[Physics] find the list of the planetary motion equations

astronomycelestial-mechanicskinematicsorbital-motionsolar system

The planetary motion equations are written in the ellipse equation format, i.e. $$x^2/a + y^2/b = 1.$$

Can anyone please tell me where I can find the list of all the planetary motion equations, (Mercury, Venus, Mars, Earth, and so on) in the form of $$(x(t)-c)^2/a + (y(t)-d)^2/b = 1~?$$ (Both $x$ and $y$ are functions of $t$.)

Best Answer

Orbits are pretty complicated. Most texts on this deal in terms of predicting positions at a specific time rather than just a simple ellipse, because that model while correct is too basic. As others mentioned position estimation makes it more complex because there are more parameters involved. I'll just pull some stuff for you on the basics. For standard Keplerian orbits you'll need the orbital elements:

  • Semi-Major Axis, $a$
  • Eccentricity, $e$
  • Inclination, $i$
  • Argument of Periapsis, $\omega$
  • Time of Periapsis Passage, $T$
  • Longitude of Ascending Node, $\Omega$

This image is Earth-centric like for a satellite, but it illustrates the basics. elements

Here is an overview and data from JPL good to 2050ad:

               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
Pluto    39.48211675      0.24882730     17.14001206      238.92903833    224.06891629    110.30393684
         -0.00031596      0.00005170      0.00004818      145.20780515     -0.04062942     -0.01183482

The method is a bit long to duplicate here. So I'll link to JPL's illustration on how to compute the planet's position based on the date and the orbital elements.

And for more reading about orbits you might like this article.

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