Orbital Computational Dynamics

Kepler's 3 Laws Governing Orbits

Orbital computational dynamics concerns the calculation of orbits, i.e. the circulation of a large object around a point. The complexity of a system can make it too complex to solve all equations by analytical approach, so numerical methods implemented on modern computers may be required. There are several techniques in use.

As an introduction to the subject, this article is going to present arguably the oldest application of orbital mechanics; Kepler's laws governing the motion of planets. They were presented as early as 1609 [1] and 1618 [2] by German Mathematician and Astronomer Johannes Kepler, working out of his observatory in the Prague [3]. They were based on huge amounts of data from the observations carried out in collaboration with Danish astronomer Tycho Brahe [4].This challenged contemporary scientific consensus of the old Copernicus model of celestial objects moving in circles around the Sun, also called the heliocentric planetary model [5] .

The laws are formulated as follows:

The motion of the planets is an elliptical orbit with the Sun as one the focal points (See Figure 1).

Remarkably, the second focal point depicted in Figure 1 is not a body of any kind, just an empty point in space. The most influential consequence of the First law is that the distance between the Sun and the planets vary [7].  The vector between any planet and the Sun sweeps out equal areas in any intervals of equal time (Figure 2).

For this to hold, the planets must move faster when they are closer to the Sun compared to when they are at a longer distance [9] . It also means that the Winter half of the year from Winter solstice to Summer solstice is slightly shorter than the Summer half by approximately 3 days [10]. Naturally, the opposite holds for the Southern Hemisphere.

The square of the orbital period of a planet is proportional to the cube of the semimajor axes of its orbit [11]
A definition of the semimajor axis term can be seen in Figure 3.

The third law controls the length of the year for each of the planets in the solar system. The formula also includes mass and the Gravitational constant. There are quite a few interactive calculators available through the WWW, for example [13] Kepler’s laws were later confirmed and explained by more general laws of gravitation exposed by Isaac Newton.

References
1. Kepler, J: Astronomia nova. Prague 1609

2. Kepler, J: Harmonice Mundi . Prague 1618

3. Johannes Kepler. https://en.wikipedia.org/wiki/Johannes_Kepler Wikipedia

4. Supernovas: Making Astronomical History. Tycho and Kepler

5. Williams, M.: What is the HelioCentric Model of the Universe? Universe Today, 2016, http://www.universetoday.com/33113/heliocentricmodel/

6. Kepler’s Law of Planetary Motion https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

7. Dictionary: Johannes Kepler: The Laws of Planetary Motion. The Spaceguard Foundation and of the Spaceguard System, http://spaceguard.rm.iasf.cnr.it/NScience/neo/dictionary/kepler.htm

8. Fix, John D.: Astronomy: Journey to the Cosmic Frontier. McGrawHill, 1999

9. Kepler’s 2nd Law: The Speeds of Planets . Windows to the Universe, National Earth Science Teachers Association, http://www.windows2universe.org/the_universe/uts/kepler2.html

10. Stern, David P.: More on Kepler’s Second Law: The Ratio of Velocities at Perigee and Apogee. From Stargazes to Starships, http://www.phy6.org/stargaze/Skepl2A.htm

11. Kepler’s Third Law. Boundless Textbooks. https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/uniformcircularmotionandgravitation5/keplerslaws56/keplersthirdlaw26711197/

12. Johannes Kepler’s 3 Laws of Planetary Motion. Buzzle.com, http://www.buzzle.com/articles/johanneskeplers3lawsofplanetarymotion.html

13. Kepler’s 3rd Law Ultra Calculator. Software Systems, http://www.1728.org/kepler3a.htm

14. Newton, I.: Philosophiæ Naturalis Principa Mathematica. 1687

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