5. A dynamical timescale td for a system of radius R and mass M may be obtained

Question

5. A dynamical timescale td for a system of radius R and mass M may be obtained by cal-
culating the time taken for a unit mass to fall freely from the surface of the system to the
center. Derive such an expression. This timescale provides an estimate of the time taken for
any small instabilities in the system to develop into system-wide instabilities (which would, for
example, then disrupt the entire system). Calculate the dynamical timescale for (i) the Earth
(ii) the Sun (iii) the Solar system (iv) a giant interstellar gas cloud 100pc across and mass 10^6Mass of sun

Explanation / Answer

i .

ma = GMm / R^2 ;

for unit mass ,

a = GM / r^2 ;

a = G ( 4 / 3 ) r^3 / r^2 ;

a = ( 4G / 3 ) r ;

a= ( 4 GM / [ 3 * ( 4 / 3 ) R^3 ] r ;

a =[  GM / R^3 ] r ;  

so acceleration is dependent on distacne from centre of planet , simple harmonic motion , of the form ,

a = - ^2 x ;   

= ( GM / R^3 ) ;

equation of motion ,

x = R cos ( t ) ;

x = R ( GM / R^3 ) cos [ ( GM / R^3 ) t ] ,

for earth

T = 2 / ,

timescale , t = 0.25 T = 0.25 * 2 / = 1.570 ( R^3 / GM )

t ( M , R ) = 1.570 ( R^3 / GM )

t ( earth) = 1153.16 s ;

t ( sun ) = 2505.256 s ;

t ( M , R ) = 1.570 ( R^3 / GM )    < -- this is the scale ,

for gas cloud and solar system the value of M and R is not clear ,

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