# Rate of planets flowing into stars: Discussion

by astrostuart
Tags: astronomy research, distribution, flow rate, flowing, planet migration, planets, rate, stars
 P: 9 I just posted a question to ask for help on the Calculus & Analysis section to determine how fast planets go into stars. I should give a little background, in addition to pointing you to my latest two short papers: http://arxiv.org/abs/1301.4229 and http://arxiv.org/abs/1211.1984 I have already done this problem numerically in those two papers, but for the full Journal paper, I want to show as much of this as I can analytically. I will be able to do my calculations much faster with analytical equations as well. I am taking the distribution of planets found using Kepler data, and using the equations for tidal migration, calculating the rate of planets migrating into the star as a function of the tidal dissipation value Q. I am also looking how the occurrence distribution changes as a function of time. There are several more interesting questions I'll be pursuing, including a correlation between the Fe in the star and the eccentricity of the planet. I'm hoping to get more discussion. Hope you head over to the other post. Thanks! Stuart Hong Kong and Sedona, Arizona
 P: 9 I have a tentative solution to my question, how fast are planets migrating inwards, posted at http://www.physicsforums.com/showthr...04#post4240604 I've posted my math in the calculus section, because it is hardcore calculus help that I want, but I sure would be grateful if some of you wanting to do some theoretical exoplanet astrophysics headed on over and looked at my math. I "end" with still more to do. Thanks Stuart
 Astronomy Sci Advisor PF Gold P: 22,674 I'm retired and not able to help any substantive way. I was impressed by your writing style: clear cogent, well organized. If I had a friend I was in touch with in the appropriate astrophysics department I could honestly recommend that he or she check out your papers. But besides encouragement and wishing you the best of luck there is little I can do. It's great you are attending all these conferences and presenting your work.
 P: 9 Eureka! I found it! The rate of migration of planets into the star Posted 2014 Mar 05 Here is the first public presentation of the equation giving the rate of planets going into the star!: $$\frac{dN}{dt}= - \frac{27}{4} \frac{ \left( 2 \pi \right)^{13/3} R^5_{\ast} M_p}{G^{5/3} M_{\ast}^{8/3} Q^{\prime}_{\ast}} k P^{\alpha - 13/3}$$ Here, $M_{\ast}$ and $M_p$, and $R_{\ast}$ and $R_{p}$ are the masses and radii of the star and planet, respectively, and $Q^{\prime}_{\ast}$ is a measure of the strength of the dissipation of energy by tides on the star called the tidal quality factor.'' The distribution of planets is determined by $k$ and $α$ in the expression describing the exponential distribution of planets in the falloff, or, separately, the distribution beyond the falloff: $$\frac{dN}{d\log[P]}= k P^{\alpha }$$ so that α is the power index of the planet distribution, and k the normalization. I have simplified this distribution from the combined form I used in previous posts. The first important point is how the dependence of the rate on the period P drops out when α equals 13/3, which means that if the power index is found to be 13/3, then this indicates that the distribution is shaped by planets tidally migrating into the star. The power index found by Howard et al. (2012) for giant and medium radii planets, as found by Kepler, in the falloff region is in fact a little above 4.0 (Using the approximation in the Howard equation for the distribution which has two regions each following a power law, that for small P, α = β+γ). The other important result is that a calculation of this rate using the Howard+ (2012) results is an inward flow of less than $10^{-12}$ giant planets/star/yr when calculated for $Q^{\prime}_{\ast}$ of $10^{7}$, or less than one planet per 1000 stars per gigayear. This means that in the 10 Gyr lifetime of a star like the sun, that supplying this migration flow would take a 1% reduction in the occurrence of giant planets further out. Not only is this easily sustainable, but so would be the larger flows required by $Q^{\prime}_{\ast}$ of even as strong as $10^{6}$. (Stronger tidal dissipation is indicated by a lower number.) I conclude that the planets in the falloff region have not been there since the formation of the planets, but in fact are part of planet migration. There is no need to say that the presence of these planets indicates that tides in the star must be as weak as $Q^{\prime}_{\ast}$ of $10^{7}$. This bears on many mysteries about the shortest period planets. The planets in the falloff may also more newly arrived than planets beyond the falloff, which means we should look to see if such planets are more likely to be inflated. This also bears on whether the correlations of iron abundance with planet and star parameters could be due to planet pollution of the stars. The most massive giant planets will produce transient events, and the majority of these planets will likely produce bloatar'' type objects such as found by Spezzi et al. This would be the first measurement of the rate of planet migration. I am writing this up for my blog (astrostuart.blogspot.com, give me a day or so) and will submit my paper, but for the paper would enjoy discussing this as a means of having an audience to help me write the paper. Stuart F. Taylor