L10 From Planetesimals to protoplanets

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August 3, 2011 Analytical solutions of the master equation August August3, 3, 2011 2011 1 Task A 1 Task A For orderly growth, and for runaway, analytical solutions to the growth equation can be found. We will work using the radius R, instead of the mass M, as the radius directly enters in the cross section. 1.1We No assume focussing, that planets ΣP constant are spherical and have a constant density. Then we can always In convert this problem, radius as into well mass as in all and other vice ones, versa. we will work using theradiusR, instead of the mass M, astheradiusdirectlyentersinthecrosssection.We 1) Orderly growth assume that (Fg=1), planets constant are spherical planetesimal and have asurface constantdensity density. Then we can The constant always surface convert density radiusapproximation into mass and vice should versa. be fine as long as M
3 Task C Analytical solutions II 2) Orderly growth (Fg=1), decreasing planetesimal surface density The accretion of the planet from a feeding zone of with BRH leads to a The accretion decreaseof of the theplanet surface from density a feeding as zone of width B RH leads to a decrease of the surface density as ΣP (t) =Σ0 − M(t) (10) 2πaBRH where Σ0 is the initial planetesimal surface density. We can express RH in where Σ0 terms is the of initial the planetesimal radius of the planet, surface density. 2 We can express RH in terms of the radius of the planet, 1/3 4πρ RH = Ra. (11) 9M∗ With these equations, we can express ΣP (t) asafunctionofR(t), and find ΣP (R) =Σ0 − k3R2 = k2 − k3R2 . 3.1 No focussing, ΣP variable Plugging ΣP (R) backinourmasterequationfordR(t)/dt, wenowgeta differential equation dR dt = k1(k2 − k3R 2 )=a − bR 2 where Σ0 is the initial planetesimal surface density. We can expres terms of the radius of the planet, 1/3 4πρ RH = Ra. 9M∗ With these equations, we can express ΣP(t) as a function of R(t), and find With these equations, we can express ΣP (t) asafunctionofR(t), ΣP (R) =Σ0 − k3R (12) where we have defined a number of constants for simple algebra. Re-arranging to separate the variables, we have dR = dt. (13) a − bR2 2 = k2 − k3R2 . 3.1 No focussing, ΣP variable Plugging ΣP (R) backinourmasterequationfordR(t)/dt, weno differential equation dR dt = k1(k2 − k3R 2 )=a − bR 2 where Σ0 is the initial planetesimal surface density. We can express RH in terms of the radius of the planet, 1/3 4πρ RH = Ra. (11) 9M∗ With these equations, we can express ΣP (t) asafunctionofR(t), and find ΣP (R) =Σ0 − k3R Plugging ΣP(R) back in our master equation for dR(t)/dt for Fg=1, we now get a differential equation where we have defined a number of constants for simple algebra. Re-a to separate the variables, we have 2 = k2 − k3R2 . 3.1 No focussing, ΣP variable Plugging ΣP (R) backinourmasterequationfordR(t)/dt, wenowgeta differential equation dR dt = k1(k2 − k3R 2 )=a − bR 2 where Σ0 is the initial planetesimal surface density. We can express RH in terms of the radius of the planet, 1/3 4πρ RH = Ra. (11) 9M∗ With these equations, we can express ΣP (t) asafunctionofR(t), and find ΣP (R) =Σ0 − k3R (12) where we where have defined we havea defined number a number of constants of constants for simple for simple algebra. algebra. Re-arranging Re-arranging to separate the variables, we to separate have the variables, we have dR = dt. (13) a − bR2 2 = k2 − k3R2 . 3.1 No focussing, ΣP variable Plugging ΣP (R) backinourmasterequationfordR(t)/dt, wenowgeta differential equation dR dt = k1(k2 − k3R 2 )=a − bR 2 (12) where we have defined a number of constants for simple algebra. Re-arranging to separate the variables, we have dR = dt. (13) a − bR2 The right side The is trivial right to side integrate, is trivialthe to left integrate, we rewrite the left as we rewrite again as
Page 1 and 2: Lecture 10 Lecture Universität Hei
Page 3: 1. Hills radius, random velocities
Page 11 and 12: 2. Collisional growth
Page 13: 2.1 Focussing factor
Page 17: 2.2 Growth rate
Page 20: 2.3 Isolation mass
Page 24 and 25: Planetesimal random velocities In r
Page 26 and 27: Viscous stirring II The timescale o
Page 29 and 30: Dynamical friction II Direct N-body
Page 31: Runaway growth The first stage of c
Page 34 and 35: 2.6 Oligarchic growth
Page 36: Ida & Makino 1993 studied when the
Page 39 and 40: 2.7 Orderly growth
Page 41: 3. Analytical solutions
Page 45 and 46: where k1 is again another constant.
Page 47 and 48: Mass [Earth Masses] 10 1 0.1 0.01 0
Page 50 and 51: 5 Task E Mass [Earth Masses] Mass [
Page 52 and 53: Inner solar system Many 0.01 to 0.1
Page 54 and 55: This method is based on the followi
Page 56: Questions?

L10 From Planetesimals to protoplanets

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