A closely packed system of low-mass, low-density planets transiting Kepler-11

NASA has published the first list of exoplanets found by Kepler satellite. In particular, the NASA team led by Jack Lissauer reports a discovery of a system of six closely packed planets (see the article in Nature) around a Sunlike star christened as Kepler-11_a located in the direction of constellation Cygnus at distance of about 2000 light years. The basic data about the six planets Kepler-11_i, i=b,c,d,e,f,g and star Kepler-11_a can be found in Wikipedia. Below I will refer to the star by Kepler-11 and planets with label i=b,c,d,e,f,g.

Lissauer regards it as quite possible that there are further planets at larger distances. The fact that the radius of planet g is only .462AU together with what we know about solar system suggests that this could be be the case. This leaves door for Earth like planet.

The conclusions from the basic data

Let us list the basic data.

1. The radius and mass and surface temperature of Kepler-11 are very near to those of Sun.

2. The orbital radii using AU as unit are given by
   (.091,.106,.159,.194,.250,.462).
   The orbital radii can be deduced quite accurately from the orbital periods by using Kepler's law stating that the squares of periods are proportional to cubes of orbital radii.The orbital periods of the five inner planets are between 10 and 47 days whereas g has a longer period of 118.37774 days (note the amazing accuracy). The orbital radii of e and f are .194 AU and .250 AU so that the temperature is expected to be much higher than at Earth so that life as we know it is not expected to be there. The average temperature of the radiation from Kepler-11 scaling as 1/r² would be 4 times the temperature at Earth. The fact that gas forms a considerable fraction of the planet's mass could however mean that this does not give a good estimate for the temperature of the planet.

3. The mass estimates using Earth mass as unit are
   (4.3,13.5,6.1,8.4,2.3 , <300).
   There are considerable uncertainties involved here, of order factor of 1/2.

4. The estimates for the radii of the planets using the radius of Earth as unit are
   (1.97, 3.15,3.43,4.52,2.61,3.66).
   The uncertainties are about 20 per cent.

5. From the estimates for the radii and mass estimates one can conclude that the estimates for the densities of the planets are considerably lower than those for Earth. Density of (e,f) is about (1/8,1/4) of that for Earth. The surface gravitation for e and f is roughly 1/2 of that at Earth. For g it is same as for Earth if g has mass roughly m≈ 15. For planet g only an upper bound 300 so that one can only that surface gravity is weaker than 20g.

The basic conclusions are following. One cannot exclude the possibility that the planetary system could contain Earth like planets. Furthermore, the distribution of the orbital radii of the planets differs dramatically from that in solar system.

How to understand the tight packing of the inner planets?

The striking aspect of the planetary system is how tightly packed it is. The ratio for the radii of g and b is about 5. This is a real puzzle for model builders with me included. TGD suggests three phenomenological approaches.

1. Titius-Bode law
   r(n)=r₀+ r₁2ⁿ
   is supported by p-adic length scale hypothesis. Stars would have onion-like structure consisting of spherical shells with inner and outer radii of the shell differing by factor two. The formation of planetary system involves condensation of matter to planets at these spherical shells. The preferred extremals of Kähler action describing stationary axially symmetric system corresponds to spherical shells containing most of the matter. A rough model for star would be in terms of this kind of spherical shells defined an onion-like structure defining a hierarchy of space-time sheets topologically condensed on each other. The value of the parameter r₀ could be vanishing in the initial situation but subsequent gravitational dynamics could make it positive reducing the ratio r(n)/r(n-1) from its value 2.

2. Bohr orbitology suggested by the proposal that gravitonic space-time sheets assigned with a given planet-star pair correspond to a gigantic value of gravitational Planck constant given by
   hbar_gr= GMm/v₀,
   where v₀ has dimensions of velocity and actually equal to the orbital velocity for the lowest Bohr orbit. For inner planets in solar system one has v₀/c≈ 2^-11.

   The physical picture is visible matter concentrates around dark matter and in this matter makes it astroscopic quantum behavior visible. The model is extremely predictive since the spectrum of orbital radii would depend only on the mass of the star and planetary systems would be much like atoms with obvious implications for the probability of Earth like systems supporting life. This model is consistent with the Titius-Bode model only if the Bohr orbitology is a late-comer in the planetary evolution.

3. The third model is based on same general assumptions as the second one but only assumes that dark matter in astrophysical length scales associated with anyonic 2-surfaces (with light-like orbits in induced metric in accordance with holography) characterized by the value of the gravitational Planck constant. In this case the hydrogen atom inspired Bohr orbitology is just the first guess and cannot be taken too seriously. What would be important would be genuinely quantal dynamics for the formation of planetary system.

Can one interpret the radii in this framework in any reasonable manner?

1. Titius-Bode predicts
   [r(n)-r(n-1)]/[r(n-1)-r(n-2)]=2
   and works excellently for c, f, and g. For b, d and e the law fails. This suggests that the four inner planets a,b,c,d, whose radii span single 2-adic octave in good approximation (!) correspond to single system which has split from single plane or will fuse to single planet distant future.

2. Hydrogenic Bohr orbitology works only if g corresponds to n=2 orbit. n=1 orbit would have radius .116AU. From the proportionality r ∝ hbar_gr² ∝ 1/v₀², one obtains that the value one must have

   R==v₀²(Kepler)/v₀²(Sun)=3.04.

   This would result in a reasonable approximation for v₀(Kepler)/v₀(Sun)=7/4 (note that the value of Planck constant are predicted to be integer multiples of the standard value) giving R=7/4² ≈ 3.06.

   Note that the planets would correspond to those missing in Earth-Sun system for which one has n=3,4,5 for the inner planets Mercury, Venus, Earth.

   One could argue that Bohr orbits result as the planets fuse to two planets at these radii. This picture is not consistent with Titius-Bode law which predicts three planets in the final situation unless n=2 planet remains unrealized. By looking the graphical representation of the orbital radii of the planet system one has tendency to say that b,c,d,e, and f form a single subsystem and could eventually collapse to single planet. The ratio of gravitational forces between g and f is larger than that between f and e for m(g) > 6m_E so that one can ask whether f could be eventually caught be g in this case. Also the fact that one has r(g)/r(f)<2 mildly suggests this.