Distribution of mass of the protoplanetary disk HL Tau/Distribucion de Masa del Disco Protoplanetario de HL Tau.
In 2014 the ALMA (Atacama Large Millimeter/Submillimeter Array) team showed a picture (Figure 1) of the star HL Tau (J2000 04:31:38.45 18:13:9.0), product of continuous emission in the band 6 (211-275 GHz).
In the image several concentric rings appear: rings of gas and dust, emitting radiation which are separated by a gap (dark area), which is due to the existence of a planet, massive enough, to capture the matter found around .
But there are difficulties with this interpretation, since the existence of gaps involves orbital resonance phenomena, which lead to stable orbits, for example, it suffices that two planets are at a greater distance to 3.5 RH (Hill's radii)  without however, the gaps are too close to each other, which implies interaction of planets each other, generating an orbital instability which would destroy the observed gaps.
An attempt was made to explain the existence of stable orbits for planets very close, with the help of the protoplanetary disk , but without the orbital migration process eventually the system becomes unstable; the simulations do not show how this situation could occur. Additionally, there should be no planetary formation processes so advanced, because the star is very young, implying that all models of planetary formation are wrong.
In most models it is considered that the protoplanetary disk is formed by a distribution of continuous material, which as it rotates around the star, acquires dynamics (stable and / or turbulent), so that originates planets through a process of accretion of planetesimals.
It is proposed that the protoplanetary disk has a defined internal structure, which gives rise to rings and gaps observed in HL-Tau system and other protoplanetary systems The internal structure originates from the existence of resonance phenomena orbital of the system, leading to a series of overlapping concentric toroids, which shapes the disc. This paper shows how to decompose the protoplanetary disk in this series of toroids, explain the observed distribution of matter, and how it gives rise to a stable planetary system.
2. Protoplanetary disk model
The disc matter distribution along the radius 0 < r [less than or equal to] Ro (AU), gives rise to the ring structure by a superposition of [R.sub.n](r)[THETA]([theta]) basis functions given by:
[rho](r, [theta]) = [summation over (n)][m.sub.n][R.sub.n](r)[THETA]([theta]) (1)
where n is the number of stable and closed orbits, and mn the mass contained in each element. The radial distribution of matter is described in ,
[mathematical expression not reproducible] (2)
being [a.sub.s] = k[a.sub.[??]], a parameter. k = [M.sub.s]/[M.sub. [??]] = 0,55  and [a.sub.[??]] = 0,0292705  see Figure 2(a). The zenithal distribution, Figure 2(b), is:
[[THETA].sub.n]([theta]) = 2[pi][[absolute value of [Y.sup.n-1.sub.n-1]([theta], [phi])].sup.2]sin[theta]. (3)
2.1. Photographic treatment
The estimated radius of protoplanetary disk of HL Tau, the image from ALMA (ESO/NAOJ/NRAO) is [R.sub.0] =117.5 AU with a total mass [M.sub.0] = 0.135 [M.sub.[??]]. Initially image of 1800 x 1800 pixels is transformed to grayscale; 0[degrees] [less than or equal to] [phi] < 360[degrees] sweep was made.
The major and minor semi-axes of the ellipse were determined by the minimum points, which are [phi] =45.26[degrees], 134.65[degrees], 225.17[degrees], 315,13[degrees]; it is known that the separation should be n = (1,3,5,7) times 45, by [phi]/n ratio obtain the position angle photography: 45.04898 [+ or -] 0.07806, giving rise to an image of 2546 x 2546 pixels. Using the expression for ellipticity cos([phi])=b/a, by minimum sweep the average inclination angle of [phi] = 46.82687[degrees] established; finally an image of 1622 x 1617 pixels, obtained (Figure 3.)
In the image a sweep of 0[degrees] [less than or equal to] [phi] < 360[degrees] is made, for each 0 < R [less than or equal to] [R.sub.0] (AU), obtaining a profile of luminosity-distance, Figure 4.
To suppress the effect of brightening the gas due to the presence of the star, a brightness-density conversion is made by the relationship [rho] ~ [L.sup.3,5] - exp(-r/12.84842) + 0.036446, which depends on the variation of temperature T ~ [r.sup.-0.43], obtaining the distribution of matter, [rho](r, [pi]/2) given by equation (1); the area under the curve corresponds to the mass of the disk, Figure 5.
An adjustment is made by least squares to obtain the terms of mass [m.sub.n] in equation (1). The red line corresponds to the setting and the masses obtained a computer simulation using the algorithm of Barnes-hut with the integrator IAS15 of rebound  develops, is that the orbits are stable for extremely long time, in the Figure 6 orbits and the Hill's radius of objects shown. The main objects are at 17.07, 20.77, 41.76, 75.98, 101.33 with mass [greater than or equal to] 1 [M.sub.J].
The observed gaps in the protoplanetary disk HL-Tau are not due to the existence of planets but is an effect of the distribution of matter in the disk, in this model the disc has a discontinuous structure as a series of concentric toroids, rather than a matter continuous distribution. The orbits of the objects obtained are stable.
We thank the Direccion de Investigaciones (DIN) of the Universidad Pedagogica y Tecnologica de Colombia (UPTC).
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Nidia Yiseth Buitrago Carreno (1), Nicanor Poveda Tejada (1) *, Nelson Vera-Villamizar (1)
(1) Grupo de Astrofisica y Cosmologia, Universidad Pedagogica y Tecnologica de Colombia, Tunja, Colombia
Received: 24 Nov 2015
Accepted: 26 Feb 2016
Available Online: 29 Feb 2016
* Corresponding Author.
Caption: Figure 1 Protoplanetary disk around the star HL-Tau (ALMA)
Caption: Figure 2 (a) Radial distribution of matter and (b) zenithal distribution.
Caption: Figure 3 Protoplanetary disk projected image.
Caption: Figure 4 Profile normalized light-distance.
Caption: Figure 5: Distance density profile.
Caption: Figure 6: Orbits and the Hill's radii of the objects