Table Of Content

An Algorithm to Fit Orbital Distances in some Exoplanetary Systems by Patrick Bruskiewich Abstract In this short paper this simple algorithm will be used to model orbital distances for recently discovered exo-planets in the HR8799, 82 G. Eridani and the Tau Ceti exoplanetary systems. In a separate paper a simple three step technique is outlined to derive an algorithm that can describe the spacing of planetary orbits. It is a variation on the Titius-Bode Rule. Introduction A simple three step-technique can be used to derive an algorithm that can describe the spacing of planetary orbits. This technique is outlined in reference 1.This algorithm is given by 1 where the distance D of a planet from the star is in some standard unit of measure n that is appropriate for the planetary system in question. In terms of our solar system you may recognize the algorithm as a variation of the standard Titus-Bode Rule In a separate paper a modern version of the algorithm is derived of the form which leads to a total variation of 7.6 % that is considerably less than a half that of the variation of 19% for the standard Titus-Bode Rule. An Outsider’s Look at the Big Planets in our Solar System Imagine you are a great distance from our solar system and that you can only see the planets Jupiter, Saturn and Uranus. What kind of an algorithm would you develop to describe the planetary spacing in our solar system? Planet Measured Distance Algorithm Variation (A.U.) Distance 2 (A.U.) Jupiter 5.2 5.1 – 0.01961 Saturn 9.5 9.7 0.016985 Uranus 19.2 18.9 – 0.01511 Total Variation: – 0.01774 Table 1: An Algorithm Fit to Jupiter, Saturn and Uranus The algorithm used to fit this data is Can we use this fit to predict the presence of other planets in this system? Yes we can when we double down so to speak with the index n. Let is shift the index by four to read n + 4. To do this we have to take the pre-factor of 4.6 and divide it by sixteen so that our new algorithm becomes 3 From whence we can predict other planets to have the following orbital characteristics n+4 Planet Measured Algorithm Variation Distance Distance (A.U.) (A.U.) 2 Mars 1.52 1.65 0.085 3 Ceres 2.55 2.80 0.098 4 Jupiter 5.2 5.1 – 0.01961 5 Saturn 9.5 9.7 0.016985 6 Uranus 19.2 18.9 – 0.01511 Total Variation: 0.13 Table 2: A Double Down Algorithm Fit to Planetary Data This doubling down approach yields meaningful results. Once the outsiders start to see the smaller inner planets they can in turn revise their algorithm to resemble a closer fit, such as 4 A Study of Some Exoplanetary Systems As we have discovered exo-planetary systems it seems fitting to see if a modern Titius-Bode algorithm can be found to fit the measured distances from the stars. Let us begin by assuming studying stars that are sun like (a G type with a mass and radius comparable to that of our sun). There are a number of exoplanetary systems within 150 light years from our solar system that appear to have planetary spacing that can be described by such an algorithm. To date most of the Exoplanets found in other star systems have been larger than the planet Jupiter. Let us consider how we might find an algorithm for a few such exoplanetary systems. HR8799 In the HR8977 system planets have been found with the following characteristics: Planet Orbital Distance A.U. e 14.5 d 24 c 38 5 b 68 Table 3: Planetary Distances in the HR8799 Exoplanetary System An algorithm to fit this exoplanetary data is with the index n = 0, 1, 2 and 3. The variation is given in the following table. N Planet Orbital Algorithm Variation Distance A.U. Distance 0 e 14.5 15.3 0.055 1 d 24 22.8 – 0.05 2 c 38 37.8 – 0.005 3 b 68 67.8 – 0.003 Total – 0.003 Variation: Table 4: Algorithm Distances in the HR8799 Exoplanetary System 6 This provides for a fit with a small total variation. Let us double down with this planetary system and see if we can predict where other planets may perhaps lie in the HR8799 system: N Planet Orbital Algorithm Variation Distance A.U. Distance n.a. undiscovered n.a 7.8 n.a. 0 undiscovered n.a 9.7 n.a. 1 undiscovered n.a. 11.6 n.a 2 e 14.5 15.3 0.055 3 d 24 22.8 – 0.05 4 c 38 37.8 – 0.005 5 b 68 67.8 – 0.003 Total – 0.003 Variation: Table 5: Algorithm Distances in the HR8799 Exoplanetary System By doubling down we can predict the possible location of three yet to be discovered planets in the HR8799 system at 7.8 A.U., 9.7 A.U. and 11.6 A.U. If 7 the construct of this exoplanetary system is similar to that of our solar system, it is possible that a belt of planetoids may be found at 11.6 A.U. from the star in the HR8977 system. It is worth noting that this star has an unusual lack of metals. This may preclude terrestrial like planets within this exoplanetary system. The HR8799 exoplanetary system is 129 Light years from our solar system and has a young age of only 30 million years. The star in the system has a mass of about 1.5 solar masses and a luminosity around 5 times that of our sun. The first three exoplanets discovered in the HR8799 system were first observed in 2008 by a Canadian astrophysicist, Dr. Christain Marois, using the Keck and Gemini telescopes in Hawaii. 82 G. Eridani For the 82 G. Eridani system planets have been found with the following characteristics (we have excluded the inner and outer disk and two small planets) Planet Orbital Distance A.U. c (unconfirmed) 0.225 d 0.364 e 0.509 f (unconfirmed) 0.875 8 Table 6: Planetary Distances in the 82 G. Eridani Exoplanetary System An algorithm to fit this exoplanetary data is with the index n = 0, 1, 2 and 3. The variation is given in the following table. N Planet Orbital Algorithm Variation Distance A.U. Distance n.a c 0.225 0.2 – 0.11 0 d 0.364 0.366 0.005 1 e 0.509 0.532 0.045 2 f 0.875 0.864 – 0.012 Total – 0.073 Variation: Table 7: Algorithm Distances in the 82 G. Eridani Exoplanetary System 82 G. Eridani is a main sequence star with a mass that is 0.7 solar masses and a luminosity that is 0.74 that of our sun. It is 20 light years away from our solar 9 system. This exoplanetary system has a disk that is between 19 and 30 A.U. out from the star. Tau Ceti In the Tau Ceti exoplanetary system there are four confirmed planets. Planet Orbital Distance A.U. b 0.105 c 0.195 d 0.374 e 0.552 undiscovered 0.8 f 1.375 Table 8: Planetary Distances in the Tau Ceti Exoplanetary System We have included an undiscovered planet in this system for reasons that will become apparent given the following algorithm: 10