Question

1. When the solar system first formed, it was characterized by a disk of dust and gas (called the "solar nebula") with some larger planetesimals embedded within it. As these planetesimals orbited in the solar nebula, they continually accreted material from the nebula, slowly growing in size. In this problem, we will estimate how long it took these planetesimals to grow into reasonably sized objects i.e. a thousand km or so in size). a) If a planetesimal has a cross-sectional area of R' (where R is the planetesimal's radius) and is sweeping through a cloud of smaller particles of fixed size with a velocity V, show that the number of collisions per second will be dn TRPVPN - m where pn=the space density (kg/m) of particles in the cloud and m=the mass of each particle. dt b) Show that if each collision results in the target particles sticking to the planetesimal, the planetesimal will gain mass at a rate of dM = ARⓇV PN 1. dt where M=the planetesimal mass. c) Show that the time to grow to radius Rois t=- 4R0 PP V PN where pe=the density of the planetesimal itself. (Assume that pn and V stay constant as particles are swept up.) d) Assuming that a reasonable value for the density of accretable material in inner part of the early solar nebula is PN = 10 kg/m", estimate the time to accrete a body of 1,000 km radius. Assume a reasonable pp. Is the timescale reasonable given the formation time of the planets (50 million years or so)? How does the timescale change if the accretion process is inefficient, so that not every particle sticks to the planetesimal?

Answer 1

Ans! a): Planetesimal has a cross-sectional area of TR . D o All the particles in the cloud that will fall in this area will collide with the planetesimal so we have to compute the noi of particles in the cross- sectional area :-

2 let an be the no. of particles in the area T.R. man Lave volume occupied by particles so AV = (TT R x Vat man AV so (TRx VAL D you can - Tavle alime - Sion, we RV at ma b. The mass of the planetesimal will increase according to - the no. of collisions: mo : original mass of planetesimalia m's mot nm 80 TR V Pin ..! di nm +mo) - TR V PAN ! . M = R Ven; menm dm = TR ven dt = TRVEN

Jest ve Siam Tve P al flo 499R IR - TVPN J dk . us les 2x10 kg x 10 m = 2 X 10 Kg.mo $0. Roya Aha looo xo x 2 x Xlor 2x/ lox 98 xiß. ...inia: 13 to 10 jó sec. - No, this time scale is not reasonable.