the calculation units were radians, as this is the preferred unit of Microsoft Excel. The worked example here is Microsoft Excel (97-2003 compatible format, it's 1/2 MB large). Please note; the calculations provided by this spreadsheet are illustrative and mostly for fun. No liability will be accepted for thier accuracy, if you are planning an interplanetary voyage, it is recommended you use a more accurate model. References: 1. Planetary position (Astronomical Almanac Data) Courtesy of the US Naval Observatory. U.S. Naval Observatory 3450 Massachusetts Avenue, NW Washington, DC 20392-5420 USA The Astronomical Almanac is a joint publication of the U. S. Nautical Almanac Office in the United States (USNO) and Her Majesty's Nautical Almanac Office (HMNAO) in the United Kingdom. http://www.usno.navy.mil/ http://asa.usno.navy.mil/SecE/2008/Osculating_Elements_2008.txt 2. Astronomical Constants referred to: http://asa.usno.navy.mil/SecK/2008/Astronomical_Constants_2008.txt 3. Approximate astronomical positions. Details of common astronomical calculations for amateur astronomers by Keith Burnett.. http://www.stargazing.net/kepler/ellipse.html Guidance note on planetary position calculations. 4. Heavens-above.com Astronomy website with useful European based data on orbital objects. The planetary summary page provides useful daily information on the planets. http://www.heavens-above.com/planetsummary.asp?lat=0&lng=0&loc=Unspecified&alt=0&tz=CET 5. Stellaris A shareware application for amateur astronomers. Version 1.3 was used for validating the results of this project. Stellaris can be contacted (and obtained) from: www.stellaris-software.com.
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