# 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.

λ(arcsec) ϕ(arcsec) ρ(1000 km) λ(arcsec) ϕ(arcsec) ρ(1000 km) 1800 AD — 2050 AD 3000 BC — 3000 AD 15 1 1 20 15 1 20 1 4 40 30 8 20 8 6 40 15 15 40 2 25 100 40 30 400 10 600 600 100 1000 600 25 1500 1000 100 4000 50 2 1000 2000 30 8000 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$$_{\rm eph}$$,

1. 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$$_{\rm eph}-2451545.0)/36525$$.
2. Compute the argument of perihelion, $$\omega$$, and the mean anomaly, $$M$$:
$\omega = \varpi - \Omega \ \ ; \ \ M = L \ - \ \varpi \ + \ b {\rm T}^2 \ + \ c \cos(f {\rm T}) \ + \ s \sin(f {\rm T})$
3. 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 $$e^{\ast} \ = \ 180/\pi \ e \ = \ 57.29578 \ e$$.
4. Compute the planet's heliocentric coordinates in its orbital plane, $${\bf r'}$$, with the $$x'$$-axis aligned from the focus to the perihelion: $x' = a ( \cos E - e ) \quad ; \quad y' = a \sqrt{1 - e^2}\ \sin E \quad ; \quad z'=0.$
5. 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 $\matrix{ x_{ecl} & = & \ (\cos \omega \cos \Omega - \sin \omega \sin \Omega \cos I) & x' & + \ (- \sin \omega \cos \Omega - \cos \omega \sin \Omega \cos I) & y' \cr y_{ecl} & = & \ (\cos \omega \sin \Omega + \sin \omega \cos \Omega \cos I) & x' & + \ (- \sin \omega \sin \Omega + \cos \omega \cos \Omega \cos I) & y' \cr z_{ecl} & = & \ ( \sin \omega \sin I) & x' & + \ (\cos \omega \sin I) & y' \cr }$
6. 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.