.3 Suppose that the solar nebula has no dust par- ticles per m3, each of radius
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Question
.3 Suppose that the solar nebula has no dust par- ticles per m3, each of radius To, volume Vo and mass mo-4/3 at 0, where is the den- sity. Assume that the particles move about with relative velocities and have essentially geometri- cal cross sections for collisions, and that whenever two particles collide, they stick together, keeping constant. As a result, the average radius r of each particle will grow with time and the average number of particles per m3 n will fall. (a) As- suming that at any time all the particles have the same radius r (a highly artificial assumption, but it allows us to solve the problem), and that the density of the particles stays constant, derive an expression relating the number density of parti cles to the radius r of the particles at any time, and to the initial values ro and no. (b) Now de- rive an approximate expression for dr/dt. (Hint approximate dt by one collision time Te, and esti- mate dr for that time interval from the fact that the volume of each particle is assumed to double in Tc. Remember that both r and n are changing. How does your expression depend on r? (c) For a particle mass density of 3000 kg m-3, an average nebular mass density at t = 0 of 10-7 kg m -3 and u ~ 2 × 10-2 m s-1, what are the numerical values of n of dr/dt? (d) How long does it take to produce particles 0.1 m in radius?Explanation / Answer
(a)As from law of conservation of mass , Total mass M is constant so initially
M= no.(4/3).(22/7).ro3
and when number is n and radius is r
M=n.(4/3).(22/7).r3 equating for total mass M we have no(4/3).(22/7).ro3 =n.(4/3).(22/7).r3 or
no.ro3 =n..r3
or n=noro3/r3
Considering Number density N as n/V=n/((4/3).(22/7).r3) Subsituting N we get
N=noro3/((4/3).(22/7).r6)
(b)As volume V doubles in after each collision time tc, V.tc=2Vor dV/dt=2 and number is halved or n=n/2 so dn/dt=1/2 and dr/dt=dr/dn.dn/dt= 1/2. rono(1/2)n(-3/2)
(c) M/V=3000kg/m3 , at t=0 average nebular mass density=10-7kg/m3 and v=2X10-2m/s
(d) rfinal=0.1m
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