1. ON THE STRUCTURAL LAW OF
EXOPLANETARY SYSTEMS
Patricia Lara, Arcadio Poveda & Christine Allen
Instituto de Astronomía, UNAM
ICNAAM 2012
Kos, Greece
19-25 September 2012

2. Titius-Bode Relation
n
a 2
3
.
0
4
.
0 


donde n = -∞ (Mercury), 0 (Venus),1 (Earth), 2,…

3. Titius-Bode relation
Johann DanielTitius
(1729-1796)
Johann Elert Bode
(1747-1826)

5. Titius-Bode Relation (TBR)
Original TBR
where n = 0, 3, 6,12 ..
Classic TBR
(modern formulation)
where n = -∞ , 0, 1, 2, …
10
4


n
a n
a 2
3
.
0
4
.
0 


Exponential TBR
where n = 1, 2, 3, …
n
n k
P
P 
 0
Dermott´s law
where n = 0, 1, 2, …
bn
n
n e
a
C
a
a 0
0 



6. TBR in the Solar System
Sistema Solar
a = 0.1912e0.5594n
R
2
= 0.992
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8 9 10
n
a
(AU)
Poveda &Lara, 2008
Solar System

8. 55 Cancri
Distance
12.34 (± 0.4) pc
Spectral Type
K0IV-V
Mass
0.905 (± 0.015) M
Age
10.2 (± 2.5) Gyr
Radius
0.943 (± 0.01) R
Metallicity [Fe/H]
0.31 (± 0.04)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
55 Cnc e b c f d
Orbital period (days) 0.7365449
± 5e-06
14.651262
± 0.0008
44.3446
± 0.007
260.7
± 1.1
5218
± 230
Eccentricity 0.057 0.0159 0.053
a (AU) [measured] 0.0156 0.1148 0.2403 0.781 - 5.76 -
a (AU) [predicted] 0.0239 0.074 0.2292 0.7103 2.2011 6.8214 21.14
exoplanet.eu

9. υ Andromeda
Distance
13.47 (± 0.13) pc
Spectral Type
F8 V
Mass
1.27 (± 0.06) M
Age
3.8 (± 1) Gyr
Radius
1.631 (± 0.014) R
Metallicity [Fe/H]
0.09 (± 0.06)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
Ups And b c d e
Orbital period (days) 4.6171
± 4.7e-05
237.7
± 0.2
1302.61 3848.86
± 0.74
Eccentricity 0.013 0.24 0.274 0.00536
a (AU) [measured] 0.059 - 0.861 2.55 5.2456 -
a (AU) [predicted] 0.069 0.218 0.683 2.145 6.732 21.13

10. μ Ara
Distance
15.3 pc
Spectral Type
G3 IV-V
Mass
1.08 (± 0.05) M
Age
6.41 Gyr
Radius
1.245 (± 0.255) R
Metallicity [Fe/H]
0.28 (± 0.04)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
mu Ara c d b e
Orbital period (days)
9.6386
± 0.0015
310.55
± 0.83
643.25
± 0.9
4205.8
± 758.9
Eccentricity 0.172 0.0666 0.128 0.0985
a (AU) [measured] 0.09094 - 0.921 1.5 5.235 -
a (AU) [predicted] 0.098 0.263 0.704 1.885 5.046 13.507

11. Gliese 876
Distance
4.7 (± 0.01) pc
Spectral Type
M4 V
Mass
0.334 (± 0.03) M
Age
2.5 (+2.5
-2.4) Gyr
Radius
0.36 R
Metallicity [Fe/H]
0.05 (± 0.2)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
Gliese 876 d c b e
Orbital period (days) 1.93778
± 2.0e-05
30.0881
± 0.0082
61.1166
± 0.0086
124.26
± 0.7
Eccentricity 0.207 0.25591 0.0324 0.055
a (AU) [measured] 0.020807 - - 0.12959 0.208317 0.3343
a (AU) [predicted] 0.022 0.038 0.067 0.117 0.205 0.36 0.63

12. KOI-730
Apparent Magnitude V
15
Mass
1.07 M
Effective Temperature
5590 K
Radius
1.1 R
n = 1 n = 2 n = 3 n = 4 n = 5
KOI-730 e c b d
Orbital period
(days)
7.3831
± 4.0e-04
9.8499
± 3.0e-04
14.7903
± 0.0002
19.7216
± 0.0004
a (AU) [measured] 0.076 0.092 0.12 0.145 -
a (AU) [predicted] 0.075 0.094 0.117 0.146 0.182

13. HR 8799
Distance
39.4 (± 0.1) pc
Spectral Type
A5V
Mass
1.56 M
Age
0.06 (+0.1
-0.03) Gyr
Radius
1.5 (± 0.3) R
Metallicity [Fe/H]
-0.47
n = 1 n = 2 n = 3 n = 4 n = 5
HR 8799 e d c b
Orbital period (days) 18000 41 054 82 145 164 250
a (AU) [measured] 14.5 27 42.9 68 -
a (AU) [predicted] 15.214 25.333 42.183 70.239 116.957

14. Gliese 581
Distance
6.21 (± 0.1) pc
Spectral Type
M2.5 V
Mass
0.31 (± 0.02) M
Age
8 (+3
-1) Gyr
Radius
0.3 (± 0.01) R
Metallicity [Fe/H]
-0.135
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
Gliese 581 e b c d
Orbital period (days)
3.14945
± 1.7e-04
5.36865
± 9e-05
12.9182
± 0.0022
66.64
± 0.14
Eccentricity 0.32 0.031 0.07
a (AU) [measured] 0.028 0.041 0.073 - - 0.22 -
a (AU) [predicted] 0.029 0.043 0.066 0.099 0.15 0.227 0.344

15. Kepler-20
Distance
290 (± 30) pc
Spectral Type
G8
Mass
0.912 (± 0.035) M
Age
8.8 (+4.7
-2.7) Gyr
Radius
0.944 (+0.06
-0.095)R
Metallicity [Fe/H]
0.02 (± 0.04)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8
Kepler-20 b e c f d
Orbital period (days) 3.6961219
6.098493
± 6.5 e-05
10.85092
± 1.3 e-05
19.57706
± 0.00052
77.61185
Eccentricity < 0.32 - < 0.4 0.09 < 0.6
a (AU) [measured] 0.04537 0.0507 0.093 0.11 - - 0.3453 -
a (AU) [predicted] 0.042 0.059 0.083 0.118 0.168 0.238 0.337 0.477

16. Kepler-33
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
Kepler-33 b c d e f
Orbital period (days)
5.66793
± 0.00012
13.17562
± 1.4 e-04
21.77596
± 1.1 e-04
31.7844
± 0.00039
41.02902
± 0.0004
a (AU) [measured] 0.0677 - 0.1189 0.1662 0.2138 0.2535 -
a (AU) [predicted] 0.07 0.091 0.12 0.157 0.206 0.27 0.354
Apparent Magnitude V
14
Mass
1.291 (+0.063
-0.121) M
Effective Temperature
5904 K
Radius
1.82 (+0.14
-0.18) R

17. HD 10180
Distance
39.4 (± 1) pc
Spectral Type
G1V
Mass
1.06 (± 0.05) M
Age
4.3 (± 0.5) Gyr
Metallicity [Fe/H]
0.08 (± 0.01)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
HD 10180 c d e f g h
Orbital period (days) 5.75979
± 0.0001
16.3579
± 0.038
49.745
± 0.022
122.76
± 0.17
601.2
± 8.1
2222
± 91
Eccentricity 0 0.088 0.026 0.135 0.19 0.08
a (AU) [measured] 0.0641 0.1286 0.2699 0.4929 1.422 3.4 -
a (AU) [predicted] 0.058 0.128 0.282 0.621 1.369 3.019 6.655

18. Kepler-11
Spectral Type
G
Mass
0.95 (± 0.1) M
Age
8 (± 2) Gyr
Radius
1.1 (± 0.1) R
Metallicity [Fe/H]
0 (± 0.1)
n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8
Kepler-11 b c d e f g
Orbital period (days)
10.30375
± 0.00016
13.02502
± 8 e-05
22.68719
± 0.00021
31.9959
± 0.00028
46.68876
± 0.00074
118.37774
± 0.00112
Eccentricity 0 0 0 0 0 0
a (AU) [measured] 0.091 0.106 0.159 0.194 0.25 - 0.462 -
a (AU) [predicted] 0.087 0.114 0.15 0.198 0.26 0.342 0.449 0.591

19. In summary…
 Exoplanetary systems with 4 observed planets
(with vacancies and extrapolations)
υ And 1 vacancy
n = 2 a = 0.218 AU
n = 6 a = 21.13 AU
KOI-730 0 vacancies
n= 5 a = 0.182 AU
Gliese 581 2 vacancies
n = 4 a = 0.099 AU
n =5 a=0.15 AU**
n= 7 a = 0.63 AU
Gliese 876 2 vacancies
n = 2 a = 0.038 AU
n = 3 a = 0.067 AU
n = 7 a = 0.63 AU
μ Ara 1 vacancy
n = 2 a = 0.263 AU
n = 6 a = 13.507 AU
HR 8799 0 vacancies
n = 5 a = 116.957 AU

20.  Exoplanetary systems with
5 planets
55 Cancri 1 vacancy
n = 5 a= 2.20 AU
n = 7 a = 21.14 AU
Kepler-20 2 vacancies
n = 5 a = 0.168 AU
n = 6 a = 0.238 AU
n = 8 a = 0.477 AU
Kepler-33 1 vacancy
n = 2 a = 0.091 AU
n = 7 a = 0.354 AU
 Exoplanetary systems with
6 planets
HD 10180 0 vacancies
n = 7 a = 6.655 AU
Kepler-11 1 vacancy
n = 6 a = 0.342 AU
n = 8 a = 0.591 AU

21. Conclusions
 The results shown here demonstrate that all 11 exoplanetary
systems currently known to harbor four or more planets obey aTB-
like structural relation .
 The two parameters entering the exponential formulation used
here characterize two different aspects of the systems. a0 is
essentially a measure of the overall compactness of the planetary
system. b, on the other hand, characterizes the separation (in units
of a0) between successive planets in the system.
 We find no relation between the value of b and the intrinsic
characteristics (angular momentum, ratio between the stellar and
planetary mass, etc.) of the systems considered here.
 This seems to suggest that the origin of theTB relation is not
related to the formation mechanisms of the systems, but rather to
their subsequent dynamical evolution.

22. Thank you!

23. For once pay attention to the widths of the planets from each other
and notice that they are distant from each other almost in a
proportion as their bodily heights increase. Given the distance from
the Sun to Saturn as 100 units, then Mercury is distant 4 such units
from the Sun,Venus 4+3=7 of the same; the Earth 4+6=10; Mars
4+12=16. But see, from Mars to Jupiter there comes forth a
departure from this so exact progression.
From Mars follows a place of 4+24=28 such units, where at the
present neither a chief nor a neighboring planet is to be seen. And
shall the Builder have left this place empty? Never! Let us
confidently wager that, without doubt, this place belongs to the as
yet still undiscovered satellites of Mars; let us add that perhaps
Jupiter also has several around itself that until now have not been
seen with any glass. Above this, to us unrevelead, position ariases
Jupiter’s domain of 4+48=52; and Saturn’s at 4+96=100 units.What
a praiseworthy relation!

24. Titius-Bode Relation in the
Solar System

25. Titius-Bode Relation in
Exoplanetary Systems