Question

Question 3: Problem 15-6 from the textbook Save S 15-6. (a) Calculate the rate of growth, dR/dt, of a planetary embryo of radius R 4000 km and mass M-1024 kg, in a planetesimal disk of surface density ơ -10 g cm-2, temperature T-300 K, and velocity dispersion v 1 km s1, at a distance of 2 AU from a star of mass 3 Mo. You may use the two-body approximation for planetesimal/ planetary embryo encounters. (b) What will halt (or at least severely slow down) the accretion of such a planetary embryo? What will its mass be at this point? (Hint: See Equation 15.24.)

Please explain

Answer 1

The planetary embryo can grow at a rate that increases as its masses increase which even may consume nearly all planetesimal in the neighborhood, while rapidly gaining mass and becoming a planetary core or protoplanet. Once its size grows large enough and the escape velocity from its surface exceeds the thermal speed of the surrounding gas. Calculate the rate of growth dt of a planetary embryo of radius R as follows, dR 5o,n dt 4p, Reduced mass is calculated according to the information given in the question as follows, 3x1 x 104 = 75 x10" kg 75x 10 x 7 P, 4x 22x(4) x10 525x 102 5632x10 = 0.0932 x 105 = 9.32x 10" kg/km = 9.32 x 10 kg/m' = 932kg / m

dR -as follows, dr Calculate the value of dR 100 x1 V3.14 932 x 4x 4000 dt 0.9775x 1000 14912000 = 6.55x10 m/s b) In the low relative velocity regime, the growth rate of an embryo is M" and thus the large the planetary embryo the faster it grows, that process is known as runaway growth. During this growth, the largest embryo grows faster than any other embryo within its accretion region.