Approximate Positions of the Planets
Introduction
Lower accuracy formulae for planetary positions have a number of important applications when one doesn’t need the full accuracy of an integrated ephemeris. They are often used in observation scheduling, telescope pointing, and prediction of certain phenomena as well as in the planning and design of spacecraft missions.
Approximate positions of the planets may be found by using Keplerian formulae with their associated elements and rates. Such elements are not intended to represent any sort of mean; they are simply the result of being adjusted for a best fit. As such, it must be noted that the elements are not valid outside the given time-interval over which they were fit.
High precision ephemerides for the planets are available via the Horizons system.
Accuracy
The table below lists nominal errors in heliocentric longitude, λ, latitude, φ, and distance, ρ, using this approximation of planetary positions.
| 1800 AD — 2050 AD | 3000 BC — 3000 AD | |||||
| λ (arcsec) |
ϕ (arcsec) |
ρ (1000 km) |
λ (arcsec) |
ϕ (arcsec) |
ρ (1000 km) |
|
|---|---|---|---|---|---|---|
| Mercury | 15 | 1 | 1 | 20 | 15 | 1 |
| Venus | 20 | 1 | 4 | 40 | 30 | 8 |
| EM Bary† | 20 | 8 | 6 | 40 | 15 | 15 |
| Mars | 40 | 2 | 25 | 100 | 40 | 30 |
| Jupiter | 400 | 10 | 600 | 600 | 100 | 1000 |
| Saturn | 600 | 25 | 1500 | 1000 | 100 | 4000 |
| Uranus | 50 | 2 | 1000 | 2000 | 30 | 8000 |
| Neptune | 10 | 1 | 200 | 400 | 15 | 4000 |
† EM Bary = Earth/Moon Barycenter
Formulae for using the Keplerian elements
Keplerian elements given in the tables below are
| \( a_o, \dot a \) | semi-major axis [au, au/century] |
| \( e_o, \dot e \) | eccentricity |
| \( I_o, \dot I \) | inclination [degrees, degrees/century] |
| \( L_o, \dot L \) | mean longitude [degrees, degrees/century] |
| \( \varpi_o, \dot \varpi \) | longitude of perihelion [degrees, degrees/century] |
| \( \Omega_o, \dot \Omega \) | longitude of the ascending node [degrees, degrees/century] |
In order to obtain the coordinates of one of the planets at a given Julian ephemeris date, T\(_{\rm eph}\),
- Compute the value of each of that planet's six elements: \( a = a_o + \dot a {\rm T} \), etc., where T, the number of centuries past J2000.0, is T = (T\(_{\rm eph}-2451545.0)/36525\).
-
Compute the argument of perihelion, \(\omega\), and the mean anomaly, \(M\):
\[ \omega = \varpi - \Omega \ \ ; \ \ M = L \ - \ \varpi \ + \ b {\rm T}^2 \ + \ c \cos(f {\rm T}) \ + \ s \sin(f {\rm T}) \] - Modulus the mean anomaly so that \(-180^{\rm o} \leq M \leq +180^{\rm o}\) and then obtain the eccentric anomaly, \(E\), from the solution of Kepler's equation (see below): \[ M \ = \ E - e^{\ast} \sin E \] where \( e^{\ast} \ = \ 180/\pi \ e \ = \ 57.29578 \ e \).
- Compute the planet's heliocentric coordinates in its orbital plane, \({\bf r'}\), with the \(x'\)-axis aligned from the focus to the perihelion: \[ x' = a ( \cos E - e ) \quad ; \quad y' = a \sqrt{1 - e^2}\ \sin E \quad ; \quad z'=0. \]
- Compute the coordinates, \({\bf r}_{ecl}\), in the J2000 ecliptic plane, with the x-axis aligned toward the equinox: \[ {\bf r}_{ecl} \ = {\cal M} {\bf r'} \ \equiv \ {\cal R}_z (-\Omega) {\cal R}_x (-I) {\cal R}_z (-\omega) {\bf r'} \] so that \[ \matrix{ x_{ecl} & = & \ (\cos \omega \cos \Omega - \sin \omega \sin \Omega \cos I) & x' & + \ (- \sin \omega \cos \Omega - \cos \omega \sin \Omega \cos I) & y' \cr y_{ecl} & = & \ (\cos \omega \sin \Omega + \sin \omega \cos \Omega \cos I) & x' & + \ (- \sin \omega \sin \Omega + \cos \omega \cos \Omega \cos I) & y' \cr z_{ecl} & = & \ ( \sin \omega \sin I) & x' & + \ (\cos \omega \sin I) & y' \cr } \]
- If desired, obtain the equatorial coordinates in the "ICRF" or "J2000 frame", \({\bf r_{eq}}\): \[ \matrix{ x_{eq} & = & x_{ecl} \cr y_{eq} & = & & + \ \cos \varepsilon & y_{ecl} & \ - \ \sin \varepsilon & z_{ecl} \cr z_{eq} & = & & + \ \sin \varepsilon & y_{ecl} & \ + \ \cos \varepsilon & z_{ecl} \cr } \] where the obliquity at J2000 is \( \varepsilon = 23\rlap .{^{\rm o} 43928} \).
Solution of Kepler's Equation
\[ M \ = \ E - e^{\ast} \sin E \] Given the mean anomaly, \(M\), and the eccentricity, \(e^{\ast}\), both in degrees, start with†† \[ E_0 = M + e^{\ast} \sin M \] and iterate the following three equations, with \(n=0,1,2,...\), until \(|\Delta E| \leq tol\): \[ \Delta M = M - (E_n - e^{\ast} \sin E_n) \ \ ; \ \ \Delta E = \Delta M / (1 - e \cos E_n) \ \ ; \ \ E_{n+1} = E_n + \Delta E . \] For the approximate formulae in this present context, \(tol = 10^{-6}\) degrees is sufficient.
†† This starting guess ensures faster convergence, but \( E_0 = M \) or \( E_0 = M - e^{\ast} \sin M \) could also be used.
Keplerian Elements and Rates
Table 1
Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000, valid for the time-interval 1800 AD - 2050 AD.
a e I L long.peri. long.node.
au, au/Cy rad, rad/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy
-----------------------------------------------------------------------------------------------------------
Mercury 0.38709927 0.20563593 7.00497902 252.25032350 77.45779628 48.33076593
0.00000037 0.00001906 -0.00594749 149472.67411175 0.16047689 -0.12534081
Venus 0.72333566 0.00677672 3.39467605 181.97909950 131.60246718 76.67984255
0.00000390 -0.00004107 -0.00078890 58517.81538729 0.00268329 -0.27769418
EM Bary 1.00000261 0.01671123 -0.00001531 100.46457166 102.93768193 0.0
0.00000562 -0.00004392 -0.01294668 35999.37244981 0.32327364 0.0
Mars 1.52371034 0.09339410 1.84969142 -4.55343205 -23.94362959 49.55953891
0.00001847 0.00007882 -0.00813131 19140.30268499 0.44441088 -0.29257343
Jupiter 5.20288700 0.04838624 1.30439695 34.39644051 14.72847983 100.47390909
-0.00011607 -0.00013253 -0.00183714 3034.74612775 0.21252668 0.20469106
Saturn 9.53667594 0.05386179 2.48599187 49.95424423 92.59887831 113.66242448
-0.00125060 -0.00050991 0.00193609 1222.49362201 -0.41897216 -0.28867794
Uranus 19.18916464 0.04725744 0.77263783 313.23810451 170.95427630 74.01692503
-0.00196176 -0.00004397 -0.00242939 428.48202785 0.40805281 0.04240589
Neptune 30.06992276 0.00859048 1.77004347 -55.12002969 44.96476227 131.78422574
0.00026291 0.00005105 0.00035372 218.45945325 -0.32241464 -0.00508664
------------------------------------------------------------------------------------------------------
EM Bary = Earth/Moon Barycenter
Table 2a
Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000, valid for the time-interval 3000 BC -- 3000 AD. NOTE: the computation of M for Jupiter through Neptune *must* be augmented by the additional terms given in Table 2b (below).
a e I L long.peri. long.node.
au, au/Cy rad, rad/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy deg, deg/Cy
------------------------------------------------------------------------------------------------------
Mercury 0.38709843 0.20563661 7.00559432 252.25166724 77.45771895 48.33961819
0.00000000 0.00002123 -0.00590158 149472.67486623 0.15940013 -0.12214182
Venus 0.72332102 0.00676399 3.39777545 181.97970850 131.76755713 76.67261496
-0.00000026 -0.00005107 0.00043494 58517.81560260 0.05679648 -0.27274174
EM Bary 1.00000018 0.01673163 -0.00054346 100.46691572 102.93005885 -5.11260389
-0.00000003 -0.00003661 -0.01337178 35999.37306329 0.31795260 -0.24123856
Mars 1.52371243 0.09336511 1.85181869 -4.56813164 -23.91744784 49.71320984
0.00000097 0.00009149 -0.00724757 19140.29934243 0.45223625 -0.26852431
Jupiter 5.20248019 0.04853590 1.29861416 34.33479152 14.27495244 100.29282654
-0.00002864 0.00018026 -0.00322699 3034.90371757 0.18199196 0.13024619
Saturn 9.54149883 0.05550825 2.49424102 50.07571329 92.86136063 113.63998702
-0.00003065 -0.00032044 0.00451969 1222.11494724 0.54179478 -0.25015002
Uranus 19.18797948 0.04685740 0.77298127 314.20276625 172.43404441 73.96250215
-0.00020455 -0.00001550 -0.00180155 428.49512595 0.09266985 0.05739699
Neptune 30.06952752 0.00895439 1.77005520 304.22289287 46.68158724 131.78635853
0.00006447 0.00000818 0.00022400 218.46515314 0.01009938 -0.00606302
------------------------------------------------------------------------------------------------------
EM Bary = Earth/Moon Barycenter
Table 2b
Additional terms which must be added to the computation of M for Jupiter through Neptune, 3000 BC to 3000 AD, as described in the related document.
b c s f
---------------------------------------------------------------
Jupiter -0.00012452 0.06064060 -0.35635438 38.35125000
Saturn 0.00025899 -0.13434469 0.87320147 38.35125000
Uranus 0.00058331 -0.97731848 0.17689245 7.67025000
Neptune -0.00041348 0.68346318 -0.10162547 7.67025000
---------------------------------------------------------------
Reference
This content is from an article written by E.M. Standish and J.G. Williams in 1992. It has been published here with permission from the author and reformatted for optimal web-based presentation. The former planet Pluto has also been removed.
