# Rate of planets flowing into stars: Discussion

 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