MATHEMATICAL FORMULA FOR THE FINDING OF RELATIVE
DISTANCES OF ORBITS OF PLANETS AND SATELLITES.
GENERALIZATION OF THE LAW OF TITIUS-BODE
Polychronis Karagkiozidis
M.Sc. Chemist – Amateur Astronomer
A study of the distances of various large satellites from their parent planets showed
that these are subject to the same remarkable harmony that applies to the distances of
most of the planets from the sun.
I came to this conclusion while searching for this harmony for the solar system’s
major satellites. I decided to include in each case an extremely small satellite as the first
member of the series, since in the Titius-Bode law the first planet, Mercury, is very small
in comparison with the gas giants.
For five satellites of Jupiter, including the four Galilean moons:
Take the geometrical series 21 42 84 168, in which each term is double the
previous one. With the addition of 0 as the first term, we have the series 0 21 42 84
168. Adding 22 to each term produces a third series: 22 43 64 106 190. If we multiply
these terms by 10,000, we arrive at the distances in kilometers of Jupiter’s four Galilean
moons and the small satellite Thebe from the planet’s centre.
220,000
Thebe
430,000
Io
640,000
Europa
1,060,000
Ganymede
1,900,000
Callisto
For five satellites of Uranus:
Take the series 46 92 184 368, in which each term is double the previous one.
With the addition of 0 as the first term, we have the series 0 46 92 184 368. Adding
86 to each term produces a third series: 86 132 178 270 454. If we multiply these
numbers by 1,000, we arrive at the distances in kilometers of the following five satellites
from the centre of Uranus:
86,000
Puck
132,000
Miranda
178,000
Ariel
270,000
Umbriel
454,000
Titania
The above appear to be an adaptation of the well-known Titius-Bode law to
these satellites. This is achieved by identifying the suitable small satellite whose
position defines the two constants of the formula that will be given below.
The Titius-Bode law is as follows:
Take the geometrical series 3 6 12 24 48 96 192, in which each term is double
the previous one. With the addition of 0 as the first term, we have the series 0 3 6 12
24 48 96 192. Adding 4 to each term produces a third series: 4 7 10 16 28 52 100
196. If we divide these numbers by 10, we arrive at the distances of most of the planets
from the Sun as expressed in astronomical units.
0.4
Mercury
0.7
Venus
1.0
Earth
1.6
Mars
2.8
Ceres
5.2
Jupiter
10.0
Saturn
19.6
Uranus
The general formula giving the distances of planets and satellites, which I first
presented at the Fourth Panhellenic Amateur Astronomy Conference, is the following:
D=λ2n +κ (formula 1) Polychronis Karagkiozidis
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For a planet’s satellite system, κ corresponds to the (orbit of the) first satellite’s
distance from the planet’s centre, and λ to the distance between the second satellite and
the first, while n is the number of the satellite in the series, beginning with 0.
Similarly for the solar system, κ corresponds to the distance of the first planet,
Mercury, from the Sun, and λ to the distance between the second and first planets orbits
(Venus − Mercury), while n is the number of the planet in the series, beginning with −∞.
For the above Jovian satellites we have:
κ=220,000 km (Thebe) and λ=210,000 km (Thebe − Io distance), which give:
D=210,000x2n+220,000 (equation 1)
Satellite
Io
Europa
Ganymede
Callisto
n
0
1
2
3
D (km)
430,000
640,000
1,060,000
1,900,000
True distance (km)
421,600
670,900
1,070,000
1,883,000
Deviation %
-1.99
4.61
0.93
-0.90
For the above satellites of Uranus we have :
Κ=86,000 km (Puck) and λ=46,000 km (Puck − Miranda distance), which give:
D=46,000x2n+86,000 (equation 2)
Satellite
Miranda
Ariel
Umbriel
Titania
n
0
1
2
3
D (km)
132,000
178,000
270,000
454,000
True distance (km)
129,780
191,240
265,970
435,840
Deviation %
-1.71
6.92
-1.52
4.17
For the solar system we have:
κ=0.4 AU (Mercury’s distance from the Sun) and λ=0.3 AU (Mercury − Venus distance’s
orbit), which give:
D=0.3x2n+0.4 (equation 3)
Planet
Venus
Earth
Mars
Ceres
Jupiter
Saturn
Uranus
n
0
1
2
3
4
5
6
D (AU)
0.7
1
1.6
2.8
5.2
10
19.6
True distance (AU)
0.72
1
1.52
2.76
5.2
9.6
19.2
Deviation %
2.87
0.00
-5.26
-1.45
0.00
-4.17
-2.08
Ceres is the largest of the asteroids or minor planets, which orbit between Mars and
Jupiter.
REFERENCES
Many authors and researchers have studied the Titius-Bode law. An indication of this
is that up to April 2005 on the Google search engine there were 9,200 references to this
law.
In many of these the law is given as d=0.4+0.3x2n (like equation 3 in the present
work), from which for n= −∞,0,1,2,3,… we derive the planets’ distances from the Sun in
astronomical units.
Here we should note that the interval from −∞ to 0 is infinite, and the harmony only
exists from 0 onwards, as from that point onward the difference is always one unit.
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Concerning the satellites of giant planets, different rules are mentioned for each
case, as on the site http://floridastars.org/9605cohe.html, which mentions the following:
Take the series 1 2 3 8 and add 1 to create a new series 2 3 5 9. If we divide
these numbers by 2 we obtain the numbers 1, 1.5, 2.5 and 4.5, which correspond to the
relative distances of the four Galilean moons from Jupiter.
For the larger satellites of Uranus, the same site gives the rule:
Take the series 1 2 3 6 8 and add 1 to create a new series 2 3 4 7 9. If we
divide these numbers by 2 we obtain the numbers 1, 1.5, 2, 3.5 and 4.5, which
correspond to the relative distances of the four large Uranian satellites from their parent
planet.
CONCLUSION – ADVANTAGE OF THE FORMULA D=λ2n+κ
What is new about the present work is that the formula D=0.3x2n+0.4 is generalised
by introducing two factors in place of the constants 0.3 and 0.4. As a result the formula
covers not only the solar system but also five Jovian satellites, including the four Galilean
moons, and five satellites of Uranus.
At first glance this formula appears to fit the Solar System less well than the classic
Titius-Bode law, as it does not include Mercury. The Sun − Mercury distance is, however,
included in the formula as the constant κ.
REFERENCES
1. Κωνσταντίνου Μαυρομάτη, Λεξικό Αστρονομίας, Εκδώσεις «Ώρες» Βόλος 2001. (Νόμος
Bode Titius σελ271)
2. Κωνσταντίνου Γαβρίλη, Μαργαρίτας Μεταξά, Παναγιώτη Νιάρχου και Κωνσταντίνου
Παπαμηχάλη. Στοιχεία Αστρονομίας και Διαστημικής. Σχολικό βιβλίο Β’ Λυκείου 1999.
(αποστάσεις δορυφόρων σελ. 62 63)
3. Πρακτικά 4ου Πανελληνίου Συνεδρίου Ερασιτεχνικής Αστρονομίας. (σελ 57-62)
4. Ian Ridpath, Dictionary of Astronomy, New York, Oxford University Press, 1997.
From the internet
5. http://www.daviddarling.info/encyclopedia/T/Titius-Bode_Law.html
6. http://www.anaconda-2.net/g_m/L001.html
7. http://steph.mathis.free.fr/curtitius.html
8. http://www.yorku.ca/sasit/sts/nats1800/lecture11a.html
9. http://www.mira.org/fts0/planets/091/text/txt001x.htm
10. http://almaak.tripod.com/temas/titius_bode_law.htm
11. http://encyclopedia.lockergnome.com/s/b/Titius-Bode_law
12. http://www.floridastars.org/9605cohe.html
Polychronis Karagkiozidis
M.Sc. Chemist
Address: Akropoleos 49
54634 Thessalonica Greece
E- mail: info@polkarag.gr
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